Subgroups — C₂×C₆₇×#3019 #30739634

Order 17152.

5118 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(5,𝔽₂)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#575×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#535×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3019	C₂	1	16	yes	maximal, direct product factor
5	C₂×#455×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#453×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#421×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂²×#100×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×H(5,𝔽₂)×C₆₇	—	2	4	no	—
10	C₂×H(5,𝔽₂)×C₆₇	—	2	8	no	—
11	C₂×H(5,𝔽₂)×C₆₇	—	2	2	no	—
12	C₂×H(5,𝔽₂)×C₆₇	—	2	4	no	—
13	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	4	yes	—
14	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	8	yes	complemented
15	C₂²×#41×C₆₇	—	2	2	no	—
16	C₂²×#41×C₆₇	C₂²	1	2	yes	—
17	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	C₂³×D₄×C₆₇	—	2	2	no	—
19	C₂³×D₄×C₆₇	—	2	2	no	—
20	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	#575×C₆₇	—	2	8	no	—
22	#575×C₆₇	C₂²	1	16	yes	complemented
23	C₂×#137×C₆₇	—	2	2	no	—
24	C₂×#137×C₆₇	C₂²	1	4	yes	complemented
25	C₂×AGL(1,ℤ/8)×C₆₇	—	2	2	no	—
26	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	complemented
27	C₂×#135×C₆₇	—	2	4	no	—
28	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
29	#535×C₆₇	—	2	4	no	—
30	#535×C₆₇	C₂²	1	8	yes	complemented
31	C₂×#127×C₆₇	C₂²	1	2	yes	complemented
32	C₂×#124×C₆₇	C₂²	1	2	yes	complemented
33	C₂×#121×C₆₇	—	2	2	no	—
34	C₂×#120×C₆₇	—	2	2	no	—
35	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#455×C₆₇	C₂²	1	32	yes	complemented
37	#453×C₆₇	C₂²	1	32	yes	complemented
38	#421×C₆₇	C₂²	1	16	yes	complemented
39	C₂×#104×C₆₇	C₂²	1	8	yes	complemented
40	C₂×#100×C₆₇	C₂²	1	4	yes	complemented
41	C₂×#100×C₆₇	C₂²	1	8	yes	complemented
42	C₂⁵×C₆₇	—	2	2	no	—
43	C₂⁵×C₆₇	D₄	1	2	yes	—
44	H(5,𝔽₂)×C₆₇	—	2	16	no	—
45	H(5,𝔽₂)×C₆₇	—	2	32	no	—
46	H(5,𝔽₂)×C₆₇	—	2	16	no	—
47	H(5,𝔽₂)×C₆₇	—	2	32	no	—
48	H(5,𝔽₂)×C₆₇	—	2	8	no	—
49	H(5,𝔽₂)×C₆₇	—	2	16	no	—
50	H(5,𝔽₂)×C₆₇	C₂³	1	16	yes	—
51	C₂×#41×C₆₇	—	2	16	no	—
52	C₂×#41×C₆₇	—	2	8	no	—
53	C₂×#41×C₆₇	—	2	8	no	—
54	C₂×#41×C₆₇	—	2	16	no	—
55	C₂×#41×C₆₇	—	2	2	no	—
56	C₂×#41×C₆₇	—	2	4	no	—
57	C₂×#41×C₆₇	—	2	2	no	—
58	C₂×#41×C₆₇	—	2	4	no	—
59	C₂×#41×C₆₇	—	2	2	no	—
60	C₂×#41×C₆₇	—	2	2	no	—
61	C₂×#41×C₆₇	—	2	2	no	—
62	C₂×#41×C₆₇	—	2	4	no	—
63	C₂×#41×C₆₇	D₄	1	4	yes	complemented
64	C₂×#41×C₆₇	D₄	1	8	yes	complemented
65	C₂×#41×C₆₇	C₂³	1	4	yes	—
66	C₂×#41×C₆₇	C₂³	1	8	yes	—
67	C₂×#41×C₆₇	C₂³	1	4	yes	complemented
68	C₂²×Q₈×C₆₇	D₄	1	2	yes	complemented
69	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic, complemented
70	C₂²×D₄×C₆₇	—	2	4	no	—
71	C₂²×D₄×C₆₇	—	2	16	no	—
72	C₂²×D₄×C₆₇	—	2	8	no	—
73	C₂²×D₄×C₆₇	—	2	16	no	—
74	C₂²×D₄×C₆₇	—	2	8	no	—
75	C₂²×D₄×C₆₇	—	2	2	no	—
76	C₂²×D₄×C₆₇	—	2	8	no	—
77	C₂²×D₄×C₆₇	—	2	16	no	—
78	C₂²×D₄×C₆₇	—	2	8	no	—
79	C₂²×D₄×C₆₇	—	2	16	no	—
80	C₂²×D₄×C₆₇	—	2	1	no	—
81	C₂²×D₄×C₆₇	—	2	4	no	—
82	C₂²×D₄×C₆₇	—	2	2	no	—
83	C₂²×D₄×C₆₇	—	2	2	no	—
84	C₂²×D₄×C₆₇	—	2	4	no	—
85	C₂²×D₄×C₆₇	D₄	1	2	yes	—
86	C₂²×D₄×C₆₇	D₄	1	4	yes	—
87	C₂²×D₄×C₆₇	D₄	1	2	yes	complemented
88	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
89	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
90	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic, complemented
91	C₂³×C₄×C₆₇	—	2	2	no	—
92	C₂³×C₄×C₆₇	—	2	2	no	—
93	C₂³×C₄×C₆₇	—	2	1	no	—
94	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
95	#137×C₆₇	—	2	8	no	—
96	#137×C₆₇	—	2	16	no	—
97	#137×C₆₇	C₂³	1	16	yes	complemented
98	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
99	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
100	AGL(1,ℤ/8)×C₆₇	C₂³	1	16	yes	complemented
101	#135×C₆₇	—	2	32	no	—
102	C₂×Q₁₆×C₆₇	—	2	4	no	—
103	C₂×SD₁₆×C₆₇	—	2	4	no	—
104	C₂×SD₁₆×C₆₇	—	2	4	no	—
105	C₂×D₈×C₆₇	—	2	4	no	—
106	C₂×M₁₆×C₆₇	—	2	2	no	—
107	C₂×M₁₆×C₆₇	—	2	4	no	—
108	C₂×M₁₆×C₆₇	C₂³	1	4	yes	complemented
109	C₂×M₁₆×C₆₇	C₂³	1	8	yes	complemented
110	C₂²×C₈×C₆₇	—	2	2	no	—
111	#127×C₆₇	—	2	4	no	—
112	#127×C₆₇	C₂³	1	8	yes	complemented
113	#124×C₆₇	—	2	4	no	—
114	#124×C₆₇	C₂³	1	8	yes	complemented
115	#121×C₆₇	—	2	16	no	—
116	#120×C₆₇	—	2	16	no	—
117	#117×C₆₇	—	2	2	no	—
118	#117×C₆₇	C₂³	1	4	yes	complemented
119	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
121	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
122	C₂×C₄²×C₆₇	C₂³	1	2	yes	complemented
123	#104×C₆₇	C₂³	1	32	yes	complemented
124	#100×C₆₇	C₂³	1	16	yes	complemented
125	C₂⁴×C₆₇	—	2	16	no	—
126	C₂⁴×C₆₇	—	2	8	no	—
127	C₂⁴×C₆₇	—	2	2	no	—
128	C₂⁴×C₆₇	—	2	2	no	—
129	C₂⁴×C₆₇	—	2	4	no	—
130	C₂⁴×C₆₇	—	2	2	no	—
131	C₂⁴×C₆₇	—	2	1	no	—
132	C₂⁴×C₆₇	—	8	4	no	—
133	C₂⁴×C₆₇	—	8	8	no	—
134	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
135	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
136	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
137	#41×C₆₇	—	2	32	no	—
138	#41×C₆₇	—	2	64	no	—
139	#41×C₆₇	—	2	32	no	—
140	#41×C₆₇	—	2	8	no	—
141	#41×C₆₇	—	2	8	no	—
142	#41×C₆₇	—	2	8	no	—
143	#41×C₆₇	—	2	4	no	—
144	#41×C₆₇	—	2	8	no	—
145	#41×C₆₇	—	2	8	no	—
146	#41×C₆₇	—	2	4	no	—
147	#41×C₆₇	—	2	8	no	—
148	#41×C₆₇	—	2	8	no	—
149	#41×C₆₇	—	4	4	no	—
150	#41×C₆₇	—	4	4	no	—
151	#41×C₆₇	C₂×D₄	1	8	yes	—
152	#41×C₆₇	C₂×D₄	1	16	yes	complemented
153	#41×C₆₇	C₂⁴	1	8	yes	—
154	C₂×Q₈×C₆₇	—	2	4	no	—
155	C₂×Q₈×C₆₇	—	2	2	no	—
156	C₂×Q₈×C₆₇	—	2	1	no	—
157	C₂×Q₈×C₆₇	—	2	2	no	—
158	C₂×Q₈×C₆₇	—	2	2	no	—
159	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
160	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	complemented
161	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	complemented
162	C₂×D₄×C₆₇	—	2	32	no	—
163	C₂×D₄×C₆₇	—	2	16	no	—
164	C₂×D₄×C₆₇	—	2	32	no	—
165	C₂×D₄×C₆₇	—	2	64	no	—
166	C₂×D₄×C₆₇	—	2	32	no	—
167	C₂×D₄×C₆₇	—	2	64	no	—
168	C₂×D₄×C₆₇	—	2	8	no	—
169	C₂×D₄×C₆₇	—	2	4	no	—
170	C₂×D₄×C₆₇	—	2	8	no	—
171	C₂×D₄×C₆₇	—	2	4	no	—
172	C₂×D₄×C₆₇	—	2	32	no	—
173	C₂×D₄×C₆₇	—	2	32	no	—
174	C₂×D₄×C₆₇	—	2	4	no	—
175	C₂×D₄×C₆₇	—	2	16	no	—
176	C₂×D₄×C₆₇	—	2	8	no	—
177	C₂×D₄×C₆₇	—	2	4	no	—
178	C₂×D₄×C₆₇	—	2	4	no	—
179	C₂×D₄×C₆₇	—	2	2	no	—
180	C₂×D₄×C₆₇	—	2	4	no	—
181	C₂×D₄×C₆₇	—	2	8	no	—
182	C₂×D₄×C₆₇	—	2	8	no	—
183	C₂×D₄×C₆₇	—	2	4	no	—
184	C₂×D₄×C₆₇	—	2	8	no	—
185	C₂×D₄×C₆₇	—	2	4	no	—
186	C₂×D₄×C₆₇	—	2	2	no	—
187	C₂×D₄×C₆₇	—	2	2	no	—
188	C₂×D₄×C₆₇	—	2	1	no	—
189	C₂×D₄×C₆₇	—	4	2	no	—
190	C₂×D₄×C₆₇	—	4	1	no	—
191	C₂×D₄×C₆₇	—	4	1	no	—
192	C₂×D₄×C₆₇	—	8	8	no	—
193	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
194	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
195	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
196	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	complemented
197	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
198	C₂×D₄×C₆₇	C₂⁴	1	2	yes	complemented
199	C₂²×C₄×C₆₇	—	2	16	no	—
200	C₂²×C₄×C₆₇	—	2	8	no	—
201	C₂²×C₄×C₆₇	—	2	2	no	—
202	C₂²×C₄×C₆₇	—	2	8	no	—
203	C₂²×C₄×C₆₇	—	2	4	no	—
204	C₂²×C₄×C₆₇	—	2	8	no	—
205	C₂²×C₄×C₆₇	—	2	4	no	—
206	C₂²×C₄×C₆₇	—	2	2	no	—
207	C₂²×C₄×C₆₇	—	2	1	no	—
208	C₂²×C₄×C₆₇	—	2	4	no	—
209	C₂²×C₄×C₆₇	—	4	2	no	—
210	C₂²×C₄×C₆₇	—	4	2	no	—
211	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
212	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
213	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
214	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
215	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
216	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
217	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
218	Q₁₆×C₆₇	—	2	16	no	—
219	SD₁₆×C₆₇	—	2	16	no	—
220	SD₁₆×C₆₇	—	2	16	no	—
221	D₈×C₆₇	—	2	16	no	—
222	M₁₆×C₆₇	—	2	8	no	—
223	M₁₆×C₆₇	—	2	4	no	—
224	M₁₆×C₆₇	C₂⁴	1	8	yes	complemented
225	C₂×C₈×C₆₇	—	2	4	no	—
226	C₂×C₈×C₆₇	—	2	8	no	—
227	C₄⋊C₄×C₆₇	—	2	4	no	—
228	C₂²⋊C₄×C₆₇	—	2	4	no	—
229	C₂²⋊C₄×C₆₇	—	2	16	no	—
230	C₄²×C₆₇	—	2	2	no	—
231	C₄²×C₆₇	C₂⁴	1	4	yes	complemented
232	C₂³×C₆₇	—	2	32	no	—
233	C₂³×C₆₇	—	2	8	no	—
234	C₂³×C₆₇	—	2	8	no	—
235	C₂³×C₆₇	—	2	4	no	—
236	C₂³×C₆₇	—	2	4	no	—
237	C₂³×C₆₇	—	2	4	no	—
238	C₂³×C₆₇	—	2	8	no	—
239	C₂³×C₆₇	—	2	2	no	—
240	C₂³×C₆₇	—	4	2	no	—
241	C₂³×C₆₇	—	4	8	no	—
242	C₂³×C₆₇	—	4	8	no	—
243	C₂³×C₆₇	—	4	8	no	—
244	C₂³×C₆₇	—	4	4	no	—
245	C₂³×C₆₇	—	4	1	no	—
246	C₂³×C₆₇	—	4	1	no	—
247	C₂³×C₆₇	—	8	4	no	—
248	C₂³×C₆₇	—	8	8	no	—
249	C₂³×C₆₇	—	8	8	no	—
250	C₂³×C₆₇	—	8	32	no	—
251	C₂³×C₆₇	—	8	16	no	—
252	C₂³×C₆₇	#120	1	1	yes	characteristic
253	C₂³×C₆₇	#120	1	1	yes	characteristic
254	C₂³×C₆₇	#120	1	2	yes	—
255	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
256	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
257	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
258	Q₈×C₆₇	—	2	4	no	—
259	Q₈×C₆₇	—	2	8	no	—
260	Q₈×C₆₇	—	2	4	no	—
261	Q₈×C₆₇	—	4	2	no	—
262	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
263	D₄×C₆₇	—	2	64	no	—
264	D₄×C₆₇	—	2	8	no	—
265	D₄×C₆₇	—	2	16	no	—
266	D₄×C₆₇	—	2	8	no	—
267	D₄×C₆₇	—	2	16	no	—
268	D₄×C₆₇	—	2	8	no	—
269	D₄×C₆₇	—	2	4	no	—
270	D₄×C₆₇	—	2	8	no	—
271	D₄×C₆₇	—	2	8	no	—
272	D₄×C₆₇	—	2	4	no	—
273	D₄×C₆₇	—	4	4	no	—
274	D₄×C₆₇	—	4	4	no	—
275	D₄×C₆₇	—	4	2	no	—
276	D₄×C₆₇	—	8	16	no	—
277	D₄×C₆₇	C₂²×D₄	1	8	yes	—
278	C₂×C₄×C₆₇	—	2	32	no	—
279	C₂×C₄×C₆₇	—	2	8	no	—
280	C₂×C₄×C₆₇	—	2	4	no	—
281	C₂×C₄×C₆₇	—	2	8	no	—
282	C₂×C₄×C₆₇	—	2	16	no	—
283	C₂×C₄×C₆₇	—	2	8	no	—
284	C₂×C₄×C₆₇	—	2	4	no	—
285	C₂×C₄×C₆₇	—	2	2	no	—
286	C₂×C₄×C₆₇	—	4	4	no	—
287	C₂×C₄×C₆₇	—	4	1	no	—
288	C₂×C₄×C₆₇	—	4	1	no	—
289	C₂×C₄×C₆₇	—	4	8	no	—
290	C₂×C₄×C₆₇	#120	1	4	yes	—
291	C₂×C₄×C₆₇	#120	1	2	yes	—
292	C₂×C₄×C₆₇	#120	1	4	yes	—
293	C₂×C₄×C₆₇	#120	1	2	yes	—
294	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
295	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
296	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
297	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
298	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
299	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
300	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
301	C₈×C₆₇	—	2	8	no	—
302	C₂²×C₆₇	—	2	2	no	—
303	C₂²×C₆₇	—	2	2	no	—
304	C₂²×C₆₇	—	2	2	no	—
305	C₂²×C₆₇	—	2	8	no	—
306	C₂²×C₆₇	—	2	8	no	—
307	C₂²×C₆₇	—	4	2	no	—
308	C₂²×C₆₇	—	4	2	no	—
309	C₂²×C₆₇	—	4	8	no	—
310	C₂²×C₆₇	—	4	16	no	—
311	C₂²×C₆₇	—	4	16	no	—
312	C₂²×C₆₇	—	4	4	no	—
313	C₂²×C₆₇	—	4	8	no	—
314	C₂²×C₆₇	—	4	8	no	—
315	C₂²×C₆₇	—	4	2	no	—
316	C₂²×C₆₇	—	8	4	no	—
317	C₂²×C₆₇	—	8	8	no	—
318	C₂²×C₆₇	—	8	2	no	—
319	C₂²×C₆₇	—	8	32	no	—
320	C₂²×C₆₇	—	8	2	no	—
321	C₂²×C₆₇	C₂×#120	1	2	yes	—
322	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
323	C₂²×C₆₇	C₂×#120	1	2	yes	—
324	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
325	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
326	C₄×C₆₇	—	2	8	no	—
327	C₄×C₆₇	—	4	4	no	—
328	C₄×C₆₇	—	4	2	no	—
329	C₄×C₆₇	C₂×#120	1	4	yes	—
330	C₄×C₆₇	C₂×#120	1	2	yes	—
331	C₄×C₆₇	C₂³×D₄	1	2	yes	—
332	C₂×#3019	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
333	C₂×C₆₇	—	2	1	no	—
334	C₂×C₆₇	—	2	2	no	—
335	C₂×C₆₇	—	2	1	no	—
336	C₂×C₆₇	—	2	2	no	—
337	C₂×C₆₇	—	4	8	no	—
338	C₂×C₆₇	—	4	8	no	—
339	C₂×C₆₇	—	8	2	no	—
340	C₂×C₆₇	#3019	1	2	yes	direct product factor
341	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
342	C₂²×H(5,𝔽₂)	C₂×C₆₇	1	1	yes	characteristic, complemented
343	C₂×#575	C₂×C₆₇	1	2	yes	complemented
344	C₂×#535	C₂×C₆₇	1	1	yes	characteristic, complemented
345	#3019	C₂×C₆₇	1	16	yes	direct product factor
346	C₂×#455	C₂×C₆₇	1	4	yes	complemented
347	C₂×#453	C₂×C₆₇	1	4	yes	complemented
348	C₂×#421	C₂×C₆₇	1	2	yes	complemented
349	C₂²×#100	C₂×C₆₇	1	1	yes	characteristic, complemented
350	C₆₇	C₂×#3019	1	1	yes	characteristic, direct product factor, Sylow
351	C₂×H(5,𝔽₂)	—	2	4	no	—
352	C₂×H(5,𝔽₂)	—	2	8	no	—
353	C₂×H(5,𝔽₂)	—	2	2	no	—
354	C₂×H(5,𝔽₂)	—	2	4	no	—
355	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	4	yes	—
356	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	8	yes	complemented
357	C₂²×#41	—	2	2	no	—
358	C₂²×#41	C₂²×C₆₇	1	2	yes	—
359	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic, complemented
360	C₂³×D₄	—	2	2	no	—
361	C₂³×D₄	—	2	2	no	—
362	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
363	#575	—	2	8	no	—
364	#575	C₂²×C₆₇	1	16	yes	complemented
365	C₂×#137	—	2	2	no	—
366	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
367	C₂×AGL(1,ℤ/8)	—	2	2	no	—
368	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	complemented
369	C₂×#135	—	2	4	no	—
370	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
371	#535	—	2	4	no	—
372	#535	C₂²×C₆₇	1	8	yes	complemented
373	C₂×#127	C₂²×C₆₇	1	2	yes	complemented
374	C₂×#124	C₂²×C₆₇	1	2	yes	complemented
375	C₂×#121	—	2	2	no	—
376	C₂×#120	—	2	2	no	—
377	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic, complemented
378	#455	C₂²×C₆₇	1	32	yes	complemented
379	#453	C₂²×C₆₇	1	32	yes	complemented
380	#421	C₂²×C₆₇	1	16	yes	complemented
381	C₂×#104	C₂²×C₆₇	1	8	yes	complemented
382	C₂×#100	C₂²×C₆₇	1	4	yes	complemented
383	C₂×#100	C₂²×C₆₇	1	8	yes	complemented
384	C₂⁵	—	2	2	no	—
385	C₂⁵	D₄×C₆₇	1	2	yes	—
386	H(5,𝔽₂)	—	2	16	no	—
387	H(5,𝔽₂)	—	2	32	no	—
388	H(5,𝔽₂)	—	2	16	no	—
389	H(5,𝔽₂)	—	2	32	no	—
390	H(5,𝔽₂)	—	2	8	no	—
391	H(5,𝔽₂)	—	2	16	no	—
392	H(5,𝔽₂)	C₂³×C₆₇	1	16	yes	—
393	C₂×#41	—	2	8	no	—
394	C₂×#41	—	2	16	no	—
395	C₂×#41	—	2	8	no	—
396	C₂×#41	—	2	16	no	—
397	C₂×#41	—	2	2	no	—
398	C₂×#41	—	2	4	no	—
399	C₂×#41	—	2	2	no	—
400	C₂×#41	—	2	4	no	—
401	C₂×#41	—	2	2	no	—
402	C₂×#41	—	2	2	no	—
403	C₂×#41	—	2	2	no	—
404	C₂×#41	—	2	4	no	—
405	C₂×#41	D₄×C₆₇	1	4	yes	complemented
406	C₂×#41	D₄×C₆₇	1	8	yes	complemented
407	C₂×#41	C₂³×C₆₇	1	4	yes	—
408	C₂×#41	C₂³×C₆₇	1	8	yes	—
409	C₂×#41	C₂³×C₆₇	1	4	yes	complemented
410	C₂²×Q₈	D₄×C₆₇	1	2	yes	complemented
411	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic, complemented
412	C₂²×D₄	—	2	4	no	—
413	C₂²×D₄	—	2	8	no	—
414	C₂²×D₄	—	2	16	no	—
415	C₂²×D₄	—	2	8	no	—
416	C₂²×D₄	—	2	16	no	—
417	C₂²×D₄	—	2	2	no	—
418	C₂²×D₄	—	2	8	no	—
419	C₂²×D₄	—	2	16	no	—
420	C₂²×D₄	—	2	8	no	—
421	C₂²×D₄	—	2	16	no	—
422	C₂²×D₄	—	2	1	no	—
423	C₂²×D₄	—	2	4	no	—
424	C₂²×D₄	—	2	2	no	—
425	C₂²×D₄	—	2	2	no	—
426	C₂²×D₄	—	2	4	no	—
427	C₂²×D₄	D₄×C₆₇	1	2	yes	—
428	C₂²×D₄	D₄×C₆₇	1	4	yes	—
429	C₂²×D₄	D₄×C₆₇	1	2	yes	complemented
430	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
431	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
432	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic, complemented
433	C₂³×C₄	—	2	2	no	—
434	C₂³×C₄	—	2	2	no	—
435	C₂³×C₄	—	2	1	no	—
436	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
437	#137	—	2	8	no	—
438	#137	—	2	16	no	—
439	#137	C₂³×C₆₇	1	16	yes	complemented
440	AGL(1,ℤ/8)	—	2	8	no	—
441	AGL(1,ℤ/8)	—	2	16	no	—
442	AGL(1,ℤ/8)	C₂³×C₆₇	1	16	yes	complemented
443	#135	—	2	32	no	—
444	C₂×Q₁₆	—	2	4	no	—
445	C₂×SD₁₆	—	2	4	no	—
446	C₂×SD₁₆	—	2	4	no	—
447	C₂×D₈	—	2	4	no	—
448	C₂×M₁₆	—	2	2	no	—
449	C₂×M₁₆	—	2	4	no	—
450	C₂×M₁₆	C₂³×C₆₇	1	4	yes	complemented
451	C₂×M₁₆	C₂³×C₆₇	1	8	yes	complemented
452	C₂²×C₈	—	2	2	no	—
453	#127	—	2	4	no	—
454	#127	C₂³×C₆₇	1	8	yes	complemented
455	#124	—	2	4	no	—
456	#124	C₂³×C₆₇	1	8	yes	complemented
457	#121	—	2	16	no	—
458	#120	—	2	16	no	—
459	#117	—	2	2	no	—
460	#117	C₂³×C₆₇	1	4	yes	complemented
461	C₂×C₄⋊C₄	—	2	1	no	—
462	C₂×C₂²⋊C₄	—	2	1	no	—
463	C₂×C₂²⋊C₄	—	2	4	no	—
464	C₂×C₄²	C₂³×C₆₇	1	2	yes	complemented
465	#104	C₂³×C₆₇	1	32	yes	complemented
466	#100	C₂³×C₆₇	1	16	yes	complemented
467	C₂⁴	—	2	8	no	—
468	C₂⁴	—	2	16	no	—
469	C₂⁴	—	2	2	no	—
470	C₂⁴	—	2	2	no	—
471	C₂⁴	—	2	2	no	—
472	C₂⁴	—	2	4	no	—
473	C₂⁴	—	2	1	no	—
474	C₂⁴	—	8	4	no	—
475	C₂⁴	—	8	8	no	—
476	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
477	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
478	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
479	#41	—	2	32	no	—
480	#41	—	2	64	no	—
481	#41	—	2	32	no	—
482	#41	—	2	8	no	—
483	#41	—	2	8	no	—
484	#41	—	2	8	no	—
485	#41	—	2	4	no	—
486	#41	—	2	8	no	—
487	#41	—	2	8	no	—
488	#41	—	2	4	no	—
489	#41	—	2	8	no	—
490	#41	—	2	8	no	—
491	#41	—	4	4	no	—
492	#41	—	4	4	no	—
493	#41	C₂×D₄×C₆₇	1	8	yes	—
494	#41	C₂×D₄×C₆₇	1	16	yes	complemented
495	#41	C₂⁴×C₆₇	1	8	yes	—
496	C₂×Q₈	—	2	2	no	—
497	C₂×Q₈	—	2	4	no	—
498	C₂×Q₈	—	2	1	no	—
499	C₂×Q₈	—	2	2	no	—
500	C₂×Q₈	—	2	2	no	—
501	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
502	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	complemented
503	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	complemented
504	C₂×D₄	—	2	16	no	—
505	C₂×D₄	—	2	32	no	—
506	C₂×D₄	—	2	32	no	—
507	C₂×D₄	—	2	64	no	—
508	C₂×D₄	—	2	32	no	—
509	C₂×D₄	—	2	64	no	—
510	C₂×D₄	—	2	4	no	—
511	C₂×D₄	—	2	8	no	—
512	C₂×D₄	—	2	4	no	—
513	C₂×D₄	—	2	8	no	—
514	C₂×D₄	—	2	32	no	—
515	C₂×D₄	—	2	32	no	—
516	C₂×D₄	—	2	4	no	—
517	C₂×D₄	—	2	16	no	—
518	C₂×D₄	—	2	4	no	—
519	C₂×D₄	—	2	8	no	—
520	C₂×D₄	—	2	2	no	—
521	C₂×D₄	—	2	4	no	—
522	C₂×D₄	—	2	4	no	—
523	C₂×D₄	—	2	8	no	—
524	C₂×D₄	—	2	4	no	—
525	C₂×D₄	—	2	8	no	—
526	C₂×D₄	—	2	8	no	—
527	C₂×D₄	—	2	4	no	—
528	C₂×D₄	—	2	2	no	—
529	C₂×D₄	—	2	2	no	—
530	C₂×D₄	—	2	1	no	—
531	C₂×D₄	—	4	1	no	—
532	C₂×D₄	—	4	1	no	—
533	C₂×D₄	—	4	2	no	—
534	C₂×D₄	—	8	8	no	—
535	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
536	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
537	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
538	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	complemented
539	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
540	C₂×D₄	C₂⁴×C₆₇	1	2	yes	complemented
541	C₂²×C₄	—	2	8	no	—
542	C₂²×C₄	—	2	16	no	—
543	C₂²×C₄	—	2	2	no	—
544	C₂²×C₄	—	2	4	no	—
545	C₂²×C₄	—	2	8	no	—
546	C₂²×C₄	—	2	4	no	—
547	C₂²×C₄	—	2	8	no	—
548	C₂²×C₄	—	2	2	no	—
549	C₂²×C₄	—	2	1	no	—
550	C₂²×C₄	—	2	4	no	—
551	C₂²×C₄	—	4	2	no	—
552	C₂²×C₄	—	4	2	no	—
553	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
554	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
555	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
556	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
557	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
558	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
559	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
560	Q₁₆	—	2	16	no	—
561	SD₁₆	—	2	16	no	—
562	SD₁₆	—	2	16	no	—
563	D₈	—	2	16	no	—
564	M₁₆	—	2	8	no	—
565	M₁₆	—	2	4	no	—
566	M₁₆	C₂⁴×C₆₇	1	8	yes	complemented
567	C₂×C₈	—	2	4	no	—
568	C₂×C₈	—	2	8	no	—
569	C₄⋊C₄	—	2	4	no	—
570	C₂²⋊C₄	—	2	4	no	—
571	C₂²⋊C₄	—	2	16	no	—
572	C₄²	—	2	2	no	—
573	C₄²	C₂⁴×C₆₇	1	4	yes	complemented
574	C₂³	—	2	32	no	—
575	C₂³	—	2	8	no	—
576	C₂³	—	2	4	no	—
577	C₂³	—	2	4	no	—
578	C₂³	—	2	8	no	—
579	C₂³	—	2	4	no	—
580	C₂³	—	2	8	no	—
581	C₂³	—	2	2	no	—
582	C₂³	—	4	2	no	—
583	C₂³	—	4	8	no	—
584	C₂³	—	4	8	no	—
585	C₂³	—	4	8	no	—
586	C₂³	—	4	4	no	—
587	C₂³	—	4	1	no	—
588	C₂³	—	4	1	no	—
589	C₂³	—	8	4	no	—
590	C₂³	—	8	8	no	—
591	C₂³	—	8	8	no	—
592	C₂³	—	8	32	no	—
593	C₂³	—	8	16	no	—
594	C₂³	#120×C₆₇	1	1	yes	characteristic
595	C₂³	#120×C₆₇	1	1	yes	characteristic
596	C₂³	#120×C₆₇	1	2	yes	—
597	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
598	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
599	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
600	Q₈	—	2	4	no	—
601	Q₈	—	2	8	no	—
602	Q₈	—	2	4	no	—
603	Q₈	—	4	2	no	—
604	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
605	D₄	—	2	64	no	—
606	D₄	—	2	8	no	—
607	D₄	—	2	16	no	—
608	D₄	—	2	8	no	—
609	D₄	—	2	16	no	—
610	D₄	—	2	8	no	—
611	D₄	—	2	4	no	—
612	D₄	—	2	8	no	—
613	D₄	—	2	8	no	—
614	D₄	—	2	4	no	—
615	D₄	—	4	4	no	—
616	D₄	—	4	4	no	—
617	D₄	—	4	2	no	—
618	D₄	—	8	16	no	—
619	D₄	C₂²×D₄×C₆₇	1	8	yes	—
620	C₂×C₄	—	2	32	no	—
621	C₂×C₄	—	2	4	no	—
622	C₂×C₄	—	2	8	no	—
623	C₂×C₄	—	2	8	no	—
624	C₂×C₄	—	2	16	no	—
625	C₂×C₄	—	2	8	no	—
626	C₂×C₄	—	2	4	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	4	4	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	4	8	no	—
632	C₂×C₄	#120×C₆₇	1	2	yes	—
633	C₂×C₄	#120×C₆₇	1	4	yes	—
634	C₂×C₄	#120×C₆₇	1	2	yes	—
635	C₂×C₄	#120×C₆₇	1	4	yes	—
636	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
637	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
638	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
639	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
640	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
641	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
642	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
643	C₈	—	2	8	no	—
644	C₂²	—	2	2	no	—
645	C₂²	—	2	2	no	—
646	C₂²	—	2	2	no	—
647	C₂²	—	2	8	no	—
648	C₂²	—	2	8	no	—
649	C₂²	—	4	2	no	—
650	C₂²	—	4	2	no	—
651	C₂²	—	4	8	no	—
652	C₂²	—	4	16	no	—
653	C₂²	—	4	16	no	—
654	C₂²	—	4	4	no	—
655	C₂²	—	4	8	no	—
656	C₂²	—	4	8	no	—
657	C₂²	—	4	2	no	—
658	C₂²	—	8	4	no	—
659	C₂²	—	8	8	no	—
660	C₂²	—	8	2	no	—
661	C₂²	—	8	32	no	—
662	C₂²	—	8	2	no	—
663	C₂²	C₂×#120×C₆₇	1	2	yes	—
664	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
665	C₂²	C₂×#120×C₆₇	1	2	yes	—
666	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
667	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
668	C₄	—	2	8	no	—
669	C₄	—	4	4	no	—
670	C₄	—	4	2	no	—
671	C₄	C₂×#120×C₆₇	1	4	yes	—
672	C₄	C₂×#120×C₆₇	1	2	yes	—
673	C₄	C₂³×D₄×C₆₇	1	2	yes	—
674	C₂	—	2	1	no	—
675	C₂	—	2	2	no	—
676	C₂	—	2	1	no	—
677	C₂	—	2	2	no	—
678	C₂	—	4	8	no	—
679	C₂	—	4	8	no	—
680	C₂	—	8	2	no	—
681	C₂	C₆₇×#3019	1	2	yes	direct product factor
682	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×H(5,𝔽₂)×C₆₇	C₂	semidirect
2. C₂×#575×C₆₇	C₂	semidirect
3. C₂×#535×C₆₇	C₂	semidirect
4. C₆₇×#3019	C₂	direct (N × H)
5. C₂×#455×C₆₇	C₂	semidirect
6. C₂×#453×C₆₇	C₂	semidirect
7. C₂×#421×C₆₇	C₂	semidirect
8. C₂²×#100×C₆₇	C₂	semidirect
14. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
17. C₂²×#41×C₆₇	C₂²	semidirect
22. #575×C₆₇	C₂²	semidirect
24. C₂×#137×C₆₇	C₂²	semidirect
26. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
28. C₂²×M₁₆×C₆₇	C₂²	semidirect
30. #535×C₆₇	C₂²	semidirect
31. C₂×#127×C₆₇	C₂²	semidirect
32. C₂×#124×C₆₇	C₂²	semidirect
35. C₂×#117×C₆₇	C₂²	semidirect
36. #455×C₆₇	C₂²	semidirect
37. #453×C₆₇	C₂²	semidirect
38. #421×C₆₇	C₂²	semidirect
39. C₂×#104×C₆₇	C₂²	semidirect
40. C₂×#100×C₆₇	C₂²	semidirect
41. C₂×#100×C₆₇	C₂²	semidirect
63. C₂×#41×C₆₇	D₄	semidirect
64. C₂×#41×C₆₇	D₄	semidirect
67. C₂×#41×C₆₇	C₂³	semidirect
68. C₂²×Q₈×C₆₇	D₄	semidirect
69. C₂²×Q₈×C₆₇	C₂³	semidirect
87. C₂²×D₄×C₆₇	D₄	semidirect
90. C₂²×D₄×C₆₇	C₂³	semidirect
97. #137×C₆₇	C₂³	semidirect
100. AGL(1,ℤ/8)×C₆₇	C₂³	semidirect
108. C₂×M₁₆×C₆₇	C₂³	semidirect
109. C₂×M₁₆×C₆₇	C₂³	semidirect
112. #127×C₆₇	C₂³	semidirect
114. #124×C₆₇	C₂³	semidirect
118. #117×C₆₇	C₂³	semidirect
122. C₂×C₄²×C₆₇	C₂³	semidirect
123. #104×C₆₇	C₂³	semidirect
124. #100×C₆₇	C₂³	semidirect
152. #41×C₆₇	C₂×D₄	semidirect
160. C₂×Q₈×C₆₇	C₂×D₄	semidirect
161. C₂×Q₈×C₆₇	C₂⁴	semidirect
196. C₂×D₄×C₆₇	C₂×D₄	semidirect
198. C₂×D₄×C₆₇	C₂⁴	semidirect
224. M₁₆×C₆₇	C₂⁴	semidirect
231. C₄²×C₆₇	C₂⁴	semidirect
332. C₂×#3019	C₆₇	direct (N × H)
340. C₂×C₆₇	#3019	direct (N × H)
342. C₂²×H(5,𝔽₂)	C₂×C₆₇	semidirect
343. C₂×#575	C₂×C₆₇	semidirect
344. C₂×#535	C₂×C₆₇	semidirect
345. #3019	C₂×C₆₇	direct (N × H)
346. C₂×#455	C₂×C₆₇	semidirect
347. C₂×#453	C₂×C₆₇	semidirect
348. C₂×#421	C₂×C₆₇	semidirect
349. C₂²×#100	C₂×C₆₇	semidirect
350. C₆₇	C₂×#3019	direct (N × H)
356. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
359. C₂²×#41	C₂²×C₆₇	semidirect
364. #575	C₂²×C₆₇	semidirect
366. C₂×#137	C₂²×C₆₇	semidirect
368. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
370. C₂²×M₁₆	C₂²×C₆₇	semidirect
372. #535	C₂²×C₆₇	semidirect
373. C₂×#127	C₂²×C₆₇	semidirect
374. C₂×#124	C₂²×C₆₇	semidirect
377. C₂×#117	C₂²×C₆₇	semidirect
378. #455	C₂²×C₆₇	semidirect
379. #453	C₂²×C₆₇	semidirect
380. #421	C₂²×C₆₇	semidirect
381. C₂×#104	C₂²×C₆₇	semidirect
382. C₂×#100	C₂²×C₆₇	semidirect
383. C₂×#100	C₂²×C₆₇	semidirect
405. C₂×#41	D₄×C₆₇	semidirect
406. C₂×#41	D₄×C₆₇	semidirect
409. C₂×#41	C₂³×C₆₇	semidirect
410. C₂²×Q₈	D₄×C₆₇	semidirect
411. C₂²×Q₈	C₂³×C₆₇	semidirect
429. C₂²×D₄	D₄×C₆₇	semidirect
432. C₂²×D₄	C₂³×C₆₇	semidirect
439. #137	C₂³×C₆₇	semidirect
442. AGL(1,ℤ/8)	C₂³×C₆₇	semidirect
450. C₂×M₁₆	C₂³×C₆₇	semidirect
451. C₂×M₁₆	C₂³×C₆₇	semidirect
454. #127	C₂³×C₆₇	semidirect
456. #124	C₂³×C₆₇	semidirect
460. #117	C₂³×C₆₇	semidirect
464. C₂×C₄²	C₂³×C₆₇	semidirect
465. #104	C₂³×C₆₇	semidirect
466. #100	C₂³×C₆₇	semidirect
494. #41	C₂×D₄×C₆₇	semidirect
502. C₂×Q₈	C₂×D₄×C₆₇	semidirect
503. C₂×Q₈	C₂⁴×C₆₇	semidirect
538. C₂×D₄	C₂×D₄×C₆₇	semidirect
540. C₂×D₄	C₂⁴×C₆₇	semidirect
566. M₁₆	C₂⁴×C₆₇	semidirect
573. C₄²	C₂⁴×C₆₇	semidirect
681. C₂	C₆₇×#3019	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×H(5,𝔽₂)×C₆₇	2
C₂×#575×C₆₇	2
C₂×#535×C₆₇	2
C₆₇×#3019	2
C₂×#455×C₆₇	2
C₂×#453×C₆₇	2
C₂×#421×C₆₇	2
C₂²×#100×C₆₇	2
C₂×#3019	67

Subgroups — C2×C67×#3019 #30739634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3019 #30739634