Subgroups — C₂×C₆₇×#3022 #30739637

Order 17152.

4742 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#585×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#137×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#536×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3022	C₂	1	16	yes	maximal, direct product factor
5	C₂×#456×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#455×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#421×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#413×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#585×C₆₇	—	2	8	no	—
10	#585×C₆₇	—	2	4	no	—
11	#585×C₆₇	C₂²	1	8	yes	—
12	C₂×2₋¹⁺⁴×C₆₇	—	2	2	no	—
13	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	—
14	C₂×H(5,𝔽₂)×C₆₇	—	2	2	no	—
15	C₂×H(5,𝔽₂)×C₆₇	—	2	2	no	—
16	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	2	yes	—
17	C₂²×#41×C₆₇	—	2	4	no	—
18	C₂²×#41×C₆₇	—	2	2	no	—
19	C₂²×#41×C₆₇	C₂²	1	2	yes	—
20	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
21	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
22	C₂×#137×C₆₇	—	2	4	no	—
23	C₂×#137×C₆₇	—	2	8	no	—
24	C₂×#137×C₆₇	C₂²	1	8	yes	complemented
25	C₂×#137×C₆₇	C₂²	1	16	yes	complemented
26	C₂²×Q₁₆×C₆₇	—	2	2	no	—
27	C₂²×SD₁₆×C₆₇	—	2	2	no	—
28	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
29	#536×C₆₇	—	2	4	no	—
30	#536×C₆₇	C₂²	1	8	yes	complemented
31	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
32	C₂×#124×C₆₇	C₂²	1	2	yes	complemented
33	C₂×#123×C₆₇	—	2	2	no	—
34	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
35	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#456×C₆₇	C₂²	1	32	yes	complemented
37	#455×C₆₇	C₂²	1	32	yes	complemented
38	#421×C₆₇	C₂²	1	16	yes	complemented
39	C₂×#104×C₆₇	C₂²	1	8	yes	complemented
40	#413×C₆₇	C₂²	1	8	yes	complemented
41	C₂×#101×C₆₇	C₂²	1	2	yes	complemented
42	C₂×#100×C₆₇	C₂²	1	2	yes	complemented
43	2₋¹⁺⁴×C₆₇	—	2	16	no	—
44	2₋¹⁺⁴×C₆₇	—	2	8	no	—
45	2₋¹⁺⁴×C₆₇	—	2	16	no	—
46	2₋¹⁺⁴×C₆₇	—	2	4	no	—
47	2₋¹⁺⁴×C₆₇	C₂³	1	8	yes	—
48	H(5,𝔽₂)×C₆₇	—	2	16	no	—
49	H(5,𝔽₂)×C₆₇	—	2	16	no	—
50	H(5,𝔽₂)×C₆₇	—	2	8	no	—
51	H(5,𝔽₂)×C₆₇	—	2	16	no	—
52	H(5,𝔽₂)×C₆₇	—	2	16	no	—
53	H(5,𝔽₂)×C₆₇	—	2	4	no	—
54	H(5,𝔽₂)×C₆₇	C₂³	1	8	yes	—
55	C₂×#41×C₆₇	—	2	4	no	—
56	C₂×#41×C₆₇	—	2	8	no	—
57	C₂×#41×C₆₇	—	2	4	no	—
58	C₂×#41×C₆₇	—	2	32	no	—
59	C₂×#41×C₆₇	—	2	16	no	—
60	C₂×#41×C₆₇	—	2	32	no	—
61	C₂×#41×C₆₇	—	2	4	no	—
62	C₂×#41×C₆₇	—	2	4	no	—
63	C₂×#41×C₆₇	—	2	2	no	—
64	C₂×#41×C₆₇	—	2	2	no	—
65	C₂×#41×C₆₇	—	2	4	no	—
66	C₂×#41×C₆₇	—	2	16	no	—
67	C₂×#41×C₆₇	—	2	2	no	—
68	C₂×#41×C₆₇	—	2	2	no	—
69	C₂×#41×C₆₇	—	2	4	no	—
70	C₂×#41×C₆₇	—	2	4	no	—
71	C₂×#41×C₆₇	—	2	4	no	—
72	C₂×#41×C₆₇	D₄	1	4	yes	—
73	C₂×#41×C₆₇	D₄	1	8	yes	—
74	C₂×#41×C₆₇	D₄	1	4	yes	—
75	C₂×#41×C₆₇	C₂³	1	4	yes	—
76	C₂×#41×C₆₇	C₂³	1	8	yes	—
77	C₂×#41×C₆₇	C₂³	1	4	yes	—
78	C₂²×Q₈×C₆₇	—	2	2	no	—
79	C₂²×Q₈×C₆₇	—	2	4	no	—
80	C₂²×Q₈×C₆₇	—	2	1	no	—
81	C₂²×Q₈×C₆₇	—	2	2	no	—
82	C₂²×Q₈×C₆₇	—	2	2	no	—
83	C₂²×Q₈×C₆₇	—	2	2	no	—
84	C₂²×Q₈×C₆₇	—	2	4	no	—
85	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
86	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
87	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₂²×Q₈×C₆₇	C₂³	1	4	yes	complemented
89	C₂²×D₄×C₆₇	—	2	8	no	—
90	C₂²×D₄×C₆₇	—	2	2	no	—
91	C₂²×D₄×C₆₇	—	2	4	no	—
92	C₂²×D₄×C₆₇	—	2	2	no	—
93	C₂²×D₄×C₆₇	—	2	4	no	—
94	C₂²×D₄×C₆₇	D₄	1	2	yes	—
95	C₂²×D₄×C₆₇	D₄	1	2	yes	—
96	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
97	C₂³×C₄×C₆₇	—	2	4	no	—
98	C₂³×C₄×C₆₇	—	2	2	no	—
99	C₂³×C₄×C₆₇	—	2	1	no	—
100	C₂³×C₄×C₆₇	D₄	1	2	yes	—
101	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
102	#137×C₆₇	—	2	16	no	—
103	#137×C₆₇	—	2	32	no	—
104	#137×C₆₇	C₂³	1	32	yes	complemented
105	C₂×Q₁₆×C₆₇	—	2	8	no	—
106	C₂×Q₁₆×C₆₇	—	2	16	no	—
107	C₂×SD₁₆×C₆₇	—	2	8	no	—
108	C₂×SD₁₆×C₆₇	—	2	16	no	—
109	C₂×M₁₆×C₆₇	—	2	2	no	—
110	C₂×M₁₆×C₆₇	—	2	4	no	—
111	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
112	C₂×M₁₆×C₆₇	C₂³	1	8	yes	complemented
113	C₂²×C₈×C₆₇	—	2	2	no	—
114	C₄⋊Q₈×C₆₇	—	2	4	no	—
115	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
116	#124×C₆₇	—	2	4	no	—
117	#124×C₆₇	C₂³	1	8	yes	complemented
118	#123×C₆₇	—	2	16	no	—
119	C₂²⋊Q₈×C₆₇	—	2	16	no	—
120	#117×C₆₇	—	2	2	no	—
121	#117×C₆₇	C₂³	1	4	yes	complemented
122	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
124	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
125	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
126	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
127	#104×C₆₇	C₂³	1	32	yes	complemented
128	#101×C₆₇	C₂³	1	8	yes	complemented
129	#100×C₆₇	C₂³	1	8	yes	complemented
130	C₂⁴×C₆₇	—	2	4	no	—
131	C₂⁴×C₆₇	—	2	2	no	—
132	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
133	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
134	#41×C₆₇	—	2	32	no	—
135	#41×C₆₇	—	2	64	no	—
136	#41×C₆₇	—	2	16	no	—
137	#41×C₆₇	—	2	16	no	—
138	#41×C₆₇	—	2	16	no	—
139	#41×C₆₇	—	2	32	no	—
140	#41×C₆₇	—	2	16	no	—
141	#41×C₆₇	—	2	16	no	—
142	#41×C₆₇	—	2	8	no	—
143	#41×C₆₇	—	2	8	no	—
144	#41×C₆₇	—	2	8	no	—
145	#41×C₆₇	—	2	8	no	—
146	#41×C₆₇	—	2	8	no	—
147	#41×C₆₇	—	2	8	no	—
148	#41×C₆₇	—	2	8	no	—
149	#41×C₆₇	—	2	16	no	—
150	#41×C₆₇	—	2	4	no	—
151	#41×C₆₇	—	2	8	no	—
152	#41×C₆₇	—	2	8	no	—
153	#41×C₆₇	—	2	4	no	—
154	#41×C₆₇	C₂×D₄	1	16	yes	—
155	#41×C₆₇	C₂×D₄	1	8	yes	—
156	#41×C₆₇	C₂⁴	1	8	yes	—
157	C₂×Q₈×C₆₇	—	2	4	no	—
158	C₂×Q₈×C₆₇	—	2	16	no	—
159	C₂×Q₈×C₆₇	—	2	16	no	—
160	C₂×Q₈×C₆₇	—	2	16	no	—
161	C₂×Q₈×C₆₇	—	2	2	no	—
162	C₂×Q₈×C₆₇	—	2	4	no	—
163	C₂×Q₈×C₆₇	—	2	2	no	—
164	C₂×Q₈×C₆₇	—	2	2	no	—
165	C₂×Q₈×C₆₇	—	2	4	no	—
166	C₂×Q₈×C₆₇	—	2	2	no	—
167	C₂×Q₈×C₆₇	—	2	4	no	—
168	C₂×Q₈×C₆₇	—	2	8	no	—
169	C₂×Q₈×C₆₇	—	2	4	no	—
170	C₂×Q₈×C₆₇	—	2	8	no	—
171	C₂×Q₈×C₆₇	—	4	2	no	—
172	C₂×Q₈×C₆₇	—	4	4	no	—
173	C₂×Q₈×C₆₇	—	4	4	no	—
174	C₂×Q₈×C₆₇	—	4	1	no	—
175	C₂×Q₈×C₆₇	—	4	1	no	—
176	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
177	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
178	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
179	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
180	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
181	C₂×D₄×C₆₇	—	2	8	no	—
182	C₂×D₄×C₆₇	—	2	4	no	—
183	C₂×D₄×C₆₇	—	2	16	no	—
184	C₂×D₄×C₆₇	—	2	64	no	—
185	C₂×D₄×C₆₇	—	2	16	no	—
186	C₂×D₄×C₆₇	—	2	16	no	—
187	C₂×D₄×C₆₇	—	2	4	no	—
188	C₂×D₄×C₆₇	—	2	4	no	—
189	C₂×D₄×C₆₇	—	2	8	no	—
190	C₂×D₄×C₆₇	—	2	4	no	—
191	C₂×D₄×C₆₇	—	2	4	no	—
192	C₂×D₄×C₆₇	—	2	16	no	—
193	C₂×D₄×C₆₇	—	2	8	no	—
194	C₂×D₄×C₆₇	—	2	16	no	—
195	C₂×D₄×C₆₇	—	2	2	no	—
196	C₂×D₄×C₆₇	—	2	4	no	—
197	C₂×D₄×C₆₇	—	2	2	no	—
198	C₂×D₄×C₆₇	—	2	2	no	—
199	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
200	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
201	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
202	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
203	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₆₇	—	2	4	no	—
205	C₂²×C₄×C₆₇	—	2	8	no	—
206	C₂²×C₄×C₆₇	—	2	32	no	—
207	C₂²×C₄×C₆₇	—	2	2	no	—
208	C₂²×C₄×C₆₇	—	2	4	no	—
209	C₂²×C₄×C₆₇	—	2	2	no	—
210	C₂²×C₄×C₆₇	—	2	8	no	—
211	C₂²×C₄×C₆₇	—	2	2	no	—
212	C₂²×C₄×C₆₇	—	2	4	no	—
213	C₂²×C₄×C₆₇	—	2	4	no	—
214	C₂²×C₄×C₆₇	—	2	8	no	—
215	C₂²×C₄×C₆₇	—	2	2	no	—
216	C₂²×C₄×C₆₇	—	2	2	no	—
217	C₂²×C₄×C₆₇	—	2	4	no	—
218	C₂²×C₄×C₆₇	—	4	2	no	—
219	C₂²×C₄×C₆₇	—	4	2	no	—
220	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
221	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
222	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
223	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
224	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
225	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
226	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
227	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
228	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
229	Q₁₆×C₆₇	—	2	32	no	—
230	SD₁₆×C₆₇	—	2	32	no	—
231	M₁₆×C₆₇	—	2	4	no	—
232	M₁₆×C₆₇	—	2	8	no	—
233	M₁₆×C₆₇	C₂⁴	1	8	yes	—
234	C₂×C₈×C₆₇	—	2	4	no	—
235	C₂×C₈×C₆₇	—	2	8	no	—
236	C₄⋊C₄×C₆₇	—	2	4	no	—
237	C₄⋊C₄×C₆₇	—	2	16	no	—
238	C₂²⋊C₄×C₆₇	—	2	4	no	—
239	C₂²⋊C₄×C₆₇	—	2	16	no	—
240	C₄²×C₆₇	—	2	2	no	—
241	C₄²×C₆₇	C₂⁴	1	4	yes	—
242	C₂³×C₆₇	—	2	16	no	—
243	C₂³×C₆₇	—	2	2	no	—
244	C₂³×C₆₇	—	2	4	no	—
245	C₂³×C₆₇	—	2	2	no	—
246	C₂³×C₆₇	—	2	8	no	—
247	C₂³×C₆₇	—	4	2	no	—
248	C₂³×C₆₇	—	4	4	no	—
249	C₂³×C₆₇	—	8	4	no	—
250	C₂³×C₆₇	—	8	8	no	—
251	C₂³×C₆₇	#120	1	2	yes	—
252	C₂³×C₆₇	#120	1	2	yes	—
253	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
254	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
255	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
256	Q₈×C₆₇	—	2	16	no	—
257	Q₈×C₆₇	—	2	8	no	—
258	Q₈×C₆₇	—	2	4	no	—
259	Q₈×C₆₇	—	2	4	no	—
260	Q₈×C₆₇	—	2	4	no	—
261	Q₈×C₆₇	—	2	8	no	—
262	Q₈×C₆₇	—	4	4	no	—
263	Q₈×C₆₇	—	4	4	no	—
264	Q₈×C₆₇	—	4	4	no	—
265	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
266	D₄×C₆₇	—	2	32	no	—
267	D₄×C₆₇	—	2	16	no	—
268	D₄×C₆₇	—	2	16	no	—
269	D₄×C₆₇	—	2	8	no	—
270	D₄×C₆₇	—	2	16	no	—
271	D₄×C₆₇	—	2	8	no	—
272	D₄×C₆₇	—	2	4	no	—
273	D₄×C₆₇	—	2	8	no	—
274	D₄×C₆₇	—	2	4	no	—
275	D₄×C₆₇	—	2	4	no	—
276	D₄×C₆₇	C₂²×D₄	1	8	yes	—
277	C₂×C₄×C₆₇	—	2	16	no	—
278	C₂×C₄×C₆₇	—	2	32	no	—
279	C₂×C₄×C₆₇	—	2	8	no	—
280	C₂×C₄×C₆₇	—	2	2	no	—
281	C₂×C₄×C₆₇	—	2	4	no	—
282	C₂×C₄×C₆₇	—	2	2	no	—
283	C₂×C₄×C₆₇	—	2	16	no	—
284	C₂×C₄×C₆₇	—	2	8	no	—
285	C₂×C₄×C₆₇	—	2	8	no	—
286	C₂×C₄×C₆₇	—	2	8	no	—
287	C₂×C₄×C₆₇	—	2	8	no	—
288	C₂×C₄×C₆₇	—	2	4	no	—
289	C₂×C₄×C₆₇	—	2	4	no	—
290	C₂×C₄×C₆₇	—	4	4	no	—
291	C₂×C₄×C₆₇	—	4	2	no	—
292	C₂×C₄×C₆₇	—	4	8	no	—
293	C₂×C₄×C₆₇	—	4	2	no	—
294	C₂×C₄×C₆₇	#120	1	4	yes	—
295	C₂×C₄×C₆₇	#120	1	2	yes	—
296	C₂×C₄×C₆₇	#120	1	4	yes	—
297	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
298	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
299	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
300	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
301	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
302	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
303	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
304	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
305	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
306	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
307	C₈×C₆₇	—	2	8	no	—
308	C₂²×C₆₇	—	2	2	no	—
309	C₂²×C₆₇	—	2	4	no	—
310	C₂²×C₆₇	—	2	4	no	—
311	C₂²×C₆₇	—	2	8	no	—
312	C₂²×C₆₇	—	4	2	no	—
313	C₂²×C₆₇	—	4	2	no	—
314	C₂²×C₆₇	—	4	4	no	—
315	C₂²×C₆₇	—	4	4	no	—
316	C₂²×C₆₇	—	4	8	no	—
317	C₂²×C₆₇	—	8	2	no	—
318	C₂²×C₆₇	—	8	8	no	—
319	C₂²×C₆₇	—	8	16	no	—
320	C₂²×C₆₇	C₂×#120	1	4	yes	—
321	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
322	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
323	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
324	C₄×C₆₇	—	2	8	no	—
325	C₄×C₆₇	—	2	4	no	—
326	C₄×C₆₇	—	4	4	no	—
327	C₄×C₆₇	—	4	4	no	—
328	C₄×C₆₇	C₂×#120	1	2	yes	—
329	C₄×C₆₇	C₂×#120	1	2	yes	—
330	C₄×C₆₇	C₂×#120	1	2	yes	—
331	C₄×C₆₇	C₂³×D₄	1	2	yes	—
332	C₂×#3022	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
333	C₂×C₆₇	—	2	1	no	—
334	C₂×C₆₇	—	2	1	no	—
335	C₂×C₆₇	—	2	4	no	—
336	C₂×C₆₇	—	4	4	no	—
337	C₂×C₆₇	—	4	8	no	—
338	C₂×C₆₇	#3022	1	2	yes	direct product factor
339	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
340	C₂×#585	C₂×C₆₇	1	1	yes	characteristic
341	C₂²×#137	C₂×C₆₇	1	2	yes	complemented
342	C₂×#536	C₂×C₆₇	1	1	yes	characteristic, complemented
343	#3022	C₂×C₆₇	1	16	yes	direct product factor
344	C₂×#456	C₂×C₆₇	1	4	yes	complemented
345	C₂×#455	C₂×C₆₇	1	4	yes	complemented
346	C₂×#421	C₂×C₆₇	1	2	yes	complemented
347	C₂×#413	C₂×C₆₇	1	1	yes	characteristic, complemented
348	C₆₇	C₂×#3022	1	1	yes	characteristic, direct product factor, Sylow
349	#585	—	2	8	no	—
350	#585	—	2	4	no	—
351	#585	C₂²×C₆₇	1	8	yes	—
352	C₂×2₋¹⁺⁴	—	2	2	no	—
353	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	2	yes	—
354	C₂×H(5,𝔽₂)	—	2	2	no	—
355	C₂×H(5,𝔽₂)	—	2	2	no	—
356	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	2	yes	—
357	C₂²×#41	—	2	4	no	—
358	C₂²×#41	—	2	2	no	—
359	C₂²×#41	C₂²×C₆₇	1	2	yes	—
360	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
361	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
362	C₂×#137	—	2	4	no	—
363	C₂×#137	—	2	8	no	—
364	C₂×#137	C₂²×C₆₇	1	8	yes	complemented
365	C₂×#137	C₂²×C₆₇	1	16	yes	complemented
366	C₂²×Q₁₆	—	2	2	no	—
367	C₂²×SD₁₆	—	2	2	no	—
368	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
369	#536	—	2	4	no	—
370	#536	C₂²×C₆₇	1	8	yes	complemented
371	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
372	C₂×#124	C₂²×C₆₇	1	2	yes	complemented
373	C₂×#123	—	2	2	no	—
374	C₂×C₂²⋊Q₈	—	2	2	no	—
375	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic, complemented
376	#456	C₂²×C₆₇	1	32	yes	complemented
377	#455	C₂²×C₆₇	1	32	yes	complemented
378	#421	C₂²×C₆₇	1	16	yes	complemented
379	C₂×#104	C₂²×C₆₇	1	8	yes	complemented
380	#413	C₂²×C₆₇	1	8	yes	complemented
381	C₂×#101	C₂²×C₆₇	1	2	yes	complemented
382	C₂×#100	C₂²×C₆₇	1	2	yes	complemented
383	2₋¹⁺⁴	—	2	16	no	—
384	2₋¹⁺⁴	—	2	8	no	—
385	2₋¹⁺⁴	—	2	16	no	—
386	2₋¹⁺⁴	—	2	4	no	—
387	2₋¹⁺⁴	C₂³×C₆₇	1	8	yes	—
388	H(5,𝔽₂)	—	2	16	no	—
389	H(5,𝔽₂)	—	2	16	no	—
390	H(5,𝔽₂)	—	2	8	no	—
391	H(5,𝔽₂)	—	2	16	no	—
392	H(5,𝔽₂)	—	2	16	no	—
393	H(5,𝔽₂)	—	2	4	no	—
394	H(5,𝔽₂)	C₂³×C₆₇	1	8	yes	—
395	C₂×#41	—	2	4	no	—
396	C₂×#41	—	2	8	no	—
397	C₂×#41	—	2	4	no	—
398	C₂×#41	—	2	32	no	—
399	C₂×#41	—	2	16	no	—
400	C₂×#41	—	2	32	no	—
401	C₂×#41	—	2	4	no	—
402	C₂×#41	—	2	4	no	—
403	C₂×#41	—	2	2	no	—
404	C₂×#41	—	2	2	no	—
405	C₂×#41	—	2	4	no	—
406	C₂×#41	—	2	16	no	—
407	C₂×#41	—	2	2	no	—
408	C₂×#41	—	2	2	no	—
409	C₂×#41	—	2	4	no	—
410	C₂×#41	—	2	4	no	—
411	C₂×#41	—	2	4	no	—
412	C₂×#41	D₄×C₆₇	1	4	yes	—
413	C₂×#41	D₄×C₆₇	1	8	yes	—
414	C₂×#41	D₄×C₆₇	1	4	yes	—
415	C₂×#41	C₂³×C₆₇	1	4	yes	—
416	C₂×#41	C₂³×C₆₇	1	8	yes	—
417	C₂×#41	C₂³×C₆₇	1	4	yes	—
418	C₂²×Q₈	—	2	2	no	—
419	C₂²×Q₈	—	2	4	no	—
420	C₂²×Q₈	—	2	1	no	—
421	C₂²×Q₈	—	2	2	no	—
422	C₂²×Q₈	—	2	2	no	—
423	C₂²×Q₈	—	2	2	no	—
424	C₂²×Q₈	—	2	4	no	—
425	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
426	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
427	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
428	C₂²×Q₈	C₂³×C₆₇	1	4	yes	complemented
429	C₂²×D₄	—	2	8	no	—
430	C₂²×D₄	—	2	2	no	—
431	C₂²×D₄	—	2	4	no	—
432	C₂²×D₄	—	2	2	no	—
433	C₂²×D₄	—	2	4	no	—
434	C₂²×D₄	D₄×C₆₇	1	2	yes	—
435	C₂²×D₄	D₄×C₆₇	1	2	yes	—
436	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
437	C₂³×C₄	—	2	4	no	—
438	C₂³×C₄	—	2	2	no	—
439	C₂³×C₄	—	2	1	no	—
440	C₂³×C₄	D₄×C₆₇	1	2	yes	—
441	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
442	#137	—	2	16	no	—
443	#137	—	2	32	no	—
444	#137	C₂³×C₆₇	1	32	yes	complemented
445	C₂×Q₁₆	—	2	8	no	—
446	C₂×Q₁₆	—	2	16	no	—
447	C₂×SD₁₆	—	2	8	no	—
448	C₂×SD₁₆	—	2	16	no	—
449	C₂×M₁₆	—	2	2	no	—
450	C₂×M₁₆	—	2	4	no	—
451	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
452	C₂×M₁₆	C₂³×C₆₇	1	8	yes	complemented
453	C₂²×C₈	—	2	2	no	—
454	C₄⋊Q₈	—	2	4	no	—
455	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
456	#124	—	2	4	no	—
457	#124	C₂³×C₆₇	1	8	yes	complemented
458	#123	—	2	16	no	—
459	C₂²⋊Q₈	—	2	16	no	—
460	#117	—	2	2	no	—
461	#117	C₂³×C₆₇	1	4	yes	complemented
462	C₂×C₄⋊C₄	—	2	1	no	—
463	C₂×C₄⋊C₄	—	2	4	no	—
464	C₂×C₂²⋊C₄	—	2	1	no	—
465	C₂×C₂²⋊C₄	—	2	4	no	—
466	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
467	#104	C₂³×C₆₇	1	32	yes	complemented
468	#101	C₂³×C₆₇	1	8	yes	complemented
469	#100	C₂³×C₆₇	1	8	yes	complemented
470	C₂⁴	—	2	4	no	—
471	C₂⁴	—	2	2	no	—
472	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
473	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
474	#41	—	2	32	no	—
475	#41	—	2	64	no	—
476	#41	—	2	16	no	—
477	#41	—	2	16	no	—
478	#41	—	2	16	no	—
479	#41	—	2	32	no	—
480	#41	—	2	16	no	—
481	#41	—	2	16	no	—
482	#41	—	2	8	no	—
483	#41	—	2	8	no	—
484	#41	—	2	8	no	—
485	#41	—	2	8	no	—
486	#41	—	2	8	no	—
487	#41	—	2	8	no	—
488	#41	—	2	8	no	—
489	#41	—	2	16	no	—
490	#41	—	2	4	no	—
491	#41	—	2	8	no	—
492	#41	—	2	8	no	—
493	#41	—	2	4	no	—
494	#41	C₂×D₄×C₆₇	1	16	yes	—
495	#41	C₂×D₄×C₆₇	1	8	yes	—
496	#41	C₂⁴×C₆₇	1	8	yes	—
497	C₂×Q₈	—	2	4	no	—
498	C₂×Q₈	—	2	16	no	—
499	C₂×Q₈	—	2	16	no	—
500	C₂×Q₈	—	2	16	no	—
501	C₂×Q₈	—	2	2	no	—
502	C₂×Q₈	—	2	4	no	—
503	C₂×Q₈	—	2	2	no	—
504	C₂×Q₈	—	2	2	no	—
505	C₂×Q₈	—	2	4	no	—
506	C₂×Q₈	—	2	2	no	—
507	C₂×Q₈	—	2	4	no	—
508	C₂×Q₈	—	2	8	no	—
509	C₂×Q₈	—	2	4	no	—
510	C₂×Q₈	—	2	8	no	—
511	C₂×Q₈	—	4	1	no	—
512	C₂×Q₈	—	4	1	no	—
513	C₂×Q₈	—	4	2	no	—
514	C₂×Q₈	—	4	4	no	—
515	C₂×Q₈	—	4	4	no	—
516	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
517	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
518	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
519	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
520	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
521	C₂×D₄	—	2	8	no	—
522	C₂×D₄	—	2	4	no	—
523	C₂×D₄	—	2	16	no	—
524	C₂×D₄	—	2	64	no	—
525	C₂×D₄	—	2	16	no	—
526	C₂×D₄	—	2	16	no	—
527	C₂×D₄	—	2	4	no	—
528	C₂×D₄	—	2	4	no	—
529	C₂×D₄	—	2	8	no	—
530	C₂×D₄	—	2	4	no	—
531	C₂×D₄	—	2	4	no	—
532	C₂×D₄	—	2	16	no	—
533	C₂×D₄	—	2	8	no	—
534	C₂×D₄	—	2	16	no	—
535	C₂×D₄	—	2	2	no	—
536	C₂×D₄	—	2	4	no	—
537	C₂×D₄	—	2	2	no	—
538	C₂×D₄	—	2	2	no	—
539	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
540	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
541	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
542	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
543	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
544	C₂²×C₄	—	2	4	no	—
545	C₂²×C₄	—	2	8	no	—
546	C₂²×C₄	—	2	32	no	—
547	C₂²×C₄	—	2	2	no	—
548	C₂²×C₄	—	2	4	no	—
549	C₂²×C₄	—	2	2	no	—
550	C₂²×C₄	—	2	8	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	4	no	—
553	C₂²×C₄	—	2	4	no	—
554	C₂²×C₄	—	2	8	no	—
555	C₂²×C₄	—	2	2	no	—
556	C₂²×C₄	—	2	2	no	—
557	C₂²×C₄	—	2	4	no	—
558	C₂²×C₄	—	4	2	no	—
559	C₂²×C₄	—	4	2	no	—
560	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
561	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
562	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
563	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
564	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
565	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
566	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
567	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
568	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
569	Q₁₆	—	2	32	no	—
570	SD₁₆	—	2	32	no	—
571	M₁₆	—	2	4	no	—
572	M₁₆	—	2	8	no	—
573	M₁₆	C₂⁴×C₆₇	1	8	yes	—
574	C₂×C₈	—	2	4	no	—
575	C₂×C₈	—	2	8	no	—
576	C₄⋊C₄	—	2	4	no	—
577	C₄⋊C₄	—	2	16	no	—
578	C₂²⋊C₄	—	2	4	no	—
579	C₂²⋊C₄	—	2	16	no	—
580	C₄²	—	2	2	no	—
581	C₄²	C₂⁴×C₆₇	1	4	yes	—
582	C₂³	—	2	16	no	—
583	C₂³	—	2	2	no	—
584	C₂³	—	2	4	no	—
585	C₂³	—	2	2	no	—
586	C₂³	—	2	8	no	—
587	C₂³	—	4	2	no	—
588	C₂³	—	4	4	no	—
589	C₂³	—	8	4	no	—
590	C₂³	—	8	8	no	—
591	C₂³	#120×C₆₇	1	2	yes	—
592	C₂³	#120×C₆₇	1	2	yes	—
593	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
594	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
595	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
596	Q₈	—	2	16	no	—
597	Q₈	—	2	8	no	—
598	Q₈	—	2	4	no	—
599	Q₈	—	2	4	no	—
600	Q₈	—	2	4	no	—
601	Q₈	—	2	8	no	—
602	Q₈	—	4	4	no	—
603	Q₈	—	4	4	no	—
604	Q₈	—	4	4	no	—
605	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
606	D₄	—	2	32	no	—
607	D₄	—	2	16	no	—
608	D₄	—	2	16	no	—
609	D₄	—	2	8	no	—
610	D₄	—	2	16	no	—
611	D₄	—	2	8	no	—
612	D₄	—	2	4	no	—
613	D₄	—	2	8	no	—
614	D₄	—	2	4	no	—
615	D₄	—	2	4	no	—
616	D₄	C₂²×D₄×C₆₇	1	8	yes	—
617	C₂×C₄	—	2	16	no	—
618	C₂×C₄	—	2	32	no	—
619	C₂×C₄	—	2	4	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	8	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	16	no	—
624	C₂×C₄	—	2	8	no	—
625	C₂×C₄	—	2	8	no	—
626	C₂×C₄	—	2	8	no	—
627	C₂×C₄	—	2	8	no	—
628	C₂×C₄	—	2	4	no	—
629	C₂×C₄	—	2	4	no	—
630	C₂×C₄	—	4	4	no	—
631	C₂×C₄	—	4	2	no	—
632	C₂×C₄	—	4	8	no	—
633	C₂×C₄	—	4	2	no	—
634	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
635	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
636	C₂×C₄	#120×C₆₇	1	4	yes	—
637	C₂×C₄	#120×C₆₇	1	2	yes	—
638	C₂×C₄	#120×C₆₇	1	4	yes	—
639	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
640	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
641	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
642	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
643	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
644	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
645	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
646	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
647	C₈	—	2	8	no	—
648	C₂²	—	2	2	no	—
649	C₂²	—	2	4	no	—
650	C₂²	—	2	4	no	—
651	C₂²	—	2	8	no	—
652	C₂²	—	4	2	no	—
653	C₂²	—	4	2	no	—
654	C₂²	—	4	4	no	—
655	C₂²	—	4	4	no	—
656	C₂²	—	4	8	no	—
657	C₂²	—	8	2	no	—
658	C₂²	—	8	8	no	—
659	C₂²	—	8	16	no	—
660	C₂²	C₂×#120×C₆₇	1	4	yes	—
661	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
662	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
663	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
664	C₄	—	2	8	no	—
665	C₄	—	2	4	no	—
666	C₄	—	4	4	no	—
667	C₄	—	4	4	no	—
668	C₄	C₂×#120×C₆₇	1	2	yes	—
669	C₄	C₂×#120×C₆₇	1	2	yes	—
670	C₄	C₂×#120×C₆₇	1	2	yes	—
671	C₄	C₂³×D₄×C₆₇	1	2	yes	—
672	C₂	—	2	1	no	—
673	C₂	—	2	1	no	—
674	C₂	—	2	4	no	—
675	C₂	—	4	4	no	—
676	C₂	—	4	8	no	—
677	C₂	C₆₇×#3022	1	2	yes	direct product factor
678	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#585×C₆₇	C₂
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
2. C₂²×#137×C₆₇	C₂	semidirect
3. C₂×#536×C₆₇	C₂	semidirect
4. C₆₇×#3022	C₂	direct (N × H)
5. C₂×#456×C₆₇	C₂	semidirect
6. C₂×#455×C₆₇	C₂	semidirect
7. C₂×#421×C₆₇	C₂	semidirect
8. C₂×#413×C₆₇	C₂	semidirect
21. C₂³×Q₈×C₆₇	C₂²	semidirect
24. C₂×#137×C₆₇	C₂²	semidirect
25. C₂×#137×C₆₇	C₂²	semidirect
28. C₂²×M₁₆×C₆₇	C₂²	semidirect
30. #536×C₆₇	C₂²	semidirect
31. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
32. C₂×#124×C₆₇	C₂²	semidirect
35. C₂×#117×C₆₇	C₂²	semidirect
36. #456×C₆₇	C₂²	semidirect
37. #455×C₆₇	C₂²	semidirect
38. #421×C₆₇	C₂²	semidirect
39. C₂×#104×C₆₇	C₂²	semidirect
40. #413×C₆₇	C₂²	semidirect
41. C₂×#101×C₆₇	C₂²	semidirect
42. C₂×#100×C₆₇	C₂²	semidirect
88. C₂²×Q₈×C₆₇	C₂³	semidirect
104. #137×C₆₇	C₂³	semidirect
112. C₂×M₁₆×C₆₇	C₂³	semidirect
115. C₄⋊Q₈×C₆₇	C₂³	semidirect
117. #124×C₆₇	C₂³	semidirect
121. #117×C₆₇	C₂³	semidirect
127. #104×C₆₇	C₂³	semidirect
128. #101×C₆₇	C₂³	semidirect
129. #100×C₆₇	C₂³	semidirect
332. C₂×#3022	C₆₇	direct (N × H)
338. C₂×C₆₇	#3022	direct (N × H)
341. C₂²×#137	C₂×C₆₇	semidirect
342. C₂×#536	C₂×C₆₇	semidirect
343. #3022	C₂×C₆₇	direct (N × H)
344. C₂×#456	C₂×C₆₇	semidirect
345. C₂×#455	C₂×C₆₇	semidirect
346. C₂×#421	C₂×C₆₇	semidirect
347. C₂×#413	C₂×C₆₇	semidirect
348. C₆₇	C₂×#3022	direct (N × H)
361. C₂³×Q₈	C₂²×C₆₇	semidirect
364. C₂×#137	C₂²×C₆₇	semidirect
365. C₂×#137	C₂²×C₆₇	semidirect
368. C₂²×M₁₆	C₂²×C₆₇	semidirect
370. #536	C₂²×C₆₇	semidirect
371. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
372. C₂×#124	C₂²×C₆₇	semidirect
375. C₂×#117	C₂²×C₆₇	semidirect
376. #456	C₂²×C₆₇	semidirect
377. #455	C₂²×C₆₇	semidirect
378. #421	C₂²×C₆₇	semidirect
379. C₂×#104	C₂²×C₆₇	semidirect
380. #413	C₂²×C₆₇	semidirect
381. C₂×#101	C₂²×C₆₇	semidirect
382. C₂×#100	C₂²×C₆₇	semidirect
428. C₂²×Q₈	C₂³×C₆₇	semidirect
444. #137	C₂³×C₆₇	semidirect
452. C₂×M₁₆	C₂³×C₆₇	semidirect
455. C₄⋊Q₈	C₂³×C₆₇	semidirect
457. #124	C₂³×C₆₇	semidirect
461. #117	C₂³×C₆₇	semidirect
467. #104	C₂³×C₆₇	semidirect
468. #101	C₂³×C₆₇	semidirect
469. #100	C₂³×C₆₇	semidirect
677. C₂	C₆₇×#3022	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#585×C₆₇	2
C₂²×#137×C₆₇	2
C₂×#536×C₆₇	2
C₆₇×#3022	2
C₂×#456×C₆₇	2
C₂×#455×C₆₇	2
C₂×#421×C₆₇	2
C₂×#413×C₆₇	2
C₂×#3022	67

Subgroups — C2×C67×#3022 #30739637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3022 #30739637