Subgroups — C₂×C₆₇×#3020 #30739635

Order 17152.

4710 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#575×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#536×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3020	C₂	1	16	yes	maximal, direct product factor
5	C₂×#456×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#454×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#421×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂²×#101×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×2₋¹⁺⁴×C₆₇	—	2	4	no	—
10	C₂×2₋¹⁺⁴×C₆₇	—	2	8	no	—
11	C₂×2₋¹⁺⁴×C₆₇	—	2	2	no	—
12	C₂×2₋¹⁺⁴×C₆₇	—	2	4	no	—
13	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	4	yes	—
14	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	8	yes	complemented
15	C₂²×#41×C₆₇	—	2	2	no	—
16	C₂²×#41×C₆₇	—	2	2	no	—
17	C₂²×#41×C₆₇	C₂²	1	2	yes	—
18	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×Q₈×C₆₇	—	2	2	no	—
20	C₂³×Q₈×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	—
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
36	#456×C₆₇	C₂²	1	32	yes	complemented
37	#454×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×#101×C₆₇	C₂²	1	4	yes	complemented
41	C₂×#101×C₆₇	C₂²	1	8	yes	complemented
42	2₋¹⁺⁴×C₆₇	—	2	32	no	—
43	2₋¹⁺⁴×C₆₇	—	2	16	no	—
44	2₋¹⁺⁴×C₆₇	—	2	16	no	—
45	2₋¹⁺⁴×C₆₇	—	2	32	no	—
46	2₋¹⁺⁴×C₆₇	—	2	8	no	—
47	2₋¹⁺⁴×C₆₇	—	2	16	no	—
48	2₋¹⁺⁴×C₆₇	C₂³	1	16	yes	—
49	C₂×#41×C₆₇	—	2	16	no	—
50	C₂×#41×C₆₇	—	2	8	no	—
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	8	no	—
56	C₂×#41×C₆₇	—	2	16	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	4	no	—
61	C₂×#41×C₆₇	—	2	2	no	—
62	C₂×#41×C₆₇	—	2	2	no	—
63	C₂×#41×C₆₇	—	2	2	no	—
64	C₂×#41×C₆₇	—	2	4	no	—
65	C₂×#41×C₆₇	D₄	1	4	yes	—
66	C₂×#41×C₆₇	D₄	1	8	yes	—
67	C₂×#41×C₆₇	C₂³	1	4	yes	—
68	C₂×#41×C₆₇	C₂³	1	8	yes	—
69	C₂×#41×C₆₇	C₂³	1	4	yes	—
70	C₂²×Q₈×C₆₇	—	2	2	no	—
71	C₂²×Q₈×C₆₇	—	2	16	no	—
72	C₂²×Q₈×C₆₇	—	2	8	no	—
73	C₂²×Q₈×C₆₇	—	2	8	no	—
74	C₂²×Q₈×C₆₇	—	2	16	no	—
75	C₂²×Q₈×C₆₇	—	2	2	no	—
76	C₂²×Q₈×C₆₇	—	2	2	no	—
77	C₂²×Q₈×C₆₇	—	2	4	no	—
78	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
79	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
80	C₂²×Q₈×C₆₇	D₄	1	4	yes	—
81	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
83	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
84	C₂²×D₄×C₆₇	—	2	2	no	—
85	C₂²×D₄×C₆₇	—	2	2	no	—
86	C₂²×D₄×C₆₇	—	2	1	no	—
87	C₂²×D₄×C₆₇	D₄	1	2	yes	—
88	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
89	C₂³×C₄×C₆₇	—	2	4	no	—
90	C₂³×C₄×C₆₇	—	2	2	no	—
91	C₂³×C₄×C₆₇	—	2	1	no	—
92	C₂³×C₄×C₆₇	D₄	1	2	yes	—
93	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
94	#137×C₆₇	—	2	8	no	—
95	#137×C₆₇	—	2	16	no	—
96	#137×C₆₇	C₂³	1	16	yes	complemented
97	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
98	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
99	AGL(1,ℤ/8)×C₆₇	C₂³	1	16	yes	complemented
100	#135×C₆₇	—	2	32	no	—
101	C₂×Q₁₆×C₆₇	—	2	4	no	—
102	C₂×SD₁₆×C₆₇	—	2	4	no	—
103	C₂×SD₁₆×C₆₇	—	2	4	no	—
104	C₂×D₈×C₆₇	—	2	4	no	—
105	C₂×M₁₆×C₆₇	—	2	2	no	—
106	C₂×M₁₆×C₆₇	—	2	4	no	—
107	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
108	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
109	C₂²×C₈×C₆₇	—	2	2	no	—
110	C₄⋊Q₈×C₆₇	—	2	4	no	—
111	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	complemented
112	#124×C₆₇	—	2	4	no	—
113	#124×C₆₇	C₂³	1	8	yes	complemented
114	#123×C₆₇	—	2	16	no	—
115	C₂²⋊Q₈×C₆₇	—	2	16	no	—
116	#117×C₆₇	—	2	2	no	—
117	#117×C₆₇	C₂³	1	4	yes	—
118	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₂×C₄⋊C₄×C₆₇	—	2	4	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	—
123	#104×C₆₇	C₂³	1	32	yes	complemented
124	#101×C₆₇	C₂³	1	16	yes	complemented
125	C₂⁴×C₆₇	—	2	2	no	—
126	C₂⁴×C₆₇	—	2	1	no	—
127	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
128	#41×C₆₇	—	2	32	no	—
129	#41×C₆₇	—	2	64	no	—
130	#41×C₆₇	—	2	32	no	—
131	#41×C₆₇	—	2	64	no	—
132	#41×C₆₇	—	2	32	no	—
133	#41×C₆₇	—	2	8	no	—
134	#41×C₆₇	—	2	8	no	—
135	#41×C₆₇	—	2	8	no	—
136	#41×C₆₇	—	2	32	no	—
137	#41×C₆₇	—	2	4	no	—
138	#41×C₆₇	—	2	8	no	—
139	#41×C₆₇	—	2	8	no	—
140	#41×C₆₇	—	2	4	no	—
141	#41×C₆₇	—	2	8	no	—
142	#41×C₆₇	—	2	8	no	—
143	#41×C₆₇	—	4	4	no	—
144	#41×C₆₇	—	4	4	no	—
145	#41×C₆₇	C₂×D₄	1	16	yes	—
146	#41×C₆₇	C₂×D₄	1	8	yes	—
147	#41×C₆₇	C₂⁴	1	8	yes	—
148	C₂×Q₈×C₆₇	—	2	16	no	—
149	C₂×Q₈×C₆₇	—	2	8	no	—
150	C₂×Q₈×C₆₇	—	2	32	no	—
151	C₂×Q₈×C₆₇	—	2	64	no	—
152	C₂×Q₈×C₆₇	—	2	32	no	—
153	C₂×Q₈×C₆₇	—	2	4	no	—
154	C₂×Q₈×C₆₇	—	2	8	no	—
155	C₂×Q₈×C₆₇	—	2	2	no	—
156	C₂×Q₈×C₆₇	—	2	4	no	—
157	C₂×Q₈×C₆₇	—	2	4	no	—
158	C₂×Q₈×C₆₇	—	2	8	no	—
159	C₂×Q₈×C₆₇	—	2	8	no	—
160	C₂×Q₈×C₆₇	—	2	4	no	—
161	C₂×Q₈×C₆₇	—	2	8	no	—
162	C₂×Q₈×C₆₇	—	2	4	no	—
163	C₂×Q₈×C₆₇	—	2	1	no	—
164	C₂×Q₈×C₆₇	—	2	2	no	—
165	C₂×Q₈×C₆₇	—	2	2	no	—
166	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
167	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
168	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	—
169	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
170	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
171	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
172	C₂×D₄×C₆₇	—	2	16	no	—
173	C₂×D₄×C₆₇	—	2	8	no	—
174	C₂×D₄×C₆₇	—	2	8	no	—
175	C₂×D₄×C₆₇	—	2	4	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	4	no	—
180	C₂×D₄×C₆₇	—	2	2	no	—
181	C₂×D₄×C₆₇	—	2	2	no	—
182	C₂×D₄×C₆₇	—	2	2	no	—
183	C₂×D₄×C₆₇	—	2	1	no	—
184	C₂×D₄×C₆₇	—	4	2	no	—
185	C₂×D₄×C₆₇	—	4	1	no	—
186	C₂×D₄×C₆₇	—	4	1	no	—
187	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
188	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
189	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₆₇	—	2	32	no	—
191	C₂²×C₄×C₆₇	—	2	16	no	—
192	C₂²×C₄×C₆₇	—	2	2	no	—
193	C₂²×C₄×C₆₇	—	2	2	no	—
194	C₂²×C₄×C₆₇	—	2	4	no	—
195	C₂²×C₄×C₆₇	—	2	2	no	—
196	C₂²×C₄×C₆₇	—	2	8	no	—
197	C₂²×C₄×C₆₇	—	2	4	no	—
198	C₂²×C₄×C₆₇	—	2	8	no	—
199	C₂²×C₄×C₆₇	—	2	4	no	—
200	C₂²×C₄×C₆₇	—	2	2	no	—
201	C₂²×C₄×C₆₇	—	2	1	no	—
202	C₂²×C₄×C₆₇	—	2	4	no	—
203	C₂²×C₄×C₆₇	—	4	2	no	—
204	C₂²×C₄×C₆₇	—	4	2	no	—
205	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
206	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
207	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
208	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
209	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
210	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
211	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
212	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
213	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
214	Q₁₆×C₆₇	—	2	16	no	—
215	SD₁₆×C₆₇	—	2	16	no	—
216	SD₁₆×C₆₇	—	2	16	no	—
217	D₈×C₆₇	—	2	16	no	—
218	M₁₆×C₆₇	—	2	8	no	—
219	M₁₆×C₆₇	—	2	4	no	—
220	M₁₆×C₆₇	C₂⁴	1	8	yes	—
221	C₂×C₈×C₆₇	—	2	4	no	—
222	C₂×C₈×C₆₇	—	2	8	no	—
223	C₄⋊C₄×C₆₇	—	2	4	no	—
224	C₄⋊C₄×C₆₇	—	2	16	no	—
225	C₂²⋊C₄×C₆₇	—	2	4	no	—
226	C₂²⋊C₄×C₆₇	—	2	16	no	—
227	C₄²×C₆₇	—	2	2	no	—
228	C₄²×C₆₇	C₂⁴	1	4	yes	—
229	C₂³×C₆₇	—	2	8	no	—
230	C₂³×C₆₇	—	2	4	no	—
231	C₂³×C₆₇	—	2	2	no	—
232	C₂³×C₆₇	—	4	2	no	—
233	C₂³×C₆₇	—	4	8	no	—
234	C₂³×C₆₇	—	4	4	no	—
235	C₂³×C₆₇	—	4	1	no	—
236	C₂³×C₆₇	—	4	1	no	—
237	C₂³×C₆₇	#120	1	1	yes	characteristic
238	C₂³×C₆₇	#120	1	1	yes	characteristic
239	C₂³×C₆₇	#120	1	2	yes	—
240	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
242	Q₈×C₆₇	—	2	32	no	—
243	Q₈×C₆₇	—	2	8	no	—
244	Q₈×C₆₇	—	2	4	no	—
245	Q₈×C₆₇	—	2	16	no	—
246	Q₈×C₆₇	—	2	8	no	—
247	Q₈×C₆₇	—	2	8	no	—
248	Q₈×C₆₇	—	2	4	no	—
249	Q₈×C₆₇	—	4	2	no	—
250	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
251	D₄×C₆₇	—	2	32	no	—
252	D₄×C₆₇	—	2	8	no	—
253	D₄×C₆₇	—	2	16	no	—
254	D₄×C₆₇	—	2	8	no	—
255	D₄×C₆₇	—	2	8	no	—
256	D₄×C₆₇	—	2	4	no	—
257	D₄×C₆₇	—	2	4	no	—
258	D₄×C₆₇	—	4	4	no	—
259	D₄×C₆₇	—	4	4	no	—
260	D₄×C₆₇	—	4	2	no	—
261	D₄×C₆₇	C₂²×D₄	1	8	yes	—
262	C₂×C₄×C₆₇	—	2	64	no	—
263	C₂×C₄×C₆₇	—	2	8	no	—
264	C₂×C₄×C₆₇	—	2	8	no	—
265	C₂×C₄×C₆₇	—	2	4	no	—
266	C₂×C₄×C₆₇	—	2	8	no	—
267	C₂×C₄×C₆₇	—	2	4	no	—
268	C₂×C₄×C₆₇	—	2	4	no	—
269	C₂×C₄×C₆₇	—	2	16	no	—
270	C₂×C₄×C₆₇	—	2	8	no	—
271	C₂×C₄×C₆₇	—	2	8	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	4	4	no	—
275	C₂×C₄×C₆₇	—	4	1	no	—
276	C₂×C₄×C₆₇	—	4	1	no	—
277	C₂×C₄×C₆₇	—	4	8	no	—
278	C₂×C₄×C₆₇	#120	1	4	yes	—
279	C₂×C₄×C₆₇	#120	1	2	yes	—
280	C₂×C₄×C₆₇	#120	1	4	yes	—
281	C₂×C₄×C₆₇	#120	1	2	yes	—
282	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
283	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
285	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
286	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
287	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
288	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
289	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
290	C₈×C₆₇	—	2	8	no	—
291	C₂²×C₆₇	—	2	2	no	—
292	C₂²×C₆₇	—	2	2	no	—
293	C₂²×C₆₇	—	2	2	no	—
294	C₂²×C₆₇	—	2	8	no	—
295	C₂²×C₆₇	—	4	2	no	—
296	C₂²×C₆₇	—	4	2	no	—
297	C₂²×C₆₇	—	4	16	no	—
298	C₂²×C₆₇	—	4	4	no	—
299	C₂²×C₆₇	—	4	8	no	—
300	C₂²×C₆₇	—	4	2	no	—
301	C₂²×C₆₇	—	8	2	no	—
302	C₂²×C₆₇	—	8	2	no	—
303	C₂²×C₆₇	C₂×#120	1	2	yes	—
304	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
305	C₂²×C₆₇	C₂×#120	1	2	yes	—
306	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
307	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
308	C₄×C₆₇	—	2	8	no	—
309	C₄×C₆₇	—	2	8	no	—
310	C₄×C₆₇	—	4	4	no	—
311	C₄×C₆₇	—	4	2	no	—
312	C₄×C₆₇	C₂×#120	1	4	yes	—
313	C₄×C₆₇	C₂×#120	1	2	yes	—
314	C₄×C₆₇	C₂³×D₄	1	2	yes	—
315	C₂×#3020	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
316	C₂×C₆₇	—	2	1	no	—
317	C₂×C₆₇	—	2	2	no	—
318	C₂×C₆₇	—	2	1	no	—
319	C₂×C₆₇	—	2	2	no	—
320	C₂×C₆₇	—	4	8	no	—
321	C₂×C₆₇	—	8	2	no	—
322	C₂×C₆₇	#3020	1	2	yes	direct product factor
323	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
324	C₂²×2₋¹⁺⁴	C₂×C₆₇	1	1	yes	characteristic, complemented
325	C₂×#575	C₂×C₆₇	1	2	yes	complemented
326	C₂×#536	C₂×C₆₇	1	1	yes	characteristic, complemented
327	#3020	C₂×C₆₇	1	16	yes	direct product factor
328	C₂×#456	C₂×C₆₇	1	4	yes	complemented
329	C₂×#454	C₂×C₆₇	1	4	yes	complemented
330	C₂×#421	C₂×C₆₇	1	2	yes	complemented
331	C₂²×#101	C₂×C₆₇	1	1	yes	characteristic, complemented
332	C₆₇	C₂×#3020	1	1	yes	characteristic, direct product factor, Sylow
333	C₂×2₋¹⁺⁴	—	2	4	no	—
334	C₂×2₋¹⁺⁴	—	2	8	no	—
335	C₂×2₋¹⁺⁴	—	2	2	no	—
336	C₂×2₋¹⁺⁴	—	2	4	no	—
337	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	4	yes	—
338	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	8	yes	complemented
339	C₂²×#41	—	2	2	no	—
340	C₂²×#41	—	2	2	no	—
341	C₂²×#41	C₂²×C₆₇	1	2	yes	—
342	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
343	C₂³×Q₈	—	2	2	no	—
344	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
345	#575	—	2	8	no	—
346	#575	C₂²×C₆₇	1	16	yes	complemented
347	C₂×#137	—	2	2	no	—
348	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
349	C₂×AGL(1,ℤ/8)	—	2	2	no	—
350	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	complemented
351	C₂×#135	—	2	4	no	—
352	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
353	#536	—	2	4	no	—
354	#536	C₂²×C₆₇	1	8	yes	complemented
355	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
356	C₂×#124	C₂²×C₆₇	1	2	yes	complemented
357	C₂×#123	—	2	2	no	—
358	C₂×C₂²⋊Q₈	—	2	2	no	—
359	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
360	#456	C₂²×C₆₇	1	32	yes	complemented
361	#454	C₂²×C₆₇	1	32	yes	complemented
362	#421	C₂²×C₆₇	1	16	yes	complemented
363	C₂×#104	C₂²×C₆₇	1	8	yes	complemented
364	C₂×#101	C₂²×C₆₇	1	4	yes	complemented
365	C₂×#101	C₂²×C₆₇	1	8	yes	complemented
366	2₋¹⁺⁴	—	2	32	no	—
367	2₋¹⁺⁴	—	2	16	no	—
368	2₋¹⁺⁴	—	2	16	no	—
369	2₋¹⁺⁴	—	2	32	no	—
370	2₋¹⁺⁴	—	2	8	no	—
371	2₋¹⁺⁴	—	2	16	no	—
372	2₋¹⁺⁴	C₂³×C₆₇	1	16	yes	—
373	C₂×#41	—	2	8	no	—
374	C₂×#41	—	2	16	no	—
375	C₂×#41	—	2	8	no	—
376	C₂×#41	—	2	16	no	—
377	C₂×#41	—	2	8	no	—
378	C₂×#41	—	2	16	no	—
379	C₂×#41	—	2	8	no	—
380	C₂×#41	—	2	16	no	—
381	C₂×#41	—	2	2	no	—
382	C₂×#41	—	2	4	no	—
383	C₂×#41	—	2	2	no	—
384	C₂×#41	—	2	4	no	—
385	C₂×#41	—	2	2	no	—
386	C₂×#41	—	2	2	no	—
387	C₂×#41	—	2	2	no	—
388	C₂×#41	—	2	4	no	—
389	C₂×#41	D₄×C₆₇	1	4	yes	—
390	C₂×#41	D₄×C₆₇	1	8	yes	—
391	C₂×#41	C₂³×C₆₇	1	4	yes	—
392	C₂×#41	C₂³×C₆₇	1	8	yes	—
393	C₂×#41	C₂³×C₆₇	1	4	yes	—
394	C₂²×Q₈	—	2	2	no	—
395	C₂²×Q₈	—	2	8	no	—
396	C₂²×Q₈	—	2	16	no	—
397	C₂²×Q₈	—	2	8	no	—
398	C₂²×Q₈	—	2	16	no	—
399	C₂²×Q₈	—	2	2	no	—
400	C₂²×Q₈	—	2	2	no	—
401	C₂²×Q₈	—	2	4	no	—
402	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
403	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
404	C₂²×Q₈	D₄×C₆₇	1	4	yes	—
405	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
406	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
407	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
408	C₂²×D₄	—	2	2	no	—
409	C₂²×D₄	—	2	2	no	—
410	C₂²×D₄	—	2	1	no	—
411	C₂²×D₄	D₄×C₆₇	1	2	yes	—
412	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
413	C₂³×C₄	—	2	4	no	—
414	C₂³×C₄	—	2	2	no	—
415	C₂³×C₄	—	2	1	no	—
416	C₂³×C₄	D₄×C₆₇	1	2	yes	—
417	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
418	#137	—	2	8	no	—
419	#137	—	2	16	no	—
420	#137	C₂³×C₆₇	1	16	yes	complemented
421	AGL(1,ℤ/8)	—	2	8	no	—
422	AGL(1,ℤ/8)	—	2	16	no	—
423	AGL(1,ℤ/8)	C₂³×C₆₇	1	16	yes	complemented
424	#135	—	2	32	no	—
425	C₂×Q₁₆	—	2	4	no	—
426	C₂×SD₁₆	—	2	4	no	—
427	C₂×SD₁₆	—	2	4	no	—
428	C₂×D₈	—	2	4	no	—
429	C₂×M₁₆	—	2	2	no	—
430	C₂×M₁₆	—	2	4	no	—
431	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
432	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
433	C₂²×C₈	—	2	2	no	—
434	C₄⋊Q₈	—	2	4	no	—
435	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	complemented
436	#124	—	2	4	no	—
437	#124	C₂³×C₆₇	1	8	yes	complemented
438	#123	—	2	16	no	—
439	C₂²⋊Q₈	—	2	16	no	—
440	#117	—	2	2	no	—
441	#117	C₂³×C₆₇	1	4	yes	—
442	C₂×C₄⋊C₄	—	2	1	no	—
443	C₂×C₄⋊C₄	—	2	4	no	—
444	C₂×C₂²⋊C₄	—	2	1	no	—
445	C₂×C₂²⋊C₄	—	2	4	no	—
446	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
447	#104	C₂³×C₆₇	1	32	yes	complemented
448	#101	C₂³×C₆₇	1	16	yes	complemented
449	C₂⁴	—	2	2	no	—
450	C₂⁴	—	2	1	no	—
451	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
452	#41	—	2	32	no	—
453	#41	—	2	64	no	—
454	#41	—	2	32	no	—
455	#41	—	2	64	no	—
456	#41	—	2	32	no	—
457	#41	—	2	8	no	—
458	#41	—	2	8	no	—
459	#41	—	2	8	no	—
460	#41	—	2	32	no	—
461	#41	—	2	4	no	—
462	#41	—	2	8	no	—
463	#41	—	2	8	no	—
464	#41	—	2	4	no	—
465	#41	—	2	8	no	—
466	#41	—	2	8	no	—
467	#41	—	4	4	no	—
468	#41	—	4	4	no	—
469	#41	C₂×D₄×C₆₇	1	16	yes	—
470	#41	C₂×D₄×C₆₇	1	8	yes	—
471	#41	C₂⁴×C₆₇	1	8	yes	—
472	C₂×Q₈	—	2	8	no	—
473	C₂×Q₈	—	2	16	no	—
474	C₂×Q₈	—	2	32	no	—
475	C₂×Q₈	—	2	64	no	—
476	C₂×Q₈	—	2	32	no	—
477	C₂×Q₈	—	2	2	no	—
478	C₂×Q₈	—	2	4	no	—
479	C₂×Q₈	—	2	4	no	—
480	C₂×Q₈	—	2	8	no	—
481	C₂×Q₈	—	2	4	no	—
482	C₂×Q₈	—	2	8	no	—
483	C₂×Q₈	—	2	4	no	—
484	C₂×Q₈	—	2	8	no	—
485	C₂×Q₈	—	2	8	no	—
486	C₂×Q₈	—	2	4	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	2	no	—
489	C₂×Q₈	—	2	2	no	—
490	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
491	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
492	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	—
493	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
494	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
495	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
496	C₂×D₄	—	2	8	no	—
497	C₂×D₄	—	2	16	no	—
498	C₂×D₄	—	2	4	no	—
499	C₂×D₄	—	2	8	no	—
500	C₂×D₄	—	2	4	no	—
501	C₂×D₄	—	2	8	no	—
502	C₂×D₄	—	2	4	no	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	2	4	no	—
505	C₂×D₄	—	2	2	no	—
506	C₂×D₄	—	2	2	no	—
507	C₂×D₄	—	2	1	no	—
508	C₂×D₄	—	4	1	no	—
509	C₂×D₄	—	4	1	no	—
510	C₂×D₄	—	4	2	no	—
511	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
512	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
513	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
514	C₂²×C₄	—	2	16	no	—
515	C₂²×C₄	—	2	32	no	—
516	C₂²×C₄	—	2	2	no	—
517	C₂²×C₄	—	2	2	no	—
518	C₂²×C₄	—	2	2	no	—
519	C₂²×C₄	—	2	4	no	—
520	C₂²×C₄	—	2	4	no	—
521	C₂²×C₄	—	2	8	no	—
522	C₂²×C₄	—	2	4	no	—
523	C₂²×C₄	—	2	8	no	—
524	C₂²×C₄	—	2	2	no	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	4	no	—
527	C₂²×C₄	—	4	2	no	—
528	C₂²×C₄	—	4	2	no	—
529	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
530	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
531	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
532	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
533	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
534	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
535	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
536	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
537	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
538	Q₁₆	—	2	16	no	—
539	SD₁₆	—	2	16	no	—
540	SD₁₆	—	2	16	no	—
541	D₈	—	2	16	no	—
542	M₁₆	—	2	8	no	—
543	M₁₆	—	2	4	no	—
544	M₁₆	C₂⁴×C₆₇	1	8	yes	—
545	C₂×C₈	—	2	4	no	—
546	C₂×C₈	—	2	8	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	16	no	—
549	C₂²⋊C₄	—	2	4	no	—
550	C₂²⋊C₄	—	2	16	no	—
551	C₄²	—	2	2	no	—
552	C₄²	C₂⁴×C₆₇	1	4	yes	—
553	C₂³	—	2	8	no	—
554	C₂³	—	2	4	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	4	2	no	—
557	C₂³	—	4	8	no	—
558	C₂³	—	4	4	no	—
559	C₂³	—	4	1	no	—
560	C₂³	—	4	1	no	—
561	C₂³	#120×C₆₇	1	1	yes	characteristic
562	C₂³	#120×C₆₇	1	1	yes	characteristic
563	C₂³	#120×C₆₇	1	2	yes	—
564	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
565	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
566	Q₈	—	2	32	no	—
567	Q₈	—	2	8	no	—
568	Q₈	—	2	4	no	—
569	Q₈	—	2	16	no	—
570	Q₈	—	2	8	no	—
571	Q₈	—	2	8	no	—
572	Q₈	—	2	4	no	—
573	Q₈	—	4	2	no	—
574	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
575	D₄	—	2	32	no	—
576	D₄	—	2	8	no	—
577	D₄	—	2	16	no	—
578	D₄	—	2	8	no	—
579	D₄	—	2	8	no	—
580	D₄	—	2	4	no	—
581	D₄	—	2	4	no	—
582	D₄	—	4	4	no	—
583	D₄	—	4	4	no	—
584	D₄	—	4	2	no	—
585	D₄	C₂²×D₄×C₆₇	1	8	yes	—
586	C₂×C₄	—	2	64	no	—
587	C₂×C₄	—	2	8	no	—
588	C₂×C₄	—	2	4	no	—
589	C₂×C₄	—	2	8	no	—
590	C₂×C₄	—	2	4	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	8	no	—
593	C₂×C₄	—	2	16	no	—
594	C₂×C₄	—	2	8	no	—
595	C₂×C₄	—	2	8	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	4	4	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	8	no	—
602	C₂×C₄	#120×C₆₇	1	2	yes	—
603	C₂×C₄	#120×C₆₇	1	4	yes	—
604	C₂×C₄	#120×C₆₇	1	2	yes	—
605	C₂×C₄	#120×C₆₇	1	4	yes	—
606	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
607	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
608	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
609	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
610	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
611	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
612	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
613	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
614	C₈	—	2	8	no	—
615	C₂²	—	2	2	no	—
616	C₂²	—	2	2	no	—
617	C₂²	—	2	2	no	—
618	C₂²	—	2	8	no	—
619	C₂²	—	4	2	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	4	16	no	—
622	C₂²	—	4	4	no	—
623	C₂²	—	4	8	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	8	2	no	—
626	C₂²	—	8	2	no	—
627	C₂²	C₂×#120×C₆₇	1	2	yes	—
628	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
629	C₂²	C₂×#120×C₆₇	1	2	yes	—
630	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
631	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
632	C₄	—	2	8	no	—
633	C₄	—	2	8	no	—
634	C₄	—	4	4	no	—
635	C₄	—	4	2	no	—
636	C₄	C₂×#120×C₆₇	1	4	yes	—
637	C₄	C₂×#120×C₆₇	1	2	yes	—
638	C₄	C₂³×D₄×C₆₇	1	2	yes	—
639	C₂	—	2	1	no	—
640	C₂	—	2	2	no	—
641	C₂	—	2	1	no	—
642	C₂	—	2	2	no	—
643	C₂	—	4	8	no	—
644	C₂	—	8	2	no	—
645	C₂	C₆₇×#3020	1	2	yes	direct product factor
646	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. C₂²×2₋¹⁺⁴×C₆₇	C₂	semidirect
2. C₂×#575×C₆₇	C₂	semidirect
3. C₂×#536×C₆₇	C₂	semidirect
4. C₆₇×#3020	C₂	direct (N × H)
5. C₂×#456×C₆₇	C₂	semidirect
6. C₂×#454×C₆₇	C₂	semidirect
7. C₂×#421×C₆₇	C₂	semidirect
8. C₂²×#101×C₆₇	C₂	semidirect
14. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
22. #575×C₆₇	C₂²	semidirect
24. C₂×#137×C₆₇	C₂²	semidirect
26. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
30. #536×C₆₇	C₂²	semidirect
31. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
32. C₂×#124×C₆₇	C₂²	semidirect
36. #456×C₆₇	C₂²	semidirect
37. #454×C₆₇	C₂²	semidirect
38. #421×C₆₇	C₂²	semidirect
39. C₂×#104×C₆₇	C₂²	semidirect
40. C₂×#101×C₆₇	C₂²	semidirect
41. C₂×#101×C₆₇	C₂²	semidirect
96. #137×C₆₇	C₂³	semidirect
99. AGL(1,ℤ/8)×C₆₇	C₂³	semidirect
111. C₄⋊Q₈×C₆₇	C₂³	semidirect
113. #124×C₆₇	C₂³	semidirect
123. #104×C₆₇	C₂³	semidirect
124. #101×C₆₇	C₂³	semidirect
315. C₂×#3020	C₆₇	direct (N × H)
322. C₂×C₆₇	#3020	direct (N × H)
324. C₂²×2₋¹⁺⁴	C₂×C₆₇	semidirect
325. C₂×#575	C₂×C₆₇	semidirect
326. C₂×#536	C₂×C₆₇	semidirect
327. #3020	C₂×C₆₇	direct (N × H)
328. C₂×#456	C₂×C₆₇	semidirect
329. C₂×#454	C₂×C₆₇	semidirect
330. C₂×#421	C₂×C₆₇	semidirect
331. C₂²×#101	C₂×C₆₇	semidirect
332. C₆₇	C₂×#3020	direct (N × H)
338. C₂×2₋¹⁺⁴	C₂²×C₆₇	semidirect
346. #575	C₂²×C₆₇	semidirect
348. C₂×#137	C₂²×C₆₇	semidirect
350. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
354. #536	C₂²×C₆₇	semidirect
355. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
356. C₂×#124	C₂²×C₆₇	semidirect
360. #456	C₂²×C₆₇	semidirect
361. #454	C₂²×C₆₇	semidirect
362. #421	C₂²×C₆₇	semidirect
363. C₂×#104	C₂²×C₆₇	semidirect
364. C₂×#101	C₂²×C₆₇	semidirect
365. C₂×#101	C₂²×C₆₇	semidirect
420. #137	C₂³×C₆₇	semidirect
423. AGL(1,ℤ/8)	C₂³×C₆₇	semidirect
435. C₄⋊Q₈	C₂³×C₆₇	semidirect
437. #124	C₂³×C₆₇	semidirect
447. #104	C₂³×C₆₇	semidirect
448. #101	C₂³×C₆₇	semidirect
645. C₂	C₆₇×#3020	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴×C₆₇	2
C₂×#575×C₆₇	2
C₂×#536×C₆₇	2
C₆₇×#3020	2
C₂×#456×C₆₇	2
C₂×#454×C₆₇	2
C₂×#421×C₆₇	2
C₂²×#101×C₆₇	2
C₂×#3020	67

Subgroups — C2×C67×#3020 #30739635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3020 #30739635