Subgroups — C₂×C₆₇×#3004 #30739606

Order 17152.

4970 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#137×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#525×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#3004	C₂	1	16	yes	maximal, direct product factor
5	C₂×#451×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#448×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#419×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#407×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×Q₈×C₆₇	—	2	16	no	—
11	C₂³×Q₈×C₆₇	—	2	2	no	—
12	C₂³×Q₈×C₆₇	—	2	1	no	—
13	C₂³×Q₈×C₆₇	—	2	4	no	—
14	C₂³×Q₈×C₆₇	C₂²	1	4	yes	—
15	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
16	C₂³×Q₈×C₆₇	C₂²	1	8	yes	complemented
17	C₂⁴×C₄×C₆₇	—	2	1	no	—
18	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#137×C₆₇	—	2	16	no	—
20	C₂×#137×C₆₇	C₂²	1	16	yes	complemented
21	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
22	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
23	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
24	#525×C₆₇	—	2	4	no	—
25	#525×C₆₇	C₂²	1	8	yes	—
26	C₂×#123×C₆₇	—	2	1	no	—
27	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
28	C₂×#121×C₆₇	C₂²	1	2	yes	—
29	C₂×#120×C₆₇	—	2	1	no	—
30	C₂×C₄×D₄×C₆₇	—	2	2	no	—
31	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
32	#451×C₆₇	C₂²	1	32	yes	complemented
33	#448×C₆₇	C₂²	1	32	yes	complemented
34	#419×C₆₇	C₂²	1	16	yes	complemented
35	C₂×#103×C₆₇	C₂²	1	8	yes	complemented
36	#407×C₆₇	C₂²	1	8	yes	complemented
37	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
38	C₂⁵×C₆₇	D₄	1	1	yes	characteristic
39	C₂×#41×C₆₇	—	2	2	no	—
40	C₂×#41×C₆₇	—	2	8	no	—
41	C₂×#41×C₆₇	C₂³	1	4	yes	—
42	C₂²×Q₈×C₆₇	—	2	32	no	—
43	C₂²×Q₈×C₆₇	—	2	64	no	—
44	C₂²×Q₈×C₆₇	—	2	16	no	—
45	C₂²×Q₈×C₆₇	—	2	32	no	—
46	C₂²×Q₈×C₆₇	—	2	64	no	—
47	C₂²×Q₈×C₆₇	—	2	8	no	—
48	C₂²×Q₈×C₆₇	—	2	8	no	—
49	C₂²×Q₈×C₆₇	—	2	4	no	—
50	C₂²×Q₈×C₆₇	—	2	4	no	—
51	C₂²×Q₈×C₆₇	—	2	2	no	—
52	C₂²×Q₈×C₆₇	—	2	16	no	—
53	C₂²×Q₈×C₆₇	—	2	8	no	—
54	C₂²×Q₈×C₆₇	D₄	1	16	yes	complemented
55	C₂²×Q₈×C₆₇	C₂³	1	16	yes	—
56	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
58	C₂²×Q₈×C₆₇	C₂³	1	8	yes	—
59	C₂²×D₄×C₆₇	—	2	2	no	—
60	C₂²×D₄×C₆₇	—	2	1	no	—
61	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	—	2	16	no	—
63	C₂³×C₄×C₆₇	—	2	4	no	—
64	C₂³×C₄×C₆₇	—	2	2	no	—
65	C₂³×C₄×C₆₇	—	2	8	no	—
66	C₂³×C₄×C₆₇	—	2	8	no	—
67	C₂³×C₄×C₆₇	—	2	1	no	—
68	C₂³×C₄×C₆₇	D₄	1	2	yes	—
69	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
70	C₂³×C₄×C₆₇	D₄	1	4	yes	—
71	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
72	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	#137×C₆₇	—	2	64	no	—
75	C₂×Q₁₆×C₆₇	—	2	16	no	—
76	C₂×Q₁₆×C₆₇	C₂³	1	16	yes	complemented
77	C₂×SD₁₆×C₆₇	—	2	16	no	—
78	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	complemented
79	C₂×M₁₆×C₆₇	—	2	8	no	—
80	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
81	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
82	#123×C₆₇	—	2	8	no	—
83	C₂²⋊Q₈×C₆₇	—	2	4	no	—
84	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
85	#121×C₆₇	—	2	4	no	—
86	#121×C₆₇	C₂³	1	8	yes	—
87	#120×C₆₇	—	2	8	no	—
88	C₄×D₄×C₆₇	—	2	16	no	—
89	#117×C₆₇	—	2	2	no	—
90	#117×C₆₇	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₄²×C₆₇	—	2	1	no	—
97	#103×C₆₇	C₂³	1	32	yes	complemented
98	C₂²⋊C₈×C₆₇	C₂³	1	16	yes	—
99	C₂⁴×C₆₇	—	2	4	no	—
100	C₂⁴×C₆₇	—	2	8	no	—
101	C₂⁴×C₆₇	—	2	1	no	—
102	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
103	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
105	#41×C₆₇	—	2	16	no	—
106	#41×C₆₇	—	4	8	no	—
107	C₂×Q₈×C₆₇	—	2	64	no	—
108	C₂×Q₈×C₆₇	—	2	128	no	—
109	C₂×Q₈×C₆₇	—	2	32	no	—
110	C₂×Q₈×C₆₇	—	2	64	no	—
111	C₂×Q₈×C₆₇	—	2	64	no	—
112	C₂×Q₈×C₆₇	—	2	16	no	—
113	C₂×Q₈×C₆₇	—	2	32	no	—
114	C₂×Q₈×C₆₇	—	2	8	no	—
115	C₂×Q₈×C₆₇	—	2	16	no	—
116	C₂×Q₈×C₆₇	—	2	16	no	—
117	C₂×Q₈×C₆₇	—	2	8	no	—
118	C₂×Q₈×C₆₇	—	2	4	no	—
119	C₂×Q₈×C₆₇	—	2	1	no	—
120	C₂×Q₈×C₆₇	—	2	4	no	—
121	C₂×Q₈×C₆₇	C₂×D₄	1	16	yes	—
122	C₂×Q₈×C₆₇	C₂×D₄	1	32	yes	complemented
123	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
124	C₂×Q₈×C₆₇	C₂⁴	1	8	yes	—
125	C₂×D₄×C₆₇	—	2	8	no	—
126	C₂×D₄×C₆₇	—	2	4	no	—
127	C₂×D₄×C₆₇	—	2	4	no	—
128	C₂×D₄×C₆₇	—	2	1	no	—
129	C₂×D₄×C₆₇	—	4	4	no	—
130	C₂×D₄×C₆₇	—	4	8	no	—
131	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
132	C₂²×C₄×C₆₇	—	2	32	no	—
133	C₂²×C₄×C₆₇	—	2	64	no	—
134	C₂²×C₄×C₆₇	—	2	16	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	—	2	16	no	—
137	C₂²×C₄×C₆₇	—	2	8	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	8	no	—
140	C₂²×C₄×C₆₇	—	2	16	no	—
141	C₂²×C₄×C₆₇	—	2	8	no	—
142	C₂²×C₄×C₆₇	—	2	16	no	—
143	C₂²×C₄×C₆₇	—	2	2	no	—
144	C₂²×C₄×C₆₇	—	2	1	no	—
145	C₂²×C₄×C₆₇	—	2	4	no	—
146	C₂²×C₄×C₆₇	—	4	4	no	—
147	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
148	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
149	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
154	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
157	Q₁₆×C₆₇	—	2	32	no	—
158	SD₁₆×C₆₇	—	2	32	no	—
159	M₁₆×C₆₇	—	2	16	no	—
160	C₂×C₈×C₆₇	—	2	8	no	—
161	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
162	C₄⋊C₄×C₆₇	—	2	8	no	—
163	C₄⋊C₄×C₆₇	—	2	2	no	—
164	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
165	C₂²⋊C₄×C₆₇	—	2	4	no	—
166	C₂²⋊C₄×C₆₇	—	2	8	no	—
167	C₂²⋊C₄×C₆₇	—	2	8	no	—
168	C₄²×C₆₇	—	2	4	no	—
169	C₂³×C₆₇	—	2	32	no	—
170	C₂³×C₆₇	—	2	8	no	—
171	C₂³×C₆₇	—	2	4	no	—
172	C₂³×C₆₇	—	2	4	no	—
173	C₂³×C₆₇	—	2	4	no	—
174	C₂³×C₆₇	—	2	16	no	—
175	C₂³×C₆₇	—	2	2	no	—
176	C₂³×C₆₇	—	4	2	no	—
177	C₂³×C₆₇	—	4	4	no	—
178	C₂³×C₆₇	#120	1	4	yes	—
179	C₂³×C₆₇	#137	1	8	yes	complemented
180	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
181	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
182	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
183	Q₈×C₆₇	—	2	64	no	—
184	Q₈×C₆₇	—	2	16	no	—
185	Q₈×C₆₇	—	2	32	no	—
186	Q₈×C₆₇	—	2	8	no	—
187	Q₈×C₆₇	—	2	4	no	—
188	Q₈×C₆₇	—	4	2	no	—
189	Q₈×C₆₇	C₂²×D₄	1	16	yes	—
190	D₄×C₆₇	—	2	4	no	—
191	D₄×C₆₇	—	4	8	no	—
192	D₄×C₆₇	—	4	16	no	—
193	D₄×C₆₇	—	4	2	no	—
194	C₂×C₄×C₆₇	—	2	64	no	—
195	C₂×C₄×C₆₇	—	2	16	no	—
196	C₂×C₄×C₆₇	—	2	32	no	—
197	C₂×C₄×C₆₇	—	2	8	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	8	no	—
200	C₂×C₄×C₆₇	—	2	16	no	—
201	C₂×C₄×C₆₇	—	2	8	no	—
202	C₂×C₄×C₆₇	—	2	8	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	2	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	4	4	no	—
207	C₂×C₄×C₆₇	—	4	8	no	—
208	C₂×C₄×C₆₇	#120	1	4	yes	—
209	C₂×C₄×C₆₇	#120	1	8	yes	—
210	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
211	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
214	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
215	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
216	C₈×C₆₇	—	2	8	no	—
217	C₂²×C₆₇	—	2	32	no	—
218	C₂²×C₆₇	—	2	4	no	—
219	C₂²×C₆₇	—	2	8	no	—
220	C₂²×C₆₇	—	2	16	no	—
221	C₂²×C₆₇	—	2	4	no	—
222	C₂²×C₆₇	—	2	4	no	—
223	C₂²×C₆₇	—	4	2	no	—
224	C₂²×C₆₇	—	4	4	no	—
225	C₂²×C₆₇	—	8	4	no	—
226	C₂²×C₆₇	C₂×#120	1	4	yes	—
227	C₂²×C₆₇	C₂×#120	1	2	yes	—
228	C₂²×C₆₇	C₂×#137	1	4	yes	—
229	C₂²×C₆₇	C₂×#137	1	8	yes	complemented
230	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
231	C₄×C₆₇	—	2	16	no	—
232	C₄×C₆₇	—	2	4	no	—
233	C₄×C₆₇	—	4	2	no	—
234	C₄×C₆₇	—	4	4	no	—
235	C₄×C₆₇	C₂×#120	1	4	yes	—
236	C₄×C₆₇	C₂×#120	1	2	yes	—
237	C₄×C₆₇	C₂³×D₄	1	2	yes	—
238	C₂×#3004	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂×C₆₇	—	2	4	no	—
240	C₂×C₆₇	—	2	8	no	—
241	C₂×C₆₇	—	8	2	no	—
242	C₂×C₆₇	#3004	1	4	yes	direct product factor
243	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
244	C₂×C₆₇	C₂²×#137	1	2	yes	—
245	C₂⁴×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
246	C₂²×#137	C₂×C₆₇	1	2	yes	complemented
247	C₂×#525	C₂×C₆₇	1	1	yes	characteristic
248	#3004	C₂×C₆₇	1	16	yes	direct product factor
249	C₂×#451	C₂×C₆₇	1	4	yes	complemented
250	C₂×#448	C₂×C₆₇	1	4	yes	complemented
251	C₂×#419	C₂×C₆₇	1	2	yes	complemented
252	C₂×#407	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₆₇	C₂×#3004	1	1	yes	characteristic, direct product factor, Sylow
254	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
255	C₂³×Q₈	—	2	16	no	—
256	C₂³×Q₈	—	2	2	no	—
257	C₂³×Q₈	—	2	1	no	—
258	C₂³×Q₈	—	2	4	no	—
259	C₂³×Q₈	C₂²×C₆₇	1	4	yes	—
260	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
261	C₂³×Q₈	C₂²×C₆₇	1	8	yes	complemented
262	C₂⁴×C₄	—	2	1	no	—
263	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×#137	—	2	16	no	—
265	C₂×#137	C₂²×C₆₇	1	16	yes	complemented
266	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
267	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
268	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
269	#525	—	2	4	no	—
270	#525	C₂²×C₆₇	1	8	yes	—
271	C₂×#123	—	2	1	no	—
272	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
273	C₂×#121	C₂²×C₆₇	1	2	yes	—
274	C₂×#120	—	2	1	no	—
275	C₂×C₄×D₄	—	2	2	no	—
276	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
277	#451	C₂²×C₆₇	1	32	yes	complemented
278	#448	C₂²×C₆₇	1	32	yes	complemented
279	#419	C₂²×C₆₇	1	16	yes	complemented
280	C₂×#103	C₂²×C₆₇	1	8	yes	complemented
281	#407	C₂²×C₆₇	1	8	yes	complemented
282	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
283	C₂⁵	D₄×C₆₇	1	1	yes	characteristic
284	C₂×#41	—	2	2	no	—
285	C₂×#41	—	2	8	no	—
286	C₂×#41	C₂³×C₆₇	1	4	yes	—
287	C₂²×Q₈	—	2	32	no	—
288	C₂²×Q₈	—	2	64	no	—
289	C₂²×Q₈	—	2	16	no	—
290	C₂²×Q₈	—	2	32	no	—
291	C₂²×Q₈	—	2	64	no	—
292	C₂²×Q₈	—	2	8	no	—
293	C₂²×Q₈	—	2	4	no	—
294	C₂²×Q₈	—	2	8	no	—
295	C₂²×Q₈	—	2	2	no	—
296	C₂²×Q₈	—	2	4	no	—
297	C₂²×Q₈	—	2	16	no	—
298	C₂²×Q₈	—	2	8	no	—
299	C₂²×Q₈	D₄×C₆₇	1	16	yes	complemented
300	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
301	C₂²×Q₈	C₂³×C₆₇	1	16	yes	—
302	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
303	C₂²×Q₈	C₂³×C₆₇	1	8	yes	—
304	C₂²×D₄	—	2	2	no	—
305	C₂²×D₄	—	2	1	no	—
306	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
307	C₂³×C₄	—	2	16	no	—
308	C₂³×C₄	—	2	4	no	—
309	C₂³×C₄	—	2	2	no	—
310	C₂³×C₄	—	2	8	no	—
311	C₂³×C₄	—	2	8	no	—
312	C₂³×C₄	—	2	1	no	—
313	C₂³×C₄	D₄×C₆₇	1	2	yes	—
314	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
315	C₂³×C₄	D₄×C₆₇	1	4	yes	—
316	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
317	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
318	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
319	#137	—	2	64	no	—
320	C₂×Q₁₆	—	2	16	no	—
321	C₂×Q₁₆	C₂³×C₆₇	1	16	yes	complemented
322	C₂×SD₁₆	—	2	16	no	—
323	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	complemented
324	C₂×M₁₆	—	2	8	no	—
325	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
326	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
327	#123	—	2	8	no	—
328	C₂²⋊Q₈	—	2	4	no	—
329	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
330	#121	—	2	4	no	—
331	#121	C₂³×C₆₇	1	8	yes	—
332	#120	—	2	8	no	—
333	C₄×D₄	—	2	16	no	—
334	#117	—	2	2	no	—
335	#117	C₂³×C₆₇	1	4	yes	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
338	C₂×C₂²⋊C₄	—	2	1	no	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₄²	—	2	1	no	—
342	#103	C₂³×C₆₇	1	32	yes	complemented
343	C₂²⋊C₈	C₂³×C₆₇	1	16	yes	—
344	C₂⁴	—	2	4	no	—
345	C₂⁴	—	2	8	no	—
346	C₂⁴	—	2	1	no	—
347	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
348	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
350	#41	—	2	16	no	—
351	#41	—	4	8	no	—
352	C₂×Q₈	—	2	64	no	—
353	C₂×Q₈	—	2	128	no	—
354	C₂×Q₈	—	2	32	no	—
355	C₂×Q₈	—	2	64	no	—
356	C₂×Q₈	—	2	64	no	—
357	C₂×Q₈	—	2	16	no	—
358	C₂×Q₈	—	2	32	no	—
359	C₂×Q₈	—	2	8	no	—
360	C₂×Q₈	—	2	16	no	—
361	C₂×Q₈	—	2	16	no	—
362	C₂×Q₈	—	2	8	no	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	4	no	—
366	C₂×Q₈	C₂×D₄×C₆₇	1	16	yes	—
367	C₂×Q₈	C₂×D₄×C₆₇	1	32	yes	complemented
368	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
369	C₂×Q₈	C₂⁴×C₆₇	1	8	yes	—
370	C₂×D₄	—	2	8	no	—
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	1	no	—
374	C₂×D₄	—	4	4	no	—
375	C₂×D₄	—	4	8	no	—
376	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
377	C₂²×C₄	—	2	32	no	—
378	C₂²×C₄	—	2	64	no	—
379	C₂²×C₄	—	2	16	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	16	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	8	no	—
384	C₂²×C₄	—	2	8	no	—
385	C₂²×C₄	—	2	16	no	—
386	C₂²×C₄	—	2	8	no	—
387	C₂²×C₄	—	2	16	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	4	4	no	—
392	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
393	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
394	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
395	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
396	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
398	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
399	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
400	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
401	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
402	Q₁₆	—	2	32	no	—
403	SD₁₆	—	2	32	no	—
404	M₁₆	—	2	16	no	—
405	C₂×C₈	—	2	8	no	—
406	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
407	C₄⋊C₄	—	2	8	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
410	C₂²⋊C₄	—	2	4	no	—
411	C₂²⋊C₄	—	2	8	no	—
412	C₂²⋊C₄	—	2	8	no	—
413	C₄²	—	2	4	no	—
414	C₂³	—	2	32	no	—
415	C₂³	—	2	8	no	—
416	C₂³	—	2	4	no	—
417	C₂³	—	2	4	no	—
418	C₂³	—	2	4	no	—
419	C₂³	—	2	16	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	—	4	4	no	—
423	C₂³	#120×C₆₇	1	4	yes	—
424	C₂³	#137×C₆₇	1	8	yes	complemented
425	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
426	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
427	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
428	Q₈	—	2	64	no	—
429	Q₈	—	2	16	no	—
430	Q₈	—	2	32	no	—
431	Q₈	—	2	8	no	—
432	Q₈	—	2	4	no	—
433	Q₈	—	4	2	no	—
434	Q₈	C₂²×D₄×C₆₇	1	16	yes	—
435	D₄	—	2	4	no	—
436	D₄	—	4	8	no	—
437	D₄	—	4	16	no	—
438	D₄	—	4	2	no	—
439	C₂×C₄	—	2	64	no	—
440	C₂×C₄	—	2	16	no	—
441	C₂×C₄	—	2	32	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	8	no	—
445	C₂×C₄	—	2	16	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	2	8	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	8	no	—
453	C₂×C₄	#120×C₆₇	1	4	yes	—
454	C₂×C₄	#120×C₆₇	1	8	yes	—
455	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
456	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
457	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
458	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
459	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
460	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
461	C₈	—	2	8	no	—
462	C₂²	—	2	32	no	—
463	C₂²	—	2	4	no	—
464	C₂²	—	2	4	no	—
465	C₂²	—	2	4	no	—
466	C₂²	—	2	8	no	—
467	C₂²	—	2	16	no	—
468	C₂²	—	4	4	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	8	4	no	—
471	C₂²	C₂×#120×C₆₇	1	4	yes	—
472	C₂²	C₂×#120×C₆₇	1	2	yes	—
473	C₂²	C₂×#137×C₆₇	1	4	yes	—
474	C₂²	C₂×#137×C₆₇	1	8	yes	complemented
475	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
476	C₄	—	2	16	no	—
477	C₄	—	2	4	no	—
478	C₄	—	4	4	no	—
479	C₄	—	4	2	no	—
480	C₄	C₂×#120×C₆₇	1	4	yes	—
481	C₄	C₂×#120×C₆₇	1	2	yes	—
482	C₄	C₂³×D₄×C₆₇	1	2	yes	—
483	C₂	—	2	4	no	—
484	C₂	—	2	8	no	—
485	C₂	—	8	2	no	—
486	C₂	C₆₇×#3004	1	4	yes	direct product factor
487	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
488	C₂	C₂²×#137×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#525×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂⁴×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Q₈×C₆₇	C₂	semidirect
2. C₂²×#137×C₆₇	C₂	semidirect
4. C₆₇×#3004	C₂	direct (N × H)
5. C₂×#451×C₆₇	C₂	semidirect
6. C₂×#448×C₆₇	C₂	semidirect
7. C₂×#419×C₆₇	C₂	semidirect
8. C₂×#407×C₆₇	C₂	semidirect
16. C₂³×Q₈×C₆₇	C₂²	semidirect
20. C₂×#137×C₆₇	C₂²	semidirect
21. C₂²×Q₁₆×C₆₇	C₂²	semidirect
22. C₂²×SD₁₆×C₆₇	C₂²	semidirect
32. #451×C₆₇	C₂²	semidirect
33. #448×C₆₇	C₂²	semidirect
34. #419×C₆₇	C₂²	semidirect
35. C₂×#103×C₆₇	C₂²	semidirect
36. #407×C₆₇	C₂²	semidirect
54. C₂²×Q₈×C₆₇	D₄	semidirect
76. C₂×Q₁₆×C₆₇	C₂³	semidirect
78. C₂×SD₁₆×C₆₇	C₂³	semidirect
97. #103×C₆₇	C₂³	semidirect
122. C₂×Q₈×C₆₇	C₂×D₄	semidirect
179. C₂³×C₆₇	#137	semidirect
229. C₂²×C₆₇	C₂×#137	semidirect
238. C₂×#3004	C₆₇	direct (N × H)
242. C₂×C₆₇	#3004	direct (N × H)
245. C₂⁴×Q₈	C₂×C₆₇	semidirect
246. C₂²×#137	C₂×C₆₇	semidirect
248. #3004	C₂×C₆₇	direct (N × H)
249. C₂×#451	C₂×C₆₇	semidirect
250. C₂×#448	C₂×C₆₇	semidirect
251. C₂×#419	C₂×C₆₇	semidirect
252. C₂×#407	C₂×C₆₇	semidirect
253. C₆₇	C₂×#3004	direct (N × H)
261. C₂³×Q₈	C₂²×C₆₇	semidirect
265. C₂×#137	C₂²×C₆₇	semidirect
266. C₂²×Q₁₆	C₂²×C₆₇	semidirect
267. C₂²×SD₁₆	C₂²×C₆₇	semidirect
277. #451	C₂²×C₆₇	semidirect
278. #448	C₂²×C₆₇	semidirect
279. #419	C₂²×C₆₇	semidirect
280. C₂×#103	C₂²×C₆₇	semidirect
281. #407	C₂²×C₆₇	semidirect
299. C₂²×Q₈	D₄×C₆₇	semidirect
321. C₂×Q₁₆	C₂³×C₆₇	semidirect
323. C₂×SD₁₆	C₂³×C₆₇	semidirect
342. #103	C₂³×C₆₇	semidirect
367. C₂×Q₈	C₂×D₄×C₆₇	semidirect
424. C₂³	#137×C₆₇	semidirect
474. C₂²	C₂×#137×C₆₇	semidirect
486. C₂	C₆₇×#3004	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₆₇	2
C₂²×#137×C₆₇	2
C₂×#525×C₆₇	2
C₆₇×#3004	2
C₂×#451×C₆₇	2
C₂×#448×C₆₇	2
C₂×#419×C₆₇	2
C₂×#407×C₆₇	2
C₂×#3004	67

Subgroups — C2×C67×#3004 #30739606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3004 #30739606