Subgroups — C₆₇×#21417 #30700668

Order 17152.

1214 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3444	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1995	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1963	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1896	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1759	C₂	1	4	yes	maximal, complemented
8	#533×C₆₇	C₄	1	2	yes	complemented
9	#533×C₆₇	C₂²	1	1	yes	characteristic
10	#532×C₆₇	C₄	1	2	yes	complemented
11	#532×C₆₇	C₂²	1	1	yes	characteristic
12	#529×C₆₇	—	2	2	no	—
13	#529×C₆₇	—	2	2	no	—
14	#529×C₆₇	C₄	1	4	yes	complemented
15	#525×C₆₇	—	2	2	no	—
16	#525×C₆₇	C₂²	1	2	yes	—
17	C₄×#41×C₆₇	—	2	2	no	—
18	C₄×#41×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	#432×C₆₇	—	2	2	no	—
22	#431×C₆₇	—	2	2	no	—
23	#423×C₆₇	C₂²	1	2	yes	—
24	C₂×#104×C₆₇	C₂²	1	8	yes	—
25	#407×C₆₇	C₂²	1	2	yes	—
26	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
27	#376×C₆₇	—	2	2	no	—
28	#376×C₆₇	C₂²	1	2	yes	—
29	#376×C₆₇	C₂²	1	2	yes	—
30	C₂×#95×C₆₇	—	2	2	no	—
31	#339×C₆₇	C₂²	1	8	yes	—
32	C₂×#41×C₆₇	—	2	2	no	—
33	C₂×#41×C₆₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₆₇	—	2	2	no	—
35	C₂³×C₄×C₆₇	D₄	1	2	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
39	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	complemented
40	#127×C₆₇	C₂×C₄	1	1	yes	characteristic, complemented
41	#126×C₆₇	—	2	4	no	—
42	#125×C₆₇	—	2	2	no	—
43	#125×C₆₇	C₂×C₄	1	2	yes	complemented
44	#124×C₆₇	—	2	2	no	—
45	#124×C₆₇	C₂×C₄	1	2	yes	complemented
46	#123×C₆₇	—	2	4	no	—
47	#123×C₆₇	—	2	2	no	—
48	C₂²⋊Q₈×C₆₇	—	2	2	no	—
49	C₂²⋊Q₈×C₆₇	—	2	2	no	—
50	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
51	#121×C₆₇	—	2	2	no	—
52	#121×C₆₇	—	2	2	no	—
53	#121×C₆₇	C₂×C₄	1	4	yes	—
54	#120×C₆₇	—	2	2	no	—
55	C₄×Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
58	C₄×D₄×C₆₇	—	2	4	no	—
59	C₄×D₄×C₆₇	—	2	4	no	—
60	C₄×D₄×C₆₇	—	2	4	no	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	—	2	2	no	—
63	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
64	#117×C₆₇	—	2	2	no	—
65	#117×C₆₇	—	2	4	no	—
66	#117×C₆₇	—	2	2	no	—
67	#117×C₆₇	C₂³	1	2	yes	—
68	C₂×C₄²×C₆₇	—	2	2	no	—
69	C₂×C₄²×C₆₇	—	2	2	no	—
70	C₂×C₄²×C₆₇	—	2	8	no	—
71	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
73	C₂×C₄²×C₆₇	D₄	1	2	yes	—
74	C₂×C₄²×C₆₇	D₄	1	2	yes	—
75	C₂×C₄²×C₆₇	D₄	1	2	yes	—
76	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₆₇	—	2	4	no	—
80	#104×C₆₇	—	2	16	no	—
81	C₂²⋊C₈×C₆₇	—	2	4	no	—
82	#97×C₆₇	—	2	2	no	—
83	C₄×C₈×C₆₇	—	2	2	no	—
84	#95×C₆₇	—	2	4	no	—
85	#95×C₆₇	—	2	4	no	—
86	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
87	#41×C₆₇	—	2	8	no	—
88	#41×C₆₇	—	4	8	no	—
89	C₂×Q₈×C₆₇	—	2	2	no	—
90	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂×D₄×C₆₇	—	2	4	no	—
92	C₂×D₄×C₆₇	—	2	2	no	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	—	2	4	no	—
96	C₂²×C₄×C₆₇	—	2	8	no	—
97	C₂²×C₄×C₆₇	—	2	4	no	—
98	C₂²×C₄×C₆₇	—	2	4	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	4	4	no	—
101	C₂²×C₄×C₆₇	—	4	8	no	—
102	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
112	M₁₆×C₆₇	—	2	8	no	—
113	C₂×C₈×C₆₇	—	2	4	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	4	no	—
117	C₄⋊C₄×C₆₇	—	2	4	no	—
118	C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
120	C₂²⋊C₄×C₆₇	—	2	4	no	—
121	C₂²⋊C₄×C₆₇	—	2	2	no	—
122	C₂²⋊C₄×C₆₇	—	2	4	no	—
123	C₂²⋊C₄×C₆₇	—	2	4	no	—
124	C₂²⋊C₄×C₆₇	—	2	4	no	—
125	C₄²×C₆₇	—	2	8	no	—
126	C₄²×C₆₇	—	2	4	no	—
127	C₄²×C₆₇	—	2	2	no	—
128	C₄²×C₆₇	—	2	4	no	—
129	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
130	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
131	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
132	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
133	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
134	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₄²×C₆₇	C₂×D₄	1	2	yes	—
137	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₆₇	#41	1	8	yes	complemented
140	C₂³×C₆₇	—	2	2	no	—
141	C₂³×C₆₇	—	2	4	no	—
142	C₂³×C₆₇	—	2	2	no	—
143	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
144	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
145	Q₈×C₆₇	—	2	4	no	—
146	Q₈×C₆₇	—	4	4	no	—
147	D₄×C₆₇	—	2	4	no	—
148	D₄×C₆₇	—	4	4	no	—
149	D₄×C₆₇	—	4	4	no	—
150	D₄×C₆₇	—	4	8	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	4	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	4	no	—
155	C₂×C₄×C₆₇	—	2	4	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	4	no	—
158	C₂×C₄×C₆₇	—	2	8	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	—	4	8	no	—
164	C₂×C₄×C₆₇	—	4	16	no	—
165	C₂×C₄×C₆₇	—	4	8	no	—
166	C₂×C₄×C₆₇	—	4	4	no	—
167	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
168	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
170	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
171	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
172	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
175	C₈×C₆₇	—	4	4	no	—
176	C₂²×C₆₇	—	2	8	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	2	2	no	—
179	C₂²×C₆₇	—	2	2	no	—
180	C₂²×C₆₇	—	4	4	no	—
181	C₂²×C₆₇	—	4	2	no	—
182	C₂²×C₆₇	#421	1	4	yes	—
183	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
184	C₂²×C₆₇	#525	1	2	yes	—
185	C₄×C₆₇	—	2	2	no	—
186	C₄×C₆₇	—	2	2	no	—
187	C₄×C₆₇	—	2	2	no	—
188	C₄×C₆₇	—	2	2	no	—
189	C₄×C₆₇	—	2	2	no	—
190	C₄×C₆₇	—	4	8	no	—
191	C₄×C₆₇	—	4	4	no	—
192	C₄×C₆₇	—	4	2	no	—
193	C₄×C₆₇	C₄×#41	1	2	yes	—
194	C₄×C₆₇	#532	1	2	yes	—
195	#21417	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂×C₆₇	—	2	2	no	—
197	C₂×C₆₇	—	2	4	no	—
198	C₂×C₆₇	—	8	2	no	—
199	C₂×C₆₇	#2297	1	1	yes	characteristic
200	C₂×C₆₇	C₂×#421	1	2	yes	—
201	#3444	C₂×C₆₇	1	1	yes	characteristic
202	#2920	C₂×C₆₇	1	1	yes	characteristic, complemented
203	C₂×#376	C₂×C₆₇	1	1	yes	characteristic, complemented
204	#1995	C₂×C₆₇	1	2	yes	complemented
205	#1963	C₂×C₆₇	1	2	yes	complemented
206	#1896	C₂×C₆₇	1	4	yes	complemented
207	#1759	C₂×C₆₇	1	4	yes	complemented
208	C₆₇	#21417	1	1	yes	characteristic, direct product factor, Sylow
209	#533	C₄×C₆₇	1	2	yes	complemented
210	#533	C₂²×C₆₇	1	1	yes	characteristic
211	#532	C₄×C₆₇	1	2	yes	complemented
212	#532	C₂²×C₆₇	1	1	yes	characteristic
213	#529	—	2	2	no	—
214	#529	—	2	2	no	—
215	#529	C₄×C₆₇	1	4	yes	complemented
216	#525	—	2	2	no	—
217	#525	C₂²×C₆₇	1	2	yes	—
218	C₄×#41	—	2	2	no	—
219	C₄×#41	C₂²×C₆₇	1	2	yes	—
220	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
221	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
222	#432	—	2	2	no	—
223	#431	—	2	2	no	—
224	#423	C₂²×C₆₇	1	2	yes	—
225	C₂×#104	C₂²×C₆₇	1	8	yes	—
226	#407	C₂²×C₆₇	1	2	yes	—
227	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
228	#376	—	2	2	no	—
229	#376	C₂²×C₆₇	1	2	yes	—
230	#376	C₂²×C₆₇	1	2	yes	—
231	C₂×#95	—	2	2	no	—
232	#339	C₂²×C₆₇	1	8	yes	—
233	C₂×#41	—	2	2	no	—
234	C₂×#41	C₂³×C₆₇	1	2	yes	—
235	C₂³×C₄	—	2	2	no	—
236	C₂³×C₄	D₄×C₆₇	1	2	yes	—
237	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
238	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
239	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
240	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	complemented
241	#127	C₂×C₄×C₆₇	1	1	yes	characteristic, complemented
242	#126	—	2	4	no	—
243	#125	—	2	2	no	—
244	#125	C₂×C₄×C₆₇	1	2	yes	complemented
245	#124	—	2	2	no	—
246	#124	C₂×C₄×C₆₇	1	2	yes	complemented
247	#123	—	2	4	no	—
248	#123	—	2	2	no	—
249	C₂²⋊Q₈	—	2	2	no	—
250	C₂²⋊Q₈	—	2	2	no	—
251	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
252	#121	—	2	2	no	—
253	#121	—	2	2	no	—
254	#121	C₂×C₄×C₆₇	1	4	yes	—
255	#120	—	2	2	no	—
256	C₄×Q₈	—	2	2	no	—
257	C₄×Q₈	—	2	2	no	—
258	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
259	C₄×D₄	—	2	4	no	—
260	C₄×D₄	—	2	4	no	—
261	C₄×D₄	—	2	4	no	—
262	C₄×D₄	—	2	2	no	—
263	C₄×D₄	—	2	2	no	—
264	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
265	#117	—	2	2	no	—
266	#117	—	2	4	no	—
267	#117	—	2	2	no	—
268	#117	C₂³×C₆₇	1	2	yes	—
269	C₂×C₄²	—	2	2	no	—
270	C₂×C₄²	—	2	2	no	—
271	C₂×C₄²	—	2	8	no	—
272	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
273	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
274	C₂×C₄²	D₄×C₆₇	1	2	yes	—
275	C₂×C₄²	D₄×C₆₇	1	2	yes	—
276	C₂×C₄²	D₄×C₆₇	1	2	yes	—
277	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
278	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
279	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
280	C₄⋊C₈	—	2	4	no	—
281	#104	—	2	16	no	—
282	C₂²⋊C₈	—	2	4	no	—
283	#97	—	2	2	no	—
284	C₄×C₈	—	2	2	no	—
285	#95	—	2	4	no	—
286	#95	—	2	4	no	—
287	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
288	#41	—	2	8	no	—
289	#41	—	4	8	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
292	C₂×D₄	—	2	4	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	8	no	—
298	C₂²×C₄	—	2	4	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	4	4	no	—
302	C₂²×C₄	—	4	8	no	—
303	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
304	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
305	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
306	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
307	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
308	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
310	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
311	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
312	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
313	M₁₆	—	2	8	no	—
314	C₂×C₈	—	2	4	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	2	4	no	—
319	C₄⋊C₄	—	2	4	no	—
320	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
321	C₂²⋊C₄	—	2	4	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₂²⋊C₄	—	2	4	no	—
324	C₂²⋊C₄	—	2	4	no	—
325	C₂²⋊C₄	—	2	4	no	—
326	C₄²	—	2	8	no	—
327	C₄²	—	2	4	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	2	4	no	—
330	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
331	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
332	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
333	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
334	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
335	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
336	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
337	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
338	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
339	C₄²	C₂×D₄×C₆₇	1	2	yes	—
340	C₄²	#41×C₆₇	1	8	yes	complemented
341	C₂³	—	2	2	no	—
342	C₂³	—	2	4	no	—
343	C₂³	—	2	2	no	—
344	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
345	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
346	Q₈	—	2	4	no	—
347	Q₈	—	4	4	no	—
348	D₄	—	2	4	no	—
349	D₄	—	4	4	no	—
350	D₄	—	4	4	no	—
351	D₄	—	4	8	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	4	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	8	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	8	no	—
365	C₂×C₄	—	4	16	no	—
366	C₂×C₄	—	4	8	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
369	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
370	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
371	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
372	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
373	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
375	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
376	C₈	—	4	4	no	—
377	C₂²	—	2	8	no	—
378	C₂²	—	2	2	no	—
379	C₂²	—	2	2	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	4	2	no	—
382	C₂²	—	4	4	no	—
383	C₂²	#421×C₆₇	1	4	yes	—
384	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
385	C₂²	#525×C₆₇	1	2	yes	—
386	C₄	—	2	2	no	—
387	C₄	—	2	2	no	—
388	C₄	—	2	2	no	—
389	C₄	—	2	2	no	—
390	C₄	—	2	2	no	—
391	C₄	—	4	8	no	—
392	C₄	—	4	2	no	—
393	C₄	—	4	4	no	—
394	C₄	C₄×#41×C₆₇	1	2	yes	—
395	C₄	#532×C₆₇	1	2	yes	—
396	C₂	—	2	2	no	—
397	C₂	—	2	4	no	—
398	C₂	—	8	2	no	—
399	C₂	C₆₇×#2297	1	1	yes	characteristic
400	C₂	C₂×#421×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₆₇×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2920	C₂	semidirect
3. C₂×#376×C₆₇	C₂	semidirect
4. C₆₇×#1995	C₂	semidirect
5. C₆₇×#1963	C₂	semidirect
6. C₆₇×#1896	C₂	semidirect
7. C₆₇×#1759	C₂	semidirect
8. #533×C₆₇	C₄	semidirect
10. #532×C₆₇	C₄	semidirect
14. #529×C₆₇	C₄	semidirect
38. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
39. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
40. #127×C₆₇	C₂×C₄	semidirect
43. #125×C₆₇	C₂×C₄	semidirect
45. #124×C₆₇	C₂×C₄	semidirect
57. C₄×Q₈×C₆₇	C₂×C₄	semidirect
63. C₄×D₄×C₆₇	C₂×C₄	semidirect
139. C₄²×C₆₇	#41	semidirect
195. #21417	C₆₇	direct (N × H)
202. #2920	C₂×C₆₇	semidirect
203. C₂×#376	C₂×C₆₇	semidirect
204. #1995	C₂×C₆₇	semidirect
205. #1963	C₂×C₆₇	semidirect
206. #1896	C₂×C₆₇	semidirect
207. #1759	C₂×C₆₇	semidirect
208. C₆₇	#21417	direct (N × H)
209. #533	C₄×C₆₇	semidirect
211. #532	C₄×C₆₇	semidirect
215. #529	C₄×C₆₇	semidirect
239. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
240. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
241. #127	C₂×C₄×C₆₇	semidirect
244. #125	C₂×C₄×C₆₇	semidirect
246. #124	C₂×C₄×C₆₇	semidirect
258. C₄×Q₈	C₂×C₄×C₆₇	semidirect
264. C₄×D₄	C₂×C₄×C₆₇	semidirect
340. C₄²	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3444	2
C₆₇×#2920	2
C₂×#376×C₆₇	2
C₆₇×#1995	2
C₆₇×#1963	2
C₆₇×#1896	2
C₆₇×#1759	2
#21417	67

Subgroups — C67×#21417 #30700668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21417 #30700668