Subgroups — C₆₇×#11392 #30690643

Order 17152.

974 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#544×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#343×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1736	C₂	1	2	yes	maximal
4	C₂×#339×C₆₇	C₂	1	2	yes	maximal
5	C₆₇×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
7	#544×C₆₇	—	2	4	no	—
8	#544×C₆₇	—	2	2	no	—
9	#544×C₆₇	C₄	1	4	yes	complemented
10	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
11	C₂×#125×C₆₇	—	2	2	no	—
12	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
13	C₂×#117×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#95×C₆₇	—	2	2	no	—
18	#343×C₆₇	—	2	2	no	—
19	#340×C₆₇	—	2	2	no	—
20	#340×C₆₇	—	2	2	no	—
21	#339×C₆₇	—	2	4	no	—
22	C₂²×Q₈×C₆₇	—	2	2	no	—
23	C₂³×C₄×C₆₇	—	2	1	no	—
24	C₂³×C₄×C₆₇	—	2	1	no	—
25	C₂³×C₄×C₆₇	D₄	1	2	yes	—
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₆₇	—	2	2	no	—
28	C₂×M₁₆×C₆₇	—	2	2	no	—
29	C₂×M₁₆×C₆₇	—	2	2	no	—
30	C₂×M₁₆×C₆₇	—	2	2	no	—
31	C₂×M₁₆×C₆₇	—	2	2	no	—
32	C₂×M₁₆×C₆₇	Q₈	1	4	yes	complemented
33	C₂²×C₈×C₆₇	—	2	2	no	—
34	C₄⋊Q₈×C₆₇	—	2	8	no	—
35	C₄⋊Q₈×C₆₇	—	2	8	no	—
36	#125×C₆₇	—	2	8	no	—
37	#125×C₆₇	—	2	8	no	—
38	C₂²⋊Q₈×C₆₇	—	2	8	no	—
39	C₂²⋊Q₈×C₆₇	—	2	8	no	—
40	#117×C₆₇	—	2	2	no	—
41	#117×C₆₇	—	2	2	no	—
42	#117×C₆₇	C₂×C₄	1	4	yes	—
43	#117×C₆₇	D₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₇	Q₈	1	2	yes	complemented
56	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄²×C₆₇	—	2	2	no	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	—	2	2	no	—
60	C₂×C₄²×C₆₇	—	2	1	no	—
61	C₂×C₄²×C₆₇	Q₈	1	2	yes	complemented
62	#95×C₆₇	—	4	4	no	—
63	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂×Q₈×C₆₇	—	2	8	no	—
65	C₂×Q₈×C₆₇	—	8	2	no	—
66	C₂×Q₈×C₆₇	—	8	2	no	—
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	—	2	2	no	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	2	no	—
74	C₂²×C₄×C₆₇	—	2	2	no	—
75	C₂²×C₄×C₆₇	—	4	1	no	—
76	C₂²×C₄×C₆₇	—	4	1	no	—
77	C₂²×C₄×C₆₇	—	4	1	no	—
78	C₂²×C₄×C₆₇	—	4	1	no	—
79	C₂²×C₄×C₆₇	—	4	2	no	—
80	C₂²×C₄×C₆₇	—	4	2	no	—
81	C₂²×C₄×C₆₇	—	4	2	no	—
82	C₂²×C₄×C₆₇	—	4	2	no	—
83	C₂²×C₄×C₆₇	—	4	1	no	—
84	C₂²×C₄×C₆₇	—	4	1	no	—
85	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	#41	1	2	yes	—
94	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
96	M₁₆×C₆₇	—	2	4	no	—
97	M₁₆×C₆₇	—	2	4	no	—
98	M₁₆×C₆₇	—	4	2	no	—
99	M₁₆×C₆₇	—	4	2	no	—
100	C₂×C₈×C₆₇	—	2	4	no	—
101	C₂×C₈×C₆₇	—	4	2	no	—
102	C₂×C₈×C₆₇	—	4	2	no	—
103	C₄⋊C₄×C₆₇	—	2	8	no	—
104	C₄⋊C₄×C₆₇	—	2	16	no	—
105	C₄⋊C₄×C₆₇	—	2	4	no	—
106	C₄⋊C₄×C₆₇	—	2	4	no	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	4	no	—
110	C₄⋊C₄×C₆₇	—	4	2	no	—
111	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
112	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
113	C₂²⋊C₄×C₆₇	—	2	4	no	—
114	C₂²⋊C₄×C₆₇	—	2	4	no	—
115	C₄²×C₆₇	—	2	4	no	—
116	C₄²×C₆₇	—	2	4	no	—
117	C₄²×C₆₇	—	2	4	no	—
118	C₄²×C₆₇	—	4	2	no	—
119	C₂³×C₆₇	—	2	1	no	—
120	C₂³×C₆₇	—	2	1	no	—
121	C₂³×C₆₇	—	2	1	no	—
122	C₂³×C₆₇	—	2	1	no	—
123	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂³×C₆₇	#120	1	1	yes	characteristic
125	C₂³×C₆₇	#120	1	1	yes	characteristic
126	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂³×C₆₇	#124	1	1	yes	characteristic
129	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
130	Q₈×C₆₇	—	8	8	no	—
131	C₂×C₄×C₆₇	—	2	4	no	—
132	C₂×C₄×C₆₇	—	2	4	no	—
133	C₂×C₄×C₆₇	—	2	4	no	—
134	C₂×C₄×C₆₇	—	2	4	no	—
135	C₂×C₄×C₆₇	—	4	2	no	—
136	C₂×C₄×C₆₇	—	4	2	no	—
137	C₂×C₄×C₆₇	—	4	2	no	—
138	C₂×C₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	4	2	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	—	4	2	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	2	no	—
145	C₂×C₄×C₆₇	—	4	2	no	—
146	C₂×C₄×C₆₇	—	4	2	no	—
147	C₂×C₄×C₆₇	—	4	4	no	—
148	C₂×C₄×C₆₇	—	4	4	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	—	8	2	no	—
151	C₂×C₄×C₆₇	—	8	2	no	—
152	C₂×C₄×C₆₇	—	8	1	no	—
153	C₂×C₄×C₆₇	—	8	1	no	—
154	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
156	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
158	C₂×C₄×C₆₇	#120	1	2	yes	—
159	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
160	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
161	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
162	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
163	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
164	C₂×C₄×C₆₇	#124	1	2	yes	—
165	C₂×C₄×C₆₇	#124	1	2	yes	—
166	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
168	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
169	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
170	C₈×C₆₇	—	4	4	no	—
171	C₂²×C₆₇	—	2	1	no	—
172	C₂²×C₆₇	—	2	1	no	—
173	C₂²×C₆₇	—	2	1	no	—
174	C₂²×C₆₇	—	2	1	no	—
175	C₂²×C₆₇	—	2	1	no	—
176	C₂²×C₆₇	—	2	1	no	—
177	C₂²×C₆₇	—	2	1	no	—
178	C₂²×C₆₇	—	2	1	no	—
179	C₂²×C₆₇	—	2	1	no	—
180	C₂²×C₆₇	—	2	1	no	—
181	C₂²×C₆₇	—	2	1	no	—
182	C₂²×C₆₇	—	2	1	no	—
183	C₂²×C₆₇	#379	1	1	yes	characteristic
184	C₂²×C₆₇	#391	1	1	yes	characteristic
185	C₂²×C₆₇	#391	1	1	yes	characteristic
186	C₂²×C₆₇	#393	1	1	yes	characteristic
187	C₂²×C₆₇	#393	1	1	yes	characteristic
188	C₂²×C₆₇	#395	1	1	yes	characteristic
189	C₂²×C₆₇	#395	1	1	yes	characteristic
190	C₂²×C₆₇	#453	1	1	yes	characteristic
191	C₂²×C₆₇	#454	1	1	yes	characteristic
192	C₂²×C₆₇	#455	1	1	yes	characteristic
193	C₂²×C₆₇	#456	1	1	yes	characteristic
194	C₄×C₆₇	—	4	4	no	—
195	C₄×C₆₇	—	4	4	no	—
196	C₄×C₆₇	—	4	4	no	—
197	C₄×C₆₇	—	4	4	no	—
198	C₄×C₆₇	—	8	2	no	—
199	C₄×C₆₇	#391	1	2	yes	—
200	C₄×C₆₇	#391	1	2	yes	—
201	C₄×C₆₇	#393	1	2	yes	—
202	C₄×C₆₇	#395	1	2	yes	—
203	#11392	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂×C₆₇	—	2	1	no	—
205	C₂×C₆₇	—	2	1	no	—
206	C₂×C₆₇	—	2	1	no	—
207	C₂×C₆₇	—	2	1	no	—
208	C₂×C₆₇	#1446	1	1	yes	characteristic
209	C₂×C₆₇	#1792	1	1	yes	characteristic
210	C₂×C₆₇	#1793	1	1	yes	characteristic
211	C₂×C₆₇	#2027	1	1	yes	characteristic
212	C₂×C₆₇	#2028	1	1	yes	characteristic
213	C₂×C₆₇	#2060	1	1	yes	characteristic
214	C₂×C₆₇	#2061	1	1	yes	characteristic
215	C₂×#544	C₂×C₆₇	1	1	yes	characteristic
216	C₂×#343	C₂×C₆₇	1	1	yes	characteristic
217	#1736	C₂×C₆₇	1	2	yes	—
218	C₂×#339	C₂×C₆₇	1	2	yes	—
219	C₄⋊#95	C₂×C₆₇	1	1	yes	characteristic
220	C₆₇	#11392	1	1	yes	characteristic, direct product factor, Sylow
221	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
222	#544	—	2	4	no	—
223	#544	—	2	2	no	—
224	#544	C₄×C₆₇	1	4	yes	complemented
225	C₂×C₄⋊Q₈	—	2	2	no	—
226	C₂×#125	—	2	2	no	—
227	C₂×C₂²⋊Q₈	—	2	2	no	—
228	C₂×#117	C₂²×C₆₇	1	2	yes	—
229	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
230	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
231	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
232	C₂×#95	—	2	2	no	—
233	#343	—	2	2	no	—
234	#340	—	2	2	no	—
235	#340	—	2	2	no	—
236	#339	—	2	4	no	—
237	C₂²×Q₈	—	2	2	no	—
238	C₂³×C₄	—	2	1	no	—
239	C₂³×C₄	—	2	1	no	—
240	C₂³×C₄	D₄×C₆₇	1	2	yes	—
241	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
242	C₂×M₁₆	—	2	2	no	—
243	C₂×M₁₆	—	2	2	no	—
244	C₂×M₁₆	—	2	2	no	—
245	C₂×M₁₆	—	2	2	no	—
246	C₂×M₁₆	—	2	2	no	—
247	C₂×M₁₆	Q₈×C₆₇	1	4	yes	complemented
248	C₂²×C₈	—	2	2	no	—
249	C₄⋊Q₈	—	2	8	no	—
250	C₄⋊Q₈	—	2	8	no	—
251	#125	—	2	8	no	—
252	#125	—	2	8	no	—
253	C₂²⋊Q₈	—	2	8	no	—
254	C₂²⋊Q₈	—	2	8	no	—
255	#117	—	2	2	no	—
256	#117	—	2	2	no	—
257	#117	C₂×C₄×C₆₇	1	4	yes	—
258	#117	D₄×C₆₇	1	4	yes	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄⋊C₄	—	2	4	no	—
261	C₂×C₄⋊C₄	—	2	2	no	—
262	C₂×C₄⋊C₄	—	2	2	no	—
263	C₂×C₄⋊C₄	—	2	2	no	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
267	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
268	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
269	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
270	C₂×C₄⋊C₄	Q₈×C₆₇	1	2	yes	complemented
271	C₂×C₂²⋊C₄	—	2	2	no	—
272	C₂×C₄²	—	2	2	no	—
273	C₂×C₄²	—	2	2	no	—
274	C₂×C₄²	—	2	2	no	—
275	C₂×C₄²	—	2	1	no	—
276	C₂×C₄²	Q₈×C₆₇	1	2	yes	complemented
277	#95	—	4	4	no	—
278	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
279	C₂×Q₈	—	2	8	no	—
280	C₂×Q₈	—	8	2	no	—
281	C₂×Q₈	—	8	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	2	2	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	—	4	2	no	—
296	C₂²×C₄	—	4	2	no	—
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	—	4	1	no	—
299	C₂²×C₄	—	4	1	no	—
300	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
301	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
302	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
303	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
304	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
305	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
306	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
307	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
308	C₂²×C₄	#41×C₆₇	1	2	yes	—
309	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
310	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
311	M₁₆	—	2	4	no	—
312	M₁₆	—	2	4	no	—
313	M₁₆	—	4	2	no	—
314	M₁₆	—	4	2	no	—
315	C₂×C₈	—	2	4	no	—
316	C₂×C₈	—	4	2	no	—
317	C₂×C₈	—	4	2	no	—
318	C₄⋊C₄	—	2	8	no	—
319	C₄⋊C₄	—	2	16	no	—
320	C₄⋊C₄	—	2	4	no	—
321	C₄⋊C₄	—	2	4	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	4	no	—
325	C₄⋊C₄	—	4	2	no	—
326	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
327	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
328	C₂²⋊C₄	—	2	4	no	—
329	C₂²⋊C₄	—	2	4	no	—
330	C₄²	—	2	4	no	—
331	C₄²	—	2	4	no	—
332	C₄²	—	2	4	no	—
333	C₄²	—	4	2	no	—
334	C₂³	—	2	1	no	—
335	C₂³	—	2	1	no	—
336	C₂³	—	2	1	no	—
337	C₂³	—	2	1	no	—
338	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
339	C₂³	#120×C₆₇	1	1	yes	characteristic
340	C₂³	#120×C₆₇	1	1	yes	characteristic
341	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
342	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
343	C₂³	#124×C₆₇	1	1	yes	characteristic
344	C₂³	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
345	Q₈	—	8	8	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	4	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	4	no	—
363	C₂×C₄	—	4	4	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	8	2	no	—
366	C₂×C₄	—	8	2	no	—
367	C₂×C₄	—	8	1	no	—
368	C₂×C₄	—	8	1	no	—
369	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
370	C₂×C₄	C₄×Q₈×C₆₇	1	2	yes	—
371	C₂×C₄	C₄×Q₈×C₆₇	1	1	yes	characteristic
372	C₂×C₄	C₄×Q₈×C₆₇	1	1	yes	characteristic
373	C₂×C₄	#120×C₆₇	1	2	yes	—
374	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
375	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
376	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
377	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
378	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
379	C₂×C₄	#124×C₆₇	1	2	yes	—
380	C₂×C₄	#124×C₆₇	1	2	yes	—
381	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
382	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
383	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
384	C₂×C₄	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
385	C₈	—	4	4	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	1	no	—
391	C₂²	—	2	1	no	—
392	C₂²	—	2	1	no	—
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	2	1	no	—
398	C₂²	#379×C₆₇	1	1	yes	characteristic
399	C₂²	#391×C₆₇	1	1	yes	characteristic
400	C₂²	#391×C₆₇	1	1	yes	characteristic
401	C₂²	#393×C₆₇	1	1	yes	characteristic
402	C₂²	#393×C₆₇	1	1	yes	characteristic
403	C₂²	#395×C₆₇	1	1	yes	characteristic
404	C₂²	#395×C₆₇	1	1	yes	characteristic
405	C₂²	#453×C₆₇	1	1	yes	characteristic
406	C₂²	#454×C₆₇	1	1	yes	characteristic
407	C₂²	#455×C₆₇	1	1	yes	characteristic
408	C₂²	#456×C₆₇	1	1	yes	characteristic
409	C₄	—	4	4	no	—
410	C₄	—	4	4	no	—
411	C₄	—	4	4	no	—
412	C₄	—	4	4	no	—
413	C₄	—	8	2	no	—
414	C₄	#391×C₆₇	1	2	yes	—
415	C₄	#391×C₆₇	1	2	yes	—
416	C₄	#393×C₆₇	1	2	yes	—
417	C₄	#395×C₆₇	1	2	yes	—
418	C₂	—	2	1	no	—
419	C₂	—	2	1	no	—
420	C₂	—	2	1	no	—
421	C₂	—	2	1	no	—
422	C₂	C₆₇×#1446	1	1	yes	characteristic
423	C₂	C₆₇×#1792	1	1	yes	characteristic
424	C₂	C₆₇×#1793	1	1	yes	characteristic
425	C₂	C₆₇×#2027	1	1	yes	characteristic
426	C₂	C₆₇×#2028	1	1	yes	characteristic
427	C₂	C₆₇×#2060	1	1	yes	characteristic
428	C₂	C₆₇×#2061	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	#120
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
9. #544×C₆₇	C₄	semidirect
32. C₂×M₁₆×C₆₇	Q₈	semidirect
55. C₂×C₄⋊C₄×C₆₇	Q₈	semidirect
61. C₂×C₄²×C₆₇	Q₈	semidirect
203. #11392	C₆₇	direct (N × H)
220. C₆₇	#11392	direct (N × H)
224. #544	C₄×C₆₇	semidirect
247. C₂×M₁₆	Q₈×C₆₇	semidirect
270. C₂×C₄⋊C₄	Q₈×C₆₇	semidirect
276. C₂×C₄²	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#544×C₆₇	2
C₂×#343×C₆₇	2
C₆₇×#1736	2
C₂×#339×C₆₇	2
C₆₇×C₄⋊#95	2
#11392	67

Subgroups — C67×#11392 #30690643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11392 #30690643