Subgroups — C₆₇×#25936 #30705187

Order 17152.

1034 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#487×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#462×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2568	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2024	C₂	1	4	yes	maximal
6	C₆₇×#1987	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1984	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1878	C₂	1	4	yes	maximal
9	C₄×#103×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
11	Q₈²×C₆₇	—	2	4	no	—
12	Q₈²×C₆₇	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
15	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	#487×C₆₇	—	2	4	no	—
23	#462×C₆₇	—	2	4	no	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#103×C₆₇	C₂²	1	2	yes	—
26	C₂×#103×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₂²	1	4	yes	—
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	#400×C₆₇	—	2	2	no	—
30	#391×C₆₇	—	2	1	no	—
31	#391×C₆₇	C₂²	1	4	yes	—
32	#382×C₆₇	—	2	1	no	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	#340×C₆₇	C₂²	1	4	yes	—
35	#340×C₆₇	C₂²	1	2	yes	—
36	#336×C₆₇	C₂²	1	4	yes	—
37	C₂²×Q₈×C₆₇	—	2	2	no	—
38	C₂²×Q₈×C₆₇	—	2	1	no	—
39	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
41	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×Q₁₆×C₆₇	—	2	4	no	—
43	C₂×Q₁₆×C₆₇	—	2	8	no	—
44	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₆₇	—	2	16	no	—
47	C₄⋊Q₈×C₆₇	—	2	8	no	—
48	C₄⋊Q₈×C₆₇	—	2	4	no	—
49	C₄⋊Q₈×C₆₇	—	2	4	no	—
50	C₄⋊Q₈×C₆₇	—	2	2	no	—
51	C₄⋊Q₈×C₆₇	—	2	2	no	—
52	C₄⋊Q₈×C₆₇	—	2	2	no	—
53	C₄⋊Q₈×C₆₇	D₄	1	4	yes	—
54	C₄×Q₈×C₆₇	—	2	8	no	—
55	C₄×Q₈×C₆₇	—	2	4	no	—
56	C₄×Q₈×C₆₇	—	2	4	no	—
57	C₄×Q₈×C₆₇	—	2	2	no	—
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	C₄×Q₈×C₆₇	D₄	1	4	yes	—
61	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	—	2	1	no	—
74	C₂×C₄²×C₆₇	—	2	1	no	—
75	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	#103×C₆₇	—	2	4	no	—
79	#103×C₆₇	—	2	4	no	—
80	#103×C₆₇	—	2	8	no	—
81	C₄×C₈×C₆₇	—	2	2	no	—
82	#95×C₆₇	—	2	1	no	—
83	#95×C₆₇	—	2	4	no	—
84	#95×C₆₇	—	2	4	no	—
85	C₂×Q₈×C₆₇	—	2	8	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	—	2	4	no	—
88	C₂×Q₈×C₆₇	—	2	2	no	—
89	C₂×Q₈×C₆₇	—	2	1	no	—
90	C₂×Q₈×C₆₇	—	2	2	no	—
91	C₂×Q₈×C₆₇	—	2	2	no	—
92	C₂×Q₈×C₆₇	—	2	2	no	—
93	C₂×Q₈×C₆₇	—	2	4	no	—
94	C₂×Q₈×C₆₇	—	2	4	no	—
95	C₂×Q₈×C₆₇	—	8	2	no	—
96	C₂×Q₈×C₆₇	—	8	4	no	—
97	C₂×Q₈×C₆₇	Q₁₆	1	4	yes	complemented
98	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
99	C₂×Q₈×C₆₇	#41	1	4	yes	—
100	C₂²×C₄×C₆₇	—	2	1	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	1	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
110	Q₁₆×C₆₇	—	4	8	no	—
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	4	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	8	no	—
117	C₄⋊C₄×C₆₇	—	2	8	no	—
118	C₄⋊C₄×C₆₇	—	2	4	no	—
119	C₄⋊C₄×C₆₇	—	2	4	no	—
120	C₄⋊C₄×C₆₇	—	2	4	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	4	2	no	—
127	C₄⋊C₄×C₆₇	—	4	2	no	—
128	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
129	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
130	C₄²×C₆₇	—	2	8	no	—
131	C₄²×C₆₇	—	2	4	no	—
132	C₄²×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	—	2	2	no	—
134	C₄²×C₆₇	—	4	2	no	—
135	C₄²×C₆₇	C₂×D₄	1	2	yes	—
136	C₄²×C₆₇	C₂×D₄	1	2	yes	—
137	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
138	Q₈×C₆₇	—	2	2	no	—
139	Q₈×C₆₇	—	2	4	no	—
140	Q₈×C₆₇	—	2	8	no	—
141	Q₈×C₆₇	—	4	2	no	—
142	Q₈×C₆₇	—	4	2	no	—
143	Q₈×C₆₇	—	4	4	no	—
144	Q₈×C₆₇	—	8	4	no	—
145	Q₈×C₆₇	—	8	8	no	—
146	Q₈×C₆₇	#124	1	4	yes	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	4	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	4	no	—
156	C₂×C₄×C₆₇	—	4	4	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	4	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	#120	1	2	yes	—
164	C₂×C₄×C₆₇	#120	1	2	yes	—
165	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
166	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
167	C₂×C₄×C₆₇	#137	1	2	yes	—
168	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
171	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
172	C₈×C₆₇	—	4	2	no	—
173	C₈×C₆₇	—	4	2	no	—
174	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
175	C₂²×C₆₇	#548	1	2	yes	—
176	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
177	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
178	C₂²×C₆₇	#577	1	2	yes	—
179	C₄×C₆₇	—	2	4	no	—
180	C₄×C₆₇	—	2	4	no	—
181	C₄×C₆₇	—	2	2	no	—
182	C₄×C₆₇	—	4	2	no	—
183	C₄×C₆₇	—	4	4	no	—
184	C₄×C₆₇	—	4	2	no	—
185	C₄×C₆₇	—	8	2	no	—
186	C₄×C₆₇	—	8	2	no	—
187	C₄×C₆₇	#451	1	4	yes	—
188	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
189	C₄×C₆₇	#536	1	1	yes	characteristic
190	C₄×C₆₇	#548	1	2	yes	—
191	#25936	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂×C₆₇	#2436	1	1	yes	characteristic
193	C₂×C₆₇	C₂×#451	1	1	yes	characteristic
194	C₂×C₆₇	#3009	1	1	yes	characteristic
195	C₂×C₆₇	#3363	1	2	yes	—
196	C₂×C₆₇	#3369	1	2	yes	—
197	C₂×Q₈²	C₂×C₆₇	1	1	yes	characteristic
198	C₂×#487	C₂×C₆₇	1	1	yes	characteristic
199	C₂×#462	C₂×C₆₇	1	1	yes	characteristic
200	#2568	C₂×C₆₇	1	1	yes	characteristic
201	#2024	C₂×C₆₇	1	4	yes	—
202	#1987	C₂×C₆₇	1	1	yes	characteristic
203	#1984	C₂×C₆₇	1	1	yes	characteristic
204	#1878	C₂×C₆₇	1	4	yes	—
205	C₄×#103	C₂×C₆₇	1	1	yes	characteristic
206	C₆₇	#25936	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
208	Q₈²	—	2	4	no	—
209	Q₈²	—	2	4	no	—
210	C₂×C₄⋊Q₈	—	2	1	no	—
211	C₂×C₄⋊Q₈	—	2	2	no	—
212	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
213	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
214	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
215	C₂×C₄×Q₈	—	2	1	no	—
216	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
217	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
218	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
219	#487	—	2	4	no	—
220	#462	—	2	4	no	—
221	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
222	C₂×#103	C₂²×C₆₇	1	4	yes	—
223	C₂×#103	C₂²×C₆₇	1	2	yes	—
224	C₂×#103	C₂²×C₆₇	1	2	yes	—
225	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
226	#400	—	2	2	no	—
227	#391	—	2	1	no	—
228	#391	C₂²×C₆₇	1	4	yes	—
229	#382	—	2	1	no	—
230	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
231	#340	C₂²×C₆₇	1	4	yes	—
232	#340	C₂²×C₆₇	1	2	yes	—
233	#336	C₂²×C₆₇	1	4	yes	—
234	C₂²×Q₈	—	2	2	no	—
235	C₂²×Q₈	—	2	1	no	—
236	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
237	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
238	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
239	C₂×Q₁₆	—	2	4	no	—
240	C₂×Q₁₆	—	2	8	no	—
241	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
242	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
243	C₄⋊Q₈	—	2	16	no	—
244	C₄⋊Q₈	—	2	8	no	—
245	C₄⋊Q₈	—	2	4	no	—
246	C₄⋊Q₈	—	2	4	no	—
247	C₄⋊Q₈	—	2	2	no	—
248	C₄⋊Q₈	—	2	2	no	—
249	C₄⋊Q₈	—	2	2	no	—
250	C₄⋊Q₈	D₄×C₆₇	1	4	yes	—
251	C₄×Q₈	—	2	8	no	—
252	C₄×Q₈	—	2	4	no	—
253	C₄×Q₈	—	2	4	no	—
254	C₄×Q₈	—	2	2	no	—
255	C₄×Q₈	—	2	2	no	—
256	C₄×Q₈	—	2	2	no	—
257	C₄×Q₈	D₄×C₆₇	1	4	yes	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	—	2	2	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
265	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
266	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
267	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
268	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
269	C₂×C₄²	—	2	2	no	—
270	C₂×C₄²	—	2	1	no	—
271	C₂×C₄²	—	2	1	no	—
272	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
273	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
274	C₄⋊C₈	—	2	2	no	—
275	#103	—	2	4	no	—
276	#103	—	2	4	no	—
277	#103	—	2	8	no	—
278	C₄×C₈	—	2	2	no	—
279	#95	—	2	1	no	—
280	#95	—	2	4	no	—
281	#95	—	2	4	no	—
282	C₂×Q₈	—	2	8	no	—
283	C₂×Q₈	—	2	2	no	—
284	C₂×Q₈	—	2	4	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	4	no	—
291	C₂×Q₈	—	2	4	no	—
292	C₂×Q₈	—	8	2	no	—
293	C₂×Q₈	—	8	4	no	—
294	C₂×Q₈	Q₁₆×C₆₇	1	4	yes	complemented
295	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
296	C₂×Q₈	#41×C₆₇	1	4	yes	—
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
303	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
304	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
305	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
306	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
307	Q₁₆	—	4	8	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	4	no	—
312	C₂×C₈	—	2	2	no	—
313	C₄⋊C₄	—	2	8	no	—
314	C₄⋊C₄	—	2	8	no	—
315	C₄⋊C₄	—	2	4	no	—
316	C₄⋊C₄	—	2	4	no	—
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	1	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	4	2	no	—
324	C₄⋊C₄	—	4	2	no	—
325	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
326	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
327	C₄²	—	2	8	no	—
328	C₄²	—	2	4	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	4	2	no	—
332	C₄²	C₂×D₄×C₆₇	1	2	yes	—
333	C₄²	C₂×D₄×C₆₇	1	2	yes	—
334	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
335	Q₈	—	2	8	no	—
336	Q₈	—	2	2	no	—
337	Q₈	—	2	4	no	—
338	Q₈	—	4	2	no	—
339	Q₈	—	4	2	no	—
340	Q₈	—	4	4	no	—
341	Q₈	—	8	4	no	—
342	Q₈	—	8	8	no	—
343	Q₈	#124×C₆₇	1	4	yes	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	4	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	4	4	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	4	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	#120×C₆₇	1	2	yes	—
361	C₂×C₄	#120×C₆₇	1	2	yes	—
362	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
363	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
364	C₂×C₄	#137×C₆₇	1	2	yes	—
365	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
367	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
368	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
369	C₈	—	4	2	no	—
370	C₈	—	4	2	no	—
371	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
372	C₂²	#548×C₆₇	1	2	yes	—
373	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
374	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
375	C₂²	#577×C₆₇	1	2	yes	—
376	C₄	—	2	2	no	—
377	C₄	—	2	4	no	—
378	C₄	—	2	4	no	—
379	C₄	—	4	2	no	—
380	C₄	—	4	4	no	—
381	C₄	—	4	2	no	—
382	C₄	—	8	2	no	—
383	C₄	—	8	2	no	—
384	C₄	#451×C₆₇	1	4	yes	—
385	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
386	C₄	#536×C₆₇	1	1	yes	characteristic
387	C₄	#548×C₆₇	1	2	yes	—
388	C₂	C₆₇×#2436	1	1	yes	characteristic
389	C₂	C₂×#451×C₆₇	1	1	yes	characteristic
390	C₂	C₆₇×#3009	1	1	yes	characteristic
391	C₂	C₆₇×#3363	1	2	yes	—
392	C₂	C₆₇×#3369	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
97. C₂×Q₈×C₆₇	Q₁₆	semidirect
191. #25936	C₆₇	direct (N × H)
206. C₆₇	#25936	direct (N × H)
294. C₂×Q₈	Q₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₆₇	2
C₂×#487×C₆₇	2
C₂×#462×C₆₇	2
C₆₇×#2568	2
C₆₇×#2024	2
C₆₇×#1987	2
C₆₇×#1984	2
C₆₇×#1878	2
C₄×#103×C₆₇	2
#25936	67

Subgroups — C67×#25936 #30705187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25936 #30705187