Subgroups — C₆₇×#33201 #30712452

Order 17152.

946 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#476×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#474×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2862	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1949	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1842	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1804	C₂	1	4	yes	maximal
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
14	C₂²×C₄²×C₆₇	—	2	1	no	—
15	#476×C₆₇	—	2	4	no	—
16	#474×C₆₇	—	2	4	no	—
17	C₂×#107×C₆₇	C₂²	1	2	yes	—
18	C₂×#106×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
22	C₂×#102×C₆₇	C₂²	1	2	yes	—
23	C₂×#102×C₆₇	C₂²	1	2	yes	—
24	#391×C₆₇	—	2	1	no	—
25	#391×C₆₇	—	2	1	no	—
26	#391×C₆₇	—	2	2	no	—
27	#382×C₆₇	C₂²	1	4	yes	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
31	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
32	C₄³×C₆₇	C₂²	1	1	yes	characteristic
33	#340×C₆₇	C₂²	1	4	yes	—
34	#340×C₆₇	C₂²	1	4	yes	—
35	C₂²×Q₈×C₆₇	—	2	1	no	—
36	C₂²×Q₈×C₆₇	—	2	1	no	—
37	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	—	2	1	no	—
39	C₂³×C₄×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
42	C₄⋊Q₈×C₆₇	—	2	2	no	—
43	C₄⋊Q₈×C₆₇	—	2	2	no	—
44	C₄⋊Q₈×C₆₇	—	2	2	no	—
45	C₄⋊Q₈×C₆₇	—	2	2	no	—
46	C₄×D₄×C₆₇	—	2	2	no	—
47	C₄×D₄×C₆₇	—	2	4	no	—
48	C₄×D₄×C₆₇	D₄	1	4	yes	—
49	C₄×D₄×C₆₇	Q₈	1	8	yes	complemented
50	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄²×C₆₇	—	2	2	no	—
59	C₂×C₄²×C₆₇	—	2	1	no	—
60	C₂×C₄²×C₆₇	—	4	4	no	—
61	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	D₄	1	2	yes	—
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
66	#107×C₆₇	—	2	4	no	—
67	#106×C₆₇	—	2	4	no	—
68	C₄⋊C₈×C₆₇	—	2	2	no	—
69	C₄⋊C₈×C₆₇	—	2	2	no	—
70	C₄⋊C₈×C₆₇	—	2	8	no	—
71	#102×C₆₇	—	2	4	no	—
72	#102×C₆₇	—	2	4	no	—
73	#95×C₆₇	—	2	1	no	—
74	#95×C₆₇	—	2	2	no	—
75	#95×C₆₇	—	2	4	no	—
76	#95×C₆₇	—	2	4	no	—
77	C₂⁴×C₆₇	—	2	1	no	—
78	C₂×Q₈×C₆₇	—	4	4	no	—
79	C₂×Q₈×C₆₇	—	4	2	no	—
80	C₂×Q₈×C₆₇	—	4	2	no	—
81	C₂×Q₈×C₆₇	—	4	2	no	—
82	C₂×Q₈×C₆₇	—	4	2	no	—
83	C₂×D₄×C₆₇	—	2	2	no	—
84	C₂×D₄×C₆₇	—	2	1	no	—
85	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	—
87	C₂²×C₄×C₆₇	—	2	1	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	—	2	8	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	4	4	no	—
94	C₂²×C₄×C₆₇	—	4	4	no	—
95	C₂²×C₄×C₆₇	—	4	2	no	—
96	C₂²×C₄×C₆₇	—	4	2	no	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₆₇	—	2	2	no	—
103	C₂×C₈×C₆₇	—	2	2	no	—
104	C₂×C₈×C₆₇	—	2	4	no	—
105	C₂×C₈×C₆₇	—	2	4	no	—
106	C₄⋊C₄×C₆₇	—	2	1	no	—
107	C₄⋊C₄×C₆₇	—	2	2	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₄⋊C₄×C₆₇	—	2	2	no	—
111	C₄⋊C₄×C₆₇	—	2	2	no	—
112	C₄⋊C₄×C₆₇	—	4	2	no	—
113	C₄⋊C₄×C₆₇	—	4	2	no	—
114	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
115	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
116	C₂²⋊C₄×C₆₇	—	2	4	no	—
117	C₂²⋊C₄×C₆₇	—	2	8	no	—
118	C₄²×C₆₇	—	2	2	no	—
119	C₄²×C₆₇	—	2	2	no	—
120	C₄²×C₆₇	—	4	2	no	—
121	C₄²×C₆₇	—	4	4	no	—
122	C₄²×C₆₇	—	4	2	no	—
123	C₄²×C₆₇	C₂×D₄	1	2	yes	—
124	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
125	C₄²×C₆₇	#41	1	2	yes	—
126	C₄²×C₆₇	#41	1	4	yes	—
127	C₂³×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	—	2	4	no	—
129	C₂³×C₆₇	—	4	1	no	—
130	C₂³×C₆₇	—	4	1	no	—
131	C₂³×C₆₇	—	4	2	no	—
132	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
133	Q₈×C₆₇	—	8	4	no	—
134	Q₈×C₆₇	—	8	4	no	—
135	D₄×C₆₇	—	2	2	no	—
136	D₄×C₆₇	—	2	4	no	—
137	D₄×C₆₇	C₄⋊Q₈	1	4	yes	—
138	C₂×C₄×C₆₇	—	2	2	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	4	no	—
145	C₂×C₄×C₆₇	—	4	16	no	—
146	C₂×C₄×C₆₇	—	4	1	no	—
147	C₂×C₄×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	4	1	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	—	4	8	no	—
151	C₂×C₄×C₆₇	—	4	2	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	—	4	4	no	—
156	C₂×C₄×C₆₇	#121	1	2	yes	—
157	C₂×C₄×C₆₇	#121	1	2	yes	—
158	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
159	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
160	C₂×C₄×C₆₇	#135	1	2	yes	—
161	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
162	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
164	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
165	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
166	C₈×C₆₇	—	4	2	no	—
167	C₈×C₆₇	—	4	2	no	—
168	C₂²×C₆₇	—	2	4	no	—
169	C₂²×C₆₇	—	4	4	no	—
170	C₂²×C₆₇	—	4	4	no	—
171	C₂²×C₆₇	—	4	2	no	—
172	C₂²×C₆₇	—	4	2	no	—
173	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
174	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
175	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
176	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂²×C₆₇	#575	1	2	yes	—
178	C₄×C₆₇	—	2	4	no	—
179	C₄×C₆₇	—	2	4	no	—
180	C₄×C₆₇	—	2	2	no	—
181	C₄×C₆₇	—	4	4	no	—
182	C₄×C₆₇	—	4	2	no	—
183	C₄×C₆₇	—	8	2	no	—
184	C₄×C₆₇	—	8	2	no	—
185	C₄×C₆₇	#463	1	4	yes	—
186	C₄×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
187	C₄×C₆₇	#532	1	1	yes	characteristic
188	C₄×C₆₇	#533	1	2	yes	—
189	#33201	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂×C₆₇	—	4	4	no	—
191	C₂×C₆₇	#2857	1	1	yes	characteristic
192	C₂×C₆₇	C₂×#463	1	1	yes	characteristic
193	C₂×C₆₇	#3040	1	1	yes	characteristic
194	C₂×C₆₇	#3076	1	2	yes	—
195	C₂×C₆₇	#3077	1	2	yes	—
196	C₂×#476	C₂×C₆₇	1	1	yes	characteristic
197	C₂×#474	C₂×C₆₇	1	1	yes	characteristic
198	#2862	C₂×C₆₇	1	1	yes	characteristic, complemented
199	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic
200	#1988	C₂×C₆₇	1	1	yes	characteristic, complemented
201	#1987	C₂×C₆₇	1	1	yes	characteristic, complemented
202	#1949	C₂×C₆₇	1	4	yes	complemented
203	#1842	C₂×C₆₇	1	1	yes	characteristic, complemented
204	#1804	C₂×C₆₇	1	4	yes	—
205	C₆₇	#33201	1	1	yes	characteristic, direct product factor, Sylow
206	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
207	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
208	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
209	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
210	C₂²×C₄²	—	2	1	no	—
211	#476	—	2	4	no	—
212	#474	—	2	4	no	—
213	C₂×#107	C₂²×C₆₇	1	2	yes	—
214	C₂×#106	C₂²×C₆₇	1	2	yes	—
215	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
216	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
217	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
218	C₂×#102	C₂²×C₆₇	1	2	yes	—
219	C₂×#102	C₂²×C₆₇	1	2	yes	—
220	#391	—	2	1	no	—
221	#391	—	2	1	no	—
222	#391	—	2	2	no	—
223	#382	C₂²×C₆₇	1	4	yes	—
224	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
225	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
226	C₄×C₂²⋊C₄	—	2	1	no	—
227	C₄×C₂²⋊C₄	—	2	2	no	—
228	C₄³	C₂²×C₆₇	1	1	yes	characteristic
229	#340	C₂²×C₆₇	1	4	yes	—
230	#340	C₂²×C₆₇	1	4	yes	—
231	C₂²×Q₈	—	2	1	no	—
232	C₂²×Q₈	—	2	1	no	—
233	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
234	C₂³×C₄	—	2	1	no	—
235	C₂³×C₄	—	2	2	no	—
236	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
237	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
238	C₄⋊Q₈	—	2	2	no	—
239	C₄⋊Q₈	—	2	2	no	—
240	C₄⋊Q₈	—	2	2	no	—
241	C₄⋊Q₈	—	2	2	no	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	—	2	4	no	—
244	C₄×D₄	D₄×C₆₇	1	4	yes	—
245	C₄×D₄	Q₈×C₆₇	1	8	yes	complemented
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
249	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
250	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
251	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
252	C₂×C₂²⋊C₄	—	2	1	no	—
253	C₂×C₂²⋊C₄	—	2	2	no	—
254	C₂×C₄²	—	2	2	no	—
255	C₂×C₄²	—	2	1	no	—
256	C₂×C₄²	—	4	4	no	—
257	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
258	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
259	C₂×C₄²	D₄×C₆₇	1	2	yes	—
260	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
261	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
262	#107	—	2	4	no	—
263	#106	—	2	4	no	—
264	C₄⋊C₈	—	2	2	no	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	—	2	8	no	—
267	#102	—	2	4	no	—
268	#102	—	2	4	no	—
269	#95	—	2	1	no	—
270	#95	—	2	2	no	—
271	#95	—	2	4	no	—
272	#95	—	2	4	no	—
273	C₂⁴	—	2	1	no	—
274	C₂×Q₈	—	4	4	no	—
275	C₂×Q₈	—	4	2	no	—
276	C₂×Q₈	—	4	2	no	—
277	C₂×Q₈	—	4	2	no	—
278	C₂×Q₈	—	4	2	no	—
279	C₂×D₄	—	2	2	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
282	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	—
283	C₂²×C₄	—	2	1	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	8	no	—
287	C₂²×C₄	—	2	2	no	—
288	C₂²×C₄	—	2	2	no	—
289	C₂²×C₄	—	4	4	no	—
290	C₂²×C₄	—	4	4	no	—
291	C₂²×C₄	—	4	2	no	—
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
294	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
295	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
296	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
297	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	4	no	—
301	C₂×C₈	—	2	4	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	4	2	no	—
309	C₄⋊C₄	—	4	2	no	—
310	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
311	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
312	C₂²⋊C₄	—	2	4	no	—
313	C₂²⋊C₄	—	2	8	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	2	2	no	—
316	C₄²	—	4	2	no	—
317	C₄²	—	4	4	no	—
318	C₄²	—	4	2	no	—
319	C₄²	C₂×D₄×C₆₇	1	2	yes	—
320	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
321	C₄²	#41×C₆₇	1	2	yes	—
322	C₄²	#41×C₆₇	1	4	yes	—
323	C₂³	—	2	2	no	—
324	C₂³	—	2	4	no	—
325	C₂³	—	4	1	no	—
326	C₂³	—	4	1	no	—
327	C₂³	—	4	2	no	—
328	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
329	Q₈	—	8	4	no	—
330	Q₈	—	8	4	no	—
331	D₄	—	2	2	no	—
332	D₄	—	2	4	no	—
333	D₄	C₄⋊Q₈×C₆₇	1	4	yes	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	4	16	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	8	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	4	no	—
352	C₂×C₄	#121×C₆₇	1	2	yes	—
353	C₂×C₄	#121×C₆₇	1	2	yes	—
354	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
355	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
356	C₂×C₄	#135×C₆₇	1	2	yes	—
357	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
358	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
359	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
360	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
361	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
362	C₈	—	4	2	no	—
363	C₈	—	4	2	no	—
364	C₂²	—	2	4	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	4	no	—
368	C₂²	—	4	4	no	—
369	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
370	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
371	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
372	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
373	C₂²	#575×C₆₇	1	2	yes	—
374	C₄	—	2	2	no	—
375	C₄	—	2	4	no	—
376	C₄	—	2	4	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	4	no	—
379	C₄	—	8	2	no	—
380	C₄	—	8	2	no	—
381	C₄	#463×C₆₇	1	4	yes	—
382	C₄	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
383	C₄	#532×C₆₇	1	1	yes	characteristic
384	C₄	#533×C₆₇	1	2	yes	—
385	C₂	—	4	4	no	—
386	C₂	C₆₇×#2857	1	1	yes	characteristic
387	C₂	C₂×#463×C₆₇	1	1	yes	characteristic
388	C₂	C₆₇×#3040	1	1	yes	characteristic
389	C₂	C₆₇×#3076	1	2	yes	—
390	C₂	C₆₇×#3077	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#2862	C₂	semidirect
5. C₆₇×#1988	C₂	semidirect
6. C₆₇×#1987	C₂	semidirect
7. C₆₇×#1949	C₂	semidirect
8. C₆₇×#1842	C₂	semidirect
49. C₄×D₄×C₆₇	Q₈	semidirect
189. #33201	C₆₇	direct (N × H)
198. #2862	C₂×C₆₇	semidirect
200. #1988	C₂×C₆₇	semidirect
201. #1987	C₂×C₆₇	semidirect
202. #1949	C₂×C₆₇	semidirect
203. #1842	C₂×C₆₇	semidirect
205. C₆₇	#33201	direct (N × H)
245. C₄×D₄	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×#476×C₆₇	2
C₂×#474×C₆₇	2
C₆₇×#2862	2
C₄²×D₄×C₆₇	2
C₆₇×#1988	2
C₆₇×#1987	2
C₆₇×#1949	2
C₆₇×#1842	2
C₆₇×#1804	2
#33201	67

Subgroups — C67×#33201 #30712452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33201 #30712452