Subgroups — C₂²×C₈×C₂²⋊D₆₇ #31719661

Order 17152.

3694 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×C₂²⋊D₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₂²⋊D₆₇	C₂	1	24	yes	maximal, direct product factor
4	C₂²×Dic₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₂²⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×(C₂×D₆₇)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁴×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₂²⋊D₆₇	C₄	1	1	yes	characteristic
10	C₂³×C₄×D₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₈×D₆₇	—	2	8	no	—
14	C₂²×C₈×D₆₇	C₂²	1	12	yes	complemented
15	C₂×C₄×C₂²⋊D₆₇	C₄	1	12	yes	—
16	C₂×C₄×C₂²⋊D₆₇	C₂²	1	12	yes	—
17	C₂²×Dic₆₇⋊C₄	C₄	1	1	yes	characteristic
18	C₂²×C₂²⋊Dic₆₇	C₄	1	1	yes	characteristic
19	C₂²×(C₂×D₆₇)⋊C₄	C₄	1	1	yes	characteristic
20	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
21	C₈×C₂²⋊D₆₇	C₂²	1	64	yes	direct product factor
22	C₂×Dic₆₇⋊C₈	C₂²	1	12	yes	complemented
23	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	12	yes	complemented
24	C₂×(C₂×D₆₇)⋊C₈	C₂²	1	12	yes	complemented
25	C₂×C₈×Dic₆₇	C₂²	1	12	yes	complemented
26	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂³×C₈×C₆₇	—	2	8	no	—
28	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂³×C₈×C₆₇	C₂²	1	12	yes	complemented
31	C₂⁴×D₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₂²⋊D₆₇	C₈	1	16	yes	direct product factor
33	C₂²×C₂²⋊D₆₇	C₂×C₄	1	12	yes	—
34	C₂³×Dic₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×Dic₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×Dic₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×D₆₇	—	2	4	no	—
38	C₂²×C₄×D₆₇	—	2	4	no	—
39	C₂²×C₄×D₆₇	C₂×C₄	1	6	yes	—
40	C₂²×C₄×D₆₇	C₂³	1	6	yes	—
41	C₂²×C₆₇⋊C₈	—	2	4	no	—
42	C₂²×C₆₇⋊C₈	—	2	4	no	—
43	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
44	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
45	C₂×C₈×D₆₇	—	2	48	no	—
46	C₂×C₈×D₆₇	C₂³	1	16	yes	complemented
47	C₄×C₂²⋊D₆₇	C₂×C₄	1	48	yes	—
48	C₄×C₂²⋊D₆₇	C₂³	1	16	yes	—
49	C₂×Dic₆₇⋊C₄	C₂×C₄	1	12	yes	—
50	C₂×C₂²⋊Dic₆₇	C₂×C₄	1	12	yes	—
51	C₂×(C₂×D₆₇)⋊C₄	C₂×C₄	1	12	yes	—
52	C₂×C₄×Dic₆₇	C₂×C₄	1	6	yes	—
53	C₂×C₄×Dic₆₇	C₂³	1	6	yes	—
54	Dic₆₇⋊C₈	C₂³	1	16	yes	complemented
55	C₂²⋊(C₆₇⋊C₈)	C₂³	1	16	yes	complemented
56	(C₂×D₆₇)⋊C₈	C₂³	1	16	yes	complemented
57	C₈×Dic₆₇	C₂³	1	16	yes	complemented
58	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	—	2	4	no	—
60	C₂³×C₄×C₆₇	—	2	4	no	—
61	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₆₇	—	2	4	no	—
66	C₂²×C₈×C₆₇	—	2	48	no	—
67	C₂²×C₈×C₆₇	D₄	1	8	yes	complemented
68	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
69	C₂²×C₈×C₆₇	C₂³	1	6	yes	—
70	C₂²×C₈×C₆₇	C₂³	1	16	yes	complemented
71	C₂³×D₆₇	—	2	4	no	—
72	C₂³×D₆₇	—	2	4	no	—
73	C₂³×D₆₇	C₂×C₈	1	8	yes	complemented
74	C₂³×D₆₇	C₂²×C₄	1	6	yes	—
75	C₂×C₂²⋊D₆₇	C₂×C₈	1	96	yes	direct product factor
76	C₂×C₂²⋊D₆₇	C₂²×C₄	1	16	yes	—
77	C₂²×Dic₆₇	—	2	4	no	—
78	C₂²×Dic₆₇	—	2	4	no	—
79	C₂²×Dic₆₇	C₂×C₈	1	8	yes	complemented
80	C₂²×Dic₆₇	C₂²×C₄	1	6	yes	—
81	C₂²×Dic₆₇	C₂²×C₄	1	6	yes	—
82	C₂²×Dic₆₇	C₂²×C₄	1	6	yes	—
83	C₂×C₄×D₆₇	—	2	24	no	—
84	C₂×C₄×D₆₇	—	2	12	no	—
85	C₂×C₄×D₆₇	—	2	12	no	—
86	C₂×C₄×D₆₇	C₂²×C₄	1	12	yes	—
87	C₂×C₄×D₆₇	C₂⁴	1	4	yes	—
88	C₂×C₆₇⋊C₈	—	2	12	no	—
89	C₂×C₆₇⋊C₈	—	2	12	no	—
90	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
91	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
92	C₈×D₆₇	—	2	32	no	—
93	Dic₆₇⋊C₄	C₂²×C₄	1	16	yes	—
94	C₂²⋊Dic₆₇	C₂²×C₄	1	16	yes	—
95	(C₂×D₆₇)⋊C₄	C₂²×C₄	1	16	yes	—
96	C₄×Dic₆₇	C₂²×C₄	1	12	yes	—
97	C₄×Dic₆₇	C₂⁴	1	4	yes	—
98	C₂⁴×C₆₇	—	2	4	no	—
99	C₂⁴×C₆₇	—	2	4	no	—
100	C₂⁴×C₆₇	C₂×C₈	1	8	yes	complemented
101	C₂⁴×C₆₇	C₂²×C₄	1	6	yes	—
102	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	—	2	4	no	—
104	C₂²×C₄×C₆₇	—	2	24	no	—
105	C₂²×C₄×C₆₇	—	2	12	no	—
106	C₂²×C₄×C₆₇	—	2	12	no	—
107	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
108	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
111	C₂²×C₄×C₆₇	#41	1	4	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
114	C₂×C₈×C₆₇	—	2	12	no	—
115	C₂×C₈×C₆₇	—	2	32	no	—
116	C₂×C₈×C₆₇	C₂×D₄	1	24	yes	complemented
117	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
118	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
119	C₂³×C₆₇	—	2	12	no	—
120	C₂³×C₆₇	—	2	12	no	—
121	C₂³×C₆₇	—	2	32	no	—
122	C₂³×C₆₇	C₄×D₄	1	4	yes	—
123	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
124	C₂³×C₆₇	C₂²×C₈	1	24	yes	complemented
125	C₂³×C₆₇	#131	1	4	yes	—
126	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
127	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
128	C₂²×D₆₇	—	2	12	no	—
129	C₂²×D₆₇	—	2	12	no	—
130	C₂²×D₆₇	—	2	32	no	—
131	C₂²×D₆₇	C₂²×C₈	1	24	yes	complemented
132	C₂²×D₆₇	C₂³×C₄	1	4	yes	—
133	C₂×C₄×C₆₇	—	2	12	no	—
134	C₂×C₄×C₆₇	—	2	24	no	—
135	C₂×C₄×C₆₇	—	2	4	no	—
136	C₂×C₄×C₆₇	—	2	4	no	—
137	C₂×C₄×C₆₇	C₄×D₄	1	12	yes	—
138	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
139	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
140	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
141	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
142	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
143	C₂²⋊D₆₇	C₂²×C₈	1	64	yes	direct product factor
144	C₂×Dic₆₇	—	2	12	no	—
145	C₂×Dic₆₇	—	2	12	no	—
146	C₂×Dic₆₇	C₂²×C₈	1	24	yes	complemented
147	C₂×Dic₆₇	C₂³×C₄	1	4	yes	—
148	C₂×Dic₆₇	C₂³×C₄	1	4	yes	—
149	C₂×Dic₆₇	C₂³×C₄	1	4	yes	—
150	C₄×D₆₇	—	2	24	no	—
151	C₄×D₆₇	—	2	4	no	—
152	C₄×D₆₇	—	2	4	no	—
153	C₈×C₆₇	—	2	4	no	—
154	C₈×C₆₇	C₂²×D₄	1	8	yes	complemented
155	C₆₇⋊C₈	—	2	4	no	—
156	C₆₇⋊C₈	—	2	4	no	—
157	C₂²×C₆₇	—	2	4	no	—
158	C₂²×C₆₇	—	2	48	no	—
159	C₂²×C₆₇	—	2	4	no	—
160	C₂²×C₆₇	C₈×D₄	1	16	yes	complemented
161	C₂²×C₆₇	C₂×C₄×D₄	1	6	yes	—
162	C₂²×C₆₇	C₂³×C₈	1	6	yes	—
163	C₂²×C₆₇	C₂³×C₈	1	8	yes	complemented
164	C₂²×C₆₇	C₂×#131	1	6	yes	—
165	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
166	C₂×D₆₇	—	2	4	no	—
167	C₂×D₆₇	—	2	48	no	—
168	C₂×D₆₇	—	2	4	no	—
169	C₂×D₆₇	C₂³×C₈	1	8	yes	complemented
170	C₄×C₆₇	—	2	4	no	—
171	C₄×C₆₇	C₂×C₄×D₄	1	6	yes	—
172	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
173	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
174	Dic₆₇	—	2	4	no	—
175	Dic₆₇	—	2	4	no	—
176	Dic₆₇	C₂³×C₈	1	8	yes	complemented
177	C₂²×C₈×D₄	—	67	1	no	maximal, Sylow
178	C₂×C₆₇	—	2	8	no	—
179	C₂×C₆₇	C₂×C₈×D₄	1	12	yes	complemented
180	C₂×C₆₇	C₂²×C₄×D₄	1	1	yes	characteristic
181	C₂×C₆₇	C₂⁴×C₈	1	1	yes	characteristic
182	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
183	D₆₇	—	2	8	no	—
184	C₂⁴×C₈	—	67	1	no	—
185	C₂⁴×C₈	D₆₇	1	1	yes	characteristic, complemented
186	C₂²×C₄×D₄	—	67	1	no	—
187	C₂×C₈×D₄	—	67	24	no	—
188	C₂²×C₄⋊C₈	—	67	1	no	—
189	C₂²×C₂²⋊C₈	—	67	1	no	—
190	C₂²×C₂²⋊C₈	—	67	1	no	—
191	C₂²×C₄×C₈	—	67	1	no	—
192	C₆₇	C₂²×C₈×D₄	1	1	yes	characteristic, complemented, Sylow
193	C₂³×D₄	—	67	1	no	—
194	C₂⁴×C₄	—	67	1	no	—
195	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
196	C₂³×C₈	—	2	8	no	—
197	C₂³×C₈	—	67	1	no	—
198	C₂³×C₈	—	67	1	no	—
199	C₂³×C₈	—	67	12	no	—
200	C₂³×C₈	—	134	8	no	—
201	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
202	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
203	C₂³×C₈	C₂×D₆₇	1	12	yes	complemented
204	C₂×C₄×D₄	—	67	12	no	—
205	C₂×C₄×D₄	—	67	12	no	—
206	C₂²×C₄⋊C₄	—	67	1	no	—
207	C₂²×C₂²⋊C₄	—	67	1	no	—
208	C₂²×C₂²⋊C₄	—	67	1	no	—
209	C₂²×C₄²	—	67	1	no	—
210	C₈×D₄	—	67	64	no	—
211	C₂×C₄⋊C₈	—	67	12	no	—
212	C₂×C₂²⋊C₈	—	67	12	no	—
213	C₂×C₂²⋊C₈	—	67	12	no	—
214	C₂×C₄×C₈	—	67	12	no	—
215	C₂⁵	—	67	1	no	—
216	C₂⁵	C₄×D₆₇	1	1	yes	characteristic
217	C₂²×D₄	—	67	16	no	—
218	C₂²×D₄	—	67	12	no	—
219	C₂³×C₄	—	2	4	no	—
220	C₂³×C₄	—	2	4	no	—
221	C₂³×C₄	—	67	1	no	—
222	C₂³×C₄	—	67	1	no	—
223	C₂³×C₄	—	67	1	no	—
224	C₂³×C₄	—	67	6	no	—
225	C₂³×C₄	—	67	6	no	—
226	C₂³×C₄	—	134	4	no	—
227	C₂³×C₄	—	134	4	no	—
228	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
229	C₂³×C₄	C₄×D₆₇	1	6	yes	—
230	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
231	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
232	C₂²×C₈	—	2	4	no	—
233	C₂²×C₈	—	2	48	no	—
234	C₂²×C₈	—	67	6	no	—
235	C₂²×C₈	—	67	6	no	—
236	C₂²×C₈	—	67	16	no	—
237	C₂²×C₈	—	134	4	no	—
238	C₂²×C₈	—	134	4	no	—
239	C₂²×C₈	—	134	48	no	—
240	C₂²×C₈	C₂²⋊D₆₇	1	8	yes	direct product factor
241	C₂²×C₈	C₂²×D₆₇	1	6	yes	—
242	C₂²×C₈	C₂²×D₆₇	1	6	yes	—
243	C₂²×C₈	C₂²×D₆₇	1	16	yes	complemented
244	C₄×D₄	—	67	48	no	—
245	C₄×D₄	—	67	16	no	—
246	C₂×C₄⋊C₄	—	67	12	no	—
247	C₂×C₂²⋊C₄	—	67	12	no	—
248	C₂×C₂²⋊C₄	—	67	12	no	—
249	C₂×C₄²	—	67	6	no	—
250	C₂×C₄²	—	67	6	no	—
251	C₄⋊C₈	—	67	16	no	—
252	C₂²⋊C₈	—	67	16	no	—
253	C₂²⋊C₈	—	67	16	no	—
254	C₄×C₈	—	67	16	no	—
255	C₂⁴	—	2	4	no	—
256	C₂⁴	—	2	4	no	—
257	C₂⁴	—	67	8	no	—
258	C₂⁴	—	67	6	no	—
259	C₂⁴	—	134	4	no	—
260	C₂⁴	—	134	4	no	—
261	C₂⁴	C₈×D₆₇	1	8	yes	complemented
262	C₂⁴	C₂×C₄×D₆₇	1	6	yes	—
263	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
264	C₂×D₄	—	67	96	no	—
265	C₂×D₄	—	67	16	no	—
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	2	24	no	—
268	C₂²×C₄	—	2	12	no	—
269	C₂²×C₄	—	2	12	no	—
270	C₂²×C₄	—	67	8	no	—
271	C₂²×C₄	—	67	6	no	—
272	C₂²×C₄	—	67	6	no	—
273	C₂²×C₄	—	67	6	no	—
274	C₂²×C₄	—	67	12	no	—
275	C₂²×C₄	—	67	4	no	—
276	C₂²×C₄	—	134	4	no	—
277	C₂²×C₄	—	134	4	no	—
278	C₂²×C₄	—	134	24	no	—
279	C₂²×C₄	—	134	12	no	—
280	C₂²×C₄	—	134	12	no	—
281	C₂²×C₄	C₂×C₄×D₆₇	1	6	yes	—
282	C₂²×C₄	C₂×C₄×D₆₇	1	12	yes	—
283	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
284	C₂²×C₄	C₂×C₂²⋊D₆₇	1	4	yes	—
285	C₂²×C₄	#11761628	1	4	yes	—
286	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
287	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
288	C₂×C₈	—	2	12	no	—
289	C₂×C₈	—	2	32	no	—
290	C₂×C₈	—	67	4	no	—
291	C₂×C₈	—	67	4	no	—
292	C₂×C₈	—	134	12	no	—
293	C₂×C₈	—	134	12	no	—
294	C₂×C₈	—	134	32	no	—
295	C₂×C₈	C₂×C₂²⋊D₆₇	1	24	yes	direct product factor
296	C₂×C₈	C₂³×D₆₇	1	4	yes	—
297	C₂×C₈	C₂³×D₆₇	1	4	yes	—
298	C₄⋊C₄	—	67	16	no	—
299	C₂²⋊C₄	—	67	16	no	—
300	C₂²⋊C₄	—	67	16	no	—
301	C₄²	—	67	12	no	—
302	C₄²	—	67	4	no	—
303	C₂³	—	2	12	no	—
304	C₂³	—	2	12	no	—
305	C₂³	—	2	32	no	—
306	C₂³	—	67	24	no	—
307	C₂³	—	67	4	no	—
308	C₂³	—	134	12	no	—
309	C₂³	—	134	12	no	—
310	C₂³	—	134	32	no	—
311	C₂³	C₄×C₂²⋊D₆₇	1	4	yes	—
312	C₂³	C₂×C₈×D₆₇	1	4	yes	—
313	C₂³	C₂×C₈×D₆₇	1	24	yes	complemented
314	C₂³	#14524049	1	4	yes	—
315	C₂³	C₂²×C₄×D₆₇	1	4	yes	—
316	C₂³	C₂²×C₄×D₆₇	1	3	yes	—
317	D₄	—	67	64	no	—
318	C₂×C₄	—	2	12	no	—
319	C₂×C₄	—	2	24	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	67	24	no	—
323	C₂×C₄	—	67	4	no	—
324	C₂×C₄	—	67	4	no	—
325	C₂×C₄	—	67	4	no	—
326	C₂×C₄	—	134	12	no	—
327	C₂×C₄	—	134	12	no	—
328	C₂×C₄	—	134	24	no	—
329	C₂×C₄	—	134	4	no	—
330	C₂×C₄	—	134	4	no	—
331	C₂×C₄	C₄×C₂²⋊D₆₇	1	12	yes	—
332	C₂×C₄	C₂²×C₄×D₆₇	1	4	yes	—
333	C₂×C₄	C₂²×C₄×D₆₇	1	3	yes	—
334	C₂×C₄	C₂²×C₂²⋊D₆₇	1	6	yes	—
335	C₂×C₄	C₂×#11761628	1	6	yes	—
336	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
337	C₈	—	2	4	no	—
338	C₈	—	134	4	no	—
339	C₈	—	134	4	no	—
340	C₈	C₂²×C₂²⋊D₆₇	1	8	yes	direct product factor
341	C₂²	—	2	4	no	—
342	C₂²	—	2	4	no	—
343	C₂²	—	2	48	no	—
344	C₂²	—	67	8	no	—
345	C₂²	—	134	4	no	—
346	C₂²	—	134	4	no	—
347	C₂²	—	134	48	no	—
348	C₂²	C₈×C₂²⋊D₆₇	1	16	yes	direct product factor
349	C₂²	C₂×C₄×C₂²⋊D₆₇	1	6	yes	—
350	C₂²	C₂²×C₈×D₆₇	1	6	yes	—
351	C₂²	C₂²×C₈×D₆₇	1	8	yes	complemented
352	C₂²	C₂×#14524049	1	6	yes	—
353	C₂²	C₂³×C₄×D₆₇	1	1	yes	characteristic
354	C₄	—	2	4	no	—
355	C₄	—	67	8	no	—
356	C₄	—	134	4	no	—
357	C₄	—	134	4	no	—
358	C₄	C₂×C₄×C₂²⋊D₆₇	1	6	yes	—
359	C₄	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
360	C₄	C₂²×#11761628	1	1	yes	characteristic
361	C₂	—	2	8	no	—
362	C₂	—	134	8	no	—
363	C₂	C₂×C₈×C₂²⋊D₆₇	1	12	yes	direct product factor
364	C₂	C₂²×C₄×C₂²⋊D₆₇	1	1	yes	characteristic
365	C₂	C₂³×C₈×D₆₇	1	1	yes	characteristic
366	C₂	C₂²×#14524049	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₆₇
commutator	C₂×C₆₇	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂⁴×C₈×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂²×C₄
prime radical	C₂³×C₂²⋊D₆₇	C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂⁴×C₈	D₆₇
hypocenter	C₆₇	C₂²×C₈×D₄
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₈ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
1. C₂³×C₈×D₆₇	C₂	semidirect
3. C₂×C₈×C₂²⋊D₆₇	C₂	direct (N × H)
4. C₂²×Dic₆₇⋊C₈	C₂	semidirect
5. C₂²×C₂²⋊(C₆₇⋊C₈)	C₂	semidirect
6. C₂²×(C₂×D₆₇)⋊C₈	C₂	semidirect
7. C₂²×C₈×Dic₆₇	C₂	semidirect
8. C₂⁴×C₈×C₆₇	C₂	semidirect
14. C₂²×C₈×D₆₇	C₂²	semidirect
21. C₈×C₂²⋊D₆₇	C₂²	direct (N × H)
22. C₂×Dic₆₇⋊C₈	C₂²	semidirect
23. C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	semidirect
24. C₂×(C₂×D₆₇)⋊C₈	C₂²	semidirect
25. C₂×C₈×Dic₆₇	C₂²	semidirect
30. C₂³×C₈×C₆₇	C₂²	semidirect
32. C₂²×C₂²⋊D₆₇	C₈	direct (N × H)
46. C₂×C₈×D₆₇	C₂³	semidirect
54. Dic₆₇⋊C₈	C₂³	semidirect
55. C₂²⋊(C₆₇⋊C₈)	C₂³	semidirect
56. (C₂×D₆₇)⋊C₈	C₂³	semidirect
57. C₈×Dic₆₇	C₂³	semidirect
67. C₂²×C₈×C₆₇	D₄	semidirect
70. C₂²×C₈×C₆₇	C₂³	semidirect
73. C₂³×D₆₇	C₂×C₈	semidirect
75. C₂×C₂²⋊D₆₇	C₂×C₈	direct (N × H)
79. C₂²×Dic₆₇	C₂×C₈	semidirect
100. C₂⁴×C₆₇	C₂×C₈	semidirect
116. C₂×C₈×C₆₇	C₂×D₄	semidirect
124. C₂³×C₆₇	C₂²×C₈	semidirect
131. C₂²×D₆₇	C₂²×C₈	semidirect
143. C₂²⋊D₆₇	C₂²×C₈	direct (N × H)
146. C₂×Dic₆₇	C₂²×C₈	semidirect
154. C₈×C₆₇	C₂²×D₄	semidirect
160. C₂²×C₆₇	C₈×D₄	semidirect
163. C₂²×C₆₇	C₂³×C₈	semidirect
169. C₂×D₆₇	C₂³×C₈	semidirect
176. Dic₆₇	C₂³×C₈	semidirect
179. C₂×C₆₇	C₂×C₈×D₄	semidirect
185. C₂⁴×C₈	D₆₇	semidirect
192. C₆₇	C₂²×C₈×D₄	semidirect
203. C₂³×C₈	C₂×D₆₇	semidirect
240. C₂²×C₈	C₂²⋊D₆₇	direct (N × H)
243. C₂²×C₈	C₂²×D₆₇	semidirect
261. C₂⁴	C₈×D₆₇	semidirect
295. C₂×C₈	C₂×C₂²⋊D₆₇	direct (N × H)
313. C₂³	C₂×C₈×D₆₇	semidirect
340. C₈	C₂²×C₂²⋊D₆₇	direct (N × H)
348. C₂²	C₈×C₂²⋊D₆₇	direct (N × H)
351. C₂²	C₂²×C₈×D₆₇	semidirect
363. C₂	C₂×C₈×C₂²⋊D₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₈×D₆₇	2
C₂²×C₄×C₂²⋊D₆₇	2
C₂×C₈×C₂²⋊D₆₇	2
C₂²×Dic₆₇⋊C₈	2
C₂²×C₂²⋊(C₆₇⋊C₈)	2
C₂²×(C₂×D₆₇)⋊C₈	2
C₂²×C₈×Dic₆₇	2
C₂⁴×C₈×C₆₇	2
C₂²×C₈×D₄	67

Subgroups — C22×C8×C22⋊D67 #31719661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×C₂²⋊D₆₇ #31719661