Subgroups — C₄×(C₂×D₆₇)⋊C₁₆ #30770859

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₆₇)⋊C₁₆	C₂	1	2	yes	maximal
3	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558286	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₆₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₆₇	C₄	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×D₆₇	—	2	2	no	—
14	C₄×C₈×D₆₇	—	2	2	no	—
15	(C₂×D₆₇)⋊C₁₆	C₄	1	8	yes	direct product factor
16	C₄×C₆₇⋊C₁₆	—	2	2	no	—
17	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×C₁₆×C₆₇	—	2	2	no	—
20	C₂²×C₄×D₆₇	C₈	1	2	yes	—
21	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₆₇	—	2	2	no	—
25	C₂×C₈×D₆₇	—	2	2	no	—
26	C₂×C₈×D₆₇	—	2	1	no	—
27	C₂×C₈×D₆₇	—	2	1	no	—
28	C₂×C₈×D₆₇	—	2	2	no	—
29	C₂×C₈×D₆₇	C₈	1	4	yes	—
30	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₆₇	C₈	1	2	yes	—
32	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×D₆₇	—	2	2	no	—
34	C₄²×D₆₇	—	2	2	no	—
35	C₂×C₆₇⋊C₁₆	—	2	2	no	—
36	C₂×C₆₇⋊C₁₆	—	2	2	no	—
37	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
38	C₄×C₆₇⋊C₈	—	2	2	no	—
39	C₈×Dic₆₇	—	2	2	no	—
40	C₈×Dic₆₇	—	2	1	no	—
41	C₈×Dic₆₇	—	2	1	no	—
42	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₆×C₆₇	—	2	2	no	—
46	C₂×C₁₆×C₆₇	—	2	2	no	—
47	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
48	C₄×C₈×C₆₇	D₄	1	2	yes	—
49	C₄×C₈×C₆₇	D₄	1	2	yes	—
50	C₂³×D₆₇	C₂×C₈	1	1	yes	characteristic
51	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
52	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
53	C₂×C₄×D₆₇	—	2	4	no	—
54	C₂×C₄×D₆₇	—	2	4	no	—
55	C₂×C₄×D₆₇	—	2	1	no	—
56	C₂×C₄×D₆₇	—	2	1	no	—
57	C₂×C₄×D₆₇	—	2	1	no	—
58	C₂×C₄×D₆₇	—	2	1	no	—
59	C₂×C₄×D₆₇	C₁₆	1	8	yes	complemented
60	C₂×C₄×D₆₇	C₄²	1	2	yes	—
61	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
62	C₂×C₆₇⋊C₈	—	2	1	no	—
63	C₂×C₆₇⋊C₈	—	2	1	no	—
64	C₂×C₆₇⋊C₈	—	2	1	no	—
65	C₂×C₆₇⋊C₈	—	2	1	no	—
66	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
67	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
68	C₈×D₆₇	—	2	8	no	—
69	C₈×D₆₇	—	2	4	no	—
70	C₈×D₆₇	—	2	4	no	—
71	C₄×Dic₆₇	—	2	4	no	—
72	C₄×Dic₆₇	—	2	1	no	—
73	C₄×Dic₆₇	—	2	1	no	—
74	C₆₇⋊C₁₆	—	2	4	no	—
75	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
80	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₆₇	#41	1	2	yes	—
84	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₆₇	M₁₆	1	2	yes	—
86	C₁₆×C₆₇	—	2	4	no	—
87	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
88	C₂²×D₆₇	—	2	1	no	—
89	C₂²×D₆₇	—	2	1	no	—
90	C₂²×D₆₇	—	2	2	no	—
91	C₂²×D₆₇	C₄×C₈	1	2	yes	—
92	C₂²×D₆₇	C₂×C₁₆	1	4	yes	—
93	C₂×C₄×C₆₇	C₂²⋊C₈	1	4	yes	—
94	C₂×C₄×C₆₇	C₂×C₁₆	1	4	yes	—
95	C₂×C₄×C₆₇	M₃₂	1	4	yes	—
96	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
102	C₂×Dic₆₇	—	2	2	no	—
103	C₂×Dic₆₇	—	2	1	no	—
104	C₂×Dic₆₇	—	2	1	no	—
105	C₂×Dic₆₇	—	2	2	no	—
106	C₂×Dic₆₇	C₄×C₈	1	2	yes	—
107	C₂×Dic₆₇	C₂×C₁₆	1	4	yes	—
108	C₄×D₆₇	—	2	16	no	—
109	C₄×D₆₇	—	2	2	no	—
110	C₄×D₆₇	—	2	2	no	—
111	C₄×D₆₇	—	2	2	no	—
112	C₄×D₆₇	—	2	2	no	—
113	C₈×C₆₇	C₂²⋊C₈	1	4	yes	—
114	C₈×C₆₇	C₄×D₄	1	2	yes	—
115	C₈×C₆₇	C₄×D₄	1	2	yes	—
116	C₆₇⋊C₈	—	2	2	no	—
117	C₆₇⋊C₈	—	2	2	no	—
118	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
119	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₆₇	#431	1	1	yes	characteristic
121	C₂²×C₆₇	C₂²×C₁₆	1	2	yes	—
122	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
123	C₂×D₆₇	—	2	2	no	—
124	C₂×D₆₇	—	2	4	no	—
125	C₂×D₆₇	—	2	4	no	—
126	C₂×D₆₇	—	2	2	no	—
127	C₂×D₆₇	C₄×C₁₆	1	4	yes	complemented
128	C₄×C₆₇	C₂²⋊C₁₆	1	8	yes	complemented
129	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
130	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
131	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
132	C₄×C₆₇	#431	1	1	yes	characteristic
133	Dic₆₇	—	2	4	no	—
134	Dic₆₇	—	2	2	no	—
135	C₄×C₂²⋊C₁₆	—	67	1	no	maximal, Sylow
136	C₂×C₆₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
137	C₂×C₆₇	C₂×C₄×C₁₆	1	1	yes	characteristic
138	C₂×C₆₇	C₄×M₃₂	1	1	yes	characteristic
139	C₂×C₆₇	C₂×C₂²⋊C₁₆	1	2	yes	—
140	C₂×C₆₇	#2162	1	2	yes	—
141	D₆₇	—	2	4	no	—
142	C₂²×C₄×C₈	—	67	1	no	—
143	C₂×C₂²⋊C₁₆	—	67	2	no	—
144	C₂×C₄×C₁₆	—	67	1	no	—
145	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic, complemented
146	#1374	—	67	2	no	—
147	C₆₇	C₄×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₈	—	67	1	no	—
149	C₂³×C₈	—	67	1	no	—
150	C₂²×C₄²	—	67	1	no	—
151	C₂²×C₁₆	—	67	2	no	—
152	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
153	C₂×C₄×C₈	—	67	1	no	—
154	C₂×C₄×C₈	—	67	1	no	—
155	C₂×C₄×C₈	—	67	1	no	—
156	C₂×C₄×C₈	—	134	2	no	—
157	C₂×C₄×C₈	—	134	2	no	—
158	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
159	C₂²⋊C₁₆	—	67	8	no	—
160	C₄×C₁₆	—	2	2	no	—
161	C₄×C₁₆	—	134	2	no	—
162	C₂³×C₄	—	67	2	no	—
163	C₂³×C₄	—	67	1	no	—
164	C₂²×C₈	—	67	1	no	—
165	C₂²×C₈	—	67	1	no	—
166	C₂²×C₈	—	67	4	no	—
167	C₂²×C₈	—	67	4	no	—
168	C₂²×C₈	—	134	2	no	—
169	C₂²×C₈	—	134	2	no	—
170	C₂²×C₈	—	134	1	no	—
171	C₂²×C₈	—	134	1	no	—
172	C₂²×C₈	—	134	2	no	—
173	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
175	C₂×C₄²	—	67	2	no	—
176	C₂×C₄²	—	67	1	no	—
177	C₂×C₄²	—	134	2	no	—
178	C₂×C₄²	—	134	2	no	—
179	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
180	C₂×C₁₆	—	2	2	no	—
181	C₂×C₁₆	—	2	2	no	—
182	C₂×C₁₆	—	67	4	no	—
183	C₂×C₁₆	—	134	2	no	—
184	C₂×C₁₆	—	134	2	no	—
185	C₂×C₁₆	C₄×D₆₇	1	4	yes	complemented
186	C₄×C₈	—	134	2	no	—
187	C₄×C₈	—	134	1	no	—
188	C₄×C₈	—	134	1	no	—
189	C₄×C₈	—	134	2	no	—
190	C₄×C₈	D₂₆₈	1	2	yes	—
191	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
192	C₂⁴	—	67	1	no	—
193	C₂²×C₄	—	67	2	no	—
194	C₂²×C₄	—	67	1	no	—
195	C₂²×C₄	—	67	8	no	—
196	C₂²×C₄	—	67	2	no	—
197	C₂²×C₄	—	67	2	no	—
198	C₂²×C₄	—	134	4	no	—
199	C₂²×C₄	—	134	4	no	—
200	C₂²×C₄	—	134	1	no	—
201	C₂²×C₄	—	134	1	no	—
202	C₂²×C₄	—	134	1	no	—
203	C₂²×C₄	—	134	1	no	—
204	C₂²×C₄	C₈×D₆₇	1	2	yes	—
205	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
206	C₂×C₈	—	67	2	no	—
207	C₂×C₈	—	67	2	no	—
208	C₂×C₈	—	134	1	no	—
209	C₂×C₈	—	134	1	no	—
210	C₂×C₈	—	134	1	no	—
211	C₂×C₈	—	134	1	no	—
212	C₂×C₈	—	134	8	no	—
213	C₂×C₈	—	134	4	no	—
214	C₂×C₈	—	134	4	no	—
215	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
216	C₂×C₈	C₈×D₆₇	1	2	yes	—
217	C₂×C₈	#11761611	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
219	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
220	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
221	C₂×C₈	#11761628	1	2	yes	—
222	C₄²	—	134	4	no	—
223	C₄²	—	134	1	no	—
224	C₄²	—	134	1	no	—
225	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
226	C₄²	#11761611	1	2	yes	—
227	C₁₆	—	2	4	no	—
228	C₁₆	—	134	4	no	—
229	C₂³	—	67	4	no	—
230	C₂³	—	67	2	no	—
231	C₂³	—	134	2	no	—
232	C₂³	—	134	1	no	—
233	C₂³	—	134	1	no	—
234	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
235	C₂×C₄	—	67	4	no	—
236	C₂×C₄	—	67	2	no	—
237	C₂×C₄	—	134	2	no	—
238	C₂×C₄	—	134	2	no	—
239	C₂×C₄	—	134	1	no	—
240	C₂×C₄	—	134	1	no	—
241	C₂×C₄	—	134	2	no	—
242	C₂×C₄	—	134	2	no	—
243	C₂×C₄	—	134	2	no	—
244	C₂×C₄	—	134	16	no	—
245	C₂×C₄	—	134	2	no	—
246	C₂×C₄	(C₂×D₆₇)⋊C₈	1	4	yes	—
247	C₂×C₄	C₁₆×D₆₇	1	4	yes	—
248	C₂×C₄	#14523982	1	4	yes	—
249	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
250	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
251	C₂×C₄	#14524000	1	1	yes	characteristic
252	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
253	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
254	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
255	C₈	—	134	2	no	—
256	C₈	—	134	2	no	—
257	C₈	(C₂×D₆₇)⋊C₈	1	4	yes	—
258	C₈	C₄×C₂²⋊D₆₇	1	2	yes	—
259	C₈	C₄×D₂₆₈	1	2	yes	—
260	C₂²	—	67	4	no	—
261	C₂²	—	134	2	no	—
262	C₂²	—	134	4	no	—
263	C₂²	—	134	4	no	—
264	C₂²	—	134	2	no	—
265	C₂²	C₄×C₈×D₆₇	1	1	yes	characteristic
266	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
267	C₂²	#16863447	1	1	yes	characteristic
268	C₂²	C₂×C₁₆×D₆₇	1	2	yes	—
269	C₂²	C₂×#14523982	1	2	yes	—
270	C₄	—	134	4	no	—
271	C₄	—	134	2	no	—
272	C₄	(C₂×D₆₇)⋊C₁₆	1	8	yes	direct product factor
273	C₄	C₄×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
274	C₄	C₄×#11761611	1	1	yes	characteristic
275	C₄	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
276	C₄	#16863447	1	1	yes	characteristic
277	C₂	—	134	4	no	—
278	C₂	C₄×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
279	C₂	C₄×C₁₆×D₆₇	1	1	yes	characteristic
280	C₂	C₄×#14523982	1	1	yes	characteristic
281	C₂	C₂×(C₂×D₆₇)⋊C₁₆	1	2	yes	—
282	C₂	#22562006	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₆₇
commutator	C₂×C₆₇	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂³×D₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	C₄×C₂²⋊C₁₆
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₆₇⋊C₁₆	C₂	semidirect
4. #22558286	C₂	semidirect
5. C₂×C₄×C₁₆×C₆₇	C₂	semidirect
15. (C₂×D₆₇)⋊C₁₆	C₄	direct (N × H)
37. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
47. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
59. C₂×C₄×D₆₇	C₁₆	semidirect
127. C₂×D₆₇	C₄×C₁₆	semidirect
128. C₄×C₆₇	C₂²⋊C₁₆	semidirect
145. C₂×C₄×C₁₆	D₆₇	semidirect
147. C₆₇	C₄×C₂²⋊C₁₆	semidirect
185. C₂×C₁₆	C₄×D₆₇	semidirect
272. C₄	(C₂×D₆₇)⋊C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₆₇	2
C₂×(C₂×D₆₇)⋊C₁₆	2
C₂×C₄×C₆₇⋊C₁₆	2
#22558286	2
C₂×C₄×C₁₆×C₆₇	2
C₄×C₂²⋊C₁₆	67

Subgroups — C4×(C2×D67)⋊C16 #30770859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×(C₂×D₆₇)⋊C₁₆ #30770859