Subgroups — C₂×D₄⋊(C₇₁⋊C₁₆) #31907151

Order 18176.

470 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊(C₇₁⋊C₁₆)	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₈×D₄×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
6	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	C₂²	1	2	yes	complemented
9	C₄×C₇₁⋊C₁₆	—	2	1	no	—
10	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	complemented
11	C₂³×C₈×C₇₁	—	2	1	no	—
12	C₂×C₄×D₄×C₇₁	C₄	1	1	yes	characteristic
13	C₈×D₄×C₇₁	—	2	1	no	—
14	C₈×D₄×C₇₁	—	2	1	no	—
15	C₈×D₄×C₇₁	C₄	1	2	yes	—
16	C₈×D₄×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₇₁	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₇₁	—	2	1	no	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₇₁⋊C₁₆	—	2	4	no	—
21	C₂×C₇₁⋊C₁₆	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	—	2	2	no	—
23	C₂×C₇₁⋊C₁₆	—	4	2	no	—
24	C₂²×D₄×C₇₁	C₈	1	1	yes	characteristic
25	C₂³×C₄×C₇₁	—	2	1	no	—
26	C₂²×C₈×C₇₁	—	2	1	no	—
27	C₂²×C₈×C₇₁	—	2	2	no	—
28	C₂²×C₈×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	—	4	2	no	—
30	C₂²×C₈×C₇₁	—	4	2	no	—
31	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
33	C₄×D₄×C₇₁	—	2	1	no	—
34	C₄×D₄×C₇₁	—	2	1	no	—
35	C₄×D₄×C₇₁	C₈	1	2	yes	—
36	C₄×D₄×C₇₁	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₇₁	C₈	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
39	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₄⋊C₈×C₇₁	—	2	1	no	—
41	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈×C₇₁	—	2	2	no	—
43	C₂²⋊C₈×C₇₁	—	2	2	no	—
44	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
48	C₇₁⋊C₁₆	—	2	4	no	—
49	C₇₁⋊C₁₆	—	4	2	no	—
50	C₂⁴×C₇₁	—	2	1	no	—
51	C₂×D₄×C₇₁	—	2	1	no	—
52	C₂×D₄×C₇₁	—	2	2	no	—
53	C₂×D₄×C₇₁	C₁₆	1	4	yes	complemented
54	C₂×D₄×C₇₁	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₇₁	—	2	1	no	—
56	C₂²×C₄×C₇₁	—	2	2	no	—
57	C₂²×C₄×C₇₁	—	2	2	no	—
58	C₂²×C₄×C₇₁	—	4	2	no	—
59	C₂²×C₄×C₇₁	—	4	2	no	—
60	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
63	C₂×C₈×C₇₁	—	2	2	no	—
64	C₂×C₈×C₇₁	—	2	2	no	—
65	C₂×C₈×C₇₁	—	4	2	no	—
66	C₂×C₈×C₇₁	—	4	4	no	—
67	C₂×C₈×C₇₁	—	4	2	no	—
68	C₂×C₈×C₇₁	—	4	2	no	—
69	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₁	D₈	1	2	yes	—
72	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
73	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₇₁	—	2	1	no	—
76	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
77	C₂²⋊C₄×C₇₁	—	2	2	no	—
78	C₂²⋊C₄×C₇₁	—	2	2	no	—
79	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
81	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
82	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
83	C₂³×C₇₁	—	2	4	no	—
84	C₂³×C₇₁	—	2	2	no	—
85	C₂³×C₇₁	—	4	2	no	—
86	C₂³×C₇₁	—	4	2	no	—
87	C₂³×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
88	D₄×C₇₁	—	2	4	no	—
89	D₄×C₇₁	—	2	2	no	—
90	D₄×C₇₁	C₂×C₁₆	1	4	yes	complemented
91	C₂×C₄×C₇₁	—	2	2	no	—
92	C₂×C₄×C₇₁	—	2	2	no	—
93	C₂×C₄×C₇₁	—	4	2	no	—
94	C₂×C₄×C₇₁	—	4	4	no	—
95	C₂×C₄×C₇₁	—	4	2	no	—
96	C₂×C₄×C₇₁	—	4	2	no	—
97	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄×C₇₁	#102	1	2	yes	—
101	C₂×C₄×C₇₁	#104	1	2	yes	—
102	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
103	C₂×C₄×C₇₁	M₃₂	1	2	yes	—
104	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
107	C₈×C₇₁	—	2	1	no	—
108	C₈×C₇₁	—	2	1	no	—
109	C₈×C₇₁	—	4	2	no	—
110	C₈×C₇₁	#102	1	2	yes	—
111	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
112	C₈×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂²×C₇₁	—	2	4	no	—
114	C₂²×C₇₁	—	4	8	no	—
115	C₂²×C₇₁	—	4	2	no	—
116	C₂²×C₇₁	—	4	2	no	—
117	C₂²×C₇₁	D₄⋊C₈	1	2	yes	—
118	C₂²×C₇₁	C₂²⋊C₁₆	1	2	yes	—
119	C₂²×C₇₁	#350	1	2	yes	—
120	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	—	2	1	no	—
123	C₄×C₇₁	—	4	2	no	—
124	C₄×C₇₁	D₄⋊C₈	1	2	yes	—
125	C₄×C₇₁	C₂²⋊C₁₆	1	2	yes	—
126	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
127	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
128	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
129	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
130	C₂×D₄⋊C₁₆	—	71	1	no	maximal, Sylow
131	C₂×C₇₁	—	4	4	no	—
132	C₂×C₇₁	D₄⋊C₁₆	1	4	yes	complemented
133	C₂×C₇₁	C₂×D₄⋊C₈	1	1	yes	characteristic
134	C₂×C₇₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
135	C₂×C₇₁	C₂×#350	1	1	yes	characteristic
136	C₂×C₈×D₄	D₇₁	1	1	yes	characteristic
137	C₂×C₄⋊C₁₆	—	71	1	no	—
138	C₂×C₄×C₁₆	—	71	1	no	—
139	D₄⋊C₁₆	—	71	4	no	—
140	C₇₁	C₂×D₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₈	—	2	1	no	—
142	C₂×C₄×D₄	Dic₇₁	1	1	yes	characteristic
143	C₂²×C₁₆	—	142	1	no	—
144	C₂²×C₁₆	—	142	1	no	—
145	C₈×D₄	—	2	1	no	—
146	C₈×D₄	—	2	1	no	—
147	C₈×D₄	Dic₇₁	1	2	yes	—
148	C₈×D₄	C₂×D₇₁	1	2	yes	—
149	C₂×C₄⋊C₈	Dic₇₁	1	1	yes	characteristic
150	C₂×C₂²⋊C₈	—	2	1	no	—
151	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
152	C₄⋊C₁₆	—	71	2	no	—
153	C₄⋊C₁₆	—	142	1	no	—
154	C₄×C₁₆	—	71	2	no	—
155	C₄×C₁₆	—	142	1	no	—
156	C₂²×D₄	C₇₁⋊C₈	1	1	yes	characteristic
157	C₂³×C₄	—	2	1	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	4	2	no	—
162	C₂²×C₈	—	4	2	no	—
163	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	C₇₁⋊C₈	1	2	yes	—
168	C₄×D₄	C₂×Dic₇₁	1	2	yes	—
169	C₂×C₄⋊C₄	C₇₁⋊C₈	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	—	2	1	no	—
171	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
172	C₂×C₁₆	—	284	2	no	—
173	C₂×C₁₆	—	142	2	no	—
174	C₂×C₁₆	—	142	4	no	—
175	C₂×C₁₆	—	142	2	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	C₂×Dic₇₁	1	2	yes	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
183	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
184	C₂⁴	—	2	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	C₇₁⋊C₁₆	1	4	yes	complemented
188	C₂×D₄	C₂×C₇₁⋊C₈	1	2	yes	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
196	C₂²×C₄	#11765849	1	1	yes	characteristic
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
205	C₂×C₈	#11765850	1	2	yes	—
206	C₂×C₈	#11765853	1	2	yes	—
207	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
208	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
214	C₄²	C₂×C₇₁⋊C₈	1	1	yes	characteristic
215	C₄²	#11765849	1	1	yes	characteristic
216	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
217	C₁₆	—	284	2	no	—
218	C₁₆	—	142	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	4	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
224	D₄	—	2	2	no	—
225	D₄	—	2	4	no	—
226	D₄	C₂×C₇₁⋊C₁₆	1	4	yes	complemented
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
234	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
235	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
236	C₂×C₄	#14556049	1	2	yes	—
237	C₂×C₄	#14556055	1	2	yes	—
238	C₂×C₄	C₂×C₇₁⋊C₁₆	1	2	yes	—
239	C₂×C₄	#14556068	1	2	yes	—
240	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
241	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
242	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	4	2	no	—
246	C₈	#14556049	1	2	yes	—
247	C₈	C₂×#11765850	1	1	yes	characteristic
248	C₈	C₂×#11765853	1	1	yes	characteristic
249	C₂²	—	2	4	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	8	no	—
253	C₂²	D₄⋊(C₇₁⋊C₈)	1	2	yes	—
254	C₂²	C₂²⋊(C₇₁⋊C₁₆)	1	2	yes	—
255	C₂²	#17985783	1	2	yes	—
256	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	D₄⋊(C₇₁⋊C₈)	1	2	yes	—
261	C₄	C₂²⋊(C₇₁⋊C₁₆)	1	2	yes	—
262	C₄	C₂×#14556049	1	1	yes	characteristic
263	C₄	C₂×#14556055	1	1	yes	characteristic
264	C₄	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
265	C₄	C₂×#14556068	1	1	yes	characteristic
266	C₂	—	4	4	no	—
267	C₂	D₄⋊(C₇₁⋊C₁₆)	1	4	yes	direct product factor
268	C₂	C₂×D₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
269	C₂	C₂×C₂²⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
270	C₂	C₂×#17985783	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₇₁
commutator	C₄×C₇₁	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₂×C₈×D₄×C₇₁	C₂
socle	C₂³×C₇₁	C₂²⋊C₈
prime radical	C₂²×D₄×C₇₁	C₈
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₂×C₈×D₄	D₇₁
hypocenter	C₇₁	C₂×D₄⋊C₁₆
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
2. C₂×C₄×C₇₁⋊C₁₆	C₂	semidirect
3. D₄⋊(C₇₁⋊C₁₆)	C₂	direct (N × H)
8. C₄⋊(C₇₁⋊C₁₆)	C₂²	semidirect
10. C₄×C₇₁⋊C₁₆	C₂²	semidirect
53. C₂×D₄×C₇₁	C₁₆	semidirect
90. D₄×C₇₁	C₂×C₁₆	semidirect
132. C₂×C₇₁	D₄⋊C₁₆	semidirect
140. C₇₁	C₂×D₄⋊C₁₆	semidirect
187. C₂×D₄	C₇₁⋊C₁₆	semidirect
226. D₄	C₂×C₇₁⋊C₁₆	semidirect
267. C₂	D₄⋊(C₇₁⋊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⋊(C₇₁⋊C₁₆)	2
C₂×C₄×C₇₁⋊C₁₆	2
D₄⋊(C₇₁⋊C₁₆)	2
C₂×C₈×D₄×C₇₁	2
C₂×D₄⋊C₁₆	71

Subgroups — C2×D4⋊(C71⋊C16) #31907151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄⋊(C₇₁⋊C₁₆) #31907151