Subgroups — C₄⋊C₈×C₇₁⋊C₈ #31888575

Order 18176.

550 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₇₁⋊C₈	C₂	1	2	yes	maximal
4	C₇₁⋊#1273	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₄	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₄	1	2	yes	—
8	C₂×C₄×C₇₁⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
12	#17985754	C₄	1	2	yes	—
13	C₈×C₇₁⋊C₈	—	2	4	no	—
14	C₂×C₄⋊C₈×C₇₁	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
16	C₄³×C₇₁	C₂²	1	1	yes	characteristic
17	#336×C₇₁	C₄	1	2	yes	—
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
21	C₄⋊(C₇₁⋊C₈)	C₈	1	8	yes	complemented
22	C₄×C₇₁⋊C₈	—	2	4	no	—
23	C₄×C₇₁⋊C₈	—	2	2	no	—
24	C₄×C₇₁⋊C₈	—	2	1	no	—
25	C₄×C₇₁⋊C₈	—	2	1	no	—
26	C₄×C₇₁⋊C₈	—	2	2	no	—
27	C₄×C₇₁⋊C₈	C₈	1	8	yes	complemented
28	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
29	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
30	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₇₁	C₈	1	8	yes	complemented
39	C₄×C₈×C₇₁	—	2	2	no	—
40	C₄×C₈×C₇₁	—	2	2	no	—
41	C₂×C₇₁⋊C₈	—	2	2	no	—
42	C₂×C₇₁⋊C₈	—	2	4	no	—
43	C₂×C₇₁⋊C₈	—	2	2	no	—
44	C₂×C₇₁⋊C₈	—	2	4	no	—
45	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
46	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
47	C₂×C₇₁⋊C₈	C₂×C₈	1	8	yes	—
48	C₂×C₇₁⋊C₈	C₂×C₈	1	4	yes	—
49	C₂×C₇₁⋊C₈	M₁₆	1	4	yes	—
50	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	—	2	4	no	—
58	C₂×C₈×C₇₁	—	2	4	no	—
59	C₂×C₈×C₇₁	C₂×C₈	1	8	yes	—
60	C₄²×C₇₁	—	2	2	no	—
61	C₄²×C₇₁	—	2	2	no	—
62	C₄²×C₇₁	—	2	2	no	—
63	C₄²×C₇₁	C₂×C₈	1	4	yes	—
64	C₄²×C₇₁	C₂×C₈	1	4	yes	—
65	C₄²×C₇₁	C₂×C₈	1	4	yes	—
66	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₇₁	#41	1	1	yes	characteristic
69	C₄²×C₇₁	#41	1	1	yes	characteristic
70	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₇₁	—	2	1	no	—
72	C₂×C₄×C₇₁	—	2	1	no	—
73	C₂×C₄×C₇₁	—	2	1	no	—
74	C₂×C₄×C₇₁	—	2	1	no	—
75	C₂×C₄×C₇₁	—	2	1	no	—
76	C₂×C₄×C₇₁	—	2	1	no	—
77	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
78	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
79	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
80	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
81	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
82	C₂×C₄×C₇₁	C₄×C₈	1	2	yes	—
83	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₇₁	#131	1	2	yes	—
94	C₂×C₄×C₇₁	#131	1	2	yes	—
95	C₈×C₇₁	—	2	8	no	—
96	C₇₁⋊C₈	—	2	4	no	—
97	C₇₁⋊C₈	—	2	8	no	—
98	C₇₁⋊C₈	C₄⋊C₈	1	8	yes	direct product factor
99	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
101	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
102	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
103	C₂²×C₇₁	C₄×M₁₆	1	1	yes	characteristic
104	C₂²×C₇₁	#405	1	1	yes	characteristic
105	C₂²×C₇₁	#405	1	1	yes	characteristic
106	C₄×C₇₁	—	2	2	no	—
107	C₄×C₇₁	—	2	2	no	—
108	C₄×C₇₁	—	2	2	no	—
109	C₄×C₇₁	C₈²	1	4	yes	complemented
110	C₄×C₇₁	C₂×C₄⋊C₈	1	2	yes	—
111	C₄×C₇₁	#431	1	2	yes	—
112	C₄×C₇₁	C₈×D₄	1	2	yes	—
113	C₄×C₇₁	C₈×D₄	1	2	yes	—
114	C₄×C₇₁	C₈×Q₈	1	2	yes	—
115	C₄×C₇₁	C₈×Q₈	1	2	yes	—
116	C₈×C₄⋊C₈	—	71	1	no	maximal, Sylow
117	C₂×C₇₁	C₂×C₈²	1	1	yes	characteristic
118	C₂×C₇₁	C₈×M₁₆	1	1	yes	characteristic
119	C₂×C₇₁	C₈×M₁₆	1	1	yes	characteristic
120	C₂×C₇₁	#1450	1	1	yes	characteristic
121	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
122	C₂×C₇₁	C₈×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₇₁	C₈×C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₄⋊C₈	—	71	1	no	—
125	C₄×C₄⋊C₈	D₇₁	1	1	yes	characteristic
126	C₄²×C₈	—	71	1	no	—
127	C₂×C₈²	—	71	2	no	—
128	#1273	—	71	2	no	—
129	C₇₁	C₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
130	C₂×C₄⋊C₈	—	71	2	no	—
131	C₂×C₄⋊C₈	Dic₇₁	1	2	yes	—
132	C₂×C₄×C₈	—	71	2	no	—
133	C₂×C₄×C₈	—	71	1	no	—
134	C₂×C₄×C₈	—	71	2	no	—
135	C₂×C₄×C₈	—	71	2	no	—
136	C₂×C₄×C₈	—	71	1	no	—
137	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
138	C₄³	C₂×D₇₁	1	1	yes	characteristic
139	#336	—	71	2	no	—
140	#336	Dic₇₁	1	2	yes	—
141	C₈²	—	142	4	no	—
142	C₂²×C₈	—	71	2	no	—
143	C₂²×C₈	—	71	4	no	—
144	C₂²×C₈	—	71	2	no	—
145	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
146	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
147	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
148	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
149	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
150	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
152	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
153	C₄⋊C₈	—	71	8	no	—
154	C₄⋊C₈	C₇₁⋊C₈	1	8	yes	direct product factor
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	71	8	no	—
158	C₄×C₈	—	71	2	no	—
159	C₄×C₈	—	71	2	no	—
160	C₄×C₈	—	142	4	no	—
161	C₄×C₈	—	142	2	no	—
162	C₄×C₈	—	142	1	no	—
163	C₄×C₈	—	142	1	no	—
164	C₄×C₈	—	142	2	no	—
165	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
166	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
167	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
168	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	71	4	no	—
175	C₂×C₈	—	71	4	no	—
176	C₂×C₈	—	71	4	no	—
177	C₂×C₈	—	71	4	no	—
178	C₂×C₈	—	71	8	no	—
179	C₂×C₈	—	142	2	no	—
180	C₂×C₈	—	142	4	no	—
181	C₂×C₈	—	142	2	no	—
182	C₂×C₈	—	142	4	no	—
183	C₂×C₈	C₂×C₇₁⋊C₈	1	8	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	C₈×D₇₁	1	4	yes	—
188	C₄²	C₈×D₇₁	1	4	yes	—
189	C₄²	C₂×C₇₁⋊C₈	1	4	yes	—
190	C₄²	D₄×D₇₁	1	1	yes	characteristic
191	C₄²	Q₈×D₇₁	1	1	yes	characteristic
192	C₄²	#11765864	1	1	yes	characteristic
193	C₄²	#11765866	1	1	yes	characteristic
194	C₂³	C₂×C₄×Dic₇₁	1	1	yes	characteristic
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
202	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
203	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
204	C₂×C₄	C₈×Dic₇₁	1	2	yes	—
205	C₂×C₄	C₄×C₇₁⋊C₈	1	2	yes	—
206	C₂×C₄	C₄×C₇₁⋊C₈	1	2	yes	—
207	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
208	C₂×C₄	#14556086	1	1	yes	characteristic
209	C₂×C₄	#14556090	1	1	yes	characteristic
210	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
211	C₂×C₄	#14556093	1	1	yes	characteristic
212	C₂×C₄	Q₈×Dic₇₁	1	1	yes	characteristic
213	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
214	C₂×C₄	C₂×C₈×D₇₁	1	2	yes	—
215	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
216	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
217	C₂×C₄	#14556135	1	2	yes	—
218	C₂×C₄	#14556136	1	2	yes	—
219	C₈	—	2	8	no	—
220	C₈	—	71	8	no	—
221	C₈	—	142	4	no	—
222	C₈	—	142	8	no	—
223	C₂²	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
224	C₂²	C₂×C₈×Dic₇₁	1	1	yes	characteristic
225	C₂²	C₂×C₈×Dic₇₁	1	1	yes	characteristic
226	C₂²	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
227	C₂²	M₁₆×Dic₇₁	1	1	yes	characteristic
228	C₂²	#17985955	1	1	yes	characteristic
229	C₂²	#17985957	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	C₈×C₇₁⋊C₈	1	4	yes	complemented
234	C₄	C₄⋊C₈×D₇₁	1	2	yes	—
235	C₄	#17986051	1	2	yes	—
236	C₄	#17986064	1	2	yes	—
237	C₄	D₄×C₇₁⋊C₈	1	2	yes	—
238	C₄	#17986123	1	2	yes	—
239	C₄	Q₈×C₇₁⋊C₈	1	2	yes	—
240	C₂	C₂×C₈×C₇₁⋊C₈	1	1	yes	characteristic
241	C₂	M₁₆×C₇₁⋊C₈	1	1	yes	characteristic
242	C₂	#22586621	1	1	yes	characteristic
243	C₂	#22586624	1	1	yes	characteristic
244	C₂	C₄⋊C₈×Dic₇₁	1	1	yes	characteristic
245	C₂	C₄⋊C₄×C₇₁⋊C₈	1	1	yes	characteristic
246	C₂	#22588205	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₇₁
commutator	C₂×C₇₁	C₂×C₈²
Fitting	C₄×C₄⋊C₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄²
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×C₄⋊C₈	D₇₁
hypocenter	C₇₁	C₈×C₄⋊C₈
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
21. C₄⋊(C₇₁⋊C₈)	C₈	semidirect
27. C₄×C₇₁⋊C₈	C₈	semidirect
38. C₄⋊C₈×C₇₁	C₈	semidirect
98. C₇₁⋊C₈	C₄⋊C₈	direct (N × H)
109. C₄×C₇₁	C₈²	semidirect
129. C₇₁	C₈×C₄⋊C₈	semidirect
154. C₄⋊C₈	C₇₁⋊C₈	direct (N × H)
233. C₄	C₈×C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₇₁⋊C₈)	2
C₄²×C₇₁⋊C₈	2
C₂×C₈×C₇₁⋊C₈	2
C₇₁⋊#1273	2
C₄×C₄⋊C₈×C₇₁	2
C₈×C₄⋊C₈	71

Subgroups — C4⋊C8×C71⋊C8 #31888575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₈×C₇₁⋊C₈ #31888575