Subgroups — 2₋¹⁺⁴×C₇₁⋊C₈ #32863489

Order 18176.

2190 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597451	C₂	1	10	yes	maximal, complemented
2	#22597421	C₂	1	5	yes	maximal, complemented
3	#41×C₇₁⋊C₈	C₂	1	10	yes	maximal, complemented
4	C₂×Q₈×C₇₁⋊C₈	C₂	1	5	yes	maximal, complemented
5	C₄×2₋¹⁺⁴×C₇₁	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₇₁⋊C₈	C₂²	1	10	yes	—
7	Q₈×C₇₁⋊C₈	C₂²	1	20	yes	—
8	Q₈×C₇₁⋊C₈	C₂²	1	10	yes	—
9	D₄×C₇₁⋊C₈	C₂²	1	30	yes	—
10	D₄×C₇₁⋊C₈	C₂²	1	10	yes	—
11	#17986052	C₂²	1	30	yes	—
12	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	15	yes	—
13	C₂×C₄×C₇₁⋊C₈	C₂²	1	15	yes	—
14	C₂×2₋¹⁺⁴×C₇₁	C₄	1	1	yes	characteristic
15	#520×C₇₁	C₄	1	10	yes	—
16	#519×C₇₁	C₄	1	5	yes	—
17	C₄×#41×C₇₁	C₂²	1	10	yes	—
18	C₂×C₄×Q₈×C₇₁	C₂²	1	5	yes	—
19	C₂²×C₇₁⋊C₈	C₂³	1	5	yes	—
20	C₂²×C₇₁⋊C₈	C₂³	1	15	yes	—
21	C₄⋊(C₇₁⋊C₈)	C₂³	1	10	yes	—
22	C₄⋊(C₇₁⋊C₈)	C₂³	1	30	yes	—
23	C₂²⋊(C₇₁⋊C₈)	C₂³	1	20	yes	—
24	C₄×C₇₁⋊C₈	C₂³	1	30	yes	—
25	C₄×C₇₁⋊C₈	C₂³	1	10	yes	—
26	2₋¹⁺⁴×C₇₁	C₈	1	16	yes	complemented
27	C₂×#41×C₇₁	C₂×C₄	1	10	yes	—
28	C₂²×Q₈×C₇₁	C₂×C₄	1	5	yes	—
29	C₄×Q₈×C₇₁	C₂×C₄	1	20	yes	—
30	C₄×Q₈×C₇₁	C₂×C₄	1	10	yes	—
31	C₄×Q₈×C₇₁	C₂³	1	10	yes	—
32	C₄×D₄×C₇₁	C₂×C₄	1	30	yes	—
33	C₄×D₄×C₇₁	C₂³	1	10	yes	—
34	#117×C₇₁	C₂×C₄	1	30	yes	—
35	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	15	yes	—
36	C₂×C₄²×C₇₁	C₂³	1	15	yes	—
37	C₂×C₇₁⋊C₈	—	2	10	no	—
38	C₂×C₇₁⋊C₈	—	4	15	no	—
39	C₂×C₇₁⋊C₈	C₂⁴	1	5	yes	—
40	C₂×C₇₁⋊C₈	C₂⁴	1	10	yes	—
41	C₂×C₇₁⋊C₈	C₂⁴	1	1	yes	characteristic
42	#41×C₇₁	C₂×C₈	1	80	yes	complemented
43	C₂×Q₈×C₇₁	C₂×C₈	1	40	yes	complemented
44	C₂×Q₈×C₇₁	C₂²×C₄	1	10	yes	—
45	C₂×D₄×C₇₁	C₂²×C₄	1	10	yes	—
46	C₂²×C₄×C₇₁	C₂²×C₄	1	15	yes	—
47	C₂²×C₄×C₇₁	C₂²×C₄	1	15	yes	—
48	C₂²×C₄×C₇₁	C₂⁴	1	5	yes	—
49	C₄⋊C₄×C₇₁	C₂²×C₄	1	30	yes	—
50	C₄⋊C₄×C₇₁	C₂²×C₄	1	10	yes	—
51	C₂²⋊C₄×C₇₁	C₂²×C₄	1	20	yes	—
52	C₄²×C₇₁	C₂²×C₄	1	30	yes	—
53	C₄²×C₇₁	C₂⁴	1	10	yes	—
54	C₂³×C₇₁	C₂³×C₄	1	5	yes	—
55	Q₈×C₇₁	C₂²×C₈	1	40	yes	—
56	D₄×C₇₁	C₂²×C₈	1	40	yes	—
57	C₂×C₄×C₇₁	—	2	5	no	—
58	C₂×C₄×C₇₁	—	2	5	no	—
59	C₂×C₄×C₇₁	—	4	15	no	—
60	C₂×C₄×C₇₁	C₂²×C₈	1	60	yes	—
61	C₂×C₄×C₇₁	C₂³×C₄	1	5	yes	—
62	C₂×C₄×C₇₁	C₂³×C₄	1	10	yes	—
63	C₂×C₄×C₇₁	C₂³×C₄	1	10	yes	—
64	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
65	C₇₁⋊C₈	—	2	5	no	—
66	C₇₁⋊C₈	—	2	10	no	—
67	C₇₁⋊C₈	2₋¹⁺⁴	1	2	yes	direct product factor
68	C₂²×C₇₁	—	2	5	no	—
69	C₂²×C₇₁	—	2	5	no	—
70	C₂²×C₇₁	C₂³×C₈	1	10	yes	—
71	C₂²×C₇₁	C₂⁴×C₄	1	1	yes	characteristic
72	C₄×C₇₁	—	2	5	no	—
73	C₄×C₇₁	—	2	10	no	—
74	C₄×C₇₁	C₂³×C₈	1	20	yes	—
75	C₄×C₇₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
76	C₄×C₇₁	#585	1	1	yes	characteristic
77	C₈×2₋¹⁺⁴	—	71	1	no	maximal, Sylow
78	C₂×C₇₁	—	2	10	no	—
79	C₂×C₇₁	C₄×2₋¹⁺⁴	1	1	yes	characteristic
80	C₂×C₇₁	C₂⁴×C₈	1	1	yes	characteristic
81	C₂×C₇₁	#3573	1	1	yes	characteristic
82	C₄×2₋¹⁺⁴	D₇₁	1	1	yes	characteristic
83	#2988	—	71	10	no	—
84	#2982	—	71	5	no	—
85	C₈×#41	—	71	10	no	—
86	C₂×C₈×Q₈	—	71	5	no	—
87	C₇₁	C₈×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
88	C₂×2₋¹⁺⁴	Dic₇₁	1	1	yes	characteristic
89	#520	Dic₇₁	1	10	yes	—
90	#519	Dic₇₁	1	5	yes	—
91	C₄×#41	C₂×D₇₁	1	10	yes	—
92	C₂×C₄×Q₈	C₂×D₇₁	1	5	yes	—
93	C₈×Q₈	—	71	10	no	—
94	C₈×Q₈	—	71	20	no	—
95	C₈×Q₈	—	71	10	no	—
96	C₈×D₄	—	71	10	no	—
97	C₈×D₄	—	71	30	no	—
98	#431	—	71	30	no	—
99	C₂×C₄⋊C₈	—	71	15	no	—
100	C₂×C₄×C₈	—	71	15	no	—
101	2₋¹⁺⁴	C₇₁⋊C₈	1	16	yes	direct product factor
102	C₂×#41	C₂×Dic₇₁	1	10	yes	—
103	C₂²×Q₈	C₂×Dic₇₁	1	5	yes	—
104	C₂²×C₈	—	71	15	no	—
105	C₂²×C₈	—	71	5	no	—
106	C₄×Q₈	C₂×Dic₇₁	1	20	yes	—
107	C₄×Q₈	C₂×Dic₇₁	1	10	yes	—
108	C₄×Q₈	C₂²×D₇₁	1	10	yes	—
109	C₄×D₄	C₂×Dic₇₁	1	30	yes	—
110	C₄×D₄	C₂²×D₇₁	1	10	yes	—
111	#117	C₂×Dic₇₁	1	30	yes	—
112	C₂×C₄⋊C₄	C₂×Dic₇₁	1	15	yes	—
113	C₂×C₄²	C₂²×D₇₁	1	15	yes	—
114	C₄⋊C₈	—	71	30	no	—
115	C₄⋊C₈	—	71	10	no	—
116	C₂²⋊C₈	—	71	20	no	—
117	C₄×C₈	—	71	10	no	—
118	C₄×C₈	—	71	30	no	—
119	#41	C₂×C₇₁⋊C₈	1	80	yes	complemented
120	C₂×Q₈	C₂×C₇₁⋊C₈	1	40	yes	complemented
121	C₂×Q₈	C₂²×Dic₇₁	1	10	yes	—
122	C₂×D₄	C₂²×Dic₇₁	1	10	yes	—
123	C₂²×C₄	C₂²×Dic₇₁	1	15	yes	—
124	C₂²×C₄	C₂²×Dic₇₁	1	15	yes	—
125	C₂²×C₄	C₂³×D₇₁	1	5	yes	—
126	C₂×C₈	—	284	15	no	—
127	C₂×C₈	—	71	1	no	—
128	C₂×C₈	—	71	5	no	—
129	C₂×C₈	—	71	10	no	—
130	C₂×C₈	—	142	10	no	—
131	C₄⋊C₄	C₂²×Dic₇₁	1	30	yes	—
132	C₄⋊C₄	C₂²×Dic₇₁	1	10	yes	—
133	C₂²⋊C₄	C₂²×Dic₇₁	1	20	yes	—
134	C₄²	C₂²×Dic₇₁	1	30	yes	—
135	C₄²	C₂³×D₇₁	1	10	yes	—
136	C₂³	C₂³×Dic₇₁	1	5	yes	—
137	Q₈	C₂²×C₇₁⋊C₈	1	40	yes	—
138	D₄	C₂²×C₇₁⋊C₈	1	40	yes	—
139	C₂×C₄	—	2	5	no	—
140	C₂×C₄	—	2	5	no	—
141	C₂×C₄	—	4	15	no	—
142	C₂×C₄	C₂²×C₇₁⋊C₈	1	60	yes	—
143	C₂×C₄	C₂³×Dic₇₁	1	5	yes	—
144	C₂×C₄	C₂³×Dic₇₁	1	10	yes	—
145	C₂×C₄	C₂³×Dic₇₁	1	10	yes	—
146	C₂×C₄	C₂⁴×D₇₁	1	1	yes	characteristic
147	C₈	—	71	2	no	—
148	C₈	—	142	5	no	—
149	C₈	—	142	10	no	—
150	C₂²	—	2	5	no	—
151	C₂²	—	2	5	no	—
152	C₂²	C₂³×C₇₁⋊C₈	1	10	yes	—
153	C₂²	C₂⁴×Dic₇₁	1	1	yes	characteristic
154	C₄	—	2	10	no	—
155	C₄	—	2	5	no	—
156	C₄	C₂³×C₇₁⋊C₈	1	20	yes	—
157	C₄	2₋¹⁺⁴×D₇₁	1	1	yes	characteristic
158	C₄	#17986836	1	1	yes	characteristic
159	C₂	—	2	10	no	—
160	C₂	2₋¹⁺⁴×Dic₇₁	1	1	yes	characteristic
161	C₂	C₂⁴×C₇₁⋊C₈	1	1	yes	characteristic
162	C₂	#22603052	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₇₁
commutator	C₂×C₇₁	C₂⁴×C₈
Fitting	C₄×2₋¹⁺⁴×C₇₁	C₂
socle	C₂²×C₇₁	C₂⁴×C₄
prime radical	C₂×2₋¹⁺⁴×C₇₁	C₄
cosocle	C₂×C₄×C₇₁	C₂⁵
hypercenter	C₄×2₋¹⁺⁴	D₇₁
hypocenter	C₇₁	C₈×2₋¹⁺⁴
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #22597451	C₂	semidirect
2. #22597421	C₂	semidirect
3. #41×C₇₁⋊C₈	C₂	semidirect
4. C₂×Q₈×C₇₁⋊C₈	C₂	semidirect
26. 2₋¹⁺⁴×C₇₁	C₈	semidirect
42. #41×C₇₁	C₂×C₈	semidirect
43. C₂×Q₈×C₇₁	C₂×C₈	semidirect
67. C₇₁⋊C₈	2₋¹⁺⁴	direct (N × H)
87. C₇₁	C₈×2₋¹⁺⁴	semidirect
101. 2₋¹⁺⁴	C₇₁⋊C₈	direct (N × H)
119. #41	C₂×C₇₁⋊C₈	semidirect
120. C₂×Q₈	C₂×C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22597451	2
#22597421	2
#41×C₇₁⋊C₈	2
C₂×Q₈×C₇₁⋊C₈	2
C₄×2₋¹⁺⁴×C₇₁	2
C₈×2₋¹⁺⁴	71

Subgroups — 2−1+4×C71⋊C8 #32863489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 2₋¹⁺⁴×C₇₁⋊C₈ #32863489