Subgroups — C₂×C₄⋊(C₇₁⋊C₃₂) #31917932

Order 18176.

230 conjugacy classes of subgroups.

The subgroup lattice has 130 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇₁⋊C₃₂	C₂	1	2	yes	maximal
2	C₄⋊(C₇₁⋊C₃₂)	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₁⋊C₃₂	—	2	4	no	—
5	C₂×C₇₁⋊C₃₂	C₂²	1	4	yes	—
6	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
9	C₄×C₁₆×C₇₁	C₄	1	2	yes	—
10	C₄×C₁₆×C₇₁	C₂²	1	2	yes	—
11	C₇₁⋊C₃₂	—	2	4	no	—
12	C₂²×C₈×C₇₁	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₇₁	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₇₁	—	2	2	no	—
16	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
19	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
20	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₇₁	C₈	1	2	yes	—
22	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₇₁	C₁₆	1	1	yes	characteristic
24	C₂²×C₄×C₇₁	C₁₆	1	1	yes	characteristic
25	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
26	C₂×C₈×C₇₁	—	2	2	no	—
27	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
31	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₇₁	C₁₆	1	2	yes	—
33	C₄²×C₇₁	C₂×C₈	1	2	yes	—
34	C₁₆×C₇₁	—	2	2	no	—
35	C₁₆×C₇₁	C₄⋊C₄	1	2	yes	—
36	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₇₁	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₁	—	2	2	no	—
40	C₂×C₄×C₇₁	C₃₂	1	4	yes	complemented
41	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
42	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
43	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂×C₁₆	1	2	yes	—
45	C₂×C₄×C₇₁	M₃₂	1	2	yes	—
46	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₇₁	—	2	2	no	—
48	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
49	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
51	C₂²×C₇₁	C₄⋊C₁₆	1	2	yes	—
52	C₂²×C₇₁	C₂×C₃₂	1	2	yes	—
53	C₂²×C₇₁	M₆₄	1	2	yes	—
54	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
55	C₄×C₇₁	—	2	2	no	—
56	C₄×C₇₁	C₄⋊C₁₆	1	2	yes	—
57	C₄×C₇₁	C₂×C₃₂	1	4	yes	complemented
58	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
59	C₄×C₇₁	C₂×M₃₂	1	1	yes	characteristic
60	C₂×C₄⋊C₃₂	—	71	1	no	maximal, Sylow
61	C₂×C₇₁	C₄⋊C₃₂	1	4	yes	complemented
62	C₂×C₇₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
63	C₂×C₇₁	C₂²×C₃₂	1	1	yes	characteristic
64	C₂×C₇₁	C₂×M₆₄	1	1	yes	characteristic
65	C₂²×C₃₂	—	71	2	no	—
66	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
67	C₄⋊C₃₂	—	71	4	no	—
68	C₇₁	C₂×C₄⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₁₆	Dic₇₁	1	1	yes	characteristic
70	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
71	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
72	C₂×C₃₂	—	71	4	no	—
73	C₂×C₃₂	—	142	4	no	—
74	C₄×C₁₆	Dic₇₁	1	2	yes	—
75	C₄×C₁₆	C₂×D₇₁	1	2	yes	—
76	C₂²×C₈	C₇₁⋊C₈	1	1	yes	characteristic
77	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
78	C₂×C₄²	C₇₁⋊C₈	1	1	yes	characteristic
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
81	C₂×C₁₆	D₂₈₄	1	2	yes	—
82	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
83	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
84	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
85	C₄×C₈	C₇₁⋊C₈	1	2	yes	—
86	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
87	C₃₂	—	142	4	no	—
88	C₂²×C₄	C₇₁⋊C₁₆	1	1	yes	characteristic
89	C₂²×C₄	C₇₁⋊C₁₆	1	1	yes	characteristic
90	C₂²×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
93	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
94	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
95	C₂×C₈	#11765849	1	2	yes	—
96	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
97	C₄²	C₇₁⋊C₁₆	1	2	yes	—
98	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
99	C₁₆	—	2	2	no	—
100	C₁₆	C₄⋊Dic₇₁	1	2	yes	—
101	C₁₆	C₂×D₂₈₄	1	1	yes	characteristic
102	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
103	C₂³	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₇₁⋊C₃₂	1	4	yes	complemented
106	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
107	C₂×C₄	C₂×C₇₁⋊C₁₆	1	2	yes	—
108	C₂×C₄	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
109	C₂×C₄	C₂×C₇₁⋊C₁₆	1	2	yes	—
110	C₂×C₄	#14556068	1	2	yes	—
111	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
114	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
115	C₈	C₂×#11765849	1	1	yes	characteristic
116	C₂²	C₄⋊(C₇₁⋊C₁₆)	1	2	yes	—
117	C₂²	C₂×C₇₁⋊C₃₂	1	2	yes	—
118	C₂²	#17985833	1	2	yes	—
119	C₂²	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	C₄⋊(C₇₁⋊C₁₆)	1	2	yes	—
122	C₄	C₂×C₇₁⋊C₃₂	1	4	yes	complemented
123	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
124	C₄	C₂×#14556068	1	1	yes	characteristic
125	C₂	C₄⋊(C₇₁⋊C₃₂)	1	4	yes	direct product factor
126	C₂	C₂×C₄⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
127	C₂	C₂²×C₇₁⋊C₃₂	1	1	yes	characteristic
128	C₂	C₂×#17985833	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₇₁
commutator	C₂×C₇₁	C₂²×C₃₂
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	C₂×C₄⋊C₃₂
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₇₁⋊C₃₂)	C₂	direct (N × H)
40. C₂×C₄×C₇₁	C₃₂	semidirect
57. C₄×C₇₁	C₂×C₃₂	semidirect
61. C₂×C₇₁	C₄⋊C₃₂	semidirect
68. C₇₁	C₂×C₄⋊C₃₂	semidirect
105. C₂×C₄	C₇₁⋊C₃₂	semidirect
122. C₄	C₂×C₇₁⋊C₃₂	semidirect
125. C₂	C₄⋊(C₇₁⋊C₃₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇₁⋊C₃₂	2
C₄⋊(C₇₁⋊C₃₂)	2
C₂×C₄×C₁₆×C₇₁	2
C₂×C₄⋊C₃₂	71

Subgroups — C2×C4⋊(C71⋊C32) #31917932

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊(C₇₁⋊C₃₂) #31917932