Group #22596891 — subgroups

Subgroups — C₂²×C₄⋊(C₇₁⋊C₈) #22596891

Order 9088.

640 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 469 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇₁⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄²×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₈	—	2	8	no	—
5	C₂²×C₇₁⋊C₈	C₂²	1	12	yes	—
6	C₄⋊(C₇₁⋊C₈)	C₂²	1	16	yes	direct product factor
7	C₂³×C₄×C₇₁	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₇₁	C₄	1	1	yes	characteristic
9	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₇₁	C₄	1	6	yes	—
11	C₂×C₄²×C₇₁	C₂²	1	6	yes	—
12	C₂×C₇₁⋊C₈	—	2	24	no	—
13	C₂×C₇₁⋊C₈	C₂³	1	8	yes	—
14	C₂⁴×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₇₁	—	2	4	no	—
16	C₂²×C₄×C₇₁	C₈	1	8	yes	complemented
17	C₂²×C₄×C₇₁	C₂×C₄	1	6	yes	—
18	C₂²×C₄×C₇₁	C₂×C₄	1	3	yes	—
19	C₂²×C₄×C₇₁	C₂×C₄	1	6	yes	—
20	C₂²×C₄×C₇₁	D₄	1	4	yes	—
21	C₂²×C₄×C₇₁	Q₈	1	4	yes	—
22	C₂²×C₄×C₇₁	C₂³	1	3	yes	—
23	C₄²×C₇₁	C₂×C₄	1	12	yes	—
24	C₄²×C₇₁	C₂³	1	4	yes	—
25	C₂³×C₇₁	C₄⋊C₄	1	4	yes	—
26	C₂³×C₇₁	C₂×C₈	1	4	yes	—
27	C₂³×C₇₁	M₁₆	1	4	yes	—
28	C₂³×C₇₁	C₂²×C₄	1	3	yes	—
29	C₂×C₄×C₇₁	—	2	12	no	—
30	C₂×C₄×C₇₁	C₄⋊C₄	1	12	yes	—
31	C₂×C₄×C₇₁	C₂×C₈	1	24	yes	complemented
32	C₂×C₄×C₇₁	C₂²×C₄	1	4	yes	—
33	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
34	C₂×C₄×C₇₁	C₂²×C₄	1	4	yes	—
35	C₂×C₄×C₇₁	C₂×D₄	1	6	yes	—
36	C₂×C₄×C₇₁	C₂×Q₈	1	6	yes	—
37	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
38	C₇₁⋊C₈	—	2	8	no	—
39	C₂²×C₇₁	C₄⋊C₈	1	16	yes	complemented
40	C₂²×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
41	C₂²×C₇₁	C₂²×C₈	1	6	yes	—
42	C₂²×C₇₁	C₂×M₁₆	1	6	yes	—
43	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
44	C₄×C₇₁	—	2	4	no	—
45	C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
46	C₄×C₇₁	C₂²×C₈	1	8	yes	complemented
47	C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
48	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
49	C₂×C₇₁	C₂×C₄⋊C₈	1	12	yes	complemented
50	C₂×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₇₁	C₂³×C₈	1	1	yes	characteristic
52	C₂×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
53	C₂²×C₄⋊C₈	—	71	1	no	maximal, Sylow
54	C₇₁	C₂²×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
55	C₂³×C₈	—	71	2	no	—
56	C₂²×C₄²	D₇₁	1	1	yes	characteristic
57	C₂×C₄⋊C₈	—	71	12	no	—
58	C₂³×C₄	Dic₇₁	1	1	yes	characteristic
59	C₂³×C₄	Dic₇₁	1	1	yes	characteristic
60	C₂³×C₄	C₂×D₇₁	1	1	yes	characteristic
61	C₂²×C₈	—	71	12	no	—
62	C₂²×C₈	—	142	8	no	—
63	C₂×C₄²	Dic₇₁	1	6	yes	—
64	C₂×C₄²	C₂×D₇₁	1	6	yes	—
65	C₄⋊C₈	—	71	16	no	—
66	C₂⁴	C₂×Dic₇₁	1	1	yes	characteristic
67	C₂²×C₄	—	2	4	no	—
68	C₂²×C₄	C₇₁⋊C₈	1	8	yes	complemented
69	C₂²×C₄	Dic₁₄₂	1	4	yes	—
70	C₂²×C₄	D₂₈₄	1	4	yes	—
71	C₂²×C₄	C₂×Dic₇₁	1	6	yes	—
72	C₂²×C₄	C₂×Dic₇₁	1	3	yes	—
73	C₂²×C₄	C₂×Dic₇₁	1	6	yes	—
74	C₂²×C₄	C₂²×D₇₁	1	3	yes	—
75	C₂×C₈	—	71	8	no	—
76	C₂×C₈	—	142	24	no	—
77	C₄²	C₂×Dic₇₁	1	12	yes	—
78	C₄²	C₂²×D₇₁	1	4	yes	—
79	C₂³	C₄⋊Dic₇₁	1	4	yes	—
80	C₂³	C₂×C₇₁⋊C₈	1	4	yes	—
81	C₂³	#11765849	1	4	yes	—
82	C₂³	C₂²×Dic₇₁	1	3	yes	—
83	C₂×C₄	—	2	12	no	—
84	C₂×C₄	C₄⋊Dic₇₁	1	12	yes	—
85	C₂×C₄	C₂×C₇₁⋊C₈	1	24	yes	complemented
86	C₂×C₄	C₂²×Dic₇₁	1	4	yes	—
87	C₂×C₄	C₂²×Dic₇₁	1	3	yes	—
88	C₂×C₄	C₂²×Dic₇₁	1	4	yes	—
89	C₂×C₄	C₂×D₂₈₄	1	6	yes	—
90	C₂×C₄	C₂×Dic₁₄₂	1	6	yes	—
91	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
92	C₈	—	142	8	no	—
93	C₂²	C₄⋊(C₇₁⋊C₈)	1	16	yes	direct product factor
94	C₂²	C₂×C₄⋊Dic₇₁	1	6	yes	—
95	C₂²	C₂²×C₇₁⋊C₈	1	6	yes	—
96	C₂²	C₂×#11765849	1	6	yes	—
97	C₂²	C₂³×Dic₇₁	1	1	yes	characteristic
98	C₄	—	2	4	no	—
99	C₄	C₂×C₄⋊Dic₇₁	1	6	yes	—
100	C₄	C₂²×C₇₁⋊C₈	1	8	yes	complemented
101	C₄	C₂²×D₂₈₄	1	1	yes	characteristic
102	C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
103	C₂	C₂×C₄⋊(C₇₁⋊C₈)	1	12	yes	direct product factor
104	C₂	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
105	C₂	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
106	C₂	C₂²×#11765849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₇₁⋊C₈

C₂

yes

maximal

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, direct product factor

C₂²×C₄²×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

—

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

direct product factor

C₂³×C₄×C₇₁

C₄

yes

characteristic

C₂³×C₄×C₇₁

C₄

yes

characteristic

C₂³×C₄×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂³

yes

—

C₂⁴×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₈

yes

complemented

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂³

yes

—

C₂³×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

C₂×C₈

yes

—

C₂³×C₇₁

M₁₆

yes

—

C₂³×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂×C₈

yes

complemented

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₄⋊C₈

yes

complemented

C₂²×C₇₁

C₂×C₄⋊C₄

yes

—

C₂²×C₇₁

C₂²×C₈

yes

—

C₂²×C₇₁

C₂×M₁₆

yes

—

C₂²×C₇₁

C₂³×C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₄

yes

—

C₄×C₇₁

C₂²×C₈

yes

complemented

C₄×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₂×C₇₁

C₂×C₄⋊C₈

yes

complemented

C₂×C₇₁

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₇₁

C₂³×C₈

yes

characteristic

C₂×C₇₁

C₂²×M₁₆

yes

characteristic

C₂²×C₄⋊C₈

—

maximal, Sylow

C₇₁

C₂²×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂²×C₄²

D₇₁

yes

characteristic

C₂×C₄⋊C₈

—

C₂³×C₄

Dic₇₁

yes

characteristic

C₂³×C₄

Dic₇₁

yes

characteristic

C₂³×C₄

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

142

—

C₂×C₄²

Dic₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

—

C₄⋊C₈

—

C₂⁴

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₇₁⋊C₈

yes

complemented

C₂²×C₄

Dic₁₄₂

yes

—

C₂²×C₄

D₂₈₄

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

—

C₂×C₈

—

C₂×C₈

—

142

—

C₄²

C₂×Dic₇₁

yes

—

C₄²

C₂²×D₇₁

yes

—

C₂³

C₄⋊Dic₇₁

yes

—

C₂³

C₂×C₇₁⋊C₈

yes

—

C₂³

#11765849

yes

—

C₂³

C₂²×Dic₇₁

yes

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₇₁

yes

—

C₂×C₄

C₂×C₇₁⋊C₈

yes

complemented

C₂×C₄

C₂²×Dic₇₁

yes

—

C₂×C₄

C₂²×Dic₇₁

yes

—

C₂×C₄

C₂²×Dic₇₁

yes

—

C₂×C₄

C₂×D₂₈₄

yes

—

C₂×C₄

C₂×Dic₁₄₂

yes

—

C₂×C₄

C₂³×D₇₁

yes

characteristic

C₈

—

142

—

C₂²

C₄⋊(C₇₁⋊C₈)

yes

direct product factor

C₂²

C₂×C₄⋊Dic₇₁

yes

—

C₂²

C₂²×C₇₁⋊C₈

yes

—

C₂²

C₂×#11765849

yes

—

C₂²

C₂³×Dic₇₁

yes

characteristic

C₄

—

C₄

C₂×C₄⋊Dic₇₁

yes

—

100

C₄

C₂²×C₇₁⋊C₈

yes

complemented

101

C₄

C₂²×D₂₈₄

yes

characteristic

102

C₄

C₂²×Dic₁₄₂

yes

characteristic

103

C₂

C₂×C₄⋊(C₇₁⋊C₈)

yes

direct product factor

104

C₂

C₂²×C₄⋊Dic₇₁

yes

characteristic

105

C₂

C₂³×C₇₁⋊C₈

yes

characteristic

106

C₂

C₂²×#11765849

yes

characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₇₁
commutator	C₂×C₇₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	C₂²×C₄²×C₇₁	C₂
socle	C₂⁴×C₇₁	C₂×C₄
hypercenter	C₂²×C₄²	D₇₁
hypocenter	C₇₁	C₂²×C₄⋊C₈
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

subgroup

G/H

center

C₂³×C₄

C₂×D₇₁

commutator

C₂×C₇₁

C₂³×C₈

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

C₂²×C₄²×C₇₁

C₂

socle

C₂⁴×C₇₁

C₂×C₄

hypercenter

C₂²×C₄²

D₇₁

hypocenter

C₇₁

C₂²×C₄⋊C₈

LCS

G ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. C₂×C₄⋊(C₇₁⋊C₈)	C₂	direct (N × H)
6. C₄⋊(C₇₁⋊C₈)	C₂²	direct (N × H)
16. C₂²×C₄×C₇₁	C₈	semidirect
31. C₂×C₄×C₇₁	C₂×C₈	semidirect
39. C₂²×C₇₁	C₄⋊C₈	semidirect
46. C₄×C₇₁	C₂²×C₈	semidirect
49. C₂×C₇₁	C₂×C₄⋊C₈	semidirect
54. C₇₁	C₂²×C₄⋊C₈	semidirect
68. C₂²×C₄	C₇₁⋊C₈	semidirect
85. C₂×C₄	C₂×C₇₁⋊C₈	semidirect
93. C₂²	C₄⋊(C₇₁⋊C₈)	direct (N × H)
100. C₄	C₂²×C₇₁⋊C₈	semidirect
103. C₂	C₂×C₄⋊(C₇₁⋊C₈)	direct (N × H)

H ≅ G/N

type

2. C₂×C₄⋊(C₇₁⋊C₈)

C₂

direct (N × H)

6. C₄⋊(C₇₁⋊C₈)

C₂²

direct (N × H)

16. C₂²×C₄×C₇₁

C₈

semidirect

31. C₂×C₄×C₇₁

C₂×C₈

semidirect

39. C₂²×C₇₁

C₄⋊C₈

semidirect

46. C₄×C₇₁

C₂²×C₈

semidirect

49. C₂×C₇₁

C₂×C₄⋊C₈

semidirect

54. C₇₁

C₂²×C₄⋊C₈

semidirect

68. C₂²×C₄

C₇₁⋊C₈

semidirect

85. C₂×C₄

C₂×C₇₁⋊C₈

semidirect

93. C₂²

C₄⋊(C₇₁⋊C₈)

direct (N × H)

100. C₄

C₂²×C₇₁⋊C₈

semidirect

103. C₂

C₂×C₄⋊(C₇₁⋊C₈)

direct (N × H)

Subgroups — C₂²×C₄⋊(C₇₁⋊C₈) #22596891

Conjugacy classes of subgroups

Complements

Subgroups — C22×C4⋊(C71⋊C8) #22596891

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×C₄⋊(C₇₁⋊C₈) #22596891