Group #22596890 — subgroups

Subgroups — C₂²×Dic₇₁⋊C₈ #22596890

Order 9088.

640 conjugacy classes of subgroups; only the 389 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 389 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	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₇₁⋊C₈	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₇₁	C₄	1	1	yes	characteristic
6	C₂³×Dic₇₁	C₄	1	1	yes	characteristic
7	C₂²×C₇₁⋊C₈	—	2	4	no	—
8	C₂²×C₇₁⋊C₈	C₂²	1	6	yes	—
9	C₂×C₄×Dic₇₁	C₄	1	6	yes	—
10	C₂×C₄×Dic₇₁	C₂²	1	6	yes	—
11	Dic₇₁⋊C₈	C₂²	1	16	yes	direct product factor
12	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₇₁	—	2	4	no	—
14	C₂²×C₈×C₇₁	C₂²	1	6	yes	—
15	C₂²×Dic₇₁	—	2	4	no	—
16	C₂²×Dic₇₁	C₈	1	8	yes	complemented
17	C₂²×Dic₇₁	C₂×C₄	1	6	yes	—
18	C₂²×Dic₇₁	C₂×C₄	1	6	yes	—
19	C₂×C₇₁⋊C₈	—	2	12	no	—
20	C₂×C₇₁⋊C₈	C₂³	1	4	yes	—
21	C₄×Dic₇₁	C₂×C₄	1	12	yes	—
22	C₄×Dic₇₁	C₂³	1	4	yes	—
23	C₂⁴×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₁	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₇₁	D₄	1	4	yes	—
26	C₂²×C₄×C₇₁	Q₈	1	4	yes	—
27	C₂²×C₄×C₇₁	C₂³	1	3	yes	—
28	C₂×C₈×C₇₁	—	2	12	no	—
29	C₂×C₈×C₇₁	C₂³	1	4	yes	—
30	C₂³×C₇₁	C₄⋊C₄	1	4	yes	—
31	C₂³×C₇₁	C₂×C₈	1	4	yes	—
32	C₂³×C₇₁	M₁₆	1	4	yes	—
33	C₂³×C₇₁	C₂²×C₄	1	3	yes	—
34	C₂×C₄×C₇₁	C₄⋊C₄	1	12	yes	—
35	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
36	C₂×C₄×C₇₁	C₂×D₄	1	6	yes	—
37	C₂×C₄×C₇₁	C₂×Q₈	1	6	yes	—
38	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
39	C₂×Dic₇₁	—	2	12	no	—
40	C₂×Dic₇₁	C₂×C₈	1	24	yes	complemented
41	C₂×Dic₇₁	C₂²×C₄	1	4	yes	—
42	C₂×Dic₇₁	C₂²×C₄	1	4	yes	—
43	C₈×C₇₁	—	2	4	no	—
44	C₇₁⋊C₈	—	2	4	no	—
45	C₂²×C₇₁	C₄⋊C₈	1	16	yes	complemented
46	C₂²×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
47	C₂²×C₇₁	C₂²×C₈	1	6	yes	—
48	C₂²×C₇₁	C₂×M₁₆	1	6	yes	—
49	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
50	C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
51	C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
52	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
53	Dic₇₁	—	2	4	no	—
54	Dic₇₁	C₂²×C₈	1	8	yes	complemented
55	C₂×C₇₁	C₂×C₄⋊C₈	1	12	yes	complemented
56	C₂×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₇₁	C₂³×C₈	1	1	yes	characteristic
58	C₂×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
59	C₂²×C₄⋊C₈	—	71	1	no	maximal, Sylow
60	C₇₁	C₂²×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
61	C₂³×C₈	—	71	1	no	—
62	C₂³×C₈	D₇₁	1	1	yes	characteristic
63	C₂²×C₄²	—	71	1	no	—
64	C₂×C₄⋊C₈	—	71	12	no	—
65	C₂³×C₄	—	71	1	no	—
66	C₂³×C₄	—	71	1	no	—
67	C₂³×C₄	C₂×D₇₁	1	1	yes	characteristic
68	C₂²×C₈	—	2	4	no	—
69	C₂²×C₈	—	71	6	no	—
70	C₂²×C₈	—	142	4	no	—
71	C₂²×C₈	C₂×D₇₁	1	6	yes	—
72	C₂×C₄²	—	71	6	no	—
73	C₂×C₄²	—	71	6	no	—
74	C₄⋊C₈	—	71	16	no	—
75	C₂⁴	C₄×D₇₁	1	1	yes	characteristic
76	C₂²×C₄	—	71	8	no	—
77	C₂²×C₄	—	71	6	no	—
78	C₂²×C₄	—	71	6	no	—
79	C₂²×C₄	—	142	4	no	—
80	C₂²×C₄	Dic₁₄₂	1	4	yes	—
81	C₂²×C₄	C₄×D₇₁	1	3	yes	—
82	C₂²×C₄	C₂²⋊D₇₁	1	4	yes	—
83	C₂²×C₄	C₂²×D₇₁	1	3	yes	—
84	C₂×C₈	—	2	12	no	—
85	C₂×C₈	—	71	4	no	—
86	C₂×C₈	—	142	12	no	—
87	C₂×C₈	C₂²×D₇₁	1	4	yes	—
88	C₄²	—	71	12	no	—
89	C₄²	—	71	4	no	—
90	C₂³	Dic₇₁⋊C₄	1	4	yes	—
91	C₂³	C₈×D₇₁	1	4	yes	—
92	C₂³	#11765848	1	4	yes	—
93	C₂³	C₂×C₄×D₇₁	1	3	yes	—
94	C₂×C₄	—	71	4	no	—
95	C₂×C₄	—	71	24	no	—
96	C₂×C₄	—	71	4	no	—
97	C₂×C₄	—	142	12	no	—
98	C₂×C₄	Dic₇₁⋊C₄	1	12	yes	—
99	C₂×C₄	C₂×C₄×D₇₁	1	3	yes	—
100	C₂×C₄	C₂×C₂²⋊D₇₁	1	6	yes	—
101	C₂×C₄	C₂×Dic₁₄₂	1	6	yes	—
102	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₈	—	142	4	no	—
105	C₂²	Dic₇₁⋊C₈	1	16	yes	direct product factor
106	C₂²	C₂×Dic₇₁⋊C₄	1	6	yes	—
107	C₂²	C₂×C₈×D₇₁	1	6	yes	—
108	C₂²	C₂×#11765848	1	6	yes	—
109	C₂²	C₂²×C₄×D₇₁	1	1	yes	characteristic
110	C₄	—	71	8	no	—
111	C₄	—	142	4	no	—
112	C₄	C₂×Dic₇₁⋊C₄	1	6	yes	—
113	C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
114	C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
115	C₂	C₂×Dic₇₁⋊C₈	1	12	yes	direct product factor
116	C₂	C₂²×Dic₇₁⋊C₄	1	1	yes	characteristic
117	C₂	C₂²×C₈×D₇₁	1	1	yes	characteristic
118	C₂	C₂²×#11765848	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₇₁⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₄×Dic₇₁

C₂

yes

maximal, characteristic

C₂×Dic₇₁⋊C₈

C₂

yes

maximal, direct product factor

C₂³×C₈×C₇₁

C₂

yes

maximal, characteristic

C₂³×Dic₇₁

C₄

yes

characteristic

C₂³×Dic₇₁

C₄

yes

characteristic

C₂²×C₇₁⋊C₈

—

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₇₁

C₄

yes

—

C₂×C₄×Dic₇₁

C₂²

yes

—

Dic₇₁⋊C₈

C₂²

yes

direct product factor

C₂³×C₄×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

C₂²

yes

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

C₈

yes

complemented

C₂²×Dic₇₁

C₂×C₄

yes

—

C₂²×Dic₇₁

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂³

yes

—

C₄×Dic₇₁

C₂×C₄

yes

—

C₄×Dic₇₁

C₂³

yes

—

C₂⁴×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×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

C₄⋊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

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂×C₈

yes

complemented

C₂×Dic₇₁

C₂²×C₄

yes

—

C₂×Dic₇₁

C₂²×C₄

yes

—

C₈×C₇₁

—

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

C₂×C₄⋊C₄

yes

—

C₄×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

C₂²×Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

C₂²×C₈

yes

complemented

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

—

C₂³×C₈

D₇₁

yes

characteristic

C₂²×C₄²

—

C₂×C₄⋊C₈

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

142

—

C₂²×C₈

C₂×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂⁴

C₄×D₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

Dic₁₄₂

yes

—

C₂²×C₄

C₄×D₇₁

yes

—

C₂²×C₄

C₂²⋊D₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

C₂²×D₇₁

yes

—

C₄²

—

C₄²

—

C₂³

Dic₇₁⋊C₄

yes

—

C₂³

C₈×D₇₁

yes

—

C₂³

#11765848

yes

—

C₂³

C₂×C₄×D₇₁

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

Dic₇₁⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₇₁

yes

—

100

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

101

C₂×C₄

C₂×Dic₁₄₂

yes

—

102

C₂×C₄

C₂³×D₇₁

yes

characteristic

103

C₈

—

104

C₈

—

142

—

105

C₂²

Dic₇₁⋊C₈

yes

direct product factor

106

C₂²

C₂×Dic₇₁⋊C₄

yes

—

107

C₂²

C₂×C₈×D₇₁

yes

—

108

C₂²

C₂×#11765848

yes

—

109

C₂²

C₂²×C₄×D₇₁

yes

characteristic

110

C₄

—

111

C₄

—

142

—

112

C₄

C₂×Dic₇₁⋊C₄

yes

—

113

C₄

C₂²×C₂²⋊D₇₁

yes

characteristic

114

C₄

C₂²×Dic₁₄₂

yes

characteristic

115

C₂

C₂×Dic₇₁⋊C₈

yes

direct product factor

116

C₂

C₂²×Dic₇₁⋊C₄

yes

characteristic

117

C₂

C₂²×C₈×D₇₁

yes

characteristic

118

C₂

C₂²×#11765848

yes

characteristic

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

subgroup

G/H

center

C₂³×C₄

C₂×D₇₁

commutator

C₂×C₇₁

C₂³×C₈

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

C₂³×C₈×C₇₁

C₂

socle

C₂⁴×C₇₁

C₂×C₄

hypercenter

C₂³×C₈

D₇₁

hypocenter

C₇₁

C₂²×C₄⋊C₈

LCS

G ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
3. C₂×Dic₇₁⋊C₈	C₂	direct (N × H)
11. Dic₇₁⋊C₈	C₂²	direct (N × H)
16. C₂²×Dic₇₁	C₈	semidirect
40. C₂×Dic₇₁	C₂×C₈	semidirect
45. C₂²×C₇₁	C₄⋊C₈	semidirect
54. Dic₇₁	C₂²×C₈	semidirect
55. C₂×C₇₁	C₂×C₄⋊C₈	semidirect
60. C₇₁	C₂²×C₄⋊C₈	semidirect
105. C₂²	Dic₇₁⋊C₈	direct (N × H)
115. C₂	C₂×Dic₇₁⋊C₈	direct (N × H)

H ≅ G/N

type

3. C₂×Dic₇₁⋊C₈

C₂

direct (N × H)

11. Dic₇₁⋊C₈

C₂²

direct (N × H)

16. C₂²×Dic₇₁

C₈

semidirect

40. C₂×Dic₇₁

C₂×C₈

semidirect

45. C₂²×C₇₁

C₄⋊C₈

semidirect

54. Dic₇₁

C₂²×C₈

semidirect

55. C₂×C₇₁

C₂×C₄⋊C₈

semidirect

60. C₇₁

C₂²×C₄⋊C₈

semidirect

105. C₂²

Dic₇₁⋊C₈

direct (N × H)

115. C₂

C₂×Dic₇₁⋊C₈

direct (N × H)

Subgroups — C₂²×Dic₇₁⋊C₈ #22596890

Conjugacy classes of subgroups

Complements

Subgroups — C22×Dic71⋊C8 #22596890

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×Dic₇₁⋊C₈ #22596890