Group #22597360 — subgroups

Subgroups — #22597360

Order 9088.

402 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 221 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	Q₈×C₇₁⋊C₈	C₂	1	2	yes	maximal, complemented
2	#17986077	C₂	1	6	yes	maximal, complemented
3	#17986056	C₂	1	3	yes	maximal, complemented
4	C₄×#11765849	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765849	C₂²	1	6	yes	—
7	C₄⋊(C₇₁⋊C₈)	C₂²	1	6	yes	—
8	C₂²⋊(C₇₁⋊C₈)	C₂²	1	6	yes	—
9	#14556039	C₂²	1	3	yes	—
10	#14556039	C₂²	1	1	yes	characteristic
11	C₄×C₇₁⋊C₈	C₂²	1	6	yes	—
12	C₂×#41×C₇₁	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
14	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₇₁	C₄	1	3	yes	—
16	C₄×D₄×C₇₁	C₂²	1	3	yes	—
17	#117×C₇₁	C₄	1	3	yes	—
18	C₂×C₄²×C₇₁	C₂²	1	3	yes	—
19	#11765849	—	2	12	no	—
20	C₂×C₇₁⋊C₈	C₂³	1	6	yes	—
21	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
22	#41×C₇₁	—	2	4	no	—
23	C₂×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₇₁	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₇₁	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₇₁	C₂×C₄	1	3	yes	—
27	C₂²×C₄×C₇₁	C₂³	1	3	yes	—
28	C₄⋊C₄×C₇₁	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₇₁	C₂×C₄	1	3	yes	—
30	C₂²⋊C₄×C₇₁	C₂×C₄	1	6	yes	—
31	C₄²×C₇₁	C₂×C₄	1	3	yes	—
32	C₄²×C₇₁	C₂×C₄	1	3	yes	—
33	C₄²×C₇₁	C₂³	1	1	yes	characteristic
34	C₄²×C₇₁	C₂³	1	3	yes	—
35	C₂³×C₇₁	C₂²×C₄	1	3	yes	—
36	Q₈×C₇₁	M₁₆	1	4	yes	complemented
37	D₄×C₇₁	—	2	6	no	—
38	C₂×C₄×C₇₁	—	2	6	no	—
39	C₂×C₄×C₇₁	—	2	6	no	—
40	C₂×C₄×C₇₁	—	4	3	no	—
41	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₇₁	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
47	C₇₁⋊C₈	—	2	6	no	—
48	C₇₁⋊C₈	#41	1	4	yes	complemented
49	C₂²×C₇₁	—	2	3	no	—
50	C₂²×C₇₁	—	2	3	no	—
51	C₂²×C₇₁	—	2	3	no	—
52	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
53	C₄×C₇₁	—	2	3	no	—
54	C₄×C₇₁	—	2	1	no	—
55	C₄×C₇₁	—	2	1	no	—
56	C₄×C₇₁	—	2	3	no	—
57	C₄×C₇₁	C₂×M₁₆	1	6	yes	—
58	C₄×C₇₁	C₂×#41	1	2	yes	—
59	C₂×C₇₁	—	4	3	no	—
60	C₂×C₇₁	C₄×#41	1	1	yes	characteristic
61	C₂×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
62	C₂×C₇₁	#568	1	1	yes	characteristic
63	#2971	—	71	1	no	maximal, Sylow
64	C₇₁	#2971	1	1	yes	characteristic, complemented, Sylow
65	C₄×#41	D₇₁	1	1	yes	characteristic
66	C₈×Q₈	—	71	2	no	—
67	#436	—	71	6	no	—
68	#432	—	71	3	no	—
69	C₄×M₁₆	—	71	3	no	—
70	C₂×#41	Dic₇₁	1	1	yes	characteristic
71	C₂×M₁₆	—	71	6	no	—
72	C₄×Q₈	Dic₇₁	1	1	yes	characteristic
73	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
74	C₄×D₄	Dic₇₁	1	3	yes	—
75	C₄×D₄	C₂×D₇₁	1	3	yes	—
76	#117	Dic₇₁	1	3	yes	—
77	C₂×C₄²	C₂×D₇₁	1	3	yes	—
78	C₄⋊C₈	—	71	6	no	—
79	C₂²⋊C₈	—	71	6	no	—
80	#97	—	71	1	no	—
81	#97	—	71	3	no	—
82	C₄×C₈	—	71	6	no	—
83	#41	—	2	4	no	—
84	C₂×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
85	C₂×D₄	C₂×Dic₇₁	1	3	yes	—
86	C₂²×C₄	C₂×Dic₇₁	1	3	yes	—
87	C₂²×C₄	C₂×Dic₇₁	1	3	yes	—
88	C₂²×C₄	C₂²×D₇₁	1	3	yes	—
89	M₁₆	—	142	12	no	—
90	C₂×C₈	—	71	2	no	—
91	C₂×C₈	—	71	6	no	—
92	C₄⋊C₄	C₂×Dic₇₁	1	3	yes	—
93	C₄⋊C₄	C₂×Dic₇₁	1	3	yes	—
94	C₂²⋊C₄	C₂×Dic₇₁	1	6	yes	—
95	C₄²	C₂×Dic₇₁	1	3	yes	—
96	C₄²	C₂×Dic₇₁	1	3	yes	—
97	C₄²	C₂²×D₇₁	1	3	yes	—
98	C₄²	C₂²×D₇₁	1	1	yes	characteristic
99	C₂³	C₂²×Dic₇₁	1	3	yes	—
100	Q₈	#11765849	1	4	yes	complemented
101	D₄	—	2	6	no	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	—	2	6	no	—
104	C₂×C₄	—	4	3	no	—
105	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
107	C₂×C₄	C₂²×Dic₇₁	1	3	yes	—
108	C₂×C₄	C₂²×Dic₇₁	1	3	yes	—
109	C₂×C₄	C₂²×Dic₇₁	1	3	yes	—
110	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
111	C₈	—	71	4	no	—
112	C₈	—	142	6	no	—
113	C₂²	—	2	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	—	2	3	no	—
116	C₂²	C₂³×Dic₇₁	1	1	yes	characteristic
117	C₄	—	2	3	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	—	2	3	no	—
121	C₄	C₂×#11765849	1	6	yes	—
122	C₄	#41×D₇₁	1	2	yes	—
123	C₂	—	4	3	no	—
124	C₂	#41×Dic₇₁	1	1	yes	characteristic
125	C₂	C₂²×#11765849	1	1	yes	characteristic
126	C₂	#17986766	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₇₁⋊C₈

C₂

yes

maximal, complemented

#17986077

C₂

yes

maximal, complemented

#17986056

C₂

yes

maximal, complemented

C₄×#11765849

C₂

yes

maximal, complemented

C₄×#41×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556039

C₂²

yes

—

#14556039

C₂²

yes

characteristic

C₄×C₇₁⋊C₈

C₂²

yes

—

C₂×#41×C₇₁

C₄

yes

characteristic

C₄×Q₈×C₇₁

C₄

yes

characteristic

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄×D₄×C₇₁

C₄

yes

—

C₄×D₄×C₇₁

C₂²

yes

—

#117×C₇₁

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

—

#11765849

—

C₂×C₇₁⋊C₈

C₂³

yes

—

C₂×C₇₁⋊C₈

C₂³

yes

—

#41×C₇₁

—

C₂×Q₈×C₇₁

C₂×C₄

yes

characteristic

C₂×D₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂²⋊C₄×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

C₂³

yes

—

C₂³×C₇₁

C₂²×C₄

yes

—

Q₈×C₇₁

M₁₆

yes

complemented

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

#41

yes

complemented

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂³×C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×M₁₆

yes

—

C₄×C₇₁

C₂×#41

yes

—

C₂×C₇₁

—

C₂×C₇₁

C₄×#41

yes

characteristic

C₂×C₇₁

C₂²×M₁₆

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₄×#41

D₇₁

yes

characteristic

C₈×Q₈

—

#436

—

#432

—

C₄×M₁₆

—

C₂×#41

Dic₇₁

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

Dic₇₁

yes

characteristic

C₄×Q₈

C₂×D₇₁

yes

characteristic

C₄×D₄

Dic₇₁

yes

—

C₄×D₄

C₂×D₇₁

yes

—

#117

Dic₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

#41

—

C₂×Q₈

C₂×Dic₇₁

yes

characteristic

C₂×D₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₇₁

yes

—

C₄⋊C₄

C₂×Dic₇₁

yes

—

C₂²⋊C₄

C₂×Dic₇₁

yes

—

C₄²

C₂×Dic₇₁

yes

—

C₄²

C₂×Dic₇₁

yes

—

C₄²

C₂²×D₇₁

yes

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

C₂²×Dic₇₁

yes

—

100

Q₈

#11765849

yes

complemented

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

106

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

107

C₂×C₄

C₂²×Dic₇₁

yes

—

108

C₂×C₄

C₂²×Dic₇₁

yes

—

109

C₂×C₄

C₂²×Dic₇₁

yes

—

110

C₂×C₄

C₂³×D₇₁

yes

characteristic

111

C₈

—

112

C₈

—

142

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₂³×Dic₇₁

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×#11765849

yes

—

122

C₄

#41×D₇₁

yes

—

123

C₂

—

124

C₂

#41×Dic₇₁

yes

characteristic

125

C₂

C₂²×#11765849

yes

characteristic

126

C₂

#17986766

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₄
Fitting	C₄×#41×C₇₁	C₂
hypercenter	C₄×#41	D₇₁
hypocenter	C₇₁	#2971
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₇₁

commutator, socle

C₂²×C₇₁

C₂³×C₄

Fitting

C₄×#41×C₇₁

C₂

hypercenter

C₄×#41

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. Q₈×C₇₁⋊C₈	C₂	semidirect
2. #17986077	C₂	semidirect
3. #17986056	C₂	semidirect
4. C₄×#11765849	C₂	semidirect
36. Q₈×C₇₁	M₁₆	semidirect
48. C₇₁⋊C₈	#41	semidirect
64. C₇₁	#2971	semidirect
100. Q₈	#11765849	semidirect

H ≅ G/N

type

1. Q₈×C₇₁⋊C₈

C₂

semidirect

2. #17986077

C₂

semidirect

3. #17986056

C₂

semidirect

4. C₄×#11765849

C₂

semidirect

36. Q₈×C₇₁

M₁₆

semidirect

48. C₇₁⋊C₈

#41

semidirect

64. C₇₁

#2971

semidirect

100. Q₈

#11765849

semidirect

Subgroups — #22597360

Conjugacy classes of subgroups

Complements