Group #22588506 — subgroups

Subgroups — #22588506

Order 9088.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂×#14556061	C₂	1	2	yes	maximal
2	C₂×#14556059	C₂	1	2	yes	maximal
3	C₂×#14556039	C₂	1	1	yes	maximal, characteristic
4	#17985865	C₂	1	1	yes	maximal, characteristic
5	#384×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
9	#14556061	—	2	4	no	—
10	#14556059	—	2	4	no	—
11	#14556039	C₄	1	4	yes	complemented
12	C₇₁⋊#95	—	2	2	no	—
13	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
14	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₇₁	—	2	1	no	—
17	C₂²×Dic₇₁	—	2	1	no	—
18	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
19	C₂×C₇₁⋊C₈	D₄	1	4	yes	—
20	C₂×C₇₁⋊C₈	Q₈	1	4	yes	—
21	C₄⋊Dic₇₁	—	2	2	no	—
22	C₄⋊Dic₇₁	—	2	2	no	—
23	C₂²×C₄×C₇₁	—	2	2	no	—
24	C₂²×C₄×C₇₁	D₄	1	2	yes	—
25	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₇₁	—	2	2	no	—
27	C₄⋊C₄×C₇₁	—	2	2	no	—
28	C₄⋊C₄×C₇₁	—	4	4	no	—
29	C₄²×C₇₁	C₂×C₄	1	2	yes	—
30	C₄²×C₇₁	Q₈	1	2	yes	—
31	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	—	2	2	no	—
33	C₂×C₄×C₇₁	—	2	2	no	—
34	C₂×C₄×C₇₁	—	4	2	no	—
35	C₂×C₄×C₇₁	—	4	4	no	—
36	C₂×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₇₁	—	4	1	no	—
42	C₂×Dic₇₁	—	4	1	no	—
43	C₂×Dic₇₁	—	4	2	no	—
44	C₂×Dic₇₁	—	4	2	no	—
45	C₇₁⋊C₈	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₇₁	#127	1	1	yes	characteristic
48	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₇₁	#137	1	2	yes	—
51	C₄×C₇₁	—	2	4	no	—
52	C₄×C₇₁	—	8	2	no	—
53	C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
55	Dic₇₁	—	8	1	no	—
56	Dic₇₁	—	8	1	no	—
57	C₂×C₇₁	#384	1	1	yes	characteristic
58	C₂×C₇₁	#428	1	2	yes	—
59	C₂×C₇₁	#496	1	1	yes	characteristic
60	C₂×C₇₁	#497	1	1	yes	characteristic
61	C₂×C₇₁	#501	1	2	yes	—
62	#1847	—	71	1	no	maximal, Sylow
63	C₇₁	#1847	1	1	yes	characteristic, complemented, Sylow
64	C₂×#107	—	71	2	no	—
65	C₂×#106	—	71	2	no	—
66	C₂×#97	—	71	1	no	—
67	#384	D₇₁	1	1	yes	characteristic
68	#382	—	71	1	no	—
69	C₂²×C₈	—	71	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	71	1	no	—
72	C₂×C₄⋊C₄	—	71	1	no	—
73	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
74	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
75	#107	—	142	4	no	—
76	#106	—	142	4	no	—
77	#97	—	71	4	no	—
78	#95	—	142	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	142	1	no	—
81	C₂²×C₄	—	142	1	no	—
82	C₂²×C₄	C₂²⋊D₇₁	1	2	yes	—
83	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
84	C₂×C₈	—	71	4	no	—
85	C₂×C₈	—	71	4	no	—
86	C₂×C₈	—	71	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	142	2	no	—
91	C₄⋊C₄	—	142	2	no	—
92	C₄²	Dic₁₄₂	1	2	yes	—
93	C₄²	C₄×D₇₁	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	284	1	no	—
100	C₂×C₄	—	284	1	no	—
101	C₂×C₄	—	284	2	no	—
102	C₂×C₄	—	284	2	no	—
103	C₂×C₄	Dic₇₁⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
105	C₂×C₄	D₄×D₇₁	1	2	yes	—
106	C₂×C₄	Q₈×D₇₁	1	2	yes	—
107	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
108	C₈	—	71	8	no	—
109	C₂²	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
110	C₂²	#14556124	1	1	yes	characteristic
111	C₂²	#14556126	1	1	yes	characteristic
112	C₂²	#14556154	1	2	yes	—
113	C₂²	#14556159	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	568	1	no	—
117	C₄	—	568	1	no	—
118	C₄	C₄⋊C₄×D₇₁	1	2	yes	—
119	C₄	#14556127	1	2	yes	—
120	C₂	#17985868	1	1	yes	characteristic
121	C₂	#17986039	1	2	yes	—
122	C₂	#17986386	1	1	yes	characteristic
123	C₂	#17986391	1	1	yes	characteristic
124	C₂	#17986405	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556061

C₂

yes

maximal

C₂×#14556059

C₂

yes

maximal

C₂×#14556039

C₂

yes

maximal, characteristic

#17985865

C₂

yes

maximal, characteristic

#384×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

—

—

C₄

yes

complemented

C₇₁⋊#95

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×C₇₁⋊C₈

Q₈

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

Q₈

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₇₁⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#127

yes

characteristic

C₂²×C₇₁

C₄⋊Q₈

yes

characteristic

C₂²×C₇₁

AGL(1,ℤ/8)

yes

—

C₂²×C₇₁

#137

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₄

yes

—

C₄×C₇₁

C₄⋊Q₈

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#384

yes

characteristic

C₂×C₇₁

#428

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

#1847

—

maximal, Sylow

C₇₁

#1847

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#106

—

C₂×#97

—

#384

D₇₁

yes

characteristic

#382

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#107

—

142

—

#106

—

142

—

#97

—

#95

—

142

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²⋊D₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

Dic₁₄₂

yes

—

C₄²

C₄×D₇₁

yes

—

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

100

C₂×C₄

—

284

—

101

C₂×C₄

—

284

—

102

C₂×C₄

—

284

—

103

C₂×C₄

Dic₇₁⋊C₄

yes

—

104

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

105

C₂×C₄

D₄×D₇₁

yes

—

106

C₂×C₄

Q₈×D₇₁

yes

—

107

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

108

C₈

—

109

C₂²

C₂×Dic₇₁⋊C₄

yes

characteristic

110

C₂²

#14556124

yes

characteristic

111

C₂²

#14556126

yes

characteristic

112

C₂²

#14556154

yes

—

113

C₂²

#14556159

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

568

—

117

C₄

—

568

—

118

C₄

C₄⋊C₄×D₇₁

yes

—

119

C₄

#14556127

yes

—

120

C₂

#17985868

yes

characteristic

121

C₂

#17986039

yes

—

122

C₂

#17986386

yes

characteristic

123

C₂

#17986391

yes

characteristic

124

C₂

#17986405

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#384×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	#384	D₇₁
hypocenter	C₇₁	#1847
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#384×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. #14556039	C₄	semidirect
45. C₇₁⋊C₈	C₄⋊C₄	semidirect
63. C₇₁	#1847	semidirect

H ≅ G/N

type

11. #14556039

C₄

semidirect

45. C₇₁⋊C₈

C₄⋊C₄

semidirect

63. C₇₁

#1847

semidirect

Subgroups — #22588506

Conjugacy classes of subgroups

Complements