Group #22587825 — subgroups

Subgroups — #22587825

Order 9088.

184 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 55 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	#17986376	C₂	1	1	yes	maximal, characteristic
2	#17986200	C₂	1	2	yes	maximal, complemented
3	#17986129	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985742	C₂	1	2	yes	maximal, complemented
5	#334×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11765854	—	2	2	no	—
7	C₂×D₅₆₈	—	2	1	no	—
8	C₂×D₅₆₈	—	2	1	no	—
9	#14556125	C₂²	1	2	yes	—
10	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
11	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
12	Dic₇₁⋊C₈	—	2	1	no	—
13	#14556050	—	2	2	no	—
14	#14556038	—	2	1	no	—
15	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
18	C₂×D₂₈₄	—	2	2	no	—
19	C₂×D₂₈₄	—	2	2	no	—
20	C₂×D₂₈₄	—	2	2	no	—
21	#11765854	—	4	4	no	—
22	D₅₆₈	—	4	2	no	—
23	D₅₆₈	—	4	2	no	—
24	C₂×C₇₁⋊C₈	—	2	1	no	—
25	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
26	Dic₇₁⋊C₄	—	2	1	no	—
27	C₄×Dic₇₁	—	2	1	no	—
28	C₂×C₈×C₇₁	—	2	2	no	—
29	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	C₂²×D₇₁	—	4	2	no	—
33	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
36	C₂×Dic₇₁	—	2	1	no	—
37	C₂×Dic₇₁	—	2	1	no	—
38	D₂₈₄	—	4	2	no	—
39	D₂₈₄	—	4	2	no	—
40	D₂₈₄	—	4	2	no	—
41	D₂₈₄	—	4	2	no	—
42	D₂₈₄	—	8	2	no	—
43	Dic₁₄₂	—	2	2	no	—
44	C₈×C₇₁	—	2	1	no	—
45	C₈×C₇₁	—	4	2	no	—
46	C₈×C₇₁	SD₁₆	1	2	yes	complemented
47	C₇₁⋊C₈	—	4	1	no	—
48	C₂²×C₇₁	#121	1	1	yes	characteristic
49	C₂×D₇₁	—	8	2	no	—
50	C₂×D₇₁	—	8	2	no	—
51	C₂×D₇₁	—	8	2	no	—
52	C₄×C₇₁	—	2	1	no	—
53	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
56	Dic₇₁	—	4	1	no	—
57	Dic₇₁	—	4	1	no	—
58	C₂×C₇₁	#460	1	1	yes	characteristic
59	C₂×C₇₁	#469	1	1	yes	characteristic
60	C₂×C₇₁	#472	1	1	yes	characteristic
61	D₇₁	—	16	2	no	—
62	#1688	—	71	1	no	maximal, Sylow
63	C₇₁	#1688	1	1	yes	characteristic, complemented, Sylow
64	#493	—	71	1	no	—
65	#460	—	71	2	no	—
66	#446	—	71	1	no	—
67	#334	D₇₁	1	1	yes	characteristic, complemented
68	#331	—	71	2	no	—
69	C₂×SD₁₆	—	142	2	no	—
70	C₂×D₈	—	142	1	no	—
71	C₂×D₈	—	142	1	no	—
72	#127	—	71	2	no	—
73	C₄×Q₈	—	71	1	no	—
74	C₄⋊C₈	—	71	1	no	—
75	C₄⋊C₈	—	142	1	no	—
76	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
77	#102	—	142	2	no	—
78	#97	—	142	1	no	—
79	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
80	C₂×Q₈	—	71	1	no	—
81	C₂×D₄	—	142	2	no	—
82	C₂×D₄	—	142	2	no	—
83	C₂×D₄	—	142	2	no	—
84	SD₁₆	—	284	4	no	—
85	D₈	—	284	2	no	—
86	D₈	—	284	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	142	1	no	—
89	C₂×C₈	D₂₈₄	1	1	yes	characteristic
90	C₂×C₈	D₂₈₄	1	1	yes	characteristic
91	C₄⋊C₄	—	71	1	no	—
92	C₄⋊C₄	—	142	1	no	—
93	C₄²	—	142	1	no	—
94	C₄²	C₂²×D₇₁	1	1	yes	characteristic
95	C₂³	—	284	2	no	—
96	Q₈	—	142	2	no	—
97	D₄	—	284	2	no	—
98	D₄	—	284	2	no	—
99	D₄	—	284	2	no	—
100	D₄	—	284	2	no	—
101	D₄	—	568	2	no	—
102	C₂×C₄	—	142	1	no	—
103	C₂×C₄	—	142	1	no	—
104	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
105	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
106	C₂×C₄	#11765866	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	4	2	no	—
109	C₈	—	284	1	no	—
110	C₈	#11765854	1	2	yes	complemented
111	C₂²	—	568	2	no	—
112	C₂²	—	568	2	no	—
113	C₂²	—	568	2	no	—
114	C₂²	#14556103	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	284	1	no	—
117	C₄	—	284	1	no	—
118	C₄	C₂×#11765854	1	1	yes	characteristic
119	C₄	#14556155	1	2	yes	—
120	C₄	#14556156	1	1	yes	characteristic
121	C₂	—	1136	2	no	—
122	C₂	#17986200	1	1	yes	characteristic
123	C₂	#17986253	1	1	yes	characteristic
124	C₂	#17986270	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986376

C₂

yes

maximal, characteristic

#17986200

C₂

yes

maximal, complemented

#17986129

C₂

yes

maximal, characteristic, complemented

#17985742

C₂

yes

maximal, complemented

#334×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765854

—

C₂×D₅₆₈

—

C₂×D₅₆₈

—

#14556125

C₂²

yes

—

C₄×Dic₁₄₂

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

Dic₇₁⋊C₈

—

#14556050

—

#14556038

—

C₄⋊C₈×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

#11765854

—

D₅₆₈

—

D₅₆₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

D₄

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

SD₁₆

yes

complemented

C₇₁⋊C₈

—

C₂²×C₇₁

#121

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₇₁

AGL(1,ℤ/8)

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1688

—

maximal, Sylow

C₇₁

#1688

yes

characteristic, complemented, Sylow

—

—

—

D₇₁

yes

characteristic, complemented

#331

—

C₂×SD₁₆

—

142

—

C₂×D₈

—

142

—

C₂×D₈

—

142

—

#127

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

C₂×D₇₁

yes

—

#102

—

142

—

#97

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

SD₁₆

—

284

—

D₈

—

284

—

D₈

—

284

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

D₂₈₄

yes

characteristic

C₂×C₈

D₂₈₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄²

—

142

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

—

284

—

Q₈

—

142

—

D₄

—

284

—

D₄

—

284

—

D₄

—

284

—

100

D₄

—

284

—

101

D₄

—

568

—

102

C₂×C₄

—

142

—

103

C₂×C₄

—

142

—

104

C₂×C₄

D₄×D₇₁

yes

characteristic

105

C₂×C₄

C₂×D₂₈₄

yes

characteristic

106

C₂×C₄

#11765866

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

284

—

110

C₈

#11765854

yes

complemented

111

C₂²

—

568

—

112

C₂²

—

568

—

113

C₂²

—

568

—

114

C₂²

#14556103

yes

characteristic

115

C₄

—

116

C₄

—

284

—

117

C₄

—

284

—

118

C₄

C₂×#11765854

yes

characteristic

119

C₄

#14556155

yes

—

120

C₄

#14556156

yes

characteristic

121

C₂

—

1136

—

122

C₂

#17986200

yes

characteristic

123

C₂

#17986253

yes

characteristic

124

C₂

#17986270

yes

characteristic

	subgroup	G/H
center	C₂²	#14556103
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#334×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#334	D₇₁
hypocenter	C₇₁	#1688
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14556103

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#334×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #17986200	C₂	semidirect
3. #17986129	C₂	semidirect
4. #17985742	C₂	semidirect
5. #334×C₇₁	C₂	semidirect
46. C₈×C₇₁	SD₁₆	semidirect
63. C₇₁	#1688	semidirect
67. #334	D₇₁	semidirect
110. C₈	#11765854	semidirect

H ≅ G/N

type

2. #17986200

C₂

semidirect

3. #17986129

C₂

semidirect

4. #17985742

C₂

semidirect

5. #334×C₇₁

C₂

semidirect

46. C₈×C₇₁

SD₁₆

semidirect

63. C₇₁

#1688

semidirect

67. #334

D₇₁

semidirect

110. C₈

#11765854

semidirect

Subgroups — #22587825

Conjugacy classes of subgroups

Complements