Group #22588015 — subgroups

Subgroups — #22588015

Order 9088.

176 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17986402	C₂	1	1	yes	maximal, characteristic
2	#17986402	C₂	1	1	yes	maximal, characteristic
3	#17986395	C₂	1	1	yes	maximal, characteristic
4	#17986351	C₂	1	1	yes	maximal, characteristic
5	#17986350	C₂	1	1	yes	maximal, characteristic
6	#17985721	C₂	1	1	yes	maximal, characteristic
7	#494×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#14556128	C₂²	1	1	yes	characteristic
9	#14556128	C₂²	1	1	yes	characteristic
10	#14556061	—	2	2	no	—
11	#14556061	—	2	2	no	—
12	#14556059	—	2	2	no	—
13	#14556053	—	2	2	no	—
14	#14556051	—	2	2	no	—
15	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₇₁	—	2	2	no	—
18	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₄₂	—	2	1	no	—
22	C₂×Dic₁₄₂	—	2	1	no	—
23	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
24	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
25	C₄⋊Dic₇₁	—	2	1	no	—
26	C₄⋊Dic₇₁	—	2	1	no	—
27	C₄⋊Dic₇₁	—	2	1	no	—
28	C₄⋊Dic₇₁	—	2	1	no	—
29	C₂×Q₈×C₇₁	—	2	1	no	—
30	C₂×Q₈×C₇₁	—	2	1	no	—
31	Q₁₆×C₇₁	—	4	4	no	—
32	C₂×C₈×C₇₁	D₄	1	2	yes	—
33	C₄⋊C₄×C₇₁	—	2	1	no	—
34	C₄⋊C₄×C₇₁	—	2	1	no	—
35	C₄⋊C₄×C₇₁	—	2	1	no	—
36	C₄⋊C₄×C₇₁	—	2	1	no	—
37	C₄²×C₇₁	C₂³	1	1	yes	characteristic
38	Q₈×C₇₁	—	4	2	no	—
39	Q₈×C₇₁	—	4	2	no	—
40	C₂×C₄×C₇₁	—	4	1	no	—
41	C₂×C₄×C₇₁	—	4	1	no	—
42	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₇₁	—	4	1	no	—
46	C₂×Dic₇₁	—	4	1	no	—
47	Dic₁₄₂	—	4	2	no	—
48	Dic₁₄₂	—	4	2	no	—
49	C₈×C₇₁	—	2	2	no	—
50	C₇₁⋊C₈	—	2	2	no	—
51	C₇₁⋊C₈	Q₁₆	1	4	yes	complemented
52	C₂²×C₇₁	#127	1	1	yes	characteristic
53	C₄×C₇₁	—	8	1	no	—
54	C₄×C₇₁	—	8	1	no	—
55	C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
56	C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₇₁	#137	1	2	yes	—
58	Dic₇₁	—	8	1	no	—
59	Dic₇₁	—	8	1	no	—
60	C₂×C₇₁	#494	1	1	yes	characteristic
61	C₂×C₇₁	#496	1	1	yes	characteristic
62	C₂×C₇₁	#497	1	1	yes	characteristic
63	#1723	—	71	1	no	maximal, Sylow
64	C₇₁	#1723	1	1	yes	characteristic, complemented, Sylow
65	#500	—	71	1	no	—
66	#500	—	71	1	no	—
67	#498	—	71	1	no	—
68	#494	D₇₁	1	1	yes	characteristic
69	#487	—	71	1	no	—
70	#487	—	71	1	no	—
71	#322	—	71	1	no	—
72	C₂×Q₁₆	—	2	2	no	—
73	C₄⋊Q₈	—	71	1	no	—
74	C₄⋊Q₈	—	71	1	no	—
75	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
77	#107	—	142	2	no	—
78	#107	—	142	2	no	—
79	#106	—	142	2	no	—
80	#103	—	142	2	no	—
81	#103	—	142	2	no	—
82	C₄×C₈	—	71	1	no	—
83	C₄×C₈	—	71	1	no	—
84	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	142	1	no	—
88	C₂×Q₈	—	142	1	no	—
89	Q₁₆	—	4	4	no	—
90	C₂×C₈	—	71	2	no	—
91	C₂×C₈	—	71	2	no	—
92	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	142	1	no	—
98	C₄⋊C₄	—	142	1	no	—
99	C₄⋊C₄	—	142	1	no	—
100	C₄⋊C₄	—	142	1	no	—
101	C₄²	C₂²×D₇₁	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	284	2	no	—
105	Q₈	—	284	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	284	1	no	—
109	C₂×C₄	—	284	1	no	—
110	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
111	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	71	4	no	—
115	C₈	—	142	2	no	—
116	C₂²	#14556124	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	568	1	no	—
120	C₄	—	568	1	no	—
121	C₄	Q₁₆×D₇₁	1	2	yes	—
122	C₄	#14556154	1	2	yes	—
123	C₄	#14556160	1	2	yes	—
124	C₂	#17986377	1	1	yes	characteristic
125	C₂	#17986386	1	1	yes	characteristic
126	C₂	#17986392	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986402

C₂

yes

maximal, characteristic

#17986402

C₂

yes

maximal, characteristic

#17986395

C₂

yes

maximal, characteristic

#17986351

C₂

yes

maximal, characteristic

#17986350

C₂

yes

maximal, characteristic

#17985721

C₂

yes

maximal, characteristic

#494×C₇₁

C₂

yes

maximal, characteristic

#14556128

C₂²

yes

characteristic

#14556128

C₂²

yes

—

—

—

—

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

Q₁₆

yes

complemented

C₂²×C₇₁

#127

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×Q₁₆

yes

—

C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₇₁

#137

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

D₇₁

yes

—

—

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

#107

—

142

—

#107

—

142

—

#106

—

142

—

#103

—

142

—

#103

—

142

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

100

C₄⋊C₄

—

142

—

101

C₄²

C₂²×D₇₁

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

284

—

105

Q₈

—

284

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

284

—

109

C₂×C₄

—

284

—

110

C₂×C₄

D₄×D₇₁

yes

characteristic

111

C₂×C₄

D₄×D₇₁

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

142

—

116

C₂²

#14556124

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

568

—

120

C₄

—

568

—

121

C₄

Q₁₆×D₇₁

yes

—

122

C₄

#14556154

yes

—

123

C₄

#14556160

yes

—

124

C₂

#17986377

yes

characteristic

125

C₂

#17986386

yes

characteristic

126

C₂

#17986392

yes

characteristic

	subgroup	G/H
center	C₂²	#14556124
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#494×C₇₁	C₂
socle	C₂²×C₇₁	#127
hypercenter	#494	D₇₁
hypocenter	C₇₁	#1723
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#14556124

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#494×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
51. C₇₁⋊C₈	Q₁₆	semidirect
64. C₇₁	#1723	semidirect

H ≅ G/N

type

51. C₇₁⋊C₈

Q₁₆

semidirect

64. C₇₁

#1723

semidirect

Subgroups — #22588015

Conjugacy classes of subgroups

Complements