Group #22588000 — subgroups

Subgroups — #22588000

Order 9088.

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

normal subgroups only

This group has 186 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	#17986397	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986379	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986350	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986346	C₂	1	1	yes	maximal, characteristic
5	#17986346	C₂	1	1	yes	maximal, characteristic
6	#17986346	C₂	1	1	yes	maximal, characteristic
7	#322×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₈₄	—	2	2	no	—
9	#14556128	C₂²	1	1	yes	characteristic
10	#14556128	C₂²	1	1	yes	characteristic
11	#14556128	C₂²	1	1	yes	characteristic
12	#14556125	C₂²	1	1	yes	characteristic
13	#14556060	—	2	2	no	—
14	#14556053	—	2	2	no	—
15	#14556050	—	2	2	no	—
16	#14556050	—	2	2	no	—
17	#14556050	—	2	2	no	—
18	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₄×C₈×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×Dic₁₄₂	—	2	1	no	—
24	C₂×D₂₈₄	—	2	1	no	—
25	C₂×D₂₈₄	—	2	1	no	—
26	C₂×D₂₈₄	—	2	1	no	—
27	Dic₂₈₄	—	4	4	no	—
28	C₄⋊Dic₇₁	—	2	1	no	—
29	C₄⋊Dic₇₁	—	2	1	no	—
30	C₄⋊Dic₇₁	—	2	1	no	—
31	C₄⋊Dic₇₁	—	2	1	no	—
32	C₄⋊Dic₇₁	—	2	1	no	—
33	C₄⋊Dic₇₁	—	2	1	no	—
34	C₂×C₈×C₇₁	D₄	1	2	yes	—
35	C₂×C₈×C₇₁	D₄	1	2	yes	—
36	C₂×C₈×C₇₁	D₄	1	2	yes	—
37	C₄²×C₇₁	C₂³	1	1	yes	characteristic
38	C₂²×D₇₁	—	4	1	no	—
39	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₇₁	—	4	1	no	—
43	C₂×Dic₇₁	—	4	1	no	—
44	C₂×Dic₇₁	—	4	1	no	—
45	D₂₈₄	—	4	2	no	—
46	D₂₈₄	—	4	2	no	—
47	D₂₈₄	—	4	2	no	—
48	Dic₁₄₂	—	4	2	no	—
49	Dic₁₄₂	—	4	2	no	—
50	Dic₁₄₂	—	4	2	no	—
51	C₈×C₇₁	—	2	2	no	—
52	C₈×C₇₁	—	2	2	no	—
53	C₈×C₇₁	—	2	2	no	—
54	C₂²×C₇₁	#127	1	1	yes	characteristic
55	C₂×D₇₁	—	8	1	no	—
56	C₂×D₇₁	—	8	1	no	—
57	C₂×D₇₁	—	8	1	no	—
58	C₄×C₇₁	#135	1	2	yes	—
59	C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
60	C₄×C₇₁	#137	1	2	yes	—
61	Dic₇₁	—	8	1	no	—
62	Dic₇₁	—	8	1	no	—
63	Dic₇₁	—	8	1	no	—
64	C₂×C₇₁	#495	1	1	yes	characteristic
65	C₂×C₇₁	#496	1	1	yes	characteristic
66	C₂×C₇₁	#497	1	1	yes	characteristic
67	D₇₁	—	16	1	no	—
68	#1720	—	71	1	no	maximal, Sylow
69	C₇₁	#1720	1	1	yes	characteristic, complemented, Sylow
70	#498	—	71	1	no	—
71	#494	—	71	1	no	—
72	#487	—	71	1	no	—
73	#486	—	71	1	no	—
74	#486	—	71	1	no	—
75	#486	—	71	1	no	—
76	#322	D₇₁	1	1	yes	characteristic, complemented
77	C₂×Q₁₆	—	142	2	no	—
78	C₄⋊Q₈	—	71	1	no	—
79	C₄⋊Q₈	—	71	1	no	—
80	C₄⋊Q₈	—	71	1	no	—
81	#127	—	71	1	no	—
82	#106	—	142	2	no	—
83	#103	—	142	2	no	—
84	#102	—	142	2	no	—
85	#102	—	142	2	no	—
86	#102	—	142	2	no	—
87	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
88	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
89	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
90	C₂×Q₈	—	142	1	no	—
91	C₂×Q₈	—	142	1	no	—
92	C₂×Q₈	—	142	1	no	—
93	C₂×D₄	—	142	1	no	—
94	C₂×D₄	—	142	1	no	—
95	C₂×D₄	—	142	1	no	—
96	Q₁₆	—	284	4	no	—
97	C₂×C₈	D₂₈₄	1	2	yes	—
98	C₂×C₈	D₂₈₄	1	2	yes	—
99	C₂×C₈	D₂₈₄	1	2	yes	—
100	C₄⋊C₄	—	142	1	no	—
101	C₄⋊C₄	—	142	1	no	—
102	C₄⋊C₄	—	142	1	no	—
103	C₄⋊C₄	—	142	1	no	—
104	C₄⋊C₄	—	142	1	no	—
105	C₄⋊C₄	—	142	1	no	—
106	C₄²	C₂²×D₇₁	1	1	yes	characteristic
107	C₂³	—	284	1	no	—
108	Q₈	—	284	2	no	—
109	Q₈	—	284	2	no	—
110	Q₈	—	284	2	no	—
111	D₄	—	284	2	no	—
112	D₄	—	284	2	no	—
113	D₄	—	284	2	no	—
114	C₂×C₄	—	284	1	no	—
115	C₂×C₄	—	284	1	no	—
116	C₂×C₄	—	284	1	no	—
117	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
118	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
119	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	2	2	no	—
122	C₈	—	2	2	no	—
123	C₂²	—	568	1	no	—
124	C₂²	—	568	1	no	—
125	C₂²	—	568	1	no	—
126	C₂²	#14556125	1	1	yes	characteristic
127	C₄	—	568	1	no	—
128	C₄	—	568	1	no	—
129	C₄	—	568	1	no	—
130	C₄	#14556150	1	2	yes	—
131	C₄	#14556156	1	2	yes	—
132	C₄	#14556161	1	2	yes	—
133	C₂	—	1136	1	no	—
134	C₂	#17986383	1	1	yes	characteristic
135	C₂	#17986388	1	1	yes	characteristic
136	C₂	#17986393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986397

C₂

yes

maximal, characteristic, complemented

#17986379

C₂

yes

maximal, characteristic, complemented

#17986350

C₂

yes

maximal, characteristic, complemented

#17986346

C₂

yes

maximal, characteristic

#17986346

C₂

yes

maximal, characteristic

#17986346

C₂

yes

maximal, characteristic

#322×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₈₄

—

#14556128

C₂²

yes

characteristic

#14556128

C₂²

yes

characteristic

#14556128

C₂²

yes

characteristic

#14556125

C₂²

yes

—

—

—

—

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

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

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

#127

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

#135

yes

—

C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₇₁

#137

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1720

—

maximal, Sylow

C₇₁

#1720

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

#106

—

142

—

#103

—

142

—

#102

—

142

—

#102

—

142

—

#102

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₄×C₈

C₂×D₇₁

yes

characteristic

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

Q₁₆

—

284

—

C₂×C₈

D₂₈₄

yes

—

C₂×C₈

D₂₈₄

yes

—

C₂×C₈

D₂₈₄

yes

—

100

C₄⋊C₄

—

142

—

101

C₄⋊C₄

—

142

—

102

C₄⋊C₄

—

142

—

103

C₄⋊C₄

—

142

—

104

C₄⋊C₄

—

142

—

105

C₄⋊C₄

—

142

—

106

C₄²

C₂²×D₇₁

yes

characteristic

107

C₂³

—

284

—

108

Q₈

—

284

—

109

Q₈

—

284

—

110

Q₈

—

284

—

111

D₄

—

284

—

112

D₄

—

284

—

113

D₄

—

284

—

114

C₂×C₄

—

284

—

115

C₂×C₄

—

284

—

116

C₂×C₄

—

284

—

117

C₂×C₄

C₂×D₂₈₄

yes

characteristic

118

C₂×C₄

C₂×D₂₈₄

yes

characteristic

119

C₂×C₄

C₂×D₂₈₄

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

568

—

124

C₂²

—

568

—

125

C₂²

—

568

—

126

C₂²

#14556125

yes

characteristic

127

C₄

—

568

—

128

C₄

—

568

—

129

C₄

—

568

—

130

C₄

#14556150

yes

—

131

C₄

#14556156

yes

—

132

C₄

#14556161

yes

—

133

C₂

—

1136

—

134

C₂

#17986383

yes

characteristic

135

C₂

#17986388

yes

characteristic

136

C₂

#17986393

yes

characteristic

	subgroup	G/H
center	C₂²	#14556125
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#322×C₇₁	C₂
socle	C₂²×C₇₁	#127
hypercenter	#322	D₇₁
hypocenter	C₇₁	#1720
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#14556125

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#322×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #17986397	C₂	semidirect
2. #17986379	C₂	semidirect
3. #17986350	C₂	semidirect
7. #322×C₇₁	C₂	semidirect
69. C₇₁	#1720	semidirect
76. #322	D₇₁	semidirect

H ≅ G/N

type

1. #17986397

C₂

semidirect

2. #17986379

C₂

semidirect

3. #17986350

C₂

semidirect

7. #322×C₇₁

C₂

semidirect

69. C₇₁

#1720

semidirect

76. #322

D₇₁

semidirect

Subgroups — #22588000

Conjugacy classes of subgroups

Complements