Group #22587890 — subgroups

Subgroups — #22587890

Order 9088.

160 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 49 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	#17986351	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986280	C₂	1	1	yes	maximal, characteristic
3	#17986201	C₂	1	1	yes	maximal, characteristic
4	#17986076	C₂	1	1	yes	maximal, characteristic
5	#17985751	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985746	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11765852	—	2	1	no	—
9	C₂×#11765848	—	2	1	no	—
10	#14556128	C₂²	1	1	yes	characteristic
11	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
12	#14556062	—	2	1	no	—
13	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
14	#14556052	—	2	1	no	—
15	#14556051	—	2	1	no	—
16	#14556051	—	2	1	no	—
17	#14556050	—	2	1	no	—
18	(C₂×D₇₁)⋊C₈	—	2	1	no	—
19	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×D₂₈₄	D₄	1	1	yes	characteristic
25	C₂×C₄×D₇₁	—	2	1	no	—
26	#11765852	—	4	2	no	—
27	#11765848	—	4	2	no	—
28	C₂×C₇₁⋊C₈	—	2	1	no	—
29	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
30	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
31	C₄⋊Dic₇₁	—	2	1	no	—
32	C₄⋊Dic₇₁	—	2	1	no	—
33	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
34	(C₂×D₇₁)⋊C₄	—	2	1	no	—
35	C₂×Q₈×C₇₁	—	2	1	no	—
36	C₂×C₈×C₇₁	—	2	1	no	—
37	C₂×C₈×C₇₁	—	2	1	no	—
38	C₄⋊C₄×C₇₁	—	2	1	no	—
39	C₄⋊C₄×C₇₁	—	2	1	no	—
40	C₄²×C₇₁	C₂³	1	1	yes	characteristic
41	C₂²×D₇₁	—	2	1	no	—
42	Q₈×C₇₁	—	4	1	no	—
43	Q₈×C₇₁	—	4	1	no	—
44	C₂×C₄×C₇₁	—	4	1	no	—
45	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
48	C₂×Dic₇₁	—	2	1	no	—
49	C₂×Dic₇₁	—	4	1	no	—
50	D₂₈₄	—	2	1	no	—
51	D₂₈₄	—	2	1	no	—
52	C₄×D₇₁	—	4	1	no	—
53	C₄×D₇₁	—	4	1	no	—
54	Dic₁₄₂	—	4	1	no	—
55	Dic₁₄₂	—	4	1	no	—
56	C₈×C₇₁	—	4	1	no	—
57	C₈×C₇₁	—	4	1	no	—
58	C₇₁⋊C₈	—	2	1	no	—
59	C₇₁⋊C₈	—	2	1	no	—
60	C₇₁⋊C₈	—	4	1	no	—
61	C₂²×C₇₁	#121	1	1	yes	characteristic
62	C₂×D₇₁	—	4	1	no	—
63	C₂×D₇₁	—	4	1	no	—
64	C₂×D₇₁	—	4	1	no	—
65	C₄×C₇₁	—	2	1	no	—
66	C₄×C₇₁	—	8	1	no	—
67	C₄×C₇₁	#135	1	1	yes	characteristic
68	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₇₁	#137	1	1	yes	characteristic
71	Dic₇₁	—	4	1	no	—
72	Dic₇₁	—	8	1	no	—
73	C₂×C₇₁	#463	1	1	yes	characteristic
74	C₂×C₇₁	#468	1	1	yes	characteristic
75	C₂×C₇₁	#473	1	1	yes	characteristic
76	D₇₁	—	8	1	no	—
77	#1699	—	71	1	no	maximal, Sylow
78	C₇₁	#1699	1	1	yes	characteristic, complemented, Sylow
79	#487	—	71	1	no	—
80	#474	—	71	1	no	—
81	#461	—	71	1	no	—
82	#436	—	71	1	no	—
83	#335	—	71	1	no	—
84	#332	—	71	1	no	—
85	#332	D₇₁	1	1	yes	characteristic, complemented
86	C₂×SD₁₆	—	142	1	no	—
87	C₂×M₁₆	—	142	1	no	—
88	C₄⋊Q₈	—	71	1	no	—
89	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
90	C₄×D₄	—	71	1	no	—
91	#107	—	142	1	no	—
92	C₄⋊C₈	—	71	1	no	—
93	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
94	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
95	#103	—	142	1	no	—
96	#103	—	142	1	no	—
97	#103	—	142	1	no	—
98	#102	—	142	1	no	—
99	C₂²⋊C₈	—	142	1	no	—
100	C₄×C₈	—	71	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	142	1	no	—
103	C₂×D₄	—	71	1	no	—
104	C₂²×C₄	—	142	1	no	—
105	SD₁₆	—	284	2	no	—
106	M₁₆	—	284	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	71	1	no	—
110	C₂×C₈	—	71	1	no	—
111	C₂×C₈	—	142	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	71	1	no	—
115	C₄⋊C₄	—	142	1	no	—
116	C₄⋊C₄	—	142	1	no	—
117	C₂²⋊C₄	—	142	1	no	—
118	C₄²	C₂²×D₇₁	1	1	yes	characteristic
119	C₂³	—	142	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	284	1	no	—
123	Q₈	—	284	1	no	—
124	D₄	—	142	1	no	—
125	D₄	—	142	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	284	1	no	—
128	C₂×C₄	—	284	1	no	—
129	C₂×C₄	—	284	1	no	—
130	C₂×C₄	—	142	1	no	—
131	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
132	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
133	C₂×C₄	#11765865	1	1	yes	characteristic
134	C₈	—	4	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	284	1	no	—
137	C₈	—	142	1	no	—
138	C₈	—	142	1	no	—
139	C₂²	—	284	1	no	—
140	C₂²	—	284	1	no	—
141	C₂²	—	284	1	no	—
142	C₂²	#14556101	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	8	1	no	—
145	C₄	—	284	1	no	—
146	C₄	—	568	1	no	—
147	C₄	#14556151	1	1	yes	characteristic
148	C₄	#14556152	1	1	yes	characteristic
149	C₄	#14556155	1	1	yes	characteristic
150	C₄	#14556157	1	1	yes	characteristic
151	C₂	—	568	1	no	—
152	C₂	#17986217	1	1	yes	characteristic
153	C₂	#17986244	1	1	yes	characteristic
154	C₂	#17986273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986351

C₂

yes

maximal, characteristic, complemented

#17986280

C₂

yes

maximal, characteristic

#17986201

C₂

yes

maximal, characteristic

#17986076

C₂

yes

maximal, characteristic

#17985751

C₂

yes

maximal, characteristic, complemented

#17985746

C₂

yes

maximal, characteristic, complemented

#332×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765852

—

C₂×#11765848

—

#14556128

C₂²

yes

characteristic

C₄×D₂₈₄

C₂²

yes

characteristic

#14556062

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

—

—

—

(C₂×D₇₁)⋊C₈

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×D₂₈₄

D₄

yes

characteristic

C₂×C₄×D₇₁

—

#11765852

—

#11765848

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

(C₂×D₇₁)⋊C₄

—

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

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

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#121

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1699

—

maximal, Sylow

C₇₁

#1699

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₂×SD₁₆

—

142

—

C₂×M₁₆

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

C₄×D₄

—

#107

—

142

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#103

—

142

—

#103

—

142

—

#103

—

142

—

#102

—

142

—

C₂²⋊C₈

—

142

—

100

C₄×C₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

142

—

103

C₂×D₄

—

104

C₂²×C₄

—

142

—

105

SD₁₆

—

284

—

106

M₁₆

—

284

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

142

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

142

—

116

C₄⋊C₄

—

142

—

117

C₂²⋊C₄

—

142

—

118

C₄²

C₂²×D₇₁

yes

characteristic

119

C₂³

—

142

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

284

—

123

Q₈

—

284

—

124

D₄

—

142

—

125

D₄

—

142

—

126

C₂×C₄

—

127

C₂×C₄

—

284

—

128

C₂×C₄

—

284

—

129

C₂×C₄

—

284

—

130

C₂×C₄

—

142

—

131

C₂×C₄

D₄×D₇₁

yes

characteristic

132

C₂×C₄

D₄×D₇₁

yes

characteristic

133

C₂×C₄

#11765865

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

284

—

137

C₈

—

142

—

138

C₈

—

142

—

139

C₂²

—

284

—

140

C₂²

—

284

—

141

C₂²

—

284

—

142

C₂²

#14556101

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

284

—

146

C₄

—

568

—

147

C₄

#14556151

yes

characteristic

148

C₄

#14556152

yes

characteristic

149

C₄

#14556155

yes

characteristic

150

C₄

#14556157

yes

characteristic

151

C₂

—

568

—

152

C₂

#17986217

yes

characteristic

153

C₂

#17986244

yes

characteristic

154

C₂

#17986273

yes

characteristic

	subgroup	G/H
center	C₂²	#14556101
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#332×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#332	D₇₁
hypocenter	C₇₁	#1699
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14556101

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#332×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #17986351	C₂	semidirect
5. #17985751	C₂	semidirect
6. #17985746	C₂	semidirect
7. #332×C₇₁	C₂	semidirect
78. C₇₁	#1699	semidirect
85. #332	D₇₁	semidirect

H ≅ G/N

type

1. #17986351

C₂

semidirect

5. #17985751

C₂

semidirect

6. #17985746

C₂

semidirect

7. #332×C₇₁

C₂

semidirect

78. C₇₁

#1699

semidirect

85. #332

D₇₁

semidirect

Subgroups — #22587890

Conjugacy classes of subgroups

Complements