Group #22587230 — subgroups

Subgroups — #22587230

Order 9088.

166 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₄×#11765852	C₂	1	1	yes	maximal, characteristic
2	#17986078	C₂	1	1	yes	maximal, characteristic
3	#17985749	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	#17985726	C₂	1	1	yes	maximal, characteristic
6	C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#446×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11765852	C₄	1	1	yes	characteristic
9	C₂×#11765848	—	2	1	no	—
10	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
11	#14556059	C₄	1	1	yes	characteristic
12	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
13	#14556052	C₄	1	1	yes	characteristic
14	#14556048	C₄	1	1	yes	characteristic
15	(C₂×D₇₁)⋊C₈	—	2	1	no	—
16	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₇₁	—	2	1	no	—
20	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
21	#97×C₇₁	—	2	1	no	—
22	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×D₂₈₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₇₁	—	2	1	no	—
25	#11765852	—	2	2	no	—
26	#11765848	—	4	2	no	—
27	C₂×C₇₁⋊C₈	—	2	1	no	—
28	C₂×C₇₁⋊C₈	—	2	1	no	—
29	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Dic₇₁	C₂×C₄	1	1	yes	characteristic
32	(C₂×D₇₁)⋊C₄	—	2	1	no	—
33	C₂×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₁	—	2	1	no	—
35	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₇₁	—	2	1	no	—
38	C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₇₁	—	2	1	no	—
40	C₄²×C₇₁	C₂³	1	1	yes	characteristic
41	C₂²×D₇₁	—	2	1	no	—
42	Q₈×C₇₁	—	2	1	no	—
43	Q₈×C₇₁	—	2	1	no	—
44	C₂×C₄×C₇₁	—	2	1	no	—
45	C₂×C₄×C₇₁	—	2	1	no	—
46	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
49	C₂×Dic₇₁	—	2	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	C₈×C₇₁	—	2	1	no	—
55	C₈×C₇₁	—	4	1	no	—
56	C₈×C₇₁	SD₁₆	1	2	yes	complemented
57	C₇₁⋊C₈	—	2	2	no	—
58	C₇₁⋊C₈	—	2	1	no	—
59	C₇₁⋊C₈	—	4	1	no	—
60	C₇₁⋊C₈	M₁₆	1	2	yes	complemented
61	C₂²×C₇₁	C₄×D₄	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	—	4	1	no	—
67	C₄×C₇₁	—	4	1	no	—
68	C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
69	C₄×C₇₁	#131	1	1	yes	characteristic
70	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
71	C₄×C₇₁	#135	1	1	yes	characteristic
72	Dic₇₁	—	4	1	no	—
73	C₂×C₇₁	#436	1	1	yes	characteristic
74	C₂×C₇₁	C₄×SD₁₆	1	1	yes	characteristic
75	C₂×C₇₁	#443	1	1	yes	characteristic
76	D₇₁	—	8	1	no	—
77	#1590	—	71	1	no	maximal, Sylow
78	C₇₁	#1590	1	1	yes	characteristic, complemented, Sylow
79	#446	D₇₁	1	1	yes	characteristic, complemented
80	C₄×SD₁₆	—	71	1	no	—
81	#436	—	71	1	no	—
82	#334	—	71	1	no	—
83	Q₈⋊C₈	—	71	1	no	—
84	D₄⋊C₈	—	71	1	no	—
85	C₈²	—	71	1	no	—
86	C₂×SD₁₆	—	71	1	no	—
87	C₂×M₁₆	—	142	1	no	—
88	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
89	C₄×D₄	—	71	1	no	—
90	#106	—	71	1	no	—
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	—	71	1	no	—
93	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
94	#103	—	71	1	no	—
95	#102	—	71	1	no	—
96	C₂²⋊C₈	—	142	1	no	—
97	#97	—	2	1	no	—
98	C₄×C₈	—	71	1	no	—
99	C₄×C₈	—	71	1	no	—
100	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
101	C₂×Q₈	C₄×D₇₁	1	1	yes	characteristic
102	C₂×D₄	—	71	1	no	—
103	C₂²×C₄	—	142	1	no	—
104	SD₁₆	—	142	2	no	—
105	M₁₆	—	284	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	71	1	no	—
108	C₂×C₈	—	71	1	no	—
109	C₂×C₈	—	142	1	no	—
110	C₂×C₈	—	142	1	no	—
111	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	71	1	no	—
115	C₄⋊C₄	C₄×D₇₁	1	1	yes	characteristic
116	C₂²⋊C₄	—	142	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂²×D₇₁	1	1	yes	characteristic
119	C₂³	—	142	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	2	1	no	—
122	D₄	—	142	1	no	—
123	D₄	—	142	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	284	1	no	—
127	C₂×C₄	—	284	1	no	—
128	C₂×C₄	—	142	1	no	—
129	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
131	C₂×C₄	#11765865	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	284	1	no	—
135	C₈	—	71	2	no	—
136	C₈	—	142	2	no	—
137	C₈	—	142	1	no	—
138	C₈	#11765852	1	2	yes	complemented
139	C₂²	—	284	1	no	—
140	C₂²	—	284	1	no	—
141	C₂²	—	284	1	no	—
142	C₂²	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	284	1	no	—
147	C₄	M₁₆×D₇₁	1	1	yes	characteristic
148	C₄	#14556136	1	1	yes	characteristic
149	C₄	C₂×#11765852	1	1	yes	characteristic
150	C₄	#14556148	1	1	yes	characteristic
151	C₂	—	568	1	no	—
152	C₂	#17986075	1	1	yes	characteristic
153	C₂	C₄×#11765852	1	1	yes	characteristic
154	C₂	#17986114	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11765852

C₂

yes

maximal, characteristic

#17986078

C₂

yes

maximal, characteristic

#17985749

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic, complemented

#17985726

C₂

yes

maximal, characteristic

C₈×C₇₁⋊C₈

C₂

yes

maximal, characteristic, complemented

#446×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765852

C₄

yes

characteristic

C₂×#11765848

—

C₄×D₂₈₄

C₂²

yes

characteristic

#14556059

C₄

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

#14556052

C₄

yes

characteristic

#14556048

C₄

yes

characteristic

(C₂×D₇₁)⋊C₈

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

#97×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×D₂₈₄

C₂×C₄

yes

characteristic

C₂×C₄×D₇₁

—

#11765852

—

#11765848

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₄⋊Dic₇₁

C₂×C₄

yes

characteristic

(C₂×D₇₁)⋊C₄

—

C₂×Q₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

SD₁₆

yes

complemented

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

M₁₆

yes

complemented

C₂²×C₇₁

C₄×D₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×M₁₆

yes

characteristic

C₄×C₇₁

#131

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

#135

yes

characteristic

Dic₇₁

—

C₂×C₇₁

#436

yes

characteristic

C₂×C₇₁

C₄×SD₁₆

yes

characteristic

C₂×C₇₁

#443

yes

characteristic

D₇₁

—

#1590

—

maximal, Sylow

C₇₁

#1590

yes

characteristic, complemented, Sylow

#446

D₇₁

yes

characteristic, complemented

C₄×SD₁₆

—

#436

—

#334

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₈²

—

C₂×SD₁₆

—

C₂×M₁₆

—

142

—

C₄×Q₈

C₂×D₇₁

yes

characteristic

C₄×D₄

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#103

—

#102

—

C₂²⋊C₈

—

142

—

#97

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₇₁

yes

characteristic

101

C₂×Q₈

C₄×D₇₁

yes

characteristic

102

C₂×D₄

—

103

C₂²×C₄

—

142

—

104

SD₁₆

—

142

—

105

M₁₆

—

284

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

142

—

110

C₂×C₈

—

142

—

111

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

112

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₄×D₇₁

yes

characteristic

116

C₂²⋊C₄

—

142

—

117

C₄²

—

118

C₄²

C₂²×D₇₁

yes

characteristic

119

C₂³

—

142

—

120

Q₈

—

121

Q₈

—

122

D₄

—

142

—

123

D₄

—

142

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

284

—

127

C₂×C₄

—

284

—

128

C₂×C₄

—

142

—

129

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

131

C₂×C₄

#11765865

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

284

—

135

C₈

—

136

C₈

—

142

—

137

C₈

—

142

—

138

C₈

#11765852

yes

complemented

139

C₂²

—

284

—

140

C₂²

—

284

—

141

C₂²

—

284

—

142

C₂²

C₄×C₂²⋊D₇₁

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

284

—

147

C₄

M₁₆×D₇₁

yes

characteristic

148

C₄

#14556136

yes

characteristic

149

C₄

C₂×#11765852

yes

characteristic

150

C₄

#14556148

yes

characteristic

151

C₂

—

568

—

152

C₂

#17986075

yes

characteristic

153

C₂

C₄×#11765852

yes

characteristic

154

C₂

#17986114

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	#446×C₇₁	C₂
socle	C₂²×C₇₁	C₄×D₄
hypercenter	#446	D₇₁
hypocenter	C₇₁	#1590
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #446

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

#446×C₇₁

C₂

socle

C₂²×C₇₁

C₄×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #446

N	H ≅ G/N	type
3. #17985749	C₂	semidirect
4. Q₈⋊(C₇₁⋊C₈)	C₂	semidirect
6. C₈×C₇₁⋊C₈	C₂	semidirect
7. #446×C₇₁	C₂	semidirect
56. C₈×C₇₁	SD₁₆	semidirect
60. C₇₁⋊C₈	M₁₆	semidirect
78. C₇₁	#1590	semidirect
79. #446	D₇₁	semidirect
138. C₈	#11765852	semidirect

H ≅ G/N

type

3. #17985749

C₂

semidirect

4. Q₈⋊(C₇₁⋊C₈)

C₂

semidirect

6. C₈×C₇₁⋊C₈

C₂

semidirect

7. #446×C₇₁

C₂

semidirect

56. C₈×C₇₁

SD₁₆

semidirect

60. C₇₁⋊C₈

M₁₆

semidirect

78. C₇₁

#1590

semidirect

79. #446

D₇₁

semidirect

138. C₈

#11765852

semidirect

Subgroups — #22587230

Conjugacy classes of subgroups

Complements