Group #22587028 — subgroups

Subgroups — #22587028

Order 9088.

216 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 89 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	#17986556	C₂	1	1	yes	maximal, characteristic
2	#17986023	C₂	1	1	yes	maximal, characteristic
3	#17985747	C₂	1	2	yes	maximal, complemented
4	#17985746	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765849	—	2	1	no	—
7	#14556128	C₂²	1	1	yes	characteristic
8	#14556128	C₂²	1	1	yes	characteristic
9	#14556120	—	2	1	no	—
10	#14556107	—	2	1	no	—
11	#14556107	C₄	1	2	yes	—
12	C₄×Dic₁₄₂	—	2	1	no	—
13	C₄×Dic₁₄₂	C₄	1	2	yes	—
14	#14556087	—	2	1	no	—
15	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
16	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
17	C₂²×C₈×C₇₁	—	2	1	no	—
18	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₇₁	—	2	1	no	—
20	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
21	C₂×Dic₁₄₂	—	2	1	no	—
22	C₂×Dic₁₄₂	C₂×C₄	1	2	yes	—
23	#11765849	—	4	2	no	—
24	C₂×C₇₁⋊C₈	—	2	2	no	—
25	C₄⋊Dic₇₁	—	2	1	no	—
26	C₄⋊Dic₇₁	—	2	1	no	—
27	C₄⋊Dic₇₁	—	2	1	no	—
28	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
29	Dic₇₁⋊C₄	—	2	2	no	—
30	Dic₇₁⋊C₄	—	2	2	no	—
31	C₂²⋊Dic₇₁	—	2	2	no	—
32	C₄×Dic₇₁	—	2	2	no	—
33	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
36	C₂×C₈×C₇₁	—	2	2	no	—
37	C₂×C₈×C₇₁	—	4	2	no	—
38	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₇₁	D₄	1	1	yes	characteristic
40	C₄²×C₇₁	D₄	1	1	yes	characteristic
41	C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	—	2	1	no	—
44	C₂×C₄×C₇₁	—	2	1	no	—
45	C₂×C₄×C₇₁	—	4	1	no	—
46	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
50	C₂×C₄×C₇₁	Q₁₆	1	2	yes	—
51	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₇₁	—	2	2	no	—
55	C₂×Dic₇₁	—	2	2	no	—
56	Dic₁₄₂	—	4	1	no	—
57	Dic₁₄₂	—	4	1	no	—
58	Dic₁₄₂	—	4	2	no	—
59	C₈×C₇₁	—	4	2	no	—
60	C₇₁⋊C₈	—	4	2	no	—
61	C₂²×C₇₁	—	2	1	no	—
62	C₂²×C₇₁	—	2	1	no	—
63	C₂²×C₇₁	#103	1	2	yes	—
64	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₇₁	—	2	1	no	—
66	C₄×C₇₁	—	2	1	no	—
67	C₄×C₇₁	—	2	1	no	—
68	C₄×C₇₁	#103	1	2	yes	—
69	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
71	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₇₁	#137	1	1	yes	characteristic
73	Dic₇₁	—	4	2	no	—
74	Dic₇₁	—	4	2	no	—
75	C₂×C₇₁	—	2	2	no	—
76	C₂×C₇₁	#413	1	1	yes	characteristic
77	C₂×C₇₁	C₂×#103	1	1	yes	characteristic
78	C₂×C₇₁	#417	1	1	yes	characteristic
79	#1550	—	71	1	no	maximal, Sylow
80	C₇₁	#1550	1	1	yes	characteristic, complemented, Sylow
81	#533	—	71	1	no	—
82	#423	—	71	1	no	—
83	C₂×C₄⋊C₈	D₇₁	1	1	yes	characteristic
84	#333	—	71	2	no	—
85	#332	—	71	2	no	—
86	C₂×M₁₆	—	142	1	no	—
87	C₂²×C₈	—	2	1	no	—
88	C₄⋊Q₈	—	71	1	no	—
89	C₄⋊Q₈	—	71	1	no	—
90	#125	—	142	1	no	—
91	C₂²⋊Q₈	—	71	2	no	—
92	C₂²⋊Q₈	—	142	1	no	—
93	C₄×Q₈	—	71	2	no	—
94	C₄×Q₈	—	142	1	no	—
95	#117	—	142	1	no	—
96	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
97	C₄⋊C₈	—	2	1	no	—
98	C₄⋊C₈	—	71	2	no	—
99	C₄⋊C₈	—	142	1	no	—
100	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
101	C₂×Q₈	—	71	2	no	—
102	C₂×Q₈	—	142	1	no	—
103	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
104	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
106	M₁₆	—	284	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	142	2	no	—
110	C₄⋊C₄	—	71	2	no	—
111	C₄⋊C₄	—	142	2	no	—
112	C₄⋊C₄	—	142	2	no	—
113	C₄⋊C₄	—	142	1	no	—
114	C₄⋊C₄	—	142	1	no	—
115	C₄⋊C₄	—	142	1	no	—
116	C₂²⋊C₄	—	142	2	no	—
117	C₄²	—	142	2	no	—
118	C₄²	C₄×D₇₁	1	1	yes	characteristic
119	C₄²	D₂₈₄	1	1	yes	characteristic
120	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
121	C₄²	C₂²×D₇₁	1	1	yes	characteristic
122	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
123	Q₈	—	284	2	no	—
124	Q₈	—	284	1	no	—
125	Q₈	—	284	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	142	2	no	—
130	C₂×C₄	—	142	2	no	—
131	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
132	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
133	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	#11765854	1	2	yes	—
135	C₂×C₄	Dic₂₈₄	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
139	C₈	—	4	2	no	—
140	C₈	—	284	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#14556053	1	2	yes	—
144	C₂²	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	284	2	no	—
149	C₄	—	284	2	no	—
150	C₄	#14556053	1	2	yes	—
151	C₄	C₂×#11765854	1	1	yes	characteristic
152	C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
153	C₄	#14556154	1	1	yes	characteristic
154	C₄	#14556159	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#17985981	1	1	yes	characteristic
157	C₂	C₂×#14556053	1	1	yes	characteristic
158	C₂	#17985998	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986556

C₂

yes

maximal, characteristic

#17986023

C₂

yes

maximal, characteristic

#17985747

C₂

yes

maximal, complemented

#17985746

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

—

#14556128

C₂²

yes

characteristic

#14556128

C₂²

yes

—

—

C₄

yes

—

C₄×Dic₁₄₂

—

C₄×Dic₁₄₂

C₄

yes

—

#14556087

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂²×C₈×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

C₂×C₄

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

Q₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#103

yes

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#103

yes

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

C₂×Q₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#533

—

#423

—

C₂×C₄⋊C₈

D₇₁

yes

characteristic

#333

—

#332

—

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

142

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

142

—

C₄×Q₈

—

C₄×Q₈

—

142

—

#117

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

100

C₄⋊C₈

C₂×D₇₁

yes

—

101

C₂×Q₈

—

102

C₂×Q₈

—

142

—

103

C₂²×C₄

C₄×D₇₁

yes

characteristic

104

C₂²×C₄

D₂₈₄

yes

characteristic

105

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

106

M₁₆

—

284

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

142

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

142

—

112

C₄⋊C₄

—

142

—

113

C₄⋊C₄

—

142

—

114

C₄⋊C₄

—

142

—

115

C₄⋊C₄

—

142

—

116

C₂²⋊C₄

—

142

—

117

C₄²

—

142

—

118

C₄²

C₄×D₇₁

yes

characteristic

119

C₄²

D₂₈₄

yes

characteristic

120

C₄²

C₂²⋊D₇₁

yes

characteristic

121

C₄²

C₂²×D₇₁

yes

characteristic

122

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

123

Q₈

—

284

—

124

Q₈

—

284

—

125

Q₈

—

284

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

142

—

130

C₂×C₄

—

142

—

131

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

132

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

133

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

134

C₂×C₄

#11765854

yes

—

135

C₂×C₄

Dic₂₈₄

yes

—

136

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

138

C₂×C₄

C₂×D₂₈₄

yes

characteristic

139

C₈

—

140

C₈

—

284

—

141

C₂²

—

142

C₂²

—

143

C₂²

#14556053

yes

—

144

C₂²

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

284

—

149

C₄

—

284

—

150

C₄

#14556053

yes

—

151

C₄

C₂×#11765854

yes

characteristic

152

C₄

C₂×Dic₂₈₄

yes

characteristic

153

C₄

#14556154

yes

characteristic

154

C₄

#14556159

yes

characteristic

155

C₂

—

156

C₂

#17985981

yes

characteristic

157

C₂

C₂×#14556053

yes

characteristic

158

C₂

#17985998

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	C₂×C₄⋊C₈×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊C₈	D₇₁
hypocenter	C₇₁	#1550
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₁)⋊C₄

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

C₂×C₄⋊C₈×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

hypercenter

C₂×C₄⋊C₈

D₇₁

hypocenter

C₇₁

#1550

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #17985747	C₂	semidirect
4. #17985746	C₂	semidirect
80. C₇₁	#1550	semidirect

H ≅ G/N

type

3. #17985747

C₂

semidirect

4. #17985746

C₂

semidirect

80. C₇₁

#1550

semidirect

Subgroups — #22587028

Conjugacy classes of subgroups

Complements