Group #22589064 — subgroups

Subgroups — #22589064

Order 9088.

248 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556039	C₂	1	1	yes	maximal, characteristic
4	#17985754	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
10	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
11	#14556039	—	2	2	no	—
12	C₂×C₄⋊C₄×C₇₁	C₄	1	2	yes	—
13	C₂×C₄²×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
15	#95×C₇₁	C₄	1	1	yes	characteristic
16	#95×C₇₁	C₄	1	1	yes	characteristic
17	C₂×C₇₁⋊C₈	—	2	1	no	—
18	C₂×C₇₁⋊C₈	—	2	1	no	—
19	C₂×C₇₁⋊C₈	—	2	1	no	—
20	C₂×C₇₁⋊C₈	—	2	1	no	—
21	C₂×C₇₁⋊C₈	—	2	2	no	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₂×C₇₁⋊C₈	—	2	2	no	—
24	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
25	C₂×C₇₁⋊C₈	Q₈	1	2	yes	—
26	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₇₁	—	2	4	no	—
32	C₄²×C₇₁	—	2	2	no	—
33	C₄²×C₇₁	—	2	2	no	—
34	C₄²×C₇₁	D₄	1	2	yes	—
35	C₄²×C₇₁	Q₈	1	2	yes	—
36	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	—	2	2	no	—
38	C₂×C₄×C₇₁	—	2	2	no	—
39	C₂×C₄×C₇₁	—	2	2	no	—
40	C₂×C₄×C₇₁	—	2	2	no	—
41	C₂×C₄×C₇₁	—	2	2	no	—
42	C₂×C₄×C₇₁	—	2	2	no	—
43	C₂×C₄×C₇₁	—	2	2	no	—
44	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₇₁	M₁₆	1	4	yes	—
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
52	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
53	C₇₁⋊C₈	—	2	2	no	—
54	C₇₁⋊C₈	—	2	2	no	—
55	C₇₁⋊C₈	—	4	2	no	—
56	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₇₁	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₇₁	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₇₁	C₂×M₁₆	1	2	yes	—
60	C₂²×C₇₁	#131	1	2	yes	—
61	C₄×C₇₁	—	2	2	no	—
62	C₄×C₇₁	—	2	2	no	—
63	C₄×C₇₁	—	2	4	no	—
64	C₄×C₇₁	—	4	2	no	—
65	C₄×C₇₁	#121	1	1	yes	characteristic
66	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
67	C₄×C₇₁	#125	1	1	yes	characteristic
68	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₇₁	#389	1	1	yes	characteristic
70	C₂×C₇₁	#423	1	1	yes	characteristic
71	C₂×C₇₁	#424	1	1	yes	characteristic
72	C₂×C₇₁	#435	1	2	yes	—
73	C₂×C₇₁	#446	1	2	yes	—
74	#1940	—	71	1	no	maximal, Sylow
75	C₇₁	#1940	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄⋊C₈	—	71	2	no	—
77	C₂×C₄⋊C₈	—	71	1	no	—
78	C₂×#97	—	71	1	no	—
79	C₄×C₄⋊C₄	D₇₁	1	1	yes	characteristic
80	#336	—	71	2	no	—
81	C₂²×C₈	—	71	1	no	—
82	C₂²×C₈	—	71	1	no	—
83	C₂²×C₈	—	71	2	no	—
84	C₂×C₄⋊C₄	Dic₇₁	1	2	yes	—
85	C₂×C₄²	C₂×D₇₁	1	2	yes	—
86	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
87	C₄⋊C₈	—	142	4	no	—
88	C₄⋊C₈	—	142	2	no	—
89	#97	—	142	2	no	—
90	#95	Dic₇₁	1	1	yes	characteristic
91	#95	Dic₇₁	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
93	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
94	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
95	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
96	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
97	C₂×C₈	—	71	2	no	—
98	C₂×C₈	—	71	2	no	—
99	C₂×C₈	—	142	2	no	—
100	C₂×C₈	—	142	2	no	—
101	C₂×C₈	—	142	2	no	—
102	C₂×C₈	—	142	1	no	—
103	C₂×C₈	—	142	1	no	—
104	C₂×C₈	—	142	1	no	—
105	C₂×C₈	—	142	1	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	Dic₁₄₂	1	2	yes	—
110	C₄²	D₂₈₄	1	2	yes	—
111	C₂³	C₂²×Dic₇₁	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	C₄⋊Dic₇₁	1	2	yes	—
120	C₂×C₄	C₄⋊Dic₇₁	1	2	yes	—
121	C₂×C₄	#11765849	1	4	yes	—
122	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
123	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
126	C₂×C₄	#11765864	1	1	yes	characteristic
127	C₂×C₄	#11765866	1	1	yes	characteristic
128	C₈	—	284	2	no	—
129	C₈	—	142	2	no	—
130	C₈	—	142	2	no	—
131	C₂²	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
132	C₂²	D₄×Dic₇₁	1	1	yes	characteristic
133	C₂²	Q₈×Dic₇₁	1	1	yes	characteristic
134	C₂²	C₂×#11765849	1	2	yes	—
135	C₂²	#14556135	1	2	yes	—
136	C₄	—	2	4	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	2	no	—
139	C₄	—	4	2	no	—
140	C₄	#14556103	1	1	yes	characteristic
141	C₄	#14556108	1	1	yes	characteristic
142	C₄	#14556119	1	1	yes	characteristic
143	C₄	#14556127	1	1	yes	characteristic
144	C₂	#17985898	1	1	yes	characteristic
145	C₂	#17986023	1	1	yes	characteristic
146	C₂	#17986026	1	1	yes	characteristic
147	C₂	#17986071	1	2	yes	—
148	C₂	#17986130	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14556039

C₂

yes

maximal, characteristic

#17985754

C₂

yes

maximal

C₄×C₄⋊C₄×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

#14556039

—

C₂×C₄⋊C₄×C₇₁

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#95×C₇₁

C₄

yes

characteristic

#95×C₇₁

C₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₂×C₇₁⋊C₈

Q₈

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

D₄

yes

—

C₄²×C₇₁

Q₈

yes

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

C₄×D₄

yes

characteristic

C₂²×C₇₁

C₄×Q₈

yes

characteristic

C₂²×C₇₁

C₂×M₁₆

yes

—

C₂²×C₇₁

#131

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#121

yes

characteristic

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

#125

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₂×C₇₁

#446

yes

—

#1940

—

maximal, Sylow

C₇₁

#1940

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₄×C₄⋊C₄

D₇₁

yes

characteristic

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

142

—

C₄⋊C₈

—

142

—

#97

—

142

—

#95

Dic₇₁

yes

characteristic

#95

Dic₇₁

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

100

C₂×C₈

—

142

—

101

C₂×C₈

—

142

—

102

C₂×C₈

—

142

—

103

C₂×C₈

—

142

—

104

C₂×C₈

—

142

—

105

C₂×C₈

—

142

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

Dic₁₄₂

yes

—

110

C₄²

D₂₈₄

yes

—

111

C₂³

C₂²×Dic₇₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄⋊Dic₇₁

yes

—

120

C₂×C₄

C₄⋊Dic₇₁

yes

—

121

C₂×C₄

#11765849

yes

—

122

C₂×C₄

D₄×D₇₁

yes

characteristic

123

C₂×C₄

C₂×D₂₈₄

yes

characteristic

124

C₂×C₄

Q₈×D₇₁

yes

characteristic

125

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

126

C₂×C₄

#11765864

yes

characteristic

127

C₂×C₄

#11765866

yes

characteristic

128

C₈

—

284

—

129

C₈

—

142

—

130

C₈

—

142

—

131

C₂²

C₂×C₄⋊Dic₇₁

yes

characteristic

132

C₂²

D₄×Dic₇₁

yes

characteristic

133

C₂²

Q₈×Dic₇₁

yes

characteristic

134

C₂²

C₂×#11765849

yes

—

135

C₂²

#14556135

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#14556103

yes

characteristic

141

C₄

#14556108

yes

characteristic

142

C₄

#14556119

yes

characteristic

143

C₄

#14556127

yes

characteristic

144

C₂

#17985898

yes

characteristic

145

C₂

#17986023

yes

characteristic

146

C₂

#17986026

yes

characteristic

147

C₂

#17986071

yes

—

148

C₂

#17986130

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₁
commutator, socle	C₂³×C₇₁	C₂²×C₄
Fitting	C₄×C₄⋊C₄×C₇₁	C₂
hypercenter	C₄×C₄⋊C₄	D₇₁
hypocenter	C₇₁	#1940
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator, socle

C₂³×C₇₁

C₂²×C₄

Fitting

C₄×C₄⋊C₄×C₇₁

C₂

hypercenter

C₄×C₄⋊C₄

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
75. C₇₁	#1940	semidirect

H ≅ G/N

type

75. C₇₁

#1940

semidirect

Subgroups — #22589064

Conjugacy classes of subgroups

Complements