Group #22587022 — subgroups

Subgroups — #22587022

Order 9088.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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×#14556128	C₂	1	1	yes	maximal, characteristic
2	#17986052	C₂	1	1	yes	maximal, characteristic
3	#17985747	C₂	1	4	yes	maximal, complemented
4	#423×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₂	C₄	1	2	yes	—
6	#14556128	—	2	1	no	—
7	#14556128	—	2	1	no	—
8	#14556128	C₄	1	2	yes	—
9	#14556128	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₇₁	—	2	1	no	—
11	C₂×C₄⋊Dic₇₁	—	2	1	no	—
12	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
13	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
14	C₄×C₇₁⋊C₈	—	2	1	no	—
15	C₂×M₁₆×C₇₁	—	2	1	no	—
16	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₇₁	—	2	1	no	—
18	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
19	C₂×Dic₁₄₂	—	2	2	no	—
20	C₂×Dic₁₄₂	—	4	4	no	—
21	C₂×Dic₁₄₂	C₂×C₄	1	4	yes	—
22	C₂²×Dic₇₁	—	2	2	no	—
23	C₂×C₇₁⋊C₈	—	2	2	no	—
24	C₄⋊Dic₇₁	—	2	2	no	—
25	C₄⋊Dic₇₁	—	2	2	no	—
26	C₄⋊Dic₇₁	—	2	2	no	—
27	C₄⋊Dic₇₁	—	2	2	no	—
28	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
31	M₁₆×C₇₁	—	4	2	no	—
32	C₂×C₈×C₇₁	—	2	2	no	—
33	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₇₁	D₄	1	1	yes	characteristic
35	C₄²×C₇₁	D₄	1	1	yes	characteristic
36	C₄²×C₇₁	C₂³	1	1	yes	characteristic
37	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	—	2	2	no	—
39	C₂×C₄×C₇₁	—	2	2	no	—
40	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
44	C₂×C₄×C₇₁	Q₁₆	1	2	yes	—
45	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₇₁	—	2	4	no	—
49	C₂×Dic₇₁	—	4	4	no	—
50	Dic₁₄₂	—	4	4	no	—
51	Dic₁₄₂	—	4	4	no	—
52	C₈×C₇₁	—	4	2	no	—
53	C₇₁⋊C₈	—	4	2	no	—
54	C₂²×C₇₁	—	2	1	no	—
55	C₂²×C₇₁	—	2	1	no	—
56	C₂²×C₇₁	#103	1	2	yes	—
57	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₇₁	—	2	1	no	—
59	C₄×C₇₁	—	2	1	no	—
60	C₄×C₇₁	#100	1	2	yes	—
61	C₄×C₇₁	#103	1	2	yes	—
62	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₇₁	#137	1	2	yes	—
65	Dic₇₁	—	4	4	no	—
66	C₂×C₇₁	—	2	2	no	—
67	C₂×C₇₁	C₂×#100	1	1	yes	characteristic
68	C₂×C₇₁	C₂×#103	1	1	yes	characteristic
69	C₂×C₇₁	#419	1	1	yes	characteristic
70	#1548	—	71	1	no	maximal, Sylow
71	C₇₁	#1548	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄⋊Q₈	—	71	1	no	—
73	#431	—	71	1	no	—
74	#423	D₇₁	1	1	yes	characteristic
75	#333	—	71	4	no	—
76	C₂²×Q₈	—	71	2	no	—
77	C₂×M₁₆	—	2	1	no	—
78	C₄⋊Q₈	—	71	2	no	—
79	C₄⋊Q₈	—	71	2	no	—
80	C₄⋊Q₈	—	142	1	no	—
81	C₄⋊Q₈	—	142	1	no	—
82	C₂×C₄⋊C₄	—	142	1	no	—
83	C₂×C₄⋊C₄	—	142	1	no	—
84	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	71	2	no	—
87	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
88	C₂²⋊C₈	—	142	1	no	—
89	C₄×C₈	—	142	1	no	—
90	C₂×Q₈	—	284	4	no	—
91	C₂×Q₈	—	71	4	no	—
92	C₂×Q₈	—	142	2	no	—
93	C₂²×C₄	—	142	2	no	—
94	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
95	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
97	M₁₆	—	4	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	142	2	no	—
100	C₄⋊C₄	—	142	2	no	—
101	C₄⋊C₄	—	142	2	no	—
102	C₄⋊C₄	—	142	2	no	—
103	C₄⋊C₄	—	142	2	no	—
104	C₄²	C₄×D₇₁	1	1	yes	characteristic
105	C₄²	D₂₈₄	1	1	yes	characteristic
106	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
107	C₄²	C₂²×D₇₁	1	1	yes	characteristic
108	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
109	Q₈	—	284	4	no	—
110	Q₈	—	284	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	284	4	no	—
114	C₂×C₄	—	142	4	no	—
115	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
116	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
117	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
118	C₂×C₄	#11765853	1	2	yes	—
119	C₂×C₄	#11765855	1	2	yes	—
120	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
122	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	284	2	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#14556051	1	2	yes	—
128	C₂²	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	284	4	no	—
132	C₄	#14556044	1	2	yes	—
133	C₄	#14556051	1	2	yes	—
134	C₄	C₂×#11765853	1	1	yes	characteristic
135	C₄	C₂×#11765855	1	1	yes	characteristic
136	C₄	#14556161	1	2	yes	—
137	C₂	—	2	2	no	—
138	C₂	C₂×#14556044	1	1	yes	characteristic
139	C₂	C₂×#14556051	1	1	yes	characteristic
140	C₂	#17986009	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556128

C₂

yes

maximal, characteristic

#17986052

C₂

yes

maximal, characteristic

#17985747

C₂

yes

maximal, complemented

#423×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₄

yes

—

#14556128

—

#14556128

—

#14556128

C₄

yes

—

#14556128

C₂²

yes

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₇₁⋊C₈)

—

C₄×C₇₁⋊C₈

—

C₂×M₁₆×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

C₂×C₄

yes

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

M₁₆×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

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

—

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

#100

yes

—

C₄×C₇₁

#103

yes

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₄×C₇₁

#137

yes

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#431

—

#423

D₇₁

yes

characteristic

#333

—

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₄⋊Q₈

—

142

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₁

yes

—

C₂²⋊C₈

—

142

—

C₄×C₈

—

142

—

C₂×Q₈

—

284

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

142

—

100

C₄⋊C₄

—

142

—

101

C₄⋊C₄

—

142

—

102

C₄⋊C₄

—

142

—

103

C₄⋊C₄

—

142

—

104

C₄²

C₄×D₇₁

yes

characteristic

105

C₄²

D₂₈₄

yes

characteristic

106

C₄²

C₂²⋊D₇₁

yes

characteristic

107

C₄²

C₂²×D₇₁

yes

characteristic

108

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

109

Q₈

—

284

—

110

Q₈

—

284

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

284

—

114

C₂×C₄

—

142

—

115

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

116

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

117

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

118

C₂×C₄

#11765853

yes

—

119

C₂×C₄

#11765855

yes

—

120

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

122

C₂×C₄

C₂×D₂₈₄

yes

characteristic

123

C₈

—

124

C₈

—

284

—

125

C₂²

—

126

C₂²

—

127

C₂²

#14556051

yes

—

128

C₂²

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

284

—

132

C₄

#14556044

yes

—

133

C₄

#14556051

yes

—

134

C₄

C₂×#11765853

yes

characteristic

135

C₄

C₂×#11765855

yes

characteristic

136

C₄

#14556161

yes

—

137

C₂

—

138

C₂

C₂×#14556044

yes

characteristic

139

C₂

C₂×#14556051

yes

characteristic

140

C₂

#17986009

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	#423×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	#423	D₇₁
hypocenter	C₇₁	#1548
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₁)⋊C₄

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

#423×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #17985747	C₂	semidirect
71. C₇₁	#1548	semidirect

H ≅ G/N

type

3. #17985747

C₂

semidirect

71. C₇₁

#1548

semidirect

Subgroups — #22587022

Conjugacy classes of subgroups

Complements