Group #22589329 — subgroups

Subgroups — #22589329

Order 9088.

260 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 101 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	#17986023	C₂	1	1	yes	maximal, characteristic
3	#17985767	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₂	C₄	1	2	yes	—
6	C₂×#11765849	C₂²	1	2	yes	—
7	#14556128	—	2	1	no	—
8	#14556128	—	2	1	no	—
9	#14556128	—	2	1	no	—
10	#14556128	—	2	1	no	—
11	C₂×C₄⋊Dic₇₁	C₄	1	2	yes	—
12	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
13	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
14	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
15	C₂×M₁₆×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
17	#97×C₇₁	—	2	1	no	—
18	C₄×C₈×C₇₁	—	2	1	no	—
19	C₂×Dic₁₄₂	—	2	2	no	—
20	C₂×Dic₁₄₂	—	2	2	no	—
21	C₂×Dic₁₄₂	—	4	4	no	—
22	C₂²×Dic₇₁	C₂×C₄	1	4	yes	—
23	#11765849	—	2	4	no	—
24	C₂×C₇₁⋊C₈	—	2	2	no	—
25	C₄⋊Dic₇₁	—	2	2	no	—
26	C₄⋊Dic₇₁	—	2	2	no	—
27	C₄⋊Dic₇₁	—	2	2	no	—
28	C₄⋊Dic₇₁	—	2	2	no	—
29	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
32	M₁₆×C₇₁	—	2	2	no	—
33	M₁₆×C₇₁	Q₈	1	4	yes	complemented
34	C₂×C₈×C₇₁	—	2	2	no	—
35	C₄²×C₇₁	D₄	1	1	yes	characteristic
36	C₄²×C₇₁	D₄	1	1	yes	characteristic
37	C₄²×C₇₁	D₄	1	1	yes	characteristic
38	C₄²×C₇₁	D₄	1	1	yes	characteristic
39	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	—	2	2	no	—
41	C₂×C₄×C₇₁	—	2	2	no	—
42	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
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×C₄×C₇₁	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₇₁	#41	1	2	yes	—
50	C₂×Dic₇₁	—	2	4	no	—
51	C₂×Dic₇₁	—	4	4	no	—
52	Dic₁₄₂	—	4	4	no	—
53	Dic₁₄₂	—	4	4	no	—
54	C₈×C₇₁	—	2	4	no	—
55	C₇₁⋊C₈	—	4	2	no	—
56	C₂²×C₇₁	—	2	1	no	—
57	C₂²×C₇₁	—	2	1	no	—
58	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₇₁	C₄×Q₈	1	2	yes	—
60	C₄×C₇₁	—	2	1	no	—
61	C₄×C₇₁	—	2	1	no	—
62	C₄×C₇₁	#100	1	2	yes	—
63	C₄×C₇₁	#101	1	2	yes	—
64	C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₇₁	#124	1	1	yes	characteristic
66	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₇₁	—	4	4	no	—
68	C₂×C₇₁	—	2	2	no	—
69	C₂×C₇₁	#391	1	1	yes	characteristic
70	C₂×C₇₁	C₂×#100	1	1	yes	characteristic
71	C₂×C₇₁	C₂×#101	1	1	yes	characteristic
72	#1997	—	71	1	no	maximal, Sylow
73	C₇₁	#1997	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊Q₈	—	71	1	no	—
75	#423	—	71	1	no	—
76	C₄×M₁₆	D₇₁	1	1	yes	characteristic
77	#343	—	71	4	no	—
78	C₂²×Q₈	—	71	2	no	—
79	C₂×M₁₆	—	71	2	no	—
80	C₂×M₁₆	C₂×D₇₁	1	2	yes	—
81	C₄⋊Q₈	—	142	1	no	—
82	C₄⋊Q₈	—	142	1	no	—
83	C₄⋊Q₈	—	142	1	no	—
84	C₄⋊Q₈	—	142	1	no	—
85	C₂×C₄⋊C₄	—	71	2	no	—
86	C₂×C₄⋊C₄	—	71	2	no	—
87	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
88	C₄⋊C₈	—	142	1	no	—
89	C₄⋊C₈	—	142	1	no	—
90	#97	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₂×Q₈	—	284	4	no	—
93	C₂×Q₈	—	142	2	no	—
94	C₂×Q₈	—	142	2	no	—
95	C₂²×C₄	—	71	4	no	—
96	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	142	4	no	—
101	M₁₆	Dic₁₄₂	1	4	yes	complemented
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	142	2	no	—
104	C₄⋊C₄	—	142	2	no	—
105	C₄⋊C₄	—	142	2	no	—
106	C₄⋊C₄	—	142	2	no	—
107	C₄⋊C₄	—	142	2	no	—
108	C₄²	D₂₈₄	1	1	yes	characteristic
109	C₄²	D₂₈₄	1	1	yes	characteristic
110	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
111	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
112	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
113	Q₈	—	284	4	no	—
114	Q₈	—	284	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	284	4	no	—
118	C₂×C₄	—	142	4	no	—
119	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
124	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₁₄₂	1	2	yes	—
126	C₂×C₄	#11765865	1	2	yes	—
127	C₈	—	2	4	no	—
128	C₈	—	284	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
132	C₂²	C₄×Dic₁₄₂	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	284	4	no	—
136	C₄	#14556044	1	2	yes	—
137	C₄	#14556046	1	2	yes	—
138	C₄	#14556107	1	2	yes	—
139	C₄	#14556116	1	1	yes	characteristic
140	C₄	#14556128	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#17985905	1	1	yes	characteristic
143	C₂	C₂×#14556044	1	1	yes	characteristic
144	C₂	C₂×#14556046	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556128

C₂

yes

maximal, characteristic

#17986023

C₂

yes

maximal, characteristic

#17985767

C₂

yes

maximal

C₄×M₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₄

yes

—

C₂×#11765849

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Dic₇₁

C₄

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

—

C₂×M₁₆×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

—

C₄×C₈×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

C₂×C₄

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

Q₈

yes

complemented

C₂×C₈×C₇₁

—

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₁

C₄×Q₈

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#100

yes

—

C₄×C₇₁

#101

yes

—

C₄×C₇₁

C₂²⋊Q₈

yes

—

C₄×C₇₁

#124

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#423

—

C₄×M₁₆

D₇₁

yes

characteristic

#343

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

—

C₄⋊Q₈

—

142

—

C₄⋊Q₈

—

142

—

C₄⋊Q₈

—

142

—

C₄⋊Q₈

—

142

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

142

—

C₄⋊C₈

—

142

—

#97

—

C₄×C₈

—

C₂×Q₈

—

284

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

100

M₁₆

—

142

—

101

M₁₆

Dic₁₄₂

yes

complemented

102

C₂×C₈

—

103

C₂×C₈

—

142

—

104

C₄⋊C₄

—

142

—

105

C₄⋊C₄

—

142

—

106

C₄⋊C₄

—

142

—

107

C₄⋊C₄

—

142

—

108

C₄²

D₂₈₄

yes

characteristic

109

C₄²

D₂₈₄

yes

characteristic

110

C₄²

C₂²⋊D₇₁

yes

characteristic

111

C₄²

C₂²⋊D₇₁

yes

characteristic

112

C₂³

C₂×C₄×D₇₁

yes

characteristic

113

Q₈

—

284

—

114

Q₈

—

284

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

284

—

118

C₂×C₄

—

142

—

119

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

120

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

124

C₂×C₄

C₂×D₂₈₄

yes

characteristic

125

C₂×C₄

C₂×Dic₁₄₂

yes

—

126

C₂×C₄

#11765865

yes

—

127

C₈

—

128

C₈

—

284

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

132

C₂²

C₄×Dic₁₄₂

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

284

—

136

C₄

#14556044

yes

—

137

C₄

#14556046

yes

—

138

C₄

#14556107

yes

—

139

C₄

#14556116

yes

characteristic

140

C₄

#14556128

yes

characteristic

141

C₂

—

142

C₂

#17985905

yes

characteristic

143

C₂

C₂×#14556044

yes

characteristic

144

C₂

C₂×#14556046

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₇₁)⋊C₄
commutator	C₂³×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₄×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	C₄×M₁₆	D₇₁
hypocenter	C₇₁	#1997
LCS	G ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×(C₂×D₇₁)⋊C₄

commutator

C₂³×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₄×M₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

hypercenter

C₄×M₁₆

D₇₁

hypocenter

C₇₁

#1997

LCS

G ▷ C₂³×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
33. M₁₆×C₇₁	Q₈	semidirect
73. C₇₁	#1997	semidirect
101. M₁₆	Dic₁₄₂	semidirect

H ≅ G/N

type

33. M₁₆×C₇₁

Q₈

semidirect

73. C₇₁

#1997

semidirect

101. M₁₆

Dic₁₄₂

semidirect

Subgroups — #22589329

Conjugacy classes of subgroups

Complements