Group #22589199 — subgroups

Subgroups — #22589199

Order 9088.

312 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556051	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985904	C₂	1	2	yes	maximal
5	C₂×#107×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₂	C₂²	1	2	yes	—
7	C₂×Dic₂₈₄	—	2	2	no	—
8	C₂×Dic₂₈₄	—	2	2	no	—
9	C₂×Dic₂₈₄	C₄	1	4	yes	complemented
10	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
12	#14556051	—	2	4	no	—
13	C₈×Dic₇₁	—	2	2	no	—
14	C₇₁⋊#95	—	2	4	no	—
15	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
17	#107×C₇₁	—	2	2	no	—
18	C₂×Dic₁₄₂	—	2	2	no	—
19	C₂×Dic₁₄₂	—	2	2	no	—
20	C₂×Dic₁₄₂	—	2	2	no	—
21	C₂×Dic₁₄₂	—	2	2	no	—
22	C₂×Dic₁₄₂	—	2	2	no	—
23	C₂×Dic₁₄₂	C₂×C₄	1	4	yes	—
24	C₂²×Dic₇₁	—	2	2	no	—
25	C₂²×Dic₇₁	D₄	1	2	yes	—
26	Dic₂₈₄	—	2	8	no	—
27	C₂×C₇₁⋊C₈	—	2	1	no	—
28	C₂×C₇₁⋊C₈	—	2	1	no	—
29	C₂×C₇₁⋊C₈	D₄	1	2	yes	—
30	C₄×Dic₇₁	—	2	1	no	—
31	C₄×Dic₇₁	—	2	1	no	—
32	C₂²×C₄×C₇₁	—	2	2	no	—
33	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₇₁	D₄	1	2	yes	—
36	C₂×C₈×C₇₁	D₄	1	2	yes	—
37	C₄⋊C₄×C₇₁	—	2	2	no	—
38	C₄⋊C₄×C₇₁	—	2	2	no	—
39	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	—	4	2	no	—
41	C₂×C₄×C₇₁	—	4	4	no	—
42	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
47	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
48	C₂×Dic₇₁	—	2	2	no	—
49	C₂×Dic₇₁	—	2	2	no	—
50	C₂×Dic₇₁	—	2	4	no	—
51	C₂×Dic₇₁	—	4	2	no	—
52	C₂×Dic₇₁	—	4	2	no	—
53	C₂×Dic₇₁	Q₁₆	1	4	yes	complemented
54	Dic₁₄₂	—	2	4	no	—
55	Dic₁₄₂	—	2	4	no	—
56	Dic₁₄₂	—	4	2	no	—
57	Dic₁₄₂	—	4	2	no	—
58	C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
59	C₇₁⋊C₈	—	2	2	no	—
60	C₂²×C₇₁	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₇₁	#121	1	1	yes	characteristic
62	C₂²×C₇₁	#127	1	1	yes	characteristic
63	C₂²×C₇₁	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₇₁	#135	1	2	yes	—
65	C₄×C₇₁	—	8	2	no	—
66	C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
68	C₄×C₇₁	#121	1	1	yes	characteristic
69	C₄×C₇₁	#124	1	1	yes	characteristic
70	Dic₇₁	—	2	4	no	—
71	Dic₇₁	—	4	4	no	—
72	C₂×C₇₁	#390	1	1	yes	characteristic
73	C₂×C₇₁	C₄×Q₁₆	1	2	yes	—
74	C₂×C₇₁	#467	1	2	yes	—
75	C₂×C₇₁	#494	1	1	yes	characteristic
76	C₂×C₇₁	#495	1	1	yes	characteristic
77	#1969	—	71	1	no	maximal, Sylow
78	C₇₁	#1969	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₁₆	—	71	1	no	—
80	C₂×#107	D₇₁	1	1	yes	characteristic
81	C₂×#103	—	71	2	no	—
82	C₂×C₄×C₈	—	71	1	no	—
83	#391	—	71	2	no	—
84	C₂²×Q₈	—	71	2	no	—
85	C₂×Q₁₆	—	71	4	no	—
86	C₂×Q₁₆	—	142	2	no	—
87	C₂×Q₁₆	—	142	2	no	—
88	C₂²×C₈	—	71	1	no	—
89	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
91	C₂×C₄²	—	71	1	no	—
92	#107	—	2	2	no	—
93	#103	—	142	4	no	—
94	C₄×C₈	—	142	2	no	—
95	#95	—	142	4	no	—
96	C₂×Q₈	—	71	4	no	—
97	C₂×Q₈	—	142	2	no	—
98	C₂×Q₈	—	142	2	no	—
99	C₂×Q₈	—	142	2	no	—
100	C₂×Q₈	—	142	2	no	—
101	C₂×Q₈	—	142	2	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	71	2	no	—
104	C₂²×C₄	—	142	2	no	—
105	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
106	Q₁₆	—	142	8	no	—
107	C₂×C₈	—	71	2	no	—
108	C₂×C₈	—	142	1	no	—
109	C₂×C₈	—	142	1	no	—
110	C₂×C₈	C₄×D₇₁	1	2	yes	—
111	C₂×C₈	D₂₈₄	1	2	yes	—
112	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄²	—	142	1	no	—
116	C₄²	—	142	1	no	—
117	C₂³	D₄×D₇₁	1	1	yes	characteristic
118	Q₈	—	284	2	no	—
119	Q₈	—	284	2	no	—
120	Q₈	—	142	4	no	—
121	Q₈	—	142	4	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	—	284	2	no	—
125	C₂×C₄	—	284	2	no	—
126	C₂×C₄	—	71	4	no	—
127	C₂×C₄	—	142	4	no	—
128	C₂×C₄	—	142	2	no	—
129	C₂×C₄	—	142	2	no	—
130	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
131	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
133	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
134	C₂×C₄	#11765866	1	1	yes	characteristic
135	C₂×C₄	#11765866	1	1	yes	characteristic
136	C₈	—	142	2	no	—
137	C₈	(C₂×D₇₁)⋊C₄	1	4	yes	—
138	C₂²	#14556090	1	1	yes	characteristic
139	C₂²	#14556103	1	1	yes	characteristic
140	C₂²	#14556124	1	1	yes	characteristic
141	C₂²	Q₁₆×D₇₁	1	2	yes	—
142	C₂²	#14556147	1	2	yes	—
143	C₄	—	8	2	no	—
144	C₄	—	284	4	no	—
145	C₄	—	142	4	no	—
146	C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
147	C₄	#14556090	1	1	yes	characteristic
148	C₄	#14556103	1	1	yes	characteristic
149	C₄	#14556117	1	1	yes	characteristic
150	C₂	#17985900	1	1	yes	characteristic
151	C₂	#17986093	1	2	yes	—
152	C₂	#17986241	1	2	yes	—
153	C₂	#17986377	1	1	yes	characteristic
154	C₂	#17986382	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₈₄

C₂

yes

maximal, characteristic

C₂×#14556051

C₂

yes

maximal

C₂×C₈×Dic₇₁

C₂

yes

maximal, characteristic

#17985904

C₂

yes

maximal

C₂×#107×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₂²

yes

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

C₄

yes

complemented

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556051

—

C₈×Dic₇₁

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

#107×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

C₂×C₄

yes

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

D₄

yes

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

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×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

Q₁₆

yes

complemented

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

C₂²⋊C₄

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₄×D₄

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

C₂×Q₁₆

yes

—

C₂²×C₇₁

#135

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

C₄×D₄

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#390

yes

characteristic

C₂×C₇₁

C₄×Q₁₆

yes

—

C₂×C₇₁

#467

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×#107

D₇₁

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

#391

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

—

142

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

—

#107

—

#103

—

142

—

C₄×C₈

—

142

—

#95

—

142

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

100

C₂×Q₈

—

142

—

101

C₂×Q₈

—

142

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

142

—

105

C₂²×C₄

C₂²×D₇₁

yes

characteristic

106

Q₁₆

—

142

—

107

C₂×C₈

—

108

C₂×C₈

—

142

—

109

C₂×C₈

—

142

—

110

C₂×C₈

C₄×D₇₁

yes

—

111

C₂×C₈

D₂₈₄

yes

—

112

C₂×C₈

C₂²⋊D₇₁

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

142

—

116

C₄²

—

142

—

117

C₂³

D₄×D₇₁

yes

characteristic

118

Q₈

—

284

—

119

Q₈

—

284

—

120

Q₈

—

142

—

121

Q₈

—

142

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

284

—

125

C₂×C₄

—

284

—

126

C₂×C₄

—

127

C₂×C₄

—

142

—

128

C₂×C₄

—

142

—

129

C₂×C₄

—

142

—

130

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

131

C₂×C₄

D₄×D₇₁

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

133

C₂×C₄

C₂×D₂₈₄

yes

characteristic

134

C₂×C₄

#11765866

yes

characteristic

135

C₂×C₄

#11765866

yes

characteristic

136

C₈

—

142

—

137

C₈

(C₂×D₇₁)⋊C₄

yes

—

138

C₂²

#14556090

yes

characteristic

139

C₂²

#14556103

yes

characteristic

140

C₂²

#14556124

yes

characteristic

141

C₂²

Q₁₆×D₇₁

yes

—

142

C₂²

#14556147

yes

—

143

C₄

—

144

C₄

—

284

—

145

C₄

—

142

—

146

C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

147

C₄

#14556090

yes

characteristic

148

C₄

#14556103

yes

characteristic

149

C₄

#14556117

yes

characteristic

150

C₂

#17985900

yes

characteristic

151

C₂

#17986093

yes

—

152

C₂

#17986241

yes

—

153

C₂

#17986377

yes

characteristic

154

C₂

#17986382

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#107×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#107	D₇₁
hypocenter	C₇₁	#1969
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#107×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
9. C₂×Dic₂₈₄	C₄	semidirect
53. C₂×Dic₇₁	Q₁₆	semidirect
78. C₇₁	#1969	semidirect

H ≅ G/N

type

9. C₂×Dic₂₈₄

C₄

semidirect

53. C₂×Dic₇₁

Q₁₆

semidirect

78. C₇₁

#1969

semidirect

Subgroups — #22589199

Conjugacy classes of subgroups

Complements