Group #22588461 — subgroups

Subgroups — #22588461

Order 9088.

172 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 73 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×#14556063	C₂	1	1	yes	maximal, characteristic
2	#17986035	C₂	1	1	yes	maximal, characteristic
3	#17985957	C₂	1	1	yes	maximal, characteristic
4	#17985770	C₂	1	1	yes	maximal, characteristic
5	#17985756	C₂	1	1	yes	maximal, characteristic
6	#17985755	C₂	1	1	yes	maximal, characteristic
7	#344×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765849	C₂²	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	#14556087	C₂²	1	1	yes	characteristic
11	#14556087	C₂²	1	1	yes	characteristic
12	#14556063	—	2	1	no	—
13	#14556063	—	2	1	no	—
14	#14556061	—	2	1	no	—
15	#14556059	—	2	1	no	—
16	#14556038	C₄	1	2	yes	complemented
17	C₈×Dic₇₁	C₄	1	2	yes	complemented
18	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	#117×C₇₁	C₂²	1	1	yes	characteristic
21	#11765849	—	2	1	no	—
22	#11765849	—	2	1	no	—
23	C₂×C₇₁⋊C₈	—	2	1	no	—
24	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₇₁⋊C₈	Q₈	1	1	yes	characteristic
29	C₂×C₇₁⋊C₈	Q₈	1	1	yes	characteristic
30	C₄⋊Dic₇₁	—	2	1	no	—
31	C₄⋊Dic₇₁	—	2	1	no	—
32	C₂²⋊Dic₇₁	—	2	1	no	—
33	C₂²⋊Dic₇₁	—	2	1	no	—
34	C₄×Dic₇₁	—	2	1	no	—
35	C₄×Dic₇₁	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
37	M₁₆×C₇₁	—	2	1	no	—
38	M₁₆×C₇₁	—	2	1	no	—
39	M₁₆×C₇₁	—	2	1	no	—
40	M₁₆×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	—	2	1	no	—
42	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₇₁	—	2	1	no	—
44	C₂²⋊C₄×C₇₁	—	2	1	no	—
45	C₄²×C₇₁	—	2	1	no	—
46	C₂³×C₇₁	#41	1	1	yes	characteristic
47	C₂×C₄×C₇₁	—	4	1	no	—
48	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂×D₄	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	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
54	C₂×Dic₇₁	—	2	2	no	—
55	C₂×Dic₇₁	—	4	1	no	—
56	C₈×C₇₁	—	2	2	no	—
57	C₈×C₇₁	—	4	1	no	—
58	C₇₁⋊C₈	—	4	1	no	—
59	C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
60	C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
61	C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₇₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₇₁	—	4	1	no	—
64	C₂²×C₇₁	#117	1	1	yes	characteristic
65	C₂²×C₇₁	#124	1	1	yes	characteristic
66	C₂²×C₇₁	#125	1	1	yes	characteristic
67	C₄×C₇₁	—	8	1	no	—
68	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₄×C₇₁	#117	1	1	yes	characteristic
70	C₄×C₇₁	#124	1	1	yes	characteristic
71	C₄×C₇₁	#125	1	1	yes	characteristic
72	Dic₇₁	—	4	2	no	—
73	Dic₇₁	—	8	1	no	—
74	C₂×C₇₁	—	2	1	no	—
75	C₂×C₇₁	—	2	1	no	—
76	C₂×C₇₁	—	2	1	no	—
77	C₂×C₇₁	#382	1	1	yes	characteristic
78	#1833	—	71	1	no	maximal, Sylow
79	C₇₁	#1833	1	1	yes	characteristic, complemented, Sylow
80	C₂×#108	—	71	1	no	—
81	#427	—	71	1	no	—
82	#405	—	71	1	no	—
83	#344	—	71	1	no	—
84	#344	D₇₁	1	1	yes	characteristic
85	H(3,ℤ/4)	—	71	1	no	—
86	H(3,ℤ/4)	—	71	1	no	—
87	C₂×M₁₆	—	71	1	no	—
88	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
89	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
90	C₂²×C₈	—	71	1	no	—
91	#117	—	71	1	no	—
92	#117	—	71	1	no	—
93	#117	C₂×D₇₁	1	1	yes	characteristic
94	#108	—	142	1	no	—
95	#108	—	142	1	no	—
96	#107	—	142	1	no	—
97	#106	—	142	1	no	—
98	#97	—	71	2	no	—
99	C₄×C₈	—	71	2	no	—
100	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
101	M₁₆	—	2	1	no	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	2	1	no	—
104	M₁₆	—	2	1	no	—
105	M₁₆	—	142	1	no	—
106	M₁₆	—	142	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	71	1	no	—
109	C₂×C₈	—	71	1	no	—
110	C₂×C₈	—	71	1	no	—
111	C₂×C₈	—	71	1	no	—
112	C₂×C₈	—	71	1	no	—
113	C₂×C₈	—	71	1	no	—
114	C₂×C₈	—	142	1	no	—
115	C₂×C₈	C₄×D₇₁	1	2	yes	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	142	1	no	—
118	C₄⋊C₄	—	142	1	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	142	1	no	—
121	C₂²⋊C₄	—	142	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	71	2	no	—
124	C₄²	—	142	1	no	—
125	C₂³	#11765865	1	1	yes	characteristic
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	284	1	no	—
128	C₂×C₄	—	142	2	no	—
129	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
130	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
131	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
132	C₂×C₄	#11765864	1	1	yes	characteristic
133	C₂×C₄	#11765864	1	1	yes	characteristic
134	C₂×C₄	#11765865	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	4	1	no	—
137	C₈	—	284	1	no	—
138	C₈	—	71	1	no	—
139	C₈	—	71	1	no	—
140	C₈	—	71	1	no	—
141	C₈	—	71	1	no	—
142	C₂²	—	4	1	no	—
143	C₂²	#14556085	1	1	yes	characteristic
144	C₂²	#14556114	1	1	yes	characteristic
145	C₂²	#14556118	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	284	2	no	—
148	C₄	—	568	1	no	—
149	C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
150	C₄	#14556084	1	1	yes	characteristic
151	C₄	#14556114	1	1	yes	characteristic
152	C₄	#14556118	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#17985863	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556063

C₂

yes

maximal, characteristic

#17986035

C₂

yes

maximal, characteristic

#17985957

C₂

yes

maximal, characteristic

#17985770

C₂

yes

maximal, characteristic

#17985756

C₂

yes

maximal, characteristic

#17985755

C₂

yes

maximal, characteristic

#344×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

#14556087

C₂²

yes

characteristic

#14556087

C₂²

yes

—

—

—

—

C₄

yes

complemented

C₈×Dic₇₁

C₄

yes

complemented

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#117×C₇₁

C₂²

yes

characteristic

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₇₁⋊C₈

Q₈

yes

characteristic

C₂×C₇₁⋊C₈

Q₈

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

#41

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₇₁⋊C₈

C₄⋊C₄

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₄

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

—

D₇₁

yes

characteristic

H(3,ℤ/4)

—

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

#117

—

#117

—

#117

C₂×D₇₁

yes

characteristic

#108

—

142

—

#108

—

142

—

#107

—

142

—

#106

—

142

—

#97

—

C₄×C₈

—

100

C₂²×C₄

C₂²×D₇₁

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

142

—

106

M₁₆

—

142

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

142

—

115

C₂×C₈

C₄×D₇₁

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

142

—

118

C₄⋊C₄

—

142

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

142

—

121

C₂²⋊C₄

—

142

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

142

—

125

C₂³

#11765865

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

284

—

128

C₂×C₄

—

142

—

129

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

130

C₂×C₄

D₄×D₇₁

yes

characteristic

131

C₂×C₄

Q₈×D₇₁

yes

characteristic

132

C₂×C₄

#11765864

yes

characteristic

133

C₂×C₄

#11765864

yes

characteristic

134

C₂×C₄

#11765865

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

284

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₂²

—

143

C₂²

#14556085

yes

characteristic

144

C₂²

#14556114

yes

characteristic

145

C₂²

#14556118

yes

characteristic

146

C₄

—

147

C₄

—

284

—

148

C₄

—

568

—

149

C₄

C₄⋊C₄×D₇₁

yes

characteristic

150

C₄

#14556084

yes

characteristic

151

C₄

#14556114

yes

characteristic

152

C₄

#14556118

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#17985863

yes

characteristic

	subgroup	G/H
center	C₄	#14556084
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#344×C₇₁	C₂
socle	C₂×C₇₁	#382
hypercenter	#344	D₇₁
hypocenter	C₇₁	#1833
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

subgroup

G/H

center

C₄

#14556084

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#344×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #344

N	H ≅ G/N	type
16. #14556038	C₄	semidirect
17. C₈×Dic₇₁	C₄	semidirect
79. C₇₁	#1833	semidirect

H ≅ G/N

type

16. #14556038

C₄

semidirect

17. C₈×Dic₇₁

C₄

semidirect

79. C₇₁

#1833

semidirect

Subgroups — #22588461

Conjugacy classes of subgroups

Complements