Group #22601229 — subgroups

Subgroups — #22601229

Order 9088.

324 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 133 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	#17986728	C₂	1	2	yes	maximal
2	#17986404	C₂	1	1	yes	maximal, characteristic
3	#17986400	C₂	1	2	yes	maximal
4	#17986308	C₂	1	4	yes	maximal
5	#17986284	C₂	1	2	yes	maximal
6	C₈×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
7	Q₁₆×Dic₇₁	C₂	1	2	yes	maximal
8	C₄×Q₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
9	#14556128	—	2	2	no	—
10	#14556128	C₂²	1	2	yes	—
11	#14556119	—	2	4	no	—
12	#14556119	C₂²	1	4	yes	—
13	Q₈×Dic₇₁	C₂²	1	4	yes	—
14	C₄×Dic₁₄₂	—	2	2	no	—
15	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
16	#14556062	C₂²	1	2	yes	—
17	#14556062	C₂²	1	2	yes	—
18	#14556059	C₂²	1	4	yes	—
19	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₇₁⋊C₈	C₂²	1	2	yes	—
21	#14556052	C₂²	1	4	yes	—
22	C₈×Dic₇₁	C₂²	1	2	yes	—
23	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₇₁	C₂²	1	2	yes	—
25	#107×C₇₁	C₂²	1	1	yes	characteristic
26	#103×C₇₁	C₂²	1	2	yes	—
27	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₄₂	—	2	2	no	—
29	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
30	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
31	C₄⋊Dic₇₁	—	2	2	no	—
32	C₄⋊Dic₇₁	—	2	4	no	—
33	C₄⋊Dic₇₁	—	2	4	no	—
34	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
35	C₄⋊Dic₇₁	C₂³	1	4	yes	—
36	Dic₇₁⋊C₄	—	2	4	no	—
37	Dic₇₁⋊C₄	D₄	1	2	yes	—
38	C₄×Dic₇₁	—	2	4	no	—
39	C₄×Dic₇₁	C₂³	1	2	yes	—
40	C₂×Q₈×C₇₁	C₂³	1	2	yes	—
41	Q₁₆×C₇₁	Q₈	1	4	yes	complemented
42	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₄×C₇₁	—	2	2	no	—
45	C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
46	C₄²×C₇₁	—	2	2	no	—
47	C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	Q₈×C₇₁	—	2	2	no	—
49	Q₈×C₇₁	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₇₁	—	2	4	no	—
56	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
57	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
58	Dic₁₄₂	—	2	2	no	—
59	Dic₁₄₂	—	4	4	no	—
60	C₈×C₇₁	—	2	1	no	—
61	C₈×C₇₁	C₂×Q₈	1	2	yes	—
62	C₇₁⋊C₈	—	2	2	no	—
63	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₇₁	—	2	4	no	—
65	C₄×C₇₁	—	2	1	no	—
66	C₄×C₇₁	—	4	2	no	—
67	C₄×C₇₁	#135	1	2	yes	—
68	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₇₁	—	2	2	no	—
71	Dic₇₁	—	2	2	no	—
72	Dic₇₁	—	4	4	no	—
73	C₂×C₇₁	D₄×Q₈	1	1	yes	characteristic
74	C₂×C₇₁	C₂×#135	1	1	yes	characteristic
75	C₂×C₇₁	#578	1	1	yes	characteristic
76	#3383	—	71	1	no	maximal, Sylow
77	C₇₁	#3383	1	1	yes	characteristic, complemented, Sylow
78	#557	—	71	2	no	—
79	#500	—	71	1	no	—
80	#499	—	71	2	no	—
81	#479	—	71	4	no	—
82	#475	—	71	2	no	—
83	C₈×Q₈	—	71	1	no	—
84	C₄×Q₁₆	—	71	2	no	—
85	C₄×Q₁₆	D₇₁	1	1	yes	characteristic
86	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
87	C₄⋊Q₈	—	71	2	no	—
88	C₄⋊Q₈	—	142	2	no	—
89	#125	—	71	4	no	—
90	#125	—	142	4	no	—
91	C₄×Q₈	—	71	1	no	—
92	C₄×Q₈	—	71	4	no	—
93	C₄×Q₈	—	142	2	no	—
94	C₄×Q₈	C₂×D₇₁	1	2	yes	—
95	#107	—	71	2	no	—
96	#107	—	71	2	no	—
97	#107	C₂×D₇₁	1	1	yes	characteristic
98	#106	—	71	4	no	—
99	C₄⋊C₈	—	71	2	no	—
100	C₄⋊C₈	—	71	1	no	—
101	#103	—	71	4	no	—
102	#103	C₂×D₇₁	1	2	yes	—
103	C₄×C₈	—	71	2	no	—
104	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
105	C₂×Q₈	—	71	1	no	—
106	C₂×Q₈	—	142	2	no	—
107	C₂×Q₈	C₂²×D₇₁	1	2	yes	—
108	Q₁₆	Dic₁₄₂	1	4	yes	complemented
109	C₂×C₈	—	71	2	no	—
110	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	71	2	no	—
114	C₄⋊C₄	—	71	1	no	—
115	C₄⋊C₄	—	71	4	no	—
116	C₄⋊C₄	—	142	4	no	—
117	C₄⋊C₄	—	142	4	no	—
118	C₄⋊C₄	—	142	4	no	—
119	C₄⋊C₄	—	142	2	no	—
120	C₄⋊C₄	C₂²×D₇₁	1	2	yes	—
121	C₄²	—	2	2	no	—
122	C₄²	—	71	2	no	—
123	C₄²	—	142	4	no	—
124	C₄²	C₂²×D₇₁	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	284	4	no	—
127	Q₈	—	142	2	no	—
128	Q₈	C₂×Dic₁₄₂	1	4	yes	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	71	2	no	—
132	C₂×C₄	—	71	2	no	—
133	C₂×C₄	—	142	4	no	—
134	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
135	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	142	2	no	—
139	C₈	C₂×Dic₁₄₂	1	2	yes	—
140	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	2	1	no	—
143	C₄	—	4	2	no	—
144	C₄	—	284	4	no	—
145	C₄	—	142	2	no	—
146	C₄	—	142	2	no	—
147	C₄	#14556151	1	2	yes	—
148	C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
149	C₄	#14556175	1	1	yes	characteristic
150	C₂	D₄×Dic₁₄₂	1	1	yes	characteristic
151	C₂	C₂×#14556151	1	1	yes	characteristic
152	C₂	#17986813	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986728

C₂

yes

maximal

#17986404

C₂

yes

maximal, characteristic

#17986400

C₂

yes

maximal

#17986308

C₂

yes

maximal

#17986284

C₂

yes

maximal

C₈×Dic₁₄₂

C₂

yes

maximal, characteristic

Q₁₆×Dic₇₁

C₂

yes

maximal

C₄×Q₁₆×C₇₁

C₂

yes

maximal, characteristic

#14556128

—

#14556128

C₂²

yes

—

#14556119

—

#14556119

C₂²

yes

—

Q₈×Dic₇₁

C₂²

yes

—

C₄×Dic₁₄₂

—

C₄×Dic₁₄₂

C₂²

yes

characteristic

#14556062

C₂²

yes

—

#14556062

C₂²

yes

—

#14556059

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

Dic₇₁⋊C₈

C₂²

yes

—

#14556052

C₂²

yes

—

C₈×Dic₇₁

C₂²

yes

—

C₂×Q₁₆×C₇₁

C₂²

yes

characteristic

C₄×Q₈×C₇₁

C₂²

yes

—

#107×C₇₁

C₂²

yes

characteristic

#103×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

D₄

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₂³

yes

—

C₂×Q₈×C₇₁

C₂³

yes

—

Q₁₆×C₇₁

Q₈

yes

complemented

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂×D₄

yes

—

C₂×Dic₇₁

C₂×D₄

yes

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×Q₈

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#135

yes

—

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

2₋¹⁺⁴

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

D₄×Q₈

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

D₇₁

yes

characteristic

C₂×Q₁₆

C₂×D₇₁

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

#125

—

#125

—

142

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

142

—

C₄×Q₈

C₂×D₇₁

yes

—

#107

—

#107

—

#107

C₂×D₇₁

yes

characteristic

#106

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#103

—

102

#103

C₂×D₇₁

yes

—

103

C₄×C₈

—

104

C₄×C₈

C₂×D₇₁

yes

characteristic

105

C₂×Q₈

—

106

C₂×Q₈

—

142

—

107

C₂×Q₈

C₂²×D₇₁

yes

—

108

Q₁₆

Dic₁₄₂

yes

complemented

109

C₂×C₈

—

110

C₂×C₈

C₂²×D₇₁

yes

characteristic

111

C₂×C₈

C₂²×D₇₁

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

142

—

117

C₄⋊C₄

—

142

—

118

C₄⋊C₄

—

142

—

119

C₄⋊C₄

—

142

—

120

C₄⋊C₄

C₂²×D₇₁

yes

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

142

—

124

C₄²

C₂²×D₇₁

yes

characteristic

125

Q₈

—

126

Q₈

—

284

—

127

Q₈

—

142

—

128

Q₈

C₂×Dic₁₄₂

yes

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

142

—

134

C₂×C₄

D₄×D₇₁

yes

characteristic

135

C₂×C₄

D₄×D₇₁

yes

characteristic

136

C₂×C₄

C₂³×D₇₁

yes

characteristic

137

C₈

—

138

C₈

—

142

—

139

C₈

C₂×Dic₁₄₂

yes

—

140

C₂²

C₂×D₄×D₇₁

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

284

—

145

C₄

—

142

—

146

C₄

—

142

—

147

C₄

#14556151

yes

—

148

C₄

C₂²×Dic₁₄₂

yes

characteristic

149

C₄

#14556175

yes

characteristic

150

C₂

D₄×Dic₁₄₂

yes

characteristic

151

C₂

C₂×#14556151

yes

characteristic

152

C₂

#17986813

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	C₄×Q₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	C₄×Q₁₆	D₇₁
hypocenter	C₇₁	#3383
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

C₄×Q₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

hypercenter

C₄×Q₁₆

D₇₁

hypocenter

C₇₁

#3383

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
41. Q₁₆×C₇₁	Q₈	semidirect
77. C₇₁	#3383	semidirect
108. Q₁₆	Dic₁₄₂	semidirect

H ≅ G/N

type

41. Q₁₆×C₇₁

Q₈

semidirect

77. C₇₁

#3383

semidirect

108. Q₁₆

Dic₁₄₂

semidirect

Subgroups — #22601229

Conjugacy classes of subgroups

Complements