Group #22587723 — subgroups

Subgroups — #22587723

Order 9088.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17986397	C₂	1	1	yes	maximal, characteristic
2	#17986295	C₂	1	1	yes	maximal, characteristic
3	#17986210	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
5	#17985747	C₂	1	1	yes	maximal, characteristic
6	#17985746	C₂	1	1	yes	maximal, characteristic
7	#334×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₈₄	—	2	1	no	—
9	#14556128	C₂²	1	1	yes	characteristic
10	#14556128	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
12	#14556062	—	2	1	no	—
13	#14556060	—	2	1	no	—
14	#14556060	—	2	1	no	—
15	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
16	Dic₇₁⋊C₈	—	2	1	no	—
17	#14556053	—	2	1	no	—
18	#14556053	—	2	1	no	—
19	C₈×Dic₇₁	—	2	1	no	—
20	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×Dic₁₄₂	—	2	1	no	—
25	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
26	Dic₂₈₄	—	4	2	no	—
27	C₂×C₇₁⋊C₈	—	2	1	no	—
28	C₄⋊Dic₇₁	—	2	1	no	—
29	C₄⋊Dic₇₁	—	2	1	no	—
30	C₄⋊Dic₇₁	—	2	1	no	—
31	C₄⋊Dic₇₁	—	2	1	no	—
32	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
33	Dic₇₁⋊C₄	—	2	1	no	—
34	C₄×Dic₇₁	—	2	1	no	—
35	C₂×C₈×C₇₁	—	2	1	no	—
36	C₂×C₈×C₇₁	—	2	1	no	—
37	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
38	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
39	C₄²×C₇₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
43	C₂×Dic₇₁	—	2	1	no	—
44	C₂×Dic₇₁	—	2	1	no	—
45	C₂×Dic₇₁	—	4	1	no	—
46	C₂×Dic₇₁	—	4	1	no	—
47	Dic₁₄₂	—	2	1	no	—
48	Dic₁₄₂	—	4	1	no	—
49	Dic₁₄₂	—	4	1	no	—
50	Dic₁₄₂	—	8	1	no	—
51	Dic₁₄₂	SD₁₆	1	2	yes	—
52	C₈×C₇₁	—	2	1	no	—
53	C₈×C₇₁	—	4	1	no	—
54	C₈×C₇₁	—	4	1	no	—
55	C₈×C₇₁	Q₁₆	1	2	yes	complemented
56	C₇₁⋊C₈	—	4	1	no	—
57	C₂²×C₇₁	#121	1	1	yes	characteristic
58	C₄×C₇₁	—	2	1	no	—
59	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
60	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₄×C₇₁	#135	1	1	yes	characteristic
62	C₄×C₇₁	#137	1	1	yes	characteristic
63	Dic₇₁	—	2	2	no	—
64	Dic₇₁	—	4	1	no	—
65	Dic₇₁	—	8	1	no	—
66	Dic₇₁	—	8	1	no	—
67	C₂×C₇₁	#462	1	1	yes	characteristic
68	C₂×C₇₁	#465	1	1	yes	characteristic
69	C₂×C₇₁	#471	1	1	yes	characteristic
70	#1672	—	71	1	no	maximal, Sylow
71	C₇₁	#1672	1	1	yes	characteristic, complemented, Sylow
72	#498	—	71	1	no	—
73	#477	—	71	1	no	—
74	#462	—	71	1	no	—
75	C₈×Q₈	—	71	1	no	—
76	#334	D₇₁	1	1	yes	characteristic
77	#333	—	71	1	no	—
78	#332	—	71	1	no	—
79	C₂×Q₁₆	—	142	1	no	—
80	C₄⋊Q₈	—	71	1	no	—
81	C₄⋊Q₈	—	71	1	no	—
82	C₄×Q₈	—	71	1	no	—
83	#107	—	142	1	no	—
84	#106	—	142	1	no	—
85	#106	—	142	1	no	—
86	C₄⋊C₈	—	71	1	no	—
87	C₄⋊C₈	—	142	1	no	—
88	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
89	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
90	#103	—	142	1	no	—
91	#103	—	142	1	no	—
92	C₄×C₈	—	142	1	no	—
93	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
94	C₂×Q₈	—	71	1	no	—
95	C₂×Q₈	—	142	1	no	—
96	C₂×Q₈	—	142	1	no	—
97	Q₁₆	—	284	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	142	1	no	—
101	C₂×C₈	D₂₈₄	1	1	yes	characteristic
102	C₂×C₈	D₂₈₄	1	1	yes	characteristic
103	C₄⋊C₄	—	71	1	no	—
104	C₄⋊C₄	—	142	1	no	—
105	C₄⋊C₄	—	142	1	no	—
106	C₄⋊C₄	—	142	1	no	—
107	C₄⋊C₄	—	142	1	no	—
108	C₄⋊C₄	—	142	1	no	—
109	C₄²	—	142	1	no	—
110	C₄²	C₂²×D₇₁	1	1	yes	characteristic
111	Q₈	—	284	1	no	—
112	Q₈	—	284	1	no	—
113	Q₈	—	568	1	no	—
114	Q₈	—	71	2	no	—
115	Q₈	—	142	1	no	—
116	C₂×C₄	—	284	1	no	—
117	C₂×C₄	—	284	1	no	—
118	C₂×C₄	—	142	1	no	—
119	C₂×C₄	—	142	1	no	—
120	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
121	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
122	C₂×C₄	#11765866	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	4	1	no	—
126	C₈	—	284	1	no	—
127	C₈	Dic₂₈₄	1	2	yes	complemented
128	C₂²	#14556103	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	284	1	no	—
131	C₄	—	568	1	no	—
132	C₄	—	568	1	no	—
133	C₄	—	142	2	no	—
134	C₄	SD₁₆×D₇₁	1	1	yes	characteristic
135	C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
136	C₄	#14556149	1	1	yes	characteristic
137	C₄	#14556161	1	1	yes	characteristic
138	C₂	#17986210	1	1	yes	characteristic
139	C₂	#17986229	1	1	yes	characteristic
140	C₂	#17986263	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986397

C₂

yes

maximal, characteristic

#17986295

C₂

yes

maximal, characteristic

#17986210

C₂

yes

maximal, characteristic

C₈×Dic₁₄₂

C₂

yes

maximal, characteristic

#17985747

C₂

yes

maximal, characteristic

#17985746

C₂

yes

maximal, characteristic

#334×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

—

#14556128

C₂²

yes

characteristic

#14556128

C₂²

yes

characteristic

C₄×Dic₁₄₂

C₂²

yes

—

—

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

Dic₇₁⋊C₈

—

#14556053

—

#14556053

—

C₈×Dic₇₁

—

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

characteristic

Dic₂₈₄

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

SD₁₆

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

Q₁₆

yes

complemented

C₇₁⋊C₈

—

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

C₂×Q₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#498

—

#477

—

#462

—

C₈×Q₈

—

#334

D₇₁

yes

characteristic

#333

—

#332

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

142

—

#106

—

142

—

#106

—

142

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#103

—

142

—

#103

—

142

—

C₄×C₈

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

Q₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

142

—

101

C₂×C₈

D₂₈₄

yes

characteristic

102

C₂×C₈

D₂₈₄

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

142

—

105

C₄⋊C₄

—

142

—

106

C₄⋊C₄

—

142

—

107

C₄⋊C₄

—

142

—

108

C₄⋊C₄

—

142

—

109

C₄²

—

142

—

110

C₄²

C₂²×D₇₁

yes

characteristic

111

Q₈

—

284

—

112

Q₈

—

284

—

113

Q₈

—

568

—

114

Q₈

—

115

Q₈

—

142

—

116

C₂×C₄

—

284

—

117

C₂×C₄

—

284

—

118

C₂×C₄

—

142

—

119

C₂×C₄

—

142

—

120

C₂×C₄

D₄×D₇₁

yes

characteristic

121

C₂×C₄

C₂×D₂₈₄

yes

characteristic

122

C₂×C₄

#11765866

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

284

—

127

C₈

Dic₂₈₄

yes

complemented

128

C₂²

#14556103

yes

characteristic

129

C₄

—

130

C₄

—

284

—

131

C₄

—

568

—

132

C₄

—

568

—

133

C₄

—

142

—

134

C₄

SD₁₆×D₇₁

yes

characteristic

135

C₄

C₂×Dic₂₈₄

yes

characteristic

136

C₄

#14556149

yes

characteristic

137

C₄

#14556161

yes

characteristic

138

C₂

#17986210

yes

characteristic

139

C₂

#17986229

yes

characteristic

140

C₂

#17986263

yes

characteristic

	subgroup	G/H
center	C₂²	#14556103
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#334×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#334	D₇₁
hypocenter	C₇₁	#1672
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14556103

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#334×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
55. C₈×C₇₁	Q₁₆	semidirect
71. C₇₁	#1672	semidirect
127. C₈	Dic₂₈₄	semidirect

H ≅ G/N

type

55. C₈×C₇₁

Q₁₆

semidirect

71. C₇₁

#1672

semidirect

127. C₈

Dic₂₈₄

semidirect

Subgroups — #22587723

Conjugacy classes of subgroups

Complements