Group #22587867 — subgroups

Subgroups — #22587867

Order 9088.

140 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 51 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	#17986402	C₂	1	1	yes	maximal, characteristic
2	#17986295	C₂	1	1	yes	maximal, characteristic
3	#17986291	C₂	1	1	yes	maximal, characteristic
4	#17986126	C₂	1	1	yes	maximal, characteristic
5	#17985749	C₂	1	1	yes	maximal, characteristic
6	#17985747	C₂	1	1	yes	maximal, characteristic
7	#333×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#14556128	C₂²	1	1	yes	characteristic
9	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
10	#14556062	—	2	1	no	—
11	#14556061	—	2	1	no	—
12	#14556061	—	2	1	no	—
13	#14556061	—	2	1	no	—
14	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₇₁⋊C₈	—	2	1	no	—
16	#14556053	—	2	1	no	—
17	#14556051	—	2	1	no	—
18	#14556038	—	2	1	no	—
19	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×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	D₄	1	1	yes	characteristic
25	C₂×C₇₁⋊C₈	—	2	1	no	—
26	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
28	C₄⋊Dic₇₁	—	2	1	no	—
29	C₄⋊Dic₇₁	—	2	1	no	—
30	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
31	Dic₇₁⋊C₄	—	2	1	no	—
32	C₄×Dic₇₁	—	2	1	no	—
33	C₂×Q₈×C₇₁	—	2	1	no	—
34	C₂×C₈×C₇₁	—	2	1	no	—
35	C₂×C₈×C₇₁	—	2	1	no	—
36	C₄⋊C₄×C₇₁	—	2	1	no	—
37	C₄⋊C₄×C₇₁	—	2	1	no	—
38	C₄²×C₇₁	C₂³	1	1	yes	characteristic
39	Q₈×C₇₁	—	8	1	no	—
40	C₂×C₄×C₇₁	—	4	1	no	—
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
44	C₂×Dic₇₁	—	2	1	no	—
45	C₂×Dic₇₁	—	2	1	no	—
46	C₂×Dic₇₁	—	4	1	no	—
47	Dic₁₄₂	—	2	1	no	—
48	Dic₁₄₂	—	2	1	no	—
49	Dic₁₄₂	—	8	1	no	—
50	C₈×C₇₁	—	4	1	no	—
51	C₈×C₇₁	—	4	1	no	—
52	C₇₁⋊C₈	—	2	1	no	—
53	C₇₁⋊C₈	—	4	1	no	—
54	C₇₁⋊C₈	Q₁₆	1	2	yes	—
55	C₂²×C₇₁	#121	1	1	yes	characteristic
56	C₄×C₇₁	—	2	1	no	—
57	C₄×C₇₁	—	8	1	no	—
58	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
59	C₄×C₇₁	#137	1	1	yes	characteristic
60	C₄×C₇₁	#137	1	1	yes	characteristic
61	C₄×C₇₁	#137	1	1	yes	characteristic
62	Dic₇₁	—	4	1	no	—
63	Dic₇₁	—	4	1	no	—
64	Dic₇₁	—	8	1	no	—
65	C₂×C₇₁	#462	1	1	yes	characteristic
66	C₂×C₇₁	#470	1	1	yes	characteristic
67	C₂×C₇₁	#472	1	1	yes	characteristic
68	#1696	—	71	1	no	maximal, Sylow
69	C₇₁	#1696	1	1	yes	characteristic, complemented, Sylow
70	#500	—	71	1	no	—
71	#477	—	71	1	no	—
72	#477	—	71	1	no	—
73	#446	—	71	1	no	—
74	#334	—	71	1	no	—
75	#333	—	71	1	no	—
76	#333	D₇₁	1	1	yes	characteristic
77	C₄⋊Q₈	—	71	1	no	—
78	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
79	C₄×Q₈	—	71	1	no	—
80	#107	—	142	1	no	—
81	#107	—	142	1	no	—
82	#107	—	142	1	no	—
83	#107	—	142	1	no	—
84	C₄⋊C₈	—	71	1	no	—
85	C₄⋊C₈	—	142	1	no	—
86	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
87	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
88	#103	—	142	1	no	—
89	#103	—	142	1	no	—
90	#97	—	142	1	no	—
91	C₄×C₈	—	71	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	71	1	no	—
94	C₂×Q₈	—	142	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	71	1	no	—
98	C₂×C₈	—	71	1	no	—
99	C₂×C₈	—	142	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	71	1	no	—
103	C₄⋊C₄	—	142	1	no	—
104	C₄⋊C₄	—	142	1	no	—
105	C₄⋊C₄	—	142	1	no	—
106	C₄²	—	142	1	no	—
107	C₄²	C₂²×D₇₁	1	1	yes	characteristic
108	Q₈	—	8	1	no	—
109	Q₈	—	568	1	no	—
110	Q₈	—	142	1	no	—
111	Q₈	—	142	1	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	284	1	no	—
114	C₂×C₄	—	142	1	no	—
115	C₂×C₄	—	142	1	no	—
116	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
117	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
118	C₂×C₄	#11765865	1	1	yes	characteristic
119	C₈	—	4	1	no	—
120	C₈	—	4	1	no	—
121	C₈	—	284	1	no	—
122	C₈	—	71	2	no	—
123	C₈	—	142	1	no	—
124	C₂²	#14556101	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	284	1	no	—
128	C₄	—	284	1	no	—
129	C₄	—	568	1	no	—
130	C₄	Q₁₆×D₇₁	1	1	yes	characteristic
131	C₄	#14556157	1	1	yes	characteristic
132	C₄	#14556157	1	1	yes	characteristic
133	C₄	#14556160	1	1	yes	characteristic
134	C₂	#17986208	1	1	yes	characteristic
135	C₂	#17986256	1	1	yes	characteristic
136	C₂	#17986268	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986402

C₂

yes

maximal, characteristic

#17986295

C₂

yes

maximal, characteristic

#17986291

C₂

yes

maximal, characteristic

#17986126

C₂

yes

maximal, characteristic

#17985749

C₂

yes

maximal, characteristic

#17985747

C₂

yes

maximal, characteristic

#333×C₇₁

C₂

yes

maximal, characteristic

#14556128

C₂²

yes

characteristic

C₄×Dic₁₄₂

C₂²

yes

—

—

—

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

Dic₇₁⋊C₈

—

#14556053

—

#14556051

—

#14556038

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

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

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

Q₁₆

yes

—

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×Q₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

C₄×Q₈

—

#107

—

142

—

#107

—

142

—

#107

—

142

—

#107

—

142

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#103

—

142

—

#103

—

142

—

#97

—

142

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

142

—

104

C₄⋊C₄

—

142

—

105

C₄⋊C₄

—

142

—

106

C₄²

—

142

—

107

C₄²

C₂²×D₇₁

yes

characteristic

108

Q₈

—

109

Q₈

—

568

—

110

Q₈

—

142

—

111

Q₈

—

142

—

112

C₂×C₄

—

113

C₂×C₄

—

284

—

114

C₂×C₄

—

142

—

115

C₂×C₄

—

142

—

116

C₂×C₄

D₄×D₇₁

yes

characteristic

117

C₂×C₄

D₄×D₇₁

yes

characteristic

118

C₂×C₄

#11765865

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

284

—

122

C₈

—

123

C₈

—

142

—

124

C₂²

#14556101

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

284

—

128

C₄

—

284

—

129

C₄

—

568

—

130

C₄

Q₁₆×D₇₁

yes

characteristic

131

C₄

#14556157

yes

characteristic

132

C₄

#14556157

yes

characteristic

133

C₄

#14556160

yes

characteristic

134

C₂

#17986208

yes

characteristic

135

C₂

#17986256

yes

characteristic

136

C₂

#17986268

yes

characteristic

	subgroup	G/H
center	C₂²	#14556101
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#333×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#333	D₇₁
hypocenter	C₇₁	#1696
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#14556101

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#333×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
69. C₇₁	#1696	semidirect

H ≅ G/N

type

69. C₇₁

#1696

semidirect

Subgroups — #22587867

Conjugacy classes of subgroups

Complements