Group #22587772 — subgroups

Subgroups — #22587772

Order 9088.

150 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 55 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	#17986351	C₂	1	1	yes	maximal, characteristic
2	#17986284	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#17985749	C₂	1	1	yes	maximal, characteristic
5	#17985745	C₂	1	1	yes	maximal, characteristic
6	#17985744	C₂	1	1	yes	maximal, characteristic
7	#462×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#14556128	C₂²	1	1	yes	characteristic
9	#14556059	—	2	1	no	—
10	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
11	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
13	#14556052	—	2	1	no	—
14	#14556051	—	2	1	no	—
15	#14556051	—	2	1	no	—
16	C₄×C₇₁⋊C₈	—	2	1	no	—
17	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₇₁	—	2	1	no	—
19	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
22	#103×C₇₁	—	2	1	no	—
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×C₇₁⋊C₈	—	2	1	no	—
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	C₂×Q₈×C₇₁	—	2	1	no	—
31	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
32	Q₁₆×C₇₁	—	4	2	no	—
33	C₂×C₈×C₇₁	—	2	1	no	—
34	C₄⋊C₄×C₇₁	—	2	1	no	—
35	C₄⋊C₄×C₇₁	—	2	1	no	—
36	C₄⋊C₄×C₇₁	—	2	1	no	—
37	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
38	C₄²×C₇₁	—	2	1	no	—
39	C₄²×C₇₁	C₂³	1	1	yes	characteristic
40	Q₈×C₇₁	—	2	1	no	—
41	Q₈×C₇₁	—	4	1	no	—
42	Q₈×C₇₁	—	4	1	no	—
43	Q₈×C₇₁	SD₁₆	1	2	yes	—
44	C₂×C₄×C₇₁	—	2	1	no	—
45	C₂×C₄×C₇₁	—	2	1	no	—
46	C₂×C₄×C₇₁	—	4	1	no	—
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
50	C₂×Dic₇₁	—	4	1	no	—
51	Dic₁₄₂	—	4	1	no	—
52	Dic₁₄₂	—	4	1	no	—
53	C₈×C₇₁	—	4	1	no	—
54	C₇₁⋊C₈	—	2	1	no	—
55	C₇₁⋊C₈	—	2	1	no	—
56	C₇₁⋊C₈	—	4	1	no	—
57	C₇₁⋊C₈	—	4	1	no	—
58	C₂²×C₇₁	#121	1	1	yes	characteristic
59	C₄×C₇₁	—	2	2	no	—
60	C₄×C₇₁	—	2	1	no	—
61	C₄×C₇₁	—	4	1	no	—
62	C₄×C₇₁	—	8	1	no	—
63	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₄×C₇₁	#135	1	1	yes	characteristic
65	C₄×C₇₁	#135	1	1	yes	characteristic
66	C₄×C₇₁	#137	1	1	yes	characteristic
67	Dic₇₁	—	8	1	no	—
68	C₂×C₇₁	#464	1	1	yes	characteristic
69	C₂×C₇₁	#465	1	1	yes	characteristic
70	C₂×C₇₁	#473	1	1	yes	characteristic
71	#1680	—	71	1	no	maximal, Sylow
72	C₇₁	#1680	1	1	yes	characteristic, complemented, Sylow
73	#487	—	71	1	no	—
74	#475	—	71	1	no	—
75	#462	D₇₁	1	1	yes	characteristic
76	C₈×Q₈	—	71	1	no	—
77	#334	—	71	1	no	—
78	#332	—	71	1	no	—
79	#332	—	71	1	no	—
80	C₂×Q₁₆	—	2	1	no	—
81	C₄⋊Q₈	—	71	1	no	—
82	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
83	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
84	#106	—	142	1	no	—
85	C₄⋊C₈	—	71	1	no	—
86	C₄⋊C₈	—	71	1	no	—
87	C₄⋊C₈	—	142	1	no	—
88	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
89	#103	—	2	1	no	—
90	#103	—	142	1	no	—
91	#103	—	142	1	no	—
92	#103	—	142	1	no	—
93	C₄×C₈	—	71	1	no	—
94	C₄×C₈	—	142	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	142	1	no	—
97	C₂×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
98	Q₁₆	—	4	2	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	71	1	no	—
101	C₂×C₈	—	71	1	no	—
102	C₂×C₈	—	142	1	no	—
103	C₂×C₈	—	142	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	142	1	no	—
108	C₄⋊C₄	—	142	1	no	—
109	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
110	C₄²	—	2	1	no	—
111	C₄²	C₂²×D₇₁	1	1	yes	characteristic
112	Q₈	—	2	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	4	1	no	—
115	Q₈	—	284	1	no	—
116	Q₈	—	284	1	no	—
117	Q₈	#11765853	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	284	1	no	—
122	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
124	C₂×C₄	#11765864	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	284	1	no	—
127	C₈	—	284	1	no	—
128	C₈	—	142	1	no	—
129	C₈	—	142	1	no	—
130	C₂²	#14556100	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	568	1	no	—
136	C₄	C₂×#11765853	1	1	yes	characteristic
137	C₄	#14556148	1	1	yes	characteristic
138	C₄	#14556151	1	1	yes	characteristic
139	C₄	#14556160	1	1	yes	characteristic
140	C₂	#17986223	1	1	yes	characteristic
141	C₂	#17986226	1	1	yes	characteristic
142	C₂	#17986272	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986351

C₂

yes

maximal, characteristic

#17986284

C₂

yes

maximal, characteristic

Q₈×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#17985749

C₂

yes

maximal, characteristic

#17985745

C₂

yes

maximal, characteristic

#17985744

C₂

yes

maximal, characteristic

#462×C₇₁

C₂

yes

maximal, characteristic

#14556128

C₂²

yes

characteristic

#14556059

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

—

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

#103×C₇₁

—

C₂×Dic₁₄₂

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

D₄

yes

characteristic

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×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

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#487

—

#475

—

#462

D₇₁

yes

characteristic

C₈×Q₈

—

#334

—

#332

—

#332

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

C₄×Q₈

C₂×D₇₁

yes

characteristic

#106

—

142

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#103

—

#103

—

142

—

#103

—

142

—

#103

—

142

—

C₄×C₈

—

C₄×C₈

—

142

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×Q₈

C₂²⋊D₇₁

yes

characteristic

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

142

—

103

C₂×C₈

—

142

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

142

—

108

C₄⋊C₄

—

142

—

109

C₄⋊C₄

C₂²⋊D₇₁

yes

characteristic

110

C₄²

—

111

C₄²

C₂²×D₇₁

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

284

—

116

Q₈

—

284

—

117

Q₈

#11765853

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

284

—

122

C₂×C₄

D₄×D₇₁

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

124

C₂×C₄

#11765864

yes

characteristic

125

C₈

—

126

C₈

—

284

—

127

C₈

—

284

—

128

C₈

—

142

—

129

C₈

—

142

—

130

C₂²

#14556100

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

568

—

136

C₄

C₂×#11765853

yes

characteristic

137

C₄

#14556148

yes

characteristic

138

C₄

#14556151

yes

characteristic

139

C₄

#14556160

yes

characteristic

140

C₂

#17986223

yes

characteristic

141

C₂

#17986226

yes

characteristic

142

C₂

#17986272

yes

characteristic

	subgroup	G/H
center	C₂²	#14556100
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#462×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#462	D₇₁
hypocenter	C₇₁	#1680
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#14556100

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#462×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
72. C₇₁	#1680	semidirect

H ≅ G/N

type

72. C₇₁

#1680

semidirect

Subgroups — #22587772

Conjugacy classes of subgroups

Complements