Group #22589902 — subgroups

Subgroups — #22589902

Order 9088.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#17985897	C₂	1	1	yes	maximal, characteristic
2	#17985761	C₂	1	2	yes	maximal
3	#17985761	C₂	1	2	yes	maximal
4	#17985754	C₂	1	1	yes	maximal, characteristic
5	#384×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₇₁⋊C₄	—	2	1	no	—
10	C₇₁⋊#95	—	2	1	no	—
11	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
12	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
13	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₇₁	—	2	1	no	—
16	C₂²×Dic₇₁	—	2	1	no	—
17	C₂×C₇₁⋊C₈	—	2	2	no	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₄⋊Dic₇₁	—	2	1	no	—
21	C₄⋊Dic₇₁	—	2	2	no	—
22	C₄⋊Dic₇₁	Q₈	1	2	yes	—
23	Dic₇₁⋊C₄	—	4	2	no	—
24	C₂²×C₄×C₇₁	—	2	2	no	—
25	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₇₁	—	2	4	no	—
29	C₄⋊C₄×C₇₁	—	4	2	no	—
30	C₄⋊C₄×C₇₁	—	4	2	no	—
31	C₄²×C₇₁	—	2	2	no	—
32	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	—	2	1	no	—
34	C₂×C₄×C₇₁	—	2	1	no	—
35	C₂×C₄×C₇₁	—	2	1	no	—
36	C₂×C₄×C₇₁	—	2	1	no	—
37	C₂×C₄×C₇₁	—	4	2	no	—
38	C₂×C₄×C₇₁	—	4	2	no	—
39	C₂×C₄×C₇₁	—	4	2	no	—
40	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
41	C₂×C₄×C₇₁	Q₁₆	1	2	yes	—
42	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₇₁	#41	1	2	yes	—
44	C₂×C₄×C₇₁	#41	1	2	yes	—
45	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
46	C₂×Dic₇₁	—	2	2	no	—
47	C₂×Dic₇₁	—	4	1	no	—
48	C₂×Dic₇₁	—	4	1	no	—
49	C₂×Dic₇₁	—	4	1	no	—
50	C₂×Dic₇₁	—	4	1	no	—
51	C₂×Dic₇₁	—	4	1	no	—
52	C₇₁⋊C₈	—	4	2	no	—
53	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₇₁	#123	1	1	yes	characteristic
55	C₂²×C₇₁	#124	1	1	yes	characteristic
56	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂²×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₄×C₇₁	—	2	2	no	—
61	C₄×C₇₁	—	2	2	no	—
62	C₄×C₇₁	—	8	2	no	—
63	C₄×C₇₁	#125	1	2	yes	—
64	C₄×C₇₁	#126	1	2	yes	—
65	Dic₇₁	—	4	2	no	—
66	Dic₇₁	—	8	1	no	—
67	C₂×C₇₁	#400	1	1	yes	characteristic
68	C₂×C₇₁	#476	1	1	yes	characteristic
69	C₂×C₇₁	#477	1	1	yes	characteristic
70	C₂×C₇₁	#481	1	1	yes	characteristic
71	C₂×C₇₁	#484	1	1	yes	characteristic
72	C₂×C₇₁	#487	1	1	yes	characteristic
73	C₂×C₇₁	#490	1	1	yes	characteristic
74	#2100	—	71	1	no	maximal, Sylow
75	C₇₁	#2100	1	1	yes	characteristic, complemented, Sylow
76	#389	—	71	1	no	—
77	#384	D₇₁	1	1	yes	characteristic
78	#340	—	71	2	no	—
79	#340	—	71	2	no	—
80	#336	—	71	1	no	—
81	C₂²×C₈	—	71	2	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	71	1	no	—
85	C₂×C₄⋊C₄	—	71	1	no	—
86	C₂×C₄⋊C₄	—	142	1	no	—
87	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
88	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
89	#95	—	142	1	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	142	1	no	—
92	C₂²×C₄	—	142	1	no	—
93	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
96	C₂×C₈	—	142	2	no	—
97	C₂×C₈	—	142	2	no	—
98	C₂×C₈	—	142	2	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	284	2	no	—
103	C₄⋊C₄	—	71	2	no	—
104	C₄⋊C₄	—	142	2	no	—
105	C₄⋊C₄	—	142	1	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	284	1	no	—
116	C₂×C₄	—	284	1	no	—
117	C₂×C₄	—	284	1	no	—
118	C₂×C₄	—	284	1	no	—
119	C₂×C₄	—	284	1	no	—
120	C₂×C₄	—	142	2	no	—
121	C₂×C₄	#11765852	1	2	yes	—
122	C₂×C₄	#11765855	1	2	yes	—
123	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
124	C₂×C₄	#11765864	1	2	yes	—
125	C₂×C₄	#11765865	1	2	yes	—
126	C₂×C₄	#11765866	1	1	yes	characteristic
127	C₈	—	284	2	no	—
128	C₂²	#14556104	1	1	yes	characteristic
129	C₂²	#14556111	1	1	yes	characteristic
130	C₂²	#14556115	1	1	yes	characteristic
131	C₂²	C₂×#11765852	1	1	yes	characteristic
132	C₂²	C₂×#11765855	1	1	yes	characteristic
133	C₂²	#14556154	1	1	yes	characteristic
134	C₂²	#14556154	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	8	2	no	—
138	C₄	—	284	2	no	—
139	C₄	—	568	1	no	—
140	C₄	#14556118	1	2	yes	—
141	C₄	#14556123	1	2	yes	—
142	C₂	#17985937	1	1	yes	characteristic
143	C₂	#17986286	1	1	yes	characteristic
144	C₂	#17986291	1	1	yes	characteristic
145	C₂	#17986317	1	1	yes	characteristic
146	C₂	#17986334	1	1	yes	characteristic
147	C₂	#17986349	1	1	yes	characteristic
148	C₂	#17986363	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17985897

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal

#17985761

C₂

yes

maximal

#17985754

C₂

yes

maximal, characteristic

#384×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

characteristic

C₂×Dic₇₁⋊C₄

—

C₇₁⋊#95

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

Q₈

yes

—

Dic₇₁⋊C₄

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

Q₁₆

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂²⋊Q₈

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

C₂×SD₁₆

yes

characteristic

C₂²×C₇₁

C₂×Q₁₆

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#125

yes

—

C₄×C₇₁

#126

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#389

—

#384

D₇₁

yes

—

—

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#95

—

142

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

284

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

142

—

105

C₄⋊C₄

—

142

—

106

C₄²

—

107

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

284

—

116

C₂×C₄

—

284

—

117

C₂×C₄

—

284

—

118

C₂×C₄

—

284

—

119

C₂×C₄

—

284

—

120

C₂×C₄

—

142

—

121

C₂×C₄

#11765852

yes

—

122

C₂×C₄

#11765855

yes

—

123

C₂×C₄

Q₈×D₇₁

yes

characteristic

124

C₂×C₄

#11765864

yes

—

125

C₂×C₄

#11765865

yes

—

126

C₂×C₄

#11765866

yes

characteristic

127

C₈

—

284

—

128

C₂²

#14556104

yes

characteristic

129

C₂²

#14556111

yes

characteristic

130

C₂²

#14556115

yes

characteristic

131

C₂²

C₂×#11765852

yes

characteristic

132

C₂²

C₂×#11765855

yes

characteristic

133

C₂²

#14556154

yes

characteristic

134

C₂²

#14556154

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

284

—

139

C₄

—

568

—

140

C₄

#14556118

yes

—

141

C₄

#14556123

yes

—

142

C₂

#17985937

yes

characteristic

143

C₂

#17986286

yes

characteristic

144

C₂

#17986291

yes

characteristic

145

C₂

#17986317

yes

characteristic

146

C₂

#17986334

yes

characteristic

147

C₂

#17986349

yes

characteristic

148

C₂

#17986363

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₁
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#384×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	#384	D₇₁
hypocenter	C₇₁	#2100
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₁

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#384×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
75. C₇₁	#2100	semidirect

H ≅ G/N

type

75. C₇₁

#2100

semidirect

Subgroups — #22589902

Conjugacy classes of subgroups

Complements