Group #21461182 — subgroups

Subgroups — #21461182

Order 7808.

202 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 60 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	#16840784	C₂	1	1	yes	maximal, characteristic
2	#16840780	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#92420	C₂	1	4	yes	maximal, complemented
4	#16840317	C₂	1	1	yes	maximal, characteristic
5	#14519844	—	2	1	no	—
6	C₄⋊#92420	C₂²	1	2	yes	—
7	#14519834	—	2	1	no	—
8	#14519834	—	2	1	no	—
9	#14519832	—	2	1	no	—
10	C₄×#92420	C₂²	1	2	yes	—
11	#14519777	C₄	1	1	yes	characteristic
12	#14519768	—	2	1	no	—
13	#14519757	—	2	1	no	—
14	#14519753	C₄	1	2	yes	—
15	Q₈×Dic₆₁	C₂²	1	2	yes	—
16	#14519742	—	2	1	no	—
17	#14519735	—	2	1	no	—
18	C₄²×D₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
23	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
24	C₂×Dic₁₂₂	—	2	1	no	—
25	C₂×#92420	—	2	2	no	—
26	C₂×#92420	—	2	2	no	—
27	C₄⋊C₄×C₆₁	—	2	1	no	—
28	C₄⋊C₄×C₆₁	—	2	1	no	—
29	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	#11758198	—	2	2	no	—
31	C₄⋊Dic₆₁	—	2	1	no	—
32	C₄⋊Dic₆₁	—	2	2	no	—
33	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
34	#11758196	—	2	2	no	—
35	C₄×Dic₆₁	—	2	2	no	—
36	C₄×Dic₆₁	D₄	1	1	yes	characteristic
37	C₄×Dic₆₁	D₄	1	1	yes	characteristic
38	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
39	C₂²×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
40	Q₈×C₆₁	—	2	4	no	—
41	C₂×C₄×C₆₁	—	2	2	no	—
42	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₆₁	—	2	2	no	—
46	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
49	C₄×D₆₁	—	2	2	no	—
50	C₄×D₆₁	—	2	2	no	—
51	C₄×D₆₁	—	4	1	no	—
52	Dic₁₂₂	—	4	2	no	—
53	#92420	—	4	2	no	—
54	#92420	—	4	2	no	—
55	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×D₆₁	—	2	1	no	—
57	C₂×D₆₁	—	2	1	no	—
58	C₂×D₆₁	—	2	1	no	—
59	C₄×C₆₁	—	2	1	no	—
60	C₄×C₆₁	—	2	1	no	—
61	C₄×C₆₁	—	4	2	no	—
62	C₄×C₆₁	#131	1	2	yes	—
63	Dic₆₁	—	2	1	no	—
64	Dic₆₁	—	2	1	no	—
65	Dic₆₁	—	4	2	no	—
66	Dic₆₁	#137	1	2	yes	—
67	#1497	—	61	1	no	maximal, Sylow
68	C₂×C₆₁	#408	1	1	yes	characteristic
69	C₂×C₆₁	#419	1	1	yes	characteristic
70	C₂×C₆₁	#421	1	1	yes	characteristic
71	D₆₁	—	4	1	no	—
72	#533	—	61	1	no	—
73	#432	—	61	1	no	—
74	#431	—	61	1	no	—
75	Q₈⋊C₈	—	61	4	no	—
76	C₆₁	#1497	1	1	yes	characteristic, complemented, Sylow
77	C₄⋊Q₈	—	61	1	no	—
78	C₄⋊Q₈	#6923	1	1	yes	characteristic
79	#125	—	122	1	no	—
80	C₂²⋊Q₈	—	61	2	no	—
81	C₂²⋊Q₈	—	122	1	no	—
82	C₄×Q₈	—	61	2	no	—
83	C₄×Q₈	—	122	1	no	—
84	#117	—	122	1	no	—
85	C₂×C₄²	—	61	1	no	—
86	C₄⋊C₈	—	61	2	no	—
87	C₄⋊C₈	—	122	1	no	—
88	C₂²⋊C₈	—	122	1	no	—
89	C₂²⋊C₈	—	122	1	no	—
90	#97	—	122	1	no	—
91	C₄×C₈	—	61	2	no	—
92	C₂×Q₈	—	122	1	no	—
93	C₂×Q₈	C₂×#6923	1	2	yes	—
94	C₂²×C₄	—	61	1	no	—
95	C₂²×C₄	—	61	1	no	—
96	C₂²×C₄	—	61	1	no	—
97	C₂×C₈	—	122	2	no	—
98	C₂×C₈	—	122	2	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	61	2	no	—
102	C₄⋊C₄	—	122	2	no	—
103	C₄⋊C₄	—	122	2	no	—
104	C₄⋊C₄	—	122	1	no	—
105	C₂²⋊C₄	—	122	2	no	—
106	C₄²	—	61	1	no	—
107	C₄²	—	61	1	no	—
108	C₄²	—	61	1	no	—
109	C₄²	—	122	2	no	—
110	C₄²	C₂×#6923	1	1	yes	characteristic
111	C₂³	—	61	1	no	—
112	Q₈	—	2	4	no	—
113	Q₈	—	244	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	61	1	no	—
116	C₂×C₄	—	61	1	no	—
117	C₂×C₄	—	61	1	no	—
118	C₂×C₄	—	122	2	no	—
119	C₂×C₄	—	122	2	no	—
120	C₂×C₄	—	122	2	no	—
121	C₂×C₄	—	244	1	no	—
122	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
124	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
125	C₈	—	244	2	no	—
126	C₈	—	244	2	no	—
127	C₂²	—	122	1	no	—
128	C₂²	—	122	1	no	—
129	C₂²	—	122	1	no	—
130	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	2	no	—
134	C₄	—	61	2	no	—
135	C₄	—	122	1	no	—
136	C₄	—	122	1	no	—
137	C₄	—	244	2	no	—
138	C₄	#14519864	1	2	yes	—
139	C₂	—	244	1	no	—
140	C₂	#16840736	1	1	yes	characteristic
141	C₂	#16840755	1	1	yes	characteristic
142	C₂	#16840761	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840784

C₂

yes

maximal, characteristic

#16840780

C₂

yes

maximal, characteristic

Q₈⋊#92420

C₂

yes

maximal, complemented

#16840317

C₂

yes

maximal, characteristic

#14519844

—

C₄⋊#92420

C₂²

yes

—

—

—

—

C₂²

yes

—

#14519777

C₄

yes

—

—

C₄

yes

—

Q₈×Dic₆₁

C₂²

yes

—

#14519742

—

#14519735

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

C₂×#92420

—

C₂×#92420

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

C₂³

yes

characteristic

C₂²×D₆₁

C₂²⋊C₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

#92420

—

#92420

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#131

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#137

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#533

—

#432

—

#431

—

Q₈⋊C₈

—

C₆₁

#1497

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

#6923

yes

characteristic

#125

—

122

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

C₄×Q₈

—

122

—

#117

—

122

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

#97

—

122

—

C₄×C₈

—

C₂×Q₈

—

122

—

C₂×Q₈

C₂×#6923

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

122

—

103

C₄⋊C₄

—

122

—

104

C₄⋊C₄

—

122

—

105

C₂²⋊C₄

—

122

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

122

—

110

C₄²

C₂×#6923

yes

characteristic

111

C₂³

—

112

Q₈

—

113

Q₈

—

244

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

122

—

119

C₂×C₄

—

122

—

120

C₂×C₄

—

122

—

121

C₂×C₄

—

244

—

122

C₂×C₄

C₂²⋊#6923

yes

characteristic

123

C₂×C₄

C₂²⋊#6923

yes

characteristic

124

C₂×C₄

C₂²×#6923

yes

characteristic

125

C₈

—

244

—

126

C₈

—

244

—

127

C₂²

—

122

—

128

C₂²

—

122

—

129

C₂²

—

122

—

130

C₂²

C₂×C₂²⋊#6923

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

122

—

136

C₄

—

122

—

137

C₄

—

244

—

138

C₄

#14519864

yes

—

139

C₂

—

244

—

140

C₂

#16840736

yes

characteristic

141

C₂

#16840755

yes

characteristic

142

C₂

#16840761

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₄⋊Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6923
hypocenter	C₆₁	#1497
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₄⋊Q₈×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₄⋊Q₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. Q₈⋊#92420	C₂	semidirect
76. C₆₁	#1497	semidirect

H ≅ G/N

type

3. Q₈⋊#92420

C₂

semidirect

76. C₆₁

#1497

semidirect

Subgroups — #21461182

Conjugacy classes of subgroups

Complements