Group #21461446 — subgroups

Subgroups — #21461446

Order 7808.

166 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 56 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	Q₈×#92420	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840796	C₂	1	1	yes	maximal, characteristic
3	#16840625	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#92420	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840610	C₂	1	1	yes	maximal, characteristic
6	#92420⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839847	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758213	—	2	1	no	—
9	C₄⋊#92420	—	2	1	no	—
10	C₄⋊#92420	C₂²	1	1	yes	characteristic
11	C₄⋊#92420	C₂²	1	1	yes	characteristic
12	#14519834	—	2	1	no	—
13	C₄×#92420	—	2	1	no	—
14	C₄×#92420	C₂²	1	1	yes	characteristic
15	C₄×#92420	C₂²	1	1	yes	characteristic
16	C₂×#11758201	C₄	1	1	yes	characteristic
17	Q₈×Dic₆₁	C₂²	1	1	yes	characteristic
18	#14519739	C₂²	1	1	yes	characteristic
19	#14519708	C₄	1	1	yes	characteristic
20	#14519699	C₄	1	1	yes	characteristic
21	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
22	#103×C₆₁	C₄	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×D₂₄₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₆₁	—	2	1	no	—
26	C₂×#92420	—	2	1	no	—
27	C₂×#92420	—	2	1	no	—
28	C₂×#92420	—	2	1	no	—
29	C₂×#92420	D₄	1	1	yes	characteristic
30	C₂×#92420	D₄	1	1	yes	characteristic
31	#11758213	—	4	2	no	—
32	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
34	#11758201	—	2	2	no	—
35	#11758198	—	2	1	no	—
36	C₄⋊Dic₆₁	—	2	1	no	—
37	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₆₁	—	2	1	no	—
39	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
40	C₂×#92418	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₆₁	—	2	1	no	—
42	Q₈×C₆₁	—	2	1	no	—
43	Q₈×C₆₁	M₁₆	1	2	yes	complemented
44	C₂×C₄×C₆₁	—	2	1	no	—
45	C₂×C₄×C₆₁	—	2	1	no	—
46	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₆₁	—	2	1	no	—
48	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₆₁	#41	1	1	yes	characteristic
50	D₂₄₄	—	2	1	no	—
51	D₂₄₄	—	2	1	no	—
52	C₄×D₆₁	—	4	1	no	—
53	C₄×D₆₁	—	4	1	no	—
54	#92420	—	2	2	no	—
55	#92420	—	2	1	no	—
56	#92420	—	2	1	no	—
57	#92420	—	4	1	no	—
58	#92420	—	4	1	no	—
59	C₈×C₆₁	—	2	1	no	—
60	#92418	—	2	1	no	—
61	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
62	C₂×D₆₁	—	4	1	no	—
63	C₂×D₆₁	—	4	1	no	—
64	C₂×D₆₁	—	4	1	no	—
65	C₄×C₆₁	—	2	2	no	—
66	C₄×C₆₁	—	4	1	no	—
67	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
68	C₄×C₆₁	#131	1	1	yes	characteristic
69	Dic₆₁	—	2	1	no	—
70	Dic₆₁	—	4	1	no	—
71	Dic₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
72	Dic₆₁	#137	1	1	yes	characteristic
73	#1588	—	61	1	no	maximal, Sylow
74	C₂×C₆₁	#435	1	1	yes	characteristic
75	C₂×C₆₁	#440	1	1	yes	characteristic
76	C₂×C₆₁	#443	1	1	yes	characteristic
77	D₆₁	—	8	1	no	—
78	C₈×Q₈	—	61	1	no	—
79	C₄×SD₁₆	—	61	1	no	—
80	#436	—	61	1	no	—
81	#335	—	61	1	no	—
82	Q₈⋊C₈	—	61	1	no	—
83	D₄⋊C₈	—	61	1	no	—
84	#322	—	61	1	no	—
85	C₆₁	#1588	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	61	1	no	—
87	C₂×M₁₆	—	122	1	no	—
88	C₄×Q₈	—	61	1	no	—
89	C₄×D₄	—	61	1	no	—
90	#106	—	61	1	no	—
91	C₄⋊C₈	—	61	1	no	—
92	C₄⋊C₈	—	61	1	no	—
93	C₄⋊C₈	—	122	1	no	—
94	#103	#6923	1	1	yes	characteristic
95	#102	—	61	1	no	—
96	C₂²⋊C₈	—	122	1	no	—
97	C₄×C₈	—	61	1	no	—
98	C₄×C₈	—	61	1	no	—
99	C₄×C₈	—	61	1	no	—
100	C₄×C₈	—	122	1	no	—
101	C₂×Q₈	C₂×#6923	1	1	yes	characteristic
102	C₂×D₄	—	61	1	no	—
103	C₂²×C₄	—	122	1	no	—
104	SD₁₆	—	122	2	no	—
105	M₁₆	—	244	2	no	—
106	C₂×C₈	—	61	1	no	—
107	C₂×C₈	—	61	1	no	—
108	C₂×C₈	—	61	1	no	—
109	C₂×C₈	—	122	1	no	—
110	C₂×C₈	—	122	1	no	—
111	C₂×C₈	—	122	1	no	—
112	C₂×C₈	C₂×#6923	1	1	yes	characteristic
113	C₄⋊C₄	—	61	1	no	—
114	C₄⋊C₄	—	122	1	no	—
115	C₄⋊C₄	C₂×#6923	1	1	yes	characteristic
116	C₂²⋊C₄	—	122	1	no	—
117	C₄²	—	61	1	no	—
118	C₄²	—	122	1	no	—
119	C₂³	—	122	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	#11758213	1	2	yes	complemented
122	D₄	—	122	1	no	—
123	D₄	—	122	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	61	1	no	—
127	C₂×C₄	—	61	1	no	—
128	C₂×C₄	—	122	1	no	—
129	C₂×C₄	—	244	1	no	—
130	C₂×C₄	—	244	1	no	—
131	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	122	1	no	—
134	C₈	—	122	1	no	—
135	C₈	—	122	2	no	—
136	C₈	—	122	1	no	—
137	C₈	—	244	1	no	—
138	C₈	—	244	1	no	—
139	C₂²	—	244	1	no	—
140	C₂²	—	244	1	no	—
141	C₂²	—	244	1	no	—
142	C₂²	D₄×#6923	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	1	no	—
145	C₄	—	61	1	no	—
146	C₄	—	61	1	no	—
147	C₄	—	122	1	no	—
148	C₄	—	244	1	no	—
149	C₄	C₂×#11758213	1	1	yes	characteristic
150	C₄	#14519864	1	1	yes	characteristic
151	C₂	—	488	1	no	—
152	C₂	#16840791	1	1	yes	characteristic
153	C₂	#16840801	1	1	yes	characteristic
154	C₂	#16840806	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#92420

C₂

yes

maximal, characteristic, complemented

#16840796

C₂

yes

maximal, characteristic

#16840625

C₂

yes

maximal, characteristic, complemented

Q₈⋊#92420

C₂

yes

maximal, characteristic, complemented

#16840610

C₂

yes

maximal, characteristic

#92420⋊C₈

C₂

yes

maximal, characteristic, complemented

#16839847

C₂

yes

maximal, characteristic

C₂×#11758213

—

C₄⋊#92420

—

C₄⋊#92420

C₂²

yes

characteristic

C₄⋊#92420

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#92420

C₂²

yes

characteristic

C₂×#11758201

C₄

yes

characteristic

Q₈×Dic₆₁

C₂²

yes

characteristic

#14519739

C₂²

yes

characteristic

#14519708

C₄

yes

characteristic

#14519699

C₄

yes

characteristic

C₈×Dic₆₁

C₂²

yes

characteristic

#103×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

C₂×C₄

yes

characteristic

C₂×D₂₄₄

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

—

—

—

—

D₄

yes

characteristic

C₂×#92420

D₄

yes

characteristic

#11758213

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₆₁

C₂×C₄

yes

characteristic

#11758201

—

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

characteristic

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂×C₄

yes

characteristic

C₂²×D₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

M₁₆

yes

complemented

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

#41

yes

characteristic

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

—

—

—

—

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₄×C₆₁

#131

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

yes

Dic₆₁

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₈×Q₈

—

C₄×SD₁₆

—

#436

—

#335

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

—

C₆₁

#1588

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×M₁₆

—

122

—

C₄×Q₈

—

C₄×D₄

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

yes

—

C₂²⋊C₈

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

122

—

101

C₂×Q₈

C₂×#6923

yes

characteristic

102

C₂×D₄

—

103

C₂²×C₄

—

122

—

104

SD₁₆

—

122

—

105

M₁₆

—

244

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

122

—

110

C₂×C₈

—

122

—

111

C₂×C₈

—

122

—

112

C₂×C₈

C₂×#6923

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

122

—

115

C₄⋊C₄

C₂×#6923

yes

characteristic

116

C₂²⋊C₄

—

122

—

117

C₄²

—

118

C₄²

—

122

—

119

C₂³

—

122

—

120

Q₈

—

121

Q₈

#11758213

yes

complemented

122

D₄

—

122

—

123

D₄

—

122

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

122

—

129

C₂×C₄

—

244

—

130

C₂×C₄

—

244

—

131

C₂×C₄

C₂²×#6923

yes

characteristic

132

C₈

—

133

C₈

—

122

—

134

C₈

—

122

—

135

C₈

—

122

—

136

C₈

—

122

—

137

C₈

—

244

—

138

C₈

—

244

—

139

C₂²

—

244

—

140

C₂²

—

244

—

141

C₂²

—

244

—

142

C₂²

D₄×#6923

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

122

—

148

C₄

—

244

—

149

C₄

C₂×#11758213

yes

characteristic

150

C₄

#14519864

yes

characteristic

151

C₂

—

488

—

152

C₂

#16840791

yes

characteristic

153

C₂

#16840801

yes

characteristic

154

C₂

#16840806

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	#103×C₆₁	C₄
socle	C₂²×C₆₁	C₄×D₄
hypercenter	#103	#6923
hypocenter	C₆₁	#1588
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

#103×C₆₁

C₄

socle

C₂²×C₆₁

C₄×D₄

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
1. Q₈×#92420	C₂	semidirect
3. #16840625	C₂	semidirect
4. Q₈⋊#92420	C₂	semidirect
6. #92420⋊C₈	C₂	semidirect
43. Q₈×C₆₁	M₁₆	semidirect
85. C₆₁	#1588	semidirect
121. Q₈	#11758213	semidirect

H ≅ G/N

type

1. Q₈×#92420

C₂

semidirect

3. #16840625

C₂

semidirect

4. Q₈⋊#92420

C₂

semidirect

6. #92420⋊C₈

C₂

semidirect

43. Q₈×C₆₁

M₁₆

semidirect

85. C₆₁

#1588

semidirect

121. Q₈

#11758213

semidirect

Subgroups — #21461446

Conjugacy classes of subgroups

Complements