Group #21461402 — subgroups

Subgroups — #21461402

Order 7808.

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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	#16840811	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×#92420	C₂	1	1	yes	maximal, characteristic
3	#16840624	C₂	1	1	yes	maximal, characteristic, complemented
4	#16840612	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊#92420	C₂	1	1	yes	maximal, characteristic
6	C₈×#92420	C₂	1	1	yes	maximal, characteristic, complemented
7	#16839853	C₂	1	1	yes	maximal, characteristic
8	C₂²×#92420	—	2	1	no	—
9	#14519844	—	2	1	no	—
10	C₄⋊#92420	C₂²	1	1	yes	characteristic
11	C₄⋊#92420	C₂²	1	1	yes	characteristic
12	C₂²⋊#92420	—	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×#11758200	C₄	1	1	yes	characteristic
17	#14519742	C₂²	1	1	yes	characteristic
18	D₄×Dic₆₁	C₂²	1	1	yes	characteristic
19	#14519708	C₄	1	1	yes	characteristic
20	#14519702	C₄	1	1	yes	characteristic
21	C₈×Dic₆₁	C₂²	1	1	yes	characteristic
22	#102×C₆₁	C₄	1	1	yes	characteristic
23	C₂×D₄×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₆₁	—	2	1	no	—
25	C₂×Dic₁₂₂	C₂×C₄	1	1	yes	characteristic
26	C₂×#92420	—	2	1	no	—
27	C₂×#92420	—	2	1	no	—
28	C₂×#92420	—	2	1	no	—
29	C₂×#92420	—	4	2	no	—
30	C₂×#92420	D₄	1	1	yes	characteristic
31	C₂×#92420	D₄	1	1	yes	characteristic
32	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₂²⋊Dic₆₁	—	2	1	no	—
35	#11758200	C₈	1	4	yes	complemented
36	C₄⋊Dic₆₁	C₂×C₄	1	1	yes	characteristic
37	#11758196	—	2	1	no	—
38	C₄×Dic₆₁	—	2	1	no	—
39	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
40	C₂×#92418	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₆₁	—	2	1	no	—
42	D₄×C₆₁	—	2	1	no	—
43	D₄×C₆₁	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₆₁	—	2	1	no	—
45	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₆₁	—	2	1	no	—
47	C₂×Dic₆₁	—	2	1	no	—
48	C₂×Dic₆₁	—	4	1	no	—
49	C₂×Dic₆₁	—	4	1	no	—
50	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₆₁	#41	1	1	yes	characteristic
52	Dic₁₂₂	—	2	1	no	—
53	Dic₁₂₂	C₂×C₈	1	2	yes	complemented
54	#92420	—	2	2	no	—
55	#92420	—	2	1	no	—
56	#92420	—	4	1	no	—
57	#92420	—	4	1	no	—
58	#92420	SD₁₆	1	2	yes	complemented
59	C₈×C₆₁	—	2	1	no	—
60	#92418	C₂×C₈	1	2	yes	complemented
61	C₂²×C₆₁	—	2	2	no	—
62	C₂²×C₆₁	—	4	1	no	—
63	C₂²×C₆₁	—	4	1	no	—
64	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
65	C₄×C₆₁	—	4	1	no	—
66	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
67	C₄×C₆₁	#131	1	1	yes	characteristic
68	Dic₆₁	—	2	2	no	—
69	Dic₆₁	—	2	1	no	—
70	Dic₆₁	—	4	1	no	—
71	Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
72	Dic₆₁	#135	1	1	yes	characteristic
73	C₈×SD₁₆	—	61	1	no	maximal, Sylow
74	C₂×C₆₁	—	4	2	no	—
75	C₂×C₆₁	C₈×D₄	1	1	yes	characteristic
76	C₂×C₆₁	C₄×SD₁₆	1	1	yes	characteristic
77	C₂×C₆₁	#443	1	1	yes	characteristic
78	C₈×Q₈	—	61	1	no	—
79	C₄×SD₁₆	—	61	1	no	—
80	C₈×D₄	—	61	1	no	—
81	#334	—	61	1	no	—
82	Q₈⋊C₈	—	61	1	no	—
83	D₄⋊C₈	—	61	1	no	—
84	C₈²	—	61	1	no	—
85	C₆₁	C₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	61	1	no	—
87	C₂²×C₈	—	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	—	61	1	no	—
95	#102	#6923	1	1	yes	characteristic
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	—	61	1	no	—
102	C₂×D₄	C₂×#6923	1	1	yes	characteristic
103	C₂²×C₄	—	122	1	no	—
104	SD₁₆	—	61	4	no	—
105	C₂×C₈	—	61	1	no	—
106	C₂×C₈	—	61	1	no	—
107	C₂×C₈	—	61	1	no	—
108	C₂×C₈	—	122	1	no	—
109	C₂×C₈	—	122	1	no	—
110	C₂×C₈	—	122	1	no	—
111	C₂×C₈	—	244	2	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	—	2	1	no	—
120	Q₈	—	61	2	no	—
121	Q₈	—	122	1	no	—
122	D₄	—	2	1	no	—
123	D₄	D₆₁⋊C₈	1	2	yes	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	61	1	no	—
126	C₂×C₄	—	61	1	no	—
127	C₂×C₄	—	122	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	—	61	2	no	—
134	C₈	—	61	2	no	—
135	C₈	—	122	1	no	—
136	C₈	—	122	2	no	—
137	C₈	—	244	1	no	—
138	C₈	—	244	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	D₄×#6923	1	1	yes	characteristic
143	C₄	—	4	1	no	—
144	C₄	—	61	1	no	—
145	C₄	—	61	1	no	—
146	C₄	—	122	2	no	—
147	C₄	—	122	1	no	—
148	C₄	—	244	1	no	—
149	C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
150	C₄	#14519864	1	1	yes	characteristic
151	C₂	—	4	2	no	—
152	C₂	#16840789	1	1	yes	characteristic
153	C₂	SD₁₆×#6923	1	1	yes	characteristic
154	C₂	#16840804	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840811

C₂

yes

maximal, characteristic, complemented

D₄×#92420

C₂

yes

maximal, characteristic

#16840624

C₂

yes

maximal, characteristic, complemented

#16840612

C₂

yes

maximal, characteristic, complemented

D₄⋊#92420

C₂

yes

maximal, characteristic

C₈×#92420

C₂

yes

maximal, characteristic, complemented

#16839853

C₂

yes

maximal, characteristic

C₂²×#92420

—

#14519844

—

C₄⋊#92420

C₂²

yes

characteristic

C₄⋊#92420

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#92420

C₂²

yes

characteristic

C₂×#11758200

C₄

yes

characteristic

#14519742

C₂²

yes

characteristic

D₄×Dic₆₁

C₂²

yes

characteristic

#14519708

C₄

yes

characteristic

#14519702

C₄

yes

characteristic

C₈×Dic₆₁

C₂²

yes

characteristic

#102×C₆₁

C₄

yes

characteristic

C₂×D₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

—

—

—

D₄

yes

characteristic

C₂×#92420

D₄

yes

characteristic

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²⋊Dic₆₁

—

#11758200

C₈

yes

complemented

C₄⋊Dic₆₁

C₂×C₄

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂×C₄

yes

characteristic

C₂³×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

#41

yes

characteristic

Dic₁₂₂

—

Dic₁₂₂

C₂×C₈

yes

—

—

—

—

SD₁₆

yes

complemented

C₈×C₆₁

—

#92418

C₂×C₈

yes

complemented

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂²×C₈

yes

characteristic

C₄×C₆₁

#131

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₂×SD₁₆

yes

characteristic

Dic₆₁

#135

yes

characteristic

C₈×SD₁₆

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₈×D₄

yes

characteristic

C₂×C₆₁

C₄×SD₁₆

yes

characteristic

C₂×C₆₁

#443

yes

characteristic

C₈×Q₈

—

C₄×SD₁₆

—

C₈×D₄

—

#334

—

Q₈⋊C₈

—

D₄⋊C₈

—

C₈²

—

C₆₁

C₈×SD₁₆

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂²×C₈

—

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

—

102

C₂×D₄

C₂×#6923

yes

characteristic

103

C₂²×C₄

—

122

—

104

SD₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

122

—

109

C₂×C₈

—

122

—

110

C₂×C₈

—

122

—

111

C₂×C₈

—

244

—

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

—

120

Q₈

—

121

Q₈

—

122

—

122

D₄

—

123

D₄

D₆₁⋊C₈

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

122

—

128

C₂×C₄

—

122

—

129

C₂×C₄

—

244

—

130

C₂×C₄

—

244

—

131

C₂×C₄

C₂²×#6923

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

122

—

136

C₈

—

122

—

137

C₈

—

244

—

138

C₈

—

244

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×#6923

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

122

—

147

C₄

—

122

—

148

C₄

—

244

—

149

C₄

C₂×D₆₁⋊C₈

yes

characteristic

150

C₄

#14519864

yes

characteristic

151

C₂

—

152

C₂

#16840789

yes

characteristic

153

C₂

SD₁₆×#6923

yes

characteristic

154

C₂

#16840804

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#6923
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6923
Fitting	#102×C₆₁	C₄
socle	C₂²×C₆₁	C₄×D₄
hypercenter	#102	#6923
hypocenter	C₆₁	C₈×SD₁₆
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×#6923

commutator

C₄×C₆₁

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6923

Fitting

#102×C₆₁

C₄

socle

C₂²×C₆₁

C₄×D₄

C₆₁

C₈×SD₁₆

LCS

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
1. #16840811	C₂	semidirect
3. #16840624	C₂	semidirect
4. #16840612	C₂	semidirect
6. C₈×#92420	C₂	semidirect
35. #11758200	C₈	semidirect
53. Dic₁₂₂	C₂×C₈	semidirect
58. #92420	SD₁₆	semidirect
60. #92418	C₂×C₈	semidirect
85. C₆₁	C₈×SD₁₆	semidirect

H ≅ G/N

type

1. #16840811

C₂

semidirect

3. #16840624

C₂

semidirect

4. #16840612

C₂

semidirect

6. C₈×#92420

C₂

semidirect

35. #11758200

C₈

semidirect

53. Dic₁₂₂

C₂×C₈

semidirect

58. #92420

SD₁₆

semidirect

60. #92418

C₂×C₈

semidirect

85. C₆₁

C₈×SD₁₆

semidirect

Subgroups — #21461402

Conjugacy classes of subgroups

Complements