Group #21461329 — subgroups

Subgroups — #21461329

Order 7808.

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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	#16840780	C₂	1	1	yes	maximal, characteristic
2	#16840767	C₂	1	1	yes	maximal, characteristic
3	#16840621	C₂	1	4	yes	maximal, complemented
4	C₄⋊Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758213	—	2	1	no	—
6	#14519844	—	2	1	no	—
7	#14519844	C₂²	1	2	yes	—
8	#14519844	C₂²	1	2	yes	—
9	#14519834	—	2	1	no	—
10	C₄×#92420	—	2	1	no	—
11	C₂×Q₈×D₆₁	—	2	1	no	—
12	#14519777	C₄	1	1	yes	characteristic
13	#14519776	—	2	1	no	—
14	#14519776	C₂²	1	2	yes	—
15	#14519775	—	2	1	no	—
16	C₄⋊C₄×D₆₁	—	2	1	no	—
17	C₄⋊C₄×D₆₁	C₄	1	2	yes	—
18	C₄²×D₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₆₁	—	2	1	no	—
21	Q₈×D₆₁	—	4	2	no	—
22	Q₈×D₆₁	—	4	2	no	—
23	C₂×C₄×D₆₁	—	2	2	no	—
24	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
26	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
27	C₂×Dic₁₂₂	—	2	1	no	—
28	C₂×Dic₁₂₂	—	2	2	no	—
29	C₂×#92420	—	2	2	no	—
30	C₂×#92420	—	2	2	no	—
31	#11758213	—	4	2	no	—
32	C₄⋊C₄×C₆₁	—	2	1	no	—
33	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
34	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
37	#11758196	—	2	2	no	—
38	#11758196	—	2	2	no	—
39	C₄×Dic₆₁	D₄	1	1	yes	characteristic
40	C₄×Dic₆₁	D₄	1	1	yes	characteristic
41	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
42	C₂²×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
43	Q₈×C₆₁	—	4	2	no	—
44	C₂×C₄×C₆₁	—	2	2	no	—
45	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₆₁	—	2	2	no	—
49	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
52	C₄×D₆₁	—	2	2	no	—
53	C₄×D₆₁	—	2	2	no	—
54	C₄×D₆₁	—	4	4	no	—
55	C₄×D₆₁	D₈	1	2	yes	—
56	C₄×D₆₁	SD₁₆	1	2	yes	—
57	Dic₁₂₂	—	4	2	no	—
58	Dic₁₂₂	—	4	2	no	—
59	Dic₁₂₂	—	4	2	no	—
60	#92420	—	4	2	no	—
61	#92420	—	4	2	no	—
62	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×D₆₁	—	2	1	no	—
64	C₂×D₆₁	—	2	1	no	—
65	C₂×D₆₁	#102	1	2	yes	—
66	C₄×C₆₁	—	2	1	no	—
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	—	4	2	no	—
69	C₄×C₆₁	#102	1	2	yes	—
70	Dic₆₁	—	4	2	no	—
71	Dic₆₁	#101	1	2	yes	—
72	Dic₆₁	C₂×D₈	1	1	yes	characteristic
73	Dic₆₁	C₂×SD₁₆	1	1	yes	characteristic
74	Dic₆₁	#137	1	2	yes	—
75	#1546	—	61	1	no	maximal, Sylow
76	C₂×C₆₁	C₂×#101	1	1	yes	characteristic
77	C₂×C₆₁	C₂×#102	1	1	yes	characteristic
78	C₂×C₆₁	#419	1	1	yes	characteristic
79	D₆₁	—	2	2	no	—
80	C₂×C₄⋊Q₈	—	61	1	no	—
81	#431	—	61	1	no	—
82	#423	—	61	1	no	—
83	#332	—	61	4	no	—
84	C₆₁	#1546	1	1	yes	characteristic, complemented, Sylow
85	C₂²×Q₈	—	122	1	no	—
86	C₂×M₁₆	—	122	1	no	—
87	C₄⋊Q₈	—	61	2	no	—
88	C₄⋊Q₈	—	61	1	no	—
89	C₄⋊Q₈	—	122	1	no	—
90	C₄⋊Q₈	—	122	1	no	—
91	C₄⋊Q₈	#6923	1	1	yes	characteristic
92	C₂×C₄⋊C₄	—	61	2	no	—
93	C₂×C₄⋊C₄	—	122	1	no	—
94	C₂×C₄²	—	61	1	no	—
95	C₄⋊C₈	—	61	2	no	—
96	C₄⋊C₈	—	61	2	no	—
97	C₄⋊C₈	—	122	1	no	—
98	C₂²⋊C₈	—	122	1	no	—
99	C₄×C₈	—	122	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	122	2	no	—
102	C₂×Q₈	—	122	1	no	—
103	C₂×Q₈	—	244	2	no	—
104	C₂×Q₈	—	244	2	no	—
105	C₂²×C₄	—	61	1	no	—
106	C₂²×C₄	—	61	1	no	—
107	C₂²×C₄	—	61	1	no	—
108	C₂²×C₄	—	122	2	no	—
109	M₁₆	—	244	2	no	—
110	C₂×C₈	—	122	2	no	—
111	C₂×C₈	—	122	2	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	61	2	no	—
114	C₄⋊C₄	—	122	2	no	—
115	C₄⋊C₄	—	122	2	no	—
116	C₄⋊C₄	—	122	1	no	—
117	C₄⋊C₄	C₂×#6923	1	2	yes	—
118	C₄²	—	61	1	no	—
119	C₄²	—	61	1	no	—
120	C₄²	—	61	1	no	—
121	C₄²	C₂×#6923	1	1	yes	characteristic
122	C₂³	—	61	1	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	244	2	no	—
125	Q₈	—	244	2	no	—
126	Q₈	—	244	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	61	1	no	—
129	C₂×C₄	—	61	1	no	—
130	C₂×C₄	—	61	1	no	—
131	C₂×C₄	—	61	2	no	—
132	C₂×C₄	—	61	2	no	—
133	C₂×C₄	—	122	2	no	—
134	C₂×C₄	—	122	2	no	—
135	C₂×C₄	—	122	2	no	—
136	C₂×C₄	—	244	4	no	—
137	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
138	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
139	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
140	C₈	—	244	2	no	—
141	C₈	—	244	2	no	—
142	C₂²	—	61	2	no	—
143	C₂²	—	122	1	no	—
144	C₂²	—	122	1	no	—
145	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	2	no	—
149	C₄	—	61	2	no	—
150	C₄	—	61	1	no	—
151	C₄	—	61	1	no	—
152	C₄	—	61	2	no	—
153	C₄	—	244	2	no	—
154	C₄	#14519839	1	2	yes	—
155	C₂	—	122	2	no	—
156	C₂	D₆₁⋊#101	1	1	yes	characteristic
157	C₂	C₂×#14519839	1	1	yes	characteristic
158	C₂	#16840755	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840780

C₂

yes

maximal, characteristic

#16840767

C₂

yes

maximal, characteristic

#16840621

C₂

yes

maximal, complemented

C₄⋊Q₈×D₆₁

C₂

yes

maximal, characteristic

C₂×#11758213

—

#14519844

—

#14519844

C₂²

yes

—

#14519844

C₂²

yes

—

#14519834

—

C₄×#92420

—

C₂×Q₈×D₆₁

—

#14519777

C₄

yes

characteristic

#14519776

—

#14519776

C₂²

yes

—

#14519775

—

C₄⋊C₄×D₆₁

—

C₄⋊C₄×D₆₁

C₄

yes

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

—

Q₈×D₆₁

—

Q₈×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×#92420

—

C₂×#92420

—

#11758213

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

—

#11758196

—

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

—

C₄×D₆₁

D₈

yes

—

C₄×D₆₁

SD₁₆

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

#92420

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#102

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#102

yes

—

Dic₆₁

—

Dic₆₁

#101

yes

—

Dic₆₁

C₂×D₈

yes

characteristic

Dic₆₁

C₂×SD₁₆

yes

characteristic

Dic₆₁

#137

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₂×C₄⋊Q₈

—

#431

—

#423

—

#332

—

C₆₁

#1546

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂×M₁₆

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

#6923

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

C₄×C₈

—

122

—

100

C₂×Q₈

—

101

C₂×Q₈

—

122

—

102

C₂×Q₈

—

122

—

103

C₂×Q₈

—

244

—

104

C₂×Q₈

—

244

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

122

—

109

M₁₆

—

244

—

110

C₂×C₈

—

122

—

111

C₂×C₈

—

122

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

122

—

115

C₄⋊C₄

—

122

—

116

C₄⋊C₄

—

122

—

117

C₄⋊C₄

C₂×#6923

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

C₂×#6923

yes

characteristic

122

C₂³

—

123

Q₈

—

124

Q₈

—

244

—

125

Q₈

—

244

—

126

Q₈

—

244

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

122

—

134

C₂×C₄

—

122

—

135

C₂×C₄

—

122

—

136

C₂×C₄

—

244

—

137

C₂×C₄

C₂²⋊#6923

yes

characteristic

138

C₂×C₄

C₂²⋊#6923

yes

characteristic

139

C₂×C₄

C₂²×#6923

yes

characteristic

140

C₈

—

244

—

141

C₈

—

244

—

142

C₂²

—

143

C₂²

—

122

—

144

C₂²

—

122

—

145

C₂²

C₂×C₂²⋊#6923

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

244

—

154

C₄

#14519839

yes

—

155

C₂

—

122

—

156

C₂

D₆₁⋊#101

yes

characteristic

157

C₂

C₂×#14519839

yes

characteristic

158

C₂

#16840755

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	#1546
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×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 ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #16840621	C₂	semidirect
84. C₆₁	#1546	semidirect

H ≅ G/N

type

3. #16840621

C₂

semidirect

84. C₆₁

#1546

semidirect

Subgroups — #21461329

Conjugacy classes of subgroups

Complements