Group #21454902 — subgroups

Subgroups — #21454902

Order 7808.

356 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 177 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	C₂×C₄×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	#16839892	C₂	1	4	yes	maximal, complemented
3	#16839889	C₂	1	4	yes	maximal, complemented
4	#16839816	C₂	1	4	yes	maximal
5	#16839815	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₂₂	C₄	1	1	yes	characteristic
8	C₄×Dic₁₂₂	C₄	1	2	yes	—
9	C₄×Dic₁₂₂	C₄	1	2	yes	—
10	C₄×Dic₁₂₂	C₂²	1	2	yes	—
11	C₄×Dic₁₂₂	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₆₁	C₄	1	1	yes	characteristic
13	C₂×#11758196	C₄	1	2	yes	—
14	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
15	C₄⋊#92418	C₂²	1	2	yes	—
16	#14519706	C₂²	1	8	yes	—
17	C₂²⋊#92418	C₂²	1	2	yes	—
18	#14519687	C₂²	1	8	yes	—
19	C₄×#92418	C₂²	1	2	yes	—
20	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
22	C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
23	C₄×C₈×C₆₁	C₂²	1	2	yes	—
24	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
27	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
28	C₂²×Dic₆₁	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₂₂	—	2	4	no	—
30	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
31	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₆₁	C₂³	1	4	yes	—
33	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₆₁	C₂³	1	1	yes	characteristic
36	C₄²×C₆₁	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
39	#11758196	C₂×C₄	1	4	yes	—
40	#11758196	C₂×C₄	1	4	yes	—
41	C₄×Dic₆₁	C₂×C₄	1	4	yes	—
42	C₄×Dic₆₁	C₂³	1	4	yes	—
43	C₂×#92418	C₂³	1	4	yes	—
44	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	—	2	2	no	—
46	C₂×C₄×C₆₁	—	4	1	no	—
47	C₂×C₄×C₆₁	M₁₆	1	4	yes	—
48	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₆₁	—	2	4	no	—
55	C₂×Dic₆₁	—	4	2	no	—
56	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
57	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
58	Dic₁₂₂	—	2	4	no	—
59	Dic₁₂₂	—	2	4	no	—
60	C₈×C₆₁	—	2	4	no	—
61	#92418	—	2	4	no	—
62	C₂²×C₆₁	—	2	1	no	—
63	C₂²×C₆₁	—	2	1	no	—
64	C₂²×C₆₁	C₂×M₁₆	1	2	yes	—
65	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₆₁	—	2	1	no	—
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	—	2	1	no	—
69	C₄×C₆₁	C₂×M₁₆	1	2	yes	—
70	C₄×C₆₁	C₂×M₁₆	1	2	yes	—
71	C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
72	Dic₆₁	—	2	4	no	—
73	Dic₆₁	—	2	4	no	—
74	#2983	—	61	1	no	maximal, Sylow
75	C₂×C₆₁	—	2	2	no	—
76	C₂×C₆₁	#519	1	1	yes	characteristic
77	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
78	C₂×C₆₁	#568	1	1	yes	characteristic
79	C₂×C₄×Q₈	—	61	1	no	—
80	#446	—	61	4	no	—
81	#446	—	61	4	no	—
82	#432	—	61	4	no	—
83	#431	—	61	1	no	—
84	#431	D₆₁	1	1	yes	characteristic
85	C₆₁	#2983	1	1	yes	characteristic, complemented, Sylow
86	C₂²×Q₈	—	61	1	no	—
87	C₄×Q₈	—	61	2	no	—
88	C₄×Q₈	—	61	2	no	—
89	C₄×Q₈	—	61	2	no	—
90	C₄×Q₈	—	61	2	no	—
91	C₂×C₄⋊C₄	—	61	2	no	—
92	C₂×C₄⋊C₄	—	61	1	no	—
93	C₂×C₄²	—	61	2	no	—
94	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
95	C₄⋊C₈	—	61	8	no	—
96	C₄⋊C₈	—	61	2	no	—
97	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
98	C₂²⋊C₈	—	61	2	no	—
99	C₂²⋊C₈	C₂×D₆₁	1	2	yes	—
100	#97	—	61	8	no	—
101	C₄×C₈	—	61	2	no	—
102	C₄×C₈	C₂×D₆₁	1	2	yes	—
103	C₂×Q₈	—	61	2	no	—
104	C₂×Q₈	—	61	2	no	—
105	C₂×Q₈	—	122	4	no	—
106	C₂²×C₄	—	61	2	no	—
107	C₂²×C₄	—	61	2	no	—
108	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
111	C₂×C₈	—	61	4	no	—
112	C₂×C₈	C₂²×D₆₁	1	4	yes	—
113	C₄⋊C₄	—	61	4	no	—
114	C₄⋊C₄	—	61	4	no	—
115	C₄⋊C₄	—	61	2	no	—
116	C₄⋊C₄	—	61	2	no	—
117	C₄²	—	61	4	no	—
118	C₄²	—	61	4	no	—
119	C₄²	C₄×D₆₁	1	1	yes	characteristic
120	C₄²	C₄×D₆₁	1	1	yes	characteristic
121	C₄²	C₂²×D₆₁	1	1	yes	characteristic
122	C₄²	C₂²×D₆₁	1	1	yes	characteristic
123	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
124	Q₈	—	122	4	no	—
125	Q₈	—	122	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	61	4	no	—
129	C₂×C₄	—	61	4	no	—
130	C₂×C₄	—	122	4	no	—
131	C₂×C₄	—	244	2	no	—
132	C₂×C₄	#11758189	1	4	yes	—
133	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	122	4	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#11758189	1	2	yes	—
144	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	122	4	no	—
149	C₄	—	122	4	no	—
150	C₄	C₂×#11758189	1	2	yes	—
151	C₄	C₂×#11758189	1	2	yes	—
152	C₄	#14519824	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#16840239	1	1	yes	characteristic
155	C₂	C₂²×#11758189	1	1	yes	characteristic
156	C₂	#16840528	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₂₂

C₂

yes

maximal, characteristic

#16839892

C₂

yes

maximal, complemented

#16839889

C₂

yes

maximal, complemented

#16839816

C₂

yes

maximal

#16839815

C₂

yes

maximal, characteristic

C₆₁×#431

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₄

yes

characteristic

C₄×Dic₁₂₂

C₄

yes

—

C₄×Dic₁₂₂

C₄

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₄

yes

characteristic

C₂×#11758196

C₄

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519706

C₂²

yes

—

C₂²⋊#92418

C₂²

yes

—

#14519687

C₂²

yes

—

C₄×#92418

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₂²⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

C₂×C₄

yes

—

#11758196

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂×M₁₆

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

2₋¹⁺⁴

yes

—

Dic₆₁

—

Dic₆₁

—

#2983

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#519

yes

characteristic

C₂×C₆₁

C₂²×M₁₆

yes

characteristic

C₂×C₆₁

#568

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

D₆₁

yes

characteristic

C₆₁

#2983

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₆₁

yes

—

100

#97

—

101

C₄×C₈

—

102

C₄×C₈

C₂×D₆₁

yes

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

122

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₆₁

yes

characteristic

109

C₂²×C₄

C₄×D₆₁

yes

characteristic

110

C₂²×C₄

C₂²×D₆₁

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₆₁

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₄×D₆₁

yes

characteristic

120

C₄²

C₄×D₆₁

yes

characteristic

121

C₄²

C₂²×D₆₁

yes

characteristic

122

C₄²

C₂²×D₆₁

yes

characteristic

123

C₂³

C₂×C₄×D₆₁

yes

characteristic

124

Q₈

—

122

—

125

Q₈

—

122

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

122

—

131

C₂×C₄

—

244

—

132

C₂×C₄

#11758189

yes

—

133

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

134

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

135

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

136

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

137

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

138

C₂×C₄

C₂³×D₆₁

yes

characteristic

139

C₈

—

140

C₈

—

122

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#11758189

yes

—

144

C₂²

C₂²×C₄×D₆₁

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

122

—

149

C₄

—

122

—

150

C₄

C₂×#11758189

yes

—

151

C₄

C₂×#11758189

yes

—

152

C₄

#14519824

yes

—

153

C₂

—

154

C₂

#16840239

yes

characteristic

155

C₂

C₂²×#11758189

yes

characteristic

156

C₂

#16840528

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₆₁×#431	C₂
hypercenter	#431	D₆₁
hypocenter	C₆₁	#2983
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₆₁×#431

C₂

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #16839892	C₂	semidirect
3. #16839889	C₂	semidirect
85. C₆₁	#2983	semidirect

H ≅ G/N

type

2. #16839892

C₂

semidirect

3. #16839889

C₂

semidirect

85. C₆₁

#2983

semidirect

Subgroups — #21454902

Conjugacy classes of subgroups

Complements