Group #21454924 — subgroups

Subgroups — #21454924

Order 7808.

314 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 157 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	#16840319	C₂	1	1	yes	maximal, characteristic
2	#16839891	C₂	1	4	yes	maximal, complemented
3	#16839890	C₂	1	4	yes	maximal, complemented
4	#16839820	C₂	1	4	yes	maximal
5	#16839819	C₂	1	1	yes	maximal, characteristic
6	C₆₁×#432	C₂	1	1	yes	maximal, characteristic
7	#14519777	C₄	1	1	yes	characteristic
8	#14519777	C₄	1	1	yes	characteristic
9	#14519769	C₄	1	2	yes	—
10	#14519756	C₄	1	2	yes	—
11	#14519756	C₄	1	2	yes	—
12	C₄×Dic₁₂₂	C₂²	1	2	yes	—
13	C₄×Dic₁₂₂	C₂²	1	2	yes	—
14	#14519736	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	#14519688	C₂²	1	2	yes	—
19	#14519687	C₂²	1	4	yes	—
20	C₈×Dic₆₁	C₂²	1	4	yes	—
21	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₁	C₂²	1	2	yes	—
23	C₂²⋊C₈×C₆₁	C₂²	1	2	yes	—
24	#97×C₆₁	C₂²	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₆₁	C₂³	1	4	yes	—
31	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₆₁	C₂³	1	1	yes	characteristic
34	C₄²×C₆₁	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₆₁	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
40	#11758196	C₂×C₄	1	4	yes	—
41	#11758196	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	—	4	1	no	—
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	C₂²×C₄	1	4	yes	—
55	C₂×Dic₆₁	C₂²×C₄	1	4	yes	—
56	Dic₁₂₂	—	4	2	no	—
57	Dic₁₂₂	—	4	2	no	—
58	C₈×C₆₁	—	2	4	no	—
59	#92418	—	2	4	no	—
60	C₂²×C₆₁	—	2	1	no	—
61	C₂²×C₆₁	—	2	1	no	—
62	C₂²×C₆₁	—	2	1	no	—
63	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
64	C₄×C₆₁	—	2	1	no	—
65	C₄×C₆₁	—	2	1	no	—
66	C₄×C₆₁	—	2	1	no	—
67	C₄×C₆₁	—	2	1	no	—
68	C₄×C₆₁	—	2	1	no	—
69	C₄×C₆₁	2₋¹⁺⁴	1	2	yes	—
70	Dic₆₁	—	2	4	no	—
71	Dic₆₁	—	2	4	no	—
72	#2987	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	—	4	1	no	—
74	C₂×C₆₁	#519	1	1	yes	characteristic
75	C₂×C₆₁	#568	1	1	yes	characteristic
76	C₂×C₆₁	#568	1	1	yes	characteristic
77	#533	—	61	1	no	—
78	#446	—	61	4	no	—
79	#446	—	61	4	no	—
80	#433	—	61	4	no	—
81	#432	—	61	1	no	—
82	#432	D₆₁	1	1	yes	characteristic
83	C₆₁	#2987	1	1	yes	characteristic, complemented, Sylow
84	C₄⋊Q₈	—	61	1	no	—
85	C₄⋊Q₈	—	61	1	no	—
86	#125	—	61	2	no	—
87	C₂²⋊Q₈	—	61	2	no	—
88	C₂²⋊Q₈	—	61	2	no	—
89	C₄×Q₈	—	61	2	no	—
90	C₄×Q₈	—	61	2	no	—
91	#117	—	61	2	no	—
92	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
93	C₄⋊C₈	—	61	8	no	—
94	C₄⋊C₈	—	61	2	no	—
95	C₄⋊C₈	C₂×D₆₁	1	2	yes	—
96	C₂²⋊C₈	—	61	2	no	—
97	C₂²⋊C₈	C₂×D₆₁	1	2	yes	—
98	#97	—	61	4	no	—
99	#97	—	61	2	no	—
100	#97	C₂×D₆₁	1	2	yes	—
101	C₄×C₈	—	61	4	no	—
102	C₂×Q₈	—	61	2	no	—
103	C₂×Q₈	—	61	2	no	—
104	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
105	C₂²×C₄	C₄×D₆₁	1	1	yes	characteristic
106	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
107	C₂×C₈	—	61	4	no	—
108	C₂×C₈	C₂²×D₆₁	1	4	yes	—
109	C₄⋊C₄	—	61	4	no	—
110	C₄⋊C₄	—	61	4	no	—
111	C₄⋊C₄	—	61	2	no	—
112	C₄⋊C₄	—	61	2	no	—
113	C₄⋊C₄	—	61	2	no	—
114	C₄⋊C₄	—	61	2	no	—
115	C₂²⋊C₄	—	61	4	no	—
116	C₄²	—	61	4	no	—
117	C₄²	C₄×D₆₁	1	1	yes	characteristic
118	C₄²	C₄×D₆₁	1	1	yes	characteristic
119	C₄²	C₂²×D₆₁	1	1	yes	characteristic
120	C₄²	C₂²×D₆₁	1	1	yes	characteristic
121	C₂³	C₂×C₄×D₆₁	1	1	yes	characteristic
122	Q₈	—	244	2	no	—
123	Q₈	—	244	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	61	4	no	—
128	C₂×C₄	—	61	4	no	—
129	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
135	C₈	—	2	4	no	—
136	C₈	—	122	4	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	C₂²×C₄×D₆₁	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	122	4	no	—
147	C₄	—	122	4	no	—
148	C₄	#14519824	1	2	yes	—
149	C₂	—	4	1	no	—
150	C₂	#16840239	1	1	yes	characteristic
151	C₂	#16840528	1	1	yes	characteristic
152	C₂	#16840528	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal, characteristic

#16839891

C₂

yes

maximal, complemented

#16839890

C₂

yes

maximal, complemented

#16839820

C₂

yes

maximal

#16839819

C₂

yes

maximal, characteristic

C₆₁×#432

C₂

yes

maximal, characteristic

#14519777

C₄

yes

characteristic

#14519777

C₄

yes

characteristic

#14519769

C₄

yes

—

#14519756

C₄

yes

—

#14519756

C₄

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

—

#14519736

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519706

C₂²

yes

—

C₂²⋊#92418

C₂²

yes

—

#14519688

C₂²

yes

—

#14519687

C₂²

yes

—

C₈×Dic₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₂²⋊C₈×C₆₁

C₂²

yes

—

#97×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

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

C₂²⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

C₂×C₄

yes

—

#11758196

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

—

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

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

2₋¹⁺⁴

yes

—

Dic₆₁

—

Dic₆₁

—

#2987

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

—

—

—

D₆₁

yes

characteristic

C₆₁

#2987

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₆₁

yes

—

#97

—

#97

—

100

#97

C₂×D₆₁

yes

—

101

C₄×C₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

C₄×D₆₁

yes

characteristic

105

C₂²×C₄

C₄×D₆₁

yes

characteristic

106

C₂²×C₄

C₂²×D₆₁

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

C₂²×D₆₁

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

C₄×D₆₁

yes

characteristic

118

C₄²

C₄×D₆₁

yes

characteristic

119

C₄²

C₂²×D₆₁

yes

characteristic

120

C₄²

C₂²×D₆₁

yes

characteristic

121

C₂³

C₂×C₄×D₆₁

yes

characteristic

122

Q₈

—

244

—

123

Q₈

—

244

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

130

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

131

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

132

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

133

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

134

C₂×C₄

C₂³×D₆₁

yes

characteristic

135

C₈

—

136

C₈

—

122

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂²×C₄×D₆₁

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

122

—

147

C₄

—

122

—

148

C₄

#14519824

yes

—

149

C₂

—

150

C₂

#16840239

yes

characteristic

151

C₂

#16840528

yes

characteristic

152

C₂

#16840528

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₆₁×#432	C₂
hypercenter	#432	D₆₁
hypocenter	C₆₁	#2987
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₆₁×#432

C₂

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #16839891	C₂	semidirect
3. #16839890	C₂	semidirect
83. C₆₁	#2987	semidirect

H ≅ G/N

type

2. #16839891

C₂

semidirect

3. #16839890

C₂

semidirect

83. C₆₁

#2987

semidirect

Subgroups — #21454924

Conjugacy classes of subgroups

Complements