Group #16840385 — subgroups

Subgroups — #16840385

Order 3904.

216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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	#14519776	C₂	1	1	yes	maximal, characteristic, complemented
2	#14519772	C₂	1	2	yes	maximal, complemented
3	#14519766	C₂	1	1	yes	maximal, characteristic, complemented
4	#14519763	C₂	1	2	yes	maximal, complemented
5	#14519759	C₂	1	2	yes	maximal, complemented
6	#14519757	C₂	1	2	yes	maximal, complemented
7	#14519751	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#14519739	C₂	1	1	yes	maximal, characteristic, complemented
10	#14519736	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₆₁	C₂²	1	2	yes	—
15	C₂×D₂₄₄	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₆₁	C₂²	1	2	yes	—
17	C₂×Dic₁₂₂	C₂²	1	2	yes	—
18	C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
20	C₂²⋊C₄×C₆₁	C₂²	1	2	yes	—
21	C₂²⋊Dic₆₁	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₆₁	C₂²	1	1	yes	characteristic
23	#11758198	C₂²	1	2	yes	—
24	#11758198	C₂²	1	2	yes	—
25	#11758198	C₂²	1	2	yes	—
26	#11758198	C₂²	1	2	yes	—
27	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
29	C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
30	C₄⋊Dic₆₁	C₂²	1	2	yes	—
31	#11758196	C₂²	1	2	yes	—
32	C₄×Dic₆₁	C₂²	1	1	yes	characteristic
33	C₄×Dic₆₁	C₂²	1	1	yes	characteristic
34	C₄×Dic₆₁	C₂²	1	2	yes	—
35	C₂³×C₆₁	C₂³	1	1	yes	characteristic
36	C₂²×D₆₁	C₂³	1	2	yes	—
37	Q₈×C₆₁	—	2	2	no	—
38	C₂×C₄×C₆₁	—	2	2	no	—
39	C₂×C₄×C₆₁	C₂³	1	2	yes	—
40	C₂×C₄×C₆₁	C₂³	1	2	yes	—
41	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
43	C₂²⋊D₆₁	—	4	2	no	—
44	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
45	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
46	C₂×Dic₆₁	C₂³	1	2	yes	—
47	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
48	C₂×Dic₆₁	C₂³	1	1	yes	characteristic
49	D₂₄₄	—	2	2	no	—
50	C₄×D₆₁	—	4	2	no	—
51	Dic₁₂₂	—	4	2	no	—
52	C₂²×C₆₁	—	2	1	no	—
53	C₂²×C₆₁	—	2	1	no	—
54	C₂²×C₆₁	—	2	1	no	—
55	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
56	C₂×D₆₁	—	2	2	no	—
57	C₂×D₆₁	—	2	2	no	—
58	C₂×D₆₁	—	2	2	no	—
59	C₄×C₆₁	—	2	1	no	—
60	C₄×C₆₁	—	2	2	no	—
61	C₄×C₆₁	—	2	2	no	—
62	C₄×C₆₁	#41	1	2	yes	—
63	Dic₆₁	—	2	1	no	—
64	Dic₆₁	—	2	2	no	—
65	Dic₆₁	—	2	1	no	—
66	Dic₆₁	—	2	1	no	—
67	Dic₆₁	—	2	1	no	—
68	C₂×C₆₁	—	4	1	no	—
69	C₂×C₆₁	C₂×#41	1	1	yes	characteristic
70	C₂×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
71	C₂×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
72	D₆₁	—	4	2	no	—
73	#542	—	61	1	no	maximal, Sylow
74	C₆₁	#542	1	1	yes	characteristic, complemented, Sylow
75	C₄⋊Q₈	—	61	1	no	—
76	#126	—	61	2	no	—
77	#124	—	61	1	no	—
78	#124	—	61	2	no	—
79	#123	—	61	2	no	—
80	C₂²⋊Q₈	—	61	2	no	—
81	C₂²⋊Q₈	D₆₁	1	1	yes	characteristic, complemented
82	#121	—	61	1	no	—
83	C₄×Q₈	—	61	1	no	—
84	C₄×D₄	—	61	1	no	—
85	#117	—	61	1	no	—
86	C₂×Q₈	—	61	2	no	—
87	C₂×Q₈	C₂×D₆₁	1	1	yes	characteristic
88	C₂×D₄	—	61	2	no	—
89	C₂×D₄	—	61	1	no	—
90	C₂²×C₄	—	61	2	no	—
91	C₂²×C₄	C₂×D₆₁	1	1	yes	characteristic
92	C₄⋊C₄	—	61	2	no	—
93	C₄⋊C₄	—	61	1	no	—
94	C₄⋊C₄	—	61	1	no	—
95	C₄⋊C₄	—	61	1	no	—
96	C₄⋊C₄	—	61	2	no	—
97	C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
99	C₂²⋊C₄	—	61	1	no	—
100	C₂²⋊C₄	—	61	1	no	—
101	C₂²⋊C₄	—	61	2	no	—
102	C₂²⋊C₄	—	61	2	no	—
103	C₂²⋊C₄	—	61	2	no	—
104	C₂²⋊C₄	—	61	2	no	—
105	C₂²⋊C₄	C₂×D₆₁	1	2	yes	—
106	C₄²	—	61	1	no	—
107	C₄²	—	61	1	no	—
108	C₄²	—	61	2	no	—
109	C₂³	—	61	2	no	—
110	C₂³	C₂²×D₆₁	1	1	yes	characteristic
111	Q₈	—	2	2	no	—
112	Q₈	—	244	2	no	—
113	D₄	—	122	2	no	—
114	D₄	—	244	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	61	1	no	—
117	C₂×C₄	—	61	1	no	—
118	C₂×C₄	—	61	1	no	—
119	C₂×C₄	—	61	1	no	—
120	C₂×C₄	—	61	2	no	—
121	C₂×C₄	—	244	2	no	—
122	C₂×C₄	C₂²×D₆₁	1	2	yes	—
123	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₆₁	1	2	yes	—
125	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	122	2	no	—
130	C₂²	—	122	2	no	—
131	C₂²	—	122	2	no	—
132	C₂²	C₂³×D₆₁	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	2	no	—
135	C₄	—	2	2	no	—
136	C₄	—	122	1	no	—
137	C₄	—	122	2	no	—
138	C₄	—	122	1	no	—
139	C₄	—	122	1	no	—
140	C₄	—	122	1	no	—
141	C₄	#11758224	1	2	yes	—
142	C₂	—	4	1	no	—
143	C₂	—	244	2	no	—
144	C₂	C₂×#11758224	1	1	yes	characteristic
145	C₂	#14519823	1	1	yes	characteristic
146	C₂	#14519825	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14519776

C₂

yes

maximal, characteristic, complemented

#14519772

C₂

yes

maximal, complemented

#14519766

C₂

yes

maximal, characteristic, complemented

#14519763

C₂

yes

maximal, complemented

#14519759

C₂

yes

maximal, complemented

#14519757

C₂

yes

maximal, complemented

#14519751

C₂

yes

maximal, characteristic

Q₈×Dic₆₁

C₂

yes

maximal, characteristic, complemented

#14519739

C₂

yes

maximal, characteristic, complemented

#14519736

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

C₂²

yes

characteristic

C₂×C₂²⋊D₆₁

C₂²

yes

—

C₂×D₂₄₄

C₂²

yes

characteristic

C₂×C₄×D₆₁

C₂²

yes

—

C₂×Dic₁₂₂

C₂²

yes

—

C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₄⋊C₄×C₆₁

C₂²

yes

—

C₂²⋊C₄×C₆₁

C₂²

yes

—

C₂²⋊Dic₆₁

C₂²

yes

characteristic

C₂²⋊Dic₆₁

C₂²

yes

characteristic

#11758198

C₂²

yes

—

#11758198

C₂²

yes

—

#11758198

C₂²

yes

—

#11758198

C₂²

yes

—

C₄⋊Dic₆₁

C₂²

yes

characteristic

C₄⋊Dic₆₁

C₂²

yes

characteristic

C₄⋊Dic₆₁

C₂²

yes

characteristic

C₄⋊Dic₆₁

C₂²

yes

—

#11758196

C₂²

yes

—

C₄×Dic₆₁

C₂²

yes

characteristic

C₄×Dic₆₁

C₂²

yes

characteristic

C₄×Dic₆₁

C₂²

yes

—

C₂³×C₆₁

C₂³

yes

characteristic

C₂²×D₆₁

C₂³

yes

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂³

yes

—

C₂×C₄×C₆₁

C₂³

yes

—

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂×C₄×C₆₁

C₂³

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

—

C₂×Dic₆₁

C₂³

yes

characteristic

C₂×Dic₆₁

C₂³

yes

characteristic

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂⁴

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#41

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₂×#41

yes

characteristic

C₂×C₆₁

H(5,𝔽₂)

yes

characteristic

C₂×C₆₁

2₋¹⁺⁴

yes

characteristic

D₆₁

—

#542

—

maximal, Sylow

C₆₁

#542

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₆₁

yes

characteristic, complemented

#121

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₆₁

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₆₁

yes

characteristic

C₄⋊C₄

C₂×D₆₁

yes

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

C₂×D₆₁

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₂³

—

110

C₂³

C₂²×D₆₁

yes

characteristic

111

Q₈

—

112

Q₈

—

244

—

113

D₄

—

122

—

114

D₄

—

244

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

244

—

122

C₂×C₄

C₂²×D₆₁

yes

—

123

C₂×C₄

C₂²×D₆₁

yes

characteristic

124

C₂×C₄

C₂²×D₆₁

yes

—

125

C₂×C₄

C₂²×D₆₁

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

122

—

130

C₂²

—

122

—

131

C₂²

—

122

—

132

C₂²

C₂³×D₆₁

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

122

—

137

C₄

—

122

—

138

C₄

—

122

—

139

C₄

—

122

—

140

C₄

—

122

—

141

C₄

#11758224

yes

—

142

C₂

—

143

C₂

—

244

—

144

C₂

C₂×#11758224

yes

characteristic

145

C₂

#14519823

yes

characteristic

146

C₂

#14519825

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	C₂²⋊Q₈×C₆₁	C₂
prime radical	#14519751	C₂
hypercenter	C₂²⋊Q₈	D₆₁
hypocenter	C₆₁	#542
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₆₁

commutator, socle, cosocle

C₂²×C₆₁

C₂⁴

Fitting

C₂²⋊Q₈×C₆₁

C₂

prime radical

#14519751

C₂

hypercenter

C₂²⋊Q₈

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #14519776	C₂	semidirect
2. #14519772	C₂	semidirect
3. #14519766	C₂	semidirect
4. #14519763	C₂	semidirect
5. #14519759	C₂	semidirect
6. #14519757	C₂	semidirect
8. Q₈×Dic₆₁	C₂	semidirect
9. #14519739	C₂	semidirect
10. #14519736	C₂	semidirect
11. C₂²⋊Q₈×C₆₁	C₂	semidirect
74. C₆₁	#542	semidirect
81. C₂²⋊Q₈	D₆₁	semidirect

H ≅ G/N

type

1. #14519776

C₂

semidirect

2. #14519772

C₂

semidirect

3. #14519766

C₂

semidirect

4. #14519763

C₂

semidirect

5. #14519759

C₂

semidirect

6. #14519757

C₂

semidirect

8. Q₈×Dic₆₁

C₂

semidirect

9. #14519739

C₂

semidirect

10. #14519736

C₂

semidirect

11. C₂²⋊Q₈×C₆₁

C₂

semidirect

74. C₆₁

#542

semidirect

81. C₂²⋊Q₈

D₆₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14519776	2
#14519772	2
#14519766	2
#14519763	2
#14519759	2
#14519757	2
#14519751	2
Q₈×Dic₆₁	2
#14519739	2
#14519736	2
C₂²⋊Q₈×C₆₁	2
#542	61

subgroup

index

Q₈×Dic₆₁

C₂²⋊Q₈×C₆₁

Subgroups — #16840385

Conjugacy classes of subgroups

Complements

Maximal subgroups