Group #21454457 — subgroups

Subgroups — #21454457

Order 7808.

392 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 213 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	#16839823	C₂	1	8	yes	maximal, complemented
2	#16839786	C₂	1	2	yes	maximal, complemented
3	#16839726	C₂	1	2	yes	maximal
4	C₄×#11758194	C₂	1	2	yes	maximal, complemented
5	C₂×#117×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758194	C₂²	1	4	yes	—
7	C₄⋊#92418	C₂²	1	8	yes	complemented
8	C₂²⋊#92418	C₂²	1	8	yes	—
9	#14519688	C₂²	1	4	yes	complemented
10	C₄×#92418	C₂²	1	4	yes	complemented
11	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
12	#117×C₆₁	C₄	1	4	yes	—
13	#117×C₆₁	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₆₁	C₄	1	2	yes	—
15	C₂×C₂²⋊C₄×C₆₁	C₄	1	2	yes	—
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	C₂⁴×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	—	2	4	no	—
20	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₆₁	C₂×C₄	1	8	yes	—
27	C₂²⋊C₄×C₆₁	C₂×C₄	1	8	yes	—
28	C₄²×C₆₁	C₂×C₄	1	2	yes	—
29	C₄²×C₆₁	C₂×C₄	1	2	yes	—
30	C₄²×C₆₁	C₂³	1	2	yes	—
31	C₄²×C₆₁	C₂³	1	2	yes	—
32	#11758194	—	2	8	no	—
33	C₂×#92418	C₂³	1	8	yes	—
34	C₂³×C₆₁	—	2	2	no	—
35	C₂³×C₆₁	—	4	2	no	—
36	C₂³×C₆₁	C₂²×C₄	1	2	yes	—
37	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	—	2	2	no	—
39	C₂×C₄×C₆₁	—	2	4	no	—
40	C₂×C₄×C₆₁	—	2	4	no	—
41	C₂×C₄×C₆₁	—	4	2	no	—
42	C₂×C₄×C₆₁	—	4	2	no	—
43	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	#41	1	4	yes	—
48	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
49	#92418	—	2	8	no	—
50	C₂²×C₆₁	—	2	2	no	—
51	C₂²×C₆₁	—	2	2	no	—
52	C₂²×C₆₁	—	2	4	no	—
53	C₂²×C₆₁	—	4	4	no	—
54	C₂²×C₆₁	#117	1	2	yes	—
55	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₆₁	—	2	2	no	—
57	C₄×C₆₁	—	2	4	no	—
58	C₄×C₆₁	—	2	4	no	—
59	C₄×C₆₁	#117	1	2	yes	—
60	C₄×C₆₁	C₂×#41	1	2	yes	—
61	#2921	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	—	2	2	no	—
63	C₂×C₆₁	—	4	2	no	—
64	C₂×C₆₁	C₂×#117	1	1	yes	characteristic
65	C₂×C₆₁	#568	1	2	yes	—
66	C₂×#117	D₆₁	1	1	yes	characteristic
67	#433	—	61	8	no	—
68	#423	—	61	2	no	—
69	#407	—	61	2	no	—
70	C₄×M₁₆	—	61	2	no	—
71	C₆₁	#2921	1	1	yes	characteristic, complemented, Sylow
72	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
73	C₂×M₁₆	—	61	4	no	—
74	#117	Dic₆₁	1	4	yes	—
75	#117	C₂×D₆₁	1	4	yes	—
76	C₂×C₄⋊C₄	Dic₆₁	1	2	yes	—
77	C₂×C₂²⋊C₄	Dic₆₁	1	2	yes	—
78	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
80	C₄⋊C₈	—	61	8	no	—
81	C₂²⋊C₈	—	61	8	no	—
82	#97	—	61	4	no	—
83	C₄×C₈	—	61	4	no	—
84	C₂⁴	C₂×Dic₆₁	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
88	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
89	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
90	C₂²×C₄	C₂²×D₆₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
92	M₁₆	—	122	8	no	—
93	C₂×C₈	—	61	8	no	—
94	C₄⋊C₄	C₂×Dic₆₁	1	8	yes	—
95	C₂²⋊C₄	C₂×Dic₆₁	1	8	yes	—
96	C₄²	C₂×Dic₆₁	1	2	yes	—
97	C₄²	C₂×Dic₆₁	1	2	yes	—
98	C₄²	C₂²×D₆₁	1	2	yes	—
99	C₄²	C₂²×D₆₁	1	2	yes	—
100	C₂³	—	2	2	no	—
101	C₂³	—	4	2	no	—
102	C₂³	C₂²×Dic₆₁	1	2	yes	—
103	C₂³	C₂²×Dic₆₁	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	#11758222	1	4	yes	—
110	C₂×C₄	C₂²×Dic₆₁	1	4	yes	—
111	C₂×C₄	C₂²×Dic₆₁	1	4	yes	—
112	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
113	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
114	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
115	C₈	—	122	8	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	4	4	no	—
120	C₂²	#14519736	1	2	yes	—
121	C₂²	C₂³×Dic₆₁	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	4	no	—
125	C₄	#14519736	1	2	yes	—
126	C₄	C₂×#11758222	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	4	2	no	—
129	C₂	C₂×#14519736	1	1	yes	characteristic
130	C₂	#16840529	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16839823

C₂

yes

maximal, complemented

#16839786

C₂

yes

maximal, complemented

#16839726

C₂

yes

maximal

C₄×#11758194

C₂

yes

maximal, complemented

C₂×#117×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

C₄⋊#92418

C₂²

yes

complemented

C₂²⋊#92418

C₂²

yes

—

#14519688

C₂²

yes

complemented

C₄×#92418

C₂²

yes

complemented

C₂³×C₄×C₆₁

C₂²

yes

characteristic

#117×C₆₁

C₄

yes

—

#117×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₄

yes

—

C₂×C₂²⋊C₄×C₆₁

C₄

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂⁴×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂²⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

—

#11758194

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂²×C₄

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#117

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#117

yes

—

C₄×C₆₁

C₂×#41

yes

—

#2921

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₂×#117

yes

characteristic

C₂×C₆₁

#568

yes

—

C₂×#117

D₆₁

yes

—

—

—

C₄×M₁₆

—

C₆₁

#2921

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₆₁

yes

characteristic

C₂×M₁₆

—

#117

Dic₆₁

yes

—

#117

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

Dic₆₁

yes

—

C₂×C₂²⋊C₄

Dic₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂⁴

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

122

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₆₁

yes

—

C₂²⋊C₄

C₂×Dic₆₁

yes

—

C₄²

C₂×Dic₆₁

yes

—

C₄²

C₂×Dic₆₁

yes

—

C₄²

C₂²×D₆₁

yes

—

C₄²

C₂²×D₆₁

yes

—

100

C₂³

—

101

C₂³

—

102

C₂³

C₂²×Dic₆₁

yes

—

103

C₂³

C₂²×Dic₆₁

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#11758222

yes

—

110

C₂×C₄

C₂²×Dic₆₁

yes

—

111

C₂×C₄

C₂²×Dic₆₁

yes

—

112

C₂×C₄

C₂²×Dic₆₁

yes

—

113

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

114

C₂×C₄

C₂³×D₆₁

yes

characteristic

115

C₈

—

122

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

#14519736

yes

—

121

C₂²

C₂³×Dic₆₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#14519736

yes

—

126

C₄

C₂×#11758222

yes

—

127

C₂

—

128

C₂

—

129

C₂

C₂×#14519736

yes

characteristic

130

C₂

#16840529

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₂×#117×C₆₁	C₂
hypercenter	C₂×#117	D₆₁
hypocenter	C₆₁	#2921
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₂×#117×C₆₁

C₂

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ C₂×#117

N	H ≅ G/N	type
1. #16839823	C₂	semidirect
2. #16839786	C₂	semidirect
4. C₄×#11758194	C₂	semidirect
7. C₄⋊#92418	C₂²	semidirect
9. #14519688	C₂²	semidirect
10. C₄×#92418	C₂²	semidirect
71. C₆₁	#2921	semidirect

H ≅ G/N

type

1. #16839823

C₂

semidirect

2. #16839786

C₂

semidirect

4. C₄×#11758194

C₂

semidirect

7. C₄⋊#92418

C₂²

semidirect

9. #14519688

C₂²

semidirect

10. C₄×#92418

C₂²

semidirect

71. C₆₁

#2921

semidirect

Subgroups — #21454457

Conjugacy classes of subgroups

Complements