Group #21462394 — subgroups

Subgroups — C₂²×#14519834 #21462394

Order 7808.

1072 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 338 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	C₂³×#92420	C₂	1	2	yes	maximal, complemented
2	C₂×#14519834	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₄×D₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#92420	—	2	8	no	—
5	C₂²×#92420	C₂²	1	12	yes	complemented
6	#14519834	C₂²	1	16	yes	direct product factor
7	C₂⁴×D₆₁	C₄	1	1	yes	characteristic
8	C₂³×Dic₆₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₆₁	—	2	8	no	—
10	C₂²×C₄×D₆₁	C₄	1	6	yes	—
11	C₂²×C₄×D₆₁	C₂²	1	6	yes	—
12	C₂³×C₄×C₆₁	C₄	1	1	yes	characteristic
13	C₂⁴×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂³×D₆₁	—	2	4	no	—
15	C₂³×D₆₁	—	2	4	no	—
16	C₂³×D₆₁	C₈	1	8	yes	complemented
17	C₂³×D₆₁	C₂×C₄	1	6	yes	—
18	C₂²×C₄×C₆₁	—	2	4	no	—
19	C₂²×C₄×C₆₁	C₂×C₄	1	6	yes	—
20	C₂²×Dic₆₁	C₂×C₄	1	3	yes	—
21	C₂²×Dic₆₁	D₄	1	8	yes	—
22	C₂²×Dic₆₁	C₂³	1	3	yes	—
23	C₂×C₄×D₆₁	—	2	24	no	—
24	C₂×C₄×D₆₁	—	2	24	no	—
25	C₂×C₄×D₆₁	C₂×C₄	1	12	yes	—
26	C₂×C₄×D₆₁	C₂³	1	4	yes	—
27	C₂×#92420	—	2	24	no	—
28	C₂×#92420	C₂³	1	8	yes	complemented
29	C₂³×C₆₁	C₂²⋊C₄	1	4	yes	—
30	C₂³×C₆₁	C₂×C₈	1	4	yes	—
31	C₂³×C₆₁	M₁₆	1	4	yes	—
32	C₂³×C₆₁	C₂²×C₄	1	3	yes	—
33	C₂²×D₆₁	—	2	32	no	—
34	C₂²×D₆₁	—	2	12	no	—
35	C₂²×D₆₁	—	2	12	no	—
36	C₂²×D₆₁	C₂×C₈	1	24	yes	complemented
37	C₂²×D₆₁	C₂²×C₄	1	4	yes	—
38	C₂×C₄×C₆₁	—	2	12	no	—
39	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
40	C₂×Dic₆₁	C₂²⋊C₄	1	12	yes	—
41	C₂×Dic₆₁	C₂²×C₄	1	3	yes	—
42	C₂×Dic₆₁	C₂×D₄	1	12	yes	—
43	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
44	C₄×D₆₁	—	2	24	no	—
45	C₄×D₆₁	—	2	8	no	—
46	#92420	—	2	8	no	—
47	C₂²×C₆₁	C₂²⋊C₈	1	16	yes	complemented
48	C₂²×C₆₁	C₂×C₂²⋊C₄	1	6	yes	—
49	C₂²×C₆₁	C₂²×C₈	1	6	yes	—
50	C₂²×C₆₁	C₂×M₁₆	1	6	yes	—
51	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
52	C₂×D₆₁	—	2	4	no	—
53	C₂×D₆₁	—	2	4	no	—
54	C₂×D₆₁	—	2	48	no	—
55	C₂×D₆₁	C₂²×C₈	1	8	yes	complemented
56	C₄×C₆₁	—	2	4	no	—
57	Dic₆₁	C₂×C₂²⋊C₄	1	6	yes	—
58	Dic₆₁	C₂²×D₄	1	2	yes	—
59	C₂²×C₂²⋊C₈	—	61	1	no	maximal, Sylow
60	C₂×C₆₁	C₂×C₂²⋊C₈	1	12	yes	complemented
61	C₂×C₆₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₆₁	C₂³×C₈	1	1	yes	characteristic
63	C₂×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
64	D₆₁	—	2	8	no	—
65	C₂⁴×C₄	—	61	1	no	—
66	C₂³×C₈	—	61	2	no	—
67	C₂×C₂²⋊C₈	—	61	12	no	—
68	C₆₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
69	C₂⁵	—	61	1	no	—
70	C₂³×C₄	—	61	1	no	—
71	C₂³×C₄	—	61	6	no	—
72	C₂³×C₄	—	61	6	no	—
73	C₂³×C₄	—	122	8	no	—
74	C₂³×C₄	#6923	1	1	yes	characteristic
75	C₂²×C₈	—	61	12	no	—
76	C₂²×C₈	—	122	8	no	—
77	C₂²⋊C₈	—	61	16	no	—
78	C₂⁴	—	61	8	no	—
79	C₂⁴	—	61	6	no	—
80	C₂⁴	—	122	4	no	—
81	C₂⁴	—	122	4	no	—
82	C₂⁴	C₂×#6923	1	1	yes	characteristic
83	C₂²×C₄	—	2	4	no	—
84	C₂²×C₄	—	61	8	no	—
85	C₂²×C₄	—	61	3	no	—
86	C₂²×C₄	—	61	3	no	—
87	C₂²×C₄	—	61	4	no	—
88	C₂²×C₄	—	61	12	no	—
89	C₂²×C₄	—	122	24	no	—
90	C₂²×C₄	—	122	24	no	—
91	C₂²×C₄	C₂×#6923	1	6	yes	—
92	C₂×C₈	—	61	8	no	—
93	C₂×C₈	—	122	24	no	—
94	C₂³	—	61	24	no	—
95	C₂³	—	61	4	no	—
96	C₂³	—	122	32	no	—
97	C₂³	—	122	12	no	—
98	C₂³	—	122	12	no	—
99	C₂³	D₆₁⋊C₈	1	4	yes	—
100	C₂³	#11758213	1	4	yes	—
101	C₂³	C₂²⋊#6923	1	4	yes	—
102	C₂³	C₂²×#6923	1	3	yes	—
103	C₂×C₄	—	2	12	no	—
104	C₂×C₄	—	61	12	no	—
105	C₂×C₄	—	61	12	no	—
106	C₂×C₄	—	61	3	no	—
107	C₂×C₄	—	61	1	no	—
108	C₂×C₄	—	122	24	no	—
109	C₂×C₄	—	122	8	no	—
110	C₂×C₄	C₂²×#6923	1	4	yes	—
111	C₈	—	122	8	no	—
112	C₂²	—	61	8	no	—
113	C₂²	—	122	48	no	—
114	C₂²	—	122	4	no	—
115	C₂²	—	122	4	no	—
116	C₂²	#14519834	1	16	yes	direct product factor
117	C₂²	C₂×C₂²⋊#6923	1	6	yes	—
118	C₂²	C₂×D₆₁⋊C₈	1	6	yes	—
119	C₂²	C₂×#11758213	1	6	yes	—
120	C₂²	C₂³×#6923	1	1	yes	characteristic
121	C₄	—	2	4	no	—
122	C₄	—	61	6	no	—
123	C₄	—	61	2	no	—
124	C₂	—	122	8	no	—
125	C₂	C₂×#14519834	1	12	yes	direct product factor
126	C₂	C₂²×C₂²⋊#6923	1	1	yes	characteristic
127	C₂	C₂²×D₆₁⋊C₈	1	1	yes	characteristic
128	C₂	C₂²×#11758213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×#92420

C₂

yes

maximal, complemented

C₂×#14519834

C₂

yes

maximal, direct product factor

C₂³×C₄×D₆₁

C₂

yes

maximal, characteristic

C₂²×#92420

—

C₂²×#92420

C₂²

yes

complemented

#14519834

C₂²

yes

direct product factor

C₂⁴×D₆₁

C₄

yes

characteristic

C₂³×Dic₆₁

C₂²

yes

characteristic

C₂²×C₄×D₆₁

—

C₂²×C₄×D₆₁

C₄

yes

—

C₂²×C₄×D₆₁

C₂²

yes

—

C₂³×C₄×C₆₁

C₄

yes

characteristic

C₂⁴×C₆₁

C₂×C₄

yes

characteristic

C₂³×D₆₁

—

C₂³×D₆₁

—

C₂³×D₆₁

C₈

yes

complemented

C₂³×D₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

D₄

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

C₂×C₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

—

C₂×#92420

—

C₂×#92420

C₂³

yes

complemented

C₂³×C₆₁

C₂²⋊C₄

yes

—

C₂³×C₆₁

C₂×C₈

yes

—

C₂³×C₆₁

M₁₆

yes

—

C₂³×C₆₁

C₂²×C₄

yes

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

C₂×C₈

yes

complemented

C₂²×D₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²⋊C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂⁴

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

#92420

—

C₂²×C₆₁

C₂²⋊C₈

yes

complemented

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₆₁

C₂²×C₈

yes

—

C₂²×C₆₁

C₂×M₁₆

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

C₂²×C₈

yes

complemented

C₄×C₆₁

—

Dic₆₁

C₂×C₂²⋊C₄

yes

—

Dic₆₁

C₂²×D₄

yes

—

C₂²×C₂²⋊C₈

—

maximal, Sylow

C₂×C₆₁

C₂×C₂²⋊C₈

yes

complemented

C₂×C₆₁

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₆₁

C₂³×C₈

yes

characteristic

C₂×C₆₁

C₂²×M₁₆

yes

characteristic

D₆₁

—

C₂⁴×C₄

—

C₂³×C₈

—

C₂×C₂²⋊C₈

—

C₆₁

C₂²×C₂²⋊C₈

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

122

—

C₂³×C₄

#6923

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

122

—

C₂²⋊C₈

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

122

—

C₂⁴

—

122

—

C₂⁴

C₂×#6923

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂×#6923

yes

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂³

—

C₂³

—

C₂³

—

122

—

C₂³

—

122

—

C₂³

—

122

—

C₂³

D₆₁⋊C₈

yes

—

100

C₂³

#11758213

yes

—

101

C₂³

C₂²⋊#6923

yes

—

102

C₂³

C₂²×#6923

yes

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

122

—

109

C₂×C₄

—

122

—

110

C₂×C₄

C₂²×#6923

yes

—

111

C₈

—

122

—

112

C₂²

—

113

C₂²

—

122

—

114

C₂²

—

122

—

115

C₂²

—

122

—

116

C₂²

#14519834

yes

direct product factor

117

C₂²

C₂×C₂²⋊#6923

yes

—

118

C₂²

C₂×D₆₁⋊C₈

yes

—

119

C₂²

C₂×#11758213

yes

—

120

C₂²

C₂³×#6923

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₂

—

122

—

125

C₂

C₂×#14519834

yes

direct product factor

126

C₂

C₂²×C₂²⋊#6923

yes

characteristic

127

C₂

C₂²×D₆₁⋊C₈

yes

characteristic

128

C₂

C₂²×#11758213

yes

characteristic

	subgroup	G/H
center	C₂⁴	C₂×#6923
commutator	C₂×C₆₁	C₂³×C₈
Frattini	C₂²	C₂³×#6923
Fitting	C₂³×C₄×C₆₁	C₄
socle	C₂⁴×C₆₁	C₂×C₄
hypercenter	C₂³×C₄	#6923
hypocenter	C₆₁	C₂²×C₂²⋊C₈
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄

subgroup

G/H

center

C₂⁴

C₂×#6923

commutator

C₂×C₆₁

C₂³×C₈

Frattini

C₂²

C₂³×#6923

Fitting

C₂³×C₄×C₆₁

C₄

socle

C₂⁴×C₆₁

C₂×C₄

hypercenter

C₂³×C₄

#6923

hypocenter

C₆₁

C₂²×C₂²⋊C₈

LCS

G ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂⁴ ◁ C₂³×C₄

N	H ≅ G/N	type
1. C₂³×#92420	C₂	semidirect
2. C₂×#14519834	C₂	direct (N × H)
5. C₂²×#92420	C₂²	semidirect
6. #14519834	C₂²	direct (N × H)
16. C₂³×D₆₁	C₈	semidirect
28. C₂×#92420	C₂³	semidirect
36. C₂²×D₆₁	C₂×C₈	semidirect
47. C₂²×C₆₁	C₂²⋊C₈	semidirect
55. C₂×D₆₁	C₂²×C₈	semidirect
60. C₂×C₆₁	C₂×C₂²⋊C₈	semidirect
68. C₆₁	C₂²×C₂²⋊C₈	semidirect
116. C₂²	#14519834	direct (N × H)
125. C₂	C₂×#14519834	direct (N × H)

H ≅ G/N

type

1. C₂³×#92420

C₂

semidirect

2. C₂×#14519834

C₂

direct (N × H)

5. C₂²×#92420

C₂²

semidirect

6. #14519834

C₂²

direct (N × H)

16. C₂³×D₆₁

C₈

semidirect

28. C₂×#92420

C₂³

semidirect

36. C₂²×D₆₁

C₂×C₈

semidirect

47. C₂²×C₆₁

C₂²⋊C₈

semidirect

55. C₂×D₆₁

C₂²×C₈

semidirect

60. C₂×C₆₁

C₂×C₂²⋊C₈

semidirect

68. C₆₁

C₂²×C₂²⋊C₈

semidirect

116. C₂²

#14519834

direct (N × H)

125. C₂

C₂×#14519834

direct (N × H)

Subgroups — C₂²×#14519834 #21462394

Conjugacy classes of subgroups

Complements

Subgroups — C22×#14519834 #21462394

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#14519834 #21462394