Group #21458994 — subgroups

Subgroups — C₂²×C₄×Dic₁₂₂ #21458994

Order 7808.

1664 conjugacy classes of subgroups; only the 981 normal subgroups are shown.

normal subgroups only

This group has 1664 conjugacy classes of subgroups — too many to draw. Only its 981 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×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₂₂	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊Dic₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11758196	C₂	1	2	yes	maximal
5	C₂²×C₄×Dic₆₁	C₂	1	2	yes	maximal
6	C₂²×C₄²×C₆₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₂₂	C₄	1	16	yes	direct product factor
8	C₂²×Dic₁₂₂	C₂²	1	12	yes	—
9	C₂³×Dic₆₁	C₂²	1	2	yes	—
10	C₂³×Dic₆₁	C₂²	1	2	yes	—
11	C₄×Dic₁₂₂	C₂²	1	64	yes	direct product factor
12	C₂×C₄⋊Dic₆₁	C₂²	1	12	yes	—
13	C₂×#11758196	C₂²	1	24	yes	—
14	C₂×C₄×Dic₆₁	C₂²	1	24	yes	—
15	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₆₁	C₂²	1	12	yes	—
19	C₂⁴×C₆₁	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	—	2	4	no	—
21	C₂²×C₄×C₆₁	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₆₁	Q₈	1	8	yes	complemented
23	C₂²×C₄×C₆₁	C₂³	1	6	yes	—
24	C₂²×C₄×C₆₁	C₂³	1	6	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	6	yes	—
26	C₂²×Dic₆₁	—	2	8	no	—
27	C₂²×Dic₆₁	C₂×C₄	1	16	yes	—
28	C₂²×Dic₆₁	C₂³	1	12	yes	—
29	C₂²×Dic₆₁	C₂³	1	12	yes	—
30	C₂×Dic₁₂₂	C₂×C₄	1	96	yes	direct product factor
31	C₂×Dic₁₂₂	C₂³	1	16	yes	—
32	C₄²×C₆₁	C₂³	1	16	yes	—
33	C₄⋊Dic₆₁	C₂³	1	16	yes	—
34	#11758196	C₂³	1	32	yes	—
35	C₄×Dic₆₁	C₂³	1	32	yes	—
36	C₂³×C₆₁	C₂²×C₄	1	4	yes	—
37	C₂³×C₆₁	C₂×Q₈	1	4	yes	—
38	C₂³×C₆₁	#41	1	4	yes	—
39	C₂³×C₆₁	C₂⁴	1	3	yes	—
40	C₂×C₄×C₆₁	—	2	12	no	—
41	C₂×C₄×C₆₁	C₂²×C₄	1	24	yes	—
42	C₂×C₄×C₆₁	C₂×Q₈	1	24	yes	complemented
43	C₂×C₄×C₆₁	C₂⁴	1	4	yes	—
44	C₂×C₄×C₆₁	C₂⁴	1	4	yes	—
45	C₂×C₄×C₆₁	C₂⁴	1	4	yes	—
46	C₂×Dic₆₁	—	2	24	no	—
47	C₂×Dic₆₁	C₂²×C₄	1	48	yes	—
48	C₂×Dic₆₁	C₂⁴	1	8	yes	—
49	C₂×Dic₆₁	C₂⁴	1	8	yes	—
50	Dic₁₂₂	C₂²×C₄	1	64	yes	direct product factor
51	C₂²×C₆₁	C₄×Q₈	1	16	yes	complemented
52	C₂²×C₆₁	C₂³×C₄	1	6	yes	—
53	C₂²×C₆₁	C₂²×Q₈	1	6	yes	—
54	C₂²×C₆₁	C₂×#41	1	6	yes	—
55	C₂²×C₆₁	C₂⁵	1	1	yes	characteristic
56	C₄×C₆₁	—	2	4	no	—
57	C₄×C₆₁	C₂³×C₄	1	8	yes	—
58	C₄×C₆₁	C₂²×Q₈	1	8	yes	complemented
59	Dic₆₁	—	2	8	no	—
60	Dic₆₁	C₂³×C₄	1	16	yes	—
61	C₂²×C₄×Q₈	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	C₂×C₄×Q₈	1	12	yes	complemented
63	C₂×C₆₁	C₂⁴×C₄	1	1	yes	characteristic
64	C₂×C₆₁	C₂³×Q₈	1	1	yes	characteristic
65	C₂×C₆₁	C₂²×#41	1	1	yes	characteristic
66	C₂³×Q₈	—	61	1	no	—
67	C₂×C₄×Q₈	—	61	24	no	—
68	C₂²×C₄⋊C₄	—	61	2	no	—
69	C₂²×C₄⋊C₄	—	61	1	no	—
70	C₂²×C₄²	—	61	2	no	—
71	C₂²×C₄²	D₆₁	1	1	yes	characteristic
72	C₆₁	C₂²×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	61	16	no	—
74	C₂²×Q₈	—	61	12	no	—
75	C₂³×C₄	—	61	2	no	—
76	C₂³×C₄	—	61	2	no	—
77	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
78	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
79	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
80	C₄×Q₈	—	61	64	no	—
81	C₂×C₄⋊C₄	—	61	24	no	—
82	C₂×C₄⋊C₄	—	61	12	no	—
83	C₂×C₄²	—	61	24	no	—
84	C₂×C₄²	C₂×D₆₁	1	12	yes	—
85	C₂⁴	C₂²×D₆₁	1	1	yes	characteristic
86	C₂×Q₈	—	61	96	no	—
87	C₂×Q₈	—	61	16	no	—
88	C₂²×C₄	—	2	4	no	—
89	C₂²×C₄	—	61	16	no	—
90	C₂²×C₄	—	61	12	no	—
91	C₂²×C₄	—	61	12	no	—
92	C₂²×C₄	—	122	8	no	—
93	C₂²×C₄	Dic₁₂₂	1	8	yes	direct product factor
94	C₂²×C₄	C₄×D₆₁	1	8	yes	—
95	C₂²×C₄	C₂²×D₆₁	1	6	yes	—
96	C₂²×C₄	C₂²×D₆₁	1	6	yes	—
97	C₂²×C₄	C₂²×D₆₁	1	6	yes	—
98	C₄⋊C₄	—	61	32	no	—
99	C₄⋊C₄	—	61	16	no	—
100	C₄²	—	61	32	no	—
101	C₄²	C₂²×D₆₁	1	16	yes	—
102	C₂³	C₂×Dic₁₂₂	1	4	yes	—
103	C₂³	C₂×C₄×D₆₁	1	4	yes	—
104	C₂³	#11758222	1	4	yes	—
105	C₂³	C₂³×D₆₁	1	3	yes	—
106	Q₈	—	61	64	no	—
107	C₂×C₄	—	2	12	no	—
108	C₂×C₄	—	61	48	no	—
109	C₂×C₄	—	61	8	no	—
110	C₂×C₄	—	61	8	no	—
111	C₂×C₄	—	122	24	no	—
112	C₂×C₄	C₂×Dic₁₂₂	1	24	yes	direct product factor
113	C₂×C₄	C₂×C₄×D₆₁	1	24	yes	—
114	C₂×C₄	C₂³×D₆₁	1	4	yes	—
115	C₂×C₄	C₂³×D₆₁	1	4	yes	—
116	C₂×C₄	C₂³×D₆₁	1	4	yes	—
117	C₂²	C₄×Dic₁₂₂	1	16	yes	direct product factor
118	C₂²	C₂²×C₄×D₆₁	1	6	yes	—
119	C₂²	C₂²×Dic₁₂₂	1	6	yes	—
120	C₂²	C₂×#11758222	1	6	yes	—
121	C₂²	C₂⁴×D₆₁	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	61	16	no	—
124	C₄	—	122	8	no	—
125	C₄	C₂²×C₄×D₆₁	1	8	yes	—
126	C₄	C₂²×Dic₁₂₂	1	8	yes	direct product factor
127	C₂	C₂×C₄×Dic₁₂₂	1	12	yes	direct product factor
128	C₂	C₂³×C₄×D₆₁	1	1	yes	characteristic
129	C₂	C₂³×Dic₁₂₂	1	1	yes	characteristic
130	C₂	C₂²×#11758222	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₁₂₂

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₂₂

C₂

yes

maximal, direct product factor

C₂²×C₄⋊Dic₆₁

C₂

yes

maximal, characteristic

C₂²×#11758196

C₂

yes

maximal

C₂²×C₄×Dic₆₁

C₂

yes

maximal

C₂²×C₄²×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₄

yes

direct product factor

C₂²×Dic₁₂₂

C₂²

yes

—

C₂³×Dic₆₁

C₂²

yes

—

C₂³×Dic₆₁

C₂²

yes

—

C₄×Dic₁₂₂

C₂²

yes

direct product factor

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

C₂³×C₄×C₆₁

C₂²

yes

characteristic

C₂³×C₄×C₆₁

C₂²

yes

characteristic

C₂³×C₄×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

—

C₂⁴×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

Q₈

yes

complemented

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×Dic₆₁

—

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂×Dic₁₂₂

C₂×C₄

yes

direct product factor

C₂×Dic₁₂₂

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂³×C₆₁

C₂²×C₄

yes

—

C₂³×C₆₁

C₂×Q₈

yes

—

C₂³×C₆₁

#41

yes

—

C₂³×C₆₁

C₂⁴

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

complemented

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂⁴

yes

—

C₂×Dic₆₁

C₂⁴

yes

—

Dic₁₂₂

C₂²×C₄

yes

direct product factor

C₂²×C₆₁

C₄×Q₈

yes

complemented

C₂²×C₆₁

C₂³×C₄

yes

—

C₂²×C₆₁

C₂²×Q₈

yes

—

C₂²×C₆₁

C₂×#41

yes

—

C₂²×C₆₁

C₂⁵

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂³×C₄

yes

—

C₄×C₆₁

C₂²×Q₈

yes

complemented

Dic₆₁

—

Dic₆₁

C₂³×C₄

yes

—

C₂²×C₄×Q₈

—

maximal, Sylow

C₂×C₆₁

C₂×C₄×Q₈

yes

complemented

C₂×C₆₁

C₂⁴×C₄

yes

characteristic

C₂×C₆₁

C₂³×Q₈

yes

characteristic

C₂×C₆₁

C₂²×#41

yes

characteristic

C₂³×Q₈

—

C₂×C₄×Q₈

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

—

C₂²×C₄²

D₆₁

yes

characteristic

C₆₁

C₂²×C₄×Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₆₁

yes

characteristic

C₂³×C₄

C₂×D₆₁

yes

characteristic

C₂³×C₄

C₂×D₆₁

yes

characteristic

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

—

C₂⁴

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

Dic₁₂₂

yes

direct product factor

C₂²×C₄

C₄×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

C₂²×D₆₁

yes

—

102

C₂³

C₂×Dic₁₂₂

yes

—

103

C₂³

C₂×C₄×D₆₁

yes

—

104

C₂³

#11758222

yes

—

105

C₂³

C₂³×D₆₁

yes

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

122

—

112

C₂×C₄

C₂×Dic₁₂₂

yes

direct product factor

113

C₂×C₄

C₂×C₄×D₆₁

yes

—

114

C₂×C₄

C₂³×D₆₁

yes

—

115

C₂×C₄

C₂³×D₆₁

yes

—

116

C₂×C₄

C₂³×D₆₁

yes

—

117

C₂²

C₄×Dic₁₂₂

yes

direct product factor

118

C₂²

C₂²×C₄×D₆₁

yes

—

119

C₂²

C₂²×Dic₁₂₂

yes

—

120

C₂²

C₂×#11758222

yes

—

121

C₂²

C₂⁴×D₆₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

122

—

125

C₄

C₂²×C₄×D₆₁

yes

—

126

C₄

C₂²×Dic₁₂₂

yes

direct product factor

127

C₂

C₂×C₄×Dic₁₂₂

yes

direct product factor

128

C₂

C₂³×C₄×D₆₁

yes

characteristic

129

C₂

C₂³×Dic₁₂₂

yes

characteristic

130

C₂

C₂²×#11758222

yes

characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₆₁
commutator	C₂×C₆₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₆₁
Fitting	C₂²×C₄²×C₆₁	C₂
socle	C₂⁴×C₆₁	C₂³
hypercenter	C₂²×C₄²	D₆₁
hypocenter	C₆₁	C₂²×C₄×Q₈
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

subgroup

G/H

center

C₂³×C₄

C₂×D₆₁

commutator

C₂×C₆₁

C₂⁴×C₄

Frattini

C₂²

C₂⁴×D₆₁

Fitting

C₂²×C₄²×C₆₁

C₂

socle

C₂⁴×C₆₁

C₂³

hypercenter

C₂²×C₄²

D₆₁

hypocenter

C₆₁

C₂²×C₄×Q₈

LCS

G ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³×C₄ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. C₂×C₄×Dic₁₂₂	C₂	direct (N × H)
7. C₂²×Dic₁₂₂	C₄	direct (N × H)
11. C₄×Dic₁₂₂	C₂²	direct (N × H)
22. C₂²×C₄×C₆₁	Q₈	semidirect
30. C₂×Dic₁₂₂	C₂×C₄	direct (N × H)
42. C₂×C₄×C₆₁	C₂×Q₈	semidirect
50. Dic₁₂₂	C₂²×C₄	direct (N × H)
51. C₂²×C₆₁	C₄×Q₈	semidirect
58. C₄×C₆₁	C₂²×Q₈	semidirect
62. C₂×C₆₁	C₂×C₄×Q₈	semidirect
72. C₆₁	C₂²×C₄×Q₈	semidirect
93. C₂²×C₄	Dic₁₂₂	direct (N × H)
112. C₂×C₄	C₂×Dic₁₂₂	direct (N × H)
117. C₂²	C₄×Dic₁₂₂	direct (N × H)
126. C₄	C₂²×Dic₁₂₂	direct (N × H)
127. C₂	C₂×C₄×Dic₁₂₂	direct (N × H)

H ≅ G/N

type

2. C₂×C₄×Dic₁₂₂

C₂

direct (N × H)

7. C₂²×Dic₁₂₂

C₄

direct (N × H)

11. C₄×Dic₁₂₂

C₂²

direct (N × H)

22. C₂²×C₄×C₆₁

Q₈

semidirect

30. C₂×Dic₁₂₂

C₂×C₄

direct (N × H)

42. C₂×C₄×C₆₁

C₂×Q₈

semidirect

50. Dic₁₂₂

C₂²×C₄

direct (N × H)

51. C₂²×C₆₁

C₄×Q₈

semidirect

58. C₄×C₆₁

C₂²×Q₈

semidirect

62. C₂×C₆₁

C₂×C₄×Q₈

semidirect

72. C₆₁

C₂²×C₄×Q₈

semidirect

93. C₂²×C₄

Dic₁₂₂

direct (N × H)

112. C₂×C₄

C₂×Dic₁₂₂

direct (N × H)

117. C₂²

C₄×Dic₁₂₂

direct (N × H)

126. C₄

C₂²×Dic₁₂₂

direct (N × H)

127. C₂

C₂×C₄×Dic₁₂₂

direct (N × H)

Subgroups — C₂²×C₄×Dic₁₂₂ #21458994

Conjugacy classes of subgroups

Complements

Subgroups — C22×C4×Dic122 #21458994

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×C₄×Dic₁₂₂ #21458994