Group #20287990 — subgroups

Subgroups — C₂²×#13133216 #20287990

Order 6784.

1624 conjugacy classes of subgroups; only the 725 normal subgroups are shown.

normal subgroups only

This group has 1624 conjugacy classes of subgroups — too many to draw. Only its 725 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×Dic₁₀₆	C₂	1	2	yes	maximal
2	C₂×#13133216	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719919	C₂	1	2	yes	maximal
5	C₂²×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₀₆	—	2	16	no	—
8	C₂²×Dic₁₀₆	C₂²	1	24	yes	—
9	C₂³×Dic₅₃	C₂²	1	2	yes	—
10	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂³×Dic₅₃	C₂²	1	1	yes	characteristic
12	#13133216	C₂²	1	64	yes	direct product factor
13	C₂×C₄⋊Dic₅₃	C₂²	1	12	yes	—
14	C₂×#11719919	C₂²	1	24	yes	—
15	C₂×C₄×Dic₅₃	C₂²	1	12	yes	—
16	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
19	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	—	2	8	no	—
21	C₂²×C₄×C₅₃	Q₈	1	8	yes	complemented
22	C₂²×C₄×C₅₃	C₂³	1	12	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
24	C₂²×Dic₅₃	—	2	8	no	—
25	C₂²×Dic₅₃	D₄	1	8	yes	—
26	C₂²×Dic₅₃	Q₈	1	8	yes	complemented
27	C₂²×Dic₅₃	C₂³	1	12	yes	—
28	C₂²×Dic₅₃	C₂³	1	6	yes	—
29	C₂²×Dic₅₃	C₂³	1	6	yes	—
30	C₂×Dic₁₀₆	—	2	96	no	—
31	C₂×Dic₁₀₆	C₂³	1	32	yes	—
32	C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
33	C₄⋊Dic₅₃	C₂³	1	16	yes	—
34	#11719919	C₂³	1	32	yes	—
35	C₄×Dic₅₃	C₂³	1	16	yes	—
36	C₂³×C₅₃	C₂×D₄	1	4	yes	—
37	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
38	C₂³×C₅₃	C₂×Q₈	1	4	yes	—
39	C₂³×C₅₃	C₂⁴	1	3	yes	—
40	C₂×C₄×C₅₃	—	2	24	no	—
41	C₂×C₄×C₅₃	C₂×Q₈	1	24	yes	complemented
42	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
43	C₂×C₄×C₅₃	C₂⁴	1	4	yes	—
44	C₂×Dic₅₃	—	2	24	no	—
45	C₂×Dic₅₃	C₂×D₄	1	24	yes	—
46	C₂×Dic₅₃	C₂×Q₈	1	24	yes	complemented
47	C₂×Dic₅₃	C₂⁴	1	8	yes	—
48	C₂×Dic₅₃	C₂⁴	1	4	yes	—
49	C₂×Dic₅₃	C₂⁴	1	4	yes	—
50	Dic₁₀₆	—	2	64	no	—
51	C₂²×C₅₃	C₄⋊Q₈	1	16	yes	complemented
52	C₂²×C₅₃	C₂²×D₄	1	6	yes	—
53	C₂²×C₅₃	C₂²×Q₈	1	6	yes	—
54	C₂²×C₅₃	C₂²×Q₈	1	6	yes	—
55	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
56	C₄×C₅₃	—	2	8	no	—
57	C₄×C₅₃	C₂²×Q₈	1	8	yes	complemented
58	Dic₅₃	—	2	8	no	—
59	Dic₅₃	C₂²×D₄	1	8	yes	—
60	Dic₅₃	C₂²×Q₈	1	8	yes	complemented
61	C₂²×C₄⋊Q₈	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	C₂×C₄⋊Q₈	1	12	yes	complemented
63	C₂×C₅₃	C₂³×D₄	1	1	yes	characteristic
64	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
65	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
66	C₂³×Q₈	—	53	2	no	—
67	C₂×C₄⋊Q₈	—	53	24	no	—
68	C₂²×C₄⋊C₄	—	53	2	no	—
69	C₂²×C₄⋊C₄	—	53	1	no	—
70	C₂²×C₄⋊C₄	D₅₃	1	1	yes	characteristic
71	C₂²×C₄²	—	53	1	no	—
72	C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	53	24	no	—
74	C₂²×Q₈	—	106	16	no	—
75	C₂³×C₄	—	53	1	no	—
76	C₂³×C₄	—	53	1	no	—
77	C₂³×C₄	—	53	2	no	—
78	C₂³×C₄	C₂×D₅₃	1	2	yes	—
79	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
80	C₄⋊Q₈	—	53	64	no	—
81	C₂×C₄⋊C₄	—	53	24	no	—
82	C₂×C₄⋊C₄	—	53	12	no	—
83	C₂×C₄⋊C₄	C₂×D₅₃	1	12	yes	—
84	C₂×C₄²	—	53	12	no	—
85	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
86	C₂×Q₈	—	53	32	no	—
87	C₂×Q₈	—	106	96	no	—
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	—	53	8	no	—
90	C₂²×C₄	—	53	8	no	—
91	C₂²×C₄	—	53	6	no	—
92	C₂²×C₄	—	53	6	no	—
93	C₂²×C₄	—	53	12	no	—
94	C₂²×C₄	—	106	8	no	—
95	C₂²×C₄	Dic₁₀₆	1	8	yes	complemented
96	C₂²×C₄	C₂²×D₅₃	1	12	yes	—
97	C₂²×C₄	C₂²×D₅₃	1	6	yes	—
98	C₄⋊C₄	—	53	32	no	—
99	C₄⋊C₄	—	53	16	no	—
100	C₄⋊C₄	C₂²×D₅₃	1	16	yes	—
101	C₄²	—	53	16	no	—
102	C₂³	C₂×Dic₁₀₆	1	4	yes	—
103	C₂³	D₄×D₅₃	1	4	yes	—
104	C₂³	Q₈×D₅₃	1	4	yes	—
105	C₂³	C₂³×D₅₃	1	3	yes	—
106	Q₈	—	106	64	no	—
107	C₂×C₄	—	2	24	no	—
108	C₂×C₄	—	53	24	no	—
109	C₂×C₄	—	53	24	no	—
110	C₂×C₄	—	53	4	no	—
111	C₂×C₄	—	53	4	no	—
112	C₂×C₄	—	53	8	no	—
113	C₂×C₄	—	106	24	no	—
114	C₂×C₄	C₂×Dic₁₀₆	1	24	yes	complemented
115	C₂×C₄	C₂³×D₅₃	1	8	yes	—
116	C₂×C₄	C₂³×D₅₃	1	4	yes	—
117	C₂²	#13133216	1	16	yes	direct product factor
118	C₂²	C₂×D₄×D₅₃	1	6	yes	—
119	C₂²	C₂×Q₈×D₅₃	1	6	yes	—
120	C₂²	C₂²×Dic₁₀₆	1	6	yes	—
121	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
122	C₄	—	2	8	no	—
123	C₄	—	53	8	no	—
124	C₄	—	53	8	no	—
125	C₄	—	106	8	no	—
126	C₄	C₂²×Dic₁₀₆	1	8	yes	complemented
127	C₂	C₂×#13133216	1	12	yes	direct product factor
128	C₂	C₂²×D₄×D₅₃	1	1	yes	characteristic
129	C₂	C₂²×Q₈×D₅₃	1	1	yes	characteristic
130	C₂	C₂³×Dic₁₀₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₁₀₆

C₂

yes

maximal

C₂×#13133216

C₂

yes

maximal, direct product factor

C₂²×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

C₂²×#11719919

C₂

yes

maximal

C₂²×C₄×Dic₅₃

C₂

yes

maximal, characteristic

C₂²×C₄⋊C₄×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

—

C₂²×Dic₁₀₆

C₂²

yes

—

C₂³×Dic₅₃

C₂²

yes

—

C₂³×Dic₅₃

C₂²

yes

characteristic

C₂³×Dic₅₃

C₂²

yes

characteristic

#13133216

C₂²

yes

direct product factor

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

Q₈

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

—

C₂²×Dic₅₃

Q₈

yes

complemented

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×Q₈

yes

complemented

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×Q₈

yes

complemented

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

Dic₁₀₆

—

C₂²×C₅₃

C₄⋊Q₈

yes

complemented

C₂²×C₅₃

C₂²×D₄

yes

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂²×Q₈

yes

complemented

Dic₅₃

—

Dic₅₃

C₂²×D₄

yes

—

Dic₅₃

C₂²×Q₈

yes

complemented

C₂²×C₄⋊Q₈

—

maximal, Sylow

C₂×C₅₃

C₂×C₄⋊Q₈

yes

complemented

C₂×C₅₃

C₂³×D₄

yes

characteristic

C₂×C₅₃

C₂³×Q₈

yes

characteristic

C₂×C₅₃

C₂³×Q₈

yes

characteristic

C₂³×Q₈

—

C₂×C₄⋊Q₈

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

D₅₃

yes

characteristic

C₂²×C₄²

—

C₅₃

C₂²×C₄⋊Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

106

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×D₅₃

yes

—

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

Dic₁₀₆

yes

complemented

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

C₂²×D₅₃

yes

—

101

C₄²

—

102

C₂³

C₂×Dic₁₀₆

yes

—

103

C₂³

D₄×D₅₃

yes

—

104

C₂³

Q₈×D₅₃

yes

—

105

C₂³

C₂³×D₅₃

yes

—

106

Q₈

—

106

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

106

—

114

C₂×C₄

C₂×Dic₁₀₆

yes

complemented

115

C₂×C₄

C₂³×D₅₃

yes

—

116

C₂×C₄

C₂³×D₅₃

yes

—

117

C₂²

#13133216

yes

direct product factor

118

C₂²

C₂×D₄×D₅₃

yes

—

119

C₂²

C₂×Q₈×D₅₃

yes

—

120

C₂²

C₂²×Dic₁₀₆

yes

—

121

C₂²

C₂⁴×D₅₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

106

—

126

C₄

C₂²×Dic₁₀₆

yes

complemented

127

C₂

C₂×#13133216

yes

direct product factor

128

C₂

C₂²×D₄×D₅₃

yes

characteristic

129

C₂

C₂²×Q₈×D₅₃

yes

characteristic

130

C₂

C₂³×Dic₁₀₆

yes

characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂²×C₄⋊C₄×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂³
hypercenter	C₂²×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂²×C₄⋊Q₈
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂⁴

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂⁵

Frattini

C₂²

C₂⁴×D₅₃

Fitting

C₂²×C₄⋊C₄×C₅₃

C₂

socle

C₂⁴×C₅₃

C₂³

hypercenter

C₂²×C₄⋊C₄

D₅₃

hypocenter

C₅₃

C₂²×C₄⋊Q₈

LCS

G ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. C₂×#13133216	C₂	direct (N × H)
12. #13133216	C₂²	direct (N × H)
21. C₂²×C₄×C₅₃	Q₈	semidirect
26. C₂²×Dic₅₃	Q₈	semidirect
41. C₂×C₄×C₅₃	C₂×Q₈	semidirect
46. C₂×Dic₅₃	C₂×Q₈	semidirect
51. C₂²×C₅₃	C₄⋊Q₈	semidirect
57. C₄×C₅₃	C₂²×Q₈	semidirect
60. Dic₅₃	C₂²×Q₈	semidirect
62. C₂×C₅₃	C₂×C₄⋊Q₈	semidirect
72. C₅₃	C₂²×C₄⋊Q₈	semidirect
95. C₂²×C₄	Dic₁₀₆	semidirect
114. C₂×C₄	C₂×Dic₁₀₆	semidirect
117. C₂²	#13133216	direct (N × H)
126. C₄	C₂²×Dic₁₀₆	semidirect
127. C₂	C₂×#13133216	direct (N × H)

H ≅ G/N

type

2. C₂×#13133216

C₂

direct (N × H)

12. #13133216

C₂²

direct (N × H)

21. C₂²×C₄×C₅₃

Q₈

semidirect

26. C₂²×Dic₅₃

Q₈

semidirect

41. C₂×C₄×C₅₃

C₂×Q₈

semidirect

46. C₂×Dic₅₃

C₂×Q₈

semidirect

51. C₂²×C₅₃

C₄⋊Q₈

semidirect

57. C₄×C₅₃

C₂²×Q₈

semidirect

60. Dic₅₃

C₂²×Q₈

semidirect

62. C₂×C₅₃

C₂×C₄⋊Q₈

semidirect

72. C₅₃

C₂²×C₄⋊Q₈

semidirect

95. C₂²×C₄

Dic₁₀₆

semidirect

114. C₂×C₄

C₂×Dic₁₀₆

semidirect

117. C₂²

#13133216

direct (N × H)

126. C₄

C₂²×Dic₁₀₆

semidirect

127. C₂

C₂×#13133216

direct (N × H)

Subgroups — C₂²×#13133216 #20287990

Conjugacy classes of subgroups

Complements

Subgroups — C22×#13133216 #20287990

Conjugacy classes of subgroups

Complements

Subgroups — C₂²×#13133216 #20287990