Group #20288291 — subgroups

Subgroups — C₂×Q₈×Dic₁₀₆ #20288291

Order 6784.

1112 conjugacy classes of subgroups; only the 621 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 621 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	Q₈×Dic₁₀₆	C₂	1	16	yes	maximal, direct product factor
2	C₂×#13133217	C₂	1	3	yes	maximal
3	C₂×#13133216	C₂	1	6	yes	maximal
4	C₂×Q₈×Dic₅₃	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₁₀₆	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₀₆	C₂²	1	6	yes	—
8	C₂²×Dic₁₀₆	C₂²	1	1	yes	characteristic
9	#13133217	C₂²	1	24	yes	—
10	#13133216	C₂²	1	48	yes	—
11	Q₈×Dic₅₃	C₂²	1	16	yes	—
12	C₄×Dic₁₀₆	C₂²	1	24	yes	—
13	C₂×C₄⋊Dic₅₃	C₂²	1	3	yes	—
14	C₂×C₄⋊Dic₅₃	C₂²	1	6	yes	—
15	C₂×#11719919	C₂²	1	6	yes	—
16	C₂×C₄×Dic₅₃	C₂²	1	6	yes	—
17	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₅₃	C₂²	1	8	yes	—
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
20	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
21	C₂×Q₈×C₅₃	Q₈	1	8	yes	complemented
22	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
26	C₂²×Dic₅₃	C₂³	1	6	yes	—
27	C₂²×Dic₅₃	C₂³	1	2	yes	—
28	C₂×Dic₁₀₆	—	4	12	no	—
29	C₂×Dic₁₀₆	Q₈	1	8	yes	direct product factor
30	C₂×Dic₁₀₆	C₂³	1	24	yes	—
31	C₂×Dic₁₀₆	C₂³	1	4	yes	—
32	C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
33	C₄²×C₅₃	C₂³	1	12	yes	—
34	C₄⋊Dic₅₃	C₂³	1	12	yes	—
35	C₄⋊Dic₅₃	C₂³	1	24	yes	—
36	#11719919	C₂³	1	24	yes	—
37	C₄×Dic₅₃	C₂³	1	24	yes	—
38	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
39	Q₈×C₅₃	C₂×Q₈	1	16	yes	complemented
40	C₂×C₄×C₅₃	—	2	6	no	—
41	C₂×C₄×C₅₃	C₂×Q₈	1	12	yes	—
42	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
43	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
44	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
45	C₂×C₄×C₅₃	C₂⁴	1	6	yes	—
46	C₂×Dic₅₃	—	2	12	no	—
47	C₂×Dic₅₃	C₂×Q₈	1	8	yes	—
48	C₂×Dic₅₃	C₂⁴	1	12	yes	—
49	C₂×Dic₅₃	C₂⁴	1	4	yes	—
50	Dic₁₀₆	—	4	24	no	—
51	Dic₁₀₆	C₂×Q₈	1	16	yes	direct product factor
52	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
53	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
54	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
55	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
56	C₄×C₅₃	—	2	6	no	—
57	C₄×C₅₃	C₂²×Q₈	1	4	yes	—
58	C₄×C₅₃	C₂²×Q₈	1	12	yes	—
59	Dic₅₃	—	2	12	no	—
60	Dic₅₃	C₂²×Q₈	1	8	yes	—
61	C₂×Q₈²	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	Q₈²	1	4	yes	complemented
63	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
64	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
65	C₂×C₅₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
66	Q₈²	—	53	16	no	—
67	C₂×C₄⋊Q₈	—	53	6	no	—
68	C₂×C₄⋊Q₈	—	53	3	no	—
69	C₂×C₄×Q₈	—	53	2	no	—
70	C₂×C₄×Q₈	—	53	3	no	—
71	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
72	C₅₃	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	53	1	no	—
74	C₂²×Q₈	—	53	6	no	—
75	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
76	C₄⋊Q₈	—	53	48	no	—
77	C₄⋊Q₈	—	53	24	no	—
78	C₄×Q₈	—	53	16	no	—
79	C₄×Q₈	—	53	24	no	—
80	C₄×Q₈	C₂×D₅₃	1	8	yes	—
81	C₂×C₄⋊C₄	—	53	6	no	—
82	C₂×C₄⋊C₄	—	53	6	no	—
83	C₂×C₄⋊C₄	—	53	3	no	—
84	C₂×C₄⋊C₄	C₂×D₅₃	1	3	yes	—
85	C₂×C₄²	—	53	6	no	—
86	C₂×C₄²	C₂×D₅₃	1	3	yes	—
87	C₂×Q₈	—	53	8	no	—
88	C₂×Q₈	—	53	4	no	—
89	C₂×Q₈	—	53	24	no	—
90	C₂×Q₈	—	212	12	no	—
91	C₂×Q₈	Dic₁₀₆	1	8	yes	direct product factor
92	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
93	C₂²×C₄	—	53	2	no	—
94	C₂²×C₄	—	53	6	no	—
95	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
96	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
97	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
98	C₄⋊C₄	—	53	24	no	—
99	C₄⋊C₄	—	53	12	no	—
100	C₄⋊C₄	—	53	24	no	—
101	C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
102	C₄²	—	53	24	no	—
103	C₄²	C₂²×D₅₃	1	12	yes	—
104	C₂³	C₂³×D₅₃	1	1	yes	characteristic
105	Q₈	—	53	16	no	—
106	Q₈	—	212	24	no	—
107	Q₈	C₂×Dic₁₀₆	1	16	yes	direct product factor
108	C₂×C₄	—	2	6	no	—
109	C₂×C₄	—	53	8	no	—
110	C₂×C₄	—	53	4	no	—
111	C₂×C₄	—	53	12	no	—
112	C₂×C₄	—	106	12	no	—
113	C₂×C₄	C₂×Dic₁₀₆	1	12	yes	—
114	C₂×C₄	Q₈×D₅₃	1	4	yes	—
115	C₂×C₄	C₂³×D₅₃	1	6	yes	—
116	C₂×C₄	C₂³×D₅₃	1	2	yes	—
117	C₂×C₄	C₂³×D₅₃	1	6	yes	—
118	C₂²	C₂×Q₈×D₅₃	1	2	yes	—
119	C₂²	C₂²×Dic₁₀₆	1	2	yes	—
120	C₂²	#13133263	1	2	yes	—
121	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
122	C₄	—	2	6	no	—
123	C₄	—	53	8	no	—
124	C₄	—	106	12	no	—
125	C₄	C₂×Q₈×D₅₃	1	4	yes	—
126	C₄	C₂²×Dic₁₀₆	1	12	yes	—
127	C₂	Q₈×Dic₁₀₆	1	4	yes	direct product factor
128	C₂	C₂²×Q₈×D₅₃	1	1	yes	characteristic
129	C₂	C₂³×Dic₁₀₆	1	1	yes	characteristic
130	C₂	C₂×#13133263	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₀₆

C₂

yes

maximal, direct product factor

C₂×#13133217

C₂

yes

maximal

C₂×#13133216

C₂

yes

maximal

C₂×Q₈×Dic₅₃

C₂

yes

maximal

C₂×C₄×Dic₁₀₆

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂²×Dic₁₀₆

C₂²

yes

characteristic

#13133217

C₂²

yes

—

#13133216

C₂²

yes

—

Q₈×Dic₅₃

C₂²

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

Q₈

yes

complemented

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

Q₈

yes

direct product factor

C₂×Dic₁₀₆

C₂³

yes

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

C₂×Q₈

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

C₂×Dic₅₃

C₂⁴

yes

—

Dic₁₀₆

—

Dic₁₀₆

C₂×Q₈

yes

direct product factor

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

C₂⁵

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂²×Q₈

yes

—

C₄×C₅₃

C₂²×Q₈

yes

—

Dic₅₃

—

Dic₅₃

C₂²×Q₈

yes

—

C₂×Q₈²

—

maximal, Sylow

C₂×C₅₃

Q₈²

yes

complemented

C₂×C₅₃

C₂³×Q₈

yes

characteristic

C₂×C₅₃

C₂³×Q₈

yes

characteristic

C₂×C₅₃

C₂×H(5,𝔽₂)

yes

characteristic

Q₈²

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₅₃

yes

characteristic

C₅₃

C₂×Q₈²

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

212

—

C₂×Q₈

Dic₁₀₆

yes

direct product factor

C₂×Q₈

C₂²×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂²×D₅₃

yes

—

102

C₄²

—

103

C₄²

C₂²×D₅₃

yes

—

104

C₂³

C₂³×D₅₃

yes

characteristic

105

Q₈

—

106

Q₈

—

212

—

107

Q₈

C₂×Dic₁₀₆

yes

direct product factor

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

106

—

113

C₂×C₄

C₂×Dic₁₀₆

yes

—

114

C₂×C₄

Q₈×D₅₃

yes

—

115

C₂×C₄

C₂³×D₅₃

yes

—

116

C₂×C₄

C₂³×D₅₃

yes

—

117

C₂×C₄

C₂³×D₅₃

yes

—

118

C₂²

C₂×Q₈×D₅₃

yes

—

119

C₂²

C₂²×Dic₁₀₆

yes

—

120

C₂²

#13133263

yes

—

121

C₂²

C₂⁴×D₅₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

106

—

125

C₄

C₂×Q₈×D₅₃

yes

—

126

C₄

C₂²×Dic₁₀₆

yes

—

127

C₂

Q₈×Dic₁₀₆

yes

direct product factor

128

C₂

C₂²×Q₈×D₅₃

yes

characteristic

129

C₂

C₂³×Dic₁₀₆

yes

characteristic

130

C₂

C₂×#13133263

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₂×C₄×Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	C₂×Q₈²
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂⁵

Frattini

C₂²

C₂⁴×D₅₃

Fitting

C₂×C₄×Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂×C₄×Q₈

D₅₃

hypocenter

C₅₃

C₂×Q₈²

LCS

G ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. Q₈×Dic₁₀₆	C₂	direct (N × H)
21. C₂×Q₈×C₅₃	Q₈	semidirect
29. C₂×Dic₁₀₆	Q₈	direct (N × H)
39. Q₈×C₅₃	C₂×Q₈	semidirect
51. Dic₁₀₆	C₂×Q₈	direct (N × H)
62. C₂×C₅₃	Q₈²	semidirect
72. C₅₃	C₂×Q₈²	semidirect
91. C₂×Q₈	Dic₁₀₆	direct (N × H)
107. Q₈	C₂×Dic₁₀₆	direct (N × H)
127. C₂	Q₈×Dic₁₀₆	direct (N × H)

H ≅ G/N

type

1. Q₈×Dic₁₀₆

C₂

direct (N × H)

21. C₂×Q₈×C₅₃

Q₈

semidirect

29. C₂×Dic₁₀₆

Q₈

direct (N × H)

39. Q₈×C₅₃

C₂×Q₈

semidirect

51. Dic₁₀₆

C₂×Q₈

direct (N × H)

62. C₂×C₅₃

Q₈²

semidirect

72. C₅₃

C₂×Q₈²

semidirect

91. C₂×Q₈

Dic₁₀₆

direct (N × H)

107. Q₈

C₂×Dic₁₀₆

direct (N × H)

127. C₂

Q₈×Dic₁₀₆

direct (N × H)

Subgroups — C₂×Q₈×Dic₁₀₆ #20288291

Conjugacy classes of subgroups

Complements

Subgroups — C2×Q8×Dic106 #20288291

Conjugacy classes of subgroups

Complements

Subgroups — C₂×Q₈×Dic₁₀₆ #20288291