Group #16811372 — subgroups

Subgroups — #16811372

Order 3392.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#13133215	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133212	C₂	1	2	yes	maximal, complemented
3	#13133211	C₂	1	2	yes	maximal, complemented
4	#13133207	C₂	1	2	yes	maximal, complemented
5	#13133206	C₂	1	1	yes	maximal, characteristic
6	#13133202	C₂	1	2	yes	maximal
7	#13133195	C₂	1	2	yes	maximal, complemented
8	#13133193	C₂	1	2	yes	maximal, complemented
9	C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×D₂₁₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₅₃	C₂²	1	2	yes	—
14	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	#11719921	C₂²	1	2	yes	—
19	#11719921	C₂²	1	2	yes	—
20	#11719921	C₂²	1	2	yes	—
21	#11719921	C₂²	1	2	yes	—
22	#11719921	C₂²	1	2	yes	—
23	C₄⋊Dic₅₃	C₂²	1	2	yes	—
24	#11719919	C₂²	1	2	yes	—
25	#11719919	C₂²	1	2	yes	—
26	#11719919	C₂²	1	2	yes	—
27	#11719919	C₂²	1	2	yes	—
28	#11719919	C₂²	1	2	yes	—
29	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
30	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
31	C₂²×D₅₃	C₂³	1	2	yes	—
32	Q₈×C₅₃	—	4	1	no	—
33	Q₈×C₅₃	—	4	1	no	—
34	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂³	1	2	yes	—
36	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂×Dic₅₃	C₂³	1	2	yes	—
40	C₂×Dic₅₃	C₂³	1	2	yes	—
41	C₂×Dic₅₃	C₂³	1	2	yes	—
42	D₂₁₂	—	4	1	no	—
43	C₄×D₅₃	—	4	2	no	—
44	Dic₁₀₆	—	4	1	no	—
45	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂×D₅₃	—	2	2	no	—
47	C₂×D₅₃	—	2	4	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	2	no	—
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	2	2	no	—
53	Dic₅₃	—	2	2	no	—
54	Dic₅₃	—	2	2	no	—
55	Dic₅₃	—	2	2	no	—
56	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₅₃	2₋¹⁺⁴	1	2	yes	—
58	D₅₃	—	4	2	no	—
59	#563	—	53	1	no	maximal, Sylow
60	C₅₃	#563	1	1	yes	characteristic, complemented, Sylow
61	C₄⋊Q₈	—	53	1	no	—
62	C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
63	#126	—	53	2	no	—
64	#126	—	53	2	no	—
65	#125	—	53	2	no	—
66	#124	—	53	1	no	—
67	#123	—	53	2	no	—
68	C₂²⋊Q₈	—	53	2	no	—
69	C₂²⋊Q₈	—	53	2	no	—
70	C₂×Q₈	—	53	1	no	—
71	C₂×Q₈	C₂×D₅₃	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₅₃	1	1	yes	characteristic
73	C₂×D₄	—	53	1	no	—
74	C₂²×C₄	—	53	2	no	—
75	C₄⋊C₄	—	53	2	no	—
76	C₄⋊C₄	—	53	2	no	—
77	C₄⋊C₄	—	53	2	no	—
78	C₄⋊C₄	—	53	2	no	—
79	C₄⋊C₄	—	53	2	no	—
80	C₄⋊C₄	—	53	2	no	—
81	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
82	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
83	C₂²⋊C₄	—	53	2	no	—
84	C₂²⋊C₄	—	53	2	no	—
85	C₂²⋊C₄	—	53	2	no	—
86	C₂²⋊C₄	—	53	2	no	—
87	C₂²⋊C₄	—	53	2	no	—
88	C₄²	—	53	1	no	—
89	C₄²	—	53	1	no	—
90	C₄²	C₂×D₅₃	1	1	yes	characteristic
91	C₂³	—	53	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	212	1	no	—
95	D₄	—	212	1	no	—
96	C₂×C₄	—	53	2	no	—
97	C₂×C₄	—	53	2	no	—
98	C₂×C₄	—	53	2	no	—
99	C₂×C₄	—	212	2	no	—
100	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₅₃	1	2	yes	—
103	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₅₃	1	2	yes	—
105	C₂²	—	106	2	no	—
106	C₂²	—	106	4	no	—
107	C₂²	C₂³×D₅₃	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	106	2	no	—
114	C₄	—	106	2	no	—
115	C₄	—	106	2	no	—
116	C₂	—	212	2	no	—
117	C₂	#13133262	1	1	yes	characteristic
118	C₂	#13133265	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13133215

C₂

yes

maximal, characteristic, complemented

#13133212

C₂

yes

maximal, complemented

#13133211

C₂

yes

maximal, complemented

#13133207

C₂

yes

maximal, complemented

#13133206

C₂

yes

maximal, characteristic

#13133202

C₂

yes

maximal

#13133195

C₂

yes

maximal, complemented

#13133193

C₂

yes

maximal, complemented

C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂²

yes

—

C₂×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

—

C₄⋊C₄×C₅₃

C₂²

yes

—

C₄²×C₅₃

C₂²

yes

characteristic

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

characteristic

C₄×Dic₅₃

C₂²

yes

characteristic

C₂²×D₅₃

C₂³

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₂²×C₅₃

C₂⁴

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

C₂×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂×C₅₃

2₋¹⁺⁴

yes

—

D₅₃

—

#563

—

maximal, Sylow

C₅₃

#563

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

D₅₃

yes

characteristic, complemented

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅₃

yes

—

C₄⋊C₄

C₂×D₅₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₅₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

100

C₂×C₄

C₂²×D₅₃

yes

characteristic

101

C₂×C₄

C₂²×D₅₃

yes

characteristic

102

C₂×C₄

C₂²×D₅₃

yes

—

103

C₂×C₄

C₂²×D₅₃

yes

characteristic

104

C₂×C₄

C₂²×D₅₃

yes

—

105

C₂²

—

106

—

106

C₂²

—

106

—

107

C₂²

C₂³×D₅₃

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

106

—

114

C₄

—

106

—

115

C₄

—

106

—

116

C₂

—

212

—

117

C₂

#13133262

yes

characteristic

118

C₂

#13133265

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	C₄⋊Q₈×C₅₃	C₂
prime radical	C₂×D₂₁₂	C₂²
hypercenter	C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#563
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₃

commutator, socle, cosocle

C₂²×C₅₃

C₂⁴

Fitting

C₄⋊Q₈×C₅₃

C₂

prime radical

C₂×D₂₁₂

C₂²

hypercenter

C₄⋊Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #13133215	C₂	semidirect
2. #13133212	C₂	semidirect
3. #13133211	C₂	semidirect
4. #13133207	C₂	semidirect
7. #13133195	C₂	semidirect
8. #13133193	C₂	semidirect
9. C₄⋊Q₈×C₅₃	C₂	semidirect
60. C₅₃	#563	semidirect
62. C₄⋊Q₈	D₅₃	semidirect

H ≅ G/N

type

1. #13133215

C₂

semidirect

2. #13133212

C₂

semidirect

3. #13133211

C₂

semidirect

4. #13133207

C₂

semidirect

7. #13133195

C₂

semidirect

8. #13133193

C₂

semidirect

9. C₄⋊Q₈×C₅₃

C₂

semidirect

60. C₅₃

#563

semidirect

62. C₄⋊Q₈

D₅₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13133215	2
#13133212	2
#13133211	2
#13133207	2
#13133206	2
#13133202	2
#13133195	2
#13133193	2
C₄⋊Q₈×C₅₃	2
#563	53

subgroup

index

C₄⋊Q₈×C₅₃

Subgroups — #16811372

Conjugacy classes of subgroups

Complements

Maximal subgroups