Group #20277805 — subgroups

Subgroups — #20277805

Order 6784.

496 conjugacy classes of subgroups; only the 261 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 261 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	#16810530	C₂	1	4	yes	maximal
2	#16810522	C₂	1	4	yes	maximal
3	C₄⋊C₄×Dic₅₃	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
5	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
10	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
11	#13133124	C₂²	1	8	yes	—
12	#13133124	C₂²	1	4	yes	—
13	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₅₃	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	4	2	no	—
18	C₂×Q₈×C₅₃	—	4	2	no	—
19	C₂×Q₈×C₅₃	C₂×C₄	1	8	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
23	C₂²×Dic₅₃	C₂³	1	4	yes	—
24	C₂²×Dic₅₃	C₂³	1	4	yes	—
25	C₄⋊C₄×C₅₃	C₂×C₄	1	16	yes	—
26	C₄²×C₅₃	C₂×C₄	1	4	yes	—
27	C₄⋊Dic₅₃	—	2	4	no	—
28	C₄⋊Dic₅₃	—	4	4	no	—
29	C₄⋊Dic₅₃	Q₈	1	8	yes	complemented
30	C₄×Dic₅₃	—	2	8	no	—
31	C₄×Dic₅₃	—	4	4	no	—
32	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
33	Q₈×C₅₃	—	4	8	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	2	1	no	—
36	C₂×C₄×C₅₃	—	2	4	no	—
37	C₂×C₄×C₅₃	—	2	4	no	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
41	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
42	C₂×C₄×C₅₃	#41	1	4	yes	—
43	C₂×Dic₅₃	—	2	4	no	—
44	C₂×Dic₅₃	—	2	4	no	—
45	C₂×Dic₅₃	—	2	4	no	—
46	C₂×Dic₅₃	—	2	4	no	—
47	C₂×Dic₅₃	—	2	4	no	—
48	C₂×Dic₅₃	C₂×Q₈	1	8	yes	—
49	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
50	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
51	C₂²×C₅₃	C₂×#41	1	2	yes	—
52	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
53	C₂²×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
54	C₄×C₅₃	—	2	8	no	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	C₄×Q₈	1	8	yes	—
57	Dic₅₃	—	2	8	no	—
58	Dic₅₃	—	4	4	no	—
59	#2369	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	C₂×C₄×Q₈	1	2	yes	—
61	C₂×C₅₃	#518	1	1	yes	characteristic
62	C₂×C₅₃	#554	1	2	yes	—
63	C₂×C₅₃	#555	1	1	yes	characteristic
64	C₂×C₅₃	Q₈²	1	1	yes	characteristic
65	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
66	#391	—	53	4	no	—
67	#389	—	53	4	no	—
68	C₄×C₄⋊C₄	—	53	4	no	—
69	C₄×C₄⋊C₄	—	53	2	no	—
70	C₅₃	#2369	1	1	yes	characteristic, complemented, Sylow
71	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
72	C₄⋊Q₈	Dic₅₃	1	8	yes	complemented
73	C₂×C₄⋊C₄	—	53	2	no	—
74	C₂×C₄⋊C₄	—	53	2	no	—
75	C₂×C₄⋊C₄	—	53	4	no	—
76	C₂×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
77	C₂×C₄²	—	53	4	no	—
78	C₂×C₄²	—	53	4	no	—
79	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
80	#95	—	53	8	no	—
81	#95	—	53	4	no	—
82	C₂×Q₈	—	4	2	no	—
83	C₂×Q₈	—	4	2	no	—
84	C₂×Q₈	C₂×Dic₅₃	1	8	yes	—
85	C₂²×C₄	—	53	4	no	—
86	C₂²×C₄	—	53	4	no	—
87	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
89	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
90	C₄⋊C₄	—	53	8	no	—
91	C₄⋊C₄	—	106	4	no	—
92	C₄⋊C₄	—	212	4	no	—
93	C₄⋊C₄	C₂×Dic₅₃	1	16	yes	—
94	C₄²	—	106	8	no	—
95	C₄²	—	212	4	no	—
96	C₄²	C₂×Dic₅₃	1	4	yes	—
97	C₂³	C₂³×D₅₃	1	1	yes	characteristic
98	Q₈	—	4	8	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	53	8	no	—
104	C₂×C₄	—	106	4	no	—
105	C₂×C₄	—	106	4	no	—
106	C₂×C₄	—	106	4	no	—
107	C₂×C₄	—	106	4	no	—
108	C₂×C₄	—	106	4	no	—
109	C₂×C₄	Q₈×D₅₃	1	4	yes	—
110	C₂×C₄	#11719949	1	4	yes	—
111	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
112	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
113	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
114	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
115	C₂²	C₂×Q₈×D₅₃	1	2	yes	—
116	C₂²	C₂×#11719949	1	2	yes	—
117	C₂²	#13133262	1	1	yes	characteristic
118	C₂²	#13133262	1	1	yes	characteristic
119	C₄	—	2	8	no	—
120	C₄	—	2	2	no	—
121	C₄	—	106	8	no	—
122	C₄	—	212	4	no	—
123	C₄	Q₈×Dic₅₃	1	8	yes	—
124	C₂	C₂×Q₈×Dic₅₃	1	2	yes	—
125	C₂	#16811098	1	1	yes	characteristic
126	C₂	#16811334	1	2	yes	—
127	C₂	#16811339	1	1	yes	characteristic
128	C₂	#16811353	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810530

C₂

yes

maximal

#16810522

C₂

yes

maximal

C₄⋊C₄×Dic₅₃

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

#13133124

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

Q₈

yes

complemented

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×Q₈

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

—

C₂²×C₅₃

C₂×#41

yes

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

#2369

—

maximal, Sylow

C₂×C₅₃

C₂×C₄×Q₈

yes

—

C₂×C₅₃

#518

yes

characteristic

C₂×C₅₃

#554

yes

—

C₂×C₅₃

#555

yes

characteristic

C₂×C₅₃

Q₈²

yes

characteristic

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

#391

—

#389

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₅₃

#2369

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

—

C₄⋊Q₈

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄²

—

106

—

C₄²

—

212

—

C₄²

C₂×Dic₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

106

—

105

C₂×C₄

—

106

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

106

—

109

C₂×C₄

Q₈×D₅₃

yes

—

110

C₂×C₄

#11719949

yes

—

111

C₂×C₄

C₂²×Dic₅₃

yes

—

112

C₂×C₄

C₂²×Dic₅₃

yes

—

113

C₂×C₄

C₂²×Dic₅₃

yes

—

114

C₂²

C₂³×Dic₅₃

yes

characteristic

115

C₂²

C₂×Q₈×D₅₃

yes

—

116

C₂²

C₂×#11719949

yes

—

117

C₂²

#13133262

yes

characteristic

118

C₂²

#13133262

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

106

—

122

C₄

—

212

—

123

C₄

Q₈×Dic₅₃

yes

—

124

C₂

C₂×Q₈×Dic₅₃

yes

—

125

C₂

#16811098

yes

characteristic

126

C₂

#16811334

yes

—

127

C₂

#16811339

yes

characteristic

128

C₂

#16811353

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#2369
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
14. C₄⋊Q₈×C₅₃	C₄	semidirect
29. C₄⋊Dic₅₃	Q₈	semidirect
70. C₅₃	#2369	semidirect
72. C₄⋊Q₈	Dic₅₃	semidirect

H ≅ G/N

type

14. C₄⋊Q₈×C₅₃

C₄

semidirect

29. C₄⋊Dic₅₃

Q₈

semidirect

70. C₅₃

#2369

semidirect

72. C₄⋊Q₈

Dic₅₃

semidirect

Subgroups — #20277805

Conjugacy classes of subgroups

Complements