Group #20291076 — subgroups

Subgroups — #20291076

Order 6784.

152 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₄⋊#11719910	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#11719910	C₂	1	2	yes	maximal
3	#16811511	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719910	C₂²	1	4	yes	—
6	C₂²×#88322	C₄	1	1	yes	characteristic
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₄×#88322	C₄	1	1	yes	characteristic
9	C₄×#88322	C₄	1	1	yes	characteristic
10	C₄×#88322	C₂²	1	1	yes	characteristic
11	C₄×#88322	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₈	1	1	yes	characteristic
15	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₄²×C₅₃	C₈	1	1	yes	characteristic
17	C₄²×C₅₃	C₈	1	1	yes	characteristic
18	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×#88322	—	2	2	no	—
21	C₂×#88322	—	2	2	no	—
22	C₂×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂×#88322	C₂³	1	1	yes	characteristic
26	#11719910	—	2	4	no	—
27	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	2	1	no	—
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	#88322	—	2	1	no	—
39	#88322	—	2	1	no	—
40	#88322	M₁₆	1	2	yes	—
41	#88322	#41	1	2	yes	—
42	C₂²×C₅₃	—	2	1	no	—
43	C₂²×C₅₃	—	2	1	no	—
44	C₂²×C₅₃	M₃₂	1	2	yes	—
45	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	—	2	1	no	—
48	C₄×C₅₃	—	2	1	no	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	M₃₂	1	2	yes	—
51	C₄×C₅₃	#117	1	1	yes	characteristic
52	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
53	#2163	—	53	1	no	maximal, Sylow
54	C₂×C₅₃	—	2	2	no	—
55	C₂×C₅₃	#431	1	1	yes	characteristic
56	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
57	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₄×C₈	—	53	1	no	—
59	C₄⋊C₁₆	—	53	2	no	—
60	C₂²⋊C₁₆	—	53	2	no	—
61	#346	—	53	2	no	—
62	C₅₃	#2163	1	1	yes	characteristic, complemented, Sylow
63	C₂²×C₈	—	53	1	no	—
64	C₂²×C₈	—	53	1	no	—
65	C₂×C₄²	#6157	1	1	yes	characteristic
66	C₂×C₁₆	—	53	4	no	—
67	C₄×C₈	—	53	1	no	—
68	C₄×C₈	—	53	1	no	—
69	C₄×C₈	—	53	1	no	—
70	C₄×C₈	—	53	1	no	—
71	C₂²×C₄	#88324	1	1	yes	characteristic
72	C₂²×C₄	#88324	1	1	yes	characteristic
73	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
74	C₂×C₈	—	53	1	no	—
75	C₂×C₈	—	53	1	no	—
76	C₂×C₈	—	53	1	no	—
77	C₂×C₈	—	53	1	no	—
78	C₂×C₈	—	106	2	no	—
79	C₂×C₈	—	106	2	no	—
80	C₄²	#88324	1	1	yes	characteristic
81	C₄²	#88324	1	1	yes	characteristic
82	C₄²	C₂×#6157	1	1	yes	characteristic
83	C₄²	C₂×#6157	1	1	yes	characteristic
84	C₁₆	—	106	4	no	—
85	C₂³	C₂×#88324	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	C₂×#88324	1	1	yes	characteristic
91	C₂×C₄	C₂×#88324	1	1	yes	characteristic
92	C₂×C₄	C₂×#88324	1	1	yes	characteristic
93	C₂×C₄	C₂×#88324	1	1	yes	characteristic
94	C₂×C₄	C₂×#88324	1	1	yes	characteristic
95	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
96	C₈	—	53	2	no	—
97	C₈	—	53	2	no	—
98	C₈	—	106	1	no	—
99	C₈	—	106	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	#13133291	1	2	yes	—
103	C₂²	C₂²×#88324	1	1	yes	characteristic
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	#13133291	1	2	yes	—
109	C₄	#13133296	1	1	yes	characteristic
110	C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
111	C₂	—	2	2	no	—
112	C₂	#16811639	1	1	yes	characteristic
113	C₂	C₂×#13133291	1	1	yes	characteristic
114	C₂	C₂×#13133291	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#11719910

C₂

yes

maximal, complemented

C₂²⋊#11719910

C₂

yes

maximal

#16811511

C₂

yes

maximal, complemented

C₂×C₄×#88322

C₂

yes

maximal, characteristic

C₂×#11719910

C₂²

yes

—

C₂²×#88322

C₄

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₄×#88322

C₄

yes

characteristic

C₄×#88322

C₄

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₈

yes

characteristic

C₂²×C₄×C₅₃

C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₈

yes

characteristic

C₄²×C₅₃

C₈

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

—

—

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

#11719910

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

—

M₁₆

yes

—

#88322

#41

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

M₃₂

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

M₃₂

yes

—

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

#2163

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#431

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₅₃

#2163

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

#6157

yes

characteristic

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₁₆

—

106

—

C₂³

C₂×#88324

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²×#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

106

—

C₈

—

106

—

100

C₂²

—

101

C₂²

—

102

C₂²

#13133291

yes

—

103

C₂²

C₂²×#88324

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#13133291

yes

—

109

C₄

#13133296

yes

characteristic

110

C₄

D₅₃⋊M₁₆

yes

characteristic

111

C₂

—

112

C₂

#16811639

yes

characteristic

113

C₂

C₂×#13133291

yes

characteristic

114

C₂

C₂×#13133291

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₂×C₄²×C₅₃	C₄
hypercenter	C₂×C₄²	#6157
hypocenter	C₅₃	#2163
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂²×C₈

Fitting

C₂×C₄²×C₅₃

C₄

C₂×C₄²

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
1. C₄⋊#11719910	C₂	semidirect
3. #16811511	C₂	semidirect
62. C₅₃	#2163	semidirect

H ≅ G/N

type

1. C₄⋊#11719910

C₂

semidirect

3. #16811511

C₂

semidirect

62. C₅₃

#2163

semidirect

Subgroups — #20291076

Conjugacy classes of subgroups

Complements