Group #20275656 — subgroups

Subgroups — #20275656

Order 6784.

280 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 121 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	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
3	#16810378	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	2	yes	—
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₄⋊#88322	—	2	2	no	—
8	#13133128	—	2	2	no	—
9	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₅₃	—	2	2	no	—
11	C₄×Q₈×C₅₃	—	2	2	no	—
12	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	—	2	2	no	—
17	C₂×Q₈×C₅₃	—	2	4	no	—
18	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₅₃	—	2	4	no	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄²×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	—	2	2	no	—
27	C₄²×C₅₃	D₄	1	2	yes	—
28	C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×#88322	—	2	2	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂×#88322	—	2	4	no	—
33	C₂×#88322	Q₈	1	4	yes	complemented
34	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₅₃	—	2	8	no	—
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₅₃	#41	1	2	yes	—
49	#88322	—	2	4	no	—
50	#88322	—	4	2	no	—
51	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
53	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
54	C₂²×C₅₃	#131	1	2	yes	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	4	no	—
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	—	4	2	no	—
59	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₅₃	#124	1	1	yes	characteristic
61	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
62	#1985	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	#391	1	1	yes	characteristic
64	C₂×C₅₃	#407	1	1	yes	characteristic
65	C₂×C₅₃	#408	1	1	yes	characteristic
66	C₂×C₅₃	#446	1	4	yes	—
67	C₂×C₄×Q₈	D₅₃	1	1	yes	characteristic
68	C₂×C₄⋊C₈	—	53	1	no	—
69	C₂×#97	—	53	1	no	—
70	#336	—	53	4	no	—
71	C₅₃	#1985	1	1	yes	characteristic, complemented, Sylow
72	C₂²×Q₈	Dic₅₃	1	1	yes	characteristic
73	C₂²×C₈	—	53	2	no	—
74	C₂²×C₈	—	53	2	no	—
75	C₄×Q₈	—	2	2	no	—
76	C₄×Q₈	—	2	2	no	—
77	C₂×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
78	C₂×C₄⋊C₄	Dic₅₃	1	2	yes	—
79	C₂×C₄²	C₂×D₅₃	1	2	yes	—
80	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
81	C₄⋊C₈	—	106	2	no	—
82	#97	—	106	2	no	—
83	C₂×Q₈	—	2	4	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
86	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
87	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
90	C₂×C₈	—	53	4	no	—
91	C₂×C₈	—	106	2	no	—
92	C₂×C₈	—	106	2	no	—
93	C₂×C₈	—	106	4	no	—
94	C₂×C₈	—	106	2	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	2	2	no	—
99	C₄²	C₂²⋊D₅₃	1	2	yes	—
100	C₄²	C₂²⋊D₅₃	1	2	yes	—
101	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
102	Q₈	—	2	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	#11719917	1	4	yes	—
110	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
111	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
112	C₂×C₄	Q₈×D₅₃	1	2	yes	—
113	C₂×C₄	#11719949	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
115	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
116	C₈	—	106	4	no	—
117	C₈	—	212	2	no	—
118	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
119	C₂²	Q₈×Dic₅₃	1	2	yes	—
120	C₂²	C₂×#11719917	1	2	yes	—
121	C₂²	#13133224	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	4	2	no	—
126	C₄	#13133193	1	2	yes	—
127	C₄	#13133206	1	1	yes	characteristic
128	C₄	#13133215	1	1	yes	characteristic
129	C₂	#16810530	1	1	yes	characteristic
130	C₂	#16810587	1	1	yes	characteristic
131	C₂	#16810590	1	1	yes	characteristic
132	C₂	#16810754	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal, characteristic

C₂×#13133128

C₂

yes

maximal, characteristic

#16810378

C₂

yes

maximal

C₂×C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

C₄⋊#88322

—

#13133128

—

C₂²×Q₈×C₅₃

C₄

yes

characteristic

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

D₄

yes

—

—

—

—

—

Q₈

yes

complemented

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×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

M₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

#88322

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

—

C₂²×C₅₃

C₂×M₁₆

yes

—

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#124

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₄×Q₈

D₅₃

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#97

—

#336

—

C₅₃

#1985

yes

characteristic, complemented, Sylow

C₂²×Q₈

Dic₅₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₅₃

yes

characteristic

C₂×C₄⋊C₄

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

#97

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₅₃

yes

—

100

C₄²

C₂²⋊D₅₃

yes

—

101

C₂³

C₂²×Dic₅₃

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#11719917

yes

—

110

C₂×C₄

C₂²⋊Dic₅₃

yes

—

111

C₂×C₄

C₂²⋊Dic₅₃

yes

—

112

C₂×C₄

Q₈×D₅₃

yes

—

113

C₂×C₄

#11719949

yes

—

114

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

115

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

116

C₈

—

106

—

117

C₈

—

212

—

118

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

119

C₂²

Q₈×Dic₅₃

yes

—

120

C₂²

C₂×#11719917

yes

—

121

C₂²

#13133224

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#13133193

yes

—

127

C₄

#13133206

yes

characteristic

128

C₄

#13133215

yes

characteristic

129

C₂

#16810530

yes

characteristic

130

C₂

#16810587

yes

characteristic

131

C₂

#16810590

yes

characteristic

132

C₂

#16810754

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₄
Fitting	C₂×C₄×Q₈×C₅₃	C₂
hypercenter	C₂×C₄×Q₈	D₅₃
hypocenter	C₅₃	#1985
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator, socle

C₂³×C₅₃

C₂²×C₄

Fitting

C₂×C₄×Q₈×C₅₃

C₂

hypercenter

C₂×C₄×Q₈

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
33. C₂×#88322	Q₈	semidirect
71. C₅₃	#1985	semidirect

H ≅ G/N

type

33. C₂×#88322

Q₈

semidirect

71. C₅₃

#1985

semidirect

Subgroups — #20275656

Conjugacy classes of subgroups

Complements