Group #28310427 — subgroups

Subgroups — #28310427

Order 13568.

358 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 168 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×#16811511	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719910	C₂	1	1	yes	maximal, characteristic
3	#20289753	C₂	1	4	yes	maximal
4	#20274402	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₂²	1	2	yes	—
6	C₂²×#11719910	C₂²	1	2	yes	—
7	#16811511	C₄	1	4	yes	complemented
8	C₄×#11719910	C₄	1	4	yes	complemented
9	C₂×#13133128	C₄	1	2	yes	—
10	C₂×#13133128	C₄	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₅₃×#376	C₄	1	1	yes	characteristic
14	C₂×#11719910	—	2	4	no	—
15	C₂×#11719910	—	2	4	no	—
16	C₂×#11719910	C₂×C₄	1	4	yes	—
17	C₂×#11719910	C₂×C₄	1	4	yes	—
18	C₂²×#88322	C₂×C₄	1	2	yes	—
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂³	1	1	yes	characteristic
21	#13133128	—	2	4	no	—
22	#13133128	C₈	1	4	yes	—
23	C₄×#88322	—	2	4	no	—
24	C₄×#88322	C₂×C₄	1	2	yes	—
25	C₄×#88322	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	#95×C₅₃	C₈	1	4	yes	—
29	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
30	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₅₃	—	2	2	no	—
33	C₄²×C₅₃	—	2	2	no	—
34	C₄²×C₅₃	C₄²	1	2	yes	—
35	C₄²×C₅₃	C₂×C₈	1	2	yes	—
36	C₂×#88322	—	2	2	no	—
37	C₂×#88322	—	2	2	no	—
38	C₂×#88322	C₄²	1	2	yes	—
39	C₂×#88322	C₂×C₈	1	4	yes	—
40	C₂×#88322	M₁₆	1	4	yes	—
41	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
42	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
43	C₂×#88322	#41	1	4	yes	—
44	#11719910	—	2	4	no	—
45	#11719910	—	2	4	no	—
46	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	4	no	—
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	—	2	8	no	—
50	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
51	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#117	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
55	#88322	—	2	4	no	—
56	#88322	#97	1	4	yes	—
57	#88322	#117	1	2	yes	—
58	#88322	#117	1	2	yes	—
59	#12592	—	53	1	no	maximal, Sylow
60	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
61	C₂²×C₅₃	#431	1	2	yes	—
62	C₂²×C₅₃	#504	1	4	yes	—
63	C₄×C₅₃	—	2	4	no	—
64	C₄×C₅₃	—	4	4	no	—
65	C₄×C₅₃	#376	1	1	yes	characteristic
66	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
67	C₄×C₅₃	#431	1	2	yes	—
68	C₂×#346	—	53	1	no	—
69	C₂×C₄×C₁₆	—	53	1	no	—
70	#1726	—	53	1	no	—
71	#1374	—	53	4	no	—
72	C₂×C₅₃	#1744	1	1	yes	characteristic
73	C₂×C₅₃	#2108	1	2	yes	—
74	C₂×C₅₃	#2164	1	4	yes	—
75	C₂²×C₁₆	—	53	2	no	—
76	C₂²×C₁₆	—	53	2	no	—
77	C₂×#97	—	53	1	no	—
78	C₂×#97	—	53	2	no	—
79	C₂×C₄×C₈	—	53	1	no	—
80	C₂×C₄×C₈	—	53	2	no	—
81	#376	#6157	1	1	yes	characteristic
82	#346	—	53	4	no	—
83	C₄×C₁₆	—	53	4	no	—
84	C₅₃	#12592	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₈	—	53	1	no	—
86	C₂²×C₈	—	53	1	no	—
87	C₂²×C₈	—	53	2	no	—
88	C₂×C₄²	C₂×#6157	1	2	yes	—
89	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
90	C₂×C₁₆	—	53	4	no	—
91	C₂×C₁₆	—	53	4	no	—
92	C₂×C₁₆	—	106	4	no	—
93	C₂×C₁₆	—	106	4	no	—
94	#97	—	53	4	no	—
95	#97	—	106	4	no	—
96	C₄×C₈	—	53	2	no	—
97	C₄×C₈	—	53	2	no	—
98	C₄×C₈	—	106	4	no	—
99	#95	#88324	1	4	yes	—
100	C₂²×C₄	C₂×#88324	1	4	yes	—
101	C₂²×C₄	C₂×#88324	1	2	yes	—
102	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
103	C₂×C₈	—	53	1	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	—	53	2	no	—
106	C₂×C₈	—	53	4	no	—
107	C₂×C₈	—	53	4	no	—
108	C₂×C₈	—	53	4	no	—
109	C₂×C₈	—	106	2	no	—
110	C₂×C₈	—	106	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	D₅₃⋊C₈	1	2	yes	—
114	C₄²	C₄×#6157	1	2	yes	—
115	C₁₆	—	106	4	no	—
116	C₁₆	—	106	4	no	—
117	C₂³	C₂²×#88324	1	1	yes	characteristic
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	C₄×#88324	1	4	yes	—
122	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
123	C₂×C₄	#13133296	1	2	yes	—
124	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
125	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
126	C₈	—	53	2	no	—
127	C₈	—	53	2	no	—
128	C₈	—	53	4	no	—
129	C₈	—	106	4	no	—
130	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
131	C₂²	#16811639	1	2	yes	—
132	C₂²	#16811681	1	4	yes	—
133	C₄	—	2	4	no	—
134	C₄	—	4	4	no	—
135	C₄	#16811559	1	1	yes	characteristic
136	C₄	#16811582	1	1	yes	characteristic
137	C₄	#16811641	1	2	yes	—
138	C₂	#20290429	1	1	yes	characteristic
139	C₂	#20290924	1	2	yes	—
140	C₂	#20291080	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811511

C₂

yes

maximal, characteristic

C₂×C₄×#11719910

C₂

yes

maximal, characteristic

#20289753

C₂

yes

maximal

#20274402

C₂

yes

maximal, characteristic

C₂²×#11719910

C₂²

yes

—

C₂²×#11719910

C₂²

yes

—

#16811511

C₄

yes

complemented

C₄×#11719910

C₄

yes

complemented

C₂×#13133128

C₄

yes

—

C₂×#13133128

C₄

yes

characteristic

C₂×C₄×#88322

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₅₃×#376

C₄

yes

—

—

C₂×C₄

yes

—

C₂×#11719910

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂³

yes

characteristic

#13133128

—

#13133128

C₈

yes

—

C₄×#88322

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#95×C₅₃

C₈

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₄²

yes

—

C₄²×C₅₃

C₂×C₈

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₄²

yes

—

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

M₁₆

yes

—

C₂×#88322

C₂²×C₄

yes

characteristic

C₂×#88322

C₂²×C₄

yes

characteristic

C₂×#88322

#41

yes

—

#11719910

—

#11719910

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

#88322

—

#88322

#97

yes

—

#88322

#117

yes

—

#88322

#117

yes

—

#12592

—

maximal, Sylow

C₂²×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂²×C₅₃

#431

yes

—

C₂²×C₅₃

#504

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#376

yes

characteristic

C₄×C₅₃

C₂×#97

yes

characteristic

C₄×C₅₃

#431

yes

—

C₂×#346

—

C₂×C₄×C₁₆

—

#1726

—

#1374

—

C₂×C₅₃

#1744

yes

characteristic

C₂×C₅₃

#2108

yes

—

C₂×C₅₃

#2164

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

yes

—

C₄×C₁₆

—

C₅₃

#12592

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×#6157

yes

—

C₂×C₄²

C₂×#6157

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

#97

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

#95

#88324

yes

—

100

C₂²×C₄

C₂×#88324

yes

—

101

C₂²×C₄

C₂×#88324

yes

—

102

C₂²×C₄

C₂²×#6157

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

106

—

111

C₄²

—

112

C₄²

—

113

C₄²

D₅₃⋊C₈

yes

—

114

C₄²

C₄×#6157

yes

—

115

C₁₆

—

106

—

116

C₁₆

—

106

—

117

C₂³

C₂²×#88324

yes

characteristic

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₄×#88324

yes

—

122

C₂×C₄

C₂×C₄×#6157

yes

characteristic

123

C₂×C₄

#13133296

yes

—

124

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

125

C₂×C₄

D₅₃⋊M₁₆

yes

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

106

—

130

C₂²

C₂×C₄×#88324

yes

characteristic

131

C₂²

#16811639

yes

—

132

C₂²

#16811681

yes

—

133

C₄

—

134

C₄

—

135

C₄

#16811559

yes

characteristic

136

C₄

#16811582

yes

characteristic

137

C₄

#16811641

yes

—

138

C₂

#20290429

yes

characteristic

139

C₂

#20290924

yes

—

140

C₂

#20291080

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₅₃×#376	C₄
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×#88322	C₂³
hypercenter	#376	#6157
hypocenter	C₅₃	#12592
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#6157

commutator

C₂²×C₅₃

C₂×C₄×C₈

Fitting

C₅₃×#376

C₄

socle, prime radical

C₂³×C₅₃

C₂²×C₈

cosocle

C₂²×#88322

C₂³

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
7. #16811511	C₄	semidirect
8. C₄×#11719910	C₄	semidirect
84. C₅₃	#12592	semidirect

H ≅ G/N

type

7. #16811511

C₄

semidirect

8. C₄×#11719910

C₄

semidirect

84. C₅₃

#12592

semidirect

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

subgroup	index
C₂×#16811511	2
C₂×C₄×#11719910	2
#20289753	2
#20274402	2
#12592	53

subgroup

index

Subgroups — #28310427

Conjugacy classes of subgroups

Complements

Maximal subgroups