Group #28311431 — subgroups

Subgroups — #28311431

Order 13568.

342 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 144 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×#16811546	C₂	1	4	yes	maximal
2	#20291047	C₂	1	2	yes	maximal
3	#20274402	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133291	C₂²	1	4	yes	—
5	#16811546	—	2	8	no	—
6	C₄⋊#11719910	—	2	4	no	—
7	C₂×#13133128	C₄	1	2	yes	—
8	C₂×#13133128	C₄	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₂×C₄×#88322	C₂²	1	2	yes	—
11	C₅₃×#376	C₄	1	1	yes	characteristic
12	#13133291	—	2	8	no	—
13	C₂×#11719910	—	2	4	no	—
14	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂²×#88322	C₂³	1	1	yes	characteristic
17	#13133128	—	2	4	no	—
18	#13133128	C₈	1	4	yes	—
19	C₄×#88322	D₄	1	2	yes	—
20	C₄×#88322	D₄	1	4	yes	—
21	C₄×#88322	Q₈	1	2	yes	—
22	C₄×#88322	Q₈	1	4	yes	—
23	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	#95×C₅₃	—	2	2	no	—
26	C₂²×C₄×C₅₃	—	2	2	no	—
27	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
28	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₅₃	—	2	2	no	—
30	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
31	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
32	C₄²×C₅₃	C₂×C₈	1	2	yes	—
33	C₂×#88322	—	2	2	no	—
34	C₂×#88322	C₄⋊C₄	1	2	yes	—
35	C₂×#88322	C₄⋊C₄	1	4	yes	—
36	C₂×#88322	C₂×C₈	1	4	yes	—
37	C₂×#88322	M₁₆	1	4	yes	—
38	C₂×#88322	C₂×D₄	1	1	yes	characteristic
39	C₂×#88322	C₂×D₄	1	2	yes	—
40	C₂×#88322	C₂×Q₈	1	2	yes	—
41	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
42	#11719910	—	4	4	no	—
43	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
44	C₂×C₄×C₅₃	—	2	4	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	—	4	4	no	—
47	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
48	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
52	#88322	—	2	2	no	—
53	#88322	C₄⋊C₈	1	8	yes	—
54	#88322	#127	1	1	yes	characteristic
55	#88322	C₄⋊Q₈	1	1	yes	characteristic
56	#88322	C₄⋊Q₈	1	2	yes	—
57	#12901	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	—	2	2	no	—
59	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₂²×C₅₃	#423	1	2	yes	—
61	C₄×C₅₃	—	2	4	no	—
62	C₄×C₅₃	—	4	4	no	—
63	C₄×C₅₃	#384	1	1	yes	characteristic
64	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
65	C₄×C₅₃	#423	1	1	yes	characteristic
66	C₂×#364	—	53	4	no	—
67	#2150	—	53	2	no	—
68	#1726	—	53	1	no	—
69	C₂×C₅₃	—	2	2	no	—
70	C₂×C₅₃	#1842	1	1	yes	characteristic
71	C₂×C₅₃	#2153	1	2	yes	—
72	C₂×M₃₂	—	53	4	no	—
73	C₂×#97	—	53	1	no	—
74	C₂×#97	—	53	2	no	—
75	C₂×C₄×C₈	—	53	2	no	—
76	C₂×C₄×C₈	—	53	1	no	—
77	#376	#6157	1	1	yes	characteristic
78	#364	—	106	8	no	—
79	C₄⋊C₁₆	—	106	4	no	—
80	C₅₃	#12901	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	53	1	no	—
82	C₂²×C₈	—	53	2	no	—
83	C₂²×C₈	—	53	1	no	—
84	C₂×C₄²	C₂×#6157	1	2	yes	—
85	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
86	M₃₂	—	106	8	no	—
87	C₂×C₁₆	—	106	4	no	—
88	#97	—	53	4	no	—
89	#97	—	106	4	no	—
90	C₄×C₈	—	53	2	no	—
91	C₄×C₈	—	53	4	no	—
92	C₄×C₈	—	53	4	no	—
93	C₄×C₈	—	53	2	no	—
94	#95	—	2	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	#11719940	1	2	yes	—
97	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
98	C₂×C₈	—	53	2	no	—
99	C₂×C₈	—	53	4	no	—
100	C₂×C₈	—	53	2	no	—
101	C₂×C₈	—	53	1	no	—
102	C₂×C₈	—	53	1	no	—
103	C₂×C₈	—	53	4	no	—
104	C₂×C₈	—	53	4	no	—
105	C₂×C₈	—	53	2	no	—
106	C₂×C₈	—	106	2	no	—
107	C₄²	—	2	2	no	—
108	C₄²	D₅₃⋊C₈	1	2	yes	—
109	C₄²	C₄⋊#6157	1	4	yes	—
110	C₄²	C₄⋊#6157	1	2	yes	—
111	C₁₆	—	212	4	no	—
112	C₂³	C₂×#11719940	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	#13133284	1	4	yes	—
117	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
119	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
120	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
121	C₈	—	53	1	no	—
122	C₈	—	53	2	no	—
123	C₈	—	53	1	no	—
124	C₈	—	53	8	no	—
125	C₈	—	106	2	no	—
126	C₂²	—	2	2	no	—
127	C₂²	C₂×#13133284	1	1	yes	characteristic
128	C₂²	#16811626	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	4	4	no	—
131	C₄	#16811569	1	1	yes	characteristic
132	C₄	D₅₃⋊(C₄⋊C₈)	1	2	yes	—
133	C₄	#16811628	1	1	yes	characteristic
134	C₂	—	2	2	no	—
135	C₂	#20290648	1	1	yes	characteristic
136	C₂	#20291058	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811546

C₂

yes

maximal

#20291047

C₂

yes

maximal

#20274402

C₂

yes

maximal, characteristic

C₂×#13133291

C₂²

yes

—

#16811546

—

C₄⋊#11719910

—

C₂×#13133128

C₄

yes

—

C₂×#13133128

C₄

yes

characteristic

C₂×C₄×#88322

C₂²

yes

characteristic

C₂×C₄×#88322

C₂²

yes

—

C₅₃×#376

C₄

yes

characteristic

#13133291

—

C₂×#11719910

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

—

C₂²×#88322

C₂³

yes

characteristic

#13133128

—

#13133128

C₈

yes

—

C₄×#88322

D₄

yes

—

C₄×#88322

D₄

yes

—

C₄×#88322

Q₈

yes

—

C₄×#88322

Q₈

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

—

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

M₁₆

yes

—

C₂×#88322

C₂×D₄

yes

characteristic

C₂×#88322

C₂×D₄

yes

—

C₂×#88322

C₂×Q₈

yes

—

C₂×#88322

C₂×Q₈

yes

characteristic

#11719910

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

#88322

—

#88322

C₄⋊C₈

yes

—

yes

C₄⋊Q₈

yes

characteristic

#88322

C₄⋊Q₈

yes

—

#12901

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₃

#423

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#384

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₈

yes

—

C₄×C₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1842

yes

characteristic

C₂×C₅₃

#2153

yes

—

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

yes

—

106

—

C₄⋊C₁₆

—

106

—

C₅₃

#12901

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×#6157

yes

—

C₂×C₄²

C₂×#6157

yes

characteristic

M₃₂

—

106

—

C₂×C₁₆

—

106

—

#97

—

#97

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

#11719940

yes

—

C₂²×C₄

C₂²×#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

106

—

107

C₄²

—

108

C₄²

D₅₃⋊C₈

yes

—

109

C₄²

C₄⋊#6157

yes

—

110

C₄²

C₄⋊#6157

yes

—

111

C₁₆

—

212

—

112

C₂³

C₂×#11719940

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#13133284

yes

—

117

C₂×C₄

C₂×C₄⋊#6157

yes

—

118

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

119

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

120

C₂×C₄

D₅₃⋊M₁₆

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

106

—

126

C₂²

—

127

C₂²

C₂×#13133284

yes

characteristic

128

C₂²

#16811626

yes

—

129

C₄

—

130

C₄

—

131

C₄

#16811569

yes

characteristic

132

C₄

D₅₃⋊(C₄⋊C₈)

yes

—

133

C₄

#16811628

yes

characteristic

134

C₂

—

135

C₂

#20290648

yes

characteristic

136

C₂

#20291058

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#376	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×#88322	C₂³
hypercenter	#376	#6157
hypocenter	C₅₃	#12901
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×C₄⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₅₃×#376

C₄

socle

C₂²×C₅₃

C₂×C₄⋊C₈

prime radical

C₂³×C₅₃

C₂×M₁₆

cosocle

C₂²×#88322

C₂³

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
80. C₅₃	#12901	semidirect

H ≅ G/N

type

80. C₅₃

#12901

semidirect

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

subgroup	index
C₂×#16811546	2
#20291047	2
#20274402	2
#12901	53

subgroup

index

Subgroups — #28311431

Conjugacy classes of subgroups

Complements

Maximal subgroups