Group #27296096 — subgroups

Subgroups — #27296096

Order 13568.

206 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 79 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×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#20273613	C₂	1	2	yes	maximal
3	#20272789	C₂	1	1	yes	maximal, characteristic
4	#20272680	C₂	1	2	yes	maximal
5	C₅₃×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₄	C₄	1	1	yes	characteristic
7	#16810753	—	2	2	no	—
8	C₄×Dic₂₁₂	C₂²	1	2	yes	—
9	#16810449	C₄	1	1	yes	characteristic
10	#16810428	C₄	1	2	yes	—
11	#16810376	C₂²	1	2	yes	—
12	#16810355	—	2	2	no	—
13	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₈²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₀₆	—	2	2	no	—
18	Dic₄₂₄	—	2	2	no	—
19	#13133151	C₂×C₄	1	2	yes	—
20	C₄⋊#88322	—	2	2	no	—
21	#13133146	—	4	2	no	—
22	#13133142	—	2	2	no	—
23	#13133127	—	4	2	no	—
24	C₂×C₁₆×C₅₃	—	2	1	no	—
25	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₀₆	—	2	2	no	—
31	C₂×C₈×C₅₃	—	2	1	no	—
32	C₂×C₈×C₅₃	D₈	1	2	yes	—
33	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
35	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₅₃	—	2	2	no	—
37	#11719919	—	4	2	no	—
38	C₄×Dic₅₃	—	4	2	no	—
39	C₂×#88322	—	4	2	no	—
40	Dic₂₁₂	—	2	2	no	—
41	Dic₂₁₂	—	2	2	no	—
42	C₁₆×C₅₃	—	2	2	no	—
43	C₁₆×C₅₃	—	2	1	no	—
44	C₁₆×C₅₃	M₁₆	1	2	yes	—
45	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
48	C₂×Dic₅₃	—	4	2	no	—
49	C₂×Dic₅₃	—	4	2	no	—
50	Dic₁₀₆	—	4	2	no	—
51	Dic₁₀₆	—	4	2	no	—
52	C₈×C₅₃	—	2	1	no	—
53	C₈×C₅₃	—	2	2	no	—
54	C₈×C₅₃	—	4	1	no	—
55	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
56	C₈×C₅₃	#131	1	1	yes	characteristic
57	#88322	—	8	2	no	—
58	#12085	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	#436	1	1	yes	characteristic
62	C₄×C₅₃	#443	1	1	yes	characteristic
63	C₄×C₅₃	#510	1	2	yes	—
64	Dic₅₃	—	8	2	no	—
65	Dic₅₃	—	8	2	no	—
66	C₄×Q₃₂	—	53	1	no	—
67	#1591	—	53	2	no	—
68	#1372	—	53	1	no	—
69	#1332	—	53	2	no	—
70	#1311	D₅₃	1	1	yes	characteristic
71	C₂×C₅₃	#1589	1	1	yes	characteristic
72	C₂×C₅₃	#2176	1	1	yes	characteristic
73	C₂×C₅₃	#2179	1	1	yes	characteristic
74	C₂×Q₃₂	—	53	1	no	—
75	#446	—	106	2	no	—
76	C₄×Q₁₆	—	53	2	no	—
77	#366	—	53	1	no	—
78	#358	—	53	2	no	—
79	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
80	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
81	#335	—	53	2	no	—
82	Q₈⋊C₈	—	106	2	no	—
83	C₈²	C₂×D₅₃	1	1	yes	characteristic
84	C₅₃	#12085	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	—	53	2	no	—
86	C₄×Q₈	—	106	2	no	—
87	Q₃₂	—	106	2	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
90	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
91	#107	—	53	2	no	—
92	C₄⋊C₈	—	106	2	no	—
93	C₄⋊C₈	—	212	2	no	—
94	#103	—	106	2	no	—
95	#97	—	212	2	no	—
96	C₄×C₈	D₂₁₂	1	1	yes	characteristic
97	C₄×C₈	D₂₁₂	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
99	C₂×Q₈	—	106	2	no	—
100	Q₁₆	—	106	2	no	—
101	Q₁₆	—	106	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	212	2	no	—
104	C₂×C₈	D₄₂₄	1	2	yes	—
105	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
106	C₂×C₈	#11719945	1	1	yes	characteristic
107	C₄⋊C₄	—	106	2	no	—
108	C₄⋊C₄	—	212	2	no	—
109	C₄²	—	212	2	no	—
110	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	#11719912	1	2	yes	—
114	Q₈	—	212	2	no	—
115	Q₈	—	212	2	no	—
116	C₂×C₄	—	212	2	no	—
117	C₂×C₄	—	212	2	no	—
118	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
120	C₂×C₄	#13133239	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	1	no	—
124	C₈	—	424	2	no	—
125	C₈	C₂×#11719912	1	1	yes	characteristic
126	C₈	#13133223	1	1	yes	characteristic
127	C₂²	C₄×D₄₂₄	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	424	2	no	—
130	C₄	—	424	2	no	—
131	C₄	#16810702	1	1	yes	characteristic
132	C₄	#16810741	1	1	yes	characteristic
133	C₄	#16811066	1	2	yes	—
134	C₂	#20273601	1	1	yes	characteristic
135	C₂	#20276587	1	1	yes	characteristic
136	C₂	#20276602	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₄₂₄

C₂

yes

maximal, characteristic

#20273613

C₂

yes

maximal

#20272789

C₂

yes

maximal, characteristic

#20272680

C₂

yes

maximal

C₅₃×#1311

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

C₄

yes

characteristic

#16810753

—

C₄×Dic₂₁₂

C₂²

yes

—

#16810449

C₄

yes

characteristic

#16810428

C₄

yes

—

#16810376

C₂²

yes

—

#16810355

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

C₂×C₄

yes

—

C₄×Dic₁₀₆

—

Dic₄₂₄

—

#13133151

C₂×C₄

yes

—

—

—

—

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×M₁₆

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

C₄×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

Dic₅₃

—

C₄×Q₃₂

—

—

—

—

D₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×Q₃₂

—

#446

—

106

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₄×C₁₆

C₂×D₅₃

yes

characteristic

#335

—

Q₈⋊C₈

—

106

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12085

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

106

—

Q₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₃

yes

characteristic

C₂×C₁₆

C₄×D₅₃

yes

characteristic

#107

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

#103

—

106

—

#97

—

212

—

C₄×C₈

D₂₁₂

yes

characteristic

C₄×C₈

D₂₁₂

yes

characteristic

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

100

Q₁₆

—

106

—

101

Q₁₆

—

106

—

102

C₂×C₈

—

103

C₂×C₈

—

212

—

104

C₂×C₈

D₄₂₄

yes

—

105

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

106

C₂×C₈

#11719945

yes

characteristic

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

—

212

—

109

C₄²

—

212

—

110

C₄²

C₂×D₂₁₂

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

#11719912

yes

—

114

Q₈

—

212

—

115

Q₈

—

212

—

116

C₂×C₄

—

212

—

117

C₂×C₄

—

212

—

118

C₂×C₄

C₄×D₂₁₂

yes

characteristic

119

C₂×C₄

C₂×D₄₂₄

yes

characteristic

120

C₂×C₄

#13133239

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

424

—

125

C₈

C₂×#11719912

yes

characteristic

126

C₈

#13133223

yes

characteristic

127

C₂²

C₄×D₄₂₄

yes

characteristic

128

C₄

—

129

C₄

—

424

—

130

C₄

—

424

—

131

C₄

#16810702

yes

characteristic

132

C₄

#16810741

yes

characteristic

133

C₄

#16811066

yes

—

134

C₂

#20273601

yes

characteristic

135

C₂

#20276587

yes

characteristic

136

C₂

#20276602

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1311	C₂
socle, prime radical	C₂²×C₅₃	C₄×D₈
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1311	D₅₃
hypocenter	C₅₃	#12085
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂×C₄

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1311

C₂

socle, prime radical

C₂²×C₅₃

C₄×D₈

cosocle

C₄×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

N	H ≅ G/N	type
84. C₅₃	#12085	semidirect

H ≅ G/N

type

84. C₅₃

#12085

semidirect

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

subgroup	index
C₄×Dic₄₂₄	2
#20273613	2
#20272789	2
#20272680	2
C₅₃×#1311	2
#12085	53

subgroup

index

C₄×Dic₄₂₄

C₅₃×#1311

Subgroups — #27296096

Conjugacy classes of subgroups

Complements

Maximal subgroups