Group #28312585 — subgroups

Subgroups — #28312585

Order 13568.

206 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 128 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×#16811555	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133267	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13133267	C₂	1	4	yes	maximal, complemented
4	#20276312	C₂	1	1	yes	maximal, characteristic
5	#16811555	—	2	2	no	—
6	C₂×#13133267	—	2	2	no	—
7	C₂×#13133267	C₂²	1	2	yes	—
8	C₂×#13133267	C₂²	1	2	yes	—
9	C₂×#13133157	C₄	1	1	yes	characteristic
10	C₂²×#11719908	C₂²	1	1	yes	characteristic
11	#16810399	C₄	1	2	yes	—
12	C₄×#11719908	C₂²	1	2	yes	—
13	C₅₃×#405	C₄	1	1	yes	characteristic
14	#13133267	—	2	2	no	—
15	#13133267	—	2	2	no	—
16	#13133157	—	2	2	no	—
17	C₂×#11719908	C₂×C₄	1	2	yes	—
18	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
19	C₂×#11719908	D₄	1	2	yes	—
20	C₂×#11719908	Q₈	1	2	yes	—
21	C₂×#11719908	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₅₃	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	#117×C₅₃	C₈	1	1	yes	characteristic
25	#97×C₅₃	C₈	1	2	yes	—
26	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
28	M₁₆×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
33	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₄⋊C₄×C₅₃	C₁₆	1	2	yes	—
35	C₂²⋊C₄×C₅₃	C₁₆	1	2	yes	—
36	C₄²×C₅₃	C₂×C₈	1	2	yes	—
37	#11719908	—	2	2	no	—
38	#11719908	C₄⋊C₄	1	2	yes	—
39	#11719908	C₂×D₄	1	1	yes	characteristic
40	#11719908	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×C₁₆	1	4	yes	—
44	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
46	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₅₃	—	2	2	no	—
48	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
49	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
51	#13628	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
54	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
55	C₄×C₅₃	—	2	4	no	—
56	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
57	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
59	C₂×M₆₄	—	53	1	no	—
60	C₂²×C₃₂	—	53	1	no	—
61	#2108	—	53	1	no	—
62	C₄⋊C₃₂	—	53	4	no	—
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₅₃	#2258	1	2	yes	—
66	C₂×M₃₂	—	53	1	no	—
67	C₂²×C₁₆	—	53	1	no	—
68	#405	#6157	1	1	yes	characteristic
69	M₆₄	—	106	2	no	—
70	C₂×C₃₂	—	53	2	no	—
71	C₂×C₃₂	—	53	2	no	—
72	C₂×C₃₂	—	106	2	no	—
73	#346	—	53	2	no	—
74	C₄×C₁₆	—	53	2	no	—
75	C₅₃	#13628	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	#88324	1	1	yes	characteristic
77	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
78	#117	#88324	1	1	yes	characteristic
79	M₃₂	—	106	2	no	—
80	C₂×C₁₆	—	53	2	no	—
81	C₂×C₁₆	—	53	2	no	—
82	C₂×C₁₆	—	53	1	no	—
83	C₂×C₁₆	—	53	1	no	—
84	C₂×C₁₆	—	53	2	no	—
85	#97	#88324	1	2	yes	—
86	C₄×C₈	C₂×#6157	1	2	yes	—
87	C₃₂	—	106	2	no	—
88	C₃₂	—	106	2	no	—
89	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	C₂×C₈	#11719936	1	2	yes	—
92	C₂×C₈	C₄⋊#6157	1	2	yes	—
93	C₂×C₈	C₂×#88324	1	2	yes	—
94	C₂×C₈	C₂×#88324	1	1	yes	characteristic
95	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
96	C₄⋊C₄	#11719910	1	2	yes	—
97	C₂²⋊C₄	#11719910	1	2	yes	—
98	C₄²	C₂×#88324	1	2	yes	—
99	C₁₆	—	53	1	no	—
100	C₁₆	—	53	2	no	—
101	C₁₆	—	53	1	no	—
102	C₁₆	—	106	2	no	—
103	C₂³	C₂×#11719910	1	1	yes	characteristic
104	C₂×C₄	C₄⋊#88324	1	2	yes	—
105	C₂×C₄	C₂×#11719910	1	4	yes	—
106	C₂×C₄	C₂×#11719910	1	1	yes	characteristic
107	C₂×C₄	#13133291	1	2	yes	—
108	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	C₄⋊#88324	1	2	yes	—
111	C₈	C₂×C₄⋊#6157	1	1	yes	characteristic
112	C₈	C₂×#11719936	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	C₄⋊#11719910	1	2	yes	—
115	C₂²	C₂²×#11719910	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	C₄⋊#11719910	1	2	yes	—
118	C₄	C₂×C₄⋊#88324	1	1	yes	characteristic
119	C₄	C₂×#13133291	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊#11719910	1	1	yes	characteristic
122	C₂	#20291138	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811555

C₂

yes

maximal, characteristic

C₂²×#13133267

C₂

yes

maximal, characteristic

C₄⋊#13133267

C₂

yes

maximal, complemented

#20276312

C₂

yes

maximal, characteristic

#16811555

—

C₂×#13133267

—

C₂×#13133267

C₂²

yes

—

C₂×#13133267

C₂²

yes

—

C₂×#13133157

C₄

yes

characteristic

C₂²×#11719908

C₂²

yes

characteristic

#16810399

C₄

yes

—

C₄×#11719908

C₂²

yes

—

C₅₃×#405

C₄

yes

—

—

—

C₂×C₄

yes

—

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

D₄

yes

—

C₂×#11719908

Q₈

yes

—

C₂×#11719908

C₂³

yes

characteristic

C₂×M₁₆×C₅₃

C₈

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

#117×C₅₃

C₈

yes

characteristic

#97×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₅₃

C₁₆

yes

—

C₂²⋊C₄×C₅₃

C₁₆

yes

—

C₄²×C₅₃

C₂×C₈

yes

—

#11719908

—

#11719908

C₄⋊C₄

yes

—

#11719908

C₂×D₄

yes

characteristic

#11719908

C₂×Q₈

yes

characteristic

C₂³×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

M₃₂

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

characteristic

#13628

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₁₆

yes

—

C₂²×C₅₃

C₂²×C₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄⋊C₁₆

yes

—

C₄×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₂×M₃₂

yes

characteristic

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

—

C₄⋊C₃₂

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₅₃

#2258

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

#6157

yes

characteristic

M₆₄

—

106

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

106

—

#346

—

C₄×C₁₆

—

C₅₃

#13628

yes

characteristic, complemented, Sylow

C₂×M₁₆

yes

C₂²×C₈

yes

yes

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

#88324

yes

—

C₄×C₈

C₂×#6157

yes

—

C₃₂

—

106

—

C₃₂

—

106

—

C₂²×C₄

C₂×#88324

yes

characteristic

M₁₆

—

C₂×C₈

#11719936

yes

—

C₂×C₈

C₄⋊#6157

yes

—

C₂×C₈

C₂×#88324

yes

—

C₂×C₈

C₂×#88324

yes

characteristic

C₂×C₈

C₂²×#6157

yes

characteristic

C₄⋊C₄

#11719910

yes

—

C₂²⋊C₄

#11719910

yes

—

C₄²

C₂×#88324

yes

—

C₁₆

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

106

—

103

C₂³

C₂×#11719910

yes

characteristic

104

C₂×C₄

C₄⋊#88324

yes

—

105

C₂×C₄

C₂×#11719910

yes

—

106

C₂×C₄

C₂×#11719910

yes

characteristic

107

C₂×C₄

#13133291

yes

—

108

C₂×C₄

C₂²×#88324

yes

characteristic

109

C₈

—

110

C₈

C₄⋊#88324

yes

—

111

C₈

C₂×C₄⋊#6157

yes

characteristic

112

C₈

C₂×#11719936

yes

characteristic

113

C₂²

—

114

C₂²

C₄⋊#11719910

yes

—

115

C₂²

C₂²×#11719910

yes

characteristic

116

C₄

—

117

C₄

C₄⋊#11719910

yes

—

118

C₄

C₂×C₄⋊#88324

yes

characteristic

119

C₄

C₂×#13133291

yes

characteristic

120

C₂

—

121

C₂

C₂×C₄⋊#11719910

yes

characteristic

122

C₂

#20291138

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₅₃×#405	C₄
prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂×#11719908	C₂³
hypercenter	#405	#6157
hypocenter	C₅₃	#13628
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂²×C₁₆

Fitting

C₅₃×#405

C₄

prime radical

C₂³×C₅₃

C₂×C₁₆

C₂³

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ #405

N	H ≅ G/N	type
3. C₄⋊#13133267	C₂	semidirect
75. C₅₃	#13628	semidirect

H ≅ G/N

type

3. C₄⋊#13133267

C₂

semidirect

75. C₅₃

#13628

semidirect

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

subgroup	index
C₂×#16811555	2
C₂²×#13133267	2
C₄⋊#13133267	2
#20276312	2
#13628	53

subgroup

index

Subgroups — #28312585

Conjugacy classes of subgroups

Complements

Maximal subgroups