Group #28509175 — subgroups

Subgroups — #28509175

Order 15104.

206 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 147 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×#16837236	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₅₉⋊C₃₂)	C₂	1	4	yes	maximal, complemented
4	C₅₉×#2108	C₂	1	1	yes	maximal, characteristic
5	#16837236	—	2	2	no	—
6	C₂×C₅₉⋊C₃₂	—	2	2	no	—
7	C₂×C₅₉⋊C₃₂	C₂²	1	2	yes	—
8	C₂×C₅₉⋊C₃₂	C₂²	1	2	yes	—
9	C₂×M₃₂×C₅₉	C₄	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
11	C₅₉×#405	C₄	1	1	yes	characteristic
12	C₅₉×#346	C₄	1	2	yes	—
13	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
14	C₅₉⋊C₃₂	—	2	2	no	—
15	C₅₉⋊C₃₂	—	2	2	no	—
16	C₂×M₁₆×C₅₉	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
18	#117×C₅₉	C₈	1	1	yes	characteristic
19	M₃₂×C₅₉	—	2	2	no	—
20	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
21	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
23	C₂×C₁₆×C₅₉	Q₈	1	2	yes	—
24	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
25	#97×C₅₉	C₈	1	2	yes	—
26	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
28	M₁₆×C₅₉	—	2	2	no	—
29	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
33	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
34	C₄⋊C₄×C₅₉	C₁₆	1	2	yes	—
35	C₂²⋊C₄×C₅₉	C₁₆	1	2	yes	—
36	C₄²×C₅₉	C₂×C₈	1	2	yes	—
37	C₁₆×C₅₉	—	2	2	no	—
38	C₁₆×C₅₉	C₄⋊C₄	1	2	yes	—
39	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₅₉	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₅₉	C₂×C₁₆	1	4	yes	—
44	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₅₉	M₃₂	1	2	yes	—
46	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₅₉	—	2	2	no	—
48	C₈×C₅₉	C₄⋊C₈	1	2	yes	—
49	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₅₉	C₂×M₁₆	1	1	yes	characteristic
51	#13628	—	59	1	no	maximal, Sylow
52	C₂²×C₅₉	—	2	2	no	—
53	C₂²×C₅₉	C₄⋊C₁₆	1	2	yes	—
54	C₂²×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
55	C₄×C₅₉	—	2	4	no	—
56	C₄×C₅₉	C₄⋊C₁₆	1	2	yes	—
57	C₄×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
58	C₄×C₅₉	C₂×M₃₂	1	1	yes	characteristic
59	C₂×M₆₄	—	59	1	no	—
60	C₂²×C₃₂	—	59	1	no	—
61	#2108	D₅₉	1	1	yes	characteristic
62	C₄⋊C₃₂	—	59	4	no	—
63	C₂×C₅₉	—	2	2	no	—
64	C₂×C₅₉	C₂×C₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₅₉	#2258	1	2	yes	—
66	C₂×M₃₂	Dic₅₉	1	1	yes	characteristic
67	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
68	#405	Dic₅₉	1	1	yes	characteristic
69	M₆₄	—	118	2	no	—
70	C₂×C₃₂	—	59	2	no	—
71	C₂×C₃₂	—	59	2	no	—
72	C₂×C₃₂	—	118	2	no	—
73	#346	Dic₅₉	1	2	yes	—
74	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
75	C₅₉	#13628	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	C₅₉⋊C₈	1	1	yes	characteristic
77	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
78	#117	C₅₉⋊C₈	1	1	yes	characteristic
79	M₃₂	—	2	2	no	—
80	C₂×C₁₆	Dic₁₁₈	1	2	yes	—
81	C₂×C₁₆	D₂₃₆	1	2	yes	—
82	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
83	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
84	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
85	#97	C₅₉⋊C₈	1	2	yes	—
86	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
87	C₃₂	—	118	2	no	—
88	C₃₂	—	118	2	no	—
89	C₂²×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
92	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
93	C₂×C₈	#11745664	1	2	yes	—
94	C₂×C₈	C₄⋊Dic₅₉	1	2	yes	—
95	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
96	C₄⋊C₄	C₅₉⋊C₁₆	1	2	yes	—
97	C₂²⋊C₄	C₅₉⋊C₁₆	1	2	yes	—
98	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
99	C₁₆	—	2	2	no	—
100	C₁₆	C₄⋊Dic₅₉	1	2	yes	—
101	C₁₆	C₂×Dic₁₁₈	1	1	yes	characteristic
102	C₁₆	C₂×D₂₃₆	1	1	yes	characteristic
103	C₂³	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
104	C₂×C₄	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
105	C₂×C₄	C₂×C₅₉⋊C₁₆	1	4	yes	—
106	C₂×C₄	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
107	C₂×C₄	#14273667	1	2	yes	—
108	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	C₄⋊(C₅₉⋊C₈)	1	2	yes	—
111	C₈	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
112	C₈	C₂×#11745664	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	C₄⋊(C₅₉⋊C₁₆)	1	2	yes	—
115	C₂²	C₂²×C₅₉⋊C₁₆	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	C₄⋊(C₅₉⋊C₁₆)	1	2	yes	—
118	C₄	C₂×C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
119	C₄	C₂×#14273667	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×C₄⋊(C₅₉⋊C₁₆)	1	1	yes	characteristic
122	C₂	#21426390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16837236

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₃₂

C₂

yes

maximal, characteristic

C₄⋊(C₅₉⋊C₃₂)

C₂

yes

maximal, complemented

C₅₉×#2108

C₂

yes

maximal, characteristic

#16837236

—

C₂×C₅₉⋊C₃₂

—

C₂×C₅₉⋊C₃₂

C₂²

yes

—

C₂×C₅₉⋊C₃₂

C₂²

yes

—

C₂×M₃₂×C₅₉

C₄

yes

characteristic

C₂²×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉×#405

C₄

yes

characteristic

C₅₉×#346

C₄

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

—

C₅₉⋊C₃₂

—

C₅₉⋊C₃₂

—

C₂×M₁₆×C₅₉

C₈

yes

characteristic

C₂²×C₈×C₅₉

C₂×C₄

yes

characteristic

#117×C₅₉

C₈

yes

characteristic

M₃₂×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

D₄

yes

—

C₂×C₁₆×C₅₉

Q₈

yes

—

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

#97×C₅₉

C₈

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

C₂×C₈

yes

characteristic

M₁₆×C₅₉

—

C₂×C₈×C₅₉

C₄⋊C₄

yes

—

C₂×C₈×C₅₉

C₂×C₈

yes

—

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

M₁₆

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₅₉

C₁₆

yes

—

C₂²⋊C₄×C₅₉

C₁₆

yes

—

C₄²×C₅₉

C₂×C₈

yes

—

C₁₆×C₅₉

—

C₁₆×C₅₉

C₄⋊C₄

yes

—

C₁₆×C₅₉

C₂×D₄

yes

characteristic

C₁₆×C₅₉

C₂×Q₈

yes

characteristic

C₂³×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₄×C₅₉

C₂×C₁₆

yes

—

C₂×C₄×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

M₃₂

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

C₄⋊C₈

yes

—

C₈×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₉

C₂×M₁₆

yes

characteristic

#13628

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

C₄⋊C₁₆

yes

—

C₂²×C₅₉

C₂²×C₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₄⋊C₁₆

yes

—

C₄×C₅₉

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₉

C₂×M₃₂

yes

characteristic

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

D₅₉

yes

characteristic

C₄⋊C₃₂

—

C₂×C₅₉

—

C₂×C₅₉

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₅₉

#2258

yes

—

C₂×M₃₂

Dic₅₉

yes

characteristic

C₂²×C₁₆

C₂×D₅₉

yes

characteristic

#405

Dic₅₉

yes

characteristic

M₆₄

—

118

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

118

—

#346

Dic₅₉

yes

—

C₄×C₁₆

C₂×D₅₉

yes

—

C₅₉

#13628

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₅₉⋊C₈

yes

characteristic

C₂²×C₈

C₂×Dic₅₉

yes

characteristic

#117

C₅₉⋊C₈

yes

characteristic

M₃₂

—

C₂×C₁₆

Dic₁₁₈

yes

—

C₂×C₁₆

D₂₃₆

yes

—

C₂×C₁₆

C₂×Dic₅₉

yes

—

C₂×C₁₆

C₂×Dic₅₉

yes

characteristic

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

#97

C₅₉⋊C₈

yes

—

C₄×C₈

C₂×Dic₅₉

yes

—

C₃₂

—

118

—

C₃₂

—

118

—

C₂²×C₄

C₂×C₅₉⋊C₈

yes

characteristic

M₁₆

—

C₂×C₈

C₂×C₅₉⋊C₈

yes

—

C₂×C₈

C₂×C₅₉⋊C₈

yes

characteristic

C₂×C₈

#11745664

yes

—

C₂×C₈

C₄⋊Dic₅₉

yes

—

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

C₄⋊C₄

C₅₉⋊C₁₆

yes

—

C₂²⋊C₄

C₅₉⋊C₁₆

yes

—

C₄²

C₂×C₅₉⋊C₈

yes

—

C₁₆

—

100

C₁₆

C₄⋊Dic₅₉

yes

—

101

C₁₆

C₂×Dic₁₁₈

yes

characteristic

102

C₁₆

C₂×D₂₃₆

yes

characteristic

103

C₂³

C₂×C₅₉⋊C₁₆

yes

characteristic

104

C₂×C₄

C₄⋊(C₅₉⋊C₈)

yes

—

105

C₂×C₄

C₂×C₅₉⋊C₁₆

yes

—

106

C₂×C₄

C₂×C₅₉⋊C₁₆

yes

characteristic

107

C₂×C₄

#14273667

yes

—

108

C₂×C₄

C₂²×C₅₉⋊C₈

yes

characteristic

109

C₈

—

110

C₈

C₄⋊(C₅₉⋊C₈)

yes

—

111

C₈

C₂×C₄⋊Dic₅₉

yes

characteristic

112

C₈

C₂×#11745664

yes

characteristic

113

C₂²

—

114

C₂²

C₄⋊(C₅₉⋊C₁₆)

yes

—

115

C₂²

C₂²×C₅₉⋊C₁₆

yes

characteristic

116

C₄

—

117

C₄

C₄⋊(C₅₉⋊C₁₆)

yes

—

118

C₄

C₂×C₄⋊(C₅₉⋊C₈)

yes

characteristic

119

C₄

C₂×#14273667

yes

characteristic

120

C₂

—

121

C₂

C₂×C₄⋊(C₅₉⋊C₁₆)

yes

characteristic

122

C₂

#21426390

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₉
commutator, socle	C₂²×C₅₉	C₂²×C₁₆
Fitting	C₅₉×#2108	C₂
prime radical	C₂³×C₅₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	#2108	D₅₉
hypocenter	C₅₉	#13628
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ #2108

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅₉

commutator, socle

C₂²×C₅₉

C₂²×C₁₆

Fitting

C₅₉×#2108

C₂

prime radical

C₂³×C₅₉

C₂×C₁₆

cosocle

C₂×C₁₆×C₅₉

C₂³

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂×C₁₆ ◁ #2108

N	H ≅ G/N	type
3. C₄⋊(C₅₉⋊C₃₂)	C₂	semidirect
75. C₅₉	#13628	semidirect

H ≅ G/N

type

3. C₄⋊(C₅₉⋊C₃₂)

C₂

semidirect

75. C₅₉

#13628

semidirect

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

subgroup	index
C₂×#16837236	2
C₂²×C₅₉⋊C₃₂	2
C₄⋊(C₅₉⋊C₃₂)	2
C₅₉×#2108	2
#13628	59

subgroup

index

C₂×#16837236

C₂²×C₅₉⋊C₃₂

C₄⋊(C₅₉⋊C₃₂)

C₅₉×#2108

#13628

Subgroups — #28509175

Conjugacy classes of subgroups

Complements

Maximal subgroups