Group #28509236 — subgroups

Subgroups — D₄×C₅₉⋊C₃₂ #28509236

Order 15104.

214 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 129 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	C₂²×C₅₉⋊C₃₂	C₂	1	2	yes	maximal, complemented
2	C₄⋊(C₅₉⋊C₃₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₅₉⋊C₃₂)	C₂	1	2	yes	maximal, complemented
4	C₄×C₅₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅₉⋊C₃₂	—	2	4	no	—
7	C₂×C₅₉⋊C₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₅₉⋊C₃₂	C₂²	1	2	yes	—
9	C₂×C₅₉⋊C₃₂	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
11	C₈×D₄×C₅₉	C₄	1	1	yes	characteristic
12	C₅₉×C₄⋊C₁₆	C₄	1	1	yes	characteristic
13	C₅₉×C₂²⋊C₁₆	C₄	1	2	yes	—
14	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
15	C₅₉⋊C₃₂	—	2	2	no	—
16	C₅₉⋊C₃₂	—	2	1	no	—
17	C₅₉⋊C₃₂	D₄	1	2	yes	direct product factor
18	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
19	C₄×D₄×C₅₉	C₈	1	1	yes	characteristic
20	C₂×C₁₆×C₅₉	—	2	2	no	—
21	C₂×C₁₆×C₅₉	—	2	2	no	—
22	C₂×C₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₅₉	C₈	1	1	yes	characteristic
26	C₂²⋊C₈×C₅₉	C₈	1	2	yes	—
27	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×D₄×C₅₉	C₁₆	1	1	yes	characteristic
29	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₅₉	—	2	2	no	—
31	C₂×C₈×C₅₉	—	2	2	no	—
32	C₂×C₈×C₅₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₅₉	C₁₆	1	1	yes	characteristic
36	C₂²⋊C₄×C₅₉	C₁₆	1	2	yes	—
37	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
38	C₁₆×C₅₉	—	2	2	no	—
39	C₁₆×C₅₉	—	2	1	no	—
40	C₁₆×C₅₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₅₉	#41	1	1	yes	characteristic
42	C₂³×C₅₉	C₂×C₁₆	1	2	yes	—
43	D₄×C₅₉	C₃₂	1	4	yes	complemented
44	C₂×C₄×C₅₉	—	2	2	no	—
45	C₂×C₄×C₅₉	—	2	2	no	—
46	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₅₉	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₅₉	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₅₉	—	2	2	no	—
51	C₈×C₅₉	—	2	1	no	—
52	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
53	C₈×C₅₉	#131	1	1	yes	characteristic
54	D₄×C₃₂	—	59	1	no	maximal, Sylow
55	C₂²×C₅₉	—	2	2	no	—
56	C₂²×C₅₉	—	2	2	no	—
57	C₂²×C₅₉	C₂×C₃₂	1	4	yes	complemented
58	C₂²×C₅₉	C₂²×C₁₆	1	1	yes	characteristic
59	C₄×C₅₉	—	2	2	no	—
60	C₄×C₅₉	—	2	1	no	—
61	C₄×C₅₉	C₂×C₃₂	1	2	yes	complemented
62	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
63	C₄×C₅₉	#504	1	1	yes	characteristic
64	C₂²×C₃₂	—	59	2	no	—
65	D₄×C₁₆	D₅₉	1	1	yes	characteristic
66	C₄⋊C₃₂	—	59	1	no	—
67	C₂²⋊C₃₂	—	59	2	no	—
68	C₄×C₃₂	—	59	1	no	—
69	C₂×C₅₉	—	2	4	no	—
70	C₂×C₅₉	D₄×C₁₆	1	1	yes	characteristic
71	C₂×C₅₉	C₂²×C₃₂	1	1	yes	characteristic
72	C₂×C₅₉	#2258	1	1	yes	characteristic
73	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
74	C₈×D₄	Dic₅₉	1	1	yes	characteristic
75	C₂×C₃₂	—	59	1	no	—
76	C₂×C₃₂	—	59	1	no	—
77	C₂×C₃₂	—	59	2	no	—
78	C₂×C₃₂	—	118	4	no	—
79	C₄⋊C₁₆	Dic₅₉	1	1	yes	characteristic
80	C₂²⋊C₁₆	Dic₅₉	1	2	yes	—
81	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
82	C₅₉	D₄×C₃₂	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
84	C₄×D₄	C₅₉⋊C₈	1	1	yes	characteristic
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	C₂×Dic₅₉	1	1	yes	characteristic
88	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
89	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
90	C₄⋊C₈	C₅₉⋊C₈	1	1	yes	characteristic
91	C₂²⋊C₈	C₅₉⋊C₈	1	2	yes	—
92	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
93	C₃₂	—	59	2	no	—
94	C₃₂	—	118	2	no	—
95	C₃₂	—	118	1	no	—
96	C₂×D₄	C₅₉⋊C₁₆	1	1	yes	characteristic
97	C₂²×C₄	C₂×C₅₉⋊C₈	1	2	yes	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×C₅₉⋊C₈	1	1	yes	characteristic
101	C₂×C₈	C₂×C₅₉⋊C₈	1	2	yes	—
102	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
103	C₄⋊C₄	C₅₉⋊C₁₆	1	1	yes	characteristic
104	C₂²⋊C₄	C₅₉⋊C₁₆	1	2	yes	—
105	C₄²	C₂×C₅₉⋊C₈	1	1	yes	characteristic
106	C₁₆	—	2	2	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	D₄×D₅₉	1	1	yes	characteristic
109	C₁₆	#11745687	1	1	yes	characteristic
110	C₂³	C₂×C₅₉⋊C₁₆	1	2	yes	—
111	D₄	C₅₉⋊C₃₂	1	4	yes	direct product factor
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
115	C₂×C₄	C₂×C₅₉⋊C₁₆	1	2	yes	—
116	C₂×C₄	C₂×C₅₉⋊C₁₆	1	1	yes	characteristic
117	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	2	1	no	—
120	C₈	D₄×Dic₅₉	1	1	yes	characteristic
121	C₈	#14273734	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂×C₅₉⋊C₃₂	1	4	yes	complemented
125	C₂²	C₂²×C₅₉⋊C₁₆	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	2	2	no	—
128	C₄	C₂×C₅₉⋊C₃₂	1	2	yes	complemented
129	C₄	D₄×C₅₉⋊C₈	1	1	yes	characteristic
130	C₄	#16837819	1	1	yes	characteristic
131	C₂	—	2	4	no	—
132	C₂	D₄×C₅₉⋊C₁₆	1	1	yes	characteristic
133	C₂	C₂²×C₅₉⋊C₃₂	1	1	yes	characteristic
134	C₂	#21426390	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅₉⋊C₃₂

C₂

yes

maximal, complemented

C₄⋊(C₅₉⋊C₃₂)

C₂

yes

maximal, characteristic, complemented

C₂²⋊(C₅₉⋊C₃₂)

C₂

yes

maximal, complemented

C₄×C₅₉⋊C₃₂

C₂

yes

maximal, characteristic, complemented

D₄×C₁₆×C₅₉

C₂

yes

maximal, characteristic

C₂×C₅₉⋊C₃₂

—

C₂×C₅₉⋊C₃₂

C₂²

yes

characteristic

C₂×C₅₉⋊C₃₂

C₂²

yes

—

C₂×C₅₉⋊C₃₂

C₂²

yes

characteristic

C₂²×C₁₆×C₅₉

C₂²

yes

—

C₈×D₄×C₅₉

C₄

yes

characteristic

C₅₉×C₄⋊C₁₆

C₄

yes

characteristic

C₅₉×C₂²⋊C₁₆

C₄

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₅₉⋊C₃₂

—

C₅₉⋊C₃₂

—

C₅₉⋊C₃₂

D₄

yes

direct product factor

C₂²×C₈×C₅₉

C₂×C₄

yes

—

C₄×D₄×C₅₉

C₈

yes

characteristic

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₄⋊C₈×C₅₉

C₈

yes

characteristic

C₂²⋊C₈×C₅₉

C₈

yes

—

C₄×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×D₄×C₅₉

C₁₆

yes

characteristic

C₂²×C₄×C₅₉

C₂×C₈

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₈

yes

characteristic

C₂×C₈×C₅₉

C₂×C₈

yes

—

C₂×C₈×C₅₉

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₅₉

C₁₆

yes

characteristic

C₂²⋊C₄×C₅₉

C₁₆

yes

—

C₄²×C₅₉

C₂×C₈

yes

characteristic

C₁₆×C₅₉

—

C₁₆×C₅₉

—

C₁₆×C₅₉

C₂×D₄

yes

characteristic

C₁₆×C₅₉

#41

yes

characteristic

C₂³×C₅₉

C₂×C₁₆

yes

—

D₄×C₅₉

C₃₂

yes

complemented

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂×C₁₆

yes

—

C₂×C₄×C₅₉

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₈

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

—

C₈×C₅₉

C₄×D₄

yes

characteristic

C₈×C₅₉

#131

yes

characteristic

D₄×C₃₂

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₃₂

yes

complemented

C₂²×C₅₉

C₂²×C₁₆

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₃₂

yes

complemented

C₄×C₅₉

C₈×D₄

yes

characteristic

C₄×C₅₉

#504

yes

characteristic

C₂²×C₃₂

—

D₄×C₁₆

D₅₉

yes

characteristic

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

C₄×C₃₂

—

C₂×C₅₉

—

C₂×C₅₉

D₄×C₁₆

yes

characteristic

C₂×C₅₉

C₂²×C₃₂

yes

characteristic

C₂×C₅₉

#2258

yes

characteristic

C₂²×C₁₆

C₂×D₅₉

yes

—

C₈×D₄

Dic₅₉

yes

characteristic

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

118

—

C₄⋊C₁₆

Dic₅₉

yes

characteristic

C₂²⋊C₁₆

Dic₅₉

yes

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₅₉

D₄×C₃₂

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₅₉

yes

—

C₄×D₄

C₅₉⋊C₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₅₉

yes

characteristic

C₂×C₁₆

C₂×Dic₅₉

yes

—

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

C₄⋊C₈

C₅₉⋊C₈

yes

characteristic

C₂²⋊C₈

C₅₉⋊C₈

yes

—

C₄×C₈

C₂×Dic₅₉

yes

characteristic

C₃₂

—

C₃₂

—

118

—

C₃₂

—

118

—

C₂×D₄

C₅₉⋊C₁₆

yes

characteristic

C₂²×C₄

C₂×C₅₉⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₂×C₅₉⋊C₈

yes

characteristic

101

C₂×C₈

C₂×C₅₉⋊C₈

yes

—

102

C₂×C₈

C₂²×Dic₅₉

yes

characteristic

103

C₄⋊C₄

C₅₉⋊C₁₆

yes

characteristic

104

C₂²⋊C₄

C₅₉⋊C₁₆

yes

—

105

C₄²

C₂×C₅₉⋊C₈

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

D₄×D₅₉

yes

characteristic

109

C₁₆

#11745687

yes

characteristic

110

C₂³

C₂×C₅₉⋊C₁₆

yes

—

111

D₄

C₅₉⋊C₃₂

yes

direct product factor

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₅₉⋊C₁₆

yes

characteristic

115

C₂×C₄

C₂×C₅₉⋊C₁₆

yes

—

116

C₂×C₄

C₂×C₅₉⋊C₁₆

yes

characteristic

117

C₂×C₄

C₂²×C₅₉⋊C₈

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

D₄×Dic₅₉

yes

characteristic

121

C₈

#14273734

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₅₉⋊C₃₂

yes

complemented

125

C₂²

C₂²×C₅₉⋊C₁₆

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

C₂×C₅₉⋊C₃₂

yes

complemented

129

C₄

D₄×C₅₉⋊C₈

yes

characteristic

130

C₄

#16837819

yes

characteristic

131

C₂

—

132

C₂

D₄×C₅₉⋊C₁₆

yes

characteristic

133

C₂

C₂²×C₅₉⋊C₃₂

yes

characteristic

134

C₂

#21426390

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₉
commutator	C₂×C₅₉	C₂²×C₃₂
Fitting	D₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×C₁₆
prime radical	C₂×D₄×C₅₉	C₁₆
cosocle	C₂×C₁₆×C₅₉	C₂³
hypercenter	D₄×C₁₆	D₅₉
hypocenter	C₅₉	D₄×C₃₂
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₁₆ ◁ D₄×C₁₆

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₅₉

commutator

C₂×C₅₉

C₂²×C₃₂

Fitting

D₄×C₁₆×C₅₉

C₂

socle

C₂²×C₅₉

C₂²×C₁₆

prime radical

C₂×D₄×C₅₉

C₁₆

cosocle

C₂×C₁₆×C₅₉

C₂³

hypercenter

D₄×C₁₆

D₅₉

hypocenter

C₅₉

D₄×C₃₂

LCS

G ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₁₆ ◁ D₄×C₁₆

N	H ≅ G/N	type
1. C₂²×C₅₉⋊C₃₂	C₂	semidirect
2. C₄⋊(C₅₉⋊C₃₂)	C₂	semidirect
3. C₂²⋊(C₅₉⋊C₃₂)	C₂	semidirect
4. C₄×C₅₉⋊C₃₂	C₂	semidirect
17. C₅₉⋊C₃₂	D₄	direct (N × H)
43. D₄×C₅₉	C₃₂	semidirect
57. C₂²×C₅₉	C₂×C₃₂	semidirect
61. C₄×C₅₉	C₂×C₃₂	semidirect
82. C₅₉	D₄×C₃₂	semidirect
111. D₄	C₅₉⋊C₃₂	direct (N × H)
124. C₂²	C₂×C₅₉⋊C₃₂	semidirect
128. C₄	C₂×C₅₉⋊C₃₂	semidirect

H ≅ G/N

type

1. C₂²×C₅₉⋊C₃₂

C₂

semidirect

2. C₄⋊(C₅₉⋊C₃₂)

C₂

semidirect

3. C₂²⋊(C₅₉⋊C₃₂)

C₂

semidirect

4. C₄×C₅₉⋊C₃₂

C₂

semidirect

17. C₅₉⋊C₃₂

D₄

direct (N × H)

43. D₄×C₅₉

C₃₂

semidirect

57. C₂²×C₅₉

C₂×C₃₂

semidirect

61. C₄×C₅₉

C₂×C₃₂

semidirect

82. C₅₉

D₄×C₃₂

semidirect

111. D₄

C₅₉⋊C₃₂

direct (N × H)

124. C₂²

C₂×C₅₉⋊C₃₂

semidirect

128. C₄

C₂×C₅₉⋊C₃₂

semidirect

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

subgroup	index
C₂²×C₅₉⋊C₃₂	2
C₄⋊(C₅₉⋊C₃₂)	2
C₂²⋊(C₅₉⋊C₃₂)	2
C₄×C₅₉⋊C₃₂	2
D₄×C₁₆×C₅₉	2
D₄×C₃₂	59

subgroup

index

C₂²×C₅₉⋊C₃₂

C₄⋊(C₅₉⋊C₃₂)

C₂²⋊(C₅₉⋊C₃₂)

C₄×C₅₉⋊C₃₂

D₄×C₁₆×C₅₉

D₄×C₃₂

Subgroups — D₄×C₅₉⋊C₃₂ #28509236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D4×C59⋊C32 #28509236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₅₉⋊C₃₂ #28509236