Group #21425949 — subgroups

Subgroups — SD₃₂×Dic₅₉ #21425949

Order 7552.

162 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 67 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	Q₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837230	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837206	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837203	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₅₉	—	2	1	no	—
9	D₄×Dic₅₉	—	2	1	no	—
10	C₂×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#14273661	C₂²	1	1	yes	characteristic
12	#14273661	C₂²	1	1	yes	characteristic
13	#14273651	—	2	1	no	—
14	#14273648	—	2	1	no	—
15	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
18	SD₃₂×C₅₉	C₄	1	4	yes	complemented
19	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₅₉	—	2	1	no	—
21	C₂×D₄×C₅₉	—	2	1	no	—
22	C₂²×Dic₅₉	—	4	1	no	—
23	Q₁₆×C₅₉	—	2	1	no	—
24	Q₁₆×C₅₉	C₂×C₄	1	2	yes	complemented
25	D₈×C₅₉	—	2	1	no	—
26	D₈×C₅₉	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	C₂²⋊Dic₅₉	—	4	1	no	—
29	C₄⋊Dic₅₉	—	2	1	no	—
30	C₄⋊Dic₅₉	—	2	1	no	—
31	C₄⋊Dic₅₉	—	4	1	no	—
32	C₄×Dic₅₉	—	4	1	no	—
33	C₄×Dic₅₉	D₄	1	1	yes	characteristic
34	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
35	C₁₆×C₅₉	C₂×C₄	1	2	yes	complemented
36	C₅₉⋊C₁₆	—	2	1	no	—
37	C₂³×C₅₉	—	4	1	no	—
38	Q₈×C₅₉	—	2	2	no	—
39	Q₈×C₅₉	—	4	1	no	—
40	D₄×C₅₉	—	2	2	no	—
41	D₄×C₅₉	—	4	1	no	—
42	C₂×C₄×C₅₉	—	4	1	no	—
43	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₅₉	—	4	1	no	—
45	C₂×Dic₅₉	—	4	1	no	—
46	C₂×Dic₅₉	—	8	2	no	—
47	C₂×Dic₅₉	D₈	1	1	yes	characteristic
48	C₂×Dic₅₉	D₈	1	1	yes	characteristic
49	C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
50	C₈×C₅₉	#41	1	1	yes	characteristic
51	C₅₉⋊C₈	—	2	1	no	—
52	C₂²×C₅₉	—	4	2	no	—
53	C₂²×C₅₉	—	8	1	no	—
54	C₂²×C₅₉	—	8	1	no	—
55	C₂²×C₅₉	C₂×D₈	1	1	yes	characteristic
56	C₄×C₅₉	—	4	2	no	—
57	C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
58	C₄×C₅₉	#135	1	1	yes	characteristic
59	Dic₅₉	—	2	1	no	—
60	Dic₅₉	—	8	1	no	—
61	Dic₅₉	—	8	1	no	—
62	Dic₅₉	SD₃₂	1	2	yes	direct product factor
63	C₄×SD₃₂	—	59	1	no	maximal, Sylow
64	C₂×C₅₉	—	8	2	no	—
65	C₂×C₅₉	C₄×D₈	1	1	yes	characteristic
66	C₂×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
67	C₂×C₅₉	#508	1	1	yes	characteristic
68	C₂×SD₃₂	D₅₉	1	1	yes	characteristic
69	C₄×Q₁₆	—	59	1	no	—
70	C₄×D₈	—	59	1	no	—
71	#367	—	59	1	no	—
72	#358	—	59	1	no	—
73	#357	—	59	1	no	—
74	C₄×C₁₆	—	59	1	no	—
75	C₅₉	C₄×SD₃₂	1	1	yes	characteristic, complemented, Sylow
76	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
77	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
78	C₄×Q₈	—	118	1	no	—
79	C₄×D₄	—	118	1	no	—
80	SD₃₂	Dic₅₉	1	4	yes	direct product factor
81	C₂×C₁₆	—	59	1	no	—
82	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
83	#107	—	59	1	no	—
84	#107	—	59	1	no	—
85	#103	—	118	1	no	—
86	#102	—	118	1	no	—
87	C₄×C₈	—	59	1	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	2	1	no	—
90	C₂²×C₄	—	236	1	no	—
91	Q₁₆	—	2	1	no	—
92	Q₁₆	C₂×Dic₅₉	1	2	yes	complemented
93	D₈	—	2	1	no	—
94	D₈	C₂×Dic₅₉	1	2	yes	—
95	C₂×C₈	—	59	1	no	—
96	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
97	C₄⋊C₄	—	118	1	no	—
98	C₄⋊C₄	—	118	1	no	—
99	C₄⋊C₄	—	236	1	no	—
100	C₂²⋊C₄	—	236	1	no	—
101	C₄²	—	59	1	no	—
102	C₄²	—	236	1	no	—
103	C₁₆	—	118	1	no	—
104	C₁₆	C₂×Dic₅₉	1	2	yes	complemented
105	C₂³	—	4	1	no	—
106	Q₈	—	2	2	no	—
107	Q₈	—	4	1	no	—
108	D₄	—	2	2	no	—
109	D₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	59	1	no	—
112	C₂×C₄	—	59	1	no	—
113	C₂×C₄	—	472	2	no	—
114	C₂×C₄	—	236	1	no	—
115	C₂×C₄	—	236	1	no	—
116	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
117	C₈	—	118	1	no	—
118	C₈	#11745687	1	1	yes	characteristic
119	C₈	C₂²×Dic₅₉	1	1	yes	characteristic
120	C₂²	—	4	2	no	—
121	C₂²	—	8	1	no	—
122	C₂²	—	8	1	no	—
123	C₂²	D₈×D₅₉	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	59	2	no	—
126	C₄	—	118	1	no	—
127	C₄	—	472	1	no	—
128	C₄	—	472	1	no	—
129	C₄	D₄×Dic₅₉	1	1	yes	characteristic
130	C₄	#14273746	1	1	yes	characteristic
131	C₂	—	8	2	no	—
132	C₂	D₈×Dic₅₉	1	1	yes	characteristic
133	C₂	SD₃₂×D₅₉	1	1	yes	characteristic
134	C₂	#16837833	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₁₆×Dic₅₉

C₂

yes

maximal, characteristic, complemented

D₈×Dic₅₉

C₂

yes

maximal, characteristic

#16837230

C₂

yes

maximal, characteristic, complemented

#16837206

C₂

yes

maximal, characteristic, complemented

#16837203

C₂

yes

maximal, characteristic

C₁₆×Dic₅₉

C₂

yes

maximal, characteristic, complemented

C₂×SD₃₂×C₅₉

C₂

yes

maximal, characteristic

Q₈×Dic₅₉

—

D₄×Dic₅₉

—

C₂×C₅₉⋊C₁₆

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273661

C₂²

yes

characteristic

#14273651

—

#14273648

—

C₈×Dic₅₉

C₂²

yes

characteristic

C₂×Q₁₆×C₅₉

C₂²

yes

characteristic

C₂×D₈×C₅₉

C₂²

yes

characteristic

SD₃₂×C₅₉

C₄

yes

complemented

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

—

C₂²×Dic₅₉

—

Q₁₆×C₅₉

—

Q₁₆×C₅₉

C₂×C₄

yes

complemented

D₈×C₅₉

—

D₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂³

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

C₄×Dic₅₉

D₄

yes

characteristic

C₂×C₅₉⋊C₈

D₄

yes

characteristic

C₁₆×C₅₉

C₂×C₄

yes

complemented

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₂×Dic₅₉

D₈

yes

characteristic

C₂×Dic₅₉

D₈

yes

characteristic

C₈×C₅₉

C₂²×C₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

C₄×D₄

yes

characteristic

C₄×C₅₉

#135

yes

characteristic

Dic₅₉

—

Dic₅₉

—

Dic₅₉

—

Dic₅₉

SD₃₂

yes

direct product factor

C₄×SD₃₂

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₄×D₈

yes

characteristic

C₂×C₅₉

C₂×SD₃₂

yes

characteristic

C₂×C₅₉

#508

yes

characteristic

C₂×SD₃₂

D₅₉

yes

characteristic

C₄×Q₁₆

—

C₄×D₈

—

#367

—

#358

—

#357

—

C₄×C₁₆

—

C₅₉

C₄×SD₃₂

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₅₉

yes

characteristic

C₂×D₈

C₂×D₅₉

yes

characteristic

C₄×Q₈

—

118

—

C₄×D₄

—

118

—

SD₃₂

Dic₅₉

yes

direct product factor

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₉

yes

—

—

—

118

—

#102

—

118

—

C₄×C₈

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

236

—

Q₁₆

—

Q₁₆

C₂×Dic₅₉

yes

complemented

D₈

—

D₈

C₂×Dic₅₉

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊C₄

—

118

—

C₄⋊C₄

—

118

—

C₄⋊C₄

—

236

—

100

C₂²⋊C₄

—

236

—

101

C₄²

—

102

C₄²

—

236

—

103

C₁₆

—

118

—

104

C₁₆

C₂×Dic₅₉

yes

complemented

105

C₂³

—

106

Q₈

—

107

Q₈

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

472

—

114

C₂×C₄

—

236

—

115

C₂×C₄

—

236

—

116

C₂×C₄

D₄×D₅₉

yes

characteristic

117

C₈

—

118

—

118

C₈

#11745687

yes

characteristic

119

C₈

C₂²×Dic₅₉

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

D₈×D₅₉

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

118

—

127

C₄

—

472

—

128

C₄

—

472

—

129

C₄

D₄×Dic₅₉

yes

characteristic

130

C₄

#14273746

yes

characteristic

131

C₂

—

132

C₂

D₈×Dic₅₉

yes

characteristic

133

C₂

SD₃₂×D₅₉

yes

characteristic

134

C₂

#16837833

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₉
commutator	C₈×C₅₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₂×SD₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂×D₈
hypercenter	C₂×SD₃₂	D₅₉
hypocenter	C₅₉	C₄×SD₃₂
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×SD₃₂

subgroup

G/H

center

C₂²

D₈×D₅₉

commutator

C₈×C₅₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₂×SD₃₂×C₅₉

C₂

socle

C₂²×C₅₉

C₂×D₈

hypercenter

C₂×SD₃₂

D₅₉

hypocenter

C₅₉

C₄×SD₃₂

LCS

G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×SD₃₂

N	H ≅ G/N	type
1. Q₁₆×Dic₅₉	C₂	semidirect
3. #16837230	C₂	semidirect
4. #16837206	C₂	semidirect
6. C₁₆×Dic₅₉	C₂	semidirect
18. SD₃₂×C₅₉	C₄	semidirect
24. Q₁₆×C₅₉	C₂×C₄	semidirect
35. C₁₆×C₅₉	C₂×C₄	semidirect
62. Dic₅₉	SD₃₂	direct (N × H)
75. C₅₉	C₄×SD₃₂	semidirect
80. SD₃₂	Dic₅₉	direct (N × H)
92. Q₁₆	C₂×Dic₅₉	semidirect
104. C₁₆	C₂×Dic₅₉	semidirect

H ≅ G/N

type

1. Q₁₆×Dic₅₉

C₂

semidirect

3. #16837230

C₂

semidirect

4. #16837206

C₂

semidirect

6. C₁₆×Dic₅₉

C₂

semidirect

18. SD₃₂×C₅₉

C₄

semidirect

24. Q₁₆×C₅₉

C₂×C₄

semidirect

35. C₁₆×C₅₉

C₂×C₄

semidirect

62. Dic₅₉

SD₃₂

direct (N × H)

75. C₅₉

C₄×SD₃₂

semidirect

80. SD₃₂

Dic₅₉

direct (N × H)

92. Q₁₆

C₂×Dic₅₉

semidirect

104. C₁₆

C₂×Dic₅₉

semidirect

Subgroups — SD₃₂×Dic₅₉ #21425949

Conjugacy classes of subgroups

Complements

Subgroups — SD32×Dic59 #21425949

Conjugacy classes of subgroups

Complements

Subgroups — SD₃₂×Dic₅₉ #21425949