Group #28576494 — subgroups

Subgroups — #28576494

Order 15104.

690 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 227 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	#21434682	C₂	1	4	yes	maximal
2	#21434610	C₂	1	2	yes	maximal
3	#21423773	C₂	1	8	yes	maximal, complemented
4	C₂×C₅₉×#322	C₂	1	1	yes	maximal, characteristic
5	#16837959	C₂²	1	4	yes	—
6	#16837807	C₂²	1	4	yes	—
7	#16837807	C₂²	1	4	yes	—
8	#16837804	—	2	4	no	—
9	#16837800	C₂²	1	4	yes	—
10	#16837782	C₂²	1	4	yes	—
11	#16837753	C₂²	1	8	yes	—
12	#16837643	—	2	8	no	—
13	C₄×Dic₂₃₆	—	2	8	no	—
14	#16837440	—	2	4	no	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
17	C₅₉×#322	C₂²	1	4	yes	—
18	C₂×Dic₂₃₆	—	2	8	no	—
19	#14273727	C₂³	1	8	yes	—
20	#14273719	—	2	4	no	—
21	#14273706	—	2	8	no	—
22	C₄×Dic₁₁₈	—	2	8	no	—
23	#14273686	—	2	4	no	—
24	#14273661	—	2	8	no	—
25	#14273661	—	2	8	no	—
26	#14273659	—	2	8	no	—
27	#14273652	—	2	16	no	—
28	C₂²×C₈×C₅₉	D₄	1	4	yes	—
29	C₂²×C₈×C₅₉	D₄	1	2	yes	—
30	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₉	D₄	1	4	yes	—
32	C₄×C₈×C₅₉	D₄	1	2	yes	—
33	C₄×C₈×C₅₉	C₂³	1	4	yes	—
34	C₄×C₈×C₅₉	C₂³	1	2	yes	—
35	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₁₁₈	—	2	8	no	—
38	C₂×C₈×C₅₉	—	4	4	no	—
39	C₂×C₈×C₅₉	Q₁₆	1	8	yes	complemented
40	C₂×C₈×C₅₉	C₂×D₄	1	8	yes	—
41	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
42	C₄²×C₅₉	C₂×D₄	1	2	yes	—
43	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
45	C₂²⋊Dic₅₉	—	4	4	no	—
46	C₄⋊Dic₅₉	—	2	8	no	—
47	C₄⋊Dic₅₉	—	2	8	no	—
48	Dic₅₉⋊C₄	—	4	8	no	—
49	C₄×Dic₅₉	—	4	4	no	—
50	Dic₂₃₆	—	4	16	no	—
51	C₂³×C₅₉	#127	1	1	yes	characteristic
52	C₂×C₄×C₅₉	—	2	4	no	—
53	C₂×C₄×C₅₉	#127	1	2	yes	—
54	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
55	C₂×C₄×C₅₉	C₂×Q₁₆	1	4	yes	—
56	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
58	C₂×Dic₅₉	—	4	8	no	—
59	Dic₁₁₈	—	4	16	no	—
60	C₈×C₅₉	—	2	8	no	—
61	C₈×C₅₉	C₂×Q₁₆	1	8	yes	—
62	#19618	—	59	1	no	maximal, Sylow
63	C₂²×C₅₉	—	2	2	no	—
64	C₂²×C₅₉	#494	1	2	yes	—
65	C₂²×C₅₉	C₂×#127	1	1	yes	characteristic
66	C₄×C₅₉	—	2	2	no	—
67	C₄×C₅₉	#494	1	2	yes	—
68	C₄×C₅₉	C₂²×Q₁₆	1	2	yes	—
69	C₄×C₅₉	#575	1	4	yes	—
70	Dic₅₉	—	8	8	no	—
71	#3170	—	59	4	no	—
72	#3160	—	59	2	no	—
73	#1723	—	59	8	no	—
74	C₂×#322	D₅₉	1	1	yes	characteristic
75	C₂×C₅₉	—	2	2	no	—
76	C₂×C₅₉	C₂×#494	1	1	yes	characteristic
77	C₂×C₅₉	#3150	1	2	yes	—
78	#533	—	59	4	no	—
79	#500	—	59	4	no	—
80	#500	—	59	4	no	—
81	#499	—	118	4	no	—
82	#498	—	59	4	no	—
83	#494	—	59	4	no	—
84	#487	—	59	8	no	—
85	#467	—	118	8	no	—
86	C₄×Q₁₆	—	118	8	no	—
87	#427	—	118	4	no	—
88	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
90	#322	C₂×D₅₉	1	4	yes	—
91	C₅₉	#19618	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	118	8	no	—
93	C₂²×C₈	D₂₃₆	1	4	yes	—
94	C₂²×C₈	D₂₃₆	1	2	yes	—
95	C₄⋊Q₈	—	59	8	no	—
96	#125	—	118	4	no	—
97	C₂²⋊Q₈	—	118	8	no	—
98	C₄×Q₈	—	118	8	no	—
99	#117	—	118	4	no	—
100	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
101	#107	—	118	8	no	—
102	#107	—	118	8	no	—
103	#106	—	118	8	no	—
104	#103	—	118	16	no	—
105	C₄×C₈	D₂₃₆	1	4	yes	—
106	C₄×C₈	D₂₃₆	1	2	yes	—
107	C₄×C₈	C₂²×D₅₉	1	4	yes	—
108	C₄×C₈	C₂²×D₅₉	1	2	yes	—
109	C₂×Q₈	—	118	8	no	—
110	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
111	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
112	Q₁₆	—	236	16	no	—
113	C₂×C₈	—	4	4	no	—
114	C₂×C₈	Dic₂₃₆	1	8	yes	complemented
115	C₂×C₈	C₂×D₂₃₆	1	8	yes	—
116	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
117	C₄⋊C₄	—	118	8	no	—
118	C₄⋊C₄	—	118	8	no	—
119	C₄⋊C₄	—	236	8	no	—
120	C₂²⋊C₄	—	236	4	no	—
121	C₄²	—	236	4	no	—
122	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
123	C₄²	C₂×D₂₃₆	1	2	yes	—
124	C₄²	C₂³×D₅₉	1	1	yes	characteristic
125	C₂³	#14273724	1	1	yes	characteristic
126	Q₈	—	236	16	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	236	8	no	—
129	C₂×C₄	#14273724	1	2	yes	—
130	C₂×C₄	#14273724	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₂₃₆	1	4	yes	—
132	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₂₃₆	1	2	yes	—
134	C₈	—	2	8	no	—
135	C₈	C₂×Dic₂₃₆	1	8	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	#16837782	1	2	yes	—
138	C₂²	C₂×#14273724	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	472	8	no	—
141	C₄	#16837782	1	2	yes	—
142	C₄	C₂²×Dic₂₃₆	1	2	yes	—
143	C₄	#16838202	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	C₂×#16837782	1	1	yes	characteristic
146	C₂	#21434542	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21434682

C₂

yes

maximal

#21434610

C₂

yes

maximal

#21423773

C₂

yes

maximal, complemented

C₂×C₅₉×#322

C₂

yes

maximal, characteristic

#16837959

C₂²

yes

—

#16837807

C₂²

yes

—

#16837807

C₂²

yes

—

#16837804

—

#16837800

C₂²

yes

—

#16837782

C₂²

yes

—

#16837753

C₂²

yes

—

#16837643

—

C₄×Dic₂₃₆

—

#16837440

—

C₂×C₄×C₈×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#322

C₂²

yes

—

C₂×Dic₂₃₆

—

#14273727

C₂³

yes

—

#14273719

—

#14273706

—

C₄×Dic₁₁₈

—

—

—

—

—

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂²×C₈×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

C₂³

yes

—

C₄×C₈×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

Q₁₆

yes

complemented

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

—

C₄²×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

C₂⁴

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

Dic₂₃₆

—

C₂³×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

#127

yes

—

C₂×C₄×C₅₉

#127

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₂×C₄×C₅₉

C₂²×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×D₄

yes

—

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

—

C₈×C₅₉

C₂×Q₁₆

yes

—

#19618

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#494

yes

—

C₂²×C₅₉

C₂×#127

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#494

yes

—

C₄×C₅₉

C₂²×Q₁₆

yes

—

C₄×C₅₉

#575

yes

—

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

C₂×#494

yes

characteristic

C₂×C₅₉

#3150

yes

—

—

—

—

—

118

—

—

—

—

—

118

—

C₄×Q₁₆

—

118

—

#427

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄×C₈

C₂×D₅₉

yes

—

#322

C₂×D₅₉

yes

—

C₅₉

#19618

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₂²×C₈

D₂₃₆

yes

—

C₂²×C₈

D₂₃₆

yes

—

C₄⋊Q₈

—

#125

—

118

—

C₂²⋊Q₈

—

118

—

C₄×Q₈

—

118

—

#117

—

118

—

100

C₂×C₄²

C₂²×D₅₉

yes

characteristic

101

#107

—

118

—

102

#107

—

118

—

103

#106

—

118

—

104

#103

—

118

—

105

C₄×C₈

D₂₃₆

yes

—

106

C₄×C₈

D₂₃₆

yes

—

107

C₄×C₈

C₂²×D₅₉

yes

—

108

C₄×C₈

C₂²×D₅₉

yes

—

109

C₂×Q₈

—

118

—

110

C₂²×C₄

C₂×D₂₃₆

yes

—

111

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

112

Q₁₆

—

236

—

113

C₂×C₈

—

114

C₂×C₈

Dic₂₃₆

yes

complemented

115

C₂×C₈

C₂×D₂₃₆

yes

—

116

C₂×C₈

C₂×D₂₃₆

yes

—

117

C₄⋊C₄

—

118

—

118

C₄⋊C₄

—

118

—

119

C₄⋊C₄

—

236

—

120

C₂²⋊C₄

—

236

—

121

C₄²

—

236

—

122

C₄²

C₂×D₂₃₆

yes

characteristic

123

C₄²

C₂×D₂₃₆

yes

—

124

C₄²

C₂³×D₅₉

yes

characteristic

125

C₂³

#14273724

yes

characteristic

126

Q₈

—

236

—

127

C₂×C₄

—

128

C₂×C₄

—

236

—

129

C₂×C₄

#14273724

yes

—

130

C₂×C₄

#14273724

yes

characteristic

131

C₂×C₄

C₂×Dic₂₃₆

yes

—

132

C₂×C₄

C₂²×D₂₃₆

yes

characteristic

133

C₂×C₄

C₂²×D₂₃₆

yes

—

134

C₈

—

135

C₈

C₂×Dic₂₃₆

yes

—

136

C₂²

—

137

C₂²

#16837782

yes

—

138

C₂²

C₂×#14273724

yes

characteristic

139

C₄

—

140

C₄

—

472

—

141

C₄

#16837782

yes

—

142

C₄

C₂²×Dic₂₃₆

yes

—

143

C₄

#16838202

yes

—

144

C₂

—

145

C₂

C₂×#16837782

yes

characteristic

146

C₂

#21434542

yes

—

	subgroup	G/H
center	C₂×C₄	#14273724
commutator, cosocle	C₄²×C₅₉	C₂⁴
Frattini	C₄²	C₂³×D₅₉
Fitting	C₂×C₅₉×#322	C₂
socle	C₂²×C₅₉	C₂×#127
prime radical	C₂³×C₅₉	#127
hypercenter	C₂×#322	D₅₉
hypocenter	C₅₉	#19618
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#14273724

commutator, cosocle

C₄²×C₅₉

C₂⁴

Frattini

C₄²

C₂³×D₅₉

Fitting

C₂×C₅₉×#322

C₂

C₂²×C₅₉

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
3. #21423773	C₂	semidirect
39. C₂×C₈×C₅₉	Q₁₆	semidirect
91. C₅₉	#19618	semidirect
114. C₂×C₈	Dic₂₃₆	semidirect

H ≅ G/N

type

3. #21423773

C₂

semidirect

39. C₂×C₈×C₅₉

Q₁₆

semidirect

91. C₅₉

#19618

semidirect

114. C₂×C₈

Dic₂₃₆

semidirect

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

subgroup	index
#21434682	2
#21434610	2
#21423773	2
C₂×C₅₉×#322	2
#19618	59

subgroup

index

C₂×C₅₉×#322

Subgroups — #28576494

Conjugacy classes of subgroups

Complements

Maximal subgroups