Group #28744191 — subgroups

Subgroups — #28744191

Order 15104.

482 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 219 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	#21434610	C₂	1	2	yes	maximal
2	#21426382	C₂	1	8	yes	maximal, complemented
3	#21425880	C₂	1	4	yes	maximal
4	C₂×C₅₉×#346	C₂	1	1	yes	maximal, characteristic
5	#16837959	—	2	2	no	—
6	#16837807	C₂²	1	4	yes	—
7	#16837804	—	2	2	no	—
8	#16837804	C₂²	1	4	yes	—
9	#16837800	—	2	2	no	—
10	#16837440	—	2	2	no	—
11	#16837440	C₂²	1	4	yes	—
12	#16837230	C₂²	1	8	yes	—
13	#16837228	C₂²	1	8	yes	—
14	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#346	C₂²	1	4	yes	—
17	#14273727	—	2	4	no	—
18	#14273719	—	2	4	no	—
19	#14273706	—	4	4	no	—
20	C₄×Dic₁₁₈	—	4	4	no	—
21	#14273686	—	2	4	no	—
22	#14273661	—	2	4	no	—
23	#14273661	C₂³	1	8	yes	—
24	#14273659	—	2	4	no	—
25	#14273659	—	2	4	no	—
26	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₅₉	Q₈	1	8	yes	complemented
30	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
31	C₄×C₈×C₅₉	D₄	1	2	yes	—
32	C₄×C₈×C₅₉	C₂³	1	2	yes	—
33	C₂²×C₄×C₅₉	D₈	1	2	yes	—
34	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₁₁₈	—	4	4	no	—
36	C₂×C₈×C₅₉	Q₁₆	1	4	yes	—
37	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₅₉	C₂×Q₈	1	4	yes	—
40	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
41	C₄²×C₅₉	D₈	1	2	yes	—
42	C₄²×C₅₉	C₂×D₄	1	2	yes	—
43	C₂²⋊Dic₅₉	—	4	4	no	—
44	C₄⋊Dic₅₉	—	2	8	no	—
45	C₄⋊Dic₅₉	—	4	4	no	—
46	Dic₅₉⋊C₄	—	4	8	no	—
47	C₄×Dic₅₉	—	4	4	no	—
48	C₁₆×C₅₉	C₂×Q₈	1	8	yes	—
49	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₅₉	—	4	2	no	—
51	C₂×C₄×C₅₉	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₅₉	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₅₉	—	4	8	no	—
57	Dic₁₁₈	—	8	8	no	—
58	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
59	C₈×C₅₉	C₂×Q₁₆	1	4	yes	—
60	C₈×C₅₉	C₂²×Q₈	1	2	yes	—
61	#33728	—	59	1	no	maximal, Sylow
62	C₂²×C₅₉	—	2	2	no	—
63	C₂²×C₅₉	#500	1	2	yes	—
64	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
65	C₄×C₅₉	—	2	4	no	—
66	C₄×C₅₉	#500	1	2	yes	—
67	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
69	Dic₅₉	—	8	8	no	—
70	#3160	—	59	2	no	—
71	#2255	—	59	8	no	—
72	#2158	—	59	4	no	—
73	C₂×#346	D₅₉	1	1	yes	characteristic
74	C₂×C₅₉	—	2	2	no	—
75	C₂×C₅₉	C₂×#500	1	1	yes	characteristic
76	C₂×C₅₉	#3414	1	2	yes	—
77	#533	—	118	2	no	—
78	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
79	#500	—	59	4	no	—
80	#499	—	59	4	no	—
81	#499	—	118	2	no	—
82	#498	—	118	2	no	—
83	#427	—	59	4	no	—
84	#427	—	118	2	no	—
85	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
86	#367	—	59	8	no	—
87	#366	—	59	8	no	—
88	#346	C₂×D₅₉	1	4	yes	—
89	C₅₉	#33728	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
92	C₄⋊Q₈	—	118	4	no	—
93	#125	—	118	4	no	—
94	C₂²⋊Q₈	—	236	4	no	—
95	C₄×Q₈	—	236	4	no	—
96	#117	—	118	4	no	—
97	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₁₈	1	8	yes	complemented
99	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
100	#107	—	59	8	no	—
101	#107	—	118	4	no	—
102	#106	—	118	4	no	—
103	#106	—	118	4	no	—
104	C₄×C₈	D₂₃₆	1	2	yes	—
105	C₄×C₈	C₂²×D₅₉	1	2	yes	—
106	C₂×Q₈	—	236	4	no	—
107	C₂²×C₄	D₄₇₂	1	2	yes	—
108	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
109	C₂×C₈	Dic₂₃₆	1	4	yes	—
110	C₂×C₈	C₂×Dic₁₁₈	1	4	yes	—
111	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
112	C₂×C₈	C₂×D₂₃₆	1	2	yes	—
113	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
114	C₄⋊C₄	—	118	8	no	—
115	C₄⋊C₄	—	236	8	no	—
116	C₄⋊C₄	—	236	4	no	—
117	C₂²⋊C₄	—	236	4	no	—
118	C₄²	—	236	4	no	—
119	C₄²	D₄₇₂	1	2	yes	—
120	C₄²	C₂×D₂₃₆	1	2	yes	—
121	C₁₆	C₂×Dic₁₁₈	1	8	yes	—
122	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
123	Q₈	—	472	8	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	236	8	no	—
126	C₂×C₄	#14273727	1	2	yes	—
127	C₂×C₄	C₂×D₄₇₂	1	4	yes	—
128	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₂₃₆	1	2	yes	—
130	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
131	C₈	#14273727	1	2	yes	—
132	C₈	C₂×Dic₂₃₆	1	4	yes	—
133	C₈	C₂²×Dic₁₁₈	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#16837807	1	2	yes	—
136	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	472	8	no	—
139	C₄	#16837807	1	2	yes	—
140	C₄	C₂×#14273727	1	1	yes	characteristic
141	C₄	C₂²×Dic₂₃₆	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#16837807	1	1	yes	characteristic
144	C₂	#21437232	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21434610

C₂

yes

maximal

#21426382

C₂

yes

maximal, complemented

#21425880

C₂

yes

maximal

C₂×C₅₉×#346

C₂

yes

maximal, characteristic

#16837959

—

#16837807

C₂²

yes

—

#16837804

—

#16837804

C₂²

yes

—

#16837800

—

#16837440

—

#16837440

C₂²

yes

—

#16837230

C₂²

yes

—

#16837228

C₂²

yes

—

C₂²×C₁₆×C₅₉

C₂²

yes

—

C₂×C₄×C₈×C₅₉

C₂²

yes

characteristic

C₅₉×#346

C₂²

yes

—

#14273727

—

#14273719

—

#14273706

—

C₄×Dic₁₁₈

—

#14273686

—

#14273661

—

#14273661

C₂³

yes

—

#14273659

—

#14273659

—

C₂²×C₈×C₅₉

D₄

yes

characteristic

C₂²×C₈×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

Q₈

yes

complemented

C₂×C₁₆×C₅₉

C₂³

yes

—

C₄×C₈×C₅₉

D₄

yes

—

C₄×C₈×C₅₉

C₂³

yes

—

C₂²×C₄×C₅₉

D₈

yes

—

C₂²×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

Q₁₆

yes

—

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

—

C₂×C₈×C₅₉

C₂×Q₈

yes

—

C₂×C₈×C₅₉

C₂⁴

yes

characteristic

C₄²×C₅₉

D₈

yes

—

C₄²×C₅₉

C₂×D₄

yes

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

Dic₅₉⋊C₄

—

C₄×Dic₅₉

—

C₁₆×C₅₉

C₂×Q₈

yes

—

C₂³×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₁₆

yes

—

C₂×C₄×C₅₉

C₂²×D₄

yes

characteristic

C₂×Dic₅₉

—

Dic₁₁₈

—

C₈×C₅₉

C₄⋊Q₈

yes

—

C₈×C₅₉

C₂×Q₁₆

yes

—

C₈×C₅₉

C₂²×Q₈

yes

—

#33728

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

#500

yes

—

C₂²×C₅₉

C₂²×D₈

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

#500

yes

—

C₄×C₅₉

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₅₉

C₂²×Q₁₆

yes

characteristic

Dic₅₉

—

—

—

—

D₅₉

yes

characteristic

C₂×C₅₉

—

C₂×C₅₉

C₂×#500

yes

characteristic

C₂×C₅₉

#3414

yes

—

#533

—

118

—

C₂²×C₁₆

C₂×D₅₉

yes

—

#500

—

#499

—

#499

—

118

—

#498

—

118

—

#427

—

#427

—

118

—

C₂×C₄×C₈

C₂×D₅₉

yes

—

—

C₂×D₅₉

yes

—

C₅₉

#33728

yes

characteristic, complemented, Sylow

C₂²×C₈

D₂₃₆

yes

characteristic

C₂²×C₈

C₂²×D₅₉

yes

characteristic

C₄⋊Q₈

—

118

—

#125

—

118

—

C₂²⋊Q₈

—

236

—

C₄×Q₈

—

236

—

#117

—

118

—

C₂×C₄²

D₂₃₆

yes

characteristic

C₂×C₁₆

Dic₁₁₈

yes

complemented

C₂×C₁₆

C₂²×D₅₉

yes

—

100

#107

—

101

#107

—

118

—

102

#106

—

118

—

103

#106

—

118

—

104

C₄×C₈

D₂₃₆

yes

—

105

C₄×C₈

C₂²×D₅₉

yes

—

106

C₂×Q₈

—

236

—

107

C₂²×C₄

D₄₇₂

yes

—

108

C₂²×C₄

C₂×D₂₃₆

yes

characteristic

109

C₂×C₈

Dic₂₃₆

yes

—

110

C₂×C₈

C₂×Dic₁₁₈

yes

—

111

C₂×C₈

C₂×D₂₃₆

yes

characteristic

112

C₂×C₈

C₂×D₂₃₆

yes

—

113

C₂×C₈

C₂³×D₅₉

yes

characteristic

114

C₄⋊C₄

—

118

—

115

C₄⋊C₄

—

236

—

116

C₄⋊C₄

—

236

—

117

C₂²⋊C₄

—

236

—

118

C₄²

—

236

—

119

C₄²

D₄₇₂

yes

—

120

C₄²

C₂×D₂₃₆

yes

—

121

C₁₆

C₂×Dic₁₁₈

yes

—

122

C₂³

C₂×D₄₇₂

yes

characteristic

123

Q₈

—

472

—

124

C₂×C₄

—

125

C₂×C₄

—

236

—

126

C₂×C₄

#14273727

yes

—

127

C₂×C₄

C₂×D₄₇₂

yes

—

128

C₂×C₄

C₂×D₄₇₂

yes

characteristic

129

C₂×C₄

C₂×Dic₂₃₆

yes

—

130

C₂×C₄

C₂²×D₂₃₆

yes

characteristic

131

C₈

#14273727

yes

—

132

C₈

C₂×Dic₂₃₆

yes

—

133

C₈

C₂²×Dic₁₁₈

yes

—

134

C₂²

—

135

C₂²

#16837807

yes

—

136

C₂²

C₂²×D₄₇₂

yes

characteristic

137

C₄

—

138

C₄

—

472

—

139

C₄

#16837807

yes

—

140

C₄

C₂×#14273727

yes

characteristic

141

C₄

C₂²×Dic₂₃₆

yes

characteristic

142

C₂

—

143

C₂

C₂×#16837807

yes

characteristic

144

C₂

#21437232

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#346	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂³×C₅₉	C₂×D₈
hypercenter	C₂×#346	D₅₉
hypocenter	C₅₉	#33728
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₄₇₂

commutator, cosocle

C₂×C₈×C₅₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₅₉

Fitting

C₂×C₅₉×#346

C₂

socle

C₂²×C₅₉

C₂²×D₈

prime radical

C₂³×C₅₉

C₂×D₈

D₅₉

C₅₉

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. #21426382	C₂	semidirect
29. C₂×C₁₆×C₅₉	Q₈	semidirect
89. C₅₉	#33728	semidirect
98. C₂×C₁₆	Dic₁₁₈	semidirect

H ≅ G/N

type

2. #21426382

C₂

semidirect

29. C₂×C₁₆×C₅₉

Q₈

semidirect

89. C₅₉

#33728

semidirect

98. C₂×C₁₆

Dic₁₁₈

semidirect

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

subgroup	index
#21434610	2
#21426382	2
#21425880	2
C₂×C₅₉×#346	2
#33728	59

subgroup

index

C₂×C₅₉×#346

Subgroups — #28744191

Conjugacy classes of subgroups

Complements

Maximal subgroups