Group #28509705 — subgroups

Subgroups — #28509705

Order 15104.

232 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 75 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×Dic₉₄₄	C₂	1	2	yes	maximal, complemented
2	C₂×#16837241	C₂	1	2	yes	maximal
3	#21426357	C₂	1	1	yes	maximal, characteristic
4	#21426349	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₉×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₇₂	—	2	1	no	—
7	C₂×Dic₄₇₂	C₂²	1	2	yes	—
8	C₂×Dic₄₇₂	C₂²	1	1	yes	characteristic
9	C₂×#14273672	—	2	1	no	—
10	C₂×D₉₄₄	C₂²	1	1	yes	characteristic
11	#16837786	—	2	1	no	—
12	#16837782	—	2	1	no	—
13	Dic₉₄₄	—	2	4	no	—
14	#16837241	—	2	4	no	—
15	C₂×C₃₂×C₅₉	C₂²	1	2	yes	—
16	C₄×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₃₆	—	2	2	no	—
18	C₂×Dic₂₃₆	—	2	1	no	—
19	C₂×Dic₂₃₆	—	4	1	no	—
20	C₂×#11745660	—	4	1	no	—
21	C₂×D₄₇₂	—	2	1	no	—
22	#14273727	—	4	1	no	—
23	#14273715	—	4	1	no	—
24	Dic₄₇₂	—	2	2	no	—
25	Dic₄₇₂	—	2	2	no	—
26	Dic₄₇₂	—	2	2	no	—
27	Dic₄₇₂	—	4	2	no	—
28	#14273672	—	4	2	no	—
29	D₉₄₄	—	2	2	no	—
30	C₂×C₁₆×C₅₉	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₅₉	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₃₂×C₅₉	D₄	1	4	yes	—
34	C₂×D₂₃₆	—	4	1	no	—
35	C₂×Dic₁₁₈	—	4	1	no	—
36	C₂×Dic₁₁₈	—	4	2	no	—
37	C₂×C₈×C₅₉	D₈	1	1	yes	characteristic
38	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₄²×C₅₉	D₈	1	1	yes	characteristic
40	(C₂×D₅₉)⋊C₄	—	8	2	no	—
41	C₄⋊Dic₅₉	—	8	2	no	—
42	Dic₂₃₆	—	4	2	no	—
43	Dic₂₃₆	—	4	1	no	—
44	Dic₂₃₆	—	4	1	no	—
45	Dic₂₃₆	—	4	2	no	—
46	Dic₂₃₆	—	8	2	no	—
47	D₄₇₂	—	4	1	no	—
48	D₄₇₂	—	4	1	no	—
49	#11745660	—	8	2	no	—
50	C₁₆×C₅₉	D₈	1	2	yes	—
51	C₁₆×C₅₉	C₂×D₄	1	2	yes	—
52	C₂²×D₅₉	—	8	1	no	—
53	C₂×C₄×C₅₉	D₁₆	1	2	yes	—
54	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₅₉	—	8	2	no	—
56	C₂×Dic₅₉	—	8	1	no	—
57	D₂₃₆	—	8	1	no	—
58	D₂₃₆	—	8	1	no	—
59	Dic₁₁₈	—	8	1	no	—
60	Dic₁₁₈	—	8	2	no	—
61	Dic₁₁₈	—	8	1	no	—
62	Dic₁₁₈	—	8	2	no	—
63	C₈×C₅₉	D₁₆	1	2	yes	—
64	C₈×C₅₉	#127	1	1	yes	characteristic
65	C₈×C₅₉	C₂×D₈	1	1	yes	characteristic
66	#13730	—	59	1	no	maximal, Sylow
67	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
68	C₂×D₅₉	—	16	1	no	—
69	C₂×D₅₉	—	16	2	no	—
70	C₄×C₅₉	—	2	2	no	—
71	C₄×C₅₉	#493	1	1	yes	characteristic
72	C₄×C₅₉	C₂×D₁₆	1	1	yes	characteristic
73	Dic₅₉	—	16	1	no	—
74	Dic₅₉	—	16	2	no	—
75	C₂×Q₆₄	—	59	2	no	—
76	C₂×SD₆₄	—	59	2	no	—
77	#2249	—	59	1	no	—
78	#2247	—	59	1	no	—
79	#1397	D₅₉	1	1	yes	characteristic, complemented
80	C₂×C₅₉	#2246	1	1	yes	characteristic
81	C₂×C₅₉	#2264	1	2	yes	—
82	D₅₉	—	32	1	no	—
83	C₂×Q₃₂	—	59	2	no	—
84	C₂×Q₃₂	—	59	1	no	—
85	C₂×Q₃₂	—	118	1	no	—
86	C₂×SD₃₂	—	118	1	no	—
87	C₂×D₁₆	—	59	1	no	—
88	#495	—	118	1	no	—
89	#494	—	118	1	no	—
90	Q₆₄	—	118	4	no	—
91	SD₆₄	—	118	4	no	—
92	C₂×C₃₂	C₂×D₅₉	1	2	yes	—
93	C₄×C₁₆	C₂×D₅₉	1	1	yes	characteristic
94	C₅₉	#13730	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₁₆	—	118	2	no	—
96	C₂×Q₁₆	—	118	1	no	—
97	C₂×Q₁₆	—	236	1	no	—
98	C₂×SD₁₆	—	236	1	no	—
99	C₂×D₈	—	118	1	no	—
100	C₄⋊Q₈	—	236	1	no	—
101	#124	—	236	1	no	—
102	Q₃₂	—	118	2	no	—
103	Q₃₂	—	118	2	no	—
104	Q₃₂	—	118	2	no	—
105	Q₃₂	—	236	2	no	—
106	SD₃₂	—	236	2	no	—
107	D₁₆	—	118	2	no	—
108	C₂×C₁₆	D₂₃₆	1	1	yes	characteristic
109	C₂×C₁₆	C₂²×D₅₉	1	1	yes	characteristic
110	C₄×C₈	D₂₃₆	1	1	yes	characteristic
111	C₃₂	D₂₃₆	1	4	yes	—
112	C₂×Q₈	—	236	1	no	—
113	C₂×Q₈	—	236	2	no	—
114	C₂×D₄	—	236	1	no	—
115	Q₁₆	—	472	2	no	—
116	Q₁₆	—	236	2	no	—
117	Q₁₆	—	236	1	no	—
118	Q₁₆	—	236	1	no	—
119	Q₁₆	—	236	2	no	—
120	SD₁₆	—	472	2	no	—
121	D₈	—	236	1	no	—
122	D₈	—	236	1	no	—
123	C₂×C₈	D₄₇₂	1	1	yes	characteristic
124	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
125	C₄⋊C₄	—	472	2	no	—
126	C₂²⋊C₄	—	472	2	no	—
127	C₄²	D₄₇₂	1	1	yes	characteristic
128	C₁₆	D₄₇₂	1	2	yes	—
129	C₁₆	C₂×D₂₃₆	1	2	yes	—
130	C₂³	—	472	1	no	—
131	Q₈	—	472	1	no	—
132	Q₈	—	472	2	no	—
133	Q₈	—	472	1	no	—
134	Q₈	—	472	2	no	—
135	D₄	—	472	1	no	—
136	D₄	—	472	1	no	—
137	C₂×C₄	—	472	1	no	—
138	C₂×C₄	—	472	2	no	—
139	C₂×C₄	D₉₄₄	1	2	yes	—
140	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
141	C₈	D₉₄₄	1	2	yes	—
142	C₈	#14273724	1	1	yes	characteristic
143	C₈	C₂×D₄₇₂	1	1	yes	characteristic
144	C₂²	—	944	1	no	—
145	C₂²	—	944	2	no	—
146	C₂²	C₂×D₉₄₄	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	944	1	no	—
149	C₄	—	944	2	no	—
150	C₄	#16837779	1	1	yes	characteristic
151	C₄	C₂×D₉₄₄	1	1	yes	characteristic
152	C₂	—	1888	1	no	—
153	C₂	#21426346	1	1	yes	characteristic
154	C₂	#21426416	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₉₄₄

C₂

yes

maximal, complemented

C₂×#16837241

C₂

yes

maximal

#21426357

C₂

yes

maximal, characteristic

#21426349

C₂

yes

maximal, characteristic, complemented

C₅₉×#1397

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₇₂

—

C₂×Dic₄₇₂

C₂²

yes

—

C₂×Dic₄₇₂

C₂²

yes

characteristic

C₂×#14273672

—

C₂×D₉₄₄

C₂²

yes

characteristic

#16837786

—

#16837782

—

Dic₉₄₄

—

#16837241

—

C₂×C₃₂×C₅₉

C₂²

yes

—

C₄×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×Dic₂₃₆

—

C₂×#11745660

—

C₂×D₄₇₂

—

#14273727

—

#14273715

—

Dic₄₇₂

—

Dic₄₇₂

—

Dic₄₇₂

—

Dic₄₇₂

—

#14273672

—

D₉₄₄

—

C₂×C₁₆×C₅₉

D₄

yes

characteristic

C₂×C₁₆×C₅₉

C₂³

yes

characteristic

C₄×C₈×C₅₉

D₄

yes

characteristic

C₃₂×C₅₉

D₄

yes

—

C₂×D₂₃₆

—

C₂×Dic₁₁₈

—

C₂×Dic₁₁₈

—

C₂×C₈×C₅₉

D₈

yes

characteristic

C₂×C₈×C₅₉

C₂×D₄

yes

characteristic

C₄²×C₅₉

D₈

yes

characteristic

(C₂×D₅₉)⋊C₄

—

C₄⋊Dic₅₉

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

Dic₂₃₆

—

D₄₇₂

—

D₄₇₂

—

#11745660

—

C₁₆×C₅₉

D₈

yes

—

C₁₆×C₅₉

C₂×D₄

yes

—

C₂²×D₅₉

—

C₂×C₄×C₅₉

D₁₆

yes

—

C₂×C₄×C₅₉

C₂×D₈

yes

characteristic

C₂×Dic₅₉

—

C₂×Dic₅₉

—

D₂₃₆

—

D₂₃₆

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

Dic₁₁₈

—

C₈×C₅₉

D₁₆

yes

—

C₈×C₅₉

#127

yes

characteristic

C₈×C₅₉

C₂×D₈

yes

characteristic

#13730

—

maximal, Sylow

C₂²×C₅₉

C₂×D₁₆

yes

characteristic

C₂×D₅₉

—

C₂×D₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#493

yes

characteristic

C₄×C₅₉

C₂×D₁₆

yes

characteristic

Dic₅₉

—

Dic₅₉

—

C₂×Q₆₄

—

C₂×SD₆₄

—

#2249

—

#2247

—

#1397

D₅₉

yes

characteristic, complemented

C₂×C₅₉

#2246

yes

characteristic

C₂×C₅₉

#2264

yes

—

D₅₉

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

118

—

C₂×SD₃₂

—

118

—

C₂×D₁₆

—

#495

—

118

—

#494

—

118

—

Q₆₄

—

118

—

SD₆₄

—

118

—

C₂×C₃₂

C₂×D₅₉

yes

—

C₄×C₁₆

C₂×D₅₉

yes

characteristic

C₅₉

#13730

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

118

—

C₂×Q₁₆

—

236

—

C₂×SD₁₆

—

236

—

C₂×D₈

—

118

—

100

C₄⋊Q₈

—

236

—

101

#124

—

236

—

102

Q₃₂

—

118

—

103

Q₃₂

—

118

—

104

Q₃₂

—

118

—

105

Q₃₂

—

236

—

106

SD₃₂

—

236

—

107

D₁₆

—

118

—

108

C₂×C₁₆

D₂₃₆

yes

characteristic

109

C₂×C₁₆

C₂²×D₅₉

yes

characteristic

110

C₄×C₈

D₂₃₆

yes

characteristic

111

C₃₂

D₂₃₆

yes

—

112

C₂×Q₈

—

236

—

113

C₂×Q₈

—

236

—

114

C₂×D₄

—

236

—

115

Q₁₆

—

472

—

116

Q₁₆

—

236

—

117

Q₁₆

—

236

—

118

Q₁₆

—

236

—

119

Q₁₆

—

236

—

120

SD₁₆

—

472

—

121

D₈

—

236

—

122

D₈

—

236

—

123

C₂×C₈

D₄₇₂

yes

characteristic

124

C₂×C₈

C₂×D₂₃₆

yes

characteristic

125

C₄⋊C₄

—

472

—

126

C₂²⋊C₄

—

472

—

127

C₄²

D₄₇₂

yes

characteristic

128

C₁₆

D₄₇₂

yes

—

129

C₁₆

C₂×D₂₃₆

yes

—

130

C₂³

—

472

—

131

Q₈

—

472

—

132

Q₈

—

472

—

133

Q₈

—

472

—

134

Q₈

—

472

—

135

D₄

—

472

—

136

D₄

—

472

—

137

C₂×C₄

—

472

—

138

C₂×C₄

—

472

—

139

C₂×C₄

D₉₄₄

yes

—

140

C₂×C₄

C₂×D₄₇₂

yes

characteristic

141

C₈

D₉₄₄

yes

—

142

C₈

#14273724

yes

characteristic

143

C₈

C₂×D₄₇₂

yes

characteristic

144

C₂²

—

944

—

145

C₂²

—

944

—

146

C₂²

C₂×D₉₄₄

yes

characteristic

147

C₄

—

148

C₄

—

944

—

149

C₄

—

944

—

150

C₄

#16837779

yes

characteristic

151

C₄

C₂×D₉₄₄

yes

characteristic

152

C₂

—

1888

—

153

C₂

#21426346

yes

characteristic

154

C₂

#21426416

yes

—

	subgroup	G/H
center	C₂²	C₂×D₉₄₄
commutator, cosocle	C₂×C₁₆×C₅₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₅₉
Fitting	C₅₉×#1397	C₂
socle	C₂²×C₅₉	C₂×D₁₆
prime radical	C₂×D₉₄₄	C₂²
hypercenter	#1397	D₅₉
hypocenter	C₅₉	#13730
LCS	G ▷ C₂×C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂²

C₂×D₉₄₄

commutator, cosocle

C₂×C₁₆×C₅₉

C₂³

Frattini

C₂×C₁₆

C₂²×D₅₉

Fitting

C₅₉×#1397

C₂

socle

C₂²×C₅₉

C₂×D₁₆

prime radical

C₂×D₉₄₄

C₂²

D₅₉

C₅₉

G ▷ C₂×C₁₆×C₅₉ ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
1. C₂×Dic₉₄₄	C₂	semidirect
4. #21426349	C₂	semidirect
5. C₅₉×#1397	C₂	semidirect
79. #1397	D₅₉	semidirect
94. C₅₉	#13730	semidirect

H ≅ G/N

type

1. C₂×Dic₉₄₄

C₂

semidirect

4. #21426349

C₂

semidirect

5. C₅₉×#1397

C₂

semidirect

79. #1397

D₅₉

semidirect

94. C₅₉

#13730

semidirect

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

subgroup	index
C₂×Dic₉₄₄	2
C₂×#16837241	2
#21426357	2
#21426349	2
C₅₉×#1397	2
#13730	59

subgroup

index

C₂×Dic₉₄₄

C₅₉×#1397

Subgroups — #28509705

Conjugacy classes of subgroups

Complements

Maximal subgroups