Group #28508373 — subgroups

Subgroups — #28508373

Order 15104.

416 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 81 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	#21424310	C₂	1	3	yes	maximal
2	#21422223	C₂	1	3	yes	maximal
3	C₅₉×#2811	C₂	1	1	yes	maximal, characteristic
4	#16837880	C₂²	1	1	yes	characteristic
5	#16837442	—	2	3	no	—
6	C₂×#14273646	C₂²	1	3	yes	—
7	#16837162	—	2	3	no	—
8	C₅₉×#395	—	2	3	no	—
9	C₅₉×#391	—	2	3	no	—
10	C₅₉×#391	C₂²	1	3	yes	—
11	C₂×#11745664	—	2	3	no	—
12	Q₈×Dic₅₉	—	4	1	no	—
13	Q₈×Dic₅₉	—	4	1	no	—
14	#14273686	—	2	3	no	—
15	#14273660	—	4	3	no	—
16	#14273658	—	4	3	no	—
17	#14273646	—	2	3	no	—
18	#14273646	Q₈	1	6	yes	complemented
19	C₂²×Q₈×C₅₉	—	2	1	no	—
20	C₂²×Q₈×C₅₉	—	2	1	no	—
21	C₂²×Q₈×C₅₉	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₅₉	—	2	3	no	—
23	C₂×C₄⋊C₄×C₅₉	D₄	1	3	yes	—
24	C₂×C₄²×C₅₉	D₄	1	3	yes	—
25	#95×C₅₉	—	2	3	no	—
26	#95×C₅₉	—	2	6	no	—
27	C₂×Q₈×C₅₉	—	2	4	no	—
28	C₂×Q₈×C₅₉	—	4	1	no	—
29	C₂×Q₈×C₅₉	—	4	1	no	—
30	C₂×Q₈×C₅₉	—	8	1	no	—
31	C₂×Q₈×C₅₉	—	8	1	no	—
32	C₂×Q₈×C₅₉	—	8	2	no	—
33	C₂×Q₈×C₅₉	—	8	1	no	—
34	C₂×Q₈×C₅₉	C₂×Q₈	1	3	yes	—
35	C₂×Q₈×C₅₉	#41	1	1	yes	characteristic
36	C₂²×C₄×C₅₉	—	2	3	no	—
37	C₂²×C₄×C₅₉	—	2	3	no	—
38	C₂²×C₄×C₅₉	C₂×D₄	1	3	yes	—
39	C₄⋊C₄×C₅₉	—	4	3	no	—
40	C₄⋊C₄×C₅₉	—	8	3	no	—
41	C₄²×C₅₉	—	4	3	no	—
42	C₂²⋊Dic₅₉	—	4	1	no	—
43	C₄⋊Dic₅₉	—	2	3	no	—
44	C₄⋊Dic₅₉	—	2	3	no	—
45	C₄×Dic₅₉	—	4	3	no	—
46	#11745664	—	2	6	no	—
47	#11745664	—	4	3	no	—
48	C₂×C₅₉⋊C₈	—	4	3	no	—
49	C₂³×C₅₉	#120	1	1	yes	characteristic
50	Q₈×C₅₉	—	4	3	no	—
51	Q₈×C₅₉	—	4	1	no	—
52	Q₈×C₅₉	—	8	4	no	—
53	Q₈×C₅₉	—	8	4	no	—
54	C₂×C₄×C₅₉	—	2	3	no	—
55	C₂×C₄×C₅₉	—	2	3	no	—
56	C₂×C₄×C₅₉	—	2	6	no	—
57	C₂×C₄×C₅₉	—	4	3	no	—
58	C₂×C₄×C₅₉	—	4	3	no	—
59	C₂×C₄×C₅₉	—	4	6	no	—
60	C₂×C₄×C₅₉	—	4	3	no	—
61	C₂×C₄×C₅₉	C₂²⋊Q₈	1	3	yes	—
62	C₂×C₄×C₅₉	C₄⋊Q₈	1	3	yes	—
63	C₂×Dic₅₉	—	4	1	no	—
64	C₂×Dic₅₉	—	4	1	no	—
65	C₅₉⋊C₈	—	4	6	no	—
66	#13464	—	59	1	no	maximal, Sylow
67	C₂²×C₅₉	—	2	1	no	—
68	C₂²×C₅₉	—	2	1	no	—
69	C₂²×C₅₉	#395	1	1	yes	characteristic
70	C₂²×C₅₉	B(4,𝔽₂)	1	1	yes	characteristic
71	C₂²×C₅₉	#458	1	1	yes	characteristic
72	C₄×C₅₉	—	2	3	no	—
73	C₄×C₅₉	—	2	3	no	—
74	C₄×C₅₉	—	4	6	no	—
75	C₄×C₅₉	—	8	3	no	—
76	Dic₅₉	—	8	1	no	—
77	Dic₅₉	—	8	1	no	—
78	#2811	D₅₉	1	1	yes	characteristic
79	#1857	—	59	3	no	—
80	#1394	—	59	3	no	—
81	C₂×C₅₉	—	2	1	no	—
82	C₂×C₅₉	—	2	1	no	—
83	C₂×C₅₉	#2062	1	1	yes	characteristic
84	C₂×C₅₉	#2202	1	1	yes	characteristic
85	C₂×C₅₉	#2205	1	1	yes	characteristic
86	#519	—	59	1	no	—
87	#428	—	118	3	no	—
88	C₂×#101	—	59	3	no	—
89	#395	—	2	3	no	—
90	#391	—	2	3	no	—
91	#391	C₂×D₅₉	1	3	yes	—
92	#339	—	118	3	no	—
93	C₅₉	#13464	1	1	yes	characteristic, complemented, Sylow
94	C₂²×Q₈	—	2	1	no	—
95	C₂²×Q₈	—	2	1	no	—
96	C₂²×Q₈	C₂²×D₅₉	1	1	yes	characteristic
97	C₂×M₁₆	—	118	3	no	—
98	C₄×Q₈	—	236	1	no	—
99	C₄×Q₈	—	236	1	no	—
100	#117	—	118	3	no	—
101	C₂×C₄⋊C₄	—	2	3	no	—
102	C₂×C₄⋊C₄	C₂²⋊D₅₉	1	3	yes	—
103	C₂×C₄²	C₂²⋊D₅₉	1	3	yes	—
104	#107	—	236	3	no	—
105	#106	—	236	3	no	—
106	#101	—	59	6	no	—
107	#101	—	118	3	no	—
108	#95	—	2	3	no	—
109	#95	—	2	6	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	4	1	no	—
112	C₂×Q₈	—	4	1	no	—
113	C₂×Q₈	—	8	1	no	—
114	C₂×Q₈	—	8	1	no	—
115	C₂×Q₈	—	8	2	no	—
116	C₂×Q₈	—	8	1	no	—
117	C₂×Q₈	Q₈×D₅₉	1	3	yes	—
118	C₂×Q₈	#11745689	1	1	yes	characteristic
119	C₂²×C₄	—	2	3	no	—
120	C₂²×C₄	—	2	3	no	—
121	C₂²×C₄	C₂×C₂²⋊D₅₉	1	3	yes	—
122	M₁₆	—	118	6	no	—
123	M₁₆	—	236	3	no	—
124	C₂×C₈	—	236	3	no	—
125	C₄⋊C₄	—	4	3	no	—
126	C₄⋊C₄	—	8	3	no	—
127	C₄⋊C₄	—	118	3	no	—
128	C₄⋊C₄	—	118	3	no	—
129	C₂²⋊C₄	—	236	1	no	—
130	C₄²	—	4	3	no	—
131	C₄²	—	236	3	no	—
132	C₂³	#14273696	1	1	yes	characteristic
133	Q₈	—	4	3	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	8	4	no	—
136	Q₈	—	8	4	no	—
137	C₂×C₄	—	2	3	no	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	4	3	no	—
141	C₂×C₄	—	4	3	no	—
142	C₂×C₄	—	4	6	no	—
143	C₂×C₄	—	4	3	no	—
144	C₂×C₄	—	236	1	no	—
145	C₂×C₄	—	236	1	no	—
146	C₂×C₄	#14273703	1	3	yes	—
147	C₂×C₄	#14273725	1	3	yes	—
148	C₈	—	236	6	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#16837321	1	1	yes	characteristic
152	C₂²	#16837589	1	1	yes	characteristic
153	C₂²	#16837592	1	1	yes	characteristic
154	C₄	—	2	3	no	—
155	C₄	—	2	3	no	—
156	C₄	—	4	6	no	—
157	C₄	—	8	3	no	—
158	C₄	—	472	1	no	—
159	C₄	—	472	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	#21425448	1	1	yes	characteristic
163	C₂	#21426113	1	1	yes	characteristic
164	C₂	#21426126	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21424310

C₂

yes

maximal

#21422223

C₂

yes

maximal

C₅₉×#2811

C₂

yes

maximal, characteristic

#16837880

C₂²

yes

characteristic

#16837442

—

C₂×#14273646

C₂²

yes

—

#16837162

—

C₅₉×#395

—

C₅₉×#391

—

C₅₉×#391

C₂²

yes

—

C₂×#11745664

—

Q₈×Dic₅₉

—

Q₈×Dic₅₉

—

—

—

—

—

Q₈

yes

complemented

C₂²×Q₈×C₅₉

—

C₂²×Q₈×C₅₉

—

C₂²×Q₈×C₅₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₅₉

—

C₂×C₄⋊C₄×C₅₉

D₄

yes

—

C₂×C₄²×C₅₉

D₄

yes

—

#95×C₅₉

—

#95×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

—

C₂×Q₈×C₅₉

C₂×Q₈

yes

—

C₂×Q₈×C₅₉

#41

yes

characteristic

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×D₄

yes

—

C₄⋊C₄×C₅₉

—

C₄⋊C₄×C₅₉

—

C₄²×C₅₉

—

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄⋊Dic₅₉

—

C₄×Dic₅₉

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

#120

yes

characteristic

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

Q₈×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₉

C₄⋊Q₈

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

—

C₅₉⋊C₈

—

#13464

—

maximal, Sylow

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₂²×C₅₉

yes

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

Dic₅₉

—

Dic₅₉

—

#2811

D₅₉

yes

characteristic

#1857

—

#1394

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

—

118

—

—

—

—

C₂×D₅₉

yes

—

#339

—

118

—

C₅₉

#13464

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂²×D₅₉

yes

characteristic

C₂×M₁₆

—

118

—

C₄×Q₈

—

236

—

C₄×Q₈

—

236

—

100

#117

—

118

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂²⋊D₅₉

yes

—

103

C₂×C₄²

C₂²⋊D₅₉

yes

—

104

#107

—

236

—

105

#106

—

236

—

106

#101

—

107

#101

—

118

—

108

#95

—

109

#95

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

Q₈×D₅₉

yes

—

118

C₂×Q₈

#11745689

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

C₂×C₂²⋊D₅₉

yes

—

122

M₁₆

—

118

—

123

M₁₆

—

236

—

124

C₂×C₈

—

236

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

118

—

128

C₄⋊C₄

—

118

—

129

C₂²⋊C₄

—

236

—

130

C₄²

—

131

C₄²

—

236

—

132

C₂³

#14273696

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

236

—

145

C₂×C₄

—

236

—

146

C₂×C₄

#14273703

yes

—

147

C₂×C₄

#14273725

yes

—

148

C₈

—

236

—

149

C₂²

—

150

C₂²

—

151

C₂²

#16837321

yes

characteristic

152

C₂²

#16837589

yes

characteristic

153

C₂²

#16837592

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

472

—

159

C₄

—

472

—

160

C₂

—

161

C₂

—

162

C₂

#21425448

yes

characteristic

163

C₂

#21426113

yes

characteristic

164

C₂

#21426126

yes

characteristic

	subgroup	G/H
center	C₂²	#16837589
commutator, cosocle	C₂²×Q₈×C₅₉	C₂³
Frattini	C₂²×Q₈	C₂²×D₅₉
Fitting	C₅₉×#2811	C₂
socle	C₂²×C₅₉	B(4,𝔽₂)
prime radical	C₂³×C₅₉	#120
hypercenter	#2811	D₅₉
hypocenter	C₅₉	#13464
LCS	G ▷ C₂²×Q₈×C₅₉ ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #2811

subgroup

G/H

center

C₂²

#16837589

commutator, cosocle

C₂²×Q₈×C₅₉

C₂³

Frattini

C₂²×Q₈

C₂²×D₅₉

Fitting

C₅₉×#2811

C₂

C₂²×C₅₉

C₂³×C₅₉

D₅₉

C₅₉

G ▷ C₂²×Q₈×C₅₉ ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #2811

N	H ≅ G/N	type
18. #14273646	Q₈	semidirect
93. C₅₉	#13464	semidirect

H ≅ G/N

type

18. #14273646

Q₈

semidirect

93. C₅₉

#13464

semidirect

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

subgroup	index
#21424310	2
#21422223	2
C₅₉×#2811	2
#13464	59

subgroup

index

#21424310

#21422223

C₅₉×#2811

#13464

Subgroups — #28508373

Conjugacy classes of subgroups

Complements

Maximal subgroups