Group #16777672 — subgroups

Subgroups — #16777672

Order 3328.

244 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 50 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	#13130087	C₂	1	2	yes	maximal
2	#13114905	C₂	1	1	yes	maximal, characteristic
3	C₂×#88180	C₂²	1	1	yes	characteristic
4	#11719128	—	2	2	no	—
5	C₂×C₁₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₂²⋊#1055	C₄	1	1	yes	characteristic
7	#11718305	C₄	1	4	yes	—
8	#11718285	—	2	2	no	—
9	#11718142	—	2	2	no	—
10	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈×C₁₃	—	2	1	no	—
12	C₂×C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
13	#88180	—	4	1	no	—
14	#88180	—	4	1	no	—
15	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₃	Q₈	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
18	C₂²×#1055	—	2	1	no	—
19	C₂²⋊#1057	—	2	4	no	—
20	#88112	—	4	2	no	—
21	#88110	—	4	2	no	—
22	C₁₃×C₂²⋊C₈	—	4	1	no	—
23	C₂²⋊#1055	—	4	1	no	—
24	#7325	—	13	1	no	maximal, Sylow
25	C₂⁴×C₁₃	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₁₃	—	2	1	no	—
27	C₂²×C₄×C₁₃	—	2	1	no	—
28	C₂²×C₄×C₁₃	—	4	2	no	—
29	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×Dic₁₃	C₂²⋊C₄	1	2	yes	—
32	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
33	C₂×#1057	—	4	4	no	—
34	C₂×C₈×C₁₃	—	4	2	no	—
35	C₂×C₈×C₁₃	—	4	1	no	—
36	C₂²⋊Dic₁₃	—	4	2	no	—
37	C₄⋊Dic₁₃	—	4	1	no	—
38	C₄⋊Dic₁₃	—	4	1	no	—
39	C₄×Dic₁₃	—	2	2	no	—
40	C₄×Dic₁₃	—	2	2	no	—
41	C₂×#1055	—	4	2	no	—
42	C₂×#1055	—	4	1	no	—
43	#1818	—	13	1	no	—
44	#1464	—	13	2	no	—
45	C₂³×C₁₃	—	2	1	no	—
46	C₂³×C₁₃	—	2	2	no	—
47	C₂³×C₁₃	—	4	1	no	—
48	C₂³×C₁₃	—	4	1	no	—
49	C₂³×C₁₃	#95	1	1	yes	characteristic
50	C₂×C₄×C₁₃	—	2	2	no	—
51	C₂×C₄×C₁₃	—	2	1	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	4	2	no	—
54	C₂×C₄×C₁₃	—	4	2	no	—
55	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
57	C₂×Dic₁₃	—	2	2	no	—
58	C₂×Dic₁₃	—	2	2	no	—
59	C₂×Dic₁₃	—	4	2	no	—
60	C₂×Dic₁₃	—	4	2	no	—
61	#1057	—	8	4	no	—
62	C₈×C₁₃	—	8	1	no	—
63	#1055	—	8	1	no	—
64	C₂×#117	—	13	1	no	—
65	#432	—	26	2	no	—
66	C₂×C₂²⋊C₈	—	13	1	no	—
67	C₂×C₂²⋊C₈	#261	1	1	yes	characteristic
68	#336	—	26	2	no	—
69	#324	—	26	2	no	—
70	#323	—	13	4	no	—
71	C₂²×C₁₃	—	2	1	no	—
72	C₂²×C₁₃	—	2	1	no	—
73	C₂²×C₁₃	—	4	1	no	—
74	C₂²×C₁₃	—	4	1	no	—
75	C₂²×C₁₃	—	4	1	no	—
76	C₂²×C₁₃	—	8	2	no	—
77	C₂²×C₁₃	#342	1	1	yes	characteristic
78	C₂²×C₁₃	#344	1	2	yes	—
79	C₄×C₁₃	—	2	1	no	—
80	C₄×C₁₃	—	2	1	no	—
81	C₄×C₁₃	—	4	1	no	—
82	Dic₁₃	—	4	2	no	—
83	Dic₁₃	—	4	2	no	—
84	C₂³×C₄	C₂×#261	1	1	yes	characteristic
85	C₂²×C₈	—	2	1	no	—
86	C₂²×C₈	—	26	1	no	—
87	#117	—	52	1	no	—
88	#117	—	52	1	no	—
89	C₂×C₄⋊C₄	—	13	1	no	—
90	C₂×C₄⋊C₄	—	13	1	no	—
91	C₂×C₂²⋊C₄	—	13	2	no	—
92	C₂×C₄²	—	13	2	no	—
93	C₄⋊C₈	—	52	2	no	—
94	C₂²⋊C₈	—	4	1	no	—
95	C₂²⋊C₈	—	26	4	no	—
96	C₂²⋊C₈	—	52	1	no	—
97	#97	—	52	2	no	—
98	C₂×C₁₃	—	2	2	no	—
99	C₂×C₁₃	—	8	1	no	—
100	C₂×C₁₃	#1306	1	1	yes	characteristic
101	C₂×C₁₃	#1391	1	1	yes	characteristic
102	C₂×C₁₃	#1392	1	1	yes	characteristic
103	C₂⁴	C₄×#261	1	1	yes	characteristic
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	4	2	no	—
107	C₂²×C₄	—	13	2	no	—
108	C₂²×C₄	—	13	2	no	—
109	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
110	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	52	2	no	—
114	C₂×C₈	—	52	4	no	—
115	C₂×C₈	—	52	1	no	—
116	C₄⋊C₄	—	52	1	no	—
117	C₄⋊C₄	—	52	1	no	—
118	C₂²⋊C₄	—	52	2	no	—
119	C₄²	—	26	2	no	—
120	C₄²	—	26	2	no	—
121	C₁₃	#7325	1	1	yes	characteristic, complemented, Sylow
122	C₂³	—	2	1	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	4	1	no	—
125	C₂³	—	4	1	no	—
126	C₂³	(C₂×#261)⋊C₄	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	26	2	no	—
133	C₂×C₄	—	26	2	no	—
134	C₂×C₄	—	52	2	no	—
135	C₂×C₄	—	52	2	no	—
136	C₂×C₄	#88107	1	1	yes	characteristic
137	C₂×C₄	#88119	1	1	yes	characteristic
138	C₈	—	8	1	no	—
139	C₈	—	104	1	no	—
140	C₈	—	104	4	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	8	2	no	—
147	C₂²	#11718239	1	2	yes	—
148	C₂²	#11718289	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	4	1	no	—
152	C₄	—	52	2	no	—
153	C₄	—	52	2	no	—
154	C₂	—	2	2	no	—
155	C₂	—	8	1	no	—
156	C₂	#13129739	1	1	yes	characteristic
157	C₂	#13129924	1	1	yes	characteristic
158	C₂	#13129928	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130087

C₂

yes

maximal

#13114905

C₂

yes

maximal, characteristic

C₂×#88180

C₂²

yes

characteristic

#11719128

—

C₂×C₁₃×C₂²⋊C₈

C₄

yes

characteristic

C₂×C₂²⋊#1055

C₄

yes

characteristic

#11718305

C₄

yes

—

#11718285

—

#11718142

—

C₂³×C₄×C₁₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₃

—

C₂×C₂²⋊Dic₁₃

C₂×C₄

yes

—

#88180

—

#88180

—

C₂×C₄⋊Dic₁₃

D₄

yes

characteristic

C₂×C₄⋊Dic₁₃

Q₈

yes

characteristic

C₂×C₄×Dic₁₃

D₄

yes

—

C₂²×#1055

—

C₂²⋊#1057

—

#88112

—

#88110

—

C₁₃×C₂²⋊C₈

—

C₂²⋊#1055

—

#7325

—

maximal, Sylow

C₂⁴×C₁₃

C₄²

yes

characteristic

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₁₃

C₂²⋊C₄

yes

—

C₂²×Dic₁₃

C₄⋊C₄

yes

—

C₂×#1057

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

C₄×Dic₁₃

—

—

—

—

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

#95

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

#1057

—

C₈×C₁₃

—

#1055

—

C₂×#117

—

#432

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

yes

—

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#342

yes

characteristic

C₂²×C₁₃

#344

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂³×C₄

C₂×#261

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₂×C₁₃

—

C₂×C₁₃

—

100

C₂×C₁₃

#1306

yes

characteristic

101

C₂×C₁₃

#1391

yes

characteristic

102

C₂×C₁₃

#1392

yes

characteristic

103

C₂⁴

C₄×#261

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₄⋊#261

yes

characteristic

110

C₂²×C₄

C₂²⋊#261

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₁₃

#7325

yes

characteristic, complemented, Sylow

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

(C₂×#261)⋊C₄

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#88107

yes

characteristic

137

C₂×C₄

#88119

yes

characteristic

138

C₈

—

139

C₈

—

104

—

140

C₈

—

104

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#11718239

yes

—

148

C₂²

#11718289

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₂

—

155

C₂

—

156

C₂

#13129739

yes

characteristic

157

C₂

#13129924

yes

characteristic

158

C₂

#13129928

yes

characteristic

	subgroup	G/H
center	C₂²	#11718289
commutator, prime radical	C₂⁴×C₁₃	C₄²
Frattini	C₂³×C₄	C₂×#261
Fitting	C₂×C₁₃×C₂²⋊C₈	C₄
socle	C₂²×C₁₃	#342
cosocle	C₂×#88180	C₂²
hypercenter	C₂×C₂²⋊C₈	#261
hypocenter	C₁₃	#7325
LCS	G ▷ C₂⁴×C₁₃ ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂²

#11718289

commutator, prime radical

C₂⁴×C₁₃

C₄²

Frattini

C₂³×C₄

C₂×#261

Fitting

C₂×C₁₃×C₂²⋊C₈

C₄

C₂²×C₁₃

C₂²

C₂×C₂²⋊C₈

C₁₃

G ▷ C₂⁴×C₁₃ ▷ C₂³×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
121. C₁₃	#7325	semidirect

H ≅ G/N

type

121. C₁₃

#7325

semidirect

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

subgroup	index
#13130087	2
#13114905	2
#7325	13

subgroup

index

#13130087

#13114905

#7325

Subgroups — #16777672

Conjugacy classes of subgroups

Complements

Maximal subgroups