Group #16777025 — subgroups

Subgroups — #16777025

Order 3328.

242 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 60 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	#13129880	C₂	1	2	yes	maximal
2	#13115637	C₂	1	1	yes	maximal, characteristic
3	C₂×#88123	C₄	1	1	yes	characteristic
4	C₂×#88106	—	2	2	no	—
5	C₁₃×#384	C₄	1	1	yes	characteristic
6	C₂×#88050	—	2	2	no	—
7	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
8	C₂²×Dic₂₆	—	2	1	no	—
9	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
10	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
11	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃	—	4	1	no	—
13	C₂²×#1055	D₄	1	2	yes	—
14	#88123	—	2	1	no	—
15	#88123	—	4	1	no	—
16	#88123	C₈	1	2	yes	—
17	#88106	—	2	4	no	—
18	#88106	—	4	2	no	—
19	C₂×#6092	—	4	2	no	—
20	#88050	—	4	2	no	—
21	#88050	—	4	2	no	—
22	C₄×#1055	—	4	1	no	—
23	#7108	—	13	1	no	maximal, Sylow
24	C₂²×C₄×C₁₃	—	2	1	no	—
25	C₂²×C₄×C₁₃	—	4	1	no	—
26	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×Dic₁₃	—	4	1	no	—
28	C₂×Dic₂₆	—	2	2	no	—
29	C₂×Dic₂₆	—	4	2	no	—
30	C₂×Dic₂₆	—	4	2	no	—
31	C₂×Dic₂₆	—	4	1	no	—
32	C₁₃×C₄⋊C₄	—	4	2	no	—
33	C₁₃×C₄⋊C₄	—	8	2	no	—
34	C₄²×C₁₃	—	2	1	no	—
35	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
37	C₄⋊Dic₁₃	—	4	2	no	—
38	C₄⋊Dic₁₃	—	4	2	no	—
39	C₂×#1055	—	2	2	no	—
40	C₂×#1055	—	2	2	no	—
41	C₂×#1055	D₈	1	2	yes	—
42	C₂×#1055	SD₁₆	1	2	yes	—
43	#6092	—	4	4	no	—
44	#1956	—	13	1	no	—
45	#1376	—	13	2	no	—
46	C₂³×C₁₃	#100	1	1	yes	characteristic
47	C₂×C₄×C₁₃	—	2	2	no	—
48	C₂×C₄×C₁₃	—	2	2	no	—
49	C₂×C₄×C₁₃	—	2	1	no	—
50	C₂×C₄×C₁₃	—	2	1	no	—
51	C₂×C₄×C₁₃	—	8	2	no	—
52	C₂×C₄×C₁₃	—	8	1	no	—
53	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#102	1	2	yes	—
56	C₂×C₄×C₁₃	#104	1	2	yes	—
57	C₂×Dic₁₃	—	4	2	no	—
58	C₂×Dic₁₃	—	8	1	no	—
59	C₂×Dic₁₃	—	8	1	no	—
60	Dic₂₆	—	4	4	no	—
61	Dic₂₆	—	8	2	no	—
62	#1055	—	2	2	no	—
63	#1055	—	2	2	no	—
64	C₂×C₄⋊Q₈	—	13	1	no	—
65	C₂×M₃₂	—	26	2	no	—
66	C₂×#103	—	26	2	no	—
67	C₂×C₄×C₈	—	13	1	no	—
68	#384	#261	1	1	yes	characteristic
69	C₂²×C₁₃	—	2	1	no	—
70	C₂²×C₁₃	—	2	1	no	—
71	C₂²×C₁₃	#323	1	1	yes	characteristic
72	C₂²×C₁₃	#329	1	1	yes	characteristic
73	C₂²×C₁₃	#331	1	1	yes	characteristic
74	C₄×C₁₃	—	2	2	no	—
75	C₄×C₁₃	—	2	2	no	—
76	C₄×C₁₃	—	16	1	no	—
77	C₄×C₁₃	D₄⋊C₈	1	2	yes	—
78	C₄×C₁₃	#327	1	2	yes	—
79	Dic₁₃	—	8	2	no	—
80	C₂²×Q₈	—	26	1	no	—
81	C₂²×C₈	—	13	2	no	—
82	C₄⋊Q₈	—	13	2	no	—
83	C₄⋊Q₈	—	26	1	no	—
84	C₄⋊Q₈	—	52	1	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₄⋊C₄	—	4	1	no	—
87	C₂×C₄⋊C₄	—	52	1	no	—
88	C₂×C₄²	C₂×#261	1	1	yes	characteristic
89	M₃₂	—	26	4	no	—
90	M₃₂	—	52	2	no	—
91	C₂×C₁₆	—	52	2	no	—
92	#103	—	52	2	no	—
93	#103	—	52	2	no	—
94	C₄×C₈	—	52	1	no	—
95	C₂×C₁₃	—	2	1	no	—
96	C₂×C₁₃	—	2	1	no	—
97	C₂×C₁₃	#1270	1	1	yes	characteristic
98	C₂×C₁₃	#1358	1	1	yes	characteristic
99	C₂×C₁₃	#1359	1	1	yes	characteristic
100	C₂×Q₈	—	26	2	no	—
101	C₂×Q₈	—	52	1	no	—
102	C₂×Q₈	—	52	2	no	—
103	C₂×Q₈	—	52	2	no	—
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	4	1	no	—
106	C₂²×C₄	—	52	1	no	—
107	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
108	C₂×C₈	—	13	2	no	—
109	C₂×C₈	—	13	2	no	—
110	C₂×C₈	—	26	2	no	—
111	C₂×C₈	—	26	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	8	2	no	—
114	C₄⋊C₄	—	52	2	no	—
115	C₄⋊C₄	—	52	2	no	—
116	C₄²	—	2	1	no	—
117	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
118	C₄²	#6118	1	1	yes	characteristic
119	C₁₆	—	52	4	no	—
120	C₁₃	#7108	1	1	yes	characteristic, complemented, Sylow
121	C₂³	#88121	1	1	yes	characteristic
122	Q₈	—	52	4	no	—
123	Q₈	—	104	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	8	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	52	2	no	—
131	C₂×C₄	—	104	1	no	—
132	C₂×C₄	—	104	1	no	—
133	C₂×C₄	#88107	1	1	yes	characteristic
134	C₂×C₄	#88111	1	1	yes	characteristic
135	C₂×C₄	#88115	1	2	yes	—
136	C₂×C₄	#88118	1	2	yes	—
137	C₈	—	26	2	no	—
138	C₈	—	26	2	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#11718287	1	1	yes	characteristic
142	C₂²	#11718297	1	1	yes	characteristic
143	C₂²	#11718299	1	1	yes	characteristic
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	16	1	no	—
147	C₄	—	104	2	no	—
148	C₄	#11718240	1	2	yes	—
149	C₄	#11718243	1	2	yes	—
150	C₂	—	2	1	no	—
151	C₂	—	2	1	no	—
152	C₂	#13129653	1	1	yes	characteristic
153	C₂	#13129845	1	1	yes	characteristic
154	C₂	#13129846	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13129880

C₂

yes

maximal

#13115637

C₂

yes

maximal, characteristic

C₂×#88123

C₄

yes

characteristic

C₂×#88106

—

C₁₃×#384

C₄

yes

characteristic

C₂×#88050

—

C₂×C₄×#1055

C₂²

yes

characteristic

C₂²×Dic₂₆

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂²×#1055

D₄

yes

—

#88123

—

#88123

—

#88123

C₈

yes

—

—

—

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

M₁₆

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

D₈

yes

—

C₂×#1055

SD₁₆

yes

—

#6092

—

#1956

—

#1376

—

C₂³×C₁₃

#100

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

—

C₂×C₄×C₁₃

#104

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

#1055

—

#1055

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

C₂×#103

—

C₂×C₄×C₈

—

#384

#261

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

D₄⋊C₈

yes

—

C₄×C₁₃

#327

yes

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×#261

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#103

—

#103

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₂²⋊#261

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

—

117

C₄²

D₁₃⋊C₈

yes

characteristic

118

C₄²

#6118

yes

characteristic

119

C₁₆

—

120

C₁₃

#7108

yes

characteristic, complemented, Sylow

121

C₂³

#88121

yes

characteristic

122

Q₈

—

123

Q₈

—

104

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

104

—

132

C₂×C₄

—

104

—

133

C₂×C₄

#88107

yes

characteristic

134

C₂×C₄

#88111

yes

characteristic

135

C₂×C₄

#88115

yes

—

136

C₂×C₄

#88118

yes

—

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

#11718287

yes

characteristic

142

C₂²

#11718297

yes

characteristic

143

C₂²

#11718299

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

104

—

148

C₄

#11718240

yes

—

149

C₄

#11718243

yes

—

150

C₂

—

151

C₂

—

152

C₂

#13129653

yes

characteristic

153

C₂

#13129845

yes

characteristic

154

C₂

#13129846

yes

characteristic

	subgroup	G/H
center	C₂²	#11718297
commutator	C₄²×C₁₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#261
Fitting	C₁₃×#384	C₄
socle	C₂²×C₁₃	#331
prime radical	C₂³×C₁₃	#100
cosocle	C₂×C₄×#1055	C₂²
hypercenter	#384	#261
hypocenter	C₁₃	#7108
LCS	G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂²

#11718297

commutator

C₄²×C₁₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#261

Fitting

C₁₃×#384

C₄

C₂²×C₁₃

C₂³×C₁₃

C₂×C₄×#1055

C₂²

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
120. C₁₃	#7108	semidirect

H ≅ G/N

type

120. C₁₃

#7108

semidirect

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

subgroup	index
#13129880	2
#13115637	2
#7108	13

subgroup

index

#13129880

#13115637

#7108

Subgroups — #16777025

Conjugacy classes of subgroups

Complements

Maximal subgroups