Group #16777747 — subgroups

Subgroups — #16777747

Order 3328.

216 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 40 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	#13129778	C₂	1	2	yes	maximal
2	#13115638	C₂	1	1	yes	maximal, characteristic
3	C₂×#88123	C₄	1	1	yes	characteristic
4	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
5	C₂×#88056	—	2	2	no	—
6	#11718596	C₂²	1	1	yes	characteristic
7	#11718285	—	2	2	no	—
8	C₂²×Dic₂₆	—	2	1	no	—
9	C₂²×C₈×C₁₃	—	2	1	no	—
10	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₃	—	4	1	no	—
12	C₂×C₄⋊Dic₁₃	—	4	1	no	—
13	C₂×C₄⋊Dic₁₃	D₄	1	2	yes	—
14	#88123	—	2	1	no	—
15	#88123	—	4	1	no	—
16	#88123	C₈	1	2	yes	complemented
17	C₂²⋊#1057	—	4	2	no	—
18	C₄×C₈×C₁₃	—	4	1	no	—
19	#88056	—	4	4	no	—
20	#7355	—	13	1	no	maximal, Sylow
21	C₂²×C₄×C₁₃	—	2	1	no	—
22	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×Dic₁₃	—	2	2	no	—
24	C₂²×Dic₁₃	—	4	1	no	—
25	C₂×Dic₂₆	—	2	2	no	—
26	C₂×Dic₂₆	—	4	2	no	—
27	C₂×Dic₂₆	—	4	2	no	—
28	C₂×Dic₂₆	—	4	1	no	—
29	C₂×#1057	—	8	2	no	—
30	C₂×C₈×C₁₃	—	2	2	no	—
31	C₂×C₈×C₁₃	—	4	1	no	—
32	C₂×C₈×C₁₃	—	4	1	no	—
33	C₄²×C₁₃	—	2	1	no	—
34	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
36	C₄⋊Dic₁₃	—	4	2	no	—
37	C₄⋊Dic₁₃	—	4	2	no	—
38	C₄⋊Dic₁₃	—	4	2	no	—
39	C₄⋊Dic₁₃	—	8	2	no	—
40	#1956	—	13	1	no	—
41	#1325	—	13	2	no	—
42	C₂³×C₁₃	#99	1	1	yes	characteristic
43	C₂×C₄×C₁₃	—	2	2	no	—
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	2	1	no	—
46	C₂×C₄×C₁₃	—	2	1	no	—
47	C₂×C₄×C₁₃	—	2	1	no	—
48	C₂×C₄×C₁₃	—	2	1	no	—
49	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
51	C₂×Dic₁₃	—	4	2	no	—
52	C₂×Dic₁₃	—	4	2	no	—
53	C₂×Dic₁₃	—	8	1	no	—
54	C₂×Dic₁₃	—	8	1	no	—
55	C₂×Dic₁₃	—	8	2	no	—
56	Dic₂₆	—	4	4	no	—
57	Dic₂₆	—	8	2	no	—
58	#1057	—	16	2	no	—
59	C₈×C₁₃	—	4	2	no	—
60	C₂×C₄⋊Q₈	—	13	1	no	—
61	C₂×#103	—	26	2	no	—
62	C₂×C₄×C₈	#261	1	1	yes	characteristic
63	#384	—	13	1	no	—
64	#324	—	26	2	no	—
65	C₂²×C₁₃	—	2	1	no	—
66	C₂²×C₁₃	—	2	1	no	—
67	C₂²×C₁₃	#323	1	1	yes	characteristic
68	C₂²×C₁₃	#353	1	1	yes	characteristic
69	C₂²×C₁₃	#355	1	1	yes	characteristic
70	C₄×C₁₃	—	2	2	no	—
71	C₄×C₁₃	—	2	2	no	—
72	C₄×C₁₃	—	2	1	no	—
73	C₄×C₁₃	—	2	1	no	—
74	Dic₁₃	—	8	2	no	—
75	Dic₁₃	—	8	2	no	—
76	C₂²×Q₈	—	26	1	no	—
77	C₂²×C₈	—	2	1	no	—
78	C₄⋊Q₈	—	13	2	no	—
79	C₄⋊Q₈	—	26	1	no	—
80	C₄⋊Q₈	—	52	1	no	—
81	C₂×C₄⋊C₄	—	13	2	no	—
82	C₂×C₄⋊C₄	—	52	1	no	—
83	C₂×C₄⋊C₄	—	52	1	no	—
84	C₂×C₄²	C₂×#261	1	1	yes	characteristic
85	#103	—	52	4	no	—
86	C₂²⋊C₈	—	52	2	no	—
87	C₄×C₈	—	4	1	no	—
88	C₂×C₁₃	—	2	1	no	—
89	C₂×C₁₃	—	2	1	no	—
90	C₂×C₁₃	#1318	1	1	yes	characteristic
91	C₂×C₁₃	#1409	1	1	yes	characteristic
92	C₂×C₁₃	#1411	1	1	yes	characteristic
93	C₂×Q₈	—	26	2	no	—
94	C₂×Q₈	—	52	2	no	—
95	C₂×Q₈	—	52	2	no	—
96	C₂×Q₈	—	52	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	26	2	no	—
99	C₂²×C₄	—	52	1	no	—
100	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	104	2	no	—
105	C₄⋊C₄	—	52	2	no	—
106	C₄⋊C₄	—	52	2	no	—
107	C₄⋊C₄	—	52	2	no	—
108	C₄⋊C₄	—	104	2	no	—
109	C₄²	—	2	1	no	—
110	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
111	C₄²	#6118	1	1	yes	characteristic
112	C₁₃	#7355	1	1	yes	characteristic, complemented, Sylow
113	C₂³	#88107	1	1	yes	characteristic
114	Q₈	—	52	4	no	—
115	Q₈	—	104	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	52	2	no	—
123	C₂×C₄	—	52	2	no	—
124	C₂×C₄	—	104	1	no	—
125	C₂×C₄	—	104	1	no	—
126	C₂×C₄	—	104	2	no	—
127	C₂×C₄	#88111	1	1	yes	characteristic
128	C₂×C₄	#88121	1	1	yes	characteristic
129	C₈	—	4	2	no	—
130	C₈	—	208	2	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#11718225	1	1	yes	characteristic
134	C₂²	#11718228	1	1	yes	characteristic
135	C₂²	#11718287	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	104	2	no	—
141	C₄	—	104	2	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#13129765	1	1	yes	characteristic
145	C₂	#13129970	1	1	yes	characteristic
146	C₂	#13129974	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13129778

C₂

yes

maximal

#13115638

C₂

yes

maximal, characteristic

C₂×#88123

C₄

yes

characteristic

C₂×C₄×C₈×C₁₃

C₄

yes

characteristic

C₂×#88056

—

#11718596

C₂²

yes

characteristic

#11718285

—

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

D₄

yes

—

#88123

—

#88123

—

#88123

C₈

yes

complemented

C₂²⋊#1057

—

C₄×C₈×C₁₃

—

#88056

—

#7355

—

maximal, Sylow

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₃

—

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×#1057

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

M₁₆

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#1956

—

#1325

—

C₂³×C₁₃

#99

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

characteristic

C₂×C₄×C₁₃

#100

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

#1057

—

C₈×C₁₃

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×C₄×C₈

yes

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

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

#103

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂²⋊#261

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

104

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

104

—

109

C₄²

—

110

C₄²

D₁₃⋊C₈

yes

characteristic

111

C₄²

#6118

yes

characteristic

112

C₁₃

#7355

yes

characteristic, complemented, Sylow

113

C₂³

#88107

yes

characteristic

114

Q₈

—

115

Q₈

—

104

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

104

—

125

C₂×C₄

—

104

—

126

C₂×C₄

—

104

—

127

C₂×C₄

#88111

yes

characteristic

128

C₂×C₄

#88121

yes

characteristic

129

C₈

—

130

C₈

—

208

—

131

C₂²

—

132

C₂²

—

133

C₂²

#11718225

yes

characteristic

134

C₂²

#11718228

yes

characteristic

135

C₂²

#11718287

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

104

—

141

C₄

—

104

—

142

C₂

—

143

C₂

—

144

C₂

#13129765

yes

characteristic

145

C₂

#13129970

yes

characteristic

146

C₂

#13129974

yes

characteristic

	subgroup	G/H
center	C₂²	#11718225
commutator	C₄²×C₁₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	#353
prime radical	C₂³×C₁₃	#99
cosocle	#11718596	C₂²
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#7355
LCS	G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²

#11718225

commutator

C₄²×C₁₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#261

Fitting

C₂×C₄×C₈×C₁₃

C₄

C₂²×C₁₃

C₂³×C₁₃

C₂²

C₂×C₄×C₈

C₁₃

G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
16. #88123	C₈	semidirect
112. C₁₃	#7355	semidirect

H ≅ G/N

type

16. #88123

C₈

semidirect

112. C₁₃

#7355

semidirect

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

subgroup	index
#13129778	2
#13115638	2
#7355	13

subgroup

index

#13129778

#13115638

#7355

Subgroups — #16777747

Conjugacy classes of subgroups

Complements

Maximal subgroups