Group #16777392 — subgroups

Subgroups — #16777392

Order 3328.

252 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 58 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	#13130718	C₂	1	2	yes	maximal
2	#13116451	C₂	1	1	yes	maximal, characteristic
3	C₂×#88155	C₂²	1	1	yes	characteristic
4	#11719116	—	2	2	no	—
5	#11719092	—	2	2	no	—
6	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic, complemented
7	C₂²×#6090	C₄	1	1	yes	characteristic, complemented
8	#11718267	C₄	1	4	yes	complemented
9	#11718099	—	2	2	no	—
10	C₂×#6127	D₄	1	1	yes	characteristic
11	#88237	—	2	4	no	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂×#6099	Q₈	1	1	yes	characteristic
14	#88155	—	4	1	no	—
15	#88155	—	4	1	no	—
16	C₂×#6094	C₂×C₄	1	2	yes	—
17	C₂×C₈×D₁₃	D₄	1	2	yes	—
18	#88119	—	4	2	no	—
19	#88107	—	4	2	no	—
20	#88102	—	4	2	no	—
21	#88101	—	4	2	no	—
22	C₁₃×M₃₂	—	4	1	no	—
23	C₂×C₁₃×C₁₆	—	2	1	no	—
24	C₂×#6090	—	2	1	no	—
25	C₂×#6090	—	4	1	no	—
26	#7228	—	13	1	no	maximal, Sylow
27	C₂²×C₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₂²⋊D₁₃	SD₁₆	1	2	yes	—
29	#6127	—	4	1	no	—
30	#6127	—	4	1	no	—
31	C₂×D₅₂	D₈	1	1	yes	characteristic
32	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
33	C₂×Dic₂₆	Q₁₆	1	1	yes	characteristic
34	C₂²⋊#261	—	4	4	no	—
35	C₄⋊#261	—	4	4	no	—
36	C₄×#261	—	4	4	no	—
37	C₂×C₈×C₁₃	—	2	1	no	—
38	C₂×C₈×C₁₃	—	2	1	no	—
39	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
41	#6099	—	4	1	no	—
42	C₂×#1055	C₄⋊C₄	1	2	yes	—
43	#6094	—	4	2	no	—
44	C₈×D₁₃	—	2	2	no	—
45	C₈×D₁₃	—	2	2	no	—
46	C₁₃×C₁₆	—	4	1	no	—
47	#6090	—	4	1	no	—
48	#2113	—	13	1	no	—
49	#1812	—	13	2	no	—
50	C₂³×C₁₃	#106	1	1	yes	characteristic
51	C₂²×D₁₃	#102	1	2	yes	—
52	C₂×#261	—	8	4	no	—
53	C₂×C₄×C₁₃	—	2	1	no	—
54	C₂×C₄×C₁₃	—	2	1	no	—
55	C₂×C₄×C₁₃	#95	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
57	C₂²⋊D₁₃	—	4	2	no	—
58	C₂×Dic₁₃	#103	1	2	yes	—
59	D₅₂	—	4	1	no	—
60	C₄×D₁₃	—	2	2	no	—
61	C₄×D₁₃	—	2	2	no	—
62	Dic₂₆	—	4	1	no	—
63	C₈×C₁₃	—	2	1	no	—
64	C₈×C₁₃	#99	1	1	yes	characteristic
65	C₈×C₁₃	#99	1	1	yes	characteristic
66	#1055	—	2	2	no	—
67	C₂×#131	—	13	1	no	—
68	C₂×M₃₂	#261	1	1	yes	characteristic, complemented
69	C₂²×C₁₆	—	13	1	no	—
70	#427	—	26	2	no	—
71	#410	—	26	2	no	—
72	C₂²⋊C₁₆	—	26	2	no	—
73	H(3,ℤ/4)	—	13	4	no	—
74	C₂²×C₁₃	—	2	1	no	—
75	C₂²×C₁₃	—	4	1	no	—
76	C₂²×C₁₃	#340	1	1	yes	characteristic
77	C₂×D₁₃	—	2	2	no	—
78	C₂×D₁₃	—	4	2	no	—
79	#261	—	16	4	no	—
80	C₄×C₁₃	—	2	1	no	—
81	C₄×C₁₃	H(3,ℤ/4)	1	1	yes	characteristic
82	C₄×C₁₃	#342	1	1	yes	characteristic
83	Dic₁₃	—	2	2	no	—
84	C₂×#41	—	13	1	no	—
85	#131	—	52	1	no	—
86	#131	—	52	1	no	—
87	C₂×M₁₆	—	13	1	no	—
88	C₂×M₁₆	—	13	2	no	—
89	C₂²×C₈	—	13	2	no	—
90	C₂²×C₈	C₂×#261	1	1	yes	characteristic
91	#117	—	26	4	no	—
92	M₃₂	—	4	1	no	—
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	—	26	1	no	—
95	C₂×C₁₆	—	52	1	no	—
96	#107	—	52	2	no	—
97	#106	—	52	2	no	—
98	#99	—	52	2	no	—
99	#99	—	52	2	no	—
100	C₂×C₁₃	—	2	1	no	—
101	C₂×C₁₃	—	4	1	no	—
102	C₂×C₁₃	#1289	1	1	yes	characteristic
103	D₁₃	—	4	2	no	—
104	#41	—	52	1	no	—
105	#41	—	52	1	no	—
106	C₂×Q₈	—	13	1	no	—
107	C₂×D₄	—	13	2	no	—
108	C₂×D₄	—	13	1	no	—
109	C₂²×C₄	—	13	2	no	—
110	C₂²×C₄	C₄⋊#261	1	1	yes	characteristic
111	M₁₆	—	52	2	no	—
112	M₁₆	—	52	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	13	2	no	—
116	C₂×C₈	—	26	2	no	—
117	C₂×C₈	—	26	2	no	—
118	C₂×C₈	C₄×#261	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊#261	1	1	yes	characteristic
120	C₄⋊C₄	—	52	4	no	—
121	C₂²⋊C₄	—	52	4	no	—
122	C₄²	—	52	4	no	—
123	C₁₆	—	4	1	no	—
124	C₁₆	—	52	1	no	—
125	C₁₃	#7228	1	1	yes	characteristic, complemented, Sylow
126	C₂³	—	13	2	no	—
127	C₂³	#88101	1	1	yes	characteristic
128	Q₈	—	52	1	no	—
129	D₄	—	52	1	no	—
130	D₄	—	52	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	13	2	no	—
134	C₂×C₄	—	26	2	no	—
135	C₂×C₄	—	26	2	no	—
136	C₂×C₄	—	104	4	no	—
137	C₂×C₄	#88102	1	1	yes	characteristic
138	C₂×C₄	(C₂×#261)⋊C₄	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	26	2	no	—
141	C₈	#88107	1	1	yes	characteristic
142	C₈	#88119	1	1	yes	characteristic
143	C₂²	—	2	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	26	2	no	—
146	C₂²	—	52	2	no	—
147	C₂²	#11718265	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	26	2	no	—
150	C₄	—	208	4	no	—
151	C₄	#11718267	1	1	yes	characteristic
152	C₄	#11718289	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	4	1	no	—
155	C₂	—	52	2	no	—
156	C₂	#13129699	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13130718

C₂

yes

maximal

#13116451

C₂

yes

maximal, characteristic

C₂×#88155

C₂²

yes

characteristic

#11719116

—

#11719092

—

C₂×C₁₃×M₃₂

C₄

yes

characteristic, complemented

C₂²×#6090

C₄

yes

characteristic, complemented

#11718267

C₄

yes

complemented

#11718099

—

C₂×#6127

D₄

yes

characteristic

#88237

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

C₂×#6099

Q₈

yes

—

—

C₂×C₄

yes

—

C₂×C₈×D₁₃

D₄

yes

—

—

—

—

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

—

—

—

—

C₂²×C₄×C₁₃

C₄⋊C₄

yes

characteristic

C₂×C₂²⋊D₁₃

SD₁₆

yes

—

#6127

—

#6127

—

C₂×D₅₂

D₈

yes

characteristic

C₂×C₄×D₁₃

C₂²⋊C₄

yes

—

C₂×Dic₂₆

Q₁₆

yes

characteristic

C₂²⋊#261

—

C₄⋊#261

—

C₄×#261

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄²

yes

characteristic

C₂×C₈×C₁₃

C₂²⋊C₄

yes

characteristic

#6099

—

C₂×#1055

C₄⋊C₄

yes

—

#6094

—

C₈×D₁₃

—

C₈×D₁₃

—

C₁₃×C₁₆

—

#6090

—

#2113

—

#1812

—

C₂³×C₁₃

#106

yes

characteristic

C₂²×D₁₃

#102

yes

—

C₂×#261

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

yes

C₂×C₄×C₁₃

yes

C₂²⋊D₁₃

—

C₂×Dic₁₃

#103

yes

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

yes

C₈×C₁₃

yes

—

—

C₂×M₃₂

#261

yes

characteristic, complemented

C₂²×C₁₆

—

#427

—

#410

—

C₂²⋊C₁₆

—

H(3,ℤ/4)

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#340

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

#261

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×#261

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

100

C₂×C₁₃

—

101

C₂×C₁₃

—

102

C₂×C₁₃

#1289

yes

characteristic

103

D₁₃

—

104

#41

—

105

#41

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄⋊#261

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

C₄×#261

yes

characteristic

119

C₂×C₈

C₂²⋊#261

yes

characteristic

120

C₄⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

—

123

C₁₆

—

124

C₁₆

—

125

C₁₃

#7228

yes

characteristic, complemented, Sylow

126

C₂³

—

127

C₂³

#88101

yes

characteristic

128

Q₈

—

129

D₄

—

130

D₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

104

—

137

C₂×C₄

#88102

yes

characteristic

138

C₂×C₄

(C₂×#261)⋊C₄

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

#88107

yes

characteristic

142

C₈

#88119

yes

characteristic

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#11718265

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

208

—

151

C₄

#11718267

yes

characteristic

152

C₄

#11718289

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

#13129699

yes

characteristic

	subgroup	G/H
center	C₄	#11718267
commutator	C₂×C₈×C₁₃	C₄²
Frattini	C₂²×C₈	C₂×#261
Fitting	C₂×C₁₃×M₃₂	C₄
socle	C₂×C₁₃	#1289
prime radical	C₂×#6127	D₄
cosocle	C₂×#88155	C₂²
hypercenter	C₂×M₃₂	#261
hypocenter	C₁₃	#7228
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₄

#11718267

commutator

C₂×C₈×C₁₃

C₄²

Frattini

C₂²×C₈

C₂×#261

Fitting

C₂×C₁₃×M₃₂

C₄

C₂×C₁₃

D₄

C₂²

C₂×M₃₂

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
6. C₂×C₁₃×M₃₂	C₄	semidirect
7. C₂²×#6090	C₄	semidirect
8. #11718267	C₄	semidirect
68. C₂×M₃₂	#261	semidirect
125. C₁₃	#7228	semidirect

H ≅ G/N

type

6. C₂×C₁₃×M₃₂

C₄

semidirect

7. C₂²×#6090

C₄

semidirect

8. #11718267

C₄

semidirect

68. C₂×M₃₂

#261

semidirect

125. C₁₃

#7228

semidirect

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

subgroup	index
#13130718	2
#13116451	2
#7228	13

subgroup

index

#13130718

#13116451

#7228

Subgroups — #16777392

Conjugacy classes of subgroups

Complements

Maximal subgroups