Group #16777079 — subgroups

Subgroups — #16777079

Order 3328.

202 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 62 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	#13115802	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#125	C₄	1	1	yes	characteristic
4	C₂×#88106	—	2	2	no	—
5	#11718597	C₄	1	1	yes	characteristic
6	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
7	#11718121	—	2	2	no	—
8	C₁₃×#125	—	2	1	no	—
9	C₁₃×#125	—	4	1	no	—
10	C₁₃×#125	C₈	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
12	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
13	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₃	—	2	1	no	—
15	C₂²×#1055	D₄	1	2	yes	—
16	#88106	—	2	4	no	—
17	#88106	—	4	2	no	—
18	C₂×#6092	—	4	2	no	—
19	#88071	—	4	1	no	—
20	C₄×#1055	—	4	1	no	—
21	#7135	—	13	1	no	maximal, Sylow
22	C₂²×C₄×C₁₃	—	2	1	no	—
23	C₂²×C₄×C₁₃	—	4	1	no	—
24	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×Dic₁₃	—	4	1	no	—
26	C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₁₃×C₄⋊C₄	—	4	2	no	—
28	C₁₃×C₄⋊C₄	—	4	2	no	—
29	C₁₃×C₄⋊C₄	—	4	1	no	—
30	C₄²×C₁₃	—	2	1	no	—
31	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
32	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
33	C₄⋊Dic₁₃	—	4	2	no	—
34	C₂×#1055	—	2	2	no	—
35	C₂×#1055	—	2	2	no	—
36	C₂×#1055	SD₁₆	1	2	yes	—
37	C₂×#1055	Q₁₆	1	2	yes	—
38	#6092	—	4	4	no	—
39	#1991	—	13	1	no	—
40	#1376	—	13	2	no	—
41	C₂³×C₁₃	#100	1	1	yes	characteristic
42	C₂×C₄×C₁₃	—	2	1	no	—
43	C₂×C₄×C₁₃	—	2	1	no	—
44	C₂×C₄×C₁₃	—	4	2	no	—
45	C₂×C₄×C₁₃	—	4	2	no	—
46	C₂×C₄×C₁₃	—	8	2	no	—
47	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
48	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
49	C₂×C₄×C₁₃	#103	1	2	yes	—
50	C₂×C₄×C₁₃	#104	1	2	yes	—
51	C₂×Dic₁₃	—	8	2	no	—
52	C₂×Dic₁₃	—	8	1	no	—
53	#1055	—	2	2	no	—
54	#1055	—	2	2	no	—
55	C₂×#125	#261	1	1	yes	characteristic
56	C₂×M₃₂	—	26	2	no	—
57	C₂×C₄×C₈	—	13	1	no	—
58	#382	—	13	1	no	—
59	#340	—	26	2	no	—
60	C₂²×C₁₃	—	2	2	no	—
61	C₂²×C₁₃	#323	1	1	yes	characteristic
62	C₂²×C₁₃	#329	1	1	yes	characteristic
63	C₂²×C₁₃	#333	1	1	yes	characteristic
64	C₄×C₁₃	—	4	1	no	—
65	C₄×C₁₃	—	4	1	no	—
66	C₄×C₁₃	—	8	2	no	—
67	C₄×C₁₃	Q₈⋊C₈	1	2	yes	—
68	C₄×C₁₃	#328	1	2	yes	—
69	Dic₁₃	—	16	1	no	—
70	C₂²×C₈	—	13	2	no	—
71	#125	—	2	1	no	—
72	#125	—	4	1	no	—
73	#125	#1057	1	2	yes	—
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄⋊C₄	—	4	1	no	—
76	C₂×C₄⋊C₄	—	26	1	no	—
77	C₂×C₄²	C₂×#261	1	1	yes	characteristic
78	M₃₂	—	26	4	no	—
79	M₃₂	—	52	2	no	—
80	C₂×C₁₆	—	52	2	no	—
81	C₄×C₈	—	52	1	no	—
82	#95	—	52	1	no	—
83	C₂×C₁₃	—	2	1	no	—
84	C₂×C₁₃	—	2	1	no	—
85	C₂×C₁₃	#1272	1	1	yes	characteristic
86	C₂×C₁₃	#1366	1	2	yes	—
87	C₂²×C₄	—	2	1	no	—
88	C₂²×C₄	—	4	1	no	—
89	C₂²×C₄	—	52	1	no	—
90	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
91	C₂×C₈	—	13	2	no	—
92	C₂×C₈	—	13	2	no	—
93	C₂×C₈	—	26	2	no	—
94	C₂×C₈	—	26	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄⋊C₄	—	52	2	no	—
100	C₄²	—	2	1	no	—
101	C₄²	C₂×#1057	1	1	yes	characteristic
102	C₄²	#6122	1	1	yes	characteristic
103	C₁₆	—	52	4	no	—
104	C₁₃	#7135	1	1	yes	characteristic, complemented, Sylow
105	C₂³	#88121	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	104	2	no	—
112	C₂×C₄	—	104	1	no	—
113	C₂×C₄	#88117	1	2	yes	—
114	C₂×C₄	#88118	1	2	yes	—
115	C₂×C₄	#88119	1	1	yes	characteristic
116	C₂×C₄	C₂²⋊#1057	1	1	yes	characteristic
117	C₈	—	26	2	no	—
118	C₈	—	26	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#11718299	1	1	yes	characteristic
121	C₂²	#11718303	1	1	yes	characteristic
122	C₂²	#11718305	1	1	yes	characteristic
123	C₄	—	4	1	no	—
124	C₄	—	4	1	no	—
125	C₄	—	8	2	no	—
126	C₄	—	208	1	no	—
127	C₄	#11718298	1	2	yes	—
128	C₄	Q₈⋊#1057	1	2	yes	—
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	#13129656	1	1	yes	characteristic
132	C₂	#13129856	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13129880

C₂

yes

maximal

#13115802

C₂

yes

maximal, characteristic

C₂×C₁₃×#125

C₄

yes

characteristic

C₂×#88106

—

#11718597

C₄

yes

characteristic

C₂×C₄×#1055

C₂²

yes

characteristic

#11718121

—

C₁₃×#125

—

C₁₃×#125

—

C₁₃×#125

C₈

yes

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₄²×C₁₃

C₂×C₄

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂²×#1055

D₄

yes

—

—

—

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

M₁₆

yes

characteristic

C₄⋊Dic₁₃

—

C₂×#1055

—

C₂×#1055

—

C₂×#1055

SD₁₆

yes

—

C₂×#1055

Q₁₆

yes

—

#6092

—

#1991

—

#1376

—

C₂³×C₁₃

#100

yes

characteristic

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

—

—

—

yes

C₂×M₃₂

—

C₂×C₄×C₈

—

#382

—

#340

—

C₂²×C₁₃

—

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

Q₈⋊C₈

yes

—

C₄×C₁₃

#328

yes

—

Dic₁₃

—

C₂²×C₈

—

—

—

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×#261

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

#95

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#1272

yes

characteristic

C₂×C₁₃

#1366

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊#261

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

C₂×#1057

yes

characteristic

102

C₄²

#6122

yes

characteristic

103

C₁₆

—

104

C₁₃

#7135

yes

characteristic, complemented, Sylow

105

C₂³

#88121

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

104

—

112

C₂×C₄

—

104

—

113

C₂×C₄

#88117

yes

—

114

C₂×C₄

#88118

yes

—

115

C₂×C₄

#88119

yes

characteristic

116

C₂×C₄

C₂²⋊#1057

yes

characteristic

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

#11718299

yes

characteristic

121

C₂²

#11718303

yes

characteristic

122

C₂²

#11718305

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

208

—

127

C₄

#11718298

yes

—

128

C₄

Q₈⋊#1057

yes

—

129

C₂

—

130

C₂

—

131

C₂

#13129656

yes

characteristic

132

C₂

#13129856

yes

—

	subgroup	G/H
center	C₂²	#11718303
commutator	C₄²×C₁₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#261
Fitting	C₂×C₁₃×#125	C₄
socle	C₂²×C₁₃	#333
prime radical	C₂³×C₁₃	#100
cosocle	C₂×C₄×#1055	C₂²
hypercenter	C₂×#125	#261
hypocenter	C₁₃	#7135
LCS	G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

#11718303

commutator

C₄²×C₁₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#261

Fitting

C₂×C₁₃×#125

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 ◁ C₂×#125

N	H ≅ G/N	type
104. C₁₃	#7135	semidirect

H ≅ G/N

type

104. C₁₃

#7135

semidirect

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

subgroup	index
#13129880	2
#13115802	2
#7135	13

subgroup

index

#13129880

#13115802

#7135

Subgroups — #16777079

Conjugacy classes of subgroups

Complements

Maximal subgroups