Group #16777761 — subgroups

Subgroups — #16777761

Order 3328.

184 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 36 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	#13129782	C₂	1	2	yes	maximal
2	#13115813	C₂	1	1	yes	maximal, characteristic
3	#11719175	C₄	1	1	yes	characteristic
4	C₄³×C₁₃	C₄	1	1	yes	characteristic
5	#11718585	C₂²	1	1	yes	characteristic
6	#11718284	—	2	2	no	—
7	#11718115	—	2	2	no	—
8	C₂×C₄²×C₁₃	—	2	1	no	—
9	C₂×C₄²×C₁₃	—	4	1	no	—
10	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
11	#88180	—	2	1	no	—
12	#88178	—	4	1	no	—
13	C₂×#6099	D₄	1	2	yes	—
14	#88124	C₈	1	2	yes	—
15	#88123	—	2	1	no	—
16	C₄×Dic₂₆	—	4	1	no	—
17	#88106	—	4	2	no	—
18	C₄⋊#1055	—	4	1	no	—
19	#7361	—	13	1	no	maximal, Sylow
20	C₂²×C₄×C₁₃	—	2	1	no	—
21	C₂²×C₄×C₁₃	—	4	1	no	—
22	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×Dic₂₆	—	4	1	no	—
24	C₄²×C₁₃	—	2	1	no	—
25	C₄²×C₁₃	—	2	2	no	—
26	C₄²×C₁₃	—	4	1	no	—
27	C₄²×C₁₃	—	4	4	no	—
28	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
29	C₄²×C₁₃	M₁₆	1	1	yes	characteristic
30	C₂²⋊Dic₁₃	—	4	1	no	—
31	C₄⋊Dic₁₃	—	2	2	no	—
32	C₄⋊Dic₁₃	—	4	1	no	—
33	#6101	—	4	2	no	—
34	C₄×Dic₁₃	—	4	1	no	—
35	#6099	—	4	2	no	—
36	C₂×#1055	—	2	2	no	—
37	#6092	—	8	2	no	—
38	#1994	—	13	1	no	—
39	#1327	—	13	2	no	—
40	C₂³×C₁₃	#100	1	1	yes	characteristic
41	C₂×C₄×C₁₃	—	2	1	no	—
42	C₂×C₄×C₁₃	—	2	1	no	—
43	C₂×C₄×C₁₃	—	2	1	no	—
44	C₂×C₄×C₁₃	—	2	1	no	—
45	C₂×C₄×C₁₃	—	4	2	no	—
46	C₂×C₄×C₁₃	—	4	2	no	—
47	C₂×C₄×C₁₃	—	4	4	no	—
48	C₂×C₄×C₁₃	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₁₃	#99	1	1	yes	characteristic
50	C₂×Dic₁₃	—	4	2	no	—
51	Dic₂₆	—	8	2	no	—
52	#1055	—	4	2	no	—
53	#533	—	13	1	no	—
54	#423	—	13	1	no	—
55	C₄³	#261	1	1	yes	characteristic
56	#349	—	26	2	no	—
57	#339	—	26	2	no	—
58	C₂²×C₁₃	—	2	1	no	—
59	C₂²×C₁₃	—	4	1	no	—
60	C₂²×C₁₃	#323	1	1	yes	characteristic
61	C₄×C₁₃	—	2	2	no	—
62	C₄×C₁₃	—	2	1	no	—
63	C₄×C₁₃	—	4	1	no	—
64	C₄×C₁₃	—	4	4	no	—
65	C₄×C₁₃	#354	1	1	yes	characteristic
66	C₄×C₁₃	#356	1	1	yes	characteristic
67	Dic₁₃	—	8	2	no	—
68	C₂×M₁₆	—	13	2	no	—
69	C₄⋊Q₈	—	26	1	no	—
70	#125	—	13	2	no	—
71	C₂²⋊Q₈	—	52	1	no	—
72	C₄×Q₈	—	52	1	no	—
73	#117	—	26	1	no	—
74	C₂×C₄²	—	2	1	no	—
75	C₂×C₄²	—	4	1	no	—
76	C₂×C₄²	C₂×#261	1	1	yes	characteristic
77	M₃₂	—	52	2	no	—
78	C₄⋊C₈	—	52	1	no	—
79	C₂×C₁₃	—	2	1	no	—
80	C₂×C₁₃	—	4	1	no	—
81	C₂×C₁₃	#1319	1	1	yes	characteristic
82	C₂×Q₈	—	52	1	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	4	1	no	—
85	C₂²×C₄	C₂²⋊#261	1	1	yes	characteristic
86	M₁₆	—	52	2	no	—
87	C₂×C₈	—	26	2	no	—
88	C₄⋊C₄	—	26	2	no	—
89	C₄⋊C₄	—	52	2	no	—
90	C₄⋊C₄	—	52	1	no	—
91	C₂²⋊C₄	—	52	1	no	—
92	C₄²	—	2	1	no	—
93	C₄²	—	2	2	no	—
94	C₄²	—	4	1	no	—
95	C₄²	—	4	4	no	—
96	C₄²	—	52	1	no	—
97	C₄²	D₁₃⋊C₈	1	1	yes	characteristic
98	C₄²	#6118	1	1	yes	characteristic
99	C₁₆	—	104	2	no	—
100	C₁₃	#7361	1	1	yes	characteristic, complemented, Sylow
101	C₂³	#88121	1	1	yes	characteristic
102	Q₈	—	104	2	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	—	52	2	no	—
111	C₂×C₄	#88107	1	1	yes	characteristic
112	C₂×C₄	#88111	1	1	yes	characteristic
113	C₈	—	52	2	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	4	1	no	—
116	C₂²	#11718287	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	1	no	—
119	C₄	—	4	1	no	—
120	C₄	—	4	4	no	—
121	C₄	—	104	2	no	—
122	C₄	#11718226	1	1	yes	characteristic
123	C₄	#11718227	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	4	1	no	—
126	C₂	#13129767	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13129782

C₂

yes

maximal

#13115813

C₂

yes

maximal, characteristic

#11719175

C₄

yes

characteristic

C₄³×C₁₃

C₄

yes

characteristic

#11718585

C₂²

yes

characteristic

#11718284

—

#11718115

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

—

C₂×C₄²×C₁₃

C₂×C₄

yes

—

—

D₄

yes

—

#88124

C₈

yes

—

#88123

—

C₄×Dic₂₆

—

—

—

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₆

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

—

C₄²×C₁₃

C₂×C₈

yes

characteristic

C₄²×C₁₃

M₁₆

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

C₄×Dic₁₃

—

—

—

—

—

—

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

—

C₂×C₄×C₁₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₁₃

#99

yes

characteristic

C₂×Dic₁₃

—

Dic₂₆

—

#1055

—

#533

—

#423

—

C₄³

yes

—

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#323

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#261

yes

characteristic

M₃₂

—

C₄⋊C₈

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#1319

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊#261

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

D₁₃⋊C₈

yes

characteristic

C₄²

#6118

yes

characteristic

C₁₆

—

104

—

100

C₁₃

#7361

yes

characteristic, complemented, Sylow

101

C₂³

#88121

yes

characteristic

102

Q₈

—

104

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#88107

yes

characteristic

112

C₂×C₄

#88111

yes

characteristic

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

#11718287

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

104

—

122

C₄

#11718226

yes

characteristic

123

C₄

#11718227

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

#13129767

yes

characteristic

	subgroup	G/H
center	C₄	#11718226
commutator	C₄²×C₁₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#261
Fitting	C₄³×C₁₃	C₄
socle	C₂×C₁₃	#1319
prime radical	C₂³×C₁₃	#100
cosocle	#11718585	C₂²
hypercenter	C₄³	#261
hypocenter	C₁₃	#7361
LCS	G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₄

#11718226

commutator

C₄²×C₁₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#261

Fitting

C₄³×C₁₃

C₄

C₂×C₁₃

C₂³×C₁₃

C₂²

C₄³

C₁₃

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

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
100. C₁₃	#7361	semidirect

H ≅ G/N

type

100. C₁₃

#7361

semidirect

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

subgroup	index
#13129782	2
#13115813	2
#7361	13

subgroup

index

#13129782

#13115813

#7361

Subgroups — #16777761

Conjugacy classes of subgroups

Complements

Maximal subgroups