Group #15777578 — subgroups

Subgroups — #15777578

Order 3328.

262 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 87 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	#13117089	C₂	1	1	yes	maximal, characteristic
2	#13117065	C₂	1	2	yes	maximal
3	#13114434	C₂	1	2	yes	maximal
4	C₁₆×#1055	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#500	C₂²	1	2	yes	—
7	#11718742	—	2	2	no	—
8	#11718740	—	2	2	no	—
9	C₂×Dic₁₀₄	—	2	2	no	—
10	#11718360	C₂²	1	2	yes	—
11	C₁₃×#366	—	2	2	no	—
12	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	#11718084	—	2	4	no	—
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	C₈×#1055	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	—	2	2	no	—
17	C₂×Dic₅₂	—	2	2	no	—
18	C₂×Dic₅₂	—	2	2	no	—
19	#88123	—	2	2	no	—
20	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
21	C₁₃×#107	—	2	2	no	—
22	C₁₃×#107	—	2	2	no	—
23	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	#88075	—	4	4	no	—
25	C₂×#6090	D₄	1	2	yes	—
26	#88048	—	4	4	no	—
27	C₄×#1055	D₄	1	1	yes	characteristic
28	C₄×#1055	D₄	1	1	yes	characteristic
29	Dic₁₀₄	—	4	4	no	—
30	#12504	—	13	1	no	maximal, Sylow
31	C₂×Q₈×C₁₃	—	4	2	no	—
32	C₂×Dic₂₆	—	4	2	no	—
33	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₁₃×C₄⋊C₄	—	4	2	no	—
36	C₁₃×C₄⋊C₄	—	4	2	no	—
37	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
38	C₄⋊Dic₁₃	—	4	2	no	—
39	C₄⋊Dic₁₃	—	4	2	no	—
40	C₂×#1055	D₈	1	2	yes	—
41	C₂×#1055	D₈	1	2	yes	—
42	Dic₅₂	—	4	4	no	—
43	Dic₅₂	—	4	4	no	—
44	C₁₃×C₁₆	SD₁₆	1	4	yes	—
45	#6090	—	2	2	no	—
46	#2252	D₁₃	1	1	yes	characteristic
47	#2247	—	13	1	no	—
48	#2241	—	13	2	no	—
49	#1707	—	13	2	no	—
50	C₈×C₁₆	—	13	1	no	—
51	Q₈×C₁₃	—	8	4	no	—
52	C₂×C₄×C₁₃	—	8	2	no	—
53	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₁₃	—	8	2	no	—
57	Dic₂₆	—	8	2	no	—
58	Dic₂₆	—	8	2	no	—
59	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
60	C₈×C₁₃	#135	1	2	yes	—
61	#1055	—	2	2	no	—
62	#1055	Q₃₂	1	4	yes	complemented
63	C₂×Q₃₂	—	26	2	no	—
64	#500	C₂×D₁₃	1	2	yes	—
65	#498	—	26	2	no	—
66	#494	—	13	2	no	—
67	#487	—	26	2	no	—
68	#366	—	2	2	no	—
69	#358	—	26	4	no	—
70	C₄×C₁₆	—	13	1	no	—
71	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
72	C₈²	—	13	1	no	—
73	C₂²×C₁₃	#493	1	1	yes	characteristic
74	C₄×C₁₃	—	16	2	no	—
75	C₄×C₁₃	#492	1	1	yes	characteristic
76	C₄×C₁₃	#495	1	1	yes	characteristic
77	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
78	C₄×C₁₃	#508	1	2	yes	—
79	Dic₁₃	—	16	2	no	—
80	C₂×Q₁₆	—	26	2	no	—
81	C₂×Q₁₆	—	26	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	26	2	no	—
84	Q₃₂	—	52	4	no	—
85	C₂×C₁₆	—	13	2	no	—
86	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
87	#107	—	2	2	no	—
88	#107	—	2	2	no	—
89	#106	—	52	4	no	—
90	#103	—	52	4	no	—
91	C₄×C₈	—	13	1	no	—
92	C₄×C₈	—	13	1	no	—
93	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
94	C₂×C₁₃	#1706	1	1	yes	characteristic
95	C₂×C₁₃	#2247	1	1	yes	characteristic
96	C₂×C₁₃	#2249	1	1	yes	characteristic
97	C₂×Q₈	—	4	2	no	—
98	C₂×Q₈	—	52	2	no	—
99	Q₁₆	—	52	4	no	—
100	Q₁₆	—	52	4	no	—
101	C₂×C₈	—	13	2	no	—
102	C₂×C₈	—	13	2	no	—
103	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
104	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	52	2	no	—
108	C₄⋊C₄	—	52	2	no	—
109	C₄²	D₄×D₁₃	1	1	yes	characteristic
110	C₁₆	—	26	2	no	—
111	C₁₆	#6106	1	4	yes	—
112	C₁₃	#12504	1	1	yes	characteristic, complemented, Sylow
113	Q₈	—	8	4	no	—
114	Q₈	—	104	2	no	—
115	Q₈	—	104	2	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	104	2	no	—
118	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
119	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
120	C₂×C₄	#88194	1	1	yes	characteristic
121	C₈	—	13	4	no	—
122	C₈	—	26	2	no	—
123	C₈	#88173	1	2	yes	—
124	C₈	C₂×#6106	1	2	yes	—
125	C₂²	#11718812	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	208	2	no	—
128	C₄	#11718573	1	2	yes	—
129	C₄	D₁₃×Q₃₂	1	2	yes	—
130	C₄	#11718746	1	1	yes	characteristic
131	C₄	#11718852	1	1	yes	characteristic
132	C₂	#13114429	1	1	yes	characteristic
133	C₂	#13117087	1	1	yes	characteristic
134	C₂	#13117094	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117089

C₂

yes

maximal, characteristic

#13117065

C₂

yes

maximal

#13114434

C₂

yes

maximal

C₁₆×#1055

C₂

yes

maximal, characteristic

C₁₃×#2252

C₂

yes

maximal, characteristic

C₁₃×#500

C₂²

yes

—

#11718742

—

#11718740

—

C₂×Dic₁₀₄

—

#11718360

C₂²

yes

—

C₁₃×#366

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

#11718084

—

C₄×#6090

C₂²

yes

characteristic

C₈×#1055

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

#88123

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₁₃×#107

—

C₁₃×#107

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88075

—

C₂×#6090

D₄

yes

—

#88048

—

C₄×#1055

D₄

yes

characteristic

C₄×#1055

D₄

yes

characteristic

Dic₁₀₄

—

#12504

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

D₈

yes

—

C₂×#1055

D₈

yes

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

SD₁₆

yes

—

#6090

—

#2252

D₁₃

yes

—

—

—

C₈×C₁₆

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

#135

yes

—

#1055

—

#1055

Q₃₂

yes

complemented

C₂×Q₃₂

—

#500

C₂×D₁₃

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₈²

—

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×Q₃₂

yes

—

C₄×C₁₃

#508

yes

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₃

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

100

Q₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

D₄×D₁₃

yes

characteristic

104

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

D₄×D₁₃

yes

characteristic

110

C₁₆

—

111

C₁₆

#6106

yes

—

112

C₁₃

#12504

yes

characteristic, complemented, Sylow

113

Q₈

—

114

Q₈

—

104

—

115

Q₈

—

104

—

116

C₂×C₄

—

117

C₂×C₄

—

104

—

118

C₂×C₄

D₈×D₁₃

yes

characteristic

119

C₂×C₄

D₈×D₁₃

yes

characteristic

120

C₂×C₄

#88194

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

#88173

yes

—

124

C₈

C₂×#6106

yes

—

125

C₂²

#11718812

yes

characteristic

126

C₄

—

127

C₄

—

208

—

128

C₄

#11718573

yes

—

129

C₄

D₁₃×Q₃₂

yes

—

130

C₄

#11718746

yes

characteristic

131

C₄

#11718852

yes

characteristic

132

C₂

#13114429

yes

characteristic

133

C₂

#13117087

yes

characteristic

134

C₂

#13117094

yes

characteristic

	subgroup	G/H
center	C₂²	#11718812
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2252	C₂
socle, prime radical	C₂²×C₁₃	#493
hypercenter	#2252	D₁₃
hypocenter	C₁₃	#12504
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#11718812

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#2252

C₂

C₂²×C₁₃

D₁₃

C₁₃

G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
62. #1055	Q₃₂	semidirect
112. C₁₃	#12504	semidirect

H ≅ G/N

type

62. #1055

Q₃₂

semidirect

112. C₁₃

#12504

semidirect

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

subgroup	index
#13117089	2
#13117065	2
#13114434	2
C₁₆×#1055	2
C₁₃×#2252	2
#12504	13

subgroup

index

#13117089

#13117065

#13114434

C₁₆×#1055

C₁₃×#2252

#12504

Subgroups — #15777578

Conjugacy classes of subgroups

Complements

Maximal subgroups