Group #15767415 — subgroups

Subgroups — #15767415

Order 3328.

406 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13115005	C₂	1	3	yes	maximal
2	#13112717	C₂	1	3	yes	maximal
3	C₁₃×#2844	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#400	—	2	3	no	—
5	C₁₃×#395	—	2	3	no	—
6	C₁₃×#382	C₂²	1	3	yes	—
7	C₂×#88048	—	2	3	no	—
8	C₂×#88047	—	2	3	no	—
9	#11718596	C₂²	1	1	yes	characteristic
10	C₂×#88043	C₂²	1	3	yes	—
11	C₂²×Q₈×C₁₃	—	2	1	no	—
12	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
13	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
14	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
15	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×C₄⋊Dic₁₃	—	2	3	no	—
17	C₂²×#1055	—	2	3	no	—
18	C₁₃×#95	—	2	3	no	—
19	C₁₃×#95	—	2	1	no	—
20	C₁₃×#95	—	2	6	no	—
21	#88048	—	4	6	no	—
22	#88047	—	4	6	no	—
23	#88043	—	2	3	no	—
24	#88043	Q₈	1	6	yes	complemented
25	#10583	—	13	1	no	maximal, Sylow
26	C₂×Q₈×C₁₃	—	8	3	no	—
27	C₂²×C₄×C₁₃	—	2	3	no	—
28	C₂²×C₄×C₁₃	—	2	3	no	—
29	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
30	C₂²×Dic₁₃	—	4	1	no	—
31	C₁₃×C₄⋊C₄	—	4	3	no	—
32	C₁₃×C₄⋊C₄	—	4	3	no	—
33	C₁₃×C₄⋊C₄	—	8	3	no	—
34	C₄²×C₁₃	C₂×Q₈	1	3	yes	—
35	C₄²×C₁₃	#41	1	1	yes	characteristic
36	C₄⋊Dic₁₃	—	4	3	no	—
37	C₄⋊Dic₁₃	—	4	3	no	—
38	C₂×#1055	—	2	6	no	—
39	C₂×#1055	—	2	6	no	—
40	C₂×#1055	—	2	6	no	—
41	#2844	D₁₃	1	1	yes	characteristic
42	#1847	—	13	3	no	—
43	#1295	—	13	3	no	—
44	C₂³×C₁₃	#120	1	1	yes	characteristic
45	Q₈×C₁₃	—	8	4	no	—
46	C₂×C₄×C₁₃	—	2	3	no	—
47	C₂×C₄×C₁₃	—	2	3	no	—
48	C₂×C₄×C₁₃	—	4	3	no	—
49	C₂×C₄×C₁₃	—	4	3	no	—
50	C₂×C₄×C₁₃	—	4	6	no	—
51	C₂×C₄×C₁₃	—	4	6	no	—
52	C₂×C₄×C₁₃	C₂²⋊Q₈	1	3	yes	—
53	C₂×C₄×C₁₃	C₄⋊Q₈	1	3	yes	—
54	C₂×Dic₁₃	—	8	3	no	—
55	#1055	—	2	12	no	—
56	C₂×#107	—	26	3	no	—
57	C₂×#106	—	26	3	no	—
58	C₂×#97	—	13	3	no	—
59	#400	—	2	3	no	—
60	#395	—	2	3	no	—
61	#384	—	13	1	no	—
62	#382	C₂×D₁₃	1	3	yes	—
63	C₂²×C₁₃	#395	1	1	yes	characteristic
64	C₂²×C₁₃	#453	1	3	yes	—
65	C₂²×C₁₃	#456	1	3	yes	—
66	C₄×C₁₃	—	2	3	no	—
67	C₄×C₁₃	—	2	3	no	—
68	C₄×C₁₃	—	8	3	no	—
69	C₄×C₁₃	—	8	3	no	—
70	Dic₁₃	—	16	1	no	—
71	C₂²×Q₈	—	2	1	no	—
72	C₂²×C₈	—	26	3	no	—
73	C₂×C₄⋊C₄	—	2	3	no	—
74	C₂×C₄⋊C₄	—	26	3	no	—
75	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	3	yes	—
76	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	3	yes	—
77	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
78	#107	—	52	6	no	—
79	#106	—	52	6	no	—
80	#97	—	13	6	no	—
81	#97	—	26	3	no	—
82	#95	—	2	3	no	—
83	#95	—	2	1	no	—
84	#95	—	2	6	no	—
85	C₂×C₁₃	#1655	1	1	yes	characteristic
86	C₂×C₁₃	#1660	1	3	yes	—
87	C₂×C₁₃	#2060	1	3	yes	—
88	C₂×Q₈	—	8	3	no	—
89	C₂²×C₄	—	2	3	no	—
90	C₂²×C₄	—	2	3	no	—
91	C₂²×C₄	—	52	1	no	—
92	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
93	C₂×C₈	—	26	6	no	—
94	C₂×C₈	—	26	6	no	—
95	C₂×C₈	—	26	6	no	—
96	C₄⋊C₄	—	4	3	no	—
97	C₄⋊C₄	—	4	3	no	—
98	C₄⋊C₄	—	8	3	no	—
99	C₄⋊C₄	—	52	3	no	—
100	C₄⋊C₄	—	52	3	no	—
101	C₄²	Q₈×D₁₃	1	3	yes	—
102	C₄²	#6131	1	1	yes	characteristic
103	C₁₃	#10583	1	1	yes	characteristic, complemented, Sylow
104	C₂³	#88207	1	1	yes	characteristic
105	Q₈	—	8	4	no	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	4	3	no	—
109	C₂×C₄	—	4	3	no	—
110	C₂×C₄	—	4	6	no	—
111	C₂×C₄	—	4	6	no	—
112	C₂×C₄	—	104	3	no	—
113	C₂×C₄	#88198	1	3	yes	—
114	C₂×C₄	#88200	1	3	yes	—
115	C₈	—	26	12	no	—
116	C₂²	#11718890	1	1	yes	characteristic
117	C₂²	#11718902	1	3	yes	—
118	C₂²	#11718907	1	3	yes	—
119	C₄	—	2	3	no	—
120	C₄	—	2	3	no	—
121	C₄	—	8	3	no	—
122	C₄	—	8	3	no	—
123	C₄	—	208	1	no	—
124	C₂	#13114125	1	1	yes	characteristic
125	C₂	#13114147	1	3	yes	—
126	C₂	#13116178	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115005

C₂

yes

maximal

#13112717

C₂

yes

maximal

C₁₃×#2844

C₂

yes

maximal, characteristic

C₁₃×#400

—

C₁₃×#395

—

C₁₃×#382

C₂²

yes

—

C₂×#88048

—

C₂×#88047

—

#11718596

C₂²

yes

characteristic

C₂×#88043

C₂²

yes

—

C₂²×Q₈×C₁₃

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

D₄

yes

—

C₂×C₁₃×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂²×#1055

—

C₁₃×#95

—

C₁₃×#95

—

C₁₃×#95

—

—

—

—

Q₈

yes

complemented

#10583

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×Dic₁₃

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×Q₈

yes

—

C₄²×C₁₃

#41

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

D₁₃

yes

characteristic

#1847

—

#1295

—

C₂³×C₁₃

#120

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×Dic₁₃

—

#1055

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

—

—

—

C₂×D₁₃

yes

—

C₂²×C₁₃

#395

yes

characteristic

C₂²×C₁₃

#453

yes

—

C₂²×C₁₃

#456

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₁₃

yes

—

C₂×C₄⋊C₄

C₂²⋊D₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

—

—

—

—

—

—

—

C₂×C₁₃

#1655

yes

characteristic

C₂×C₁₃

#1660

yes

—

C₂×C₁₃

#2060

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

Q₈×D₁₃

yes

—

102

C₄²

#6131

yes

characteristic

103

C₁₃

#10583

yes

characteristic, complemented, Sylow

104

C₂³

#88207

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

104

—

113

C₂×C₄

#88198

yes

—

114

C₂×C₄

#88200

yes

—

115

C₈

—

116

C₂²

#11718890

yes

characteristic

117

C₂²

#11718902

yes

—

118

C₂²

#11718907

yes

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

208

—

124

C₂

#13114125

yes

characteristic

125

C₂

#13114147

yes

—

126

C₂

#13116178

yes

—

	subgroup	G/H
center	C₂³	#88207
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2844	C₂
socle, prime radical	C₂³×C₁₃	#120
hypercenter	#2844	D₁₃
hypocenter	C₁₃	#10583
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

subgroup

G/H

center

C₂³

#88207

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#2844

C₂

C₂³×C₁₃

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

N	H ≅ G/N	type
24. #88043	Q₈	semidirect
103. C₁₃	#10583	semidirect

H ≅ G/N

type

24. #88043

Q₈

semidirect

103. C₁₃

#10583

semidirect

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

subgroup	index
#13115005	2
#13112717	2
C₁₃×#2844	2
#10583	13

subgroup

index

#13115005

#13112717

C₁₃×#2844

#10583

Subgroups — #15767415

Conjugacy classes of subgroups

Complements

Maximal subgroups