Group #16015485 — subgroups

Subgroups — #16015485

Order 3328.

1006 conjugacy classes of subgroups; only the 459 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 459 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	C₂×#11718341	C₂	1	4	yes	maximal
2	#13123234	C₂	1	2	yes	maximal
3	#13114993	C₂	1	8	yes	maximal
4	C₄²×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	—	2	4	no	—
6	C₂×#88123	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	C₂×#88057	C₂²	1	16	yes	—
11	#11718596	—	2	4	no	—
12	#11718596	C₂²	1	8	yes	—
13	#11718341	—	2	16	no	—
14	C₂²×Dic₂₆	—	2	4	no	—
15	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
16	C₂×C₄²×C₁₃	D₄	1	4	yes	—
17	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
19	C₂×C₄⋊Dic₁₃	—	2	16	no	—
20	C₂×C₄⋊Dic₁₃	—	2	4	no	—
21	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
22	#88123	—	2	16	no	—
23	#88123	—	4	16	no	—
24	C₄×C₈×C₁₃	D₄	1	8	yes	—
25	C₄×C₈×C₁₃	Q₈	1	16	yes	complemented
26	#88057	—	2	32	no	—
27	#33310	—	13	1	no	maximal, Sylow
28	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
29	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
30	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
31	C₂²×Dic₁₃	—	2	8	no	—
32	C₂×Dic₂₆	—	4	8	no	—
33	C₂×Dic₂₆	—	4	16	no	—
34	C₂×C₈×C₁₃	C₂×D₄	1	8	yes	—
35	C₂×C₈×C₁₃	C₂×Q₈	1	16	yes	—
36	C₄²×C₁₃	SD₁₆	1	16	yes	—
37	C₄²×C₁₃	C₂×D₄	1	4	yes	—
38	C₄²×C₁₃	C₂×Q₈	1	8	yes	—
39	C₄⋊Dic₁₃	—	2	16	no	—
40	C₄⋊Dic₁₃	—	4	8	no	—
41	C₄⋊Dic₁₃	—	4	32	no	—
42	C₂×#498	—	13	4	no	—
43	#2868	—	13	2	no	—
44	#1844	—	13	8	no	—
45	C₄²×C₈	D₁₃	1	1	yes	characteristic
46	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₃	#127	1	4	yes	—
48	C₂×C₄×C₁₃	C₄⋊Q₈	1	8	yes	—
49	C₂×C₄×C₁₃	C₂×SD₁₆	1	8	yes	—
50	C₂×C₄×C₁₃	C₂×SD₁₆	1	16	yes	—
51	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
52	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
53	C₂×Dic₁₃	—	4	8	no	—
54	C₂×Dic₁₃	—	4	16	no	—
55	Dic₂₆	—	8	16	no	—
56	C₈×C₁₃	C₄⋊Q₈	1	16	yes	complemented
57	C₂×C₄⋊Q₈	—	13	4	no	—
58	C₂×C₄⋊Q₈	—	26	4	no	—
59	#498	—	26	16	no	—
60	C₂×#106	—	13	16	no	—
61	C₂×C₄×C₈	C₂×D₁₃	1	4	yes	—
62	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
63	#384	—	13	8	no	—
64	#384	—	26	4	no	—
65	C₄³	C₂×D₁₃	1	1	yes	characteristic
66	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
67	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
68	C₂²×C₁₃	C₂²×SD₁₆	1	4	yes	—
69	C₄×C₁₃	#492	1	8	yes	—
70	C₄×C₁₃	#498	1	16	yes	complemented
71	C₄×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
72	Dic₁₃	—	8	8	no	—
73	C₂²×Q₈	—	26	4	no	—
74	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
75	C₄⋊Q₈	—	26	16	no	—
76	C₄⋊Q₈	—	52	16	no	—
77	C₂×C₄⋊C₄	—	13	8	no	—
78	C₂×C₄⋊C₄	—	26	4	no	—
79	C₂×C₄⋊C₄	—	26	16	no	—
80	C₂×C₄²	D₅₂	1	4	yes	—
81	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
82	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
83	#106	—	26	32	no	—
84	C₄×C₈	Dic₂₆	1	16	yes	complemented
85	C₄×C₈	D₅₂	1	8	yes	—
86	C₂×C₁₃	#2868	1	1	yes	characteristic
87	C₂×C₁₃	C₂×#492	1	2	yes	—
88	C₂×C₁₃	C₂×#498	1	4	yes	—
89	C₂×Q₈	—	52	8	no	—
90	C₂×Q₈	—	52	16	no	—
91	C₂²×C₄	—	26	8	no	—
92	C₂²×C₄	C₂×D₅₂	1	4	yes	—
93	C₂²×C₄	C₂×D₅₂	1	2	yes	—
94	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
95	C₂×C₈	C₂×Dic₂₆	1	16	yes	—
96	C₂×C₈	C₂×D₅₂	1	8	yes	—
97	C₄⋊C₄	—	26	16	no	—
98	C₄⋊C₄	—	52	8	no	—
99	C₄⋊C₄	—	52	32	no	—
100	C₄²	#6095	1	16	yes	—
101	C₄²	C₂×Dic₂₆	1	8	yes	—
102	C₄²	C₂×D₅₂	1	4	yes	—
103	C₁₃	#33310	1	1	yes	characteristic, complemented, Sylow
104	C₂³	C₂²×D₅₂	1	1	yes	characteristic
105	Q₈	—	104	16	no	—
106	C₂×C₄	—	52	8	no	—
107	C₂×C₄	—	52	16	no	—
108	C₂×C₄	#88123	1	8	yes	—
109	C₂×C₄	#88128	1	4	yes	—
110	C₂×C₄	C₂×#6095	1	16	yes	—
111	C₂×C₄	C₂×#6095	1	8	yes	—
112	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
113	C₂×C₄	C₂²×D₅₂	1	2	yes	—
114	C₈	#88123	1	16	yes	complemented
115	C₂²	C₂×#88123	1	2	yes	—
116	C₂²	C₂×#88128	1	1	yes	characteristic
117	C₂²	C₂²×#6095	1	4	yes	—
118	C₄	—	104	8	no	—
119	C₄	#11718341	1	16	yes	complemented
120	C₄	#11718354	1	8	yes	—
121	C₄	C₂×#88123	1	4	yes	—
122	C₂	#13123234	1	1	yes	characteristic
123	C₂	C₂×#11718354	1	2	yes	—
124	C₂	C₂×#11718341	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718341

C₂

yes

maximal

#13123234

C₂

yes

maximal

#13114993

C₂

yes

maximal

C₄²×C₈×C₁₃

C₂

yes

maximal, characteristic

C₂×#88123

—

C₂×#88123

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₄³×C₁₃

C₂²

yes

characteristic

C₂×#88057

C₂²

yes

—

#11718596

—

#11718596

C₂²

yes

—

#11718341

—

C₂²×Dic₂₆

—

C₂²×C₈×C₁₃

C₂³

yes

—

C₂×C₄²×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄²×C₁₃

C₂³

yes

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

C₂³

yes

—

#88123

—

#88123

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×C₄×C₁₃

C₂⁴

yes

characteristic

C₂²×Dic₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₂×C₈×C₁₃

C₂×Q₈

yes

—

C₄²×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#498

—

#2868

—

#1844

—

C₄²×C₈

D₁₃

yes

characteristic

C₂³×C₁₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₁₃

#127

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

C₂²×D₄

yes

—

C₂×C₄×C₁₃

C₂²×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

C₄⋊Q₈

yes

complemented

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

#498

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

#384

—

#384

—

C₄³

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

C₂×#127

yes

characteristic

C₂²×C₁₃

C₂×C₄⋊Q₈

yes

—

C₂²×C₁₃

C₂²×SD₁₆

yes

—

C₄×C₁₃

#492

yes

—

C₄×C₁₃

#498

yes

complemented

C₄×C₁₃

C₂×C₄⋊Q₈

yes

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×C₈

C₂²×D₁₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₅₂

yes

—

C₂×C₄²

C₂²×D₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

#106

—

C₄×C₈

Dic₂₆

yes

complemented

C₄×C₈

D₅₂

yes

—

C₂×C₁₃

#2868

yes

characteristic

C₂×C₁₃

C₂×#492

yes

—

C₂×C₁₃

C₂×#498

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₅₂

yes

—

C₂²×C₄

C₂×D₅₂

yes

—

C₂²×C₄

C₂³×D₁₃

yes

characteristic

C₂×C₈

C₂×Dic₂₆

yes

—

C₂×C₈

C₂×D₅₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

#6095

yes

—

101

C₄²

C₂×Dic₂₆

yes

—

102

C₄²

C₂×D₅₂

yes

—

103

C₁₃

#33310

yes

characteristic, complemented, Sylow

104

C₂³

C₂²×D₅₂

yes

characteristic

105

Q₈

—

104

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#88123

yes

—

109

C₂×C₄

#88128

yes

—

110

C₂×C₄

C₂×#6095

yes

—

111

C₂×C₄

C₂×#6095

yes

—

112

C₂×C₄

C₂²×Dic₂₆

yes

—

113

C₂×C₄

C₂²×D₅₂

yes

—

114

C₈

#88123

yes

complemented

115

C₂²

C₂×#88123

yes

—

116

C₂²

C₂×#88128

yes

characteristic

117

C₂²

C₂²×#6095

yes

—

118

C₄

—

104

—

119

C₄

#11718341

yes

complemented

120

C₄

#11718354

yes

—

121

C₄

C₂×#88123

yes

—

122

C₂

#13123234

yes

characteristic

123

C₂

C₂×#11718354

yes

—

124

C₂

C₂×#11718341

yes

—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator, cosocle	C₂²×C₄×C₁₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄²×C₈×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	C₂²×D₄
hypercenter	C₄²×C₈	D₁₃
hypocenter	C₁₃	#33310
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈

subgroup

G/H

center

C₂³

C₂²×D₅₂

commutator, cosocle

C₂²×C₄×C₁₃

C₂⁴

Frattini

C₂²×C₄

C₂³×D₁₃

Fitting

C₄²×C₈×C₁₃

C₂

socle, prime radical

C₂³×C₁₃

C₂²×D₄

hypercenter

C₄²×C₈

D₁₃

hypocenter

C₁₃

#33310

LCS

G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈

N	H ≅ G/N	type
25. C₄×C₈×C₁₃	Q₈	semidirect
56. C₈×C₁₃	C₄⋊Q₈	semidirect
70. C₄×C₁₃	#498	semidirect
84. C₄×C₈	Dic₂₆	semidirect
103. C₁₃	#33310	semidirect
114. C₈	#88123	semidirect
119. C₄	#11718341	semidirect

H ≅ G/N

type

25. C₄×C₈×C₁₃

Q₈

semidirect

56. C₈×C₁₃

C₄⋊Q₈

semidirect

70. C₄×C₁₃

#498

semidirect

84. C₄×C₈

Dic₂₆

semidirect

103. C₁₃

#33310

semidirect

114. C₈

#88123

semidirect

119. C₄

#11718341

semidirect

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

subgroup	index
C₂×#11718341	2
#13123234	2
#13114993	2
C₄²×C₈×C₁₃	2
#33310	13

subgroup

index

C₄²×C₈×C₁₃

Subgroups — #16015485

Conjugacy classes of subgroups

Complements

Maximal subgroups