Group #13128463 — subgroups

Subgroups — #13128463

Order 1664.

1202 conjugacy classes of subgroups; only the 491 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 491 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	Q₈²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719439	C₂	1	9	yes	maximal, complemented
3	#11719430	C₂	1	9	yes	maximal, complemented
4	#11719282	C₂	1	6	yes	maximal, complemented
5	C₄×#6131	C₂	1	6	yes	maximal, complemented
6	C₂×#6131	C₂²	1	6	yes	—
7	C₂×#6131	C₂²	1	6	yes	—
8	C₁₃×C₄⋊Q₈	C₂²	1	9	yes	—
9	C₄×Q₈×C₁₃	C₂²	1	6	yes	—
10	#88201	C₂²	1	18	yes	—
11	Q₈×Dic₁₃	C₂²	1	2	yes	—
12	#88149	C₂²	1	36	yes	—
13	#88147	C₂²	1	18	yes	—
14	#88146	C₂²	1	18	yes	—
15	#88128	C₂²	1	9	yes	—
16	C₄×D₅₂	C₂²	1	18	yes	—
17	C₄²×D₁₃	C₂²	1	9	yes	—
18	C₂×Q₈×C₁₃	C₂³	1	6	yes	—
19	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
20	#6131	—	2	24	no	—
21	#6131	—	4	12	no	—
22	C₂×D₅₂	C₂³	1	18	yes	—
23	C₂×D₅₂	C₂³	1	18	yes	—
24	C₂×C₄×D₁₃	C₂³	1	18	yes	—
25	C₂×C₄×D₁₃	C₂³	1	9	yes	—
26	C₁₃×C₄⋊C₄	C₂³	1	18	yes	—
27	C₄²×C₁₃	C₂³	1	9	yes	—
28	#6103	C₂³	1	36	yes	—
29	C₄⋊Dic₁₃	C₂³	1	6	yes	—
30	C₄×Dic₁₃	C₂³	1	6	yes	—
31	C₄×Dic₁₃	C₂³	1	9	yes	—
32	#3483	—	13	1	no	maximal, Sylow
33	C₂²×D₁₃	C₂⁴	1	9	yes	—
34	Q₈×C₁₃	—	4	6	no	—
35	Q₈×C₁₃	#41	1	8	yes	complemented
36	C₂×C₄×C₁₃	C₂⁴	1	9	yes	—
37	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
38	C₂×Dic₁₃	C₂⁴	1	6	yes	—
39	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
40	D₅₂	—	2	36	no	—
41	D₅₂	—	4	18	no	—
42	C₄×D₁₃	—	2	36	no	—
43	C₄×D₁₃	—	2	18	no	—
44	Q₈²	D₁₃	1	1	yes	characteristic, complemented
45	#555	—	13	9	no	—
46	#550	—	13	6	no	—
47	#532	—	13	9	no	—
48	C₄×#41	—	13	6	no	—
49	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
50	C₂×D₁₃	—	2	18	no	—
51	C₂×D₁₃	—	2	9	no	—
52	C₄×C₁₃	—	2	9	no	—
53	C₄×C₁₃	C₂×#41	1	12	yes	—
54	Dic₁₃	—	2	6	no	—
55	Dic₁₃	H(5,𝔽₂)	1	2	yes	—
56	C₂×#41	—	13	6	no	—
57	C₂×#41	—	13	6	no	—
58	C₄⋊Q₈	C₂×D₁₃	1	9	yes	—
59	#127	—	13	9	no	—
60	#124	—	13	18	no	—
61	#121	—	13	36	no	—
62	C₄×Q₈	—	13	2	no	—
63	C₄×Q₈	C₂×D₁₃	1	6	yes	—
64	C₄×D₄	—	13	18	no	—
65	C₄×D₄	—	13	18	no	—
66	#117	—	13	18	no	—
67	C₂×C₄²	—	13	9	no	—
68	C₂×C₁₃	C₂²×#41	1	2	yes	—
69	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
70	D₁₃	—	4	9	no	—
71	#41	—	26	24	no	—
72	#41	—	52	12	no	—
73	C₂×Q₈	C₂²×D₁₃	1	6	yes	—
74	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
75	C₂×D₄	—	13	18	no	—
76	C₂×D₄	—	13	18	no	—
77	C₂²×C₄	—	13	18	no	—
78	C₂²×C₄	—	13	9	no	—
79	C₄⋊C₄	—	13	6	no	—
80	C₄⋊C₄	C₂²×D₁₃	1	18	yes	—
81	C₂²⋊C₄	—	13	36	no	—
82	C₄²	—	13	6	no	—
83	C₄²	—	13	9	no	—
84	C₄²	C₂²×D₁₃	1	9	yes	—
85	C₁₃	#3483	1	1	yes	characteristic, complemented, Sylow
86	C₂³	—	13	9	no	—
87	Q₈	—	4	6	no	—
88	Q₈	#6131	1	8	yes	complemented
89	D₄	—	26	36	no	—
90	D₄	—	52	18	no	—
91	C₂×C₄	—	13	1	no	—
92	C₂×C₄	—	13	6	no	—
93	C₂×C₄	—	26	18	no	—
94	C₂×C₄	—	26	36	no	—
95	C₂×C₄	C₂³×D₁₃	1	9	yes	—
96	C₂×C₄	C₂³×D₁₃	1	6	yes	—
97	C₂²	—	26	18	no	—
98	C₂²	—	26	9	no	—
99	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
100	C₄	—	2	9	no	—
101	C₄	—	13	2	no	—
102	C₄	—	26	6	no	—
103	C₄	C₂×#6131	1	12	yes	—
104	C₂	—	52	9	no	—
105	C₂	C₂²×#6131	1	2	yes	—
106	C₂	D₁₃×H(5,𝔽₂)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈²×C₁₃

C₂

yes

maximal, characteristic, complemented

#11719439

C₂

yes

maximal, complemented

#11719430

C₂

yes

maximal, complemented

#11719282

C₂

yes

maximal, complemented

C₄×#6131

C₂

yes

maximal, complemented

C₂×#6131

C₂²

yes

—

C₂×#6131

C₂²

yes

—

C₁₃×C₄⋊Q₈

C₂²

yes

—

C₄×Q₈×C₁₃

C₂²

yes

—

#88201

C₂²

yes

—

Q₈×Dic₁₃

C₂²

yes

—

#88149

C₂²

yes

—

#88147

C₂²

yes

—

#88146

C₂²

yes

—

#88128

C₂²

yes

—

C₄×D₅₂

C₂²

yes

—

C₄²×D₁₃

C₂²

yes

—

C₂×Q₈×C₁₃

C₂³

yes

—

C₂×Q₈×C₁₃

C₂³

yes

—

#6131

—

#6131

—

C₂×D₅₂

C₂³

yes

—

C₂×D₅₂

C₂³

yes

—

C₂×C₄×D₁₃

C₂³

yes

—

C₂×C₄×D₁₃

C₂³

yes

—

C₁₃×C₄⋊C₄

C₂³

yes

—

C₄²×C₁₃

C₂³

yes

—

#6103

C₂³

yes

—

C₄⋊Dic₁₃

C₂³

yes

—

C₄×Dic₁₃

C₂³

yes

—

C₄×Dic₁₃

C₂³

yes

—

#3483

—

maximal, Sylow

C₂²×D₁₃

C₂⁴

yes

—

Q₈×C₁₃

—

Q₈×C₁₃

#41

yes

complemented

C₂×C₄×C₁₃

C₂⁴

yes

—

C₂×C₄×C₁₃

C₂⁴

yes

—

C₂×Dic₁₃

C₂⁴

yes

—

C₂×Dic₁₃

C₂⁴

yes

characteristic

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

Q₈²

D₁₃

yes

characteristic, complemented

#555

—

#550

—

#532

—

C₄×#41

—

C₂²×C₁₃

C₂⁵

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₂×#41

yes

—

Dic₁₃

—

Dic₁₃

H(5,𝔽₂)

yes

—

C₂×#41

—

C₂×#41

—

C₄⋊Q₈

C₂×D₁₃

yes

—

#127

—

#124

—

#121

—

C₄×Q₈

—

C₄×Q₈

C₂×D₁₃

yes

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₂×C₁₃

C₂²×#41

yes

—

C₂×C₁₃

C₂×H(5,𝔽₂)

yes

characteristic

D₁₃

—

#41

—

#41

—

C₂×Q₈

C₂²×D₁₃

yes

—

C₂×Q₈

C₂²×D₁₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₁₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₁₃

yes

—

C₁₃

#3483

yes

characteristic, complemented, Sylow

C₂³

—

Q₈

—

Q₈

#6131

yes

complemented

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂³×D₁₃

yes

—

C₂×C₄

C₂³×D₁₃

yes

—

C₂²

—

C₂²

—

C₂²

C₂⁴×D₁₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

C₂×#6131

yes

—

104

C₂

—

105

C₂

C₂²×#6131

yes

—

106

C₂

D₁₃×H(5,𝔽₂)

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	Q₈²×C₁₃	C₂
hypercenter	Q₈²	D₁₃
hypocenter	C₁₃	#3483
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ Q₈²

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₃

commutator, socle, cosocle

C₂²×C₁₃

C₂⁵

Fitting

Q₈²×C₁₃

C₂

hypercenter

Q₈²

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂² ◁ Q₈²

N	H ≅ G/N	type
1. Q₈²×C₁₃	C₂	semidirect
2. #11719439	C₂	semidirect
3. #11719430	C₂	semidirect
4. #11719282	C₂	semidirect
5. C₄×#6131	C₂	semidirect
35. Q₈×C₁₃	#41	semidirect
44. Q₈²	D₁₃	semidirect
85. C₁₃	#3483	semidirect
88. Q₈	#6131	semidirect

H ≅ G/N

type

1. Q₈²×C₁₃

C₂

semidirect

2. #11719439

C₂

semidirect

3. #11719430

C₂

semidirect

4. #11719282

C₂

semidirect

5. C₄×#6131

C₂

semidirect

35. Q₈×C₁₃

#41

semidirect

44. Q₈²

D₁₃

semidirect

85. C₁₃

#3483

semidirect

88. Q₈

#6131

semidirect

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

subgroup	index
Q₈²×C₁₃	2
#11719439	2
#11719430	2
#11719282	2
C₄×#6131	2
#3483	13

subgroup

index

Q₈²×C₁₃

Subgroups — #13128463

Conjugacy classes of subgroups

Complements

Maximal subgroups