Group #36469139 — subgroups

Subgroups — #36469139

Order 22784.

398 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 103 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	#26022501	C₂	1	4	yes	maximal, complemented
2	#26019867	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1712	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#15667953	—	2	8	no	—
6	#19151975	—	2	2	no	—
7	#19151975	C₂²	1	2	yes	—
8	C₄×#13083549	C₂²	1	2	yes	—
9	#493×C₈₉	—	2	2	no	—
10	#493×C₈₉	C₂²	1	2	yes	—
11	C₈²×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×D₇₁₂	—	2	4	no	—
13	C₂×D₇₁₂	—	4	4	no	—
14	#15668009	—	2	2	no	—
15	#15667953	—	4	16	no	—
16	C₂×#13083549	D₄	1	4	yes	—
17	C₂×D₈×C₈₉	—	2	4	no	—
18	C₂×D₈×C₈₉	—	4	4	no	—
19	#127×C₈₉	—	2	2	no	—
20	C₄×C₈×C₈₉	D₄	1	2	yes	—
21	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
22	C₂×D₃₅₆	—	4	2	no	—
23	C₂×D₃₅₆	—	4	4	no	—
24	D₇₁₂	—	4	8	no	—
25	D₇₁₂	—	8	8	no	—
26	#13083549	D₈	1	8	yes	complemented
27	C₂×D₄×C₈₉	—	4	2	no	—
28	C₂×D₄×C₈₉	—	4	4	no	—
29	D₈×C₈₉	—	4	8	no	—
30	D₈×C₈₉	—	8	8	no	—
31	C₂×C₈×C₈₉	D₈	1	4	yes	—
32	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
33	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₂³×C₈₉	—	8	2	no	—
35	C₂²×D₈₉	—	8	2	no	—
36	D₄×C₈₉	—	8	4	no	—
37	D₄×C₈₉	—	8	8	no	—
38	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
39	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
40	D₃₅₆	—	8	4	no	—
41	D₃₅₆	—	8	8	no	—
42	C₈×C₈₉	D₁₆	1	8	yes	complemented
43	C₈×C₈₉	C₂×D₈	1	4	yes	—
44	C₂²×C₈₉	—	16	2	no	—
45	C₂²×C₈₉	—	16	4	no	—
46	C₂²×C₈₉	#493	1	1	yes	characteristic
47	C₂×D₈₉	—	16	2	no	—
48	C₂×D₈₉	—	16	4	no	—
49	C₄×C₈₉	#493	1	2	yes	—
50	C₄×C₈₉	C₂×D₁₆	1	4	yes	—
51	#12515	—	89	1	no	maximal, Sylow
52	C₂×C₈₉	—	32	2	no	—
53	C₂×C₈₉	#1712	1	1	yes	characteristic
54	C₂×C₈₉	#2246	1	2	yes	—
55	D₈₉	—	32	2	no	—
56	#2246	—	89	4	no	—
57	#1712	—	89	1	no	—
58	#1712	D₈₉	1	1	yes	characteristic, complemented
59	C₈×C₁₆	—	89	1	no	—
60	C₈₉	#12515	1	1	yes	characteristic, complemented, Sylow
61	C₂×D₁₆	—	178	8	no	—
62	#493	—	2	2	no	—
63	#493	—	89	2	no	—
64	#493	—	178	2	no	—
65	#493	C₂×D₈₉	1	2	yes	—
66	C₄×C₁₆	—	89	2	no	—
67	C₈²	C₂×D₈₉	1	1	yes	characteristic
68	C₂×D₈	—	2	4	no	—
69	C₂×D₈	—	4	4	no	—
70	C₂×D₈	—	356	4	no	—
71	C₂×D₈	—	178	4	no	—
72	#127	—	2	2	no	—
73	#127	—	178	2	no	—
74	D₁₆	—	356	16	no	—
75	C₂×C₁₆	—	89	4	no	—
76	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
77	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
78	C₂×D₄	—	4	2	no	—
79	C₂×D₄	—	4	4	no	—
80	C₂×D₄	—	356	2	no	—
81	C₂×D₄	—	356	4	no	—
82	D₈	—	4	8	no	—
83	D₈	—	8	8	no	—
84	D₈	—	356	8	no	—
85	D₈	—	712	8	no	—
86	C₂×C₈	#13083559	1	4	yes	—
87	C₂×C₈	D₄×D₈₉	1	2	yes	—
88	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
89	C₁₆	—	89	8	no	—
90	C₂³	—	8	2	no	—
91	C₂³	—	712	2	no	—
92	D₄	—	8	4	no	—
93	D₄	—	8	8	no	—
94	D₄	—	712	4	no	—
95	D₄	—	712	8	no	—
96	C₂×C₄	#15668008	1	1	yes	characteristic
97	C₂×C₄	C₂×#13083559	1	2	yes	—
98	C₈	#15667953	1	8	yes	complemented
99	C₈	D₈×D₈₉	1	4	yes	—
100	C₂²	—	16	2	no	—
101	C₂²	—	16	4	no	—
102	C₂²	—	1424	2	no	—
103	C₂²	—	1424	4	no	—
104	C₂²	#19151974	1	1	yes	characteristic
105	C₄	#19151973	1	2	yes	—
106	C₄	C₂×#15667953	1	4	yes	—
107	C₂	—	32	2	no	—
108	C₂	—	2848	2	no	—
109	C₂	#26019866	1	1	yes	characteristic
110	C₂	#26022501	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022501

C₂

yes

maximal, complemented

#26019867

C₂

yes

maximal, characteristic, complemented

C₈×#13083549

C₂

yes

maximal, characteristic, complemented

C₈₉×#1712

C₂

yes

maximal, characteristic, complemented

C₂×#15667953

—

#19151975

—

#19151975

C₂²

yes

—

C₄×#13083549

C₂²

yes

—

#493×C₈₉

—

#493×C₈₉

C₂²

yes

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668009

—

#15667953

—

C₂×#13083549

D₄

yes

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

—

#127×C₈₉

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

D₇₁₂

—

D₇₁₂

—

#13083549

D₈

yes

complemented

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

D₁₆

yes

complemented

C₈×C₈₉

C₂×D₈

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#493

yes

—

C₄×C₈₉

C₂×D₁₆

yes

—

#12515

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

#1712

yes

characteristic

C₂×C₈₉

#2246

yes

—

D₈₉

—

#2246

—

#1712

—

#1712

D₈₉

yes

characteristic, complemented

C₈×C₁₆

—

C₈₉

#12515

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

178

—

#493

—

#493

—

#493

—

178

—

#493

C₂×D₈₉

yes

—

C₄×C₁₆

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

356

—

C₂×D₈

—

178

—

#127

—

#127

—

178

—

D₁₆

—

356

—

C₂×C₁₆

—

C₄×C₈

C₂²⋊D₈₉

yes

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

D₈

—

D₈

—

D₈

—

356

—

D₈

—

712

—

C₂×C₈

#13083559

yes

—

C₂×C₈

D₄×D₈₉

yes

—

C₄²

C₂×C₂²⋊D₈₉

yes

characteristic

C₁₆

—

C₂³

—

C₂³

—

712

—

D₄

—

D₄

—

D₄

—

712

—

D₄

—

712

—

C₂×C₄

#15668008

yes

characteristic

C₂×C₄

C₂×#13083559

yes

—

C₈

#15667953

yes

complemented

C₈

D₈×D₈₉

yes

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

1424

—

103

C₂²

—

1424

—

104

C₂²

#19151974

yes

characteristic

105

C₄

#19151973

yes

—

106

C₄

C₂×#15667953

yes

—

107

C₂

—

108

C₂

—

2848

—

109

C₂

#26019866

yes

characteristic

110

C₂

#26022501

yes

—

	subgroup	G/H
center	C₂²	#19151974
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1712	C₂
socle	C₂²×C₈₉	#493
hypercenter	#1712	D₈₉
hypocenter	C₈₉	#12515
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#19151974

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1712

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #26022501	C₂	semidirect
2. #26019867	C₂	semidirect
3. C₈×#13083549	C₂	semidirect
4. C₈₉×#1712	C₂	semidirect
26. #13083549	D₈	semidirect
42. C₈×C₈₉	D₁₆	semidirect
58. #1712	D₈₉	semidirect
60. C₈₉	#12515	semidirect
98. C₈	#15667953	semidirect

H ≅ G/N

type

1. #26022501

C₂

semidirect

2. #26019867

C₂

semidirect

3. C₈×#13083549

C₂

semidirect

4. C₈₉×#1712

C₂

semidirect

26. #13083549

D₈

semidirect

42. C₈×C₈₉

D₁₆

semidirect

58. #1712

D₈₉

semidirect

60. C₈₉

#12515

semidirect

98. C₈

#15667953

semidirect

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

subgroup	index
#26022501	2
#26019867	2
C₈×#13083549	2
C₈₉×#1712	2
#12515	89

subgroup

index

C₈₉×#1712

Subgroups — #36469139

Conjugacy classes of subgroups

Complements

Maximal subgroups