Group #26022516 — subgroups

Subgroups — #26022516

Order 11392.

198 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 57 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×#15667956	C₂	1	2	yes	maximal, complemented
2	C₂×#15667953	C₂	1	2	yes	maximal, complemented
3	#19151982	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151374	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
7	C₂×#13083563	—	2	1	no	—
8	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
9	#15668000	—	2	1	no	—
10	#15667956	—	2	4	no	—
11	#15667953	—	2	4	no	—
12	C₂×#13083549	C₂²	1	2	yes	—
13	C₂×D₈×C₈₉	—	2	1	no	—
14	C₂×D₈×C₈₉	C₂²	1	2	yes	—
15	#127×C₈₉	—	2	1	no	—
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₈	—	2	1	no	—
18	C₂×D₃₅₆	—	2	1	no	—
19	Dic₃₅₆	—	2	2	no	—
20	#13083563	—	4	2	no	—
21	D₇₁₂	—	2	2	no	—
22	#13083551	—	4	2	no	—
23	#13083549	D₄	1	4	yes	complemented
24	C₂×D₄×C₈₉	—	2	2	no	—
25	C₂×D₄×C₈₉	—	4	2	no	—
26	D₈×C₈₉	—	2	2	no	—
27	D₈×C₈₉	—	2	2	no	—
28	D₈×C₈₉	—	4	2	no	—
29	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₄²×C₈₉	D₄	1	1	yes	characteristic
32	C₂³×C₈₉	—	4	2	no	—
33	C₂²×D₈₉	—	4	1	no	—
34	D₄×C₈₉	—	4	2	no	—
35	D₄×C₈₉	—	4	2	no	—
36	D₄×C₈₉	—	8	4	no	—
37	C₂×C₄×C₈₉	D₈	1	2	yes	—
38	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₈₉	—	4	1	no	—
40	D₃₅₆	—	4	1	no	—
41	D₃₅₆	—	4	1	no	—
42	Dic₁₇₈	—	4	1	no	—
43	Dic₁₇₈	—	4	1	no	—
44	C₈×C₈₉	D₈	1	2	yes	—
45	C₈×C₈₉	C₂×D₄	1	2	yes	—
46	C₂²×C₈₉	—	8	2	no	—
47	C₂²×C₈₉	—	8	4	no	—
48	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
49	C₂×D₈₉	—	8	1	no	—
50	C₂×D₈₉	—	8	2	no	—
51	C₄×C₈₉	—	2	2	no	—
52	C₄×C₈₉	#127	1	1	yes	characteristic
53	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
54	Dic₈₉	—	8	1	no	—
55	C₂×C₈₉	—	16	2	no	—
56	C₂×C₈₉	#493	1	1	yes	characteristic
57	C₂×C₈₉	#509	1	2	yes	—
58	D₈₉	—	16	1	no	—
59	#2250	—	89	1	no	maximal, Sylow
60	C₈₉	#2250	1	1	yes	characteristic, complemented, Sylow
61	C₂×SD₃₂	—	89	2	no	—
62	C₂×D₁₆	—	89	2	no	—
63	#495	—	89	1	no	—
64	#493	D₈₉	1	1	yes	characteristic, complemented
65	#346	—	89	1	no	—
66	C₂×Q₁₆	—	89	1	no	—
67	C₂×SD₁₆	—	178	1	no	—
68	C₂×D₈	—	2	1	no	—
69	C₂×D₈	—	89	1	no	—
70	C₂×D₈	C₂×D₈₉	1	2	yes	—
71	#127	—	2	1	no	—
72	#124	—	178	1	no	—
73	SD₃₂	—	178	4	no	—
74	D₁₆	—	178	4	no	—
75	C₂×C₁₆	—	89	2	no	—
76	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
77	C₂×Q₈	—	178	1	no	—
78	C₂×D₄	—	2	2	no	—
79	C₂×D₄	—	4	2	no	—
80	C₂×D₄	—	178	1	no	—
81	Q₁₆	—	178	2	no	—
82	SD₁₆	—	356	2	no	—
83	D₈	—	2	2	no	—
84	D₈	—	2	2	no	—
85	D₈	—	4	2	no	—
86	D₈	—	178	2	no	—
87	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
88	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
89	C₂²⋊C₄	—	356	2	no	—
90	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
91	C₁₆	—	89	4	no	—
92	C₂³	—	4	2	no	—
93	C₂³	—	356	1	no	—
94	Q₈	—	356	1	no	—
95	Q₈	—	356	1	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	8	4	no	—
99	D₄	—	356	1	no	—
100	D₄	—	356	1	no	—
101	C₂×C₄	—	356	1	no	—
102	C₂×C₄	#13083559	1	2	yes	—
103	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
104	C₈	#13083559	1	2	yes	—
105	C₈	D₄×D₈₉	1	2	yes	—
106	C₂²	—	8	2	no	—
107	C₂²	—	8	4	no	—
108	C₂²	—	712	1	no	—
109	C₂²	—	712	2	no	—
110	C₂²	C₂×#13083559	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	712	1	no	—
113	C₄	#15668008	1	1	yes	characteristic
114	C₄	C₂×#13083559	1	1	yes	characteristic
115	C₂	—	16	2	no	—
116	C₂	—	1424	1	no	—
117	C₂	#19151974	1	1	yes	characteristic
118	C₂	#19152034	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667956

C₂

yes

maximal, complemented

C₂×#15667953

C₂

yes

maximal, complemented

#19151982

C₂

yes

maximal, characteristic, complemented

#19151374

C₂

yes

maximal, characteristic, complemented

#493×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₅₆

C₂²

yes

characteristic

C₂×#13083563

—

C₂×D₇₁₂

C₂²

yes

—

—

—

C₂²

yes

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

C₂²

yes

—

#127×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

Dic₃₅₆

—

#13083563

—

D₇₁₂

—

#13083551

—

#13083549

D₄

yes

complemented

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₂³×C₈₉

—

C₂²×D₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

D₈

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

D₈

yes

—

C₈×C₈₉

C₂×D₄

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#127

yes

characteristic

C₄×C₈₉

C₂×D₈

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#493

yes

characteristic

C₂×C₈₉

#509

yes

—

D₈₉

—

#2250

—

maximal, Sylow

C₈₉

#2250

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂×D₁₆

—

#495

—

#493

D₈₉

yes

characteristic, complemented

#346

—

C₂×Q₁₆

—

C₂×SD₁₆

—

178

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

C₂×D₈₉

yes

—

#127

—

#124

—

178

—

SD₃₂

—

178

—

D₁₆

—

178

—

C₂×C₁₆

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

178

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

178

—

Q₁₆

—

178

—

SD₁₆

—

356

—

D₈

—

D₈

—

D₈

—

D₈

—

178

—

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₂²⋊C₄

—

356

—

C₄²

C₂²⋊D₈₉

yes

characteristic

C₁₆

—

C₂³

—

C₂³

—

356

—

Q₈

—

356

—

Q₈

—

356

—

D₄

—

D₄

—

D₄

—

D₄

—

356

—

100

D₄

—

356

—

101

C₂×C₄

—

356

—

102

C₂×C₄

#13083559

yes

—

103

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

104

C₈

#13083559

yes

—

105

C₈

D₄×D₈₉

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

712

—

109

C₂²

—

712

—

110

C₂²

C₂×#13083559

yes

characteristic

111

C₄

—

112

C₄

—

712

—

113

C₄

#15668008

yes

characteristic

114

C₄

C₂×#13083559

yes

characteristic

115

C₂

—

116

C₂

—

1424

—

117

C₂

#19151974

yes

characteristic

118

C₂

#19152034

yes

—

	subgroup	G/H
center	C₂²	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#493×C₈₉	C₂
socle	C₂²×C₈₉	C₂×D₈
hypercenter	#493	D₈₉
hypocenter	C₈₉	#2250
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂²

C₂×#13083559

commutator, cosocle

C₂×C₈×C₈₉

C₂³

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

#493×C₈₉

C₂

socle

C₂²×C₈₉

C₂×D₈

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. C₂×#15667956	C₂	semidirect
2. C₂×#15667953	C₂	semidirect
3. #19151982	C₂	semidirect
4. #19151374	C₂	semidirect
5. #493×C₈₉	C₂	semidirect
23. #13083549	D₄	semidirect
60. C₈₉	#2250	semidirect
64. #493	D₈₉	semidirect

H ≅ G/N

type

1. C₂×#15667956

C₂

semidirect

2. C₂×#15667953

C₂

semidirect

3. #19151982

C₂

semidirect

4. #19151374

C₂

semidirect

5. #493×C₈₉

C₂

semidirect

23. #13083549

D₄

semidirect

60. C₈₉

#2250

semidirect

64. #493

D₈₉

semidirect

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

subgroup	index
C₂×#15667956	2
C₂×#15667953	2
#19151982	2
#19151374	2
#493×C₈₉	2
#2250	89

subgroup

index

#493×C₈₉

Subgroups — #26022516

Conjugacy classes of subgroups

Complements

Maximal subgroups