Group #26019918 — subgroups

Subgroups — #26019918

Order 11392.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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	#19151975	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151975	C₂	1	2	yes	maximal, complemented
3	#19151971	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151945	C₂	1	2	yes	maximal, complemented
5	#322×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083563	—	2	2	no	—
7	C₂×D₇₁₂	—	2	4	no	—
8	C₂×D₇₁₂	—	2	2	no	—
9	#15668012	C₂²	1	1	yes	characteristic
10	#15668009	C₂²	1	2	yes	—
11	#15668009	C₂²	1	1	yes	characteristic
12	#15667934	—	2	4	no	—
13	C₄×C₈×C₈₉	C₂²	1	2	yes	—
14	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₇₈	—	2	1	no	—
16	C₂×D₃₅₆	—	2	2	no	—
17	C₂×D₃₅₆	—	2	2	no	—
18	C₂×D₃₅₆	—	2	2	no	—
19	C₂×D₃₅₆	—	2	1	no	—
20	C₂×D₃₅₆	—	2	2	no	—
21	#13083563	—	4	4	no	—
22	D₇₁₂	—	4	8	no	—
23	D₇₁₂	—	4	4	no	—
24	C₄⋊Dic₈₉	—	2	2	no	—
25	C₂×C₈×C₈₉	D₄	1	4	yes	—
26	C₂×C₈×C₈₉	D₄	1	2	yes	—
27	C₄²×C₈₉	C₂³	1	1	yes	characteristic
28	C₂²×D₈₉	—	4	1	no	—
29	C₂²×D₈₉	—	4	2	no	—
30	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
32	C₂×Dic₈₉	—	4	1	no	—
33	D₃₅₆	—	4	4	no	—
34	D₃₅₆	—	4	2	no	—
35	D₃₅₆	—	4	4	no	—
36	D₃₅₆	—	4	4	no	—
37	D₃₅₆	—	4	4	no	—
38	Dic₁₇₈	—	4	2	no	—
39	C₈×C₈₉	—	2	4	no	—
40	C₈×C₈₉	D₈	1	4	yes	complemented
41	C₂²×C₈₉	#127	1	1	yes	characteristic
42	C₂×D₈₉	—	8	2	no	—
43	C₂×D₈₉	—	8	1	no	—
44	C₂×D₈₉	—	8	2	no	—
45	C₂×D₈₉	—	8	2	no	—
46	C₂×D₈₉	—	8	2	no	—
47	C₄×C₈₉	C₂×D₈	1	2	yes	—
48	C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₈₉	—	8	1	no	—
50	C₂×C₈₉	#493	1	1	yes	characteristic
51	C₂×C₈₉	#496	1	2	yes	—
52	D₈₉	—	16	2	no	—
53	D₈₉	—	16	1	no	—
54	#1721	—	89	1	no	maximal, Sylow
55	C₈₉	#1721	1	1	yes	characteristic, complemented, Sylow
56	#493	—	89	2	no	—
57	#493	—	89	1	no	—
58	#492	—	89	1	no	—
59	#486	—	89	2	no	—
60	#322	D₈₉	1	1	yes	characteristic, complemented
61	C₂×SD₁₆	—	178	2	no	—
62	C₂×D₈	—	178	4	no	—
63	C₂×D₈	—	178	2	no	—
64	C₄⋊Q₈	—	89	1	no	—
65	#127	—	89	2	no	—
66	#127	—	89	1	no	—
67	#102	—	178	4	no	—
68	C₄×C₈	C₂×D₈₉	1	2	yes	—
69	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
70	C₂×Q₈	—	178	1	no	—
71	C₂×D₄	—	178	2	no	—
72	C₂×D₄	—	178	2	no	—
73	C₂×D₄	—	178	2	no	—
74	C₂×D₄	—	178	2	no	—
75	C₂×D₄	—	178	1	no	—
76	SD₁₆	—	356	4	no	—
77	D₈	—	356	8	no	—
78	D₈	—	356	4	no	—
79	C₂×C₈	D₃₅₆	1	4	yes	—
80	C₂×C₈	D₃₅₆	1	2	yes	—
81	C₄⋊C₄	—	178	2	no	—
82	C₄²	C₂²×D₈₉	1	1	yes	characteristic
83	C₂³	—	356	2	no	—
84	C₂³	—	356	1	no	—
85	Q₈	—	356	2	no	—
86	D₄	—	356	4	no	—
87	D₄	—	356	2	no	—
88	D₄	—	356	4	no	—
89	D₄	—	356	4	no	—
90	D₄	—	356	4	no	—
91	C₂×C₄	—	356	1	no	—
92	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
93	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	D₇₁₂	1	4	yes	complemented
96	C₂²	—	712	2	no	—
97	C₂²	—	712	1	no	—
98	C₂²	—	712	2	no	—
99	C₂²	—	712	2	no	—
100	C₂²	—	712	2	no	—
101	C₂²	#15668009	1	1	yes	characteristic
102	C₄	—	712	1	no	—
103	C₄	C₂×D₇₁₂	1	2	yes	—
104	C₄	#15668040	1	4	yes	—
105	C₂	—	1424	2	no	—
106	C₂	—	1424	1	no	—
107	C₂	#19151975	1	1	yes	characteristic
108	C₂	#19151987	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151975

C₂

yes

maximal, characteristic, complemented

#19151975

C₂

yes

maximal, complemented

#19151971

C₂

yes

maximal, characteristic, complemented

#19151945

C₂

yes

maximal, complemented

#322×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083563

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668012

C₂²

yes

characteristic

#15668009

C₂²

yes

—

#15668009

C₂²

yes

characteristic

#15667934

—

C₄×C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

#13083563

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

D₄

yes

—

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

D₈

yes

complemented

C₂²×C₈₉

#127

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

C₂×D₈

yes

—

C₄×C₈₉

AGL(1,ℤ/8)

yes

—

Dic₈₉

—

C₂×C₈₉

#493

yes

characteristic

C₂×C₈₉

#496

yes

—

D₈₉

—

D₈₉

—

#1721

—

maximal, Sylow

C₈₉

#1721

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic, complemented

C₂×SD₁₆

—

178

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

#127

—

#127

—

#102

—

178

—

C₄×C₈

C₂×D₈₉

yes

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

C₂×D₄

—

178

—

SD₁₆

—

356

—

D₈

—

356

—

D₈

—

356

—

C₂×C₈

D₃₅₆

yes

—

C₂×C₈

D₃₅₆

yes

—

C₄⋊C₄

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

356

—

C₂³

—

356

—

Q₈

—

356

—

D₄

—

356

—

D₄

—

356

—

D₄

—

356

—

D₄

—

356

—

D₄

—

356

—

C₂×C₄

—

356

—

C₂×C₄

C₂×D₃₅₆

yes

characteristic

C₂×C₄

C₂×D₃₅₆

yes

—

C₈

—

C₈

D₇₁₂

yes

complemented

C₂²

—

712

—

C₂²

—

712

—

C₂²

—

712

—

C₂²

—

712

—

100

C₂²

—

712

—

101

C₂²

#15668009

yes

characteristic

102

C₄

—

712

—

103

C₄

C₂×D₇₁₂

yes

—

104

C₄

#15668040

yes

—

105

C₂

—

1424

—

106

C₂

—

1424

—

107

C₂

#19151975

yes

characteristic

108

C₂

#19151987

yes

—

	subgroup	G/H
center	C₂²	#15668009
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#322×C₈₉	C₂
socle	C₂²×C₈₉	#127
hypercenter	#322	D₈₉
hypocenter	C₈₉	#1721
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#15668009

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#322×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #19151975	C₂	semidirect
2. #19151975	C₂	semidirect
3. #19151971	C₂	semidirect
4. #19151945	C₂	semidirect
5. #322×C₈₉	C₂	semidirect
40. C₈×C₈₉	D₈	semidirect
55. C₈₉	#1721	semidirect
60. #322	D₈₉	semidirect
95. C₈	D₇₁₂	semidirect

H ≅ G/N

type

1. #19151975

C₂

semidirect

2. #19151975

C₂

semidirect

3. #19151971

C₂

semidirect

4. #19151945

C₂

semidirect

5. #322×C₈₉

C₂

semidirect

40. C₈×C₈₉

D₈

semidirect

55. C₈₉

#1721

semidirect

60. #322

D₈₉

semidirect

95. C₈

D₇₁₂

semidirect

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

subgroup	index
#19151975	2
#19151971	2
#19151945	2
#322×C₈₉	2
#1721	89

subgroup

index

#322×C₈₉

Subgroups — #26019918

Conjugacy classes of subgroups

Complements

Maximal subgroups