Group #26019165 — subgroups

Subgroups — #26019165

Order 11392.

176 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 63 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	C₄×D₇₁₂	C₂	1	1	yes	maximal, characteristic
2	#19151675	C₂	1	2	yes	maximal, complemented
3	#19151325	C₂	1	2	yes	maximal, complemented
4	#19151319	C₂	1	1	yes	maximal, characteristic, complemented
5	#334×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₁₂	C₄	1	1	yes	characteristic
7	C₂×#13083557	—	2	2	no	—
8	C₄×D₃₅₆	C₂²	1	2	yes	—
9	#15667946	C₄	1	1	yes	characteristic
10	#15667934	C₄	1	2	yes	—
11	#15667924	—	2	2	no	—
12	C₄×#10623494	C₂²	1	1	yes	characteristic
13	C₄×#10623494	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
15	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×D₃₅₆	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₈₉	—	2	2	no	—
18	D₇₁₂	—	2	2	no	—
19	#13083557	—	4	4	no	—
20	C₂×#10623494	—	2	1	no	—
21	C₂×#10623494	D₄	1	2	yes	—
22	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
23	#13083551	—	2	2	no	—
24	C₂×C₈×C₈₉	—	2	2	no	—
25	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₈₉	C₂³	1	1	yes	characteristic
28	C₂²×D₈₉	—	2	2	no	—
29	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
32	C₂×Dic₈₉	—	2	2	no	—
33	D₃₅₆	—	2	2	no	—
34	D₃₅₆	—	2	2	no	—
35	C₄×D₈₉	—	4	2	no	—
36	C₄×D₈₉	—	4	2	no	—
37	C₈×C₈₉	—	2	1	no	—
38	C₈×C₈₉	—	4	2	no	—
39	C₈×C₈₉	M₁₆	1	2	yes	complemented
40	#10623494	—	2	2	no	—
41	#10623494	—	4	1	no	—
42	C₂²×C₈₉	C₄×D₄	1	1	yes	characteristic
43	C₂×D₈₉	—	4	2	no	—
44	C₂×D₈₉	—	4	2	no	—
45	C₂×D₈₉	—	4	2	no	—
46	C₄×C₈₉	—	2	1	no	—
47	C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
48	C₄×C₈₉	#131	1	1	yes	characteristic
49	C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
50	Dic₈₉	—	4	2	no	—
51	C₂×C₈₉	#436	1	1	yes	characteristic
52	C₂×C₈₉	#442	1	1	yes	characteristic
53	C₂×C₈₉	#444	1	1	yes	characteristic
54	D₈₉	—	8	2	no	—
55	#1594	—	89	1	no	maximal, Sylow
56	C₈₉	#1594	1	1	yes	characteristic, complemented, Sylow
57	C₄×D₈	—	89	1	no	—
58	#436	—	89	2	no	—
59	#334	D₈₉	1	1	yes	characteristic, complemented
60	D₄⋊C₈	—	89	2	no	—
61	#322	—	89	1	no	—
62	C₂×D₈	—	89	1	no	—
63	C₂×M₁₆	—	178	2	no	—
64	C₄×D₄	—	89	2	no	—
65	#107	—	89	1	no	—
66	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
67	#102	—	89	2	no	—
68	C₂²⋊C₈	—	178	2	no	—
69	C₄×C₈	—	89	1	no	—
70	C₄×C₈	—	89	1	no	—
71	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
72	C₂×D₄	—	89	2	no	—
73	C₂²×C₄	—	178	2	no	—
74	D₈	—	178	2	no	—
75	M₁₆	—	356	4	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	89	2	no	—
78	C₂×C₈	—	178	1	no	—
79	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
80	C₂×C₈	C₄×D₈₉	1	1	yes	characteristic
81	C₄⋊C₄	—	89	2	no	—
82	C₂²⋊C₄	—	178	2	no	—
83	C₄²	C₂²×D₈₉	1	1	yes	characteristic
84	C₂³	—	178	2	no	—
85	D₄	—	178	2	no	—
86	D₄	—	178	2	no	—
87	C₂×C₄	—	356	2	no	—
88	C₂×C₄	—	356	2	no	—
89	C₂×C₄	—	178	2	no	—
90	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
91	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
92	C₂×C₄	#13083575	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	4	2	no	—
95	C₈	—	356	1	no	—
96	C₈	—	178	2	no	—
97	C₈	#13083557	1	2	yes	complemented
98	C₂²	—	356	2	no	—
99	C₂²	—	356	2	no	—
100	C₂²	—	356	2	no	—
101	C₂²	#15667974	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	356	2	no	—
104	C₄	C₂×#13083557	1	1	yes	characteristic
105	C₄	#15668020	1	1	yes	characteristic
106	C₄	#15668039	1	2	yes	—
107	C₂	—	712	2	no	—
108	C₂	#19151675	1	1	yes	characteristic
109	C₂	#19151709	1	1	yes	characteristic
110	C₂	#19151719	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₇₁₂

C₂

yes

maximal, characteristic

#19151675

C₂

yes

maximal, complemented

#19151325

C₂

yes

maximal, complemented

#19151319

C₂

yes

maximal, characteristic, complemented

#334×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×D₇₁₂

C₄

yes

characteristic

C₂×#13083557

—

C₄×D₃₅₆

C₂²

yes

—

#15667946

C₄

yes

characteristic

#15667934

C₄

yes

—

#15667924

—

C₄×#10623494

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×D₃₅₆

C₂×C₄

yes

—

C₂×C₄×D₈₉

—

D₇₁₂

—

#13083557

—

C₂×#10623494

—

C₂×#10623494

D₄

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

#13083551

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

M₁₆

yes

complemented

#10623494

—

#10623494

—

C₂²×C₈₉

C₄×D₄

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×M₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#1594

—

maximal, Sylow

C₈₉

#1594

yes

characteristic, complemented, Sylow

C₄×D₈

—

#436

—

#334

D₈₉

yes

characteristic, complemented

D₄⋊C₈

—

#322

—

C₂×D₈

—

C₂×M₁₆

—

178

—

C₄×D₄

—

#107

—

C₄⋊C₈

C₂×D₈₉

yes

—

#102

—

C₂²⋊C₈

—

178

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

178

—

D₈

—

178

—

M₁₆

—

356

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

C₄×D₈₉

yes

characteristic

C₂×C₈

C₄×D₈₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

—

178

—

D₄

—

178

—

D₄

—

178

—

C₂×C₄

—

356

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

D₄×D₈₉

yes

characteristic

C₂×C₄

#13083575

yes

characteristic

C₈

—

C₈

—

C₈

—

356

—

C₈

—

178

—

C₈

#13083557

yes

complemented

C₂²

—

356

—

C₂²

—

356

—

100

C₂²

—

356

—

101

C₂²

#15667974

yes

characteristic

102

C₄

—

103

C₄

—

356

—

104

C₄

C₂×#13083557

yes

characteristic

105

C₄

#15668020

yes

characteristic

106

C₄

#15668039

yes

—

107

C₂

—

712

—

108

C₂

#19151675

yes

characteristic

109

C₂

#19151709

yes

characteristic

110

C₂

#19151719

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₄²	C₂²×D₈₉
Fitting	#334×C₈₉	C₂
socle	C₂²×C₈₉	C₄×D₄
prime radical	C₂×D₇₁₂	C₄
cosocle	C₄²×C₈₉	C₂³
hypercenter	#334	D₈₉
hypocenter	C₈₉	#1594
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₈₉

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₄²

C₂²×D₈₉

Fitting

#334×C₈₉

C₂

socle

C₂²×C₈₉

C₄×D₄

prime radical

C₂×D₇₁₂

C₄

cosocle

C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #19151675	C₂	semidirect
3. #19151325	C₂	semidirect
4. #19151319	C₂	semidirect
5. #334×C₈₉	C₂	semidirect
39. C₈×C₈₉	M₁₆	semidirect
56. C₈₉	#1594	semidirect
59. #334	D₈₉	semidirect
97. C₈	#13083557	semidirect

H ≅ G/N

type

2. #19151675

C₂

semidirect

3. #19151325

C₂

semidirect

4. #19151319

C₂

semidirect

5. #334×C₈₉

C₂

semidirect

39. C₈×C₈₉

M₁₆

semidirect

56. C₈₉

#1594

semidirect

59. #334

D₈₉

semidirect

97. C₈

#13083557

semidirect

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

subgroup	index
C₄×D₇₁₂	2
#19151675	2
#19151325	2
#19151319	2
#334×C₈₉	2
#1594	89

subgroup

index

C₄×D₇₁₂

#334×C₈₉

Subgroups — #26019165

Conjugacy classes of subgroups

Complements

Maximal subgroups