Group #26019023 — subgroups

Subgroups — #26019023

Order 11392.

168 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 97 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	#19151622	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
3	#19151349	C₂	1	4	yes	maximal, complemented
4	#431×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13083558	—	2	1	no	—
6	C₂²×#10623494	—	2	1	no	—
7	C₄⋊#10623494	—	2	1	no	—
8	C₄⋊#10623494	—	2	1	no	—
9	C₄⋊#10623494	C₂²	1	2	yes	—
10	C₄⋊#10623494	C₂²	1	2	yes	—
11	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₈₉	C₄	1	2	yes	—
13	C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
14	C₄×C₈×C₈₉	C₂²	1	2	yes	—
15	#13083558	—	4	2	no	—
16	C₂×#10623494	—	2	2	no	—
17	C₂×#10623494	—	2	2	no	—
18	C₂×#10623494	—	4	2	no	—
19	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₈₉	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₈₉	C₂×C₄	1	4	yes	—
23	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₈₉	D₄	1	1	yes	characteristic
25	C₄²×C₈₉	Q₈	1	1	yes	characteristic
26	C₄²×C₈₉	C₂³	1	1	yes	characteristic
27	C₂³×C₈₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₈₉	—	2	2	no	—
29	C₂×C₄×C₈₉	—	4	1	no	—
30	C₂×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₈₉	SD₁₆	1	2	yes	—
34	C₂×C₄×C₈₉	SD₁₆	1	2	yes	—
35	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₈₉	—	2	4	no	—
39	#10623494	—	4	2	no	—
40	#10623494	—	4	2	no	—
41	C₂²×C₈₉	—	2	1	no	—
42	C₂²×C₈₉	—	2	1	no	—
43	C₂²×C₈₉	#106	1	2	yes	—
44	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₈₉	—	2	1	no	—
46	C₄×C₈₉	—	2	1	no	—
47	C₄×C₈₉	—	2	1	no	—
48	C₄×C₈₉	#106	1	2	yes	—
49	C₄×C₈₉	#131	1	2	yes	—
50	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×C₈₉	—	2	2	no	—
53	C₂×C₈₉	#424	1	1	yes	characteristic
54	C₂×C₈₉	C₂×#106	1	1	yes	characteristic
55	C₂×C₈₉	#430	1	1	yes	characteristic
56	#1570	—	89	1	no	maximal, Sylow
57	C₈₉	#1570	1	1	yes	characteristic, complemented, Sylow
58	#431	D₈₉	1	1	yes	characteristic
59	#423	—	89	1	no	—
60	C₂×C₄⋊C₈	—	89	1	no	—
61	#334	—	89	4	no	—
62	C₂×M₁₆	—	178	1	no	—
63	C₂²×C₈	—	178	1	no	—
64	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
65	C₄⋊C₈	—	89	2	no	—
66	C₄⋊C₈	—	89	2	no	—
67	C₄⋊C₈	—	178	1	no	—
68	C₄⋊C₈	—	178	1	no	—
69	C₄⋊C₈	Dic₈₉	1	2	yes	—
70	C₂²⋊C₈	Dic₈₉	1	2	yes	—
71	C₄×C₈	C₂×D₈₉	1	2	yes	—
72	C₂²×C₄	Dic₁₇₈	1	1	yes	characteristic
73	C₂²×C₄	D₃₅₆	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₈₉	1	1	yes	characteristic
75	M₁₆	—	356	2	no	—
76	C₂×C₈	—	356	2	no	—
77	C₂×C₈	—	178	2	no	—
78	C₂×C₈	—	178	2	no	—
79	C₂×C₈	C₂×Dic₈₉	1	4	yes	—
80	C₄²	Dic₁₇₈	1	1	yes	characteristic
81	C₄²	D₃₅₆	1	1	yes	characteristic
82	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
83	C₄²	C₂²×D₈₉	1	1	yes	characteristic
84	C₂³	C₄⋊Dic₈₉	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
88	C₂×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
90	C₂×C₄	#13083563	1	2	yes	—
91	C₂×C₄	#13083563	1	2	yes	—
92	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
93	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	356	2	no	—
97	C₈	—	356	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#15667944	1	2	yes	—
101	C₂²	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#15667944	1	2	yes	—
106	C₄	#15668019	1	2	yes	—
107	C₄	C₂×#13083563	1	1	yes	characteristic
108	C₄	C₂×#13083563	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#19151625	1	1	yes	characteristic
111	C₂	C₂×#15667944	1	1	yes	characteristic
112	C₂	#19151647	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151622

C₂

yes

maximal, characteristic

C₂×C₄⋊#10623494

C₂

yes

maximal, characteristic

#19151349

C₂

yes

maximal, complemented

#431×C₈₉

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₄

yes

—

C₂²⋊C₈×C₈₉

C₄

yes

—

C₄×C₈×C₈₉

C₂²

yes

—

—

—

—

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

Q₈

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

Q₈

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

SD₁₆

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#106

yes

—

C₂²×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#106

yes

—

C₄×C₈₉

#131

yes

—

C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#431

D₈₉

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

#334

—

C₂×M₁₆

—

178

—

C₂²×C₈

—

178

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

178

—

C₄⋊C₈

—

178

—

C₄⋊C₈

Dic₈₉

yes

—

C₂²⋊C₈

Dic₈₉

yes

—

C₄×C₈

C₂×D₈₉

yes

—

C₂²×C₄

Dic₁₇₈

yes

characteristic

C₂²×C₄

D₃₅₆

yes

characteristic

C₂²×C₄

C₂×Dic₈₉

yes

characteristic

M₁₆

—

356

—

C₂×C₈

—

356

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

C₂×Dic₈₉

yes

—

C₄²

Dic₁₇₈

yes

characteristic

C₄²

D₃₅₆

yes

characteristic

C₄²

C₂×Dic₈₉

yes

characteristic

C₄²

C₂²×D₈₉

yes

characteristic

C₂³

C₄⋊Dic₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₈₉

yes

characteristic

C₂×C₄

C₄⋊Dic₈₉

yes

characteristic

C₂×C₄

C₄⋊Dic₈₉

yes

characteristic

C₂×C₄

#13083563

yes

—

C₂×C₄

#13083563

yes

—

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

C₂×C₄

C₂×D₃₅₆

yes

characteristic

C₂×C₄

C₂×Dic₁₇₈

yes

characteristic

C₈

—

C₈

—

356

—

C₈

—

356

—

C₂²

—

C₂²

—

100

C₂²

#15667944

yes

—

101

C₂²

C₂×C₄⋊Dic₈₉

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#15667944

yes

—

106

C₄

#15668019

yes

—

107

C₄

C₂×#13083563

yes

characteristic

108

C₄

C₂×#13083563

yes

characteristic

109

C₂

—

110

C₂

#19151625

yes

characteristic

111

C₂

C₂×#15667944

yes

characteristic

112

C₂

#19151647

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₄²	C₂²×D₈₉
Fitting	#431×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄⋊C₄
prime radical	C₂³×C₈₉	C₄⋊C₄
cosocle	C₄²×C₈₉	C₂³
hypercenter	#431	D₈₉
hypocenter	C₈₉	#1570
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₃₅₆

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₄²

C₂²×D₈₉

Fitting

#431×C₈₉

C₂

socle

C₂²×C₈₉

C₂×C₄⋊C₄

prime radical

C₂³×C₈₉

C₄⋊C₄

cosocle

C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #19151349	C₂	semidirect
57. C₈₉	#1570	semidirect

H ≅ G/N

type

3. #19151349

C₂

semidirect

57. C₈₉

#1570

semidirect

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

subgroup	index
#19151622	2
C₂×C₄⋊#10623494	2
#19151349	2
#431×C₈₉	2
#1570	89

subgroup

index

#431×C₈₉

Subgroups — #26019023

Conjugacy classes of subgroups

Complements

Maximal subgroups