Group #26020109 — subgroups

Subgroups — #26020109

Order 11392.

260 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 141 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×C₄⋊#10623494	C₂	1	2	yes	maximal
2	C₂×#15667923	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
4	#19151353	C₂	1	2	yes	maximal
5	#376×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10623494	C₂²	1	2	yes	—
7	C₂²×#10623494	C₂²	1	2	yes	—
8	C₄⋊#10623494	C₄	1	8	yes	complemented
9	#15667923	—	2	2	no	—
10	C₄×#10623494	—	2	2	no	—
11	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	#95×C₈₉	C₄	1	4	yes	—
15	C₂×#10623494	—	2	2	no	—
16	C₂×#10623494	—	2	2	no	—
17	C₂×#10623494	—	2	2	no	—
18	C₂×#10623494	—	2	2	no	—
19	C₂×#10623494	C₂×C₄	1	4	yes	—
20	C₂×#10623494	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
25	C₄²×C₈₉	—	2	2	no	—
26	C₄²×C₈₉	C₂×C₄	1	4	yes	—
27	C₄²×C₈₉	D₄	1	2	yes	—
28	C₄²×C₈₉	Q₈	1	2	yes	—
29	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈₉	—	2	4	no	—
31	C₂×C₄×C₈₉	—	2	4	no	—
32	C₂×C₄×C₈₉	—	2	2	no	—
33	C₂×C₄×C₈₉	—	2	4	no	—
34	C₂×C₄×C₈₉	C₄²	1	4	yes	—
35	C₂×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₈₉	#41	1	2	yes	—
40	#10623494	—	2	4	no	—
41	#10623494	—	2	4	no	—
42	C₂²×C₈₉	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₈₉	#117	1	1	yes	characteristic
45	C₂²×C₈₉	#131	1	4	yes	—
46	C₄×C₈₉	—	2	4	no	—
47	C₄×C₈₉	—	2	4	no	—
48	C₄×C₈₉	—	4	2	no	—
49	C₄×C₈₉	C₄×D₄	1	2	yes	—
50	C₄×C₈₉	C₄×Q₈	1	2	yes	—
51	C₂×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈₉	#405	1	2	yes	—
53	C₂×C₈₉	#424	1	2	yes	—
54	C₂×C₈₉	#433	1	2	yes	—
55	#1768	—	89	1	no	maximal, Sylow
56	C₈₉	#1768	1	1	yes	characteristic, complemented, Sylow
57	C₂×C₄⋊C₈	—	89	2	no	—
58	C₂×#97	—	89	1	no	—
59	C₂×C₄×C₈	—	89	1	no	—
60	#376	D₈₉	1	1	yes	characteristic
61	#336	—	89	2	no	—
62	C₂²×C₈	—	89	2	no	—
63	C₂²×C₈	—	89	2	no	—
64	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
67	C₄⋊C₈	—	89	8	no	—
68	#97	—	178	2	no	—
69	C₄×C₈	—	178	2	no	—
70	#95	Dic₈₉	1	4	yes	—
71	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
72	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
73	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
74	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
75	C₂×C₈	—	89	4	no	—
76	C₂×C₈	—	89	4	no	—
77	C₂×C₈	—	178	2	no	—
78	C₂×C₈	—	178	2	no	—
79	C₂×C₈	—	178	2	no	—
80	C₂×C₈	—	178	2	no	—
81	C₄²	—	2	2	no	—
82	C₄²	Dic₁₇₈	1	2	yes	—
83	C₄²	C₄×D₈₉	1	4	yes	—
84	C₄²	D₃₅₆	1	2	yes	—
85	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄×Dic₈₉	1	4	yes	—
91	C₂×C₄	C₄⋊Dic₈₉	1	4	yes	—
92	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
93	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
95	C₂×C₄	#13083574	1	2	yes	—
96	C₈	—	178	4	no	—
97	C₈	—	178	4	no	—
98	C₂²	C₂×C₄×Dic₈₉	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
100	C₂²	#15667971	1	1	yes	characteristic
101	C₂²	#15668019	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	C₄×D₃₅₆	1	2	yes	—
106	C₄	C₄×Dic₁₇₈	1	2	yes	—
107	C₂	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
108	C₂	#19151554	1	2	yes	—
109	C₂	#19151625	1	2	yes	—
110	C₂	#19151659	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#10623494

C₂

yes

maximal

C₂×#15667923

C₂

yes

maximal, characteristic

C₂×C₄×#10623494

C₂

yes

maximal, characteristic

#19151353

C₂

yes

maximal

#376×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

—

C₂²×#10623494

C₂²

yes

—

C₄⋊#10623494

C₄

yes

complemented

#15667923

—

C₄×#10623494

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₄

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

D₄

yes

—

C₄²×C₈₉

Q₈

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₄²

yes

—

C₂×C₄×C₈₉

C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

C₂×C₄²

yes

characteristic

C₂²×C₈₉

C₂×C₄⋊C₄

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

C₄×Q₈

yes

—

C₂×C₈₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₈₉

#405

yes

—

C₂×C₈₉

#424

yes

—

C₂×C₈₉

#433

yes

—

#1768

—

maximal, Sylow

C₈₉

#1768

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#376

D₈₉

yes

characteristic

#336

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

#97

—

178

—

C₄×C₈

—

178

—

#95

Dic₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₂×C₈

—

178

—

C₄²

—

C₄²

Dic₁₇₈

yes

—

C₄²

C₄×D₈₉

yes

—

C₄²

D₃₅₆

yes

—

C₂³

C₂²×Dic₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₈₉

yes

—

C₂×C₄

C₄⋊Dic₈₉

yes

—

C₂×C₄

C₂×C₄×D₈₉

yes

—

C₂×C₄

C₂×D₃₅₆

yes

characteristic

C₂×C₄

C₂×Dic₁₇₈

yes

characteristic

C₂×C₄

#13083574

yes

—

C₈

—

178

—

C₈

—

178

—

C₂²

C₂×C₄×Dic₈₉

yes

characteristic

C₂²

C₂×C₄⋊Dic₈₉

yes

characteristic

100

C₂²

#15667971

yes

characteristic

101

C₂²

#15668019

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×D₃₅₆

yes

—

106

C₄

C₄×Dic₁₇₈

yes

—

107

C₂

C₄×C₄⋊Dic₈₉

yes

characteristic

108

C₂

#19151554

yes

—

109

C₂

#19151625

yes

—

110

C₂

#19151659

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂×C₄²
Fitting	#376×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#376	D₈₉
hypocenter	C₈₉	#1768
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₈₉

commutator

C₂²×C₈₉

C₂×C₄²

Fitting

#376×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

C₂²×C₄

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
8. C₄⋊#10623494	C₄	semidirect
56. C₈₉	#1768	semidirect

H ≅ G/N

type

8. C₄⋊#10623494

C₄

semidirect

56. C₈₉

#1768

semidirect

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

subgroup	index
C₂×C₄⋊#10623494	2
C₂×#15667923	2
C₂×C₄×#10623494	2
#19151353	2
#376×C₈₉	2
#1768	89

subgroup

index

#376×C₈₉

Subgroups — #26020109

Conjugacy classes of subgroups

Complements

Maximal subgroups