Group #26019390 — subgroups

Subgroups — #26019390

Order 11392.

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

normal subgroups only

This group has 206 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	#19152328	C₂	1	1	yes	maximal, characteristic
2	#19151890	C₂	1	2	yes	maximal
3	#19151347	C₂	1	1	yes	maximal, characteristic
4	#19151330	C₂	1	2	yes	maximal
5	#494×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#15668011	—	2	2	no	—
7	#15668010	—	2	2	no	—
8	C₄×Dic₁₇₈	C₂²	1	2	yes	—
9	#15667977	—	2	2	no	—
10	#15667945	—	2	2	no	—
11	C₄⋊#10623494	C₂²	1	2	yes	—
12	#15667935	—	2	2	no	—
13	C₂×Q₁₆×C₈₉	—	2	2	no	—
14	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₈	—	2	2	no	—
18	C₂×Dic₁₇₈	D₄	1	2	yes	—
19	C₂×#10623494	—	2	2	no	—
20	C₄⋊Dic₈₉	D₄	1	2	yes	—
21	#13083553	—	2	4	no	—
22	#13083553	—	2	2	no	—
23	C₄×Dic₈₉	—	2	2	no	—
24	C₄×Dic₈₉	—	2	2	no	—
25	C₂×Q₈×C₈₉	—	2	1	no	—
26	C₂×Q₈×C₈₉	D₄	1	2	yes	—
27	Q₁₆×C₈₉	—	4	4	no	—
28	C₂×C₈×C₈₉	—	2	1	no	—
29	C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₄⋊C₄×C₈₉	—	2	2	no	—
31	C₄²×C₈₉	C₂³	1	1	yes	characteristic
32	Q₈×C₈₉	—	4	2	no	—
33	Q₈×C₈₉	—	4	2	no	—
34	C₂×C₄×C₈₉	—	2	2	no	—
35	C₂×C₄×C₈₉	—	4	1	no	—
36	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₈₉	—	2	2	no	—
39	C₂×Dic₈₉	—	2	2	no	—
40	Dic₁₇₈	—	2	2	no	—
41	Dic₁₇₈	—	8	2	no	—
42	Dic₁₇₈	Q₁₆	1	4	yes	complemented
43	C₈×C₈₉	—	2	2	no	—
44	#10623494	—	4	2	no	—
45	C₂²×C₈₉	#120	1	1	yes	characteristic
46	C₄×C₈₉	—	2	1	no	—
47	C₄×C₈₉	—	4	2	no	—
48	C₄×C₈₉	—	8	1	no	—
49	C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
50	C₄×C₈₉	#137	1	2	yes	—
51	Dic₈₉	—	2	4	no	—
52	Dic₈₉	—	4	2	no	—
53	C₂×C₈₉	#451	1	2	yes	—
54	C₂×C₈₉	#453	1	1	yes	characteristic
55	#1633	—	89	1	no	maximal, Sylow
56	C₈₉	#1633	1	1	yes	characteristic, complemented, Sylow
57	Q₈²	—	89	1	no	—
58	#494	D₈₉	1	1	yes	characteristic
59	#477	—	89	2	no	—
60	#333	—	89	1	no	—
61	Q₈⋊C₈	—	89	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	178	2	no	—
64	C₄⋊Q₈	—	178	2	no	—
65	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
67	C₄×Q₈	—	89	2	no	—
68	C₄×Q₈	—	178	2	no	—
69	#107	—	178	2	no	—
70	C₄⋊C₈	—	89	2	no	—
71	#103	—	178	2	no	—
72	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	89	2	no	—
75	C₂×Q₈	—	178	2	no	—
76	C₂×Q₈	C₂²⋊D₈₉	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	178	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	89	2	no	—
83	C₄⋊C₄	—	178	4	no	—
84	C₄⋊C₄	—	178	2	no	—
85	C₄²	—	178	2	no	—
86	C₄²	—	178	2	no	—
87	C₄²	C₂²×D₈₉	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	89	4	no	—
91	Q₈	—	178	2	no	—
92	Q₈	—	712	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	178	2	no	—
96	C₂×C₄	—	178	2	no	—
97	C₂×C₄	D₄×D₈₉	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	356	2	no	—
101	C₂²	#15667980	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	356	2	no	—
106	C₄	—	178	4	no	—
107	C₄	Q₁₆×D₈₉	1	2	yes	—
108	C₄	#15668044	1	2	yes	—
109	C₂	#19151752	1	2	yes	—
110	C₂	#19151765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152328

C₂

yes

maximal, characteristic

#19151890

C₂

yes

maximal

#19151347

C₂

yes

maximal, characteristic

#19151330

C₂

yes

maximal

#494×C₈₉

C₂

yes

maximal, characteristic

#15668011

—

#15668010

—

C₄×Dic₁₇₈

C₂²

yes

—

#15667977

—

#15667945

—

C₄⋊#10623494

C₂²

yes

—

#15667935

—

C₂×Q₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊Q₈×C₈₉

C₂²

yes

characteristic

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

D₄

yes

—

C₂×#10623494

—

C₄⋊Dic₈₉

D₄

yes

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

Q₁₆

yes

complemented

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#120

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×Q₁₆

yes

—

C₄×C₈₉

#137

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#451

yes

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

Q₈²

—

#494

D₈₉

yes

characteristic

#477

—

#333

—

Q₈⋊C₈

—

C₂×Q₁₆

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

—

178

—

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

C₂×D₈₉

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

178

—

#107

—

178

—

C₄⋊C₈

—

#103

—

178

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

178

—

C₂×Q₈

C₂²⋊D₈₉

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

178

—

C₄²

—

178

—

C₄²

C₂²×D₈₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

178

—

Q₈

—

712

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

D₄×D₈₉

yes

—

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

C₈

—

100

C₈

—

356

—

101

C₂²

#15667980

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

356

—

106

C₄

—

178

—

107

C₄

Q₁₆×D₈₉

yes

—

108

C₄

#15668044

yes

—

109

C₂

#19151752

yes

—

110

C₂

#19151765

yes

characteristic

	subgroup	G/H
center	C₂²	#15667980
commutator, cosocle	C₄²×C₈₉	C₂³
Frattini	C₄²	C₂²×D₈₉
Fitting	#494×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#120
hypercenter	#494	D₈₉
hypocenter	C₈₉	#1633
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#15667980

commutator, cosocle

C₄²×C₈₉

C₂³

Frattini

C₄²

C₂²×D₈₉

Fitting

#494×C₈₉

C₂

C₂²×C₈₉

D₈₉

C₈₉

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
42. Dic₁₇₈	Q₁₆	semidirect
56. C₈₉	#1633	semidirect

H ≅ G/N

type

42. Dic₁₇₈

Q₁₆

semidirect

56. C₈₉

#1633

semidirect

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

subgroup	index
#19152328	2
#19151890	2
#19151347	2
#19151330	2
#494×C₈₉	2
#1633	89

subgroup

index

#494×C₈₉

Subgroups — #26019390

Conjugacy classes of subgroups

Complements

Maximal subgroups