Group #26022302 — subgroups

Subgroups — #26022302

Order 11392.

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

normal subgroups only

This group has 154 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	C₂×#15667958	C₂	1	2	yes	maximal
2	#19151978	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13083549	C₂	1	1	yes	maximal, characteristic
4	#19151401	C₂	1	2	yes	maximal
5	C₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₅₆	—	2	1	no	—
7	C₂×Dic₃₅₆	C₂²	1	2	yes	—
8	#15668012	—	2	1	no	—
9	#15667958	—	2	4	no	—
10	C₂×#13083549	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
12	C₄×Q₈×C₈₉	—	2	1	no	—
13	#107×C₈₉	C₂²	1	1	yes	characteristic
14	#103×C₈₉	—	2	1	no	—
15	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₇₈	—	2	2	no	—
17	Dic₃₅₆	—	2	2	no	—
18	Dic₃₅₆	—	2	2	no	—
19	Dic₃₅₆	—	4	2	no	—
20	C₄⋊Dic₈₉	—	4	1	no	—
21	C₄⋊Dic₈₉	—	4	1	no	—
22	#13083549	—	2	2	no	—
23	C₂×Q₈×C₈₉	—	2	1	no	—
24	Q₁₆×C₈₉	—	2	1	no	—
25	Q₁₆×C₈₉	D₄	1	2	yes	—
26	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₈₉	—	2	1	no	—
29	C₄⋊C₄×C₈₉	—	4	1	no	—
30	C₄²×C₈₉	—	4	1	no	—
31	C₄²×C₈₉	D₄	1	1	yes	characteristic
32	Q₈×C₈₉	—	2	2	no	—
33	Q₈×C₈₉	—	4	1	no	—
34	C₂×C₄×C₈₉	—	4	1	no	—
35	C₂×C₄×C₈₉	—	4	1	no	—
36	C₂×C₄×C₈₉	D₈	1	1	yes	characteristic
37	C₂×C₄×C₈₉	D₈	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₈₉	—	4	2	no	—
40	Dic₁₇₈	—	4	2	no	—
41	Dic₁₇₈	—	4	2	no	—
42	C₈×C₈₉	—	2	1	no	—
43	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
44	C₈×C₈₉	#41	1	1	yes	characteristic
45	C₂²×C₈₉	C₂×D₈	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	Q₃₂	1	2	yes	complemented
50	C₄×C₈₉	#121	1	1	yes	characteristic
51	C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
52	Dic₈₉	—	8	2	no	—
53	C₂×C₈₉	#459	1	1	yes	characteristic
54	C₂×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
55	C₂×C₈₉	#510	1	1	yes	characteristic
56	#2209	—	89	1	no	maximal, Sylow
57	C₈₉	#2209	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₃₂	—	89	2	no	—
59	#494	—	89	1	no	—
60	C₄×Q₁₆	D₈₉	1	1	yes	characteristic
61	C₄⋊C₁₆	—	89	1	no	—
62	#358	—	89	2	no	—
63	C₂×Q₁₆	—	89	2	no	—
64	C₂×Q₁₆	—	178	1	no	—
65	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
66	C₄⋊Q₈	—	178	1	no	—
67	C₄×Q₈	—	2	1	no	—
68	Q₃₂	—	178	4	no	—
69	C₂×C₁₆	—	89	2	no	—
70	#107	C₂×D₈₉	1	1	yes	characteristic
71	#103	—	2	1	no	—
72	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	178	2	no	—
75	Q₁₆	—	2	1	no	—
76	Q₁₆	—	356	2	no	—
77	Q₁₆	—	178	2	no	—
78	Q₁₆	—	178	2	no	—
79	Q₁₆	D₃₅₆	1	2	yes	—
80	C₂×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
82	C₄⋊C₄	—	2	1	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₄⋊C₄	—	356	1	no	—
85	C₄⋊C₄	—	356	1	no	—
86	C₄²	—	4	1	no	—
87	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
88	C₁₆	—	178	2	no	—
89	Q₈	—	2	2	no	—
90	Q₈	—	4	1	no	—
91	Q₈	—	356	2	no	—
92	Q₈	—	356	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	356	2	no	—
96	C₂×C₄	#13083559	1	1	yes	characteristic
97	C₂×C₄	#13083559	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	C₂×D₃₅₆	1	1	yes	characteristic
101	C₈	#13083574	1	1	yes	characteristic
102	C₂²	C₂×#13083559	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	8	1	no	—
106	C₄	—	712	2	no	—
107	C₄	#15667958	1	2	yes	complemented
108	C₄	#15667986	1	1	yes	characteristic
109	C₄	#15668037	1	1	yes	characteristic
110	C₂	#19151793	1	1	yes	characteristic
111	C₂	C₂×#15667958	1	1	yes	characteristic
112	C₂	#19152038	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15667958

C₂

yes

maximal

#19151978

C₂

yes

maximal, characteristic

C₄⋊#13083549

C₂

yes

maximal, characteristic

#19151401

C₂

yes

maximal

C₄×Q₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

—

C₂×Dic₃₅₆

C₂²

yes

—

#15668012

—

#15667958

—

C₂×#13083549

C₂²

yes

—

C₂×Q₁₆×C₈₉

C₂²

yes

characteristic

C₄×Q₈×C₈₉

—

#107×C₈₉

C₂²

yes

characteristic

#103×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

—

Dic₃₅₆

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083549

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

D₄

yes

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

D₄

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

D₈

yes

characteristic

C₂×C₄×C₈₉

D₈

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₄

yes

characteristic

C₈×C₈₉

#41

yes

characteristic

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

Q₃₂

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

Dic₈₉

—

C₂×C₈₉

#459

yes

characteristic

C₂×C₈₉

C₂×Q₃₂

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

#494

—

C₄×Q₁₆

D₈₉

yes

characteristic

C₄⋊C₁₆

—

#358

—

C₂×Q₁₆

—

C₂×Q₁₆

—

178

—

C₂×Q₁₆

C₂×D₈₉

yes

characteristic

C₄⋊Q₈

—

178

—

C₄×Q₈

—

Q₃₂

—

178

—

C₂×C₁₆

—

#107

C₂×D₈₉

yes

characteristic

#103

—

C₄×C₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

178

—

Q₁₆

—

Q₁₆

—

356

—

Q₁₆

—

178

—

Q₁₆

—

178

—

Q₁₆

D₃₅₆

yes

—

C₂×C₈

C₂²⋊D₈₉

yes

characteristic

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄²

—

C₄²

C₂²⋊D₈₉

yes

characteristic

C₁₆

—

178

—

Q₈

—

Q₈

—

Q₈

—

356

—

Q₈

—

356

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

C₈

—

100

C₈

C₂×D₃₅₆

yes

characteristic

101

C₈

#13083574

yes

characteristic

102

C₂²

C₂×#13083559

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

712

—

107

C₄

#15667958

yes

complemented

108

C₄

#15667986

yes

characteristic

109

C₄

#15668037

yes

characteristic

110

C₂

#19151793

yes

characteristic

111

C₂

C₂×#15667958

yes

characteristic

112

C₂

#19152038

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	C₄×Q₁₆×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₈
hypercenter	C₄×Q₁₆	D₈₉
hypocenter	C₈₉	#2209
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×#13083559

commutator, cosocle

C₂×C₈×C₈₉

C₂³

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

C₄×Q₁₆×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

C₂×D₈

hypercenter

C₄×Q₁₆

D₈₉

hypocenter

C₈₉

#2209

LCS

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

N	H ≅ G/N	type
49. C₄×C₈₉	Q₃₂	semidirect
57. C₈₉	#2209	semidirect
107. C₄	#15667958	semidirect

H ≅ G/N

type

49. C₄×C₈₉

Q₃₂

semidirect

57. C₈₉

#2209

semidirect

107. C₄

#15667958

semidirect

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

subgroup	index
C₂×#15667958	2
#19151978	2
C₄⋊#13083549	2
#19151401	2
C₄×Q₁₆×C₈₉	2
#2209	89

subgroup

index

C₄×Q₁₆×C₈₉

Subgroups — #26022302

Conjugacy classes of subgroups

Complements

Maximal subgroups