Group #26022415 — subgroups

Subgroups — #26022415

Order 11392.

124 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 47 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	#19151999	C₂	1	1	yes	maximal, characteristic
2	#19151692	C₂	1	1	yes	maximal, characteristic
3	#19151425	C₂	1	1	yes	maximal, characteristic
4	#19151416	C₂	1	1	yes	maximal, characteristic
5	#19151403	C₂	1	1	yes	maximal, characteristic
6	#19151401	C₂	1	1	yes	maximal, characteristic
7	#366×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
9	#15668003	—	2	1	no	—
10	#15667977	—	2	1	no	—
11	C₂×#13083549	C₂²	1	1	yes	characteristic
12	#15667946	C₂²	1	1	yes	characteristic
13	#15667943	—	2	1	no	—
14	#15667935	—	2	1	no	—
15	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	#107×C₈₉	C₂²	1	1	yes	characteristic
18	#107×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₇₈	—	2	1	no	—
20	Dic₃₅₆	—	2	1	no	—
21	Dic₃₅₆	Q₈	1	2	yes	—
22	C₂×#10623494	D₄	1	1	yes	characteristic
23	C₄⋊Dic₈₉	—	2	1	no	—
24	C₄⋊Dic₈₉	—	4	1	no	—
25	#13083553	—	4	1	no	—
26	#13083553	—	4	1	no	—
27	C₄×Dic₈₉	—	4	1	no	—
28	C₄×Dic₈₉	D₄	1	1	yes	characteristic
29	#13083549	—	2	1	no	—
30	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₈₉	—	2	1	no	—
32	C₄⋊C₄×C₈₉	—	2	1	no	—
33	C₁₆×C₈₉	—	2	1	no	—
34	C₂×C₄×C₈₉	—	4	1	no	—
35	C₂×C₄×C₈₉	—	4	1	no	—
36	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₈₉	—	4	1	no	—
38	C₂×Dic₈₉	—	4	1	no	—
39	C₂×Dic₈₉	D₈	1	1	yes	characteristic
40	C₂×Dic₈₉	D₈	1	1	yes	characteristic
41	Dic₁₇₈	—	2	2	no	—
42	Dic₁₇₈	—	4	1	no	—
43	C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₈₉	#41	1	1	yes	characteristic
45	#10623494	—	2	1	no	—
46	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
47	C₄×C₈₉	—	8	1	no	—
48	C₄×C₈₉	—	8	1	no	—
49	C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₄×C₈₉	#137	1	1	yes	characteristic
51	Dic₈₉	—	2	1	no	—
52	Dic₈₉	—	2	1	no	—
53	Dic₈₉	—	4	2	no	—
54	Dic₈₉	—	8	1	no	—
55	C₂×C₈₉	#474	1	1	yes	characteristic
56	C₂×C₈₉	#508	1	1	yes	characteristic
57	C₂×C₈₉	#510	1	1	yes	characteristic
58	#2229	—	89	1	no	maximal, Sylow
59	C₈₉	#2229	1	1	yes	characteristic, complemented, Sylow
60	#499	—	89	1	no	—
61	C₄×Q₁₆	—	89	1	no	—
62	#367	—	89	1	no	—
63	#366	D₈₉	1	1	yes	characteristic
64	C₄⋊C₁₆	—	89	1	no	—
65	#358	—	89	1	no	—
66	#358	—	89	1	no	—
67	C₂×Q₁₆	—	89	1	no	—
68	#125	—	178	1	no	—
69	C₄×Q₈	—	178	1	no	—
70	C₂×C₁₆	—	89	1	no	—
71	C₂×C₁₆	C₂×D₈₉	1	1	yes	characteristic
72	#107	—	89	1	no	—
73	#107	C₂×D₈₉	1	1	yes	characteristic
74	#107	C₂×D₈₉	1	1	yes	characteristic
75	#106	—	178	1	no	—
76	#103	—	178	1	no	—
77	C₄×C₈	—	89	1	no	—
78	C₂×Q₈	—	178	1	no	—
79	Q₁₆	—	89	2	no	—
80	Q₁₆	—	178	1	no	—
81	C₂×C₈	—	89	1	no	—
82	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	2	1	no	—
85	C₄⋊C₄	—	356	1	no	—
86	C₄⋊C₄	—	356	1	no	—
87	C₄⋊C₄	—	356	1	no	—
88	C₄⋊C₄	—	178	1	no	—
89	C₄²	—	89	1	no	—
90	C₄²	—	356	1	no	—
91	C₁₆	—	2	1	no	—
92	C₁₆	—	178	1	no	—
93	Q₈	—	356	1	no	—
94	Q₈	—	178	2	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	89	1	no	—
98	C₂×C₄	—	89	1	no	—
99	C₂×C₄	—	356	1	no	—
100	C₂×C₄	—	356	1	no	—
101	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
102	C₈	—	178	1	no	—
103	C₈	Q₈×D₈₉	1	1	yes	characteristic
104	C₈	#13083574	1	1	yes	characteristic
105	C₂²	D₈×D₈₉	1	1	yes	characteristic
106	C₄	—	8	1	no	—
107	C₄	—	8	1	no	—
108	C₄	—	356	2	no	—
109	C₄	—	178	1	no	—
110	C₄	—	178	1	no	—
111	C₄	—	712	1	no	—
112	C₄	#15667990	1	1	yes	characteristic
113	C₄	#15668044	1	1	yes	characteristic
114	C₂	#19151877	1	1	yes	characteristic
115	C₂	#19152027	1	1	yes	characteristic
116	C₂	#19152040	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151999

C₂

yes

maximal, characteristic

#19151692

C₂

yes

maximal, characteristic

#19151425

C₂

yes

maximal, characteristic

#19151416

C₂

yes

maximal, characteristic

#19151403

C₂

yes

maximal, characteristic

#19151401

C₂

yes

maximal, characteristic

#366×C₈₉

C₂

yes

maximal, characteristic

C₂×Dic₃₅₆

C₂²

yes

—

—

C₂²

yes

characteristic

#15667946

C₂²

yes

characteristic

#15667943

—

#15667935

—

C₈×Dic₈₉

C₂²

yes

characteristic

C₂×C₁₆×C₈₉

C₂²

yes

characteristic

#107×C₈₉

C₂²

yes

characteristic

#107×C₈₉

C₂²

yes

characteristic

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

Q₈

yes

—

C₂×#10623494

D₄

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₄

yes

characteristic

#13083549

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

D₈

yes

characteristic

C₂×Dic₈₉

D₈

yes

characteristic

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

C₂×Q₈

yes

C₈×C₈₉

yes

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

characteristic

C₄×C₈₉

#137

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#499

—

C₄×Q₁₆

—

#367

—

#366

D₈₉

yes

characteristic

C₄⋊C₁₆

—

#358

—

#358

—

C₂×Q₁₆

—

#125

—

178

—

C₄×Q₈

—

178

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₉

yes

characteristic

#107

—

#107

C₂×D₈₉

yes

characteristic

#107

C₂×D₈₉

yes

characteristic

#106

—

178

—

#103

—

178

—

C₄×C₈

—

C₂×Q₈

—

178

—

Q₁₆

—

Q₁₆

—

178

—

C₂×C₈

—

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄²

—

C₄²

—

356

—

C₁₆

—

C₁₆

—

178

—

Q₈

—

356

—

Q₈

—

178

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

356

—

100

C₂×C₄

—

356

—

101

C₂×C₄

D₄×D₈₉

yes

characteristic

102

C₈

—

178

—

103

C₈

Q₈×D₈₉

yes

characteristic

104

C₈

#13083574

yes

characteristic

105

C₂²

D₈×D₈₉

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

356

—

109

C₄

—

178

—

110

C₄

—

178

—

111

C₄

—

712

—

112

C₄

#15667990

yes

characteristic

113

C₄

#15668044

yes

characteristic

114

C₂

#19151877

yes

characteristic

115

C₂

#19152027

yes

characteristic

116

C₂

#19152040

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#366×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₈
hypercenter	#366	D₈₉
hypocenter	C₈₉	#2229
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×D₈₉

commutator, cosocle

C₂×C₈×C₈₉

C₂³

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

#366×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

C₂×D₈

D₈₉

C₈₉

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

N	H ≅ G/N	type
59. C₈₉	#2229	semidirect

H ≅ G/N

type

59. C₈₉

#2229

semidirect

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

subgroup	index
#19151999	2
#19151692	2
#19151425	2
#19151416	2
#19151403	2
#19151401	2
#366×C₈₉	2
#2229	89

subgroup

index

#366×C₈₉

Subgroups — #26022415

Conjugacy classes of subgroups

Complements

Maximal subgroups