Group #26022414 — subgroups

Subgroups — #26022414

Order 11392.

124 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 49 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	Q₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151426	C₂	1	1	yes	maximal, characteristic
4	#19151423	C₂	1	1	yes	maximal, characteristic
5	#19151416	C₂	1	1	yes	maximal, characteristic
6	#19151402	C₂	1	1	yes	maximal, characteristic
7	#358×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#15668003	—	2	1	no	—
9	Q₈×Dic₈₉	—	2	1	no	—
10	C₂×#13083549	C₂²	1	1	yes	characteristic
11	#15667946	C₂²	1	1	yes	characteristic
12	#15667946	C₂²	1	1	yes	characteristic
13	#15667943	—	2	1	no	—
14	#15667936	—	2	1	no	—
15	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	#107×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×#10623494	D₄	1	1	yes	characteristic
20	C₄⋊Dic₈₉	—	2	1	no	—
21	C₄⋊Dic₈₉	—	2	1	no	—
22	C₄⋊Dic₈₉	—	4	1	no	—
23	C₄⋊Dic₈₉	—	4	1	no	—
24	#13083553	—	4	1	no	—
25	C₄×Dic₈₉	—	4	1	no	—
26	C₄×Dic₈₉	D₄	1	1	yes	characteristic
27	#13083549	—	2	1	no	—
28	C₂×Q₈×C₈₉	—	2	1	no	—
29	Q₁₆×C₈₉	—	2	1	no	—
30	Q₁₆×C₈₉	Q₈	1	2	yes	—
31	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₈₉	—	2	1	no	—
33	C₁₆×C₈₉	—	2	1	no	—
34	Q₈×C₈₉	—	2	2	no	—
35	Q₈×C₈₉	—	4	1	no	—
36	C₂×C₄×C₈₉	—	4	1	no	—
37	C₂×C₄×C₈₉	—	4	1	no	—
38	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₈₉	—	4	1	no	—
40	C₂×Dic₈₉	—	4	1	no	—
41	C₂×Dic₈₉	D₈	1	1	yes	characteristic
42	C₂×Dic₈₉	D₈	1	1	yes	characteristic
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	—	4	2	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	—	8	1	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	—	89	1	no	—
64	C₄⋊C₁₆	—	89	1	no	—
65	#358	—	89	1	no	—
66	#358	D₈₉	1	1	yes	characteristic
67	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
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	—	2	1	no	—
79	Q₁₆	—	2	1	no	—
80	Q₁₆	Dic₁₇₈	1	2	yes	—
81	C₂×C₈	—	89	1	no	—
82	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	356	1	no	—
85	C₄⋊C₄	—	356	1	no	—
86	C₄⋊C₄	—	356	1	no	—
87	C₄⋊C₄	—	178	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	—	2	2	no	—
94	Q₈	—	4	1	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	C₂×Dic₁₇₈	1	1	yes	characteristic
104	C₈	#13083573	1	1	yes	characteristic
105	C₂²	D₈×D₈₉	1	1	yes	characteristic
106	C₄	—	4	2	no	—
107	C₄	—	8	1	no	—
108	C₄	—	178	1	no	—
109	C₄	—	178	1	no	—
110	C₄	—	712	1	no	—
111	C₄	—	712	1	no	—
112	C₄	#15667989	1	1	yes	characteristic
113	C₄	#15668041	1	1	yes	characteristic
114	C₂	#19151876	1	1	yes	characteristic
115	C₂	#19152029	1	1	yes	characteristic
116	C₂	#19152040	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151999

C₂

yes

maximal, characteristic

Q₁₆×Dic₈₉

C₂

yes

maximal, characteristic

#19151426

C₂

yes

maximal, characteristic

#19151423

C₂

yes

maximal, characteristic

#19151416

C₂

yes

maximal, characteristic

#19151402

C₂

yes

maximal, characteristic

#358×C₈₉

C₂

yes

maximal, characteristic

#15668003

—

Q₈×Dic₈₉

—

C₂×#13083549

C₂²

yes

characteristic

#15667946

C₂²

yes

characteristic

#15667946

C₂²

yes

characteristic

#15667943

—

#15667936

—

C₈×Dic₈₉

C₂²

yes

characteristic

C₂×Q₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₁₆×C₈₉

C₂²

yes

characteristic

#107×C₈₉

C₂²

yes

characteristic

C₂×#10623494

D₄

yes

characteristic

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

C₄×Dic₈₉

D₄

yes

characteristic

#13083549

—

C₂×Q₈×C₈₉

—

Q₁₆×C₈₉

—

Q₁₆×C₈₉

Q₈

yes

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₁₆×C₈₉

—

Q₈×C₈₉

—

Q₈×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

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

—

C₄⋊C₁₆

—

#358

—

#358

D₈₉

yes

characteristic

C₂×Q₁₆

C₂×D₈₉

yes

characteristic

#125

—

178

—

C₄×Q₈

—

178

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₈₉

yes

—

—

C₂×D₈₉

yes

characteristic

#106

—

178

—

#103

—

178

—

C₄×C₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

Dic₁₇₈

yes

—

C₂×C₈

—

C₂×C₈

C₂²×D₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄²

—

C₄²

—

356

—

C₁₆

—

C₁₆

—

178

—

Q₈

—

Q₈

—

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

C₂×Dic₁₇₈

yes

characteristic

104

C₈

#13083573

yes

characteristic

105

C₂²

D₈×D₈₉

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

178

—

109

C₄

—

178

—

110

C₄

—

712

—

111

C₄

—

712

—

112

C₄

#15667989

yes

characteristic

113

C₄

#15668041

yes

characteristic

114

C₂

#19151876

yes

characteristic

115

C₂

#19152029

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	#358×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₈
hypercenter	#358	D₈₉
hypocenter	C₈₉	#2229
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₈₉

commutator, cosocle

C₂×C₈×C₈₉

C₂³

Frattini

C₂×C₈

C₂²×D₈₉

Fitting

#358×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 ◁ #358

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
Q₁₆×Dic₈₉	2
#19151426	2
#19151423	2
#19151416	2
#19151402	2
#358×C₈₉	2
#2229	89

subgroup

index

Q₁₆×Dic₈₉

#358×C₈₉

Subgroups — #26022414

Conjugacy classes of subgroups

Complements

Maximal subgroups