Group #26027960 — subgroups

Subgroups — #26027960

Order 11392.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19151515	C₂	1	3	yes	maximal
2	#19151509	C₂	1	2	yes	maximal, complemented
3	#19151490	C₂	1	6	yes	maximal, complemented
4	#19151463	C₂	1	3	yes	maximal, complemented
5	#395×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₈₉	C₂²	1	3	yes	—
8	C₂×#13083551	C₂²	1	6	yes	—
9	C₂×#13083551	C₂²	1	6	yes	—
10	C₂×C₄×Dic₈₉	C₂²	1	3	yes	—
11	#15667919	C₂²	1	3	yes	—
12	#15667919	C₂²	1	6	yes	—
13	C₂²×Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
15	#95×C₈₉	C₂²	1	3	yes	—
16	C₂³×D₈₉	C₂³	1	2	yes	—
17	C₂×D₃₅₆	—	4	3	no	—
18	C₂²×Dic₈₉	C₂³	1	6	yes	—
19	C₄⋊Dic₈₉	—	4	3	no	—
20	#13083551	—	2	12	no	—
21	#13083551	—	4	6	no	—
22	C₄×Dic₈₉	—	2	6	no	—
23	C₂×Q₈×C₈₉	—	4	3	no	—
24	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
26	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
27	C₄⋊C₄×C₈₉	—	4	3	no	—
28	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
29	C₂²×D₈₉	—	2	6	no	—
30	C₂²×D₈₉	—	2	6	no	—
31	C₂²×D₈₉	—	2	2	no	—
32	Q₈×C₈₉	—	4	4	no	—
33	C₂×C₄×C₈₉	—	2	3	no	—
34	C₂×C₄×C₈₉	—	2	3	no	—
35	C₂×C₄×C₈₉	—	2	3	no	—
36	C₂×C₄×C₈₉	—	2	3	no	—
37	C₂×C₄×C₈₉	—	2	6	no	—
38	C₂×C₄×C₈₉	#41	1	6	yes	—
39	C₂×Dic₈₉	—	2	6	no	—
40	C₂×Dic₈₉	—	2	6	no	—
41	C₂×Dic₈₉	—	2	6	no	—
42	D₃₅₆	—	4	4	no	—
43	C₂²×C₈₉	C₂×#41	1	3	yes	—
44	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
45	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
46	C₂×D₈₉	—	4	6	no	—
47	C₂×D₈₉	—	4	6	no	—
48	C₂×D₈₉	—	4	2	no	—
49	C₄×C₈₉	—	2	6	no	—
50	C₄×C₈₉	—	4	1	no	—
51	C₄×C₈₉	—	4	3	no	—
52	Dic₈₉	—	4	6	no	—
53	C₂×C₈₉	#539	1	3	yes	—
54	C₂×C₈₉	#555	1	3	yes	—
55	C₂×C₈₉	H(3,𝔽₄)	1	1	yes	characteristic
56	D₈₉	—	8	2	no	—
57	#2785	—	89	1	no	maximal, Sylow
58	C₈₉	#2785	1	1	yes	characteristic, complemented, Sylow
59	#395	D₈₉	1	1	yes	characteristic, complemented
60	#394	—	89	3	no	—
61	#393	—	89	2	no	—
62	#388	—	89	6	no	—
63	#382	—	89	3	no	—
64	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
65	C₂²×D₄	—	89	1	no	—
66	C₂×C₄⋊C₄	—	89	3	no	—
67	C₂×C₄⋊C₄	C₂×D₈₉	1	3	yes	—
68	C₂×C₂²⋊C₄	—	89	6	no	—
69	C₂×C₂²⋊C₄	—	89	6	no	—
70	C₂×C₄²	—	89	3	no	—
71	#95	—	89	3	no	—
72	#95	—	89	6	no	—
73	#95	C₂×D₈₉	1	3	yes	—
74	C₂⁴	—	89	2	no	—
75	C₂×Q₈	—	4	3	no	—
76	C₂×D₄	—	356	3	no	—
77	C₂²×C₄	—	89	6	no	—
78	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
80	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
81	C₄⋊C₄	—	4	3	no	—
82	C₄⋊C₄	—	356	3	no	—
83	C₂²⋊C₄	—	356	6	no	—
84	C₂²⋊C₄	—	178	12	no	—
85	C₄²	—	178	6	no	—
86	C₂³	—	178	6	no	—
87	C₂³	—	178	6	no	—
88	C₂³	—	178	2	no	—
89	C₂³	C₂³×D₈₉	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	D₄	—	356	4	no	—
92	C₂×C₄	—	2	3	no	—
93	C₂×C₄	—	2	3	no	—
94	C₂×C₄	—	2	6	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	3	no	—
97	C₂×C₄	—	178	6	no	—
98	C₂×C₄	—	178	6	no	—
99	C₂×C₄	—	178	6	no	—
100	C₂×C₄	#13083575	1	6	yes	—
101	C₂²	—	356	6	no	—
102	C₂²	—	356	6	no	—
103	C₂²	—	356	2	no	—
104	C₂²	C₂×#13083575	1	3	yes	—
105	C₂²	#15668057	1	3	yes	—
106	C₂²	#15668059	1	1	yes	characteristic
107	C₄	—	2	6	no	—
108	C₄	—	4	1	no	—
109	C₄	—	4	3	no	—
110	C₄	—	356	6	no	—
111	C₂	—	712	2	no	—
112	C₂	#19152197	1	3	yes	—
113	C₂	#19152314	1	3	yes	—
114	C₂	#19152341	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151515

C₂

yes

maximal

#19151509

C₂

yes

maximal, complemented

#19151490

C₂

yes

maximal, complemented

#19151463

C₂

yes

maximal, complemented

#395×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂²×Q₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

—

C₂³×D₈₉

C₂³

yes

—

C₂×D₃₅₆

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

#13083551

—

#13083551

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₄⋊C₄×C₈₉

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

C₂×C₈₉

#539

yes

—

C₂×C₈₉

#555

yes

—

C₂×C₈₉

H(3,𝔽₄)

yes

characteristic

D₈₉

—

#2785

—

maximal, Sylow

C₈₉

#2785

yes

characteristic, complemented, Sylow

#395

D₈₉

yes

characteristic, complemented

—

—

—

—

C₂²×Q₈

C₂×D₈₉

yes

characteristic

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

C₂×D₈₉

yes

—

C₂⁴

—

C₂×Q₈

—

C₂×D₄

—

356

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₂²⋊C₄

—

356

—

C₂²⋊C₄

—

178

—

C₄²

—

178

—

C₂³

—

178

—

C₂³

—

178

—

C₂³

—

178

—

C₂³

C₂³×D₈₉

yes

characteristic

Q₈

—

D₄

—

356

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

100

C₂×C₄

#13083575

yes

—

101

C₂²

—

356

—

102

C₂²

—

356

—

103

C₂²

—

356

—

104

C₂²

C₂×#13083575

yes

—

105

C₂²

#15668057

yes

—

106

C₂²

#15668059

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

356

—

111

C₂

—

712

—

112

C₂

#19152197

yes

—

113

C₂

#19152314

yes

—

114

C₂

#19152341

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#395×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	#395	D₈₉
hypocenter	C₈₉	#2785
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#395×C₈₉

C₂

prime radical

C₂²×D₃₅₆

C₂²

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
2. #19151509	C₂	semidirect
3. #19151490	C₂	semidirect
4. #19151463	C₂	semidirect
5. #395×C₈₉	C₂	semidirect
58. C₈₉	#2785	semidirect
59. #395	D₈₉	semidirect

H ≅ G/N

type

2. #19151509

C₂

semidirect

3. #19151490

C₂

semidirect

4. #19151463

C₂

semidirect

5. #395×C₈₉

C₂

semidirect

58. C₈₉

#2785

semidirect

59. #395

D₈₉

semidirect

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

subgroup	index
#19151515	2
#19151509	2
#19151490	2
#19151463	2
#395×C₈₉	2
#2785	89

subgroup

index

#395×C₈₉

Subgroups — #26027960

Conjugacy classes of subgroups

Complements

Maximal subgroups