Group #26028244 — subgroups

Subgroups — #26028244

Order 11392.

432 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 149 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	#19151518	C₂	1	3	yes	maximal
2	#19151517	C₂	1	2	yes	maximal
3	#19151503	C₂	1	3	yes	maximal
4	#19151502	C₂	1	6	yes	maximal
5	#395×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₇₈	C₂²	1	3	yes	—
8	C₂×#13083553	C₂²	1	6	yes	—
9	C₂×C₄×Dic₈₉	C₂²	1	3	yes	—
10	#15667919	C₂²	1	6	yes	—
11	#15667919	C₂²	1	6	yes	—
12	#15667919	C₂²	1	3	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	C₂×Dic₁₇₈	—	4	3	no	—
17	C₂×Dic₁₇₈	—	4	3	no	—
18	C₂×Dic₁₇₈	—	4	3	no	—
19	C₂×Dic₁₇₈	Q₈	1	12	yes	complemented
20	C₂²×Dic₈₉	C₂³	1	2	yes	—
21	C₂²×Dic₈₉	C₂³	1	6	yes	—
22	#13083553	—	4	6	no	—
23	C₄×Dic₈₉	—	4	3	no	—
24	C₂×Q₈×C₈₉	—	4	3	no	—
25	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
27	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
28	C₄⋊C₄×C₈₉	—	4	3	no	—
29	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
30	Q₈×C₈₉	—	4	4	no	—
31	C₂×C₄×C₈₉	—	2	3	no	—
32	C₂×C₄×C₈₉	—	2	3	no	—
33	C₂×C₄×C₈₉	—	2	3	no	—
34	C₂×C₄×C₈₉	—	2	3	no	—
35	C₂×C₄×C₈₉	—	2	6	no	—
36	C₂×C₄×C₈₉	C₂×Q₈	1	6	yes	—
37	C₂×Dic₈₉	—	2	6	no	—
38	C₂×Dic₈₉	—	2	6	no	—
39	C₂×Dic₈₉	—	2	6	no	—
40	C₂×Dic₈₉	C₂×Q₈	1	12	yes	—
41	Dic₁₇₈	—	4	12	no	—
42	Dic₁₇₈	—	4	4	no	—
43	C₂²×C₈₉	C₂²×Q₈	1	3	yes	—
44	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
45	C₂²×C₈₉	H(5,𝔽₂)	1	3	yes	—
46	C₄×C₈₉	—	2	6	no	—
47	C₄×C₈₉	—	4	1	no	—
48	C₄×C₈₉	—	4	3	no	—
49	Dic₈₉	—	2	12	no	—
50	Dic₈₉	—	4	2	no	—
51	C₂×C₈₉	#543	1	3	yes	—
52	C₂×C₈₉	Q₈²	1	3	yes	—
53	C₂×C₈₉	H(3,𝔽₄)	1	1	yes	characteristic
54	#2811	—	89	1	no	maximal, Sylow
55	C₈₉	#2811	1	1	yes	characteristic, complemented, Sylow
56	#395	—	89	2	no	—
57	#395	—	89	3	no	—
58	#395	D₈₉	1	1	yes	characteristic
59	#391	—	89	3	no	—
60	#391	—	89	6	no	—
61	C₂²×Q₈	—	89	3	no	—
62	C₂²×Q₈	—	89	1	no	—
63	C₂²×Q₈	C₂×D₈₉	1	1	yes	characteristic
64	C₂×C₄⋊C₄	—	89	6	no	—
65	C₂×C₄⋊C₄	C₂×D₈₉	1	3	yes	—
66	C₂×C₄²	—	89	3	no	—
67	#95	—	89	3	no	—
68	#95	—	89	6	no	—
69	#95	—	89	6	no	—
70	#95	C₂×D₈₉	1	3	yes	—
71	C₂×Q₈	—	4	3	no	—
72	C₂×Q₈	—	89	12	no	—
73	C₂×Q₈	—	356	3	no	—
74	C₂×Q₈	—	356	3	no	—
75	C₂×Q₈	—	356	3	no	—
76	C₂²×C₄	—	89	6	no	—
77	C₂²×C₄	—	89	2	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	6	no	—
83	C₄²	—	356	3	no	—
84	C₂³	C₂³×D₈₉	1	1	yes	characteristic
85	Q₈	—	4	4	no	—
86	Q₈	—	356	12	no	—
87	Q₈	—	356	4	no	—
88	C₂×C₄	—	2	3	no	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	2	3	no	—
91	C₂×C₄	—	2	3	no	—
92	C₂×C₄	—	2	6	no	—
93	C₂×C₄	—	89	12	no	—
94	C₂×C₄	—	178	6	no	—
95	C₂×C₄	—	178	6	no	—
96	C₂×C₄	—	178	6	no	—
97	C₂×C₄	Q₈×D₈₉	1	6	yes	—
98	C₂²	C₂×Q₈×D₈₉	1	3	yes	—
99	C₂²	#15668057	1	3	yes	—
100	C₂²	#15668058	1	1	yes	characteristic
101	C₄	—	2	6	no	—
102	C₄	—	4	3	no	—
103	C₄	—	4	1	no	—
104	C₄	—	356	2	no	—
105	C₄	—	178	12	no	—
106	C₂	#19152223	1	3	yes	—
107	C₂	#19152328	1	3	yes	—
108	C₂	#19152341	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151518

C₂

yes

maximal

#19151517

C₂

yes

maximal

#19151503

C₂

yes

maximal

#19151502

C₂

yes

maximal

#395×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

C₂²×Dic₁₇₈

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂²×Q₈×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

#95×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

Q₈

yes

complemented

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

#13083553

—

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

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

C₂×Q₈

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×Q₈

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#543

yes

—

C₂×C₈₉

Q₈²

yes

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#395

—

#395

—

#395

D₈₉

yes

characteristic

#391

—

#391

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

C₂×C₄²

—

—

—

—

C₂×D₈₉

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

C₂×Q₈

—

356

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂²×C₄

C₂²×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

356

—

C₄²

—

356

—

C₂³

C₂³×D₈₉

yes

characteristic

Q₈

—

Q₈

—

356

—

Q₈

—

356

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

Q₈×D₈₉

yes

—

C₂²

C₂×Q₈×D₈₉

yes

—

C₂²

#15668057

yes

—

100

C₂²

#15668058

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

356

—

105

C₄

—

178

—

106

C₂

#19152223

yes

—

107

C₂

#19152328

yes

—

108

C₂

#19152341

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#395×C₈₉	C₂
hypercenter	#395	D₈₉
hypocenter	C₈₉	#2811
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #395

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

#395×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ #395

N	H ≅ G/N	type
19. C₂×Dic₁₇₈	Q₈	semidirect
55. C₈₉	#2811	semidirect

H ≅ G/N

type

19. C₂×Dic₁₇₈

Q₈

semidirect

55. C₈₉

#2811

semidirect

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

subgroup	index
#19151518	2
#19151517	2
#19151503	2
#19151502	2
#395×C₈₉	2
#2811	89

subgroup

index

#395×C₈₉

Subgroups — #26028244

Conjugacy classes of subgroups

Complements

Maximal subgroups