Group #19152347 — subgroups

Subgroups — #19152347

Order 5696.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#15668010	C₂	1	1	yes	maximal, characteristic, complemented
2	#15668007	C₂	1	2	yes	maximal, complemented
3	#15668006	C₂	1	2	yes	maximal, complemented
4	#15668002	C₂	1	2	yes	maximal, complemented
5	#15668001	C₂	1	1	yes	maximal, characteristic
6	#15667997	C₂	1	2	yes	maximal
7	#15667990	C₂	1	2	yes	maximal, complemented
8	#15667988	C₂	1	2	yes	maximal, complemented
9	C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₁₇₈	C₂²	1	1	yes	characteristic
11	C₂×D₃₅₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₈₉	C₂²	1	2	yes	—
13	C₄⋊Dic₈₉	C₂²	1	2	yes	—
14	#13083553	C₂²	1	2	yes	—
15	#13083553	C₂²	1	2	yes	—
16	#13083553	C₂²	1	2	yes	—
17	#13083553	C₂²	1	2	yes	—
18	#13083553	C₂²	1	2	yes	—
19	#13083551	C₂²	1	2	yes	—
20	#13083551	C₂²	1	2	yes	—
21	#13083551	C₂²	1	2	yes	—
22	#13083551	C₂²	1	2	yes	—
23	#13083551	C₂²	1	2	yes	—
24	C₄×Dic₈₉	C₂²	1	1	yes	characteristic
25	C₄×Dic₈₉	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
29	C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
30	C₄²×C₈₉	C₂²	1	1	yes	characteristic
31	C₂²×D₈₉	C₂³	1	2	yes	—
32	Q₈×C₈₉	—	4	1	no	—
33	Q₈×C₈₉	—	4	1	no	—
34	C₂×C₄×C₈₉	C₂³	1	2	yes	—
35	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂³	1	2	yes	—
39	C₂×Dic₈₉	C₂³	1	2	yes	—
40	C₂×Dic₈₉	C₂³	1	2	yes	—
41	C₂×Dic₈₉	C₂³	1	2	yes	—
42	D₃₅₆	—	4	1	no	—
43	C₄×D₈₉	—	4	2	no	—
44	Dic₁₇₈	—	4	1	no	—
45	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
46	C₂×D₈₉	—	2	2	no	—
47	C₂×D₈₉	—	2	4	no	—
48	C₄×C₈₉	—	2	1	no	—
49	C₄×C₈₉	—	2	2	no	—
50	C₄×C₈₉	—	2	1	no	—
51	C₄×C₈₉	—	2	1	no	—
52	C₄×C₈₉	—	2	2	no	—
53	Dic₈₉	—	2	2	no	—
54	Dic₈₉	—	2	2	no	—
55	Dic₈₉	—	2	2	no	—
56	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₈₉	2₋¹⁺⁴	1	2	yes	—
58	D₈₉	—	4	2	no	—
59	C₈₉	#563	1	1	yes	characteristic, complemented, Sylow
60	#563	—	89	1	no	maximal, Sylow
61	C₄⋊Q₈	—	89	1	no	—
62	C₄⋊Q₈	D₈₉	1	1	yes	characteristic, complemented
63	#126	—	89	2	no	—
64	#126	—	89	2	no	—
65	#125	—	89	2	no	—
66	#124	—	89	1	no	—
67	#123	—	89	2	no	—
68	C₂²⋊Q₈	—	89	2	no	—
69	C₂²⋊Q₈	—	89	2	no	—
70	C₂×Q₈	—	89	1	no	—
71	C₂×Q₈	C₂×D₈₉	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₈₉	1	1	yes	characteristic
73	C₂×D₄	—	89	1	no	—
74	C₂²×C₄	—	89	2	no	—
75	C₄⋊C₄	—	89	2	no	—
76	C₄⋊C₄	—	89	2	no	—
77	C₄⋊C₄	—	89	2	no	—
78	C₄⋊C₄	—	89	2	no	—
79	C₄⋊C₄	—	89	2	no	—
80	C₄⋊C₄	—	89	2	no	—
81	C₄⋊C₄	C₂×D₈₉	1	2	yes	—
82	C₄⋊C₄	C₂×D₈₉	1	2	yes	—
83	C₂²⋊C₄	—	89	2	no	—
84	C₂²⋊C₄	—	89	2	no	—
85	C₂²⋊C₄	—	89	2	no	—
86	C₂²⋊C₄	—	89	2	no	—
87	C₂²⋊C₄	—	89	2	no	—
88	C₄²	—	89	1	no	—
89	C₄²	—	89	1	no	—
90	C₄²	C₂×D₈₉	1	1	yes	characteristic
91	C₂³	—	89	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	356	1	no	—
95	D₄	—	356	1	no	—
96	C₂×C₄	—	89	2	no	—
97	C₂×C₄	—	89	2	no	—
98	C₂×C₄	—	89	2	no	—
99	C₂×C₄	—	356	2	no	—
100	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₈₉	1	2	yes	—
103	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₈₉	1	2	yes	—
105	C₂²	—	178	2	no	—
106	C₂²	—	178	4	no	—
107	C₂²	C₂³×D₈₉	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	178	2	no	—
114	C₄	—	178	2	no	—
115	C₄	—	178	2	no	—
116	C₂	—	356	2	no	—
117	C₂	#15668057	1	1	yes	characteristic
118	C₂	#15668060	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15668010

C₂

yes

maximal, characteristic, complemented

#15668007

C₂

yes

maximal, complemented

#15668006

C₂

yes

maximal, complemented

#15668002

C₂

yes

maximal, complemented

#15668001

C₂

yes

maximal, characteristic

#15667997

C₂

yes

maximal

#15667990

C₂

yes

maximal, complemented

#15667988

C₂

yes

maximal, complemented

C₄⋊Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₇₈

C₂²

yes

characteristic

C₂×D₃₅₆

C₂²

yes

characteristic

C₂×C₄×D₈₉

C₂²

yes

—

C₄⋊Dic₈₉

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

C₄×Dic₈₉

C₂²

yes

characteristic

C₄×Dic₈₉

C₂²

yes

characteristic

C₂×Q₈×C₈₉

C₂²

yes

characteristic

C₂×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

—

C₄⋊C₄×C₈₉

C₂²

yes

—

C₄²×C₈₉

C₂²

yes

characteristic

C₂²×D₈₉

C₂³

yes

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

D₃₅₆

—

C₄×D₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

C₂⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂×C₈₉

2₋¹⁺⁴

yes

—

D₈₉

—

C₈₉

#563

yes

characteristic, complemented, Sylow

#563

—

maximal, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

D₈₉

yes

characteristic, complemented

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₈₉

yes

characteristic

C₂×Q₈

C₂×D₈₉

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₈₉

yes

—

C₄⋊C₄

C₂×D₈₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₈₉

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

356

—

D₄

—

356

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

356

—

100

C₂×C₄

C₂²×D₈₉

yes

characteristic

101

C₂×C₄

C₂²×D₈₉

yes

characteristic

102

C₂×C₄

C₂²×D₈₉

yes

—

103

C₂×C₄

C₂²×D₈₉

yes

characteristic

104

C₂×C₄

C₂²×D₈₉

yes

—

105

C₂²

—

178

—

106

C₂²

—

178

—

107

C₂²

C₂³×D₈₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

178

—

114

C₄

—

178

—

115

C₄

—

178

—

116

C₂

—

356

—

117

C₂

#15668057

yes

characteristic

118

C₂

#15668060

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	C₄⋊Q₈×C₈₉	C₂
hypercenter	C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#563
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂⁴

Fitting

C₄⋊Q₈×C₈₉

C₂

hypercenter

C₄⋊Q₈

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #15668010	C₂	semidirect
2. #15668007	C₂	semidirect
3. #15668006	C₂	semidirect
4. #15668002	C₂	semidirect
7. #15667990	C₂	semidirect
8. #15667988	C₂	semidirect
9. C₄⋊Q₈×C₈₉	C₂	semidirect
59. C₈₉	#563	semidirect
62. C₄⋊Q₈	D₈₉	semidirect

H ≅ G/N

type

1. #15668010

C₂

semidirect

2. #15668007

C₂

semidirect

3. #15668006

C₂

semidirect

4. #15668002

C₂

semidirect

7. #15667990

C₂

semidirect

8. #15667988

C₂

semidirect

9. C₄⋊Q₈×C₈₉

C₂

semidirect

59. C₈₉

#563

semidirect

62. C₄⋊Q₈

D₈₉

semidirect

Subgroups — #19152347

Conjugacy classes of subgroups

Complements