Group #19152308 — subgroups

Subgroups — #19152308

Order 5696.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#15668012	C₂	1	1	yes	maximal, characteristic, complemented
2	#15668004	C₂	1	2	yes	maximal, complemented
3	#15667990	C₂	1	4	yes	maximal, complemented
4	#15667988	C₂	1	2	yes	maximal, complemented
5	C₄×D₃₅₆	C₂	1	2	yes	maximal
6	C₄×D₃₅₆	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×D₈₉	C₂	1	2	yes	maximal, complemented
8	C₄×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₁₇₈	C₂²	1	2	yes	—
10	C₂×D₃₅₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₈₉	C₂²	1	2	yes	—
12	C₂×C₄×D₈₉	C₂²	1	4	yes	—
13	C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₈₉	C₂²	1	2	yes	—
15	C₄⋊Dic₈₉	C₂²	1	2	yes	—
16	#13083553	C₂²	1	4	yes	—
17	#13083553	C₂²	1	4	yes	—
18	#13083551	C₂²	1	4	yes	—
19	#13083551	C₂²	1	2	yes	—
20	C₂×Q₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
22	C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
23	C₄²×C₈₉	C₂²	1	2	yes	—
24	C₄²×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×D₈₉	C₂³	1	2	yes	—
26	Q₈×C₈₉	—	2	2	no	—
27	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₈₉	C₂³	1	2	yes	—
29	C₂×C₄×C₈₉	C₂³	1	2	yes	—
30	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₈₉	C₂³	1	1	yes	characteristic
32	C₂×Dic₈₉	C₂³	1	2	yes	—
33	C₂×Dic₈₉	C₂³	1	4	yes	—
34	D₃₅₆	Q₈	1	4	yes	complemented
35	C₄×D₈₉	—	2	4	no	—
36	C₄×D₈₉	—	4	4	no	—
37	Dic₁₇₈	—	4	2	no	—
38	C₂²×C₈₉	C₂⁴	1	1	yes	characteristic
39	C₂×D₈₉	—	2	2	no	—
40	C₂×D₈₉	—	2	2	no	—
41	C₂×D₈₉	C₂×Q₈	1	4	yes	—
42	C₄×C₈₉	—	2	2	no	—
43	C₄×C₈₉	—	2	1	no	—
44	C₄×C₈₉	—	2	2	no	—
45	C₄×C₈₉	C₂×Q₈	1	2	yes	—
46	C₄×C₈₉	#41	1	2	yes	—
47	Dic₈₉	—	2	2	no	—
48	Dic₈₉	—	2	4	no	—
49	C₂×C₈₉	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₈₉	C₂×#41	1	1	yes	characteristic
51	C₂×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
52	D₈₉	—	2	4	no	—
53	C₈₉	#554	1	1	yes	characteristic, complemented, Sylow
54	#554	—	89	1	no	maximal, Sylow
55	C₄⋊Q₈	—	89	1	no	—
56	#125	—	89	2	no	—
57	C₂²⋊Q₈	—	89	2	no	—
58	C₂²⋊Q₈	—	89	4	no	—
59	C₄×Q₈	D₈₉	1	1	yes	characteristic, complemented
60	C₄×D₄	—	89	2	no	—
61	C₄×D₄	—	89	1	no	—
62	C₂×C₄⋊C₄	—	89	2	no	—
63	C₂×Q₈	—	89	2	no	—
64	C₂×Q₈	C₂×D₈₉	1	1	yes	characteristic
65	C₂×D₄	—	89	1	no	—
66	C₂²×C₄	—	89	4	no	—
67	C₂²×C₄	—	89	2	no	—
68	C₄⋊C₄	—	89	4	no	—
69	C₄⋊C₄	—	89	2	no	—
70	C₄⋊C₄	—	89	1	no	—
71	C₄⋊C₄	—	89	2	no	—
72	C₄⋊C₄	—	89	4	no	—
73	C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
74	C₄⋊C₄	C₂×D₈₉	1	2	yes	—
75	C₂²⋊C₄	—	89	2	no	—
76	C₂²⋊C₄	—	89	4	no	—
77	C₄²	C₂×D₈₉	1	2	yes	—
78	C₄²	C₂×D₈₉	1	1	yes	characteristic
79	C₂³	—	89	2	no	—
80	Q₈	—	2	2	no	—
81	Q₈	—	356	2	no	—
82	D₄	—	89	4	no	—
83	C₂×C₄	—	89	2	no	—
84	C₂×C₄	—	89	4	no	—
85	C₂×C₄	—	356	4	no	—
86	C₂×C₄	—	178	4	no	—
87	C₂×C₄	C₂²×D₈₉	1	2	yes	—
88	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₈₉	1	2	yes	—
90	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
91	C₂×C₄	C₂²×D₈₉	1	1	yes	characteristic
92	C₂²	—	89	4	no	—
93	C₂²	—	178	2	no	—
94	C₂²	—	178	2	no	—
95	C₂²	C₂³×D₈₉	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	2	2	no	—
98	C₄	—	2	1	no	—
99	C₄	—	178	2	no	—
100	C₄	—	178	4	no	—
101	C₄	Q₈×D₈₉	1	2	yes	—
102	C₄	#13083574	1	2	yes	—
103	C₂	—	178	4	no	—
104	C₂	C₂×Q₈×D₈₉	1	1	yes	characteristic
105	C₂	C₂×#13083574	1	1	yes	characteristic
106	C₂	#15668058	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15668012

C₂

yes

maximal, characteristic, complemented

#15668004

C₂

yes

maximal, complemented

#15667990

C₂

yes

maximal, complemented

#15667988

C₂

yes

maximal, complemented

C₄×D₃₅₆

C₂

yes

maximal

C₄×D₃₅₆

C₂

yes

maximal, characteristic

C₄⋊C₄×D₈₉

C₂

yes

maximal, complemented

C₄×Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₇₈

C₂²

yes

—

C₂×D₃₅₆

C₂²

yes

characteristic

C₂×C₄×D₈₉

C₂²

yes

—

C₂×C₄×D₈₉

C₂²

yes

—

C₄⋊Dic₈₉

C₂²

yes

characteristic

C₄⋊Dic₈₉

C₂²

yes

—

C₄⋊Dic₈₉

C₂²

yes

—

#13083553

C₂²

yes

—

#13083553

C₂²

yes

—

#13083551

C₂²

yes

—

#13083551

C₂²

yes

—

C₂×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₄×C₈₉

C₂²

yes

—

C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₄²×C₈₉

C₂²

yes

—

C₄²×C₈₉

C₂²

yes

characteristic

C₂²×D₈₉

C₂³

yes

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

—

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₄×C₈₉

C₂³

yes

characteristic

C₂×Dic₈₉

C₂³

yes

—

C₂×Dic₈₉

C₂³

yes

—

D₃₅₆

Q₈

yes

complemented

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

C₂²×C₈₉

C₂⁴

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

C₂×Q₈

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×Q₈

yes

—

C₄×C₈₉

#41

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂²×Q₈

yes

characteristic

C₂×C₈₉

C₂×#41

yes

characteristic

C₂×C₈₉

H(5,𝔽₂)

yes

characteristic

D₈₉

—

C₈₉

#554

yes

characteristic, complemented, Sylow

#554

—

maximal, Sylow

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

D₈₉

yes

characteristic, complemented

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×Q₈

C₂×D₈₉

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₄⋊C₄

C₂×D₈₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×D₈₉

yes

—

C₄²

C₂×D₈₉

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

356

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

C₂²×D₈₉

yes

—

C₂×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₄

C₂²×D₈₉

yes

—

C₂×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₄

C₂²×D₈₉

yes

characteristic

C₂²

—

C₂²

—

178

—

C₂²

—

178

—

C₂²

C₂³×D₈₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

178

—

100

C₄

—

178

—

101

C₄

Q₈×D₈₉

yes

—

102

C₄

#13083574

yes

—

103

C₂

—

178

—

104

C₂

C₂×Q₈×D₈₉

yes

characteristic

105

C₂

C₂×#13083574

yes

characteristic

106

C₂

#15668058

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂⁴
Fitting	C₄×Q₈×C₈₉	C₂
hypercenter	C₄×Q₈	D₈₉
hypocenter	C₈₉	#554
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. #15668012	C₂	semidirect
2. #15668004	C₂	semidirect
3. #15667990	C₂	semidirect
4. #15667988	C₂	semidirect
7. C₄⋊C₄×D₈₉	C₂	semidirect
8. C₄×Q₈×C₈₉	C₂	semidirect
34. D₃₅₆	Q₈	semidirect
53. C₈₉	#554	semidirect
59. C₄×Q₈	D₈₉	semidirect

H ≅ G/N

type

1. #15668012

C₂

semidirect

2. #15668004

C₂

semidirect

3. #15667990

C₂

semidirect

4. #15667988

C₂

semidirect

7. C₄⋊C₄×D₈₉

C₂

semidirect

8. C₄×Q₈×C₈₉

C₂

semidirect

34. D₃₅₆

Q₈

semidirect

53. C₈₉

#554

semidirect

59. C₄×Q₈

D₈₉

semidirect

Subgroups — #19152308

Conjugacy classes of subgroups

Complements