Group #23749776 — subgroups

Subgroups — #23749776

Order 9536.

188 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 51 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	C₂×#14558724	C₂	1	2	yes	maximal, complemented
2	#18011890	C₂	1	1	yes	maximal, characteristic, complemented
3	#18011832	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#11924746	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₁₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₉₈	C₂²	1	1	yes	characteristic, complemented
7	#14558724	—	2	4	no	—
8	C₂×#11924746	C₂²	1	2	yes	complemented
9	C₄⋊Dic₁₄₉	C₂²	1	1	yes	characteristic, complemented
10	#14558715	—	2	1	no	—
11	C₂²⋊Dic₁₄₉	—	2	1	no	—
12	C₂⁴×C₁₄₉	—	2	1	no	—
13	C₂×D₄×C₁₄₉	—	2	4	no	—
14	C₂×D₄×C₁₄₉	—	2	1	no	—
15	C₂×D₄×C₁₄₉	C₂²	1	2	yes	—
16	C₂²×C₄×C₁₄₉	C₂²	1	1	yes	characteristic
17	Dic₂₉₈	—	2	2	no	—
18	C₂×Dic₁₄₉	—	2	1	no	—
19	C₂×Dic₁₄₉	—	2	1	no	—
20	#11924746	—	2	2	no	—
21	C₂³×C₁₄₉	—	2	4	no	—
22	C₂³×C₁₄₉	—	2	2	no	—
23	C₂³×C₁₄₉	—	4	2	no	—
24	C₂³×C₁₄₉	—	4	2	no	—
25	C₂³×C₁₄₉	D₄	1	1	yes	characteristic
26	D₄×C₁₄₉	—	2	4	no	—
27	D₄×C₁₄₉	—	2	2	no	—
28	D₄×C₁₄₉	D₄	1	4	yes	—
29	C₂×C₄×C₁₄₉	—	2	2	no	—
30	C₂×C₄×C₁₄₉	D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₄₉	C₂³	1	1	yes	characteristic
32	C₂²×C₁₄₉	—	2	1	no	—
33	C₂²×C₁₄₉	—	2	1	no	—
34	C₂²×C₁₄₉	—	2	4	no	—
35	C₂²×C₁₄₉	—	4	8	no	—
36	C₂²×C₁₄₉	—	4	2	no	—
37	C₂²×C₁₄₉	—	4	2	no	—
38	C₂²×C₁₄₉	SD₁₆	1	2	yes	complemented
39	C₂²×C₁₄₉	C₂×D₄	1	1	yes	characteristic
40	C₄×C₁₄₉	—	2	1	no	—
41	C₄×C₁₄₉	C₂×D₄	1	2	yes	—
42	Dic₁₄₉	—	4	1	no	—
43	Dic₁₄₉	—	4	1	no	—
44	C₂×C₁₄₉	—	2	2	no	—
45	C₂×C₁₄₉	—	4	4	no	—
46	C₂×C₁₄₉	#120	1	1	yes	characteristic
47	C₂×C₁₄₉	C₂×SD₁₆	1	1	yes	characteristic
48	C₂×C₁₄₉	AGL(1,ℤ/8)	1	1	yes	characteristic
49	C₁₄₉	#450	1	1	yes	characteristic, complemented, Sylow
50	#450	—	149	1	no	maximal, Sylow
51	C₂²×D₄	D₁₄₉	1	1	yes	characteristic
52	C₂×SD₁₆	—	149	2	no	—
53	C₂²⋊Q₈	—	149	1	no	—
54	#102	—	149	2	no	—
55	C₂²⋊C₈	—	149	1	no	—
56	C₂⁴	—	2	1	no	—
57	C₂×Q₈	—	149	1	no	—
58	C₂×D₄	—	2	4	no	—
59	C₂×D₄	—	2	1	no	—
60	C₂×D₄	C₂×D₁₄₉	1	2	yes	—
61	C₂²×C₄	C₂×D₁₄₉	1	1	yes	characteristic
62	SD₁₆	—	298	4	no	—
63	C₂×C₈	—	149	2	no	—
64	C₄⋊C₄	—	298	1	no	—
65	C₄⋊C₄	—	149	1	no	—
66	C₂²⋊C₄	—	298	1	no	—
67	C₂³	—	2	4	no	—
68	C₂³	—	2	2	no	—
69	C₂³	—	4	2	no	—
70	C₂³	—	4	2	no	—
71	C₂³	C₂²⋊D₁₄₉	1	1	yes	characteristic
72	Q₈	—	298	2	no	—
73	D₄	—	2	4	no	—
74	D₄	—	2	2	no	—
75	D₄	C₂²⋊D₁₄₉	1	4	yes	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	298	1	no	—
78	C₂×C₄	—	298	1	no	—
79	C₂×C₄	C₂²⋊D₁₄₉	1	1	yes	characteristic
80	C₂×C₄	C₂²×D₁₄₉	1	1	yes	characteristic
81	C₈	—	298	2	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	4	8	no	—
86	C₂²	—	4	2	no	—
87	C₂²	—	4	2	no	—
88	C₂²	#14558724	1	2	yes	complemented
89	C₂²	C₂×C₂²⋊D₁₄₉	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	596	1	no	—
92	C₄	—	596	1	no	—
93	C₄	C₂×C₂²⋊D₁₄₉	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	—	4	4	no	—
96	C₂	#18011880	1	1	yes	characteristic
97	C₂	C₂×#14558724	1	1	yes	characteristic
98	C₂	#18011937	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14558724

C₂

yes

maximal, complemented

#18011890

C₂

yes

maximal, characteristic, complemented

#18011832

C₂

yes

maximal, complemented

C₂²⋊#11924746

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₄₉

C₂

yes

maximal, characteristic

C₂×Dic₂₉₈

C₂²

yes

characteristic, complemented

#14558724

—

C₂×#11924746

C₂²

yes

complemented

C₄⋊Dic₁₄₉

C₂²

yes

characteristic, complemented

#14558715

—

C₂²⋊Dic₁₄₉

—

C₂⁴×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

—

C₂×D₄×C₁₄₉

C₂²

yes

—

C₂²×C₄×C₁₄₉

C₂²

yes

characteristic

Dic₂₉₈

—

C₂×Dic₁₄₉

—

C₂×Dic₁₄₉

—

#11924746

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

—

C₂³×C₁₄₉

D₄

yes

characteristic

D₄×C₁₄₉

—

D₄×C₁₄₉

—

D₄×C₁₄₉

D₄

yes

—

C₂×C₄×C₁₄₉

—

C₂×C₄×C₁₄₉

D₄

yes

characteristic

C₂×C₄×C₁₄₉

C₂³

yes

characteristic

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

—

C₂²×C₁₄₉

SD₁₆

yes

complemented

C₂²×C₁₄₉

C₂×D₄

yes

characteristic

C₄×C₁₄₉

—

C₄×C₁₄₉

C₂×D₄

yes

—

Dic₁₄₉

—

Dic₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

—

C₂×C₁₄₉

#120

yes

characteristic

C₂×C₁₄₉

C₂×SD₁₆

yes

C₂×C₁₄₉

yes

C₁₄₉

yes

characteristic, complemented, Sylow

#450

—

149

maximal, Sylow

C₂²×D₄

D₁₄₉

yes

characteristic

C₂×SD₁₆

—

149

—

C₂²⋊Q₈

—

149

—

#102

—

149

—

C₂²⋊C₈

—

149

—

C₂⁴

—

C₂×Q₈

—

149

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₁₄₉

yes

—

C₂²×C₄

C₂×D₁₄₉

yes

characteristic

SD₁₆

—

298

—

C₂×C₈

—

149

—

C₄⋊C₄

—

298

—

C₄⋊C₄

—

149

—

C₂²⋊C₄

—

298

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊D₁₄₉

yes

characteristic

Q₈

—

298

—

D₄

—

D₄

—

D₄

C₂²⋊D₁₄₉

yes

—

C₂×C₄

—

C₂×C₄

—

298

—

C₂×C₄

—

298

—

C₂×C₄

C₂²⋊D₁₄₉

yes

characteristic

C₂×C₄

C₂²×D₁₄₉

yes

characteristic

C₈

—

298

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#14558724

yes

complemented

C₂²

C₂×C₂²⋊D₁₄₉

yes

characteristic

C₄

—

C₄

—

596

—

C₄

—

596

—

C₄

C₂×C₂²⋊D₁₄₉

yes

—

C₂

—

C₂

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₄₉
commutator, cosocle	C₂×C₄×C₁₄₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₄₉
Fitting, prime radical	C₂²×D₄×C₁₄₉	C₂
socle	C₂²×C₁₄₉	C₂×D₄
hypercenter	C₂²×D₄	D₁₄₉
hypocenter	C₁₄₉	#450
LCS	G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₄₉

commutator, cosocle

C₂×C₄×C₁₄₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₄₉

Fitting, prime radical

C₂²×D₄×C₁₄₉

C₂

socle

C₂²×C₁₄₉

C₂×D₄

hypercenter

C₂²×D₄

D₁₄₉

hypocenter

C₁₄₉

#450

LCS

G ▷ C₂×C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂×#14558724	C₂	semidirect
2. #18011890	C₂	semidirect
3. #18011832	C₂	semidirect
4. C₂²⋊#11924746	C₂	semidirect
6. C₂×Dic₂₉₈	C₂²	semidirect
8. C₂×#11924746	C₂²	semidirect
9. C₄⋊Dic₁₄₉	C₂²	semidirect
38. C₂²×C₁₄₉	SD₁₆	semidirect
49. C₁₄₉	#450	semidirect
88. C₂²	#14558724	semidirect

H ≅ G/N

type

1. C₂×#14558724

C₂

semidirect

2. #18011890

C₂

semidirect

3. #18011832

C₂

semidirect

4. C₂²⋊#11924746

C₂

semidirect

6. C₂×Dic₂₉₈

C₂²

semidirect

8. C₂×#11924746

C₂²

semidirect

9. C₄⋊Dic₁₄₉

C₂²

semidirect

38. C₂²×C₁₄₉

SD₁₆

semidirect

49. C₁₄₉

#450

semidirect

88. C₂²

#14558724

semidirect

Subgroups — #23749776

Conjugacy classes of subgroups

Complements