Subgroups — C₂³×#15668424 #37546814

Order 23168.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15668424	C₂	1	14	yes	maximal, direct product factor
2	C₂³×C₈×C₁₈₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#15668424	C₂²	1	28	yes	direct product factor
4	C₂³×C₄×C₁₈₁	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₁₈₁	C₄	1	7	yes	—
6	C₂²×C₈×C₁₈₁	C₂²	1	7	yes	—
7	#15668424	C₂³	1	8	yes	direct product factor
8	C₂⁴×C₁₈₁	C₈	1	1	yes	characteristic
9	C₂²×C₄×C₁₈₁	C₈	1	7	yes	—
10	C₂²×C₄×C₁₈₁	C₂×C₄	1	7	yes	—
11	C₂×C₈×C₁₈₁	C₂×C₄	1	21	yes	—
12	C₂×C₈×C₁₈₁	C₂³	1	7	yes	—
13	C₂³×C₁₈₁	C₁₆	1	8	yes	complemented
14	C₂³×C₁₈₁	C₂×C₈	1	7	yes	—
15	C₂×C₄×C₁₈₁	C₂×C₈	1	21	yes	—
16	C₂×C₄×C₁₈₁	C₂²×C₄	1	7	yes	—
17	C₈×C₁₈₁	C₂²×C₄	1	7	yes	—
18	C₈×C₁₈₁	C₂⁴	1	1	yes	characteristic
19	C₂²×C₁₈₁	C₂×C₁₆	1	28	yes	complemented
20	C₂²×C₁₈₁	C₂²×C₈	1	7	yes	—
21	C₄×C₁₈₁	C₂²×C₈	1	7	yes	—
22	C₄×C₁₈₁	C₂³×C₄	1	1	yes	characteristic
23	C₂×C₁₈₁	C₂²×C₁₆	1	14	yes	complemented
24	C₂×C₁₈₁	C₂³×C₈	1	1	yes	characteristic
25	C₁₈₁	C₂³×C₁₆	1	1	yes	characteristic, complemented, Sylow
26	C₂³×C₁₆	—	181	1	no	maximal, Sylow
27	C₂³×C₈	D₁₈₁	1	1	yes	characteristic
28	C₂²×C₁₆	—	181	14	no	—
29	C₂³×C₄	Dic₁₈₁	1	1	yes	characteristic
30	C₂²×C₈	Dic₁₈₁	1	7	yes	—
31	C₂²×C₈	C₂×D₁₈₁	1	7	yes	—
32	C₂×C₁₆	—	181	28	no	—
33	C₂⁴	#13089678	1	1	yes	characteristic
34	C₂²×C₄	#13089678	1	7	yes	—
35	C₂²×C₄	C₂×Dic₁₈₁	1	7	yes	—
36	C₂×C₈	C₂×Dic₁₈₁	1	21	yes	—
37	C₂×C₈	C₂²×D₁₈₁	1	7	yes	—
38	C₁₆	—	181	8	no	—
39	C₂³	#15668424	1	8	yes	direct product factor
40	C₂³	C₂×#13089678	1	7	yes	—
41	C₂×C₄	C₂×#13089678	1	21	yes	—
42	C₂×C₄	C₂²×Dic₁₈₁	1	7	yes	—
43	C₈	C₂²×Dic₁₈₁	1	7	yes	—
44	C₈	C₂³×D₁₈₁	1	1	yes	characteristic
45	C₂²	C₂×#15668424	1	28	yes	direct product factor
46	C₂²	C₂²×#13089678	1	7	yes	—
47	C₄	C₂²×#13089678	1	7	yes	—
48	C₄	C₂³×Dic₁₈₁	1	1	yes	characteristic
49	C₂	C₂²×#15668424	1	14	yes	direct product factor
50	C₂	C₂³×#13089678	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₈	D₁₈₁
commutator, hypocenter	C₁₈₁	C₂³×C₁₆
Frattini	C₈	C₂³×D₁₈₁
Fitting	C₂³×C₈×C₁₈₁	C₂
socle, prime radical	C₂⁴×C₁₈₁	C₈
cosocle	C₈×C₁₈₁	C₂⁴
LCS	G ▷ C₁₈₁
UCS	C₁ ◁ C₂³×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#15668424	C₂	direct (N × H)
3. C₂×#15668424	C₂²	direct (N × H)
7. #15668424	C₂³	direct (N × H)
13. C₂³×C₁₈₁	C₁₆	semidirect
19. C₂²×C₁₈₁	C₂×C₁₆	semidirect
23. C₂×C₁₈₁	C₂²×C₁₆	semidirect
25. C₁₈₁	C₂³×C₁₆	semidirect
39. C₂³	#15668424	direct (N × H)
45. C₂²	C₂×#15668424	direct (N × H)
49. C₂	C₂²×#15668424	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15668424	2
C₂³×C₈×C₁₈₁	2
C₂³×C₁₆	181

Subgroups — C23×#15668424 #37546814

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#15668424 #37546814