Subgroups — #37542113

Order 23168.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041170	C₂	1	7	yes	maximal
2	#26041163	C₂	1	7	yes	maximal, complemented
3	C₄³×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#15668428	C₂²	1	7	yes	—
5	C₂×#15668426	C₂²	1	7	yes	—
6	#19153723	C₂²	1	14	yes	—
7	C₂×C₄²×C₁₈₁	C₂²	1	7	yes	—
8	C₂³×D₁₈₁	C₂³	1	1	yes	characteristic
9	C₂²×Dic₁₈₁	C₂³	1	7	yes	—
10	#15668428	—	2	14	no	—
11	#15668426	—	2	14	no	—
12	C₂²×C₄×C₁₈₁	C₂³	1	7	yes	—
13	C₄²×C₁₈₁	—	2	14	no	—
14	C₂²×D₁₈₁	—	2	7	no	—
15	C₂×Dic₁₈₁	—	2	7	no	—
16	C₂×Dic₁₈₁	—	2	14	no	—
17	C₂³×C₁₈₁	C₂⁴	1	1	yes	characteristic
18	C₂×C₄×C₁₈₁	—	2	14	no	—
19	C₂×C₄×C₁₈₁	#41	1	14	yes	—
20	C₂²×C₁₈₁	C₂×#41	1	7	yes	—
21	C₂×D₁₈₁	—	4	7	no	—
22	C₄×C₁₈₁	—	2	14	no	—
23	Dic₁₈₁	—	4	7	no	—
24	C₂×C₁₈₁	#529	1	7	yes	—
25	D₁₈₁	—	8	1	no	—
26	C₁₈₁	#2866	1	1	yes	characteristic, complemented, Sylow
27	#2866	—	181	1	no	maximal, Sylow
28	#388	—	181	7	no	—
29	#387	—	181	7	no	—
30	C₄³	D₁₈₁	1	1	yes	characteristic, complemented
31	C₂×C₄⋊C₄	—	181	7	no	—
32	C₂×C₂²⋊C₄	—	181	7	no	—
33	C₂×C₄²	C₂×D₁₈₁	1	7	yes	—
34	#95	—	181	14	no	—
35	C₂⁴	—	181	1	no	—
36	C₂²×C₄	—	181	7	no	—
37	C₂²×C₄	C₂²×D₁₈₁	1	7	yes	—
38	C₄⋊C₄	—	362	14	no	—
39	C₂²⋊C₄	—	362	14	no	—
40	C₄²	—	2	14	no	—
41	C₂³	—	362	7	no	—
42	C₂³	C₂³×D₁₈₁	1	1	yes	characteristic
43	C₂×C₄	—	2	14	no	—
44	C₂×C₄	—	362	7	no	—
45	C₂×C₄	—	362	14	no	—
46	C₂×C₄	#15668449	1	14	yes	—
47	C₂²	—	724	7	no	—
48	C₂²	C₂×#15668449	1	7	yes	—
49	C₄	—	2	14	no	—
50	C₄	—	724	7	no	—
51	C₂	—	1448	1	no	—
52	C₂	#26041813	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₈₁
commutator, socle, cosocle	C₂³×C₁₈₁	C₂⁴
Fitting	C₄³×C₁₈₁	C₂
prime radical	C₂³×D₁₈₁	C₂³
hypercenter	C₄³	D₁₈₁
hypocenter	C₁₈₁	#2866
LCS	G ▷ C₂³×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
2. #26041163	C₂	semidirect
3. C₄³×C₁₈₁	C₂	semidirect
26. C₁₈₁	#2866	semidirect
30. C₄³	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041170	2
#26041163	2
C₄³×C₁₈₁	2
#2866	181