Subgroups — C₂×#26014984 #36378309

Order 22592.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26014984	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₃₅₃	C₂	1	2	yes	maximal
3	C₂×#19150716	C₂	1	4	yes	maximal
4	C₂×C₄²×C₃₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₅₃	C₂²	1	4	yes	—
6	C₄⋊Dic₃₅₃	C₂²	1	8	yes	—
7	#19150716	C₂²	1	16	yes	—
8	C₂²×C₄×C₃₅₃	C₂²	1	2	yes	—
9	C₂²×C₄×C₃₅₃	C₂²	1	1	yes	characteristic
10	C₄²×C₃₅₃	C₂²	1	4	yes	—
11	C₂×Dic₃₅₃	—	2	8	no	—
12	C₂×Dic₃₅₃	C₂³	1	8	yes	—
13	C₂³×C₃₅₃	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₃₅₃	—	2	4	no	—
15	C₂×C₄×C₃₅₃	Q₈	1	4	yes	—
16	C₂×C₄×C₃₅₃	C₂³	1	4	yes	—
17	C₂×C₄×C₃₅₃	C₂³	1	2	yes	—
18	C₂²×C₃₅₃	C₂×Q₈	1	2	yes	—
19	C₂²×C₃₅₃	#41	1	4	yes	—
20	C₂²×C₃₅₃	C₂⁴	1	1	yes	characteristic
21	C₄×C₃₅₃	—	2	4	no	—
22	C₄×C₃₅₃	C₂×Q₈	1	4	yes	—
23	Dic₃₅₃	—	2	8	no	—
24	C₂×C₃₅₃	#125	1	4	yes	complemented
25	C₂×C₃₅₃	C₂²×Q₈	1	1	yes	characteristic
26	C₂×C₃₅₃	C₂×#41	1	2	yes	—
27	C₃₅₃	C₂×#125	1	1	yes	characteristic, complemented, Sylow
28	C₂×#125	—	353	1	no	maximal, Sylow
29	#125	—	353	8	no	—
30	C₂×C₄⋊C₄	—	353	4	no	—
31	C₂×C₄⋊C₄	—	353	2	no	—
32	C₂×C₄²	D₃₅₃	1	1	yes	characteristic
33	C₂²×C₄	—	353	4	no	—
34	C₂²×C₄	C₂×D₃₅₃	1	2	yes	—
35	C₂²×C₄	C₂×D₃₅₃	1	1	yes	characteristic
36	C₄⋊C₄	—	353	16	no	—
37	C₄⋊C₄	—	353	8	no	—
38	C₄²	C₂×D₃₅₃	1	4	yes	—
39	C₂³	C₂²×D₃₅₃	1	1	yes	characteristic
40	C₂×C₄	—	2	4	no	—
41	C₂×C₄	—	353	8	no	—
42	C₂×C₄	—	706	8	no	—
43	C₂×C₄	Dic₇₀₆	1	4	yes	—
44	C₂×C₄	C₂²×D₃₅₃	1	4	yes	—
45	C₂×C₄	C₂²×D₃₅₃	1	2	yes	—
46	C₂²	C₂×Dic₇₀₆	1	2	yes	—
47	C₂²	#19150737	1	4	yes	—
48	C₂²	C₂³×D₃₅₃	1	1	yes	characteristic
49	C₄	—	2	4	no	—
50	C₄	—	706	8	no	—
51	C₄	C₂×Dic₇₀₆	1	4	yes	—
52	C₂	#26014984	1	4	yes	direct product factor
53	C₂	C₂²×Dic₇₀₆	1	1	yes	characteristic
54	C₂	C₂×#19150737	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₅₃
commutator, cosocle	C₂²×C₃₅₃	C₂⁴
Frattini	C₂²	C₂³×D₃₅₃
Fitting	C₂×C₄²×C₃₅₃	C₂
socle, prime radical	C₂³×C₃₅₃	C₂³
hypercenter	C₂×C₄²	D₃₅₃
hypocenter	C₃₅₃	C₂×#125
LCS	G ▷ C₂²×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #26014984	C₂	direct (N × H)
24. C₂×C₃₅₃	#125	semidirect
27. C₃₅₃	C₂×#125	semidirect
52. C₂	#26014984	direct (N × H)

Maximal subgroups

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

subgroup	index
#26014984	2
C₂×C₄⋊Dic₃₅₃	2
C₂×#19150716	2
C₂×C₄²×C₃₅₃	2
C₂×#125	353

Subgroups — C2×#26014984 #36378309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26014984 #36378309