Subgroups — C₂×#125×C₁₉₉ #27163677

Order 12736.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×C₁₉₉	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊C₄×C₁₉₉	C₂	1	4	yes	maximal
3	C₂×C₄⋊C₄×C₁₉₉	C₂	1	2	yes	maximal
4	C₂×C₄²×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×C₁₉₉	C₂²	1	2	yes	—
6	C₂²×C₄×C₁₉₉	C₂²	1	4	yes	—
7	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₁₉₉	C₂²	1	16	yes	—
9	C₄⋊C₄×C₁₉₉	C₂²	1	8	yes	—
10	C₄²×C₁₉₉	C₂²	1	4	yes	—
11	C₂³×C₁₉₉	C₂³	1	1	yes	characteristic
12	C₂×C₄×C₁₉₉	—	2	4	no	—
13	C₂×C₄×C₁₉₉	—	2	8	no	—
14	C₂×C₄×C₁₉₉	Q₈	1	4	yes	—
15	C₂×C₄×C₁₉₉	C₂³	1	4	yes	—
16	C₂×C₄×C₁₉₉	C₂³	1	8	yes	—
17	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
18	C₂²×C₁₉₉	C₂×Q₈	1	2	yes	—
19	C₂²×C₁₉₉	#41	1	4	yes	—
20	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
21	C₄×C₁₉₉	—	2	4	no	—
22	C₄×C₁₉₉	—	2	8	no	—
23	C₄×C₁₉₉	C₂×Q₈	1	4	yes	—
24	C₂×C₁₉₉	#125	1	4	yes	direct product factor
25	C₂×C₁₉₉	C₂²×Q₈	1	1	yes	characteristic
26	C₂×C₁₉₉	C₂×#41	1	2	yes	—
27	C₁₉₉	C₂×#125	1	1	yes	characteristic, direct product factor, Sylow
28	C₂×#125	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	#125	C₂×C₁₉₉	1	8	yes	direct product factor
30	C₂×C₄⋊C₄	C₂×C₁₉₉	1	4	yes	—
31	C₂×C₄⋊C₄	C₂×C₁₉₉	1	2	yes	—
32	C₂×C₄²	C₂×C₁₉₉	1	1	yes	characteristic
33	C₂²×C₄	C₂²×C₁₉₉	1	2	yes	—
34	C₂²×C₄	C₂²×C₁₉₉	1	4	yes	—
35	C₂²×C₄	C₂²×C₁₉₉	1	1	yes	characteristic
36	C₄⋊C₄	C₂²×C₁₉₉	1	16	yes	—
37	C₄⋊C₄	C₂²×C₁₉₉	1	8	yes	—
38	C₄²	C₂²×C₁₉₉	1	4	yes	—
39	C₂³	C₂³×C₁₉₉	1	1	yes	characteristic
40	C₂×C₄	—	2	4	no	—
41	C₂×C₄	—	2	8	no	—
42	C₂×C₄	Q₈×C₁₉₉	1	4	yes	—
43	C₂×C₄	C₂³×C₁₉₉	1	4	yes	—
44	C₂×C₄	C₂³×C₁₉₉	1	8	yes	—
45	C₂×C₄	C₂³×C₁₉₉	1	2	yes	—
46	C₂²	C₂×Q₈×C₁₉₉	1	2	yes	—
47	C₂²	#41×C₁₉₉	1	4	yes	—
48	C₂²	C₂⁴×C₁₉₉	1	1	yes	characteristic
49	C₄	—	2	4	no	—
50	C₄	—	2	8	no	—
51	C₄	C₂×Q₈×C₁₉₉	1	4	yes	—
52	C₂	#125×C₁₉₉	1	4	yes	direct product factor
53	C₂	C₂²×Q₈×C₁₉₉	1	1	yes	characteristic
54	C₂	C₂×#41×C₁₉₉	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉₉	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₉₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #125×C₁₉₉	C₂	direct (N × H)
24. C₂×C₁₉₉	#125	direct (N × H)
27. C₁₉₉	C₂×#125	direct (N × H)
28. C₂×#125	C₁₉₉	direct (N × H)
29. #125	C₂×C₁₉₉	direct (N × H)
52. C₂	#125×C₁₉₉	direct (N × H)

Maximal subgroups

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

subgroup	index
#125×C₁₉₉	2
C₂×C₄⋊C₄×C₁₉₉	2
C₂×C₄²×C₁₉₉	2
C₂×#125	199

Subgroups — C2×#125×C199 #27163677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#125×C₁₉₉ #27163677