Subgroups — #423×C₁₉₉ #27163573

Order 12736.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×C₁₉₉	C₂	1	2	yes	maximal
2	C₂×C₄²×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×C₁₉₉	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₁₉₉	C₄	1	2	yes	—
5	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
6	M₁₆×C₁₉₉	—	2	4	no	—
7	C₂×C₈×C₁₉₉	C₂²	1	4	yes	—
8	C₄²×C₁₉₉	C₄	1	1	yes	characteristic
9	C₄²×C₁₉₉	C₄	1	1	yes	characteristic
10	C₄²×C₁₉₉	C₂²	1	2	yes	—
11	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₁₉₉	—	2	4	no	—
13	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
17	C₂×C₄×C₁₉₉	Q₈	1	2	yes	—
18	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
19	C₈×C₁₉₉	—	2	4	no	—
20	C₂²×C₁₉₉	—	2	2	no	—
21	C₂²×C₁₉₉	C₄⋊C₄	1	2	yes	—
22	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
23	C₄×C₁₉₉	—	2	2	no	—
24	C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
25	C₄×C₁₉₉	M₁₆	1	4	yes	complemented
26	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
27	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₁₉₉	—	2	2	no	—
29	C₂×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₁₉₉	C₂×M₁₆	1	2	yes	—
31	C₁₉₉	#423	1	1	yes	characteristic, direct product factor, Sylow
32	#423	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×M₁₆	C₂×C₁₉₉	1	2	yes	—
34	C₂×C₄²	C₂×C₁₉₉	1	1	yes	characteristic
35	C₄⋊C₈	C₂×C₁₉₉	1	4	yes	complemented
36	C₂²×C₄	C₄×C₁₉₉	1	2	yes	—
37	C₂²×C₄	C₂²×C₁₉₉	1	1	yes	characteristic
38	M₁₆	—	2	4	no	—
39	C₂×C₈	C₂²×C₁₉₉	1	4	yes	—
40	C₄²	C₄×C₁₉₉	1	1	yes	characteristic
41	C₄²	C₄×C₁₉₉	1	1	yes	characteristic
42	C₄²	C₂²×C₁₉₉	1	2	yes	—
43	C₂³	C₂×C₄×C₁₉₉	1	1	yes	characteristic
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	C₂×C₄×C₁₉₉	1	2	yes	—
46	C₂×C₄	C₂×C₄×C₁₉₉	1	1	yes	characteristic
47	C₂×C₄	C₂×C₄×C₁₉₉	1	2	yes	—
48	C₂×C₄	D₄×C₁₉₉	1	2	yes	—
49	C₂×C₄	Q₈×C₁₉₉	1	2	yes	—
50	C₂×C₄	C₂³×C₁₉₉	1	1	yes	characteristic
51	C₈	—	2	4	no	—
52	C₂²	—	2	2	no	—
53	C₂²	C₄⋊C₄×C₁₉₉	1	2	yes	—
54	C₂²	C₂²×C₄×C₁₉₉	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	C₄⋊C₄×C₁₉₉	1	2	yes	—
57	C₄	M₁₆×C₁₉₉	1	4	yes	complemented
58	C₄	C₂×D₄×C₁₉₉	1	1	yes	characteristic
59	C₄	C₂×Q₈×C₁₉₉	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄⋊C₄×C₁₉₉	1	1	yes	characteristic
62	C₂	C₂×M₁₆×C₁₉₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₉₉	C₂³
commutator	C₂²	C₂²×C₄×C₁₉₉
Frattini, cosocle	C₂×C₄	C₂³×C₁₉₉
socle	C₂²×C₁₉₉	C₂²×C₄
prime radical	C₂³×C₁₉₉	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄⋊C₈×C₁₉₉	C₂	semidirect
25. C₄×C₁₉₉	M₁₆	semidirect
31. C₁₉₉	#423	direct (N × H)
32. #423	C₁₉₉	direct (N × H)
35. C₄⋊C₈	C₂×C₁₉₉	semidirect
57. C₄	M₁₆×C₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×C₁₉₉	2
C₂×C₄²×C₁₉₉	2
C₄⋊C₈×C₁₉₉	2
#423	199

Subgroups — #423×C199 #27163573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #423×C₁₉₉ #27163573