Subgroups — #475×C₁₉₉ #27163625

Order 12736.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#106×C₁₉₉	C₂	1	2	yes	maximal
4	C₄⋊C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	#103×C₁₉₉	C₂	1	2	yes	maximal
6	C₂×Q₈×C₁₉₉	—	2	1	no	—
7	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₁₉₉	C₂²	1	2	yes	—
9	C₄⋊C₄×C₁₉₉	—	2	1	no	—
10	C₄⋊C₄×C₁₉₉	—	2	1	no	—
11	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
12	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₄²×C₁₉₉	—	2	1	no	—
14	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	Q₈×C₁₉₉	—	2	1	no	—
16	Q₈×C₁₉₉	—	4	2	no	—
17	Q₈×C₁₉₉	Q₈	1	2	yes	complemented
18	C₂×C₄×C₁₉₉	—	2	1	no	—
19	C₂×C₄×C₁₉₉	—	2	1	no	—
20	C₂×C₄×C₁₉₉	—	2	2	no	—
21	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
24	C₈×C₁₉₉	—	2	2	no	—
25	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
26	C₄×C₁₉₉	—	2	2	no	—
27	C₄×C₁₉₉	—	2	1	no	—
28	C₄×C₁₉₉	—	4	2	no	—
29	C₄×C₁₉₉	—	4	1	no	—
30	C₄×C₁₉₉	SD₁₆	1	2	yes	—
31	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
32	C₄×C₁₉₉	#41	1	1	yes	characteristic
33	C₂×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
35	C₂×C₁₉₉	#137	1	1	yes	characteristic
36	C₁₉₉	#475	1	1	yes	characteristic, direct product factor, Sylow
37	#475	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₄⋊Q₈	C₂×C₁₉₉	1	1	yes	characteristic
39	C₄×Q₈	C₂×C₁₉₉	1	1	yes	characteristic
40	#106	C₂×C₁₉₉	1	2	yes	—
41	C₄⋊C₈	C₂×C₁₉₉	1	1	yes	characteristic
42	#103	C₂×C₁₉₉	1	2	yes	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×Q₈	C₂²×C₁₉₉	1	1	yes	characteristic
45	C₂×C₈	C₂²×C₁₉₉	1	2	yes	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	2	1	no	—
48	C₄⋊C₄	C₂²×C₁₉₉	1	2	yes	—
49	C₄⋊C₄	C₂²×C₁₉₉	1	1	yes	characteristic
50	C₄²	—	2	1	no	—
51	C₄²	C₂²×C₁₉₉	1	1	yes	characteristic
52	Q₈	—	2	1	no	—
53	Q₈	—	4	2	no	—
54	Q₈	Q₈×C₁₉₉	1	2	yes	complemented
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	D₄×C₁₉₉	1	1	yes	characteristic
59	C₂×C₄	D₄×C₁₉₉	1	1	yes	characteristic
60	C₂×C₄	C₂³×C₁₉₉	1	1	yes	characteristic
61	C₈	—	2	2	no	—
62	C₂²	C₂×D₄×C₁₉₉	1	1	yes	characteristic
63	C₄	—	2	2	no	—
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	—	4	1	no	—
67	C₄	SD₁₆×C₁₉₉	1	2	yes	—
68	C₄	C₂×Q₈×C₁₉₉	1	1	yes	characteristic
69	C₄	#41×C₁₉₉	1	1	yes	characteristic
70	C₂	C₂²⋊Q₈×C₁₉₉	1	1	yes	characteristic
71	C₂	C₂×SD₁₆×C₁₉₉	1	1	yes	characteristic
72	C₂	#137×C₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₉₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₁₉₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉₉ ◁ C₄²×C₁₉₉ ◁ G

Complements

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

N	H ≅ G/N	type
17. Q₈×C₁₉₉	Q₈	semidirect
36. C₁₉₉	#475	direct (N × H)
37. #475	C₁₉₉	direct (N × H)
54. Q₈	Q₈×C₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₁₉₉	2
C₄×Q₈×C₁₉₉	2
#106×C₁₉₉	2
C₄⋊C₈×C₁₉₉	2
#103×C₁₉₉	2
#475	199

Subgroups — #475×C199 #27163625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #475×C₁₉₉ #27163625