Subgroups — #27164426

Order 12736.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263954	C₂	1	1	yes	maximal, characteristic
2	#20263946	C₂	1	1	yes	maximal, characteristic
3	#20263888	C₂	1	1	yes	maximal, characteristic
4	#20263887	C₂	1	1	yes	maximal, characteristic
5	#20263886	C₂	1	1	yes	maximal, characteristic
6	#20263865	C₂	1	1	yes	maximal, characteristic
7	#107×C₁₉₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₉₈	—	2	1	no	—
9	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₉₉	—	2	1	no	—
11	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
13	Dic₁₉₉⋊C₄	—	2	1	no	—
14	Dic₁₉₉⋊C₄	—	2	1	no	—
15	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
19	Dic₃₉₈	—	4	2	no	—
20	C₂×Dic₁₉₉	—	2	1	no	—
21	C₂×Dic₁₉₉	—	2	1	no	—
22	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
23	C₂×Dic₁₉₉	D₄	1	1	yes	characteristic
24	C₁₉₉⋊C₈	Q₈	1	2	yes	complemented
25	C₂×C₄×C₁₉₉	—	2	1	no	—
26	C₂×C₄×C₁₉₉	—	2	1	no	—
27	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
28	C₈×C₁₉₉	Q₈	1	2	yes	complemented
29	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
30	C₄×C₁₉₉	—	4	1	no	—
31	C₄×C₁₉₉	—	4	1	no	—
32	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
34	Dic₁₉₉	—	2	1	no	—
35	Dic₁₉₉	—	2	1	no	—
36	Dic₁₉₉	—	4	1	no	—
37	Dic₁₉₉	—	4	1	no	—
38	C₂×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
40	C₂×C₁₉₉	#137	1	1	yes	characteristic
41	C₁₉₉	#501	1	1	yes	characteristic, complemented, Sylow
42	#501	—	199	1	no	maximal, Sylow
43	C₄⋊Q₈	—	199	1	no	—
44	#125	—	199	1	no	—
45	#107	—	199	1	no	—
46	#107	D₁₉₉	1	1	yes	characteristic
47	#106	—	199	1	no	—
48	#106	—	199	1	no	—
49	#97	—	199	1	no	—
50	C₂×Q₈	—	398	1	no	—
51	C₂×C₈	—	199	1	no	—
52	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
53	C₄⋊C₄	—	398	1	no	—
54	C₄⋊C₄	—	398	1	no	—
55	C₄⋊C₄	—	398	1	no	—
56	C₄⋊C₄	—	199	1	no	—
57	C₄⋊C₄	—	199	1	no	—
58	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
59	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
60	C₄²	—	199	1	no	—
61	Q₈	—	796	2	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	398	1	no	—
65	C₂×C₄	—	398	1	no	—
66	C₂×C₄	—	199	1	no	—
67	C₂×C₄	—	199	1	no	—
68	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
69	C₈	—	199	2	no	—
70	C₈	Dic₃₉₈	1	2	yes	complemented
71	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	796	1	no	—
75	C₄	—	796	1	no	—
76	C₄	—	398	1	no	—
77	C₄	—	398	1	no	—
78	C₄	Q₈×D₁₉₉	1	1	yes	characteristic
79	C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
80	C₂	#20263954	1	1	yes	characteristic
81	C₂	#20263979	1	1	yes	characteristic
82	C₂	#20263987	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#107×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
hypercenter	#107	D₁₉₉
hypocenter	C₁₉₉	#501
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
24. C₁₉₉⋊C₈	Q₈	semidirect
28. C₈×C₁₉₉	Q₈	semidirect
41. C₁₉₉	#501	semidirect
70. C₈	Dic₃₉₈	semidirect

Maximal subgroups

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

subgroup	index
#20263954	2
#20263946	2
#20263888	2
#20263887	2
#20263886	2
#20263865	2
#107×C₁₉₉	2
#501	199