Subgroups — #27164835

Order 12736.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20264002	C₂	1	2	yes	maximal, complemented
2	#20263988	C₂	1	6	yes	maximal, complemented
3	#20263977	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₇₉₆	C₂	1	3	yes	maximal, complemented
5	#131×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15691886	C₂²	1	6	yes	—
7	#15691885	—	2	6	no	—
8	C₂×Dic₃₉₈	C₂²	1	6	yes	—
9	Q₈×D₁₉₉	—	2	2	no	—
10	Dic₇₉₆	C₂²	1	3	yes	—
11	Dic₇₉₆	C₂²	1	6	yes	—
12	#15691875	C₂²	1	6	yes	—
13	D₁₅₉₂	C₂²	1	1	yes	characteristic
14	#41×C₁₉₉	C₂²	1	1	yes	characteristic
15	M₁₆×C₁₉₉	C₂²	1	3	yes	—
16	C₂×C₈×C₁₉₉	C₂²	1	3	yes	—
17	Dic₃₉₈	—	2	6	no	—
18	Dic₃₉₈	C₂³	1	6	yes	—
19	D₇₉₆	C₂³	1	2	yes	—
20	C₂²⋊D₁₉₉	—	2	6	no	—
21	C₂×Dic₁₉₉	—	2	6	no	—
22	C₄×D₁₉₉	—	2	6	no	—
23	Q₈×C₁₉₉	D₄	1	1	yes	characteristic
24	D₄×C₁₉₉	D₄	1	3	yes	—
25	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
26	C₈×C₁₉₉	C₂³	1	1	yes	characteristic
27	C₈×C₁₉₉	C₂³	1	3	yes	—
28	C₂²×C₁₉₉	C₂×D₄	1	3	yes	—
29	C₂×D₁₉₉	—	2	2	no	—
30	C₄×C₁₉₉	C₂×D₄	1	3	yes	—
31	C₄×C₁₉₉	C₂⁴	1	1	yes	characteristic
32	Dic₁₉₉	—	2	6	no	—
33	C₂×C₁₉₉	—	2	3	no	—
34	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
35	D₁₉₉	—	4	2	no	—
36	C₁₉₉	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	199	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	199	2	no	—
39	#137	—	199	6	no	—
40	#135	—	199	3	no	—
41	C₂×Q₁₆	—	199	3	no	—
42	#131	D₁₉₉	1	1	yes	characteristic, complemented
43	#41	—	398	6	no	—
44	#41	—	199	6	no	—
45	#41	C₂×D₁₉₉	1	1	yes	characteristic
46	C₂×Q₈	—	398	2	no	—
47	C₂×Q₈	—	199	6	no	—
48	Q₁₆	—	199	3	no	—
49	Q₁₆	—	199	6	no	—
50	SD₁₆	—	199	6	no	—
51	D₈	—	199	1	no	—
52	M₁₆	C₂×D₁₉₉	1	3	yes	—
53	C₂×C₈	C₂×D₁₉₉	1	3	yes	—
54	Q₈	—	398	6	no	—
55	Q₈	—	199	6	no	—
56	Q₈	D₇₉₆	1	1	yes	characteristic
57	D₄	—	398	6	no	—
58	D₄	—	199	2	no	—
59	D₄	D₇₉₆	1	3	yes	—
60	C₂×C₄	—	398	6	no	—
61	C₂×C₄	—	398	6	no	—
62	C₂×C₄	C₂²×D₁₉₉	1	3	yes	—
63	C₈	C₂²×D₁₉₉	1	1	yes	characteristic
64	C₈	C₂²×D₁₉₉	1	3	yes	—
65	C₂²	—	398	2	no	—
66	C₂²	C₂×D₇₉₆	1	3	yes	—
67	C₄	—	398	6	no	—
68	C₄	C₂×D₇₉₆	1	3	yes	—
69	C₄	C₂³×D₁₉₉	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	796	2	no	—
72	C₂	C₂²×D₇₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₇₉₆
commutator, cosocle	C₄×C₁₉₉	C₂⁴
Frattini	C₄	C₂³×D₁₉₉
Fitting	#131×C₁₉₉	C₂
socle	C₂×C₁₉₉	C₂²×D₄
hypercenter	#131	D₁₉₉
hypocenter	C₁₉₉	#578
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #20264002	C₂	semidirect
2. #20263988	C₂	semidirect
3. #20263977	C₂	semidirect
4. C₂×Dic₇₉₆	C₂	semidirect
5. #131×C₁₉₉	C₂	semidirect
36. C₁₉₉	#578	semidirect
42. #131	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20264002	2
#20263988	2
#20263977	2
C₂×Dic₇₉₆	2
#131×C₁₉₉	2
#578	199