Subgroups — #27163992

Order 12736.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263914	C₂	1	2	yes	maximal, complemented
2	#20263870	C₂	1	4	yes	maximal, complemented
3	C₂×#41×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₁₉₉	—	2	2	no	—
5	C₂²⋊Dic₁₉₉	C₂²	1	4	yes	—
6	C₄×Dic₁₉₉	—	2	2	no	—
7	#41×C₁₉₉	—	2	2	no	—
8	#41×C₁₉₉	—	2	2	no	—
9	C₂×Q₈×C₁₉₉	C₄	1	1	yes	characteristic, complemented
10	C₂×D₄×C₁₉₉	C₄	1	1	yes	characteristic, complemented
11	C₂×D₄×C₁₉₉	C₂²	1	2	yes	—
12	C₂²×C₄×C₁₉₉	C₄	1	2	yes	complemented
13	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₉₉	—	2	4	no	—
15	C₂³×C₁₉₉	C₂×C₄	1	2	yes	—
16	C₂³×C₁₉₉	C₂³	1	1	yes	characteristic
17	Q₈×C₁₉₉	—	2	2	no	—
18	D₄×C₁₉₉	—	2	4	no	—
19	D₄×C₁₉₉	—	2	2	no	—
20	C₂×C₄×C₁₉₉	—	2	2	no	—
21	C₂×C₄×C₁₉₉	—	2	2	no	—
22	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
26	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
27	C₂²×C₁₉₉	—	2	2	no	—
28	C₂²×C₁₉₉	—	4	1	no	—
29	C₂²×C₁₉₉	—	4	2	no	—
30	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₁₉₉	C₂×D₄	1	2	yes	—
32	C₄×C₁₉₉	—	2	2	no	—
33	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
35	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₁₉₉	—	4	4	no	—
37	C₂×C₁₉₉	—	2	2	no	—
38	C₂×C₁₉₉	—	2	1	no	—
39	C₂×C₁₉₉	—	4	2	no	—
40	C₂×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₁₉₉	#410	1	1	yes	characteristic, complemented, Sylow
42	#410	—	199	1	no	maximal, Sylow
43	C₂×#41	D₁₉₉	1	1	yes	characteristic
44	#117	—	199	2	no	—
45	#99	—	199	4	no	—
46	#41	—	2	2	no	—
47	#41	—	2	2	no	—
48	C₂×Q₈	Dic₁₉₉	1	1	yes	characteristic, complemented
49	C₂×D₄	Dic₁₉₉	1	1	yes	characteristic, complemented
50	C₂×D₄	C₂×D₁₉₉	1	2	yes	—
51	C₂²×C₄	Dic₁₉₉	1	2	yes	complemented
52	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
53	C₄⋊C₄	—	398	2	no	—
54	C₂²⋊C₄	—	199	4	no	—
55	C₄²	—	398	2	no	—
56	C₂³	C₂×Dic₁₉₉	1	2	yes	—
57	C₂³	C₂²×D₁₉₉	1	1	yes	characteristic
58	Q₈	—	2	2	no	—
59	D₄	—	2	4	no	—
60	D₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	398	4	no	—
64	C₂×C₄	C₂×Dic₁₉₉	1	2	yes	—
65	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
66	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
67	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
68	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
69	C₂²	—	2	2	no	—
70	C₂²	—	4	1	no	—
71	C₂²	—	4	2	no	—
72	C₂²	C₂²×Dic₁₉₉	1	1	yes	characteristic
73	C₂²	C₂×C₂²⋊D₁₉₉	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	796	4	no	—
76	C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
77	C₄	C₂²⋊Dic₁₉₉	1	2	yes	—
78	C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
79	C₂	—	2	2	no	—
80	C₂	—	2	1	no	—
81	C₂	—	4	2	no	—
82	C₂	C₂×C₂²⋊Dic₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊Dic₁₉₉
commutator	C₂²×C₁₉₉	C₂²×C₄
Frattini	C₂³	C₂²×D₁₉₉
Fitting, prime radical	C₂×#41×C₁₉₉	C₂
socle	C₂×C₁₉₉	C₂×C₂²⋊C₄
cosocle	C₂³×C₁₉₉	C₂³
hypercenter	C₂×#41	D₁₉₉
hypocenter	C₁₉₉	#410
LCS	G ▷ C₂²×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #20263914	C₂	semidirect
2. #20263870	C₂	semidirect
9. C₂×Q₈×C₁₉₉	C₄	semidirect
10. C₂×D₄×C₁₉₉	C₄	semidirect
12. C₂²×C₄×C₁₉₉	C₄	semidirect
41. C₁₉₉	#410	semidirect
48. C₂×Q₈	Dic₁₉₉	semidirect
49. C₂×D₄	Dic₁₉₉	semidirect
51. C₂²×C₄	Dic₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263914	2
#20263870	2
C₂×#41×C₁₉₉	2
#410	199