Subgroups — #27163774

Order 12736.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691870	C₂	1	1	yes	maximal, characteristic
2	#20263914	C₂	1	1	yes	maximal, characteristic
3	#117×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	#15691870	—	2	1	no	—
5	#15691870	—	2	1	no	—
6	C₂×C₁₉₉⋊C₈	C₄	1	2	yes	complemented
7	C₄⋊Dic₁₉₉	—	2	1	no	—
8	C₂²⋊Dic₁₉₉	—	2	1	no	—
9	C₄×Dic₁₉₉	C₄	1	2	yes	complemented
10	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₁₉₉	—	2	1	no	—
12	C₂²⋊C₄×C₁₉₉	—	2	1	no	—
13	C₄²×C₁₉₉	C₄	1	2	yes	complemented
14	C₂×Dic₁₉₉	—	2	2	no	—
15	C₁₉₉⋊C₈	—	2	2	no	—
16	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₉₉	—	2	2	no	—
18	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₉	Q₈	1	1	yes	characteristic
24	C₂²×C₁₉₉	—	4	1	no	—
25	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
28	C₄×C₁₉₉	—	4	2	no	—
29	C₄×C₁₉₉	C₄²	1	1	yes	characteristic
30	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
32	C₄×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
33	Dic₁₉₉	—	4	2	no	—
34	C₂×C₁₉₉	—	2	1	no	—
35	C₂×C₁₉₉	—	2	1	no	—
36	C₂×C₁₉₉	—	2	1	no	—
37	C₂×C₁₉₉	#95	1	1	yes	characteristic
38	C₁₉₉	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
39	H(3,ℤ/4)	—	199	1	no	maximal, Sylow
40	C₂×M₁₆	—	199	1	no	—
41	#117	—	199	1	no	—
42	#117	D₁₉₉	1	1	yes	characteristic
43	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
44	M₁₆	—	398	1	no	—
45	M₁₆	—	398	1	no	—
46	C₂×C₈	—	199	2	no	—
47	C₄⋊C₄	—	2	1	no	—
48	C₄⋊C₄	—	398	1	no	—
49	C₂²⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	—	398	1	no	—
51	C₄²	—	199	2	no	—
52	C₄²	Dic₁₉₉	1	2	yes	complemented
53	C₂³	C₂²⋊D₁₉₉	1	1	yes	characteristic
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	398	2	no	—
56	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
57	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
58	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
60	C₂×C₄	D₇₉₆	1	1	yes	characteristic
61	C₂×C₄	Dic₃₉₈	1	1	yes	characteristic
62	C₈	—	398	2	no	—
63	C₂²	—	4	1	no	—
64	C₂²	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
65	C₂²	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
66	C₂²	Dic₁₉₉⋊C₄	1	1	yes	characteristic
67	C₄	—	4	2	no	—
68	C₄	—	796	2	no	—
69	C₄	C₄×Dic₁₉₉	1	1	yes	characteristic
70	C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
71	C₄	Dic₁₉₉⋊C₄	1	1	yes	characteristic
72	C₄	C₄⋊Dic₁₉₉	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	2	1	no	—
75	C₂	—	2	1	no	—
76	C₂	C₁₉₉⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×Dic₁₉₉
commutator	C₄×C₁₉₉	C₄²
Frattini	C₂²×C₄	C₂×D₁₉₉
Fitting	#117×C₁₉₉	C₂
socle	C₂×C₁₉₉	#95
prime radical	C₂³×C₁₉₉	D₄
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	#117	D₁₉₉
hypocenter	C₁₉₉	H(3,ℤ/4)
LCS	G ▷ C₄×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
6. C₂×C₁₉₉⋊C₈	C₄	semidirect
9. C₄×Dic₁₉₉	C₄	semidirect
13. C₄²×C₁₉₉	C₄	semidirect
38. C₁₉₉	H(3,ℤ/4)	semidirect
52. C₄²	Dic₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15691870	2
#20263914	2
#117×C₁₉₉	2
H(3,ℤ/4)	199