Subgroups — #27163783

Order 12736.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691870	C₂	1	2	yes	maximal
2	C₂²×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#15691870	—	2	2	no	—
4	#15691870	C₄	1	4	yes	—
5	C₂×C₁₉₉⋊C₈	—	2	2	no	—
6	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₁₉₉	—	2	2	no	—
8	C₂×C₈×C₁₉₉	C₄	1	2	yes	—
9	C₁₉₉⋊C₈	—	2	4	no	—
10	C₂³×C₁₉₉	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
14	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
15	C₈×C₁₉₉	—	2	2	no	—
16	C₂²×C₁₉₉	—	2	2	no	—
17	C₂²×C₁₉₉	C₄⋊C₄	1	2	yes	—
18	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
19	C₄×C₁₉₉	C₄²	1	1	yes	characteristic
20	C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
21	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₁₉₉	—	2	2	no	—
23	C₂×C₁₉₉	#95	1	1	yes	characteristic
24	C₂×C₁₉₉	#108	1	2	yes	—
25	C₁₉₉	#341	1	1	yes	characteristic, complemented, Sylow
26	#341	—	199	1	no	maximal, Sylow
27	C₂×M₁₆	—	199	2	no	—
28	C₂²×C₈	D₁₉₉	1	1	yes	characteristic
29	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
30	M₁₆	—	398	2	no	—
31	M₁₆	—	199	4	no	—
32	C₂×C₈	—	2	2	no	—
33	C₂×C₈	—	398	2	no	—
34	C₂×C₈	Dic₁₉₉	1	2	yes	—
35	C₂³	Dic₃₉₈	1	1	yes	characteristic
36	C₂×C₄	C₄×D₁₉₉	1	2	yes	—
37	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
38	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
39	C₂×C₄	D₇₉₆	1	1	yes	characteristic
40	C₈	—	2	2	no	—
41	C₈	—	398	4	no	—
42	C₂²	—	2	2	no	—
43	C₂²	Dic₁₉₉⋊C₄	1	2	yes	—
44	C₂²	C₄⋊Dic₁₉₉	1	1	yes	characteristic
45	C₄	C₄×Dic₁₉₉	1	1	yes	characteristic
46	C₄	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
47	C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
48	C₂	—	2	2	no	—
49	C₂	C₁₉₉⋊#95	1	1	yes	characteristic
50	C₂	#20263891	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₇₉₆
commutator	C₄×C₁₉₉	C₄²
Frattini	C₂²×C₄	C₂×D₁₉₉
Fitting	C₂²×C₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₄⋊C₄
prime radical	C₂³×C₁₉₉	Q₈
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	C₂²×C₈	D₁₉₉
hypocenter	C₁₉₉	#341
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
25. C₁₉₉	#341	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
C₂²×C₈×C₁₉₉	2
#341	199