Subgroups — #27163787

Order 12736.

98 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	C₂×C₄⋊C₄×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#15691870	—	2	2	no	—
4	#15691870	C₄	1	4	yes	complemented
5	C₂×C₁₉₉⋊C₈	—	2	2	no	—
6	C₂²×C₄×C₁₉₉	C₄	1	2	yes	—
7	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
8	C₄⋊C₄×C₁₉₉	—	2	2	no	—
9	C₁₉₉⋊C₈	—	2	4	no	—
10	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₁₉₉	—	2	2	no	—
12	C₂×C₄×C₁₉₉	—	4	2	no	—
13	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
15	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₉₉	Q₈	1	1	yes	characteristic
17	C₂²×C₁₉₉	—	2	1	no	—
18	C₂²×C₁₉₉	—	2	1	no	—
19	C₂²×C₁₉₉	C₄²	1	1	yes	characteristic
20	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
22	C₄×C₁₉₉	—	4	2	no	—
23	C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
24	C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
25	C₂×C₁₉₉	—	2	1	no	—
26	C₂×C₁₉₉	—	2	1	no	—
27	C₂×C₁₉₉	#95	1	1	yes	characteristic
28	C₂×C₁₉₉	#100	1	1	yes	characteristic
29	C₂×C₁₉₉	#101	1	1	yes	characteristic
30	C₁₉₉	#343	1	1	yes	characteristic, complemented, Sylow
31	#343	—	199	1	no	maximal, Sylow
32	C₂×M₁₆	—	199	2	no	—
33	C₂×C₄⋊C₄	D₁₉₉	1	1	yes	characteristic
34	C₂²×C₄	Dic₁₉₉	1	2	yes	—
35	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
36	M₁₆	—	398	2	no	—
37	M₁₆	—	199	4	no	—
38	C₂×C₈	—	398	2	no	—
39	C₄⋊C₄	—	2	2	no	—
40	C₂³	C₂×Dic₁₉₉	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	4	2	no	—
43	C₂×C₄	C₄×D₁₉₉	1	2	yes	—
44	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
45	C₂×C₄	D₇₉₆	1	1	yes	characteristic
46	C₂×C₄	Dic₃₉₈	1	1	yes	characteristic
47	C₈	—	398	4	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₄×Dic₁₉₉	1	1	yes	characteristic
51	C₂²	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
52	C₂²	C₄⋊Dic₁₉₉	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
55	C₄	Dic₁₉₉⋊C₄	1	2	yes	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₁₉₉⋊#95	1	1	yes	characteristic
59	C₂	#20263872	1	1	yes	characteristic
60	C₂	#20263874	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₁₉₉
commutator	C₂²×C₁₉₉	C₄²
Frattini	C₂²×C₄	C₂×D₁₉₉
Fitting	C₂×C₄⋊C₄×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂²⋊C₄
prime radical	C₂³×C₁₉₉	C₂×C₄
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	C₂×C₄⋊C₄	D₁₉₉
hypocenter	C₁₉₉	#343
LCS	G ▷ C₂²×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. #15691870	C₄	semidirect
30. C₁₉₉	#343	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
C₂×C₄⋊C₄×C₁₉₉	2
#343	199