Subgroups — #27163782

Order 12736.

90 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	C₂²×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	#15691870	—	2	1	no	—
5	#15691870	C₄	1	2	yes	—
6	C₂×C₁₉₉⋊C₈	—	2	1	no	—
7	C₂×C₁₉₉⋊C₈	—	2	1	no	—
8	C₂×C₁₉₉⋊C₈	—	2	1	no	—
9	C₂×C₁₉₉⋊C₈	C₄	1	2	yes	—
10	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
11	M₁₆×C₁₉₉	—	2	1	no	—
12	M₁₆×C₁₉₉	C₄	1	2	yes	—
13	C₂×C₈×C₁₉₉	—	2	1	no	—
14	C₁₉₉⋊C₈	—	2	2	no	—
15	C₁₉₉⋊C₈	—	2	2	no	—
16	C₂³×C₁₉₉	Q₈	1	1	yes	characteristic
17	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
23	C₈×C₁₉₉	—	2	2	no	—
24	C₂²×C₁₉₉	—	2	2	no	—
25	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
28	C₄×C₁₉₉	C₄²	1	1	yes	characteristic
29	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₁₉₉	—	2	1	no	—
33	C₂×C₁₉₉	—	2	1	no	—
34	C₂×C₁₉₉	#95	1	1	yes	characteristic
35	C₂×C₁₉₉	#108	1	2	yes	—
36	C₁₉₉	#341	1	1	yes	characteristic, complemented, Sylow
37	#341	—	199	1	no	maximal, Sylow
38	C₂×M₁₆	—	199	1	no	—
39	C₂×M₁₆	D₁₉₉	1	1	yes	characteristic
40	C₂²×C₈	—	199	1	no	—
41	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
42	M₁₆	—	2	1	no	—
43	M₁₆	—	398	1	no	—
44	M₁₆	—	199	2	no	—
45	M₁₆	Dic₁₉₉	1	2	yes	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	398	1	no	—
48	C₂×C₈	—	398	1	no	—
49	C₂×C₈	—	398	1	no	—
50	C₂×C₈	—	199	2	no	—
51	C₂³	Dic₃₉₈	1	1	yes	characteristic
52	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
53	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
54	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
55	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
57	C₂×C₄	D₇₉₆	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₈	—	398	2	no	—
60	C₈	—	398	2	no	—
61	C₂²	—	2	2	no	—
62	C₂²	Dic₁₉₉⋊C₄	1	1	yes	characteristic
63	C₂²	Dic₁₉₉⋊C₄	1	1	yes	characteristic
64	C₂²	C₄⋊Dic₁₉₉	1	1	yes	characteristic
65	C₄	C₄×Dic₁₉₉	1	1	yes	characteristic
66	C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
67	C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
68	C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	C₁₉₉⋊#95	1	1	yes	characteristic
72	C₂	#20263890	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₁₉₉
commutator	C₄×C₁₉₉	C₄²
Frattini	C₂²×C₄	C₂×D₁₉₉
Fitting	C₂×M₁₆×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₄⋊C₄
prime radical	C₂³×C₁₉₉	Q₈
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	C₂×M₁₆	D₁₉₉
hypocenter	C₁₉₉	#341
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
36. 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×C₁₉₉⋊C₈	2
C₂×M₁₆×C₁₉₉	2
#341	199