Subgroups — #27163788

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	C₂×#15691870	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	#15691870	—	2	1	no	—
7	#15691870	C₄	1	2	yes	complemented
8	C₂×C₁₉₉⋊C₈	—	2	1	no	—
9	C₂×C₁₉₉⋊C₈	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	C₄	1	2	yes	—
13	C₄²×C₁₉₉	—	2	1	no	—
14	C₁₉₉⋊C₈	—	2	2	no	—
15	C₁₉₉⋊C₈	—	2	2	no	—
16	C₂³×C₁₉₉	C₂×C₄	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	D₄	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	C₄²	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	C₂²⋊C₄	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	C₂×C₁₉₉	—	2	1	no	—
34	C₂×C₁₉₉	—	2	1	no	—
35	C₂×C₁₉₉	—	2	1	no	—
36	C₂×C₁₉₉	#95	1	1	yes	characteristic
37	C₁₉₉	#344	1	1	yes	characteristic, complemented, Sylow
38	#344	—	199	1	no	maximal, Sylow
39	C₂×M₁₆	—	199	1	no	—
40	C₂×M₁₆	—	199	1	no	—
41	#117	D₁₉₉	1	1	yes	characteristic
42	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
43	M₁₆	—	398	1	no	—
44	M₁₆	—	398	1	no	—
45	M₁₆	—	398	1	no	—
46	M₁₆	—	199	2	no	—
47	C₂×C₈	—	398	1	no	—
48	C₂×C₈	—	199	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	Dic₁₉₉	1	2	yes	—
51	C₄²	—	2	1	no	—
52	C₂³	C₂×Dic₁₉₉	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
55	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
57	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
58	C₂×C₄	D₇₉₆	1	1	yes	characteristic
59	C₂×C₄	Dic₃₉₈	1	1	yes	characteristic
60	C₈	—	398	2	no	—
61	C₈	—	398	2	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₄×Dic₁₉₉	1	1	yes	characteristic
64	C₂²	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
65	C₂²	C₄⋊Dic₁₉₉	1	1	yes	characteristic
66	C₄	—	4	2	no	—
67	C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
68	C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
69	C₄	Dic₁₉₉⋊C₄	1	1	yes	characteristic
70	C₄	Dic₁₉₉⋊C₄	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	2	1	no	—
73	C₂	—	2	1	no	—
74	C₂	C₁₉₉⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	(C₂×D₁₉₉)⋊C₄
commutator	C₂²×C₁₉₉	C₄²
Frattini	C₂²×C₄	C₂×D₁₉₉
Fitting	#117×C₁₉₉	C₂
socle	C₂×C₁₉₉	#95
prime radical	C₂³×C₁₉₉	C₂×C₄
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	#117	D₁₉₉
hypocenter	C₁₉₉	#344
LCS	G ▷ C₂²×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
7. #15691870	C₄	semidirect
9. C₂×C₁₉₉⋊C₈	C₄	semidirect
37. C₁₉₉	#344	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
#117×C₁₉₉	2
#344	199