Subgroups — #27163786

Order 12736.

98 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×C₄⋊Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₂×M₁₆×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₁₉₉	C₄	1	2	yes	—
5	#15691870	—	2	1	no	—
6	#15691870	C₄	1	2	yes	complemented
7	C₂×C₁₉₉⋊C₈	—	2	1	no	—
8	C₄⋊Dic₁₉₉	—	2	1	no	—
9	C₄⋊Dic₁₉₉	—	2	1	no	—
10	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
11	M₁₆×C₁₉₉	—	2	1	no	—
12	M₁₆×C₁₉₉	C₄	1	2	yes	complemented
13	C₂×C₈×C₁₉₉	—	2	1	no	—
14	C₂×Dic₁₉₉	—	2	2	no	—
15	C₂×Dic₁₉₉	—	4	2	no	—
16	C₁₉₉⋊C₈	—	2	2	no	—
17	C₂³×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	C₂×C₄×C₁₉₉	Q₈	1	1	yes	characteristic
24	C₈×C₁₉₉	—	2	2	no	—
25	C₂²×C₁₉₉	—	2	1	no	—
26	C₂²×C₁₉₉	—	2	1	no	—
27	C₂²×C₁₉₉	C₄²	1	1	yes	characteristic
28	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
34	Dic₁₉₉	—	4	2	no	—
35	C₂×C₁₉₉	—	2	1	no	—
36	C₂×C₁₉₉	—	2	1	no	—
37	C₂×C₁₉₉	#95	1	1	yes	characteristic
38	C₂×C₁₉₉	#100	1	1	yes	characteristic
39	C₂×C₁₉₉	#101	1	1	yes	characteristic
40	C₁₉₉	#343	1	1	yes	characteristic, complemented, Sylow
41	#343	—	199	1	no	maximal, Sylow
42	C₂×M₁₆	—	199	1	no	—
43	C₂×M₁₆	D₁₉₉	1	1	yes	characteristic
44	C₂×C₄⋊C₄	—	199	1	no	—
45	C₂²×C₄	—	199	2	no	—
46	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
47	M₁₆	—	2	1	no	—
48	M₁₆	—	398	1	no	—
49	M₁₆	—	199	2	no	—
50	M₁₆	Dic₁₉₉	1	2	yes	complemented
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	398	1	no	—
53	C₄⋊C₄	—	398	1	no	—
54	C₄⋊C₄	—	398	1	no	—
55	C₂³	C₄×D₁₉₉	1	1	yes	characteristic
56	C₂×C₄	—	796	2	no	—
57	C₂×C₄	—	398	2	no	—
58	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
59	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
60	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
62	C₂×C₄	D₇₉₆	1	1	yes	characteristic
63	C₂×C₄	Dic₃₉₈	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₈	—	398	2	no	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	C₄×Dic₁₉₉	1	1	yes	characteristic
69	C₂²	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
70	C₂²	Dic₁₉₉⋊C₄	1	1	yes	characteristic
71	C₄	—	796	2	no	—
72	C₄	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
73	C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
74	C₄	Dic₁₉₉⋊C₄	1	1	yes	characteristic
75	C₄	C₄⋊Dic₁₉₉	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	2	1	no	—
78	C₂	C₁₉₉⋊#95	1	1	yes	characteristic
79	C₂	#20263871	1	1	yes	characteristic
80	C₂	#20263873	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₉₉)⋊C₄
commutator	C₂²×C₁₉₉	C₄²
Frattini	C₂²×C₄	C₂×D₁₉₉
Fitting	C₂×M₁₆×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂²⋊C₄
prime radical	C₂³×C₁₉₉	C₂×C₄
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	C₂×M₁₆	D₁₉₉
hypocenter	C₁₉₉	#343
LCS	G ▷ C₂²×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
6. #15691870	C₄	semidirect
12. M₁₆×C₁₉₉	C₄	semidirect
40. C₁₉₉	#343	semidirect
50. M₁₆	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
C₂×C₄⋊Dic₁₉₉	2
C₂×M₁₆×C₁₉₉	2
#343	199