Subgroups — #27164042

Order 12736.

126 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	C₄⋊(C₁₉₉⋊C₈)	C₂	1	4	yes	maximal, complemented
3	C₂×C₄²×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	#15691870	—	2	4	no	—
5	C₂×C₁₉₉⋊C₈	C₂²	1	4	yes	—
6	C₂²×C₄×C₁₉₉	C₄	1	2	yes	—
7	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
8	C₄²×C₁₉₉	C₄	1	1	yes	characteristic
9	C₄²×C₁₉₉	C₄	1	1	yes	characteristic
10	C₄²×C₁₉₉	C₂²	1	2	yes	—
11	C₁₉₉⋊C₈	—	2	4	no	—
12	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₉₉	—	2	4	no	—
14	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
18	C₂×C₄×C₁₉₉	Q₈	1	2	yes	—
19	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
20	C₂²×C₁₉₉	—	2	2	no	—
21	C₂²×C₁₉₉	C₄⋊C₄	1	2	yes	—
22	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
23	C₄×C₁₉₉	—	2	2	no	—
24	C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
25	C₄×C₁₉₉	M₁₆	1	4	yes	complemented
26	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
27	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₁₉₉	—	2	2	no	—
29	C₂×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₁₉₉	C₂×M₁₆	1	2	yes	—
31	C₁₉₉	#423	1	1	yes	characteristic, complemented, Sylow
32	#423	—	199	1	no	maximal, Sylow
33	C₂×M₁₆	—	199	2	no	—
34	C₂×C₄²	D₁₉₉	1	1	yes	characteristic
35	C₄⋊C₈	—	199	4	no	—
36	C₂²×C₄	Dic₁₉₉	1	2	yes	—
37	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
38	M₁₆	—	398	4	no	—
39	C₂×C₈	—	199	4	no	—
40	C₄²	Dic₁₉₉	1	1	yes	characteristic
41	C₄²	Dic₁₉₉	1	1	yes	characteristic
42	C₄²	C₂×D₁₉₉	1	2	yes	—
43	C₂³	C₂×Dic₁₉₉	1	1	yes	characteristic
44	C₂×C₄	—	2	4	no	—
45	C₂×C₄	C₂×Dic₁₉₉	1	2	yes	—
46	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
47	C₂×C₄	C₂×Dic₁₉₉	1	2	yes	—
48	C₂×C₄	D₇₉₆	1	2	yes	—
49	C₂×C₄	Dic₃₉₈	1	2	yes	—
50	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
51	C₈	—	398	4	no	—
52	C₂²	—	2	2	no	—
53	C₂²	C₄⋊Dic₁₉₉	1	2	yes	—
54	C₂²	C₂²×Dic₁₉₉	1	1	yes	characteristic
55	C₄	—	2	2	no	—
56	C₄	C₄⋊Dic₁₉₉	1	2	yes	—
57	C₄	#15691870	1	4	yes	complemented
58	C₄	C₂×D₇₉₆	1	1	yes	characteristic
59	C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×C₄⋊Dic₁₉₉	1	1	yes	characteristic
62	C₂	C₂×#15691870	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉₉
commutator, socle	C₂²×C₁₉₉	C₂²×C₄
Fitting	C₂×C₄²×C₁₉₉	C₂
prime radical	C₂³×C₁₉₉	C₂×C₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₂×C₄²	D₁₉₉
hypocenter	C₁₉₉	#423
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₁₉₉⋊C₈)	C₂	semidirect
25. C₄×C₁₉₉	M₁₆	semidirect
31. C₁₉₉	#423	semidirect
57. C₄	#15691870	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
C₄⋊(C₁₉₉⋊C₈)	2
C₂×C₄²×C₁₉₉	2
#423	199