Subgroups — #27164056

Order 12736.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263914	C₂	1	2	yes	maximal
2	#20263889	C₂	1	2	yes	maximal, complemented
3	#20263887	C₂	1	2	yes	maximal, complemented
4	C₂²×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₁₉₉	C₂²	1	4	yes	—
6	C₂²⋊Dic₁₉₉	—	2	2	no	—
7	C₄×Dic₁₉₉	—	2	2	no	—
8	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
9	C₂×C₈×C₁₉₉	C₄	1	4	yes	complemented
10	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₉₉	—	2	4	no	—
13	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
14	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
16	C₂×C₄×C₁₉₉	Q₈	1	2	yes	—
17	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
18	C₈×C₁₉₉	C₂×C₄	1	2	yes	—
19	C₈×C₁₉₉	C₂×C₄	1	2	yes	—
20	C₂²×C₁₉₉	—	2	2	no	—
21	C₂²×C₁₉₉	C₄⋊C₄	1	2	yes	—
22	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
23	C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
24	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
26	Dic₁₉₉	—	4	4	no	—
27	C₂×C₁₉₉	—	2	2	no	—
28	C₂×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₁₉₉	#135	1	2	yes	—
30	C₁₉₉	#427	1	1	yes	characteristic, complemented, Sylow
31	#427	—	199	1	no	maximal, Sylow
32	C₂²×C₈	D₁₉₉	1	1	yes	characteristic
33	#117	—	199	2	no	—
34	#107	—	199	2	no	—
35	#106	—	199	2	no	—
36	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
37	C₂×C₈	Dic₁₉₉	1	4	yes	complemented
38	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
39	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
40	C₄⋊C₄	—	199	4	no	—
41	C₂²⋊C₄	—	398	2	no	—
42	C₄²	—	398	2	no	—
43	C₂³	D₇₉₆	1	1	yes	characteristic
44	C₂×C₄	—	398	4	no	—
45	C₂×C₄	C₂×Dic₁₉₉	1	2	yes	—
46	C₂×C₄	D₇₉₆	1	1	yes	characteristic
47	C₂×C₄	Dic₃₉₈	1	2	yes	—
48	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
49	C₈	C₂×Dic₁₉₉	1	2	yes	—
50	C₈	C₂×Dic₁₉₉	1	2	yes	—
51	C₂²	—	2	2	no	—
52	C₂²	C₄⋊Dic₁₉₉	1	2	yes	—
53	C₂²	C₂×D₇₉₆	1	1	yes	characteristic
54	C₄	—	796	4	no	—
55	C₄	C₄⋊Dic₁₉₉	1	2	yes	—
56	C₄	C₂²×Dic₁₉₉	1	1	yes	characteristic
57	C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	C₂×C₄⋊Dic₁₉₉	1	1	yes	characteristic
60	C₂	#20263977	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₇₉₆
commutator	C₄×C₁₉₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₂²×C₈×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	C₂³×C₁₉₉	D₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₂²×C₈	D₁₉₉
hypocenter	C₁₉₉	#427
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. #20263889	C₂	semidirect
3. #20263887	C₂	semidirect
9. C₂×C₈×C₁₉₉	C₄	semidirect
30. C₁₉₉	#427	semidirect
37. C₂×C₈	Dic₁₉₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20263914	2
#20263889	2
#20263887	2
C₂²×C₈×C₁₉₉	2
#427	199