Subgroups — #27164384

Order 12736.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263952	C₂	1	1	yes	maximal, characteristic
2	#20263875	C₂	1	4	yes	maximal, complemented
3	#20263866	C₂	1	1	yes	maximal, characteristic, complemented
4	#125×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₇₉₆	—	2	2	no	—
6	C₂×D₇₉₆	C₂²	1	2	yes	—
7	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
8	C₄⋊C₄×C₁₉₉	—	2	2	no	—
9	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
10	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂²×D₁₉₉	—	2	2	no	—
12	D₇₉₆	—	2	4	no	—
13	D₇₉₆	—	4	4	no	—
14	C₁₉₉⋊C₈	—	2	2	no	—
15	C₂×C₄×C₁₉₉	—	2	2	no	—
16	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
17	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
18	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
19	C₂×D₁₉₉	—	4	2	no	—
20	C₂×D₁₉₉	—	4	4	no	—
21	C₄×C₁₉₉	—	2	2	no	—
22	C₄×C₁₉₉	—	4	2	no	—
23	C₄×C₁₉₉	#41	1	2	yes	—
24	C₂×C₁₉₉	#124	1	1	yes	characteristic
25	C₂×C₁₉₉	AGL(1,ℤ/8)	1	2	yes	—
26	D₁₉₉	—	8	2	no	—
27	C₁₉₉	#490	1	1	yes	characteristic, complemented, Sylow
28	#490	—	199	1	no	maximal, Sylow
29	#127	—	199	1	no	—
30	#125	D₁₉₉	1	1	yes	characteristic, complemented
31	#102	—	199	4	no	—
32	#97	—	199	1	no	—
33	C₂×D₄	—	398	2	no	—
34	C₂×D₄	—	199	2	no	—
35	C₂×C₈	—	199	2	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
38	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
39	C₂³	—	398	2	no	—
40	D₄	—	796	4	no	—
41	D₄	—	398	4	no	—
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
44	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
45	C₈	—	398	2	no	—
46	C₂²	—	796	2	no	—
47	C₂²	—	796	4	no	—
48	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
49	C₄	—	2	2	no	—
50	C₄	—	4	2	no	—
51	C₄	#15691887	1	2	yes	—
52	C₂	—	1592	2	no	—
53	C₂	#20263944	1	1	yes	characteristic
54	C₂	#20263980	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#125×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	#20263952	C₂
hypercenter	#125	D₁₉₉
hypocenter	C₁₉₉	#490
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #20263875	C₂	semidirect
3. #20263866	C₂	semidirect
4. #125×C₁₉₉	C₂	semidirect
27. C₁₉₉	#490	semidirect
30. #125	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263952	2
#20263875	2
#20263866	2
#125×C₁₉₉	2
#490	199