Subgroups — #27163757

Order 12736.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263943	C₂	1	1	yes	maximal, characteristic
2	#20263866	C₂	1	1	yes	maximal, characteristic, complemented
3	#97×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₃₉₈	C₄	1	1	yes	characteristic
5	C₂×D₇₉₆	C₄	1	1	yes	characteristic
6	C₂×C₁₉₉⋊C₈	—	2	1	no	—
7	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
8	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
9	C₂×C₈×C₁₉₉	—	2	1	no	—
10	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂²×D₁₉₉	—	2	1	no	—
12	Dic₃₉₈	—	4	1	no	—
13	D₇₉₆	—	4	1	no	—
14	C₂×Dic₁₉₉	—	2	1	no	—
15	C₁₉₉⋊C₈	—	2	2	no	—
16	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
19	C₈×C₁₉₉	—	2	2	no	—
20	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×D₁₉₉	—	4	1	no	—
22	C₂×D₁₉₉	—	4	1	no	—
23	C₂×D₁₉₉	—	4	1	no	—
24	C₄×C₁₉₉	—	2	1	no	—
25	C₄×C₁₉₉	—	2	1	no	—
26	C₄×C₁₉₉	—	2	1	no	—
27	Dic₁₉₉	—	4	1	no	—
28	C₂×C₁₉₉	#100	1	1	yes	characteristic
29	C₂×C₁₉₉	#104	1	1	yes	characteristic
30	C₂×C₁₉₉	#104	1	1	yes	characteristic
31	D₁₉₉	—	8	1	no	—
32	C₁₉₉	#329	1	1	yes	characteristic, complemented, Sylow
33	#329	—	199	1	no	maximal, Sylow
34	#124	—	199	1	no	—
35	#97	—	199	1	no	—
36	#97	D₁₉₉	1	1	yes	characteristic, complemented
37	C₂×Q₈	—	199	1	no	—
38	C₂×D₄	—	199	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	398	1	no	—
41	C₂²⋊C₄	—	398	1	no	—
42	C₂²⋊C₄	—	398	1	no	—
43	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
44	C₂³	—	398	1	no	—
45	Q₈	—	796	1	no	—
46	D₄	—	796	1	no	—
47	C₂×C₄	—	398	1	no	—
48	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
50	C₂×C₄	D₇₉₆	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	398	2	no	—
53	C₂²	—	796	1	no	—
54	C₂²	—	796	1	no	—
55	C₂²	—	796	1	no	—
56	C₂²	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	—	2	1	no	—
59	C₄	—	2	1	no	—
60	C₄	—	796	1	no	—
61	C₂	—	1592	1	no	—
62	C₂	#20263871	1	1	yes	characteristic
63	C₂	#20263881	1	1	yes	characteristic
64	C₂	#20263883	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₉₉)⋊C₄
commutator	C₂×C₄×C₁₉₉	C₂×C₄
Frattini	C₄²	C₂×D₁₉₉
Fitting	#97×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂²⋊C₄
prime radical	C₂×D₇₉₆	C₄
cosocle	C₄²×C₁₉₉	C₂²
hypercenter	#97	D₁₉₉
hypocenter	C₁₉₉	#329
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. #20263866	C₂	semidirect
3. #97×C₁₉₉	C₂	semidirect
32. C₁₉₉	#329	semidirect
36. #97	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263943	2
#20263866	2
#97×C₁₉₉	2
#329	199