Subgroups — #27164378

Order 12736.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263955	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263943	C₂	1	1	yes	maximal, characteristic
3	#20263880	C₂	1	2	yes	maximal, complemented
4	#20263877	C₂	1	2	yes	maximal
5	#97×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₉₈	—	2	1	no	—
7	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
8	C₂×D₇₉₆	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₉₉	—	2	1	no	—
10	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
11	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
12	(C₂×D₁₉₉)⋊C₄	—	2	1	no	—
13	C₂×C₈×C₁₉₉	C₂²	1	2	yes	—
14	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	C₂²×D₁₉₉	—	2	1	no	—
16	Dic₃₉₈	—	2	2	no	—
17	Dic₃₉₈	—	4	2	no	—
18	D₇₉₆	—	2	2	no	—
19	C₂×Dic₁₉₉	—	2	1	no	—
20	C₂×Dic₁₉₉	—	2	2	no	—
21	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
24	C₈×C₁₉₉	—	2	2	no	—
25	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
26	C₂×D₁₉₉	—	4	1	no	—
27	C₂×D₁₉₉	—	4	1	no	—
28	C₂×D₁₉₉	—	4	1	no	—
29	C₄×C₁₉₉	—	2	1	no	—
30	C₄×C₁₉₉	—	2	1	no	—
31	C₄×C₁₉₉	#41	1	2	yes	—
32	Dic₁₉₉	—	4	1	no	—
33	Dic₁₉₉	—	4	2	no	—
34	C₂×C₁₉₉	#124	1	1	yes	characteristic
35	C₂×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₁₉₉	#137	1	1	yes	characteristic
37	D₁₉₉	—	8	1	no	—
38	C₁₉₉	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	199	1	no	maximal, Sylow
40	C₄⋊Q₈	—	199	1	no	—
41	#124	—	199	1	no	—
42	#103	—	199	2	no	—
43	#102	—	199	2	no	—
44	#97	D₁₉₉	1	1	yes	characteristic, complemented
45	C₂×Q₈	—	398	1	no	—
46	C₂×Q₈	—	199	1	no	—
47	C₂×D₄	—	199	1	no	—
48	C₂×C₈	C₂×D₁₉₉	1	2	yes	—
49	C₄⋊C₄	—	398	1	no	—
50	C₄⋊C₄	—	199	2	no	—
51	C₂²⋊C₄	—	398	1	no	—
52	C₂²⋊C₄	—	398	1	no	—
53	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
54	C₂³	—	398	1	no	—
55	Q₈	—	796	2	no	—
56	Q₈	—	398	2	no	—
57	D₄	—	398	2	no	—
58	C₂×C₄	—	398	1	no	—
59	C₂×C₄	—	398	2	no	—
60	C₂×C₄	D₇₉₆	1	1	yes	characteristic
61	C₂×C₄	D₇₉₆	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₂²	—	796	1	no	—
65	C₂²	—	796	1	no	—
66	C₂²	—	796	1	no	—
67	C₂²	C₂×D₇₉₆	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	796	1	no	—
71	C₄	—	796	2	no	—
72	C₄	#15691886	1	2	yes	—
73	C₂	—	1592	1	no	—
74	C₂	#20263943	1	1	yes	characteristic
75	C₂	#20263983	1	1	yes	characteristic
76	C₂	#20263988	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₇₉₆
commutator, cosocle	C₂×C₄×C₁₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#97×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	C₂×D₇₉₆	C₂²
hypercenter	#97	D₁₉₉
hypocenter	C₁₉₉	#489
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. #20263955	C₂	semidirect
3. #20263880	C₂	semidirect
5. #97×C₁₉₉	C₂	semidirect
38. C₁₉₉	#489	semidirect
44. #97	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263955	2
#20263943	2
#20263880	2
#20263877	2
#97×C₁₉₉	2
#489	199