Subgroups — #27164061

Order 12736.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691870	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#20263888	C₂	1	2	yes	maximal, complemented
4	#20263886	C₂	1	2	yes	maximal, complemented
5	#117×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₉₉	—	2	1	no	—
7	#15691870	C₄	1	4	yes	complemented
8	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₉₉	—	2	1	no	—
10	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₁₉₉	—	2	1	no	—
15	C₄²×C₁₉₉	—	2	1	no	—
16	C₂×Dic₁₉₉	—	2	2	no	—
17	C₂×Dic₁₉₉	—	4	2	no	—
18	C₁₉₉⋊C₈	C₂×C₄	1	4	yes	complemented
19	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₉	—	2	1	no	—
21	C₂×C₄×C₁₉₉	—	2	1	no	—
22	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₉₉	Q₈	1	2	yes	—
25	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
26	C₂²×C₁₉₉	—	2	1	no	—
27	C₂²×C₁₉₉	—	2	1	no	—
28	C₂²×C₁₉₉	C₄⋊C₄	1	2	yes	—
29	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
30	C₄×C₁₉₉	—	4	1	no	—
31	C₄×C₁₉₉	—	4	1	no	—
32	C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
33	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
35	Dic₁₉₉	—	4	2	no	—
36	C₂×C₁₉₉	—	2	2	no	—
37	C₂×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
39	C₂×C₁₉₉	#137	1	1	yes	characteristic
40	C₁₉₉	#428	1	1	yes	characteristic, complemented, Sylow
41	#428	—	199	1	no	maximal, Sylow
42	C₂×M₁₆	—	199	1	no	—
43	#117	D₁₉₉	1	1	yes	characteristic
44	C₂×C₄⋊C₄	—	199	1	no	—
45	#107	—	199	2	no	—
46	#106	—	199	2	no	—
47	C₂²×C₄	—	398	1	no	—
48	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
49	M₁₆	—	199	4	no	—
50	C₂×C₈	—	199	2	no	—
51	C₄⋊C₄	—	398	1	no	—
52	C₄⋊C₄	—	199	2	no	—
53	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
54	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
55	C₂²⋊C₄	—	2	1	no	—
56	C₄²	—	2	1	no	—
57	C₂³	C₂²⋊D₁₉₉	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	796	2	no	—
61	C₂×C₄	—	398	2	no	—
62	C₂×C₄	C₄×D₁₉₉	1	2	yes	—
63	C₂×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
64	C₂×C₄	Dic₃₉₈	1	2	yes	—
65	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
66	C₈	—	199	4	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	Dic₁₉₉⋊C₄	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	796	2	no	—
74	C₄	Dic₁₉₉⋊C₄	1	2	yes	—
75	C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
76	C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×Dic₁₉₉⋊C₄	1	1	yes	characteristic
79	C₂	#20263980	1	1	yes	characteristic
80	C₂	#20263985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₉₉
commutator	C₄×C₁₉₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	#117×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂×D₄
prime radical	C₂³×C₁₉₉	D₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	#117	D₁₉₉
hypocenter	C₁₉₉	#428
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #20263888	C₂	semidirect
4. #20263886	C₂	semidirect
7. #15691870	C₄	semidirect
18. C₁₉₉⋊C₈	C₂×C₄	semidirect
40. C₁₉₉	#428	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
C₂×C₄⋊Dic₁₉₉	2
#20263888	2
#20263886	2
#117×C₁₉₉	2
#428	199