Subgroups — #27164460

Order 12736.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263978	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263901	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263900	C₂	1	1	yes	maximal, characteristic, complemented
5	#20263898	C₂	1	1	yes	maximal, characteristic, complemented
6	#20263894	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₃₂×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#15691887	—	2	1	no	—
9	Q₈×D₁₉₉	—	2	1	no	—
10	Dic₇₉₆	C₂²	1	1	yes	characteristic
11	#15691876	—	2	1	no	—
12	#15691873	—	2	1	no	—
13	D₁₅₉₂	C₂²	1	1	yes	characteristic
14	C₈×D₁₉₉	C₂²	1	1	yes	characteristic
15	C₁₉₉⋊C₁₆	C₂²	1	1	yes	characteristic
16	Q₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
17	Q₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
18	C₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
19	Dic₃₉₈	—	2	1	no	—
20	Dic₃₉₈	—	4	1	no	—
21	D₇₉₆	—	2	1	no	—
22	D₇₉₆	—	4	1	no	—
23	C₄×D₁₉₉	—	4	1	no	—
24	C₄×D₁₉₉	—	4	1	no	—
25	C₄×D₁₉₉	D₄	1	1	yes	characteristic
26	C₁₉₉⋊C₈	D₄	1	1	yes	characteristic
27	Q₈×C₁₉₉	—	2	1	no	—
28	Q₈×C₁₉₉	—	2	1	no	—
29	C₈×C₁₉₉	C₂³	1	1	yes	characteristic
30	C₂×D₁₉₉	—	4	1	no	—
31	C₂×D₁₉₉	D₈	1	1	yes	characteristic
32	C₄×C₁₉₉	—	4	1	no	—
33	C₄×C₁₉₉	—	4	1	no	—
34	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
35	Dic₁₉₉	—	4	1	no	—
36	Dic₁₉₉	D₈	1	1	yes	characteristic
37	C₂×C₁₉₉	C₂×D₈	1	1	yes	characteristic
38	D₁₉₉	—	2	1	no	—
39	D₁₉₉	—	8	1	no	—
40	C₁₉₉	#510	1	1	yes	characteristic, complemented, Sylow
41	#510	—	199	1	no	maximal, Sylow
42	#135	—	199	1	no	—
43	C₂×Q₁₆	—	199	1	no	—
44	Q₃₂	—	199	1	no	—
45	Q₃₂	D₁₉₉	1	1	yes	characteristic, complemented
46	SD₃₂	—	199	1	no	—
47	SD₃₂	—	199	1	no	—
48	M₃₂	—	199	1	no	—
49	#41	—	398	1	no	—
50	C₂×Q₈	—	398	1	no	—
51	Q₁₆	—	398	1	no	—
52	Q₁₆	—	199	1	no	—
53	Q₁₆	C₂×D₁₉₉	1	1	yes	characteristic
54	Q₁₆	C₂×D₁₉₉	1	1	yes	characteristic
55	SD₁₆	—	398	1	no	—
56	D₈	—	199	1	no	—
57	C₂×C₈	—	199	1	no	—
58	C₁₆	—	199	1	no	—
59	C₁₆	C₂×D₁₉₉	1	1	yes	characteristic
60	Q₈	—	2	1	no	—
61	Q₈	—	2	1	no	—
62	Q₈	—	796	1	no	—
63	Q₈	—	398	1	no	—
64	D₄	—	796	1	no	—
65	D₄	—	398	1	no	—
66	C₂×C₄	—	796	1	no	—
67	C₂×C₄	—	796	1	no	—
68	C₂×C₄	—	199	1	no	—
69	C₈	—	199	1	no	—
70	C₈	C₂²×D₁₉₉	1	1	yes	characteristic
71	C₂²	—	796	1	no	—
72	C₂²	—	199	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	796	1	no	—
76	C₄	—	199	1	no	—
77	C₄	D₄×D₁₉₉	1	1	yes	characteristic
78	C₂	—	1592	1	no	—
79	C₂	—	398	1	no	—
80	C₂	D₈×D₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₁₉₉
commutator, cosocle	C₈×C₁₉₉	C₂³
Frattini	C₈	C₂²×D₁₉₉
Fitting	Q₃₂×C₁₉₉	C₂
socle	C₂×C₁₉₉	C₂×D₈
prime radical	#20263978	C₂
hypercenter	Q₃₂	D₁₉₉
hypocenter	C₁₉₉	#510
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×D₁₉₉	C₂	semidirect
3. #20263901	C₂	semidirect
4. #20263900	C₂	semidirect
5. #20263898	C₂	semidirect
6. #20263894	C₂	semidirect
7. Q₃₂×C₁₉₉	C₂	semidirect
40. C₁₉₉	#510	semidirect
45. Q₃₂	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263978	2
Q₁₆×D₁₉₉	2
#20263901	2
#20263900	2
#20263898	2
#20263894	2
Q₃₂×C₁₉₉	2
#510	199