Subgroups — #27164461

Order 12736.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263977	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₉₆	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₅₉₂	C₂	1	2	yes	maximal, complemented
4	#20263900	C₂	1	2	yes	maximal, complemented
5	M₃₂×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#15691886	—	2	1	no	—
7	C₂×Dic₃₉₈	—	2	1	no	—
8	Dic₇₉₆	—	2	1	no	—
9	Dic₇₉₆	C₂²	1	2	yes	—
10	Dic₇₉₆	C₂²	1	1	yes	characteristic
11	#15691875	—	2	1	no	—
12	D₁₅₉₂	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₁₆×C₁₉₉	C₂²	1	2	yes	—
15	Dic₃₉₈	—	2	1	no	—
16	Dic₃₉₈	—	2	2	no	—
17	Dic₃₉₈	—	4	1	no	—
18	D₇₉₆	—	2	1	no	—
19	C₂²⋊D₁₉₉	—	4	1	no	—
20	C₂×Dic₁₉₉	—	4	1	no	—
21	C₄×D₁₉₉	—	4	1	no	—
22	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
23	C₈×C₁₉₉	D₄	1	1	yes	characteristic
24	C₈×C₁₉₉	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₉	D₈	1	1	yes	characteristic
26	C₂×D₁₉₉	—	4	1	no	—
27	C₄×C₁₉₉	D₈	1	1	yes	characteristic
28	C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
29	Dic₁₉₉	—	4	1	no	—
30	Dic₁₉₉	—	4	2	no	—
31	C₂×C₁₉₉	—	2	1	no	—
32	C₂×C₁₉₉	C₂×D₈	1	1	yes	characteristic
33	D₁₉₉	—	8	1	no	—
34	C₁₉₉	#510	1	1	yes	characteristic, complemented, Sylow
35	#510	—	199	1	no	maximal, Sylow
36	#135	—	199	1	no	—
37	C₂×Q₁₆	—	199	1	no	—
38	Q₃₂	—	199	2	no	—
39	SD₃₂	—	199	2	no	—
40	M₃₂	D₁₉₉	1	1	yes	characteristic, complemented
41	#41	—	398	1	no	—
42	C₂×Q₈	—	398	1	no	—
43	Q₁₆	—	398	1	no	—
44	Q₁₆	—	199	2	no	—
45	Q₁₆	—	199	1	no	—
46	SD₁₆	—	398	1	no	—
47	D₈	—	199	1	no	—
48	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
49	C₁₆	C₂×D₁₉₉	1	2	yes	—
50	Q₈	—	796	1	no	—
51	Q₈	—	398	1	no	—
52	Q₈	—	398	2	no	—
53	D₄	—	796	1	no	—
54	D₄	—	398	1	no	—
55	C₂×C₄	—	796	1	no	—
56	C₂×C₄	—	796	1	no	—
57	C₂×C₄	D₇₉₆	1	1	yes	characteristic
58	C₈	D₇₉₆	1	1	yes	characteristic
59	C₈	C₂²×D₁₉₉	1	1	yes	characteristic
60	C₂²	—	796	1	no	—
61	C₂²	D₁₅₉₂	1	1	yes	characteristic
62	C₄	—	796	1	no	—
63	C₄	—	796	2	no	—
64	C₄	D₁₅₉₂	1	1	yes	characteristic
65	C₄	C₂×D₇₉₆	1	1	yes	characteristic
66	C₂	—	2	1	no	—
67	C₂	—	1592	1	no	—
68	C₂	C₂×D₁₅₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₅₉₂
commutator, cosocle	C₈×C₁₉₉	C₂³
Frattini	C₈	C₂²×D₁₉₉
Fitting	M₃₂×C₁₉₉	C₂
socle	C₂×C₁₉₉	C₂×D₈
prime radical	#20263977	C₂
hypercenter	M₃₂	D₁₉₉
hypocenter	C₁₉₉	#510
LCS	G ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇₉₆	C₂	semidirect
3. Dic₁₅₉₂	C₂	semidirect
4. #20263900	C₂	semidirect
5. M₃₂×C₁₉₉	C₂	semidirect
34. C₁₉₉	#510	semidirect
40. M₃₂	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#20263977	2
C₂×Dic₇₉₆	2
Dic₁₅₉₂	2
#20263900	2
M₃₂×C₁₉₉	2
#510	199