Subgroups — #29567904

Order 15616.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₉₅₂	C₂	1	1	yes	maximal, characteristic
2	#21444023	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₉₇₆	—	2	1	no	—
5	D₁₉₅₂	—	2	1	no	—
6	D₁₉₅₂	C₄	1	2	yes	—
7	#16839592	—	2	1	no	—
8	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
9	C₆₁×C₆₄	—	2	1	no	—
10	C₂×D₄₈₈	—	4	1	no	—
11	D₉₇₆	—	2	2	no	—
12	D₉₇₆	—	4	1	no	—
13	#14519713	—	4	1	no	—
14	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
15	C₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₆₁	D₄	1	1	yes	characteristic
17	C₂×D₂₄₄	—	8	1	no	—
18	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
19	#11758194	—	8	1	no	—
20	D₄₈₈	—	4	2	no	—
21	D₄₈₈	—	8	1	no	—
22	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₆₁	SD₁₆	1	1	yes	characteristic
24	C₂²×D₆₁	—	16	1	no	—
25	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
26	D₂₄₄	—	8	2	no	—
27	D₂₄₄	—	16	1	no	—
28	C₈×C₆₁	#102	1	1	yes	characteristic
29	C₈×C₆₁	SD₃₂	1	1	yes	characteristic
30	#92418	—	16	1	no	—
31	#7541	—	61	1	no	maximal, Sylow
32	C₂²×C₆₁	SD₆₄	1	1	yes	characteristic
33	C₂×D₆₁	—	16	2	no	—
34	C₂×D₆₁	—	32	2	no	—
35	C₄×C₆₁	#357	1	1	yes	characteristic
36	C₄×C₆₁	D₃₂	1	1	yes	characteristic
37	C₂×D₃₂	—	61	1	no	—
38	M₁₂₈	D₆₁	1	1	yes	characteristic, complemented
39	#1425	—	61	1	no	—
40	C₂×C₆₁	—	2	1	no	—
41	C₂×C₆₁	#1415	1	1	yes	characteristic
42	D₆₁	—	32	2	no	—
43	C₂×D₁₆	—	122	1	no	—
44	D₃₂	—	61	2	no	—
45	D₃₂	—	122	1	no	—
46	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
47	#368	—	122	1	no	—
48	C₆₄	—	2	1	no	—
49	C₆₁	#7541	1	1	yes	characteristic, complemented, Sylow
50	C₂×D₈	—	244	1	no	—
51	D₁₆	—	122	2	no	—
52	D₁₆	—	244	1	no	—
53	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
54	#108	—	244	1	no	—
55	C₃₂	C₄×D₆₁	1	1	yes	characteristic
56	C₃₂	C₂²⋊D₆₁	1	1	yes	characteristic
57	C₂×D₄	—	488	1	no	—
58	D₈	—	488	1	no	—
59	D₈	—	244	2	no	—
60	M₁₆	—	488	1	no	—
61	C₂×C₈	D₄₈₈	1	1	yes	characteristic
62	C₁₆	#11758190	1	1	yes	characteristic
63	C₁₆	#11758198	1	1	yes	characteristic
64	C₂³	—	976	1	no	—
65	D₄	—	976	1	no	—
66	D₄	—	488	2	no	—
67	C₂×C₄	D₉₇₆	1	1	yes	characteristic
68	C₈	—	976	1	no	—
69	C₈	#14519699	1	1	yes	characteristic
70	C₈	#14519722	1	1	yes	characteristic
71	C₂²	—	1952	2	no	—
72	C₂²	—	976	2	no	—
73	C₂²	#16839601	1	1	yes	characteristic
74	C₄	#16839564	1	1	yes	characteristic
75	C₄	D₁₉₅₂	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	1952	2	no	—
78	C₂	#21443998	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21443998
commutator	C₃₂×C₆₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₆₁
Fitting	C₆₁×M₁₂₈	C₂
socle	C₂×C₆₁	#1415
prime radical	C₂×D₁₉₅₂	C₂
cosocle	C₂×C₃₂×C₆₁	C₂²
hypercenter	M₁₂₈	D₆₁
hypocenter	C₆₁	#7541
LCS	G ▷ C₃₂×C₆₁ ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #21444023	C₂	semidirect
3. C₆₁×M₁₂₈	C₂	semidirect
38. M₁₂₈	D₆₁	semidirect
49. C₆₁	#7541	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₉₅₂	2
#21444023	2
C₆₁×M₁₂₈	2
#7541	61