Subgroups — #29596427

Order 15616.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11758185	C₂	1	6	yes	maximal
2	C₆₁×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11758185	C₂²	1	4	yes	—
4	C₄⋊#11758185	—	2	12	no	—
5	C₂×#97×C₆₁	C₄	1	3	yes	—
6	C₂×C₄×C₈×C₆₁	C₂²	1	3	yes	—
7	C₆₁×#376	C₄	1	1	yes	characteristic
8	C₂×#11758185	—	2	12	no	—
9	C₂²×C₈×C₆₁	C₂×C₄	1	3	yes	—
10	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
11	C₂×C₄²×C₆₁	C₂×C₄	1	3	yes	—
12	#97×C₆₁	—	2	6	no	—
13	C₄×C₈×C₆₁	D₄	1	6	yes	—
14	C₄×C₈×C₆₁	Q₈	1	6	yes	—
15	#95×C₆₁	C₈	1	4	yes	—
16	C₂²×C₄×C₆₁	C₂×C₈	1	6	yes	—
17	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₈×C₆₁	—	2	6	no	—
19	C₂×C₈×C₆₁	C₄⋊C₄	1	6	yes	—
20	C₂×C₈×C₆₁	C₂×D₄	1	3	yes	—
21	C₂×C₈×C₆₁	C₂×Q₈	1	3	yes	—
22	C₄²×C₆₁	C₄⋊C₄	1	6	yes	—
23	C₄²×C₆₁	M₁₆	1	6	yes	—
24	#11758185	—	4	4	no	—
25	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
26	C₂×C₄×C₆₁	—	2	12	no	—
27	C₂×C₄×C₆₁	C₄⋊C₈	1	12	yes	—
28	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
29	C₂×C₄×C₆₁	C₂×M₁₆	1	3	yes	—
30	C₈×C₆₁	—	2	6	no	—
31	C₈×C₆₁	#127	1	1	yes	characteristic
32	C₈×C₆₁	C₄⋊Q₈	1	3	yes	—
33	#12899	—	61	1	no	maximal, Sylow
34	C₂²×C₆₁	C₂×C₄⋊C₈	1	3	yes	—
35	C₂²×C₆₁	#504	1	4	yes	—
36	C₄×C₆₁	—	2	12	no	—
37	C₄×C₆₁	#384	1	1	yes	characteristic
38	C₄×C₆₁	#423	1	3	yes	—
39	C₂×C₄⋊C₁₆	—	61	6	no	—
40	#1726	D₆₁	1	1	yes	characteristic
41	C₂×C₆₁	#1842	1	1	yes	characteristic
42	C₂×C₆₁	#2151	1	6	yes	—
43	C₂²×C₁₆	—	61	4	no	—
44	C₂×#97	Dic₆₁	1	3	yes	—
45	C₂×C₄×C₈	C₂×D₆₁	1	3	yes	—
46	#376	Dic₆₁	1	1	yes	characteristic
47	C₄⋊C₁₆	—	122	12	no	—
48	C₆₁	#12899	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₈	C₂×Dic₆₁	1	3	yes	—
50	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
51	C₂×C₄²	C₂×Dic₆₁	1	3	yes	—
52	C₂×C₁₆	—	122	12	no	—
53	#97	—	2	6	no	—
54	C₄×C₈	Dic₁₂₂	1	6	yes	—
55	C₄×C₈	D₂₄₄	1	6	yes	—
56	#95	#92418	1	4	yes	—
57	C₂²×C₄	C₂×#92418	1	6	yes	—
58	C₂²×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
59	C₂×C₈	—	2	6	no	—
60	C₂×C₈	C₄⋊Dic₆₁	1	6	yes	—
61	C₂×C₈	C₂×Dic₁₂₂	1	3	yes	—
62	C₂×C₈	C₂×D₂₄₄	1	3	yes	—
63	C₄²	#11758194	1	6	yes	—
64	C₄²	C₄⋊Dic₆₁	1	6	yes	—
65	C₁₆	—	244	4	no	—
66	C₂³	C₂²×#92418	1	1	yes	characteristic
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	C₄⋊#92418	1	12	yes	—
69	C₂×C₄	C₂×C₄⋊Dic₆₁	1	3	yes	—
70	C₂×C₄	C₂×#11758194	1	3	yes	—
71	C₈	—	2	6	no	—
72	C₈	#14519774	1	1	yes	characteristic
73	C₈	#14519777	1	3	yes	—
74	C₂²	C₂×C₄⋊#92418	1	3	yes	—
75	C₂²	#16840179	1	4	yes	—
76	C₄	—	2	12	no	—
77	C₄	#16839632	1	1	yes	characteristic
78	C₄	#16839786	1	3	yes	—
79	C₂	#21445971	1	1	yes	characteristic
80	C₂	#21447578	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₁
commutator, socle, prime radical	C₂³×C₆₁	C₂²×C₈
Fitting	C₆₁×#1726	C₂
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	#1726	D₆₁
hypocenter	C₆₁	#12899
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
48. C₆₁	#12899	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11758185	2
C₆₁×#1726	2
#12899	61