Subgroups — #30607345

Order 15616.

370 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11758187	C₂	1	6	yes	maximal
2	#21445506	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11758187	C₂²	1	4	yes	—
4	C₄⋊#11758187	—	2	12	no	—
5	C₂×#14519688	C₄	1	3	yes	—
6	C₂×C₄×#92418	C₂²	1	3	yes	—
7	C₆₁×#376	C₄	1	1	yes	characteristic
8	C₂×#11758187	—	2	12	no	—
9	C₂²×#92418	C₂×C₄	1	3	yes	—
10	C₂²×#92418	C₂³	1	1	yes	characteristic
11	#14519688	—	2	6	no	—
12	C₄×#92418	D₄	1	6	yes	—
13	C₄×#92418	Q₈	1	6	yes	—
14	C₂×C₄²×C₆₁	C₂×C₄	1	3	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	C₄²×C₆₁	C₄⋊C₄	1	6	yes	—
19	C₄²×C₆₁	M₁₆	1	6	yes	—
20	C₂×#92418	—	2	6	no	—
21	C₂×#92418	C₄⋊C₄	1	6	yes	—
22	C₂×#92418	C₂×D₄	1	3	yes	—
23	C₂×#92418	C₂×Q₈	1	3	yes	—
24	#11758187	—	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	#92418	—	2	6	no	—
31	#92418	#127	1	1	yes	characteristic
32	#92418	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	—	61	1	no	—
41	C₂×C₆₁	#1842	1	1	yes	characteristic
42	C₂×C₆₁	#2151	1	6	yes	—
43	C₂²×C₁₆	—	61	4	no	—
44	C₂×#97	—	61	3	no	—
45	C₂×C₄×C₈	—	61	3	no	—
46	#376	#6923	1	1	yes	characteristic
47	C₄⋊C₁₆	—	122	12	no	—
48	C₆₁	#12899	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₈	—	61	1	no	—
50	C₂²×C₈	—	61	3	no	—
51	C₂×C₄²	C₂×#6923	1	3	yes	—
52	C₂×C₁₆	—	122	12	no	—
53	#97	—	122	6	no	—
54	C₄×C₈	—	61	6	no	—
55	C₄×C₈	—	61	6	no	—
56	#95	#92420	1	4	yes	—
57	C₂²×C₄	C₂×#92420	1	6	yes	—
58	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
59	C₂×C₈	—	61	3	no	—
60	C₂×C₈	—	61	3	no	—
61	C₂×C₈	—	61	6	no	—
62	C₂×C₈	—	122	6	no	—
63	C₄²	#11758213	1	6	yes	—
64	C₄²	C₄⋊#6923	1	6	yes	—
65	C₁₆	—	244	4	no	—
66	C₂³	C₂²×#92420	1	1	yes	characteristic
67	C₂×C₄	—	2	12	no	—
68	C₂×C₄	C₄⋊#92420	1	12	yes	—
69	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
70	C₂×C₄	C₂×#11758213	1	3	yes	—
71	C₈	—	61	1	no	—
72	C₈	—	61	3	no	—
73	C₈	—	122	6	no	—
74	C₂²	C₂×C₄⋊#92420	1	3	yes	—
75	C₂²	#16840820	1	4	yes	—
76	C₄	—	2	12	no	—
77	C₄	#16840708	1	1	yes	characteristic
78	C₄	#16840766	1	3	yes	—
79	C₂	#21461751	1	1	yes	characteristic
80	C₂	#21462154	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6923
commutator, socle, prime radical	C₂³×C₆₁	C₂²×C₈
Fitting	C₆₁×#376	C₄
cosocle	C₂²×#92418	C₂³
hypercenter	#376	#6923
hypocenter	C₆₁	#12899
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ #376

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⋊#11758187	2
#21445506	2
#12899	61