Subgroups — #29600527

Order 15616.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839592	C₂	1	2	yes	maximal
2	#21444023	C₂	1	4	yes	maximal, complemented
3	C₂×C₆₁×M₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519713	—	2	2	no	—
5	#16839592	—	2	2	no	—
6	#16839592	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×M₆₄	C₄	1	4	yes	—
9	C₂×C₃₂×C₆₁	C₂²	1	2	yes	—
10	C₂×#11758194	—	4	2	no	—
11	#14519713	—	2	4	no	—
12	#14519713	—	4	2	no	—
13	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
15	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₆₁	Q₈	1	2	yes	—
17	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
18	C₃₂×C₆₁	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
20	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
21	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
22	C₂×C₈×C₆₁	Q₁₆	1	2	yes	—
23	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
24	#11758194	—	4	4	no	—
25	#11758194	—	8	2	no	—
26	C₂×#92418	—	8	2	no	—
27	C₁₆×C₆₁	C₄⋊C₄	1	2	yes	—
28	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
29	C₁₆×C₆₁	C₂×Q₈	1	1	yes	characteristic
30	C₂³×C₆₁	Q₃₂	1	1	yes	characteristic
31	C₂×C₄×C₆₁	#107	1	2	yes	—
32	C₂×C₄×C₆₁	D₁₆	1	2	yes	—
33	C₂×C₄×C₆₁	Q₃₂	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
35	C₈×C₆₁	#107	1	2	yes	—
36	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
38	#92418	—	8	4	no	—
39	#13636	—	61	1	no	maximal, Sylow
40	C₂²×C₆₁	—	4	1	no	—
41	C₂²×C₆₁	#366	1	2	yes	—
42	C₂²×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
43	C₄×C₆₁	#366	1	2	yes	—
44	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
45	C₄×C₆₁	C₂×D₁₆	1	1	yes	characteristic
46	C₂×M₆₄	D₆₁	1	1	yes	characteristic
47	C₂×#368	—	61	2	no	—
48	#1425	—	61	4	no	—
49	C₂×C₆₁	—	2	2	no	—
50	C₂×C₆₁	—	2	1	no	—
51	C₂×C₆₁	C₂×#366	1	1	yes	characteristic
52	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
53	C₂×#108	—	122	2	no	—
54	M₆₄	Dic₆₁	1	4	yes	—
55	C₂×C₃₂	C₂×D₆₁	1	2	yes	—
56	#368	—	61	4	no	—
57	#368	—	122	2	no	—
58	C₆₁	#13636	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	—	244	2	no	—
60	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
61	C₂×C₁₆	Dic₁₂₂	1	2	yes	—
62	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
63	C₂×C₁₆	C₂×Dic₆₁	1	2	yes	—
64	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
65	#108	—	122	4	no	—
66	#108	—	244	2	no	—
67	C₃₂	C₂×Dic₆₁	1	4	yes	—
68	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
69	M₁₆	—	488	2	no	—
70	M₁₆	—	244	4	no	—
71	C₂×C₈	—	488	2	no	—
72	C₂×C₈	D₄₈₈	1	1	yes	characteristic
73	C₂×C₈	Dic₂₄₄	1	2	yes	—
74	C₂×C₈	C₄⋊Dic₆₁	1	2	yes	—
75	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
76	C₁₆	C₄⋊Dic₆₁	1	2	yes	—
77	C₁₆	C₂×Dic₁₂₂	1	1	yes	characteristic
78	C₁₆	C₂²×Dic₆₁	1	1	yes	characteristic
79	C₂³	Dic₄₈₈	1	1	yes	characteristic
80	C₂×C₄	#14519711	1	2	yes	—
81	C₂×C₄	D₉₇₆	1	2	yes	—
82	C₂×C₄	Dic₄₈₈	1	1	yes	characteristic
83	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
84	C₈	—	488	4	no	—
85	C₈	#14519711	1	2	yes	—
86	C₈	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
87	C₈	C₂×Dic₂₄₄	1	1	yes	characteristic
88	C₂²	—	4	1	no	—
89	C₂²	#16839588	1	2	yes	—
90	C₂²	C₂×Dic₄₈₈	1	1	yes	characteristic
91	C₄	#16839588	1	2	yes	—
92	C₄	C₂×#14519711	1	1	yes	characteristic
93	C₄	C₂×D₉₇₆	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	—	2	1	no	—
96	C₂	C₂×#16839588	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₉₇₆
commutator	C₁₆×C₆₁	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₂×C₆₁×M₆₄	C₂
socle	C₂×C₆₁	C₂×#366
prime radical	C₂³×C₆₁	Q₃₂
cosocle	C₂×C₁₆×C₆₁	C₂³
hypercenter	C₂×M₆₄	D₆₁
hypocenter	C₆₁	#13636
LCS	G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. #21444023	C₂	semidirect
58. C₆₁	#13636	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839592	2
#21444023	2
C₂×C₆₁×M₆₄	2
#13636	61