Subgroups — #29601008

Order 15616.

270 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₉₅₂	C₂	1	4	yes	maximal, complemented
2	#21448069	C₂	1	2	yes	maximal, complemented
3	C₄×C₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₉₇₆	—	2	2	no	—
5	C₂×D₉₇₆	C₂²	1	4	yes	—
6	#16840139	—	2	2	no	—
7	D₁₉₅₂	—	2	8	no	—
8	C₂×C₃₂×C₆₁	C₂²	1	2	yes	—
9	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×D₄₈₈	—	2	4	no	—
11	C₂×D₄₈₈	—	4	2	no	—
12	#14519774	—	4	2	no	—
13	D₉₇₆	—	2	8	no	—
14	D₉₇₆	—	4	4	no	—
15	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
17	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
18	C₃₂×C₆₁	D₄	1	4	yes	complemented
19	C₂×D₂₄₄	—	4	4	no	—
20	C₂×D₂₄₄	—	8	4	no	—
21	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
22	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
23	C₄²×C₆₁	D₈	1	1	yes	characteristic
24	D₄₈₈	—	4	4	no	—
25	D₄₈₈	—	4	4	no	—
26	D₄₈₈	—	8	4	no	—
27	C₁₆×C₆₁	D₈	1	2	yes	—
28	C₁₆×C₆₁	C₂×D₄	1	2	yes	—
29	C₂²×D₆₁	—	8	4	no	—
30	C₂×C₄×C₆₁	D₁₆	1	2	yes	—
31	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
32	D₂₄₄	—	8	4	no	—
33	D₂₄₄	—	8	4	no	—
34	D₂₄₄	—	16	8	no	—
35	C₈×C₆₁	D₁₆	1	2	yes	—
36	C₈×C₆₁	#127	1	1	yes	characteristic
37	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
38	#13725	—	61	1	no	maximal, Sylow
39	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
40	C₂×D₆₁	—	16	4	no	—
41	C₂×D₆₁	—	16	8	no	—
42	C₄×C₆₁	D₃₂	1	4	yes	complemented
43	C₄×C₆₁	#493	1	1	yes	characteristic
44	C₄×C₆₁	C₂×D₁₆	1	1	yes	characteristic
45	C₂×D₃₂	—	61	4	no	—
46	#2246	—	61	2	no	—
47	C₄×C₃₂	D₆₁	1	1	yes	characteristic, complemented
48	C₂×C₆₁	#2246	1	1	yes	characteristic
49	C₂×C₆₁	C₂×D₃₂	1	2	yes	—
50	D₆₁	—	32	4	no	—
51	C₂×D₁₆	—	61	4	no	—
52	C₂×D₁₆	—	122	2	no	—
53	#493	—	122	2	no	—
54	D₃₂	—	122	8	no	—
55	C₂×C₃₂	C₂×D₆₁	1	2	yes	—
56	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
57	C₆₁	#13725	1	1	yes	characteristic, complemented, Sylow
58	C₂×D₈	—	122	4	no	—
59	C₂×D₈	—	244	2	no	—
60	#127	—	244	2	no	—
61	D₁₆	—	122	8	no	—
62	D₁₆	—	244	4	no	—
63	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
64	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
65	C₄×C₈	D₂₄₄	1	1	yes	characteristic
66	C₃₂	D₂₄₄	1	4	yes	complemented
67	C₂×D₄	—	488	4	no	—
68	C₂×D₄	—	244	4	no	—
69	D₈	—	488	4	no	—
70	D₈	—	244	4	no	—
71	D₈	—	244	4	no	—
72	C₂×C₈	D₄₈₈	1	1	yes	characteristic
73	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
74	C₄²	D₄₈₈	1	1	yes	characteristic
75	C₁₆	D₄₈₈	1	2	yes	—
76	C₁₆	C₂×D₂₄₄	1	2	yes	—
77	C₂³	—	488	4	no	—
78	D₄	—	976	8	no	—
79	D₄	—	488	4	no	—
80	D₄	—	488	4	no	—
81	C₂×C₄	D₉₇₆	1	2	yes	—
82	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
83	C₈	D₉₇₆	1	2	yes	—
84	C₈	#14519774	1	1	yes	characteristic
85	C₈	C₂×D₄₈₈	1	1	yes	characteristic
86	C₂²	—	976	4	no	—
87	C₂²	—	976	8	no	—
88	C₂²	C₂×D₉₇₆	1	1	yes	characteristic
89	C₄	D₁₉₅₂	1	4	yes	complemented
90	C₄	#16840139	1	1	yes	characteristic
91	C₄	C₂×D₉₇₆	1	1	yes	characteristic
92	C₂	—	1952	4	no	—
93	C₂	#21448069	1	1	yes	characteristic
94	C₂	C₂×D₁₉₅₂	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₉₇₆
commutator, cosocle	C₂×C₁₆×C₆₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₄×C₃₂×C₆₁	C₂
socle	C₂²×C₆₁	C₂×D₁₆
hypercenter	C₄×C₃₂	D₆₁
hypocenter	C₆₁	#13725
LCS	G ▷ C₂×C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₉₅₂	C₂	semidirect
2. #21448069	C₂	semidirect
3. C₄×C₃₂×C₆₁	C₂	semidirect
18. C₃₂×C₆₁	D₄	semidirect
42. C₄×C₆₁	D₃₂	semidirect
47. C₄×C₃₂	D₆₁	semidirect
57. C₆₁	#13725	semidirect
66. C₃₂	D₂₄₄	semidirect
89. C₄	D₁₉₅₂	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
#21448069	2
C₄×C₃₂×C₆₁	2
#13725	61