Subgroups — #16821742

Order 3712.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₆₄	C₂	1	4	yes	maximal, complemented
2	#14271540	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₃₂	—	2	2	no	—
5	C₂×D₂₃₂	C₂²	1	4	yes	—
6	#11745236	—	2	2	no	—
7	C₂×C₁₆×C₂₉	C₂²	1	2	yes	—
8	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
9	D₄₆₄	—	2	8	no	—
10	C₂×D₁₁₆	—	2	4	no	—
11	C₂×D₁₁₆	—	4	4	no	—
12	C₂×C₈×C₂₉	D₄	1	1	yes	characteristic
13	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
14	C₄²×C₂₉	D₄	1	1	yes	characteristic
15	D₂₃₂	—	2	8	no	—
16	D₂₃₂	—	4	4	no	—
17	C₁₆×C₂₉	D₄	1	4	yes	complemented
18	C₂²×D₂₉	—	4	4	no	—
19	C₂×C₄×C₂₉	D₈	1	2	yes	—
20	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
21	D₁₁₆	—	4	4	no	—
22	D₁₁₆	—	4	4	no	—
23	D₁₁₆	—	8	8	no	—
24	C₈×C₂₉	D₈	1	2	yes	—
25	C₈×C₂₉	C₂×D₄	1	2	yes	—
26	#2246	—	29	1	no	maximal, Sylow
27	C₂²×C₂₉	C₂×D₈	1	1	yes	characteristic
28	C₂×D₂₉	—	8	4	no	—
29	C₂×D₂₉	—	8	8	no	—
30	C₄×C₂₉	D₁₆	1	4	yes	complemented
31	C₄×C₂₉	#127	1	1	yes	characteristic
32	C₄×C₂₉	C₂×D₈	1	1	yes	characteristic
33	C₂×D₁₆	—	29	4	no	—
34	#493	—	29	2	no	—
35	C₄×C₁₆	D₂₉	1	1	yes	characteristic, complemented
36	C₂×C₂₉	#493	1	1	yes	characteristic
37	C₂×C₂₉	C₂×D₁₆	1	2	yes	—
38	D₂₉	—	16	4	no	—
39	C₂×D₈	—	29	4	no	—
40	C₂×D₈	—	58	2	no	—
41	#127	—	58	2	no	—
42	D₁₆	—	58	8	no	—
43	C₂×C₁₆	C₂×D₂₉	1	2	yes	—
44	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
45	C₂₉	#2246	1	1	yes	characteristic, complemented, Sylow
46	C₂×D₄	—	58	4	no	—
47	C₂×D₄	—	116	4	no	—
48	D₈	—	58	8	no	—
49	D₈	—	116	4	no	—
50	C₂×C₈	D₁₁₆	1	1	yes	characteristic
51	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
52	C₄²	D₁₁₆	1	1	yes	characteristic
53	C₁₆	D₁₁₆	1	4	yes	complemented
54	C₂³	—	116	4	no	—
55	D₄	—	116	4	no	—
56	D₄	—	116	4	no	—
57	D₄	—	232	8	no	—
58	C₂×C₄	D₂₃₂	1	2	yes	—
59	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
60	C₈	D₂₃₂	1	2	yes	—
61	C₈	C₂×D₁₁₆	1	2	yes	—
62	C₂²	—	232	4	no	—
63	C₂²	—	232	8	no	—
64	C₂²	C₂×D₂₃₂	1	1	yes	characteristic
65	C₄	D₄₆₄	1	4	yes	complemented
66	C₄	#11745236	1	1	yes	characteristic
67	C₄	C₂×D₂₃₂	1	1	yes	characteristic
68	C₂	—	464	4	no	—
69	C₂	#14271540	1	1	yes	characteristic
70	C₂	C₂×D₄₆₄	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₂₃₂
commutator, cosocle	C₂×C₈×C₂₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₉
Fitting	C₄×C₁₆×C₂₉	C₂
socle	C₂²×C₂₉	C₂×D₈
hypercenter	C₄×C₁₆	D₂₉
hypocenter	C₂₉	#2246
LCS	G ▷ C₂×C₈×C₂₉ ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₄₆₄	C₂	semidirect
2. #14271540	C₂	semidirect
3. C₄×C₁₆×C₂₉	C₂	semidirect
17. C₁₆×C₂₉	D₄	semidirect
30. C₄×C₂₉	D₁₆	semidirect
35. C₄×C₁₆	D₂₉	semidirect
45. C₂₉	#2246	semidirect
53. C₁₆	D₁₁₆	semidirect
65. C₄	D₄₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₆₄	2
#14271540	2
C₄×C₁₆×C₂₉	2
#2246	29