Subgroups — #14271601

Order 1856.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11745266	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₉×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745148	C₂	1	2	yes	maximal, complemented
5	D₄₆₄	C₂	1	2	yes	maximal, complemented
6	#90805	—	2	1	no	—
7	C₂×D₁₁₆	—	2	1	no	—
8	C₂×C₈×C₂₉	C₂²	1	1	yes	characteristic
9	Dic₁₁₆	C₂²	1	1	yes	characteristic, complemented
10	D₂₃₂	—	2	1	no	—
11	D₂₃₂	C₂²	1	2	yes	—
12	D₂₃₂	C₂²	1	1	yes	characteristic, complemented
13	#90773	—	2	1	no	—
14	C₁₆×C₂₉	C₂²	1	2	yes	complemented
15	C₂²×D₂₉	—	4	1	no	—
16	C₂×C₄×C₂₉	D₄	1	1	yes	characteristic
17	C₂²⋊D₂₉	—	4	1	no	—
18	D₁₁₆	—	2	1	no	—
19	D₁₁₆	—	2	2	no	—
20	D₁₁₆	—	4	1	no	—
21	C₄×D₂₉	—	4	1	no	—
22	Dic₅₈	—	2	1	no	—
23	C₈×C₂₉	D₄	1	1	yes	characteristic
24	C₈×C₂₉	C₂³	1	1	yes	characteristic
25	C₂²×C₂₉	D₈	1	1	yes	characteristic
26	C₂×D₂₉	—	4	1	no	—
27	C₂×D₂₉	—	4	2	no	—
28	C₂×D₂₉	—	8	2	no	—
29	C₄×C₂₉	D₈	1	1	yes	characteristic
30	C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
31	Dic₂₉	—	4	1	no	—
32	#509	—	29	1	no	maximal, Sylow
33	C₂×C₂₉	—	2	1	no	—
34	C₂×C₂₉	C₂×D₈	1	1	yes	characteristic
35	D₂₉	—	8	1	no	—
36	D₂₉	—	8	2	no	—
37	#135	—	29	1	no	—
38	C₂×D₈	—	29	1	no	—
39	SD₃₂	—	29	2	no	—
40	D₁₆	—	29	2	no	—
41	M₃₂	D₂₉	1	1	yes	characteristic, complemented
42	C₂₉	#509	1	1	yes	characteristic, complemented, Sylow
43	#41	—	58	1	no	—
44	C₂×D₄	—	58	1	no	—
45	Q₁₆	—	29	1	no	—
46	SD₁₆	—	58	1	no	—
47	D₈	—	29	1	no	—
48	D₈	—	29	2	no	—
49	D₈	—	58	1	no	—
50	C₂×C₈	C₂×D₂₉	1	1	yes	characteristic
51	C₁₆	C₂×D₂₉	1	2	yes	complemented
52	C₂³	—	116	1	no	—
53	Q₈	—	58	1	no	—
54	D₄	—	58	2	no	—
55	D₄	—	58	1	no	—
56	D₄	—	116	1	no	—
57	D₄	—	116	1	no	—
58	C₂×C₄	—	116	1	no	—
59	C₂×C₄	D₁₁₆	1	1	yes	characteristic
60	C₈	D₁₁₆	1	1	yes	characteristic
61	C₈	C₂²×D₂₉	1	1	yes	characteristic
62	C₂²	—	116	1	no	—
63	C₂²	—	116	2	no	—
64	C₂²	—	232	2	no	—
65	C₂²	D₂₃₂	1	1	yes	characteristic
66	C₄	—	116	1	no	—
67	C₄	D₂₃₂	1	1	yes	characteristic
68	C₄	C₂×D₁₁₆	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	232	2	no	—
71	C₂	—	232	1	no	—
72	C₂	C₂×D₂₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₃₂
commutator, cosocle	C₈×C₂₉	C₂³
Frattini	C₈	C₂²×D₂₉
Fitting	C₂₉×M₃₂	C₂
socle	C₂×C₂₉	C₂×D₈
hypercenter	M₃₂	D₂₉
hypocenter	C₂₉	#509
LCS	G ▷ C₈×C₂₉ ▷ C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. #11745266	C₂	semidirect
2. C₂×D₂₃₂	C₂	semidirect
3. C₂₉×M₃₂	C₂	semidirect
4. #11745148	C₂	semidirect
5. D₄₆₄	C₂	semidirect
9. Dic₁₁₆	C₂²	semidirect
12. D₂₃₂	C₂²	semidirect
14. C₁₆×C₂₉	C₂²	semidirect
41. M₃₂	D₂₉	semidirect
42. C₂₉	#509	semidirect
51. C₁₆	C₂×D₂₉	semidirect

Maximal subgroups

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

subgroup	index
#11745266	2
C₂×D₂₃₂	2
C₂₉×M₃₂	2
#11745148	2
D₄₆₄	2
#509	29