Subgroups — #10597790

Order 624.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#64625	C₂	1	2	yes	maximal, complemented
3	D₁₅₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₃₉	C₂	1	2	yes	maximal, complemented
5	Dic₇₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#6127	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×D₃₉	C₂²	1	2	yes	—
9	C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	Dic₃₉	C₂²	1	2	yes	—
12	C₂×C₄×C₁₃	D₃	1	1	yes	characteristic, complemented
13	C₂²⋊D₁₃	—	3	2	no	—
14	D₅₂	—	3	1	no	—
15	C₄×D₁₃	—	3	2	no	—
16	Dic₂₆	—	3	1	no	—
17	C₂×C₃×C₁₃	—	2	1	no	—
18	C₂×C₃×C₁₃	C₂³	1	1	yes	characteristic
19	D₃₉	—	2	2	no	—
20	C₂²×C₁₃	C₂×D₃	1	1	yes	characteristic
21	C₂×D₁₃	—	3	2	no	—
22	C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
23	C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
24	Dic₁₃	—	3	2	no	—
25	#235	—	13	1	no	maximal, Hall
26	C₃×C₁₃	#41	1	1	yes	characteristic, complemented, Hall
27	C₂×C₁₃	—	2	1	no	—
28	C₂×C₁₃	C₂²×D₃	1	1	yes	characteristic
29	D₁₃	—	6	2	no	—
30	C₂×C₃×C₄	D₁₃	1	1	yes	characteristic, complemented
31	#67	—	13	2	no	—
32	D₁₂	—	13	1	no	—
33	C₄×D₃	—	13	2	no	—
34	Dic₆	—	13	1	no	—
35	#41	—	39	1	no	Sylow
36	C₁₃	#235	1	1	yes	characteristic, complemented, Sylow
37	C₂²×C₃	C₂×D₁₃	1	1	yes	characteristic
38	C₂×D₃	—	13	2	no	—
39	C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
40	C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
41	Dic₃	—	13	2	no	—
42	Q₈	—	39	1	no	—
43	D₄	—	39	2	no	—
44	D₄	—	39	1	no	—
45	C₂×C₄	—	39	2	no	—
46	C₂×C₄	D₃₉	1	1	yes	characteristic, complemented
47	C₂×C₃	—	2	1	no	—
48	C₂×C₃	C₂²×D₁₃	1	1	yes	characteristic
49	D₃	—	26	2	no	—
50	C₂²	—	39	2	no	—
51	C₂²	C₂×D₃₉	1	1	yes	characteristic
52	C₄	—	39	2	no	—
53	C₄	C₂×D₃₉	1	1	yes	characteristic
54	C₄	C₂×D₃₉	1	1	yes	characteristic
55	C₃	#6127	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	2	1	no	—
57	C₂	—	78	2	no	—
58	C₂	C₂²×D₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₃₉
commutator, socle, cosocle	C₂×C₃×C₁₃	C₂³
Frattini	C₂	C₂²×D₃₉
Fitting	C₂×C₃×C₄×C₁₃	C₂
hypercenter	C₂×C₄	D₃₉
hypocenter	C₃×C₁₃	#41
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₁₃	C₂	semidirect
2. #64625	C₂	semidirect
3. D₁₅₆	C₂	semidirect
4. C₄×D₃₉	C₂	semidirect
5. Dic₇₈	C₂	semidirect
12. C₂×C₄×C₁₃	D₃	semidirect
26. C₃×C₁₃	#41	semidirect
30. C₂×C₃×C₄	D₁₃	semidirect
36. C₁₃	#235	semidirect
46. C₂×C₄	D₃₉	semidirect
55. C₃	#6127	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₁₃	2
#64625	2
D₁₅₆	2
C₄×D₃₉	2
Dic₇₈	2
#6127	3
#235	13