Subgroups — #11928937

Order 1248.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#10597666	C₂	1	4	yes	maximal, complemented
5	#88129	—	3	1	no	maximal, Hall
6	C₂²×D₃₉	C₂²	1	2	yes	—
7	C₂×C₃×C₄×C₁₃	C₂²	1	2	yes	—
8	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×Dic₃₉	C₂²	1	2	yes	—
10	D₁₅₆	—	2	2	no	—
11	Dic₇₈	—	2	2	no	—
12	C₂×D₅₂	—	3	1	no	—
13	C₂×Dic₂₆	—	3	1	no	—
14	C₄²×C₁₃	D₃	1	1	yes	characteristic, complemented
15	#6103	—	3	4	no	—
16	C₂²×C₃×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×D₃₉	—	2	2	no	—
18	C₂×D₃₉	—	2	4	no	—
19	C₃×C₄×C₁₃	—	2	2	no	—
20	C₃×C₄×C₁₃	D₄	1	2	yes	—
21	Dic₃₉	—	2	2	no	—
22	C₂²×D₁₃	—	3	2	no	—
23	C₂×C₄×C₁₃	C₂×D₃	1	2	yes	—
24	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
25	C₂×Dic₁₃	—	3	2	no	—
26	D₅₂	—	6	2	no	—
27	Dic₂₆	—	6	2	no	—
28	#875	—	13	1	no	maximal, Hall
29	C₂×C₃×C₁₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₃×C₁₃	#41	1	2	yes	—
31	D₃₉	—	4	2	no	—
32	C₂²×C₁₃	C₂²×D₃	1	1	yes	characteristic
33	C₂×D₁₃	—	6	2	no	—
34	C₂×D₁₃	—	6	4	no	—
35	C₄×C₁₃	—	2	2	no	—
36	C₄×C₁₃	D₁₂	1	2	yes	—
37	Dic₁₃	—	6	2	no	—
38	C₂×D₁₂	—	13	1	no	—
39	C₂×Dic₆	—	13	1	no	—
40	C₃×C₄²	D₁₃	1	1	yes	characteristic, complemented
41	(C₂×D₃)⋊C₄	—	13	4	no	—
42	C₃×C₁₃	#124	1	1	yes	characteristic, complemented, Hall
43	#124	—	39	1	no	Sylow
44	C₂×C₁₃	C₂×D₁₂	1	1	yes	characteristic
45	C₂×C₁₃	#235	1	2	yes	—
46	D₁₃	—	12	2	no	—
47	C₂²×D₃	—	13	2	no	—
48	C₂×C₃×C₄	C₂×D₁₃	1	2	yes	—
49	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
50	C₂×Dic₃	—	13	2	no	—
51	D₁₂	—	26	2	no	—
52	Dic₆	—	26	2	no	—
53	C₂×Q₈	—	39	1	no	—
54	C₂×D₄	—	39	1	no	—
55	C₂²⋊C₄	—	39	4	no	—
56	C₄²	D₃₉	1	1	yes	characteristic, complemented
57	C₁₃	#875	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₃	C₂²×D₁₃	1	1	yes	characteristic
59	C₂×D₃	—	26	2	no	—
60	C₂×D₃	—	26	4	no	—
61	C₃×C₄	—	2	2	no	—
62	C₃×C₄	D₅₂	1	2	yes	—
63	Dic₃	—	26	2	no	—
64	C₂³	—	39	2	no	—
65	Q₈	—	78	2	no	—
66	D₄	—	78	2	no	—
67	C₂×C₄	—	39	2	no	—
68	C₂×C₄	C₂×D₃₉	1	2	yes	—
69	C₂×C₄	C₂×D₃₉	1	1	yes	characteristic
70	C₂×C₃	C₂×D₅₂	1	1	yes	characteristic
71	C₂×C₃	#6127	1	2	yes	—
72	D₃	—	52	2	no	—
73	C₂²	—	78	2	no	—
74	C₂²	—	78	4	no	—
75	C₂²	C₂²×D₃₉	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	78	2	no	—
78	C₄	D₁₅₆	1	2	yes	—
79	C₃	#88129	1	1	yes	characteristic, complemented, Sylow
80	C₂	—	156	2	no	—
81	C₂	C₂×D₁₅₆	1	1	yes	characteristic
82	C₂	#10597790	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₉
commutator, socle, cosocle	C₂²×C₃×C₁₃	C₂³
Fitting	C₃×C₄²×C₁₃	C₂
prime radical	C₂×D₁₅₆	C₂
hypercenter	C₄²	D₃₉
hypocenter	C₃×C₁₃	#124
LCS	G ▷ C₂²×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇₈	C₂	semidirect
3. C₃×C₄²×C₁₃	C₂	semidirect
4. #10597666	C₂	semidirect
14. C₄²×C₁₃	D₃	semidirect
40. C₃×C₄²	D₁₃	semidirect
42. C₃×C₁₃	#124	semidirect
56. C₄²	D₃₉	semidirect
57. C₁₃	#875	semidirect
79. C₃	#88129	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₅₆	2
C₂×Dic₇₈	2
C₃×C₄²×C₁₃	2
#10597666	2
#88129	3
#875	13