Subgroups — #11928122

Order 1248.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊#1055	C₂	1	1	yes	maximal, characteristic
4	#88050	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₁₃	—	2	1	no	—
6	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₉	—	2	1	no	—
8	Dic₇₈	—	2	1	no	—
9	Dic₇₈	C₄	1	2	yes	complemented
10	C₃⋊#1055	—	2	1	no	—
11	C₂×Dic₂₆	—	3	1	no	—
12	C₁₃×C₄⋊C₄	D₃	1	1	yes	characteristic
13	C₂×#1055	—	3	1	no	—
14	C₂²×C₃×C₁₃	D₄	1	1	yes	characteristic
15	C₃×C₄×C₁₃	—	4	1	no	—
16	C₃×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₃×C₄×C₁₃	D₄	1	1	yes	characteristic
18	Dic₃₉	—	2	2	no	—
19	C₂×C₄×C₁₃	—	2	1	no	—
20	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
21	C₂×Dic₁₃	—	6	1	no	—
22	Dic₂₆	—	3	2	no	—
23	Dic₂₆	—	6	1	no	—
24	#1055	—	6	1	no	—
25	#810	—	13	1	no	maximal, Hall
26	C₂×C₃×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₃×C₁₃	SD₁₆	1	1	yes	characteristic
28	C₂×C₃×C₁₃	Q₁₆	1	1	yes	characteristic
29	C₂²×C₁₃	#67	1	1	yes	characteristic
30	C₄×C₁₃	—	4	1	no	—
31	C₄×C₁₃	C₄×D₃	1	1	yes	characteristic
32	C₄×C₁₃	D₁₂	1	1	yes	characteristic
33	Dic₁₃	—	6	2	no	—
34	C₂×Dic₆	—	13	1	no	—
35	C₃×C₄⋊C₄	D₁₃	1	1	yes	characteristic
36	C₂×C₃⋊C₈	—	13	1	no	—
37	C₃×C₁₃	#103	1	1	yes	characteristic, complemented, Hall
38	#103	—	39	1	no	Sylow
39	C₂×C₁₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
40	C₂×C₁₃	#214	1	1	yes	characteristic
41	C₂×C₁₃	#216	1	1	yes	characteristic
42	C₂×C₃×C₄	—	2	1	no	—
43	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
44	C₂×Dic₃	—	26	1	no	—
45	Dic₆	—	13	2	no	—
46	Dic₆	—	26	1	no	—
47	C₃⋊C₈	—	26	1	no	—
48	C₂×Q₈	—	39	1	no	—
49	C₂×C₈	—	39	1	no	—
50	C₄⋊C₄	D₃₉	1	1	yes	characteristic
51	C₁₃	#810	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₃	C₂²⋊D₁₃	1	1	yes	characteristic
53	C₃×C₄	—	4	1	no	—
54	C₃×C₄	C₄×D₁₃	1	1	yes	characteristic
55	C₃×C₄	D₅₂	1	1	yes	characteristic
56	Dic₃	—	26	2	no	—
57	Q₈	—	39	2	no	—
58	Q₈	—	78	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	78	1	no	—
61	C₂×C₄	C₂×D₃₉	1	1	yes	characteristic
62	C₈	—	78	1	no	—
63	C₂×C₃	#6103	1	1	yes	characteristic
64	C₂×C₃	#6105	1	1	yes	characteristic
65	C₂×C₃	#6107	1	1	yes	characteristic
66	C₂²	#64625	1	1	yes	characteristic
67	C₄	—	4	1	no	—
68	C₄	—	78	2	no	—
69	C₄	C₄×D₃₉	1	1	yes	characteristic
70	C₄	D₁₅₆	1	1	yes	characteristic
71	C₃	#88050	1	1	yes	characteristic, complemented, Sylow
72	C₂	#10597666	1	1	yes	characteristic
73	C₂	#10597668	1	1	yes	characteristic
74	C₂	#10597670	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#64625
commutator	C₃×C₄×C₁₃	C₂×C₄
Frattini	C₂×C₄	C₂×D₃₉
Fitting	C₃×C₁₃×C₄⋊C₄	C₂
socle, prime radical	C₂²×C₃×C₁₃	D₄
cosocle	C₂×C₃×C₄×C₁₃	C₂²
hypercenter	C₄⋊C₄	D₃₉
hypocenter	C₃×C₁₃	#103
LCS	G ▷ C₃×C₄×C₁₃ ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. Dic₇₈	C₄	semidirect
37. C₃×C₁₃	#103	semidirect
51. C₁₃	#810	semidirect
71. C₃	#88050	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₈	2
C₃×C₁₃×C₄⋊C₄	2
C₂×C₃⋊#1055	2
#88050	3
#810	13