Subgroups — #11929142

Order 1248.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₁₃⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₁₃⋊C₃	C₂	1	2	yes	maximal, complemented
3	#6129⋊C₃	C₂	1	6	yes	maximal, complemented
4	#6127⋊C₃	C₂	1	3	yes	maximal, complemented
5	C₂×(C₁₃⋊C₃)⋊Q₈	C₂	1	3	yes	maximal, complemented
6	#88257	C₃	1	1	yes	maximal, characteristic, complemented, Hall
7	Q₈×C₁₃⋊C₃	C₂²	1	1	yes	characteristic
8	D₄×C₁₃⋊C₃	C₂²	1	3	yes	—
9	C₂×C₄×C₁₃⋊C₃	C₂²	1	3	yes	—
10	(C₂²⋊D₁₃)⋊C₃	C₂²	1	6	yes	—
11	C₂×Dic₁₃⋊C₃	C₂²	1	6	yes	—
12	C₄⋊(D₁₃⋊C₃)	C₂²	1	1	yes	characteristic
13	C₄×D₁₃⋊C₃	C₂²	1	6	yes	—
14	(C₁₃⋊C₃)⋊Q₈	C₂²	1	3	yes	—
15	(C₁₃⋊C₃)⋊Q₈	C₂²	1	6	yes	—
16	C₁₃×#41	C₂×C₃	1	1	yes	characteristic, complemented
17	Q₈×D₁₃	C₂×C₃	1	2	yes	complemented
18	#6129	C₂×C₃	1	6	yes	complemented
19	#6127	C₂×C₃	1	3	yes	complemented
20	C₂×Dic₂₆	C₂×C₃	1	3	yes	complemented
21	C₂²×C₁₃⋊C₃	C₂³	1	3	yes	—
22	C₂×D₁₃⋊C₃	C₂³	1	2	yes	—
23	C₄×C₁₃⋊C₃	C₂³	1	1	yes	characteristic
24	C₄×C₁₃⋊C₃	C₂³	1	3	yes	—
25	Dic₁₃⋊C₃	C₂³	1	6	yes	—
26	Q₈×C₁₃	C₂²×C₃	1	1	yes	characteristic
27	D₄×C₁₃	C₂²×C₃	1	3	yes	—
28	C₂×C₄×C₁₃	C₂²×C₃	1	3	yes	—
29	C₂²⋊D₁₃	C₂²×C₃	1	6	yes	—
30	C₂×Dic₁₃	C₂²×C₃	1	6	yes	—
31	D₅₂	C₂²×C₃	1	1	yes	characteristic
32	C₄×D₁₃	C₂²×C₃	1	6	yes	—
33	Dic₂₆	C₂²×C₃	1	3	yes	—
34	Dic₂₆	C₂²×C₃	1	6	yes	—
35	C₃×2₋¹⁺⁴	—	13	1	no	maximal, Hall
36	C₂×C₁₃⋊C₃	—	2	3	no	—
37	C₂×C₁₃⋊C₃	C₂⁴	1	1	yes	characteristic
38	D₁₃⋊C₃	—	2	2	no	—
39	C₂²×C₁₃	C₂³×C₃	1	3	yes	—
40	C₂×D₁₃	C₂³×C₃	1	2	yes	—
41	C₄×C₁₃	C₂³×C₃	1	1	yes	characteristic
42	C₄×C₁₃	C₂³×C₃	1	3	yes	—
43	Dic₁₃	C₂³×C₃	1	6	yes	—
44	C₃×#41	—	13	3	no	—
45	C₃×#41	—	13	6	no	—
46	C₃×#41	—	13	1	no	—
47	C₂×C₃×Q₈	—	13	3	no	—
48	C₂×C₃×Q₈	—	13	2	no	—
49	C₁₃⋊C₃	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
50	2₋¹⁺⁴	—	13	1	no	Sylow
51	C₂×C₁₃	—	2	3	no	—
52	C₂×C₁₃	C₂⁴×C₃	1	1	yes	characteristic
53	D₁₃	—	2	2	no	—
54	C₃×Q₈	—	13	3	no	—
55	C₃×Q₈	—	13	6	no	—
56	C₃×Q₈	—	13	1	no	—
57	C₃×D₄	—	13	1	no	—
58	C₃×D₄	—	13	6	no	—
59	C₃×D₄	—	13	3	no	—
60	C₂×C₃×C₄	—	13	6	no	—
61	C₂×C₃×C₄	—	13	6	no	—
62	C₂×C₃×C₄	—	13	3	no	—
63	#41	—	13	6	no	—
64	#41	—	13	3	no	—
65	#41	D₁₃⋊C₃	1	1	yes	characteristic, complemented
66	C₂×Q₈	—	13	2	no	—
67	C₂×Q₈	—	13	3	no	—
68	C₁₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₃	—	13	2	no	—
70	C₂²×C₃	—	13	3	no	—
71	C₃×C₄	—	13	6	no	—
72	C₃×C₄	—	13	3	no	—
73	C₃×C₄	—	13	1	no	—
74	Q₈	—	13	6	no	—
75	Q₈	—	13	3	no	—
76	Q₈	C₂×D₁₃⋊C₃	1	1	yes	characteristic
77	D₄	—	13	6	no	—
78	D₄	—	13	1	no	—
79	D₄	C₂×D₁₃⋊C₃	1	3	yes	—
80	C₂×C₄	—	13	6	no	—
81	C₂×C₄	—	13	6	no	—
82	C₂×C₄	C₂×D₁₃⋊C₃	1	3	yes	—
83	C₂×C₃	—	13	1	no	—
84	C₂×C₃	—	26	3	no	—
85	C₂×C₃	—	26	2	no	—
86	C₂²	—	13	2	no	—
87	C₂²	C₂²×D₁₃⋊C₃	1	3	yes	—
88	C₄	—	13	6	no	—
89	C₄	C₂²×D₁₃⋊C₃	1	3	yes	—
90	C₄	C₂²×D₁₃⋊C₃	1	1	yes	characteristic
91	C₃	—	13	1	no	Sylow
92	C₂	—	2	3	no	—
93	C₂	—	26	2	no	—
94	C₂	C₂³×D₁₃⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₁₃⋊C₃
commutator, socle, cosocle	C₂×C₁₃	C₂⁴×C₃
Fitting	C₁₃×#41	C₂×C₃
hypercenter	#41	D₁₃⋊C₃
hypocenter	C₁₃	C₃×2₋¹⁺⁴
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #41×C₁₃⋊C₃	C₂	semidirect
2. Q₈×D₁₃⋊C₃	C₂	semidirect
3. #6129⋊C₃	C₂	semidirect
4. #6127⋊C₃	C₂	semidirect
5. C₂×(C₁₃⋊C₃)⋊Q₈	C₂	semidirect
6. #88257	C₃	semidirect
16. C₁₃×#41	C₂×C₃	semidirect
17. Q₈×D₁₃	C₂×C₃	semidirect
18. #6129	C₂×C₃	semidirect
19. #6127	C₂×C₃	semidirect
20. C₂×Dic₂₆	C₂×C₃	semidirect
49. C₁₃⋊C₃	2₋¹⁺⁴	semidirect
65. #41	D₁₃⋊C₃	semidirect
68. C₁₃	C₃×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
#41×C₁₃⋊C₃	2
Q₈×D₁₃⋊C₃	2
#6129⋊C₃	2
#6127⋊C₃	2
C₂×(C₁₃⋊C₃)⋊Q₈	2
#88257	3
C₃×2₋¹⁺⁴	13