Subgroups — #13131251

Order 1664.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#1055	C₂	1	1	yes	maximal, characteristic
2	#11718231	C₂	1	2	yes	maximal
3	C₄⋊#6092	C₂	1	2	yes	maximal
4	C₄⋊#6092	C₂	1	1	yes	maximal, characteristic
5	C₄×#6092	C₂	1	1	yes	maximal, characteristic
6	C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×#6092	C₂²	1	1	yes	characteristic
8	C₂×#6092	C₂²	1	2	yes	—
9	C₂×#6092	C₂²	1	1	yes	characteristic
10	C₄⋊#1055	C₄	1	2	yes	—
11	C₄⋊#1055	C₄	1	1	yes	characteristic
12	C₄×#1055	C₂²	1	1	yes	characteristic
13	C₄×#1055	C₂²	1	2	yes	—
14	C₂×Q₈×C₁₃	C₈	1	1	yes	characteristic
15	C₁₃×C₄⋊C₄	C₈	1	2	yes	—
16	C₁₃×C₄⋊C₄	C₈	1	1	yes	characteristic
17	C₄²×C₁₃	C₂×C₄	1	2	yes	—
18	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×#1055	C₂×C₄	1	1	yes	characteristic
20	C₂×#1055	C₂×C₄	1	2	yes	—
21	C₂×#1055	C₂³	1	1	yes	characteristic
22	#6092	—	2	1	no	—
23	#6092	—	2	2	no	—
24	#6092	Q₈	1	2	yes	complemented
25	#2183	—	13	1	no	maximal, Sylow
26	Q₈×C₁₃	—	2	2	no	—
27	C₂×C₄×C₁₃	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₃	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
32	#1055	—	2	1	no	—
33	#1055	—	2	2	no	—
34	#1055	C₂×Q₈	1	1	yes	characteristic
35	#1055	#41	1	1	yes	characteristic
36	C₈×Q₈	—	13	1	no	—
37	C₄⋊C₁₆	—	13	1	no	—
38	C₄⋊C₁₆	—	13	2	no	—
39	#346	—	13	2	no	—
40	C₄×C₁₆	—	13	1	no	—
41	C₂²×C₁₃	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₃	—	2	2	no	—
43	C₄×C₁₃	—	2	2	no	—
44	C₄×C₁₃	—	2	1	no	—
45	C₄×C₁₃	M₃₂	1	2	yes	—
46	C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
47	C₄×C₁₃	#131	1	1	yes	characteristic
48	C₄×Q₈	#261	1	1	yes	characteristic
49	C₂×C₁₆	—	13	1	no	—
50	C₂×C₁₆	—	13	1	no	—
51	C₂×C₁₆	—	13	2	no	—
52	C₄⋊C₈	—	13	2	no	—
53	C₄⋊C₈	—	13	1	no	—
54	C₄×C₈	—	13	1	no	—
55	C₄×C₈	—	13	2	no	—
56	C₂×C₁₃	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₁₃	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₁₃	#504	1	1	yes	characteristic
59	C₂×Q₈	#1057	1	1	yes	characteristic
60	C₂×C₈	—	13	1	no	—
61	C₂×C₈	—	13	1	no	—
62	C₂×C₈	—	13	2	no	—
63	C₄⋊C₄	#1057	1	2	yes	—
64	C₄⋊C₄	#1057	1	1	yes	characteristic
65	C₄²	C₂×#261	1	2	yes	—
66	C₄²	C₂×#261	1	1	yes	characteristic
67	C₁₆	—	13	2	no	—
68	C₁₆	—	26	1	no	—
69	C₁₆	—	26	2	no	—
70	C₁₃	#2183	1	1	yes	characteristic, complemented, Sylow
71	Q₈	—	2	2	no	—
72	C₂×C₄	C₂×#1057	1	2	yes	—
73	C₂×C₄	C₂×#1057	1	1	yes	characteristic
74	C₂×C₄	C₂×#1057	1	2	yes	—
75	C₂×C₄	C₂×#1057	1	1	yes	characteristic
76	C₂×C₄	C₂²×#261	1	1	yes	characteristic
77	C₈	—	13	1	no	—
78	C₈	—	13	1	no	—
79	C₈	—	26	1	no	—
80	C₈	—	26	2	no	—
81	C₂²	C₂²×#1057	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	2	2	no	—
84	C₄	—	2	1	no	—
85	C₄	#88106	1	2	yes	—
86	C₄	#88240	1	1	yes	characteristic
87	C₄	Q₈×#261	1	1	yes	characteristic
88	C₂	C₂×#88106	1	1	yes	characteristic
89	C₂	Q₈×#1057	1	1	yes	characteristic
90	C₂	#11719162	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#261
commutator, socle, prime radical	C₂²×C₁₃	C₂²×C₈
Fitting	C₄×Q₈×C₁₃	C₄
cosocle	C₂×#1055	C₂³
hypercenter	C₄×Q₈	#261
hypocenter	C₁₃	#2183
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
24. #6092	Q₈	semidirect
70. C₁₃	#2183	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#1055	2
#11718231	2
C₄⋊#6092	2
C₄×#6092	2
#2183	13