Subgroups — #13129969

Order 1664.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11718746	C₂	1	1	yes	maximal, characteristic
2	#11718225	C₂	1	2	yes	maximal
3	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
4	C₂×#6106	—	2	2	no	—
5	#88128	C₂²	1	1	yes	characteristic
6	#88107	—	2	2	no	—
7	C₄×#1055	C₄	1	1	yes	characteristic, complemented
8	C₂×Q₈×C₁₃	—	2	1	no	—
9	C₂×D₅₂	—	4	1	no	—
10	C₂×D₅₂	D₄	1	2	yes	—
11	C₂²⋊#261	—	4	2	no	—
12	C₁₃×C₄⋊C₄	—	4	1	no	—
13	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
14	#6106	—	4	4	no	—
15	C₂×#1055	—	2	1	no	—
16	#1409	—	13	1	no	maximal, Sylow
17	C₂²×D₁₃	—	2	2	no	—
18	C₂×#261	—	8	2	no	—
19	Q₈×C₁₃	—	4	2	no	—
20	C₂×C₄×C₁₃	—	2	1	no	—
21	C₂×C₄×C₁₃	—	4	1	no	—
22	C₂×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
23	D₅₂	—	4	2	no	—
24	D₅₂	—	8	2	no	—
25	#1055	—	2	2	no	—
26	#492	—	13	1	no	—
27	#353	—	13	2	no	—
28	C₂²×C₁₃	#99	1	1	yes	characteristic
29	C₂×D₁₃	—	4	2	no	—
30	C₂×D₁₃	—	8	2	no	—
31	#261	—	16	2	no	—
32	C₄×C₁₃	—	2	2	no	—
33	C₄×C₁₃	—	2	1	no	—
34	C₄×C₁₃	—	8	1	no	—
35	C₂×SD₁₆	—	26	2	no	—
36	C₄⋊Q₈	#261	1	1	yes	characteristic, complemented
37	#127	—	13	1	no	—
38	#99	—	26	2	no	—
39	C₄×C₈	—	13	1	no	—
40	C₂×C₁₃	—	2	1	no	—
41	C₂×C₁₃	#353	1	1	yes	characteristic
42	D₁₃	—	8	2	no	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×D₄	—	13	2	no	—
45	C₂×D₄	—	52	1	no	—
46	SD₁₆	—	52	4	no	—
47	C₂×C₈	—	26	1	no	—
48	C₄⋊C₄	—	4	1	no	—
49	C₂²⋊C₄	—	52	2	no	—
50	C₄²	C₂×#261	1	1	yes	characteristic
51	C₁₃	#1409	1	1	yes	characteristic, complemented, Sylow
52	C₂³	—	26	2	no	—
53	Q₈	—	4	2	no	—
54	D₄	—	52	2	no	—
55	D₄	—	104	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	4	1	no	—
58	C₂×C₄	—	104	2	no	—
59	C₂×C₄	C₂²⋊#261	1	1	yes	characteristic
60	C₈	—	26	2	no	—
61	C₂²	—	52	2	no	—
62	C₂²	—	104	2	no	—
63	C₂²	#88119	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	2	1	no	—
66	C₄	—	8	1	no	—
67	C₄	—	208	2	no	—
68	C₂	—	2	1	no	—
69	C₂	—	104	2	no	—
70	C₂	#11718294	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11718294
commutator	C₄²×C₁₃	C₂×C₄
Frattini	C₄²	C₂×#261
Fitting	C₁₃×C₄⋊Q₈	C₄
socle	C₂×C₁₃	#353
prime radical, cosocle	#88128	C₂²
hypercenter	C₄⋊Q₈	#261
hypocenter	C₁₃	#1409
LCS	G ▷ C₄²×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₁₃×C₄⋊Q₈	C₄	semidirect
7. C₄×#1055	C₄	semidirect
36. C₄⋊Q₈	#261	semidirect
51. C₁₃	#1409	semidirect

Maximal subgroups

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

subgroup	index
#11718746	2
#11718225	2
#1409	13