Subgroups — #72259254

Order 5408.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72229205	C₂	1	1	yes	maximal, characteristic
2	#72229188	C₂	1	1	yes	maximal, characteristic
3	#72229144	C₂	1	1	yes	maximal, characteristic
4	C₂×#10621728	—	2	1	no	—
5	C₄×C₁₃⋊D₁₃	C₂²	1	1	yes	characteristic
6	C₁₃×Dic₂₆	C₄	1	2	yes	complemented
7	#13062580	—	2	1	no	—
8	#13062577	—	2	1	no	—
9	#13062574	—	2	1	no	—
10	C₂×C₁₃⋊D₁₃	D₄	1	1	yes	characteristic
11	#10621728	—	4	1	no	—
12	C₄×C₁₃²	C₂×C₄	1	1	yes	characteristic
13	#10621725	D₄	1	1	yes	characteristic
14	C₁₃×Dic₁₃	—	2	2	no	—
15	#88117	—	13	2	no	maximal
16	C₂×C₁₃²	C₂²⋊C₄	1	1	yes	characteristic
17	C₁₃⋊D₁₃	SD₁₆	1	1	yes	characteristic
18	C₁₃⋊D₁₃	Q₁₆	1	1	yes	characteristic
19	Q₈×D₁₃	—	13	2	no	—
20	C₄⋊#261	—	13	2	no	—
21	C₄⋊#261	—	26	6	no	—
22	D₁₃⋊C₈	—	13	2	no	—
23	C₁₃²	#103	1	1	yes	characteristic, complemented, Sylow
24	C₂×#261	—	26	2	no	—
25	C₂×#261	—	26	12	no	—
26	Q₈×C₁₃	—	13	2	no	—
27	C₄×D₁₃	—	13	2	no	—
28	C₄×D₁₃	—	26	2	no	—
29	C₄×D₁₃	—	26	6	no	—
30	Dic₂₆	—	26	2	no	—
31	Dic₂₆	#261	1	2	yes	complemented
32	#1057	—	26	2	no	—
33	C₂×D₁₃	—	13	2	no	—
34	C₂×D₁₃	—	26	6	no	—
35	#261	—	52	2	no	—
36	#261	—	52	12	no	—
37	C₄×C₁₃	—	2	6	no	—
38	C₄×C₁₃	—	26	2	no	—
39	C₄×C₁₃	C₂×#261	1	2	yes	—
40	Dic₁₃	—	2	2	no	—
41	Dic₁₃	—	13	2	no	—
42	Dic₁₃	—	26	6	no	—
43	#103	—	169	1	no	Sylow
44	C₂×C₁₃	—	2	6	no	—
45	C₂×C₁₃	C₂²⋊#261	1	2	yes	—
46	D₁₃	—	13	2	no	—
47	D₁₃	—	13	2	no	—
48	D₁₃	—	26	6	no	—
49	D₁₃	—	26	6	no	—
50	C₂×Q₈	—	169	1	no	—
51	C₂×C₈	—	169	1	no	—
52	C₄⋊C₄	—	169	1	no	—
53	C₁₃	—	2	6	no	—
54	C₁₃	#88117	1	2	yes	complemented
55	Q₈	—	13	2	no	—
56	Q₈	—	338	1	no	—
57	C₂×C₄	—	338	1	no	—
58	C₂×C₄	—	338	1	no	—
59	C₂×C₄	—	169	1	no	—
60	C₈	—	338	1	no	—
61	C₂²	—	169	1	no	—
62	C₄	—	26	2	no	—
63	C₄	—	676	1	no	—
64	C₄	—	169	1	no	—
65	C₄	D₁₃⋊#261	1	1	yes	characteristic
66	C₂	—	169	1	no	—
67	C₂	—	169	1	no	—
68	C₂	#72214177	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#72214177
commutator, Fitting	C₄×C₁₃²	C₂×C₄
Frattini, hypercenter	C₄	D₁₃⋊#261
socle	C₂×C₁₃²	C₂²⋊C₄
prime radical	C₂×C₁₃⋊D₁₃	D₄
cosocle	C₄×C₁₃⋊D₁₃	C₂²
hypocenter	C₁₃²	#103
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
6. C₁₃×Dic₂₆	C₄	semidirect
23. C₁₃²	#103	semidirect
31. Dic₂₆	#261	semidirect
54. C₁₃	#88117	semidirect

Maximal subgroups

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

subgroup	index
#72229205	2
#72229188	2
#72229144	2
#88117	13