Subgroups — #28405604

Order 14464.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14268650	C₂	1	2	yes	maximal
2	C₈×#11744751	C₂	1	1	yes	maximal, characteristic
3	C₂×#14268650	—	2	2	no	—
4	C₄×#11744751	C₄	1	1	yes	characteristic
5	C₄×#11744751	C₂²	1	1	yes	characteristic
6	C₄×C₈×C₁₁₃	C₄	1	1	yes	characteristic
7	#14268650	—	4	2	no	—
8	C₂×#11744751	C₈	1	1	yes	characteristic
9	C₂×#11744751	C₈	1	1	yes	characteristic
10	C₂×#11744751	D₄	1	1	yes	characteristic
11	C₂×#11744751	Q₈	1	1	yes	characteristic
12	C₂×C₈×C₁₁₃	—	2	1	no	—
13	C₄²×C₁₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₁₃	C₂×C₈	1	1	yes	characteristic
16	C₂×C₄×C₁₁₃	M₁₆	1	1	yes	characteristic
17	C₈×C₁₁₃	—	2	2	no	—
18	#11744751	—	2	1	no	—
19	#11744751	—	2	1	no	—
20	#11744751	C₁₆	1	2	yes	complemented
21	#11744751	D₈	1	1	yes	characteristic
22	#11744751	Q₁₆	1	1	yes	characteristic
23	C₂²×C₁₁₃	C₄⋊C₈	1	1	yes	characteristic
24	C₄×C₁₁₃	—	2	1	no	—
25	C₄×C₁₁₃	#107	1	1	yes	characteristic
26	C₄×C₁₁₃	#108	1	1	yes	characteristic
27	C₄×C₁₁₃	C₂×C₁₆	1	1	yes	characteristic
28	C₄×C₁₁₃	M₃₂	1	1	yes	characteristic
29	C₂×C₁₁₃	#335	1	1	yes	characteristic
30	C₂×C₁₁₃	C₄⋊C₁₆	1	1	yes	characteristic
31	C₂×C₁₁₃	#364	1	1	yes	characteristic
32	#1370	—	113	1	no	maximal, Sylow
33	C₁₁₃	#1370	1	1	yes	characteristic, complemented, Sylow
34	C₄⋊C₁₆	—	113	2	no	—
35	C₈²	—	113	1	no	—
36	C₂×C₁₆	—	226	2	no	—
37	C₄×C₈	—	113	1	no	—
38	C₄×C₈	—	113	1	no	—
39	C₄×C₈	#90673	1	1	yes	characteristic
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	113	1	no	—
42	C₂×C₈	—	113	1	no	—
43	C₂×C₈	—	113	1	no	—
44	C₂×C₈	—	113	1	no	—
45	C₄²	C₂×#90673	1	1	yes	characteristic
46	C₁₆	—	452	2	no	—
47	C₂×C₄	C₄⋊#90673	1	1	yes	characteristic
48	C₂×C₄	#14268655	1	1	yes	characteristic
49	C₂×C₄	#14268656	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	113	1	no	—
52	C₈	—	113	1	no	—
53	C₈	—	113	2	no	—
54	C₈	—	226	1	no	—
55	C₈	—	226	1	no	—
56	C₂²	#16813689	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	#16813691	1	1	yes	characteristic
59	C₄	#16813692	1	1	yes	characteristic
60	C₄	#16813695	1	1	yes	characteristic
61	C₄	#16813697	1	1	yes	characteristic
62	C₂	#20299462	1	1	yes	characteristic
63	C₂	#20299519	1	1	yes	characteristic
64	C₂	#20299521	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#90673
commutator	C₄×C₁₁₃	C₂×C₁₆
Frattini	C₄²	C₂×#90673
Fitting	C₄×C₈×C₁₁₃	C₄
socle, prime radical	C₂²×C₁₁₃	C₄⋊C₈
cosocle	C₄×#11744751	C₂²
hypercenter	C₄×C₈	#90673
hypocenter	C₁₁₃	#1370
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
20. #11744751	C₁₆	semidirect
33. C₁₁₃	#1370	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14268650	2
C₈×#11744751	2
#1370	113