Subgroups — #44397917

Order 29056.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28409464	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₂₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#368×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#20300310	—	2	1	no	—
5	Dic₁₈₁₆	C₄	1	1	yes	characteristic
6	#20300233	—	2	1	no	—
7	D₃₆₃₂	C₄	1	1	yes	characteristic
8	C₂₂₇⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
10	#108×C₂₂₇	—	2	1	no	—
11	#16813884	—	4	1	no	—
12	Dic₉₀₈	—	2	1	no	—
13	#16813873	—	4	1	no	—
14	D₁₈₁₆	—	2	1	no	—
15	M₁₆×C₂₂₇	—	4	1	no	—
16	C₂×C₈×C₂₂₇	D₄	1	1	yes	characteristic
17	C₁₆×C₂₂₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₂₂₇	D₄	1	1	yes	characteristic
19	Dic₄₅₄	—	4	1	no	—
20	D₉₀₈	—	4	1	no	—
21	C₂²⋊D₂₂₇	—	8	1	no	—
22	C₄×D₂₂₇	—	8	1	no	—
23	C₂×C₄×C₂₂₇	D₈	1	1	yes	characteristic
24	C₈×C₂₂₇	—	8	1	no	—
25	C₈×C₂₂₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₂₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₂₂₇	SD₃₂	1	1	yes	characteristic
28	C₂×D₂₂₇	—	8	1	no	—
29	C₄×C₂₂₇	#102	1	1	yes	characteristic
30	C₄×C₂₂₇	D₁₆	1	1	yes	characteristic
31	Dic₂₂₇	—	8	1	no	—
32	C₂×C₂₂₇	—	2	1	no	—
33	C₂×C₂₂₇	#357	1	1	yes	characteristic
34	D₂₂₇	—	16	1	no	—
35	C₂₂₇	#1417	1	1	yes	characteristic, complemented, Sylow
36	#1417	—	227	1	no	maximal, Sylow
37	#508	—	227	1	no	—
38	C₂×C₃₂	—	227	1	no	—
39	#368	D₂₂₇	1	1	yes	characteristic, complemented
40	#135	—	454	1	no	—
41	Q₃₂	—	227	1	no	—
42	SD₃₂	—	454	1	no	—
43	D₁₆	—	227	1	no	—
44	C₂×C₁₆	C₂×D₂₂₇	1	1	yes	characteristic
45	#108	—	2	1	no	—
46	C₃₂	—	454	1	no	—
47	#41	—	908	1	no	—
48	Q₁₆	—	454	1	no	—
49	SD₁₆	—	908	1	no	—
50	D₈	—	454	1	no	—
51	M₁₆	—	4	1	no	—
52	C₂×C₈	C₂²⋊D₂₂₇	1	1	yes	characteristic
53	C₁₆	C₄×D₂₂₇	1	1	yes	characteristic
54	C₁₆	D₉₀₈	1	1	yes	characteristic
55	Q₈	—	908	1	no	—
56	D₄	—	1816	1	no	—
57	D₄	—	908	1	no	—
58	C₂×C₄	—	1816	1	no	—
59	C₂×C₄	#16813869	1	1	yes	characteristic
60	C₈	—	8	1	no	—
61	C₈	(C₂×D₂₂₇)⋊C₄	1	1	yes	characteristic
62	C₈	#16813871	1	1	yes	characteristic
63	C₂²	—	1816	1	no	—
64	C₂²	#20300231	1	1	yes	characteristic
65	C₄	—	1816	1	no	—
66	C₄	#20300208	1	1	yes	characteristic
67	C₄	#20300229	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	3632	1	no	—
70	C₂	#28408832	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#20300229
commutator	C₁₆×C₂₂₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₂₇
Fitting	#368×C₂₂₇	C₂
socle	C₂×C₂₂₇	#357
prime radical	#28409464	C₂
cosocle	C₂×C₁₆×C₂₂₇	C₂²
hypercenter	#368	D₂₂₇
hypocenter	C₂₂₇	#1417
LCS	G ▷ C₁₆×C₂₂₇ ▷ C₈×C₂₂₇ ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #368

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₂₇⋊C₃₂	C₂	semidirect
3. #368×C₂₂₇	C₂	semidirect
35. C₂₂₇	#1417	semidirect
39. #368	D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
#28409464	2
C₂×C₂₂₇⋊C₃₂	2
#368×C₂₂₇	2
#1417	227