Subgroups — #44401977

Order 29056.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28409409	C₂	1	1	yes	maximal, characteristic
2	#28408832	C₂	1	4	yes	maximal, complemented
3	#28408808	C₂	1	1	yes	maximal, characteristic, complemented
4	#499×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₈₁₆	—	2	1	no	—
6	C₂×D₁₈₁₆	C₂²	1	2	yes	—
7	#20300285	—	2	1	no	—
8	C₂×C₂₂₇⋊C₁₆	C₂²	1	2	yes	—
9	#125×C₂₂₇	—	2	1	no	—
10	#107×C₂₂₇	C₂²	1	2	yes	—
11	#106×C₂₂₇	—	2	1	no	—
12	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
13	C₂×D₉₀₈	—	2	2	no	—
14	C₂×D₉₀₈	—	4	2	no	—
15	D₁₈₁₆	—	2	4	no	—
16	D₁₈₁₆	—	4	2	no	—
17	C₂₂₇⋊C₁₆	—	2	2	no	—
18	C₂×C₈×C₂₂₇	D₄	1	1	yes	characteristic
19	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₂₂₇	—	2	2	no	—
21	C₄⋊C₄×C₂₂₇	—	4	2	no	—
22	C₄²×C₂₂₇	D₄	1	1	yes	characteristic
23	C₂²×D₂₂₇	—	4	2	no	—
24	D₉₀₈	—	4	2	no	—
25	D₉₀₈	—	4	2	no	—
26	D₉₀₈	—	8	4	no	—
27	C₂×C₄×C₂₂₇	—	4	2	no	—
28	C₂×C₄×C₂₂₇	D₈	1	2	yes	—
29	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
30	C₈×C₂₂₇	SD₁₆	1	2	yes	—
31	C₈×C₂₂₇	#41	1	2	yes	—
32	C₂²×C₂₂₇	C₂×D₈	1	1	yes	characteristic
33	C₂×D₂₂₇	—	8	2	no	—
34	C₂×D₂₂₇	—	8	4	no	—
35	C₄×C₂₂₇	—	2	2	no	—
36	C₄×C₂₂₇	—	8	2	no	—
37	C₄×C₂₂₇	#124	1	1	yes	characteristic
38	C₄×C₂₂₇	C₂×SD₁₆	1	1	yes	characteristic
39	C₂×C₂₂₇	#486	1	1	yes	characteristic
40	C₂×C₂₂₇	#509	1	2	yes	—
41	D₂₂₇	—	16	2	no	—
42	C₂₂₇	#2244	1	1	yes	characteristic, complemented, Sylow
43	#2244	—	227	1	no	maximal, Sylow
44	#499	D₂₂₇	1	1	yes	characteristic, complemented
45	#493	—	227	1	no	—
46	#357	—	227	4	no	—
47	#346	—	227	1	no	—
48	C₂×D₈	—	454	1	no	—
49	C₂×D₈	—	227	2	no	—
50	#127	—	454	1	no	—
51	#125	—	2	1	no	—
52	C₂×C₁₆	—	227	2	no	—
53	#107	C₂×D₂₂₇	1	2	yes	—
54	#106	—	2	1	no	—
55	C₄×C₈	C₂×D₂₂₇	1	1	yes	characteristic
56	C₂×D₄	—	908	2	no	—
57	C₂×D₄	—	454	2	no	—
58	D₈	—	908	2	no	—
59	D₈	—	454	4	no	—
60	C₂×C₈	C₂²⋊D₂₂₇	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄²	C₂²⋊D₂₂₇	1	1	yes	characteristic
65	C₁₆	—	454	2	no	—
66	C₂³	—	908	2	no	—
67	D₄	—	1816	4	no	—
68	D₄	—	908	2	no	—
69	D₄	—	908	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	#16813869	1	2	yes	—
72	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
73	C₈	#16813871	1	2	yes	—
74	C₈	#16813885	1	2	yes	—
75	C₂²	—	1816	2	no	—
76	C₂²	—	1816	4	no	—
77	C₂²	C₂×#16813869	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	8	2	no	—
80	C₄	#20300277	1	1	yes	characteristic
81	C₄	C₂×#16813871	1	1	yes	characteristic
82	C₂	—	3632	2	no	—
83	C₂	#28409380	1	1	yes	characteristic
84	C₂	#28409467	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#16813869
commutator, cosocle	C₂×C₈×C₂₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₂₂₇
Fitting	#499×C₂₂₇	C₂
socle	C₂²×C₂₂₇	C₂×D₈
prime radical	#28409409	C₂
hypercenter	#499	D₂₂₇
hypocenter	C₂₂₇	#2244
LCS	G ▷ C₂×C₈×C₂₂₇ ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
2. #28408832	C₂	semidirect
3. #28408808	C₂	semidirect
4. #499×C₂₂₇	C₂	semidirect
42. C₂₂₇	#2244	semidirect
44. #499	D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
#28409409	2
#28408832	2
#28408808	2
#499×C₂₂₇	2
#2244	227