Subgroups — #44401981

Order 29056.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28409412	C₂	1	1	yes	maximal, characteristic
2	#28408835	C₂	1	4	yes	maximal
3	#28408808	C₂	1	1	yes	maximal, characteristic
4	#499×C₂₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₉₀₈	—	2	1	no	—
6	C₂×Dic₉₀₈	C₂²	1	2	yes	—
7	#20300288	—	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×Dic₄₅₄	—	2	2	no	—
14	Dic₉₀₈	—	2	4	no	—
15	Dic₉₀₈	—	4	2	no	—
16	C₄⋊Dic₂₂₇	—	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	Dic₄₅₄	—	4	2	no	—
24	Dic₄₅₄	—	4	2	no	—
25	C₂×Dic₂₂₇	—	4	2	no	—
26	C₂×C₄×C₂₂₇	—	4	2	no	—
27	C₂×C₄×C₂₂₇	D₈	1	2	yes	—
28	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
29	C₈×C₂₂₇	SD₁₆	1	2	yes	—
30	C₈×C₂₂₇	#41	1	2	yes	—
31	C₂²×C₂₂₇	C₂×D₈	1	1	yes	characteristic
32	C₄×C₂₂₇	—	2	2	no	—
33	C₄×C₂₂₇	—	8	2	no	—
34	C₄×C₂₂₇	#124	1	1	yes	characteristic
35	C₄×C₂₂₇	C₂×SD₁₆	1	1	yes	characteristic
36	Dic₂₂₇	—	8	2	no	—
37	C₂×C₂₂₇	#486	1	1	yes	characteristic
38	C₂×C₂₂₇	#510	1	2	yes	—
39	C₂₂₇	#2245	1	1	yes	characteristic, complemented, Sylow
40	#2245	—	227	1	no	maximal, Sylow
41	#499	D₂₂₇	1	1	yes	characteristic
42	#494	—	227	1	no	—
43	#358	—	227	4	no	—
44	#346	—	227	1	no	—
45	C₂×Q₁₆	—	454	1	no	—
46	C₂×Q₁₆	—	227	2	no	—
47	C₄⋊Q₈	—	454	1	no	—
48	#125	—	2	1	no	—
49	C₂×C₁₆	—	227	2	no	—
50	#107	C₂×D₂₂₇	1	2	yes	—
51	#106	—	2	1	no	—
52	C₄×C₈	C₂×D₂₂₇	1	1	yes	characteristic
53	C₂×Q₈	—	454	2	no	—
54	Q₁₆	—	908	2	no	—
55	Q₁₆	—	454	4	no	—
56	C₂×C₈	C₂²⋊D₂₂₇	1	1	yes	characteristic
57	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
58	C₄⋊C₄	—	2	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	908	2	no	—
61	C₄²	C₂²⋊D₂₂₇	1	1	yes	characteristic
62	C₁₆	—	454	2	no	—
63	Q₈	—	908	2	no	—
64	Q₈	—	908	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	908	2	no	—
67	C₂×C₄	#16813869	1	2	yes	—
68	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
69	C₈	#16813871	1	2	yes	—
70	C₈	#16813885	1	2	yes	—
71	C₂²	C₂×#16813869	1	1	yes	characteristic
72	C₄	—	2	2	no	—
73	C₄	—	8	2	no	—
74	C₄	—	1816	2	no	—
75	C₄	#20300277	1	1	yes	characteristic
76	C₄	C₂×#16813871	1	1	yes	characteristic
77	C₂	#28409380	1	1	yes	characteristic
78	C₂	#28409472	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, prime radical	C₂²×C₂₂₇	C₂×D₈
hypercenter	#499	D₂₂₇
hypocenter	C₂₂₇	#2245
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
39. C₂₂₇	#2245	semidirect

Maximal subgroups

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

subgroup	index
#28409412	2
#28408835	2
#28408808	2
#499×C₂₂₇	2
#2245	227