Subgroups — #28409124

Order 14528.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813874	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
3	#20300253	C₂	1	1	yes	maximal, characteristic
4	#20300221	C₂	1	1	yes	maximal, characteristic
5	#20300213	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
7	#103×C₂₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₅₄	C₂²	1	1	yes	characteristic
9	#16813874	C₄	1	4	yes	complemented
10	C₂×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₂₇	—	2	1	no	—
12	C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
13	Dic₂₂₇⋊C₄	—	2	1	no	—
14	C₄×Dic₂₂₇	—	2	1	no	—
15	C₄×Dic₂₂₇	—	2	1	no	—
16	C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
20	Dic₄₅₄	—	2	1	no	—
21	Dic₄₅₄	C₂×C₄	1	2	yes	—
22	C₂×Dic₂₂₇	—	2	1	no	—
23	C₂×Dic₂₂₇	—	2	1	no	—
24	C₂×Dic₂₂₇	D₄	1	1	yes	characteristic
25	C₂×Dic₂₂₇	D₄	1	1	yes	characteristic
26	C₂₂₇⋊C₈	C₂×C₄	1	2	yes	—
27	Q₈×C₂₂₇	—	2	1	no	—
28	Q₈×C₂₂₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₂₂₇	—	2	1	no	—
30	C₂×C₄×C₂₂₇	—	2	1	no	—
31	C₂×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
32	C₈×C₂₂₇	—	2	1	no	—
33	C₂²×C₂₂₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₂₂₇	—	2	2	no	—
35	C₄×C₂₂₇	—	4	1	no	—
36	C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
37	C₄×C₂₂₇	#41	1	1	yes	characteristic
38	Dic₂₂₇	—	2	2	no	—
39	Dic₂₂₇	—	2	1	no	—
40	Dic₂₂₇	—	4	1	no	—
41	Dic₂₂₇	Q₁₆	1	2	yes	complemented
42	C₂×C₂₂₇	C₄×D₄	1	1	yes	characteristic
43	C₂×C₂₂₇	C₂×Q₁₆	1	1	yes	characteristic
44	C₂×C₂₂₇	#135	1	1	yes	characteristic
45	C₂₂₇	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₁₆	—	227	1	no	maximal, Sylow
47	C₂×Q₁₆	—	227	1	no	—
48	C₄×Q₈	—	227	1	no	—
49	C₄×Q₈	—	227	1	no	—
50	#107	—	227	1	no	—
51	#103	—	227	1	no	—
52	#103	D₂₂₇	1	1	yes	characteristic
53	C₄×C₈	—	227	1	no	—
54	C₂×Q₈	—	227	1	no	—
55	C₂×Q₈	C₂×D₂₂₇	1	1	yes	characteristic
56	Q₁₆	—	227	4	no	—
57	C₂×C₈	—	227	1	no	—
58	C₂×C₈	C₂×D₂₂₇	1	1	yes	characteristic
59	C₄⋊C₄	—	454	1	no	—
60	C₄⋊C₄	—	454	1	no	—
61	C₄⋊C₄	—	227	1	no	—
62	C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
63	C₄²	—	454	1	no	—
64	C₄²	—	454	1	no	—
65	C₄²	—	227	1	no	—
66	Q₈	—	2	1	no	—
67	Q₈	—	454	1	no	—
68	Q₈	—	227	2	no	—
69	Q₈	C₄×D₂₂₇	1	2	yes	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	454	1	no	—
73	C₂×C₄	—	454	1	no	—
74	C₂×C₄	—	227	1	no	—
75	C₂×C₄	—	227	1	no	—
76	C₂×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	227	2	no	—
79	C₂²	D₄×D₂₂₇	1	1	yes	characteristic
80	C₄	—	2	2	no	—
81	C₄	—	4	1	no	—
82	C₄	—	908	1	no	—
83	C₄	—	454	1	no	—
84	C₄	—	454	2	no	—
85	C₄	—	227	2	no	—
86	C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
87	C₄	#16813883	1	1	yes	characteristic
88	C₂	(C₂²⋊D₂₂₇)⋊C₄	1	1	yes	characteristic
89	C₂	Q₁₆×D₂₂₇	1	1	yes	characteristic
90	C₂	#20300309	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₂₇
commutator	C₄×C₂₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₇
Fitting	#103×C₂₂₇	C₂
socle, prime radical	C₂²×C₂₂₇	C₂×D₄
cosocle	C₂×C₄×C₂₂₇	C₂³
hypercenter	#103	D₂₂₇
hypocenter	C₂₂₇	C₄×Q₁₆
LCS	G ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
9. #16813874	C₄	semidirect
41. Dic₂₂₇	Q₁₆	semidirect
45. C₂₂₇	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813874	2
Q₈×Dic₂₂₇	2
#20300253	2
#20300221	2
#20300213	2
C₈×Dic₂₂₇	2
#103×C₂₂₇	2
C₄×Q₁₆	227