Subgroups — #28409368

Order 14528.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20300267	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₂₂₇	C₂	1	1	yes	maximal, characteristic
3	#20300222	C₂	1	2	yes	maximal, complemented
4	#20300213	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₅₄	C₂²	1	1	yes	characteristic
7	C₂²×Dic₂₂₇	—	2	1	no	—
8	C₄⋊Dic₂₂₇	—	2	1	no	—
9	C₄⋊Dic₂₂₇	C₂²	1	2	yes	—
10	C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
11	Dic₂₂₇⋊C₄	—	2	1	no	—
12	C₂²⋊Dic₂₂₇	—	2	1	no	—
13	C₂²×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₂₂₇	C₂²	1	2	yes	—
15	Dic₄₅₄	—	2	2	no	—
16	C₂×Dic₂₂₇	—	2	1	no	—
17	C₂×Dic₂₂₇	—	2	1	no	—
18	C₂×Dic₂₂₇	—	2	2	no	—
19	C₂×Dic₂₂₇	—	4	2	no	—
20	C₂³×C₂₂₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₇	—	2	2	no	—
22	C₂×C₄×C₂₂₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
24	C₈×C₂₂₇	—	2	2	no	—
25	C₂²×C₂₂₇	—	2	1	no	—
26	C₂²×C₂₂₇	—	2	1	no	—
27	C₂²×C₂₂₇	Q₁₆	1	2	yes	—
28	C₂²×C₂₂₇	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₂₇	—	2	1	no	—
30	C₄×C₂₂₇	#41	1	2	yes	—
31	Dic₂₂₇	—	4	1	no	—
32	Dic₂₂₇	—	4	1	no	—
33	Dic₂₂₇	—	4	2	no	—
34	C₂×C₂₂₇	—	2	2	no	—
35	C₂×C₂₂₇	#123	1	1	yes	characteristic
36	C₂×C₂₂₇	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₂₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
38	C₂₂₇	#484	1	1	yes	characteristic, complemented, Sylow
39	#484	—	227	1	no	maximal, Sylow
40	C₂²⋊Q₈	—	227	1	no	—
41	C₂×C₄⋊C₄	—	227	1	no	—
42	#107	—	227	2	no	—
43	#103	—	227	2	no	—
44	C₂²⋊C₈	D₂₂₇	1	1	yes	characteristic
45	C₂×Q₈	—	227	1	no	—
46	C₂²×C₄	—	454	1	no	—
47	C₂²×C₄	C₂×D₂₂₇	1	1	yes	characteristic
48	C₂×C₈	C₂×D₂₂₇	1	2	yes	—
49	C₄⋊C₄	—	454	1	no	—
50	C₄⋊C₄	—	454	1	no	—
51	C₄⋊C₄	—	227	2	no	—
52	C₄⋊C₄	—	227	1	no	—
53	C₂²⋊C₄	—	454	1	no	—
54	C₂³	D₉₀₈	1	1	yes	characteristic
55	Q₈	—	454	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	908	2	no	—
58	C₂×C₄	—	454	1	no	—
59	C₂×C₄	—	454	1	no	—
60	C₂×C₄	—	454	2	no	—
61	C₂×C₄	D₉₀₈	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	Dic₉₀₈	1	2	yes	—
67	C₂²	C₂×D₉₀₈	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	908	2	no	—
70	C₄	—	908	1	no	—
71	C₄	—	908	1	no	—
72	C₄	#16813883	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	#20300272	1	1	yes	characteristic
75	C₂	C₂×Dic₉₀₈	1	1	yes	characteristic
76	C₂	#20300316	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₉₀₈
commutator, cosocle	C₂×C₄×C₂₂₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₇
Fitting	C₂²⋊C₈×C₂₂₇	C₂
socle	C₂²×C₂₂₇	C₂×D₄
prime radical	C₂³×C₂₂₇	D₄
hypercenter	C₂²⋊C₈	D₂₂₇
hypocenter	C₂₂₇	#484
LCS	G ▷ C₂×C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
3. #20300222	C₂	semidirect
4. #20300213	C₂	semidirect
38. C₂₂₇	#484	semidirect

Maximal subgroups

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

subgroup	index
#20300267	2
C₂×C₄⋊Dic₂₂₇	2
#20300222	2
#20300213	2
C₂²⋊C₈×C₂₂₇	2
#484	227