Subgroups — #28409148

Order 14528.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20300310	C₂	1	1	yes	maximal, characteristic
2	#20300296	C₂	1	2	yes	maximal, complemented
3	#20300214	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813884	C₂²	1	2	yes	—
7	Dic₉₀₈	C₄	1	1	yes	characteristic, complemented
8	#16813873	C₄	1	2	yes	complemented
9	D₁₈₁₆	C₄	1	1	yes	characteristic, complemented
10	#16813867	—	2	2	no	—
11	C₂×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₈×D₂₂₇	—	2	2	no	—
13	C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
14	M₁₆×C₂₂₇	C₂²	1	2	yes	—
15	C₂×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
16	Dic₄₅₄	C₂×C₄	1	2	yes	—
17	D₉₀₈	C₂×C₄	1	2	yes	—
18	C₂²⋊D₂₂₇	—	2	2	no	—
19	C₂×Dic₂₂₇	—	2	1	no	—
20	C₄×D₂₂₇	—	2	2	no	—
21	C₂₂₇⋊C₈	D₄	1	1	yes	characteristic
22	C₂₂₇⋊C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
24	C₈×C₂₂₇	—	2	2	no	—
25	C₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
26	C₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₂₂₇	#41	1	1	yes	characteristic
28	C₂×D₂₂₇	—	2	2	no	—
29	C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
31	Dic₂₂₇	—	2	2	no	—
32	Dic₂₂₇	—	2	2	no	—
33	C₂×C₂₂₇	—	2	1	no	—
34	C₂×C₂₂₇	C₄×D₄	1	1	yes	characteristic
35	D₂₂₇	—	4	2	no	—
36	C₂₂₇	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	227	1	no	maximal, Sylow
38	#135	—	227	1	no	—
39	#131	—	227	2	no	—
40	#108	D₂₂₇	1	1	yes	characteristic, complemented
41	#104	—	227	2	no	—
42	C₄×C₈	—	227	1	no	—
43	#41	—	227	2	no	—
44	Q₁₆	—	227	1	no	—
45	SD₁₆	—	227	2	no	—
46	D₈	—	227	1	no	—
47	M₁₆	—	454	2	no	—
48	M₁₆	C₂×D₂₂₇	1	2	yes	—
49	C₂×C₈	—	454	2	no	—
50	C₂×C₈	—	227	1	no	—
51	C₂×C₈	C₂×D₂₂₇	1	1	yes	characteristic
52	C₄²	—	227	1	no	—
53	Q₈	—	227	2	no	—
54	D₄	—	454	2	no	—
55	D₄	—	227	2	no	—
56	C₂×C₄	—	454	1	no	—
57	C₂×C₄	—	454	2	no	—
58	C₂×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	227	1	no	—
61	C₈	—	227	1	no	—
62	C₈	C₄×D₂₂₇	1	1	yes	characteristic
63	C₈	C₄×D₂₂₇	1	1	yes	characteristic
64	C₂²	—	454	2	no	—
65	C₂²	#16813885	1	1	yes	characteristic
66	C₄	—	454	2	no	—
67	C₄	—	454	2	no	—
68	C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
69	C₄	D₄×D₂₂₇	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	908	2	no	—
72	C₂	#20300250	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₂₇
commutator	C₄×C₂₂₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₇
Fitting	#108×C₂₂₇	C₂
socle	C₂×C₂₂₇	C₄×D₄
prime radical	#20300310	C₂
cosocle	C₂×C₄×C₂₂₇	C₂³
hypercenter	#108	D₂₂₇
hypocenter	C₂₂₇	#443
LCS	G ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #20300296	C₂	semidirect
3. #20300214	C₂	semidirect
4. C₈×Dic₂₂₇	C₂	semidirect
5. #108×C₂₂₇	C₂	semidirect
7. Dic₉₀₈	C₄	semidirect
8. #16813873	C₄	semidirect
9. D₁₈₁₆	C₄	semidirect
36. C₂₂₇	#443	semidirect
40. #108	D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
#20300310	2
#20300296	2
#20300214	2
C₈×Dic₂₂₇	2
#108×C₂₂₇	2
#443	227