Subgroups — #30638342

Order 16256.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550227	C₂	1	3	yes	maximal, complemented
2	#22550223	C₂	1	3	yes	maximal
3	C₈²×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861209	C₂²	1	3	yes	—
5	#16861206	C₂²	1	1	yes	characteristic
6	#16861134	—	2	6	no	—
7	#16861131	—	2	6	no	—
8	C₄×C₈×C₁₂₇	C₂²	1	3	yes	—
9	C₂×Dic₂₅₄	—	2	3	no	—
10	C₂×D₅₀₈	—	2	3	no	—
11	C₄⋊Dic₁₂₇	—	2	3	no	—
12	C₄⋊Dic₁₂₇	—	2	3	no	—
13	C₂×C₈×C₁₂₇	D₄	1	6	yes	—
14	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
15	C₂²×D₁₂₇	—	4	1	no	—
16	Dic₂₅₄	—	4	6	no	—
17	D₅₀₈	—	4	6	no	—
18	C₂×Dic₁₂₇	—	4	3	no	—
19	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
20	C₈×C₁₂₇	—	2	6	no	—
21	C₂²×C₁₂₇	#127	1	1	yes	characteristic
22	C₂×D₁₂₇	—	8	3	no	—
23	C₄×C₁₂₇	#135	1	6	yes	—
24	Dic₁₂₇	—	8	3	no	—
25	C₂×C₁₂₇	#495	1	3	yes	—
26	D₁₂₇	—	16	1	no	—
27	#1711	—	127	1	no	maximal, Sylow
28	C₁₂₇	#1711	1	1	yes	characteristic, complemented, Sylow
29	#487	—	127	3	no	—
30	#486	—	127	3	no	—
31	C₈²	D₁₂₇	1	1	yes	characteristic, complemented
32	C₄⋊Q₈	—	127	3	no	—
33	#127	—	127	1	no	—
34	#103	—	254	6	no	—
35	#102	—	254	6	no	—
36	C₄×C₈	C₂×D₁₂₇	1	3	yes	—
37	C₂×Q₈	—	254	3	no	—
38	C₂×D₄	—	254	3	no	—
39	C₂×C₈	D₅₀₈	1	6	yes	—
40	C₄⋊C₄	—	254	3	no	—
41	C₄⋊C₄	—	254	3	no	—
42	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
43	C₂³	—	508	1	no	—
44	Q₈	—	508	6	no	—
45	D₄	—	508	6	no	—
46	C₂×C₄	—	508	3	no	—
47	C₂×C₄	C₂×D₅₀₈	1	3	yes	—
48	C₈	—	2	6	no	—
49	C₂²	—	1016	3	no	—
50	C₂²	#16861206	1	1	yes	characteristic
51	C₄	—	1016	3	no	—
52	C₄	#16861231	1	6	yes	—
53	C₂	—	2032	1	no	—
54	C₂	#22550260	1	3	yes	—

	subgroup	G/H
center	C₂²	#16861206
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	C₈²×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#127
prime radical	#16861206	C₂²
hypercenter	C₈²	D₁₂₇
hypocenter	C₁₂₇	#1711
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #22550227	C₂	semidirect
3. C₈²×C₁₂₇	C₂	semidirect
28. C₁₂₇	#1711	semidirect
31. C₈²	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550227	2
#22550223	2
C₈²×C₁₂₇	2
#1711	127