Subgroups — #30640514

Order 16256.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549914	C₂	1	2	yes	maximal
2	#22549704	C₂	1	2	yes	maximal, complemented
3	#22549702	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#16861168	—	2	2	no	—
6	#16861143	C₂²	1	4	yes	—
7	#16861141	—	2	2	no	—
8	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₂₇	C₄	1	4	yes	complemented
10	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₂₇	—	2	4	no	—
13	C₂²⋊Dic₁₂₇	—	4	2	no	—
14	C₄×Dic₁₂₇	—	4	2	no	—
15	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
19	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂×Dic₁₂₇	—	4	4	no	—
23	C₂³×C₁₂₇	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
26	C₂×C₄×C₁₂₇	Q₁₆	1	2	yes	—
27	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
28	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
29	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
30	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₁₂₇	—	2	2	no	—
32	C₂²×C₁₂₇	#107	1	2	yes	—
33	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
34	C₄×C₁₂₇	#107	1	2	yes	—
35	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
37	Dic₁₂₇	—	8	4	no	—
38	C₂×C₁₂₇	—	2	2	no	—
39	C₂×C₁₂₇	C₂×#107	1	1	yes	characteristic
40	C₂×C₁₂₇	#508	1	2	yes	—
41	#2158	—	127	1	no	maximal, Sylow
42	C₁₂₇	#2158	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₁₆	D₁₂₇	1	1	yes	characteristic
44	#427	—	127	2	no	—
45	#367	—	127	2	no	—
46	#366	—	127	2	no	—
47	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
48	#117	—	254	2	no	—
49	C₂×C₁₆	Dic₁₂₇	1	4	yes	complemented
50	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
51	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
52	#107	—	127	4	no	—
53	#106	—	254	2	no	—
54	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
55	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
56	C₂×C₈	D₅₀₈	1	1	yes	characteristic
57	C₂×C₈	Dic₂₅₄	1	2	yes	—
58	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
59	C₄⋊C₄	—	254	4	no	—
60	C₂²⋊C₄	—	508	2	no	—
61	C₄²	—	508	2	no	—
62	C₁₆	C₂×Dic₁₂₇	1	2	yes	—
63	C₁₆	C₂×Dic₁₂₇	1	2	yes	—
64	C₂³	D₁₀₁₆	1	1	yes	characteristic
65	C₂×C₄	—	508	4	no	—
66	C₂×C₄	C₄⋊Dic₁₂₇	1	2	yes	—
67	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
68	C₂×C₄	Dic₅₀₈	1	2	yes	—
69	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
70	C₈	C₄⋊Dic₁₂₇	1	2	yes	—
71	C₈	C₂²×Dic₁₂₇	1	1	yes	characteristic
72	C₈	C₂×Dic₂₅₄	1	1	yes	characteristic
73	C₂²	—	2	2	no	—
74	C₂²	#16861143	1	2	yes	—
75	C₂²	C₂×D₁₀₁₆	1	1	yes	characteristic
76	C₄	—	1016	4	no	—
77	C₄	#16861143	1	2	yes	—
78	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
79	C₄	C₂×Dic₅₀₈	1	1	yes	characteristic
80	C₂	—	2	2	no	—
81	C₂	C₂×#16861143	1	1	yes	characteristic
82	C₂	#22550308	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₁₀₁₆
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	C₂²×C₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₈
prime radical	C₂³×C₁₂₇	D₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂²×C₁₆	D₁₂₇
hypocenter	C₁₂₇	#2158
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22549704	C₂	semidirect
3. #22549702	C₂	semidirect
9. C₂×C₁₆×C₁₂₇	C₄	semidirect
42. C₁₂₇	#2158	semidirect
49. C₂×C₁₆	Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22549914	2
#22549704	2
#22549702	2
C₂²×C₁₆×C₁₂₇	2
#2158	127