Subgroups — #30640588

Order 16256.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₁₆	C₂	1	1	yes	maximal, characteristic
2	#22549970	C₂	1	2	yes	maximal
3	#22549679	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#366×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₀₈	C₂²	1	2	yes	—
7	#16861174	—	2	2	no	—
8	Dic₁₀₁₆	C₄	1	4	yes	complemented
9	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	#16861132	—	2	2	no	—
11	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	#107×C₁₂₇	C₂²	1	2	yes	—
14	C₂×Dic₂₅₄	—	2	2	no	—
15	Dic₅₀₈	—	2	2	no	—
16	Dic₅₀₈	C₂×C₄	1	4	yes	—
17	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
18	Dic₁₂₇⋊C₄	—	4	2	no	—
19	C₄×Dic₁₂₇	—	4	2	no	—
20	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
21	C₁₂₇⋊C₁₆	—	2	1	no	—
22	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₂₇	—	2	2	no	—
24	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
25	Dic₂₅₄	—	2	4	no	—
26	Dic₂₅₄	—	4	2	no	—
27	C₂×Dic₁₂₇	—	4	2	no	—
28	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
29	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
30	C₁₂₇⋊C₈	—	2	1	no	—
31	C₂×C₄×C₁₂₇	—	4	2	no	—
32	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₂₇	#41	1	1	yes	characteristic
35	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₂₇	—	8	2	no	—
37	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
38	C₄×C₁₂₇	#135	1	1	yes	characteristic
39	Dic₁₂₇	—	2	1	no	—
40	Dic₁₂₇	—	4	4	no	—
41	Dic₁₂₇	Q₃₂	1	2	yes	complemented
42	C₂×C₁₂₇	C₄×D₈	1	1	yes	characteristic
43	C₂×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
44	C₂×C₁₂₇	#508	1	1	yes	characteristic
45	C₄×Q₃₂	—	127	1	no	maximal, Sylow
46	C₁₂₇	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₃₂	—	127	1	no	—
48	C₄×Q₁₆	—	127	2	no	—
49	#366	D₁₂₇	1	1	yes	characteristic
50	#358	—	127	2	no	—
51	C₄×C₁₆	—	127	1	no	—
52	C₂×Q₁₆	—	127	2	no	—
53	C₄×Q₈	—	254	2	no	—
54	Q₃₂	—	127	4	no	—
55	C₂×C₁₆	—	127	1	no	—
56	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
57	#107	C₂×D₁₂₇	1	2	yes	—
58	#103	—	254	2	no	—
59	C₄×C₈	—	127	1	no	—
60	C₂×Q₈	—	254	2	no	—
61	Q₁₆	—	127	4	no	—
62	Q₁₆	—	254	2	no	—
63	C₂×C₈	—	127	1	no	—
64	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	508	2	no	—
67	C₄²	—	127	1	no	—
68	C₄²	—	508	2	no	—
69	C₁₆	—	254	1	no	—
70	C₁₆	C₄×D₁₂₇	1	2	yes	—
71	Q₈	—	508	2	no	—
72	Q₈	—	254	4	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	127	1	no	—
75	C₂×C₄	—	127	1	no	—
76	C₂×C₄	—	508	2	no	—
77	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
78	C₈	—	254	1	no	—
79	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
80	C₈	#14523249	1	1	yes	characteristic
81	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	127	2	no	—
84	C₄	—	508	4	no	—
85	C₄	—	254	1	no	—
86	C₄	#16861171	1	1	yes	characteristic
87	C₄	#16861232	1	1	yes	characteristic
88	C₂	#22549958	1	1	yes	characteristic
89	C₂	Q₃₂×D₁₂₇	1	1	yes	characteristic
90	C₂	#22550305	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#366×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#366	D₁₂₇
hypocenter	C₁₂₇	C₄×Q₃₂
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
8. Dic₁₀₁₆	C₄	semidirect
41. Dic₁₂₇	Q₃₂	semidirect
46. C₁₂₇	C₄×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₀₁₆	2
#22549970	2
#22549679	2
C₁₆×Dic₁₂₇	2
#366×C₁₂₇	2
C₄×Q₃₂	127