Subgroups — #30640981

Order 16256.

176 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	C₂×#16861155	C₂	1	2	yes	maximal
3	#22550254	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₀₈	C₂²	1	2	yes	—
7	C₂×#14523238	—	2	2	no	—
8	#16861207	—	2	2	no	—
9	Dic₁₀₁₆	—	2	2	no	—
10	#16861155	—	2	4	no	—
11	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
14	Q₃₂×C₁₂₇	—	2	2	no	—
15	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₅₄	—	2	2	no	—
17	Dic₅₀₈	—	2	2	no	—
18	Dic₅₀₈	—	2	2	no	—
19	#14523238	—	4	4	no	—
20	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
21	Dic₁₂₇⋊C₄	—	4	4	no	—
22	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
23	C₁₂₇⋊C₁₆	D₄	1	2	yes	—
24	C₂×Q₈×C₁₂₇	—	2	2	no	—
25	Q₁₆×C₁₂₇	—	2	2	no	—
26	Q₁₆×C₁₂₇	—	2	2	no	—
27	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
28	C₁₆×C₁₂₇	D₄	1	2	yes	—
29	Dic₂₅₄	—	4	2	no	—
30	Dic₂₅₄	—	4	2	no	—
31	C₂×Dic₁₂₇	—	4	2	no	—
32	C₂×Dic₁₂₇	D₈	1	2	yes	—
33	C₁₂₇⋊C₈	D₈	1	2	yes	—
34	Q₈×C₁₂₇	—	4	2	no	—
35	Q₈×C₁₂₇	—	4	2	no	—
36	C₂×C₄×C₁₂₇	—	4	2	no	—
37	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
39	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
41	C₄×C₁₂₇	—	8	2	no	—
42	C₄×C₁₂₇	#127	1	1	yes	characteristic
43	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
44	Dic₁₂₇	—	8	2	no	—
45	Dic₁₂₇	Q₃₂	1	4	yes	complemented
46	C₂×C₁₂₇	#493	1	1	yes	characteristic
47	C₂×C₁₂₇	C₂×Q₃₂	1	2	yes	—
48	#2247	—	127	1	no	maximal, Sylow
49	C₁₂₇	#2247	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₃₂	—	127	2	no	—
51	C₂×Q₃₂	—	127	1	no	—
52	C₂×Q₃₂	D₁₂₇	1	1	yes	characteristic
53	#494	—	127	2	no	—
54	C₄×C₁₆	—	127	1	no	—
55	C₂×Q₁₆	—	127	2	no	—
56	C₂×Q₁₆	—	254	2	no	—
57	C₂×Q₁₆	C₂×D₁₂₇	1	2	yes	—
58	C₄⋊Q₈	—	254	2	no	—
59	Q₃₂	—	2	2	no	—
60	Q₃₂	—	254	4	no	—
61	Q₃₂	—	254	2	no	—
62	C₂×C₁₆	—	127	1	no	—
63	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
64	C₄×C₈	—	127	1	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	254	2	no	—
67	Q₁₆	—	2	2	no	—
68	Q₁₆	—	2	2	no	—
69	Q₁₆	—	508	4	no	—
70	Q₁₆	—	254	2	no	—
71	Q₁₆	—	254	2	no	—
72	C₂×C₈	—	127	1	no	—
73	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
74	C₄⋊C₄	—	508	4	no	—
75	C₄²	—	127	1	no	—
76	C₁₆	—	127	2	no	—
77	C₁₆	C₂²⋊D₁₂₇	1	2	yes	—
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	Q₈	—	508	2	no	—
81	Q₈	—	508	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	127	2	no	—
84	C₂×C₄	—	508	2	no	—
85	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
86	C₈	—	127	2	no	—
87	C₈	D₄×D₁₂₇	1	1	yes	characteristic
88	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
89	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
90	C₄	—	8	2	no	—
91	C₄	—	127	4	no	—
92	C₄	—	1016	2	no	—
93	C₄	#16861205	1	1	yes	characteristic
94	C₄	D₈×D₁₂₇	1	1	yes	characteristic
95	C₂	#22550251	1	1	yes	characteristic
96	C₂	Q₃₂×D₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	C₂×Q₃₂×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
hypercenter	C₂×Q₃₂	D₁₂₇
hypocenter	C₁₂₇	#2247
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
45. Dic₁₂₇	Q₃₂	semidirect
49. C₁₂₇	#2247	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
C₂×#16861155	2
#22550254	2
C₁₆×Dic₁₂₇	2
C₂×Q₃₂×C₁₂₇	2
#2247	127