Subgroups — #30640601

Order 16256.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×Dic₁₂₇	C₂	1	2	yes	maximal
2	#22549704	C₂	1	1	yes	maximal, characteristic
3	#22549680	C₂	1	2	yes	maximal
4	#22549651	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₂₇	—	2	2	no	—
7	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	#16861143	C₂²	1	2	yes	—
9	#16861133	—	2	2	no	—
10	C₈×Dic₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
12	Q₃₂×C₁₂₇	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₂₇	—	2	2	no	—
16	C₄⋊Dic₁₂₇	—	4	2	no	—
17	C₄×Dic₁₂₇	—	4	2	no	—
18	C₄×Dic₁₂₇	D₄	1	1	yes	characteristic
19	C₁₂₇⋊C₁₆	—	2	1	no	—
20	C₂×Q₈×C₁₂₇	—	2	2	no	—
21	Q₁₆×C₁₂₇	—	2	2	no	—
22	Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
24	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₂₇	—	4	2	no	—
26	C₂×Dic₁₂₇	D₈	1	2	yes	—
27	C₁₂₇⋊C₈	—	2	1	no	—
28	Q₈×C₁₂₇	—	2	4	no	—
29	Q₈×C₁₂₇	—	4	2	no	—
30	C₂×C₄×C₁₂₇	—	4	2	no	—
31	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
32	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₂₇	#41	1	1	yes	characteristic
34	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
35	C₄×C₁₂₇	—	4	4	no	—
36	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
37	C₄×C₁₂₇	#135	1	1	yes	characteristic
38	Dic₁₂₇	—	2	2	no	—
39	Dic₁₂₇	—	8	2	no	—
40	C₂×C₁₂₇	C₄×D₈	1	1	yes	characteristic
41	C₂×C₁₂₇	#510	1	2	yes	—
42	#2176	—	127	1	no	maximal, Sylow
43	C₁₂₇	#2176	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₃₂	D₁₂₇	1	1	yes	characteristic
45	C₄×Q₁₆	—	127	2	no	—
46	#367	—	127	1	no	—
47	#358	—	127	2	no	—
48	#346	—	127	1	no	—
49	C₂×Q₁₆	C₂×D₁₂₇	1	2	yes	—
50	C₄×Q₈	—	254	2	no	—
51	Q₃₂	Dic₁₂₇	1	4	yes	complemented
52	C₂×C₁₆	—	127	1	no	—
53	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
54	#107	—	127	2	no	—
55	#103	—	254	2	no	—
56	C₄×C₈	—	127	1	no	—
57	C₂×Q₈	—	2	2	no	—
58	Q₁₆	—	2	2	no	—
59	Q₁₆	C₂×Dic₁₂₇	1	4	yes	—
60	C₂×C₈	—	127	1	no	—
61	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
62	C₄⋊C₄	—	508	2	no	—
63	C₄⋊C₄	—	254	2	no	—
64	C₄²	—	127	1	no	—
65	C₄²	—	508	2	no	—
66	C₁₆	—	254	1	no	—
67	C₁₆	C₂×Dic₁₂₇	1	2	yes	—
68	Q₈	—	2	4	no	—
69	Q₈	—	4	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	127	2	no	—
72	C₂×C₄	—	508	2	no	—
73	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
74	C₈	—	254	1	no	—
75	C₈	C₂²×Dic₁₂₇	1	1	yes	characteristic
76	C₈	#14523247	1	1	yes	characteristic
77	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
78	C₄	—	4	4	no	—
79	C₄	—	1016	2	no	—
80	C₄	—	254	2	no	—
81	C₄	D₄×Dic₁₂₇	1	1	yes	characteristic
82	C₄	#16861228	1	1	yes	characteristic
83	C₂	D₈×Dic₁₂₇	1	1	yes	characteristic
84	C₂	#22550318	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	C₂×Q₃₂×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂×Q₃₂	D₁₂₇
hypocenter	C₁₂₇	#2176
LCS	G ▷ 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
12. Q₃₂×C₁₂₇	C₄	semidirect
43. C₁₂₇	#2176	semidirect
51. Q₃₂	Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×Dic₁₂₇	2
#22549704	2
#22549680	2
#22549651	2
C₂×Q₃₂×C₁₂₇	2
#2176	127