Subgroups — #30640638

Order 16256.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₂₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	Dic₁₂₇⋊C₁₆	C₂	1	2	yes	maximal
4	#22549651	C₂	1	2	yes	maximal
5	C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₅₄	C₄	1	1	yes	characteristic
7	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
8	C₄⋊(C₁₂₇⋊C₈)	C₄	1	1	yes	characteristic
9	Dic₁₂₇⋊C₈	C₄	1	2	yes	—
10	C₈×Dic₁₂₇	C₂²	1	2	yes	—
11	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₅₄	C₈	1	1	yes	characteristic
15	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₂₇	C₈	1	1	yes	characteristic
17	Dic₁₂₇⋊C₄	C₈	1	2	yes	—
18	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
19	C₁₂₇⋊C₁₆	—	2	2	no	—
20	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₁₂₇	—	2	1	no	—
24	C₁₆×C₁₂₇	Q₈	1	2	yes	complemented
25	Dic₂₅₄	—	2	2	no	—
26	C₂×Dic₁₂₇	C₂×C₈	1	2	yes	—
27	C₂×Dic₁₂₇	C₂×C₈	1	2	yes	—
28	C₁₂₇⋊C₈	—	2	2	no	—
29	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₂₇	—	2	1	no	—
33	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₁₂₇	#41	1	1	yes	characteristic
35	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₁₂₇	—	2	1	no	—
37	C₄×C₁₂₇	M₃₂	1	2	yes	—
38	C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₁₂₇	#131	1	1	yes	characteristic
40	Dic₁₂₇	—	2	2	no	—
41	Dic₁₂₇	—	2	2	no	—
42	C₂×C₁₂₇	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
44	C₂×C₁₂₇	#504	1	1	yes	characteristic
45	#2183	—	127	1	no	maximal, Sylow
46	C₁₂₇	#2183	1	1	yes	characteristic, complemented, Sylow
47	C₈×Q₈	—	127	1	no	—
48	C₄⋊C₁₆	—	127	2	no	—
49	C₄⋊C₁₆	—	127	1	no	—
50	#346	—	127	2	no	—
51	C₄×C₁₆	D₁₂₇	1	1	yes	characteristic
52	C₄×Q₈	—	127	1	no	—
53	C₂×C₁₆	—	127	2	no	—
54	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
55	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
56	C₄⋊C₈	—	127	2	no	—
57	C₄⋊C₈	—	127	1	no	—
58	C₄×C₈	—	127	2	no	—
59	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
60	C₂×Q₈	—	127	1	no	—
61	C₂×C₈	—	127	2	no	—
62	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
64	C₄⋊C₄	—	127	2	no	—
65	C₄⋊C₄	—	127	1	no	—
66	C₄²	—	127	2	no	—
67	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	—	254	2	no	—
70	C₁₆	Dic₂₅₄	1	2	yes	complemented
71	Q₈	—	254	2	no	—
72	C₂×C₄	—	127	2	no	—
73	C₂×C₄	—	127	2	no	—
74	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
75	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	254	2	no	—
79	C₈	C₂×Dic₂₅₄	1	1	yes	characteristic
80	C₈	#14523248	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	254	2	no	—
84	C₄	—	254	2	no	—
85	C₄	#16861148	1	2	yes	—
86	C₄	C₄×Dic₂₅₄	1	1	yes	characteristic
87	C₄	#16861215	1	1	yes	characteristic
88	C₂	C₈×Dic₂₅₄	1	1	yes	characteristic
89	C₂	C₂×#16861148	1	1	yes	characteristic
90	C₂	#22550292	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₂₇
commutator, socle, prime radical	C₂²×C₁₂₇	C₂²×C₈
Fitting	C₄×C₁₆×C₁₂₇	C₂
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₄×C₁₆	D₁₂₇
hypocenter	C₁₂₇	#2183
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
24. C₁₆×C₁₂₇	Q₈	semidirect
46. C₁₂₇	#2183	semidirect
70. C₁₆	Dic₂₅₄	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₂₅₄	2
C₄⋊(C₁₂₇⋊C₁₆)	2
Dic₁₂₇⋊C₁₆	2
#22549651	2
C₄×C₁₆×C₁₂₇	2
#2183	127