Subgroups — #30641048

Order 16256.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550308	C₂	1	1	yes	maximal, characteristic
2	#22549716	C₂	1	2	yes	maximal, complemented
3	#22549713	C₂	1	2	yes	maximal, complemented
4	#22549710	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861231	—	2	1	no	—
7	Dic₁₀₁₆	C₂²	1	1	yes	characteristic
8	#16861154	—	2	1	no	—
9	D₂₀₃₂	C₂²	1	1	yes	characteristic
10	C₁₂₇⋊C₃₂	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₁₂₇	—	2	1	no	—
12	Q₃₂×C₁₂₇	—	2	1	no	—
13	Q₃₂×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
15	#14523248	—	4	1	no	—
16	Dic₅₀₈	—	2	1	no	—
17	#14523237	—	4	1	no	—
18	D₁₀₁₆	—	2	1	no	—
19	C₂×Q₈×C₁₂₇	—	4	1	no	—
20	Q₁₆×C₁₂₇	—	2	2	no	—
21	Q₁₆×C₁₂₇	—	4	1	no	—
22	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
23	C₁₆×C₁₂₇	D₄	1	1	yes	characteristic
24	C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
25	Dic₂₅₄	—	4	1	no	—
26	D₅₀₈	—	4	1	no	—
27	C₂²⋊D₁₂₇	—	8	1	no	—
28	C₄×D₁₂₇	—	8	1	no	—
29	Q₈×C₁₂₇	—	4	2	no	—
30	Q₈×C₁₂₇	—	8	1	no	—
31	C₂×C₄×C₁₂₇	—	8	1	no	—
32	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
33	C₈×C₁₂₇	D₈	1	1	yes	characteristic
34	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₂₇	D₁₆	1	1	yes	characteristic
36	C₂×D₁₂₇	—	8	1	no	—
37	C₄×C₁₂₇	—	8	2	no	—
38	C₄×C₁₂₇	D₁₆	1	1	yes	characteristic
39	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
40	Dic₁₂₇	—	8	1	no	—
41	C₂×C₁₂₇	—	2	1	no	—
42	C₂×C₁₂₇	C₂×D₁₆	1	1	yes	characteristic
43	D₁₂₇	—	16	1	no	—
44	#2264	—	127	1	no	maximal, Sylow
45	C₁₂₇	#2264	1	1	yes	characteristic, complemented, Sylow
46	#508	—	127	1	no	—
47	C₂×Q₃₂	D₁₂₇	1	1	yes	characteristic, complemented
48	Q₆₄	—	127	2	no	—
49	SD₆₄	—	127	2	no	—
50	M₆₄	—	127	1	no	—
51	#135	—	254	1	no	—
52	C₂×Q₁₆	—	2	1	no	—
53	Q₃₂	—	2	1	no	—
54	Q₃₂	—	127	1	no	—
55	Q₃₂	C₂×D₁₂₇	1	2	yes	—
56	SD₃₂	—	254	1	no	—
57	D₁₆	—	127	1	no	—
58	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
59	C₃₂	—	127	2	no	—
60	#41	—	508	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	Q₁₆	—	2	2	no	—
63	Q₁₆	—	4	1	no	—
64	Q₁₆	—	254	1	no	—
65	SD₁₆	—	508	1	no	—
66	D₈	—	254	1	no	—
67	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
68	C₁₆	C₂²⋊D₁₂₇	1	1	yes	characteristic
69	C₁₆	C₂²×D₁₂₇	1	1	yes	characteristic
70	Q₈	—	4	2	no	—
71	Q₈	—	8	1	no	—
72	Q₈	—	508	1	no	—
73	D₄	—	1016	1	no	—
74	D₄	—	508	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	—	1016	1	no	—
77	C₂×C₄	#14523233	1	1	yes	characteristic
78	C₈	#14523233	1	1	yes	characteristic
79	C₈	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
80	C₂²	—	1016	1	no	—
81	C₂²	#16861150	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	1016	1	no	—
84	C₄	#16861150	1	1	yes	characteristic
85	C₄	C₂×#14523233	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2032	1	no	—
88	C₂	C₂×#16861150	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16861150
commutator, cosocle	C₁₆×C₁₂₇	C₂³
Frattini	C₁₆	C₂²×D₁₂₇
Fitting	C₂×Q₃₂×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×D₁₆
prime radical	#22550308	C₂
hypercenter	C₂×Q₃₂	D₁₂₇
hypocenter	C₁₂₇	#2264
LCS	G ▷ C₁₆×C₁₂₇ ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #22549716	C₂	semidirect
3. #22549713	C₂	semidirect
4. #22549710	C₂	semidirect
5. C₂×Q₃₂×C₁₂₇	C₂	semidirect
45. C₁₂₇	#2264	semidirect
47. C₂×Q₃₂	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550308	2
#22549716	2
#22549713	2
#22549710	2
C₂×Q₃₂×C₁₂₇	2
#2264	127