Subgroups — #30640607

Order 16256.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₀₃₂	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₀₁₆	C₂	1	2	yes	maximal, complemented
3	#22549704	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549678	C₂	1	2	yes	maximal, complemented
5	#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₀₁₆	C₂²	1	2	yes	—
7	C₄×D₅₀₈	—	2	2	no	—
8	D₂₀₃₂	C₄	1	4	yes	complemented
9	#16861143	C₂²	1	2	yes	—
10	#16861131	—	2	2	no	—
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×D₅₀₈	—	2	2	no	—
15	C₂×C₄×D₁₂₇	—	4	2	no	—
16	D₁₀₁₆	—	2	2	no	—
17	D₁₀₁₆	C₂×C₄	1	4	yes	complemented
18	C₄⋊Dic₁₂₇	—	2	2	no	—
19	(C₂×D₁₂₇)⋊C₄	—	4	2	no	—
20	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
23	C₁₆×C₁₂₇	—	2	1	no	—
24	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	complemented
25	C₂²×D₁₂₇	—	4	2	no	—
26	D₅₀₈	—	2	4	no	—
27	D₅₀₈	—	4	2	no	—
28	C₂×Dic₁₂₇	—	4	2	no	—
29	C₄×D₁₂₇	—	8	4	no	—
30	C₂×C₄×C₁₂₇	D₈	1	2	yes	—
31	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
32	C₈×C₁₂₇	—	2	1	no	—
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	C₂×D₁₂₇	—	4	4	no	—
37	C₂×D₁₂₇	—	8	2	no	—
38	C₂×D₁₂₇	—	8	2	no	—
39	C₄×C₁₂₇	—	2	2	no	—
40	C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
41	C₄×C₁₂₇	#135	1	1	yes	characteristic
42	Dic₁₂₇	—	8	2	no	—
43	C₂×C₁₂₇	C₄×D₈	1	1	yes	characteristic
44	C₂×C₁₂₇	#509	1	2	yes	—
45	D₁₂₇	—	8	4	no	—
46	#2177	—	127	1	no	maximal, Sylow
47	C₁₂₇	#2177	1	1	yes	characteristic, complemented, Sylow
48	C₂×D₁₆	—	127	1	no	—
49	C₄×D₈	—	127	2	no	—
50	#367	—	127	1	no	—
51	#357	—	127	2	no	—
52	#346	D₁₂₇	1	1	yes	characteristic, complemented
53	C₂×D₈	—	127	2	no	—
54	C₄×D₄	—	254	2	no	—
55	D₁₆	—	127	4	no	—
56	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
57	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
58	#107	—	127	2	no	—
59	#102	—	254	2	no	—
60	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
61	C₂×D₄	—	254	2	no	—
62	C₂²×C₄	—	508	2	no	—
63	D₈	—	127	4	no	—
64	D₈	—	254	2	no	—
65	C₂×C₈	D₅₀₈	1	1	yes	characteristic
66	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
67	C₄⋊C₄	—	254	2	no	—
68	C₂²⋊C₄	—	508	2	no	—
69	C₄²	D₅₀₈	1	1	yes	characteristic
70	C₁₆	—	2	1	no	—
71	C₁₆	C₄×D₁₂₇	1	2	yes	complemented
72	C₂³	—	508	2	no	—
73	D₄	—	508	2	no	—
74	D₄	—	254	4	no	—
75	C₂×C₄	—	1016	4	no	—
76	C₂×C₄	—	508	2	no	—
77	C₂×C₄	D₁₀₁₆	1	2	yes	—
78	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
81	C₈	#14523248	1	1	yes	characteristic
82	C₂²	—	1016	2	no	—
83	C₂²	—	1016	2	no	—
84	C₂²	—	508	4	no	—
85	C₂²	C₂×D₁₀₁₆	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	1016	2	no	—
88	C₄	C₄×D₅₀₈	1	1	yes	characteristic
89	C₄	#16861231	1	1	yes	characteristic
90	C₂	—	1016	4	no	—
91	C₂	C₄×D₁₀₁₆	1	1	yes	characteristic
92	C₂	#22550314	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₀₁₆
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#346×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₈
prime radical	C₂×D₂₀₃₂	C₂
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#346	D₁₂₇
hypocenter	C₁₂₇	#2177
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₄×D₁₀₁₆	C₂	semidirect
3. #22549704	C₂	semidirect
4. #22549678	C₂	semidirect
5. #346×C₁₂₇	C₂	semidirect
8. D₂₀₃₂	C₄	semidirect
17. D₁₀₁₆	C₂×C₄	semidirect
24. C₁₆×C₁₂₇	C₂×C₄	semidirect
47. C₁₂₇	#2177	semidirect
52. #346	D₁₂₇	semidirect
71. C₁₆	C₄×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₀₃₂	2
C₄×D₁₀₁₆	2
#22549704	2
#22549678	2
#346×C₁₂₇	2
#2177	127