Subgroups — #30640523

Order 16256.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861148	C₂	1	1	yes	maximal, characteristic
2	#108×D₁₂₇	C₂	1	2	yes	maximal
3	#22549706	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549705	C₂	1	2	yes	maximal, complemented
5	#368×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	M₁₆×D₁₂₇	—	2	2	no	—
7	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
8	#16861148	C₄	1	4	yes	—
9	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	#16861145	C₂²	1	2	yes	—
11	#16861144	—	2	2	no	—
12	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	#108×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
15	#14523232	—	2	2	no	—
16	#14523231	—	4	2	no	—
17	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
18	C₈×D₁₂₇	—	4	2	no	—
19	C₈×D₁₂₇	C₂×C₄	1	2	yes	—
20	C₈×D₁₂₇	Q₈	1	2	yes	—
21	C₁₂₇⋊C₁₆	C₂×C₄	1	2	yes	—
22	M₁₆×C₁₂₇	—	2	2	no	—
23	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
24	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₂²×D₁₂₇	Q₁₆	1	1	yes	characteristic
26	C₂×Dic₁₂₇	Q₁₆	1	1	yes	characteristic
27	C₄×D₁₂₇	C₄⋊C₄	1	2	yes	—
28	C₄×D₁₂₇	D₈	1	2	yes	—
29	C₁₂₇⋊C₈	—	4	2	no	—
30	C₁₂₇⋊C₈	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
32	C₈×C₁₂₇	—	4	2	no	—
33	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
36	C₂×D₁₂₇	—	4	1	no	—
37	C₂×D₁₂₇	#107	1	2	yes	—
38	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
40	Dic₁₂₇	#107	1	2	yes	—
41	C₂×C₁₂₇	—	2	1	no	—
42	C₂×C₁₂₇	C₂×#107	1	1	yes	characteristic
43	D₁₂₇	—	2	2	no	—
44	#2161	—	127	1	no	maximal, Sylow
45	C₁₂₇	#2161	1	1	yes	characteristic, complemented, Sylow
46	C₂×M₃₂	—	127	1	no	—
47	C₂×#108	—	127	2	no	—
48	#368	—	127	1	no	—
49	#368	—	127	2	no	—
50	#368	D₁₂₇	1	1	yes	characteristic, complemented
51	C₂×M₁₆	—	254	2	no	—
52	C₂²×C₈	—	127	1	no	—
53	M₃₂	—	127	4	no	—
54	C₂×C₁₆	—	127	1	no	—
55	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
56	#108	—	127	2	no	—
57	#108	—	254	2	no	—
58	#108	C₂×D₁₂₇	1	2	yes	—
59	C₂²×C₄	—	127	1	no	—
60	M₁₆	—	2	2	no	—
61	M₁₆	—	508	2	no	—
62	M₁₆	—	254	2	no	—
63	C₂×C₈	—	127	1	no	—
64	C₂×C₈	—	127	2	no	—
65	C₂×C₈	—	127	2	no	—
66	C₂×C₈	—	508	2	no	—
67	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
68	C₁₆	—	127	2	no	—
69	C₁₆	C₄×D₁₂₇	1	2	yes	—
70	C₂³	—	127	1	no	—
71	C₂×C₄	—	127	1	no	—
72	C₂×C₄	—	127	2	no	—
73	C₂×C₄	—	127	2	no	—
74	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
75	C₈	—	4	2	no	—
76	C₈	—	127	2	no	—
77	C₈	—	508	2	no	—
78	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
79	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
80	C₂²	—	127	2	no	—
81	C₂²	—	508	1	no	—
82	C₂²	Q₁₆×D₁₂₇	1	1	yes	characteristic
83	C₄	—	127	2	no	—
84	C₄	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
85	C₄	D₈×D₁₂₇	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	254	2	no	—
88	C₂	#107×D₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₁₂₇
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#368×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×#107
prime radical	C₂²×D₁₂₇	Q₁₆
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#368	D₁₂₇
hypocenter	C₁₂₇	#2161
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
3. #22549706	C₂	semidirect
4. #22549705	C₂	semidirect
5. #368×C₁₂₇	C₂	semidirect
45. C₁₂₇	#2161	semidirect
50. #368	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861148	2
#108×D₁₂₇	2
#22549706	2
#22549705	2
#368×C₁₂₇	2
#2161	127