Subgroups — #30636834

Order 16256.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861145	C₂	1	2	yes	maximal
2	C₂×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#14523232	—	2	2	no	—
4	#16861145	—	2	2	no	—
5	#16861145	C₄	1	4	yes	—
6	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
7	M₃₂×C₁₂₇	—	2	2	no	—
8	C₂×C₁₆×C₁₂₇	C₄	1	2	yes	—
9	#14523232	—	2	4	no	—
10	#14523232	—	4	2	no	—
11	C₂×C₁₂₇⋊C₈	—	4	2	no	—
12	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
16	C₂×C₈×C₁₂₇	Q₈	1	1	yes	characteristic
17	C₁₆×C₁₂₇	—	2	2	no	—
18	C₁₂₇⋊C₈	—	4	4	no	—
19	C₂³×C₁₂₇	Q₁₆	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
21	C₂×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
23	C₂×C₄×C₁₂₇	SD₁₆	1	2	yes	—
24	C₈×C₁₂₇	C₄²	1	1	yes	characteristic
25	C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
27	C₂²×C₁₂₇	—	4	1	no	—
28	C₂²×C₁₂₇	#103	1	2	yes	—
29	C₂²×C₁₂₇	#107	1	1	yes	characteristic
30	C₄×C₁₂₇	#95	1	1	yes	characteristic
31	C₄×C₁₂₇	#102	1	2	yes	—
32	C₄×C₁₂₇	#106	1	1	yes	characteristic
33	C₂×C₁₂₇	—	2	2	no	—
34	C₂×C₁₂₇	—	2	1	no	—
35	C₂×C₁₂₇	#340	1	1	yes	characteristic
36	#1385	—	127	1	no	maximal, Sylow
37	C₁₂₇	#1385	1	1	yes	characteristic, complemented, Sylow
38	C₂×M₃₂	D₁₂₇	1	1	yes	characteristic
39	C₂×#108	—	127	2	no	—
40	C₂×M₁₆	—	254	2	no	—
41	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
42	M₃₂	—	2	2	no	—
43	C₂×C₁₆	Dic₁₂₇	1	2	yes	—
44	#108	—	127	4	no	—
45	#108	—	254	2	no	—
46	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
47	M₁₆	—	508	2	no	—
48	M₁₆	—	254	4	no	—
49	C₂×C₈	—	508	2	no	—
50	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
51	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
52	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
53	C₂×C₈	Dic₂₅₄	1	1	yes	characteristic
54	C₁₆	—	2	2	no	—
55	C₂³	Dic₅₀₈	1	1	yes	characteristic
56	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
57	C₂×C₄	C₄⋊Dic₁₂₇	1	1	yes	characteristic
58	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
59	C₂×C₄	#14523237	1	2	yes	—
60	C₈	—	508	4	no	—
61	C₈	C₄×Dic₁₂₇	1	1	yes	characteristic
62	C₈	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
63	C₈	Dic₁₂₇⋊C₄	1	2	yes	—
64	C₂²	—	4	1	no	—
65	C₂²	#16861134	1	2	yes	—
66	C₂²	#16861143	1	1	yes	characteristic
67	C₄	C₁₂₇⋊#95	1	1	yes	characteristic
68	C₄	#16861131	1	2	yes	—
69	C₄	#16861141	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	—	2	1	no	—
72	C₂	#22549639	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16861141
commutator	C₈×C₁₂₇	C₄²
Frattini	C₂²×C₈	C₂×D₁₂₇
Fitting	C₂×M₃₂×C₁₂₇	C₂
socle	C₂×C₁₂₇	#340
prime radical	C₂³×C₁₂₇	Q₁₆
cosocle	C₂²×C₈×C₁₂₇	C₂²
hypercenter	C₂×M₃₂	D₁₂₇
hypocenter	C₁₂₇	#1385
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
37. C₁₂₇	#1385	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861145	2
C₂×M₃₂×C₁₂₇	2
#1385	127