Subgroups — #30640132

Order 16256.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549924	C₂	1	3	yes	maximal
2	C₂×#16861127	C₂	1	3	yes	maximal
3	#338×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₅₄	C₂²	1	1	yes	characteristic
5	C₂×#14523232	C₂²	1	3	yes	—
6	#16861144	—	2	6	no	—
7	#16861127	—	2	6	no	—
8	C₂×M₁₆×C₁₂₇	C₂²	1	3	yes	—
9	C₂×Dic₂₅₄	—	2	3	no	—
10	C₂×Dic₂₅₄	—	2	3	no	—
11	C₂²×Dic₁₂₇	Q₈	1	2	yes	—
12	#14523232	—	2	3	no	—
13	#14523232	D₄	1	6	yes	—
14	C₂×C₁₂₇⋊C₈	—	2	3	no	—
15	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
16	M₁₆×C₁₂₇	—	2	3	no	—
17	M₁₆×C₁₂₇	—	2	3	no	—
18	C₂×C₈×C₁₂₇	—	2	3	no	—
19	Dic₂₅₄	—	4	3	no	—
20	Dic₂₅₄	—	4	1	no	—
21	C₂×Dic₁₂₇	—	2	6	no	—
22	C₁₂₇⋊C₈	—	2	6	no	—
23	C₂³×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
25	C₂×C₄×C₁₂₇	#41	1	3	yes	—
26	C₈×C₁₂₇	—	4	3	no	—
27	C₂²×C₁₂₇	—	4	1	no	—
28	C₂²×C₁₂₇	C₂²⋊Q₈	1	3	yes	—
29	C₄×C₁₂₇	#121	1	3	yes	—
30	C₄×C₁₂₇	#126	1	1	yes	characteristic
31	Dic₁₂₇	—	4	2	no	—
32	C₂×C₁₂₇	—	2	3	no	—
33	C₂×C₁₂₇	#396	1	1	yes	characteristic
34	#2070	—	127	1	no	maximal, Sylow
35	C₁₂₇	#2070	1	1	yes	characteristic, complemented, Sylow
36	#430	—	127	3	no	—
37	C₂×#101	—	127	3	no	—
38	#338	D₁₂₇	1	1	yes	characteristic
39	C₂²×Q₈	—	127	1	no	—
40	C₂×M₁₆	—	127	3	no	—
41	C₂×M₁₆	C₂×D₁₂₇	1	3	yes	—
42	#108	—	254	6	no	—
43	#101	—	254	6	no	—
44	C₂×Q₈	—	254	3	no	—
45	C₂×Q₈	—	254	3	no	—
46	C₂²×C₄	—	127	2	no	—
47	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
48	M₁₆	—	2	3	no	—
49	M₁₆	—	2	3	no	—
50	M₁₆	—	127	6	no	—
51	M₁₆	—	254	3	no	—
52	C₂×C₈	—	2	3	no	—
53	C₂×C₈	—	254	3	no	—
54	C₂³	Q₈×D₁₂₇	1	1	yes	characteristic
55	Q₈	—	508	1	no	—
56	Q₈	—	508	3	no	—
57	C₂×C₄	—	254	6	no	—
58	C₂×C₄	D₄×D₁₂₇	1	3	yes	—
59	C₂×C₄	#14523248	1	3	yes	—
60	C₈	—	4	3	no	—
61	C₈	—	254	6	no	—
62	C₂²	—	4	1	no	—
63	C₂²	#16861187	1	3	yes	—
64	C₄	—	508	2	no	—
65	C₄	#16861182	1	3	yes	—
66	C₄	#16861203	1	1	yes	characteristic
67	C₂	—	2	3	no	—
68	C₂	#22549800	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549800
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#338×C₁₂₇	C₂
socle	C₂×C₁₂₇	#396
prime radical	C₂³×C₁₂₇	C₂×Q₈
hypercenter	#338	D₁₂₇
hypocenter	C₁₂₇	#2070
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
35. C₁₂₇	#2070	semidirect

Maximal subgroups

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

subgroup	index
#22549924	2
C₂×#16861127	2
#338×C₁₂₇	2
#2070	127