Subgroups — #30640247

Order 16256.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549925	C₂	1	3	yes	maximal, complemented
2	#22549839	C₂	1	3	yes	maximal
3	#338×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₁₂₇	C₂²	1	1	yes	characteristic
5	C₂×#14523232	C₂²	1	3	yes	—
6	#16861145	—	2	6	no	—
7	(C₂×D₁₂₇)⋊C₈	—	2	6	no	—
8	C₂×M₁₆×C₁₂₇	C₂²	1	3	yes	—
9	C₂³×D₁₂₇	Q₈	1	1	yes	characteristic
10	C₂²×Dic₁₂₇	Q₈	1	1	yes	characteristic
11	C₂×C₄×D₁₂₇	—	2	3	no	—
12	C₂×C₄×D₁₂₇	—	2	3	no	—
13	#14523232	—	2	6	no	—
14	C₂×C₁₂₇⋊C₈	—	2	3	no	—
15	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
16	M₁₆×C₁₂₇	—	2	6	no	—
17	C₂×C₈×C₁₂₇	D₄	1	6	yes	—
18	C₂²×D₁₂₇	—	2	3	no	—
19	C₂²×D₁₂₇	—	4	2	no	—
20	C₂×Dic₁₂₇	—	2	3	no	—
21	C₄×D₁₂₇	—	4	3	no	—
22	C₄×D₁₂₇	—	4	1	no	—
23	C₁₂₇⋊C₈	—	4	3	no	—
24	C₂³×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
26	C₂×C₄×C₁₂₇	#41	1	3	yes	—
27	C₈×C₁₂₇	—	2	6	no	—
28	C₂²×C₁₂₇	—	4	1	no	—
29	C₂²×C₁₂₇	C₂²⋊Q₈	1	3	yes	—
30	C₂×D₁₂₇	—	2	6	no	—
31	C₂×D₁₂₇	—	4	3	no	—
32	C₂×D₁₂₇	—	4	1	no	—
33	C₄×C₁₂₇	#123	1	3	yes	—
34	C₄×C₁₂₇	#127	1	1	yes	characteristic
35	Dic₁₂₇	—	4	1	no	—
36	C₂×C₁₂₇	—	2	3	no	—
37	C₂×C₁₂₇	#399	1	1	yes	characteristic
38	D₁₂₇	—	4	2	no	—
39	#2091	—	127	1	no	maximal, Sylow
40	C₁₂₇	#2091	1	1	yes	characteristic, complemented, Sylow
41	#430	—	127	3	no	—
42	#407	—	127	3	no	—
43	#338	D₁₂₇	1	1	yes	characteristic, complemented
44	C₂³×C₄	—	127	1	no	—
45	C₂×M₁₆	—	127	3	no	—
46	C₂×M₁₆	C₂×D₁₂₇	1	3	yes	—
47	#108	—	254	6	no	—
48	C₂²⋊C₈	—	254	6	no	—
49	C₂⁴	—	127	1	no	—
50	C₂²×C₄	—	127	1	no	—
51	C₂²×C₄	—	254	3	no	—
52	C₂²×C₄	—	254	3	no	—
53	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
54	M₁₆	—	2	6	no	—
55	M₁₆	—	254	6	no	—
56	C₂×C₈	—	254	3	no	—
57	C₂×C₈	D₅₀₈	1	6	yes	—
58	C₂³	—	508	2	no	—
59	C₂³	—	254	3	no	—
60	C₂³	Q₈×D₁₂₇	1	1	yes	characteristic
61	C₂×C₄	—	508	3	no	—
62	C₂×C₄	—	508	1	no	—
63	C₂×C₄	—	254	3	no	—
64	C₂×C₄	C₂×D₅₀₈	1	3	yes	—
65	C₂×C₄	#14523247	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	—	508	3	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	508	3	no	—
70	C₂²	—	508	1	no	—
71	C₂²	—	254	6	no	—
72	C₂²	#16861189	1	3	yes	—
73	C₄	—	508	1	no	—
74	C₄	#16861193	1	3	yes	—
75	C₄	#16861206	1	1	yes	characteristic
76	C₂	—	2	3	no	—
77	C₂	—	508	2	no	—
78	C₂	#22549813	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16861206
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#338×C₁₂₇	C₂
socle	C₂×C₁₂₇	#399
prime radical	C₂³×D₁₂₇	Q₈
hypercenter	#338	D₁₂₇
hypocenter	C₁₂₇	#2091
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #22549925	C₂	semidirect
3. #338×C₁₂₇	C₂	semidirect
40. C₁₂₇	#2091	semidirect
43. #338	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22549925	2
#22549839	2
#338×C₁₂₇	2
#2091	127