Subgroups — #30638841

Order 16256.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861126	C₂	1	2	yes	maximal
2	#22549646	C₂	1	4	yes	maximal, complemented
3	#518×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523232	C₂²	1	4	yes	—
5	#16861126	—	2	4	no	—
6	C₂²×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₄×D₄×C₁₂₇	—	2	4	no	—
8	#117×C₁₂₇	C₂²	1	2	yes	—
9	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	4	yes	—
10	#14523232	—	2	4	no	—
11	#14523232	—	2	4	no	—
12	C₂×C₁₂₇⋊C₈	—	2	4	no	—
13	C₂⁴×C₁₂₇	C₂×C₄	1	2	yes	—
14	C₂×D₄×C₁₂₇	—	2	4	no	—
15	C₂×D₄×C₁₂₇	D₄	1	4	yes	—
16	C₂²×C₄×C₁₂₇	—	2	4	no	—
17	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₁₂₇	—	2	2	no	—
19	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
20	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
21	C₄²×C₁₂₇	—	2	2	no	—
22	C₁₂₇⋊C₈	—	4	4	no	—
23	C₂³×C₁₂₇	—	2	4	no	—
24	C₂³×C₁₂₇	—	8	2	no	—
25	C₂³×C₁₂₇	C₂²⋊C₄	1	4	yes	—
26	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
27	D₄×C₁₂₇	—	2	8	no	—
28	C₂×C₄×C₁₂₇	—	2	4	no	—
29	C₂×C₄×C₁₂₇	—	8	4	no	—
30	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
31	C₂×C₄×C₁₂₇	#41	1	4	yes	—
32	C₂²×C₁₂₇	—	2	4	no	—
33	C₂²×C₁₂₇	—	4	1	no	—
34	C₂²×C₁₂₇	—	4	4	no	—
35	C₂²×C₁₂₇	—	8	4	no	—
36	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₁₂₇	#117	1	2	yes	—
38	C₄×C₁₂₇	—	4	4	no	—
39	C₄×C₁₂₇	#123	1	4	yes	—
40	C₂×C₁₂₇	—	2	1	no	—
41	C₂×C₁₂₇	—	2	2	no	—
42	C₂×C₁₂₇	—	4	4	no	—
43	C₂×C₁₂₇	#381	1	1	yes	characteristic
44	#1829	—	127	1	no	maximal, Sylow
45	C₁₂₇	#1829	1	1	yes	characteristic, complemented, Sylow
46	#518	D₁₂₇	1	1	yes	characteristic
47	C₂×#100	—	127	2	no	—
48	#344	—	127	4	no	—
49	C₂²×D₄	C₂×D₁₂₇	1	1	yes	characteristic
50	C₂×M₁₆	—	127	4	no	—
51	C₄×D₄	—	2	4	no	—
52	#117	C₂×D₁₂₇	1	2	yes	—
53	C₂×C₂²⋊C₄	Dic₁₂₇	1	4	yes	—
54	#100	—	254	4	no	—
55	C₂⁴	C₂×Dic₁₂₇	1	2	yes	—
56	C₂×D₄	—	2	4	no	—
57	C₂×D₄	C₂²⋊D₁₂₇	1	4	yes	—
58	C₂²×C₄	—	2	4	no	—
59	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
60	M₁₆	—	254	4	no	—
61	M₁₆	—	254	4	no	—
62	C₂×C₈	—	254	4	no	—
63	C₄⋊C₄	—	2	2	no	—
64	C₂²⋊C₄	—	2	4	no	—
65	C₂²⋊C₄	C₂×Dic₁₂₇	1	4	yes	—
66	C₄²	—	2	2	no	—
67	C₂³	—	2	4	no	—
68	C₂³	—	8	2	no	—
69	C₂³	C₂²⋊Dic₁₂₇	1	4	yes	—
70	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
71	D₄	—	2	8	no	—
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	—	8	4	no	—
74	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	2	yes	—
75	C₂×C₄	#14523248	1	4	yes	—
76	C₈	—	508	4	no	—
77	C₂²	—	2	4	no	—
78	C₂²	—	4	1	no	—
79	C₂²	—	4	4	no	—
80	C₂²	—	8	4	no	—
81	C₂²	C₂×C₂²⋊Dic₁₂₇	1	1	yes	characteristic
82	C₂²	#16861168	1	2	yes	—
83	C₄	—	4	4	no	—
84	C₄	#16861192	1	4	yes	—
85	C₂	—	2	1	no	—
86	C₂	—	2	2	no	—
87	C₂	—	4	4	no	—
88	C₂	#22549739	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549739
commutator	C₂³×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#518×C₁₂₇	C₂
socle	C₂×C₁₂₇	#381
prime radical	C₂²×D₄×C₁₂₇	C₂²
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#518	D₁₂₇
hypocenter	C₁₂₇	#1829
LCS	G ▷ C₂³×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #518

Complements

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

N	H ≅ G/N	type
2. #22549646	C₂	semidirect
45. C₁₂₇	#1829	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861126	2
#22549646	2
#518×C₁₂₇	2
#1829	127