Subgroups — #30638479

Order 16256.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861120	C₂	1	2	yes	maximal
2	#22549631	C₂	1	4	yes	maximal
3	C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₂₇⋊C₈	C₂²	1	4	yes	—
5	#16861120	C₄	1	8	yes	complemented
6	C₂×C₄²×C₁₂₇	C₄	1	4	yes	—
7	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₂₇⋊C₈	—	2	8	no	—
10	C₂×C₁₂₇⋊C₈	C₂×C₄	1	8	yes	—
11	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
12	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
13	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
14	C₄²×C₁₂₇	—	2	8	no	—
15	C₄²×C₁₂₇	—	2	4	no	—
16	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
17	C₁₂₇⋊C₈	—	2	8	no	—
18	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₂₇	—	2	4	no	—
20	C₂×C₄×C₁₂₇	—	2	4	no	—
21	C₂×C₄×C₁₂₇	—	2	4	no	—
22	C₂×C₄×C₁₂₇	C₄²	1	4	yes	—
23	C₂×C₄×C₁₂₇	M₁₆	1	8	yes	—
24	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
25	C₂×C₄×C₁₂₇	#41	1	4	yes	—
26	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
27	C₂²×C₁₂₇	#117	1	2	yes	—
28	C₂²×C₁₂₇	C₂×M₁₆	1	4	yes	—
29	C₄×C₁₂₇	—	2	4	no	—
30	C₄×C₁₂₇	—	2	8	no	—
31	C₄×C₁₂₇	#117	1	4	yes	—
32	C₂×C₁₂₇	#376	1	1	yes	characteristic
33	C₂×C₁₂₇	C₄×M₁₆	1	2	yes	—
34	C₂×C₁₂₇	#432	1	4	yes	—
35	#1745	—	127	1	no	maximal, Sylow
36	C₁₂₇	#1745	1	1	yes	characteristic, complemented, Sylow
37	C₂×#97	—	127	2	no	—
38	C₄³	D₁₂₇	1	1	yes	characteristic
39	#336	—	127	4	no	—
40	C₂²×C₈	—	127	4	no	—
41	C₂×C₄²	Dic₁₂₇	1	4	yes	—
42	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
43	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
44	#97	—	127	8	no	—
45	C₂²×C₄	C₂×Dic₁₂₇	1	2	yes	—
46	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
47	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
48	C₂×C₈	—	127	8	no	—
49	C₂×C₈	—	254	8	no	—
50	C₄²	—	2	8	no	—
51	C₄²	—	2	4	no	—
52	C₄²	C₄×D₁₂₇	1	4	yes	—
53	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	C₄×Dic₁₂₇	1	4	yes	—
58	C₂×C₄	#14523232	1	8	yes	—
59	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
60	C₂×C₄	#14523248	1	4	yes	—
61	C₈	—	254	8	no	—
62	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
63	C₂²	#16861168	1	2	yes	—
64	C₂²	C₂×#14523232	1	4	yes	—
65	C₄	—	2	4	no	—
66	C₄	—	2	8	no	—
67	C₄	#16861166	1	4	yes	—
68	C₂	#22549722	1	1	yes	characteristic
69	C₂	C₄×#14523232	1	2	yes	—
70	C₂	#22549933	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂×C₄²
Fitting	C₄³×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₄³	D₁₂₇
hypocenter	C₁₂₇	#1745
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
5. #16861120	C₄	semidirect
36. C₁₂₇	#1745	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861120	2
#22549631	2
C₄³×C₁₂₇	2
#1745	127