Subgroups — #30638478

Order 16256.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861120	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#22549630	C₂	1	4	yes	maximal
4	C₂×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₂₇	C₄	1	4	yes	—
7	C₂×C₄×Dic₁₂₇	C₂²	1	2	yes	—
8	#16861120	C₄	1	4	yes	complemented
9	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	#97×C₁₂₇	C₄	1	4	yes	complemented
12	C₂²×Dic₁₂₇	C₂×C₄	1	4	yes	—
13	C₂×C₁₂₇⋊C₈	—	2	4	no	—
14	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
15	C₄×Dic₁₂₇	—	2	8	no	—
16	C₄×Dic₁₂₇	—	2	4	no	—
17	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	—	2	4	no	—
20	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
21	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂×Dic₁₂₇	—	2	4	no	—
24	C₂×Dic₁₂₇	—	2	4	no	—
25	C₂×Dic₁₂₇	M₁₆	1	8	yes	—
26	C₁₂₇⋊C₈	—	2	4	no	—
27	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	—	2	4	no	—
29	C₂×C₄×C₁₂₇	C₄²	1	4	yes	—
30	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	#41	1	4	yes	—
33	C₈×C₁₂₇	—	2	4	no	—
34	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
35	C₂²×C₁₂₇	#117	1	2	yes	—
36	C₂²×C₁₂₇	C₂×M₁₆	1	4	yes	—
37	C₄×C₁₂₇	—	2	4	no	—
38	C₄×C₁₂₇	#117	1	2	yes	—
39	C₄×C₁₂₇	#117	1	2	yes	—
40	Dic₁₂₇	—	2	8	no	—
41	C₂×C₁₂₇	#376	1	1	yes	characteristic
42	C₂×C₁₂₇	C₄×M₁₆	1	2	yes	—
43	C₂×C₁₂₇	#432	1	4	yes	—
44	#1745	—	127	1	no	maximal, Sylow
45	C₁₂₇	#1745	1	1	yes	characteristic, complemented, Sylow
46	C₂×#97	—	127	1	no	—
47	C₂×#97	D₁₂₇	1	1	yes	characteristic
48	C₄³	—	127	1	no	—
49	#336	—	127	4	no	—
50	C₂²×C₈	—	127	2	no	—
51	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
52	C₂×C₄²	—	127	4	no	—
53	C₂×C₄²	—	127	2	no	—
54	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
55	#97	—	127	4	no	—
56	#97	Dic₁₂₇	1	4	yes	complemented
57	C₂²×C₄	—	127	4	no	—
58	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
59	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	127	4	no	—
62	C₂×C₈	—	254	4	no	—
63	C₂×C₈	C₂×Dic₁₂₇	1	4	yes	—
64	C₄²	—	254	8	no	—
65	C₄²	—	254	4	no	—
66	C₄²	C₄×D₁₂₇	1	2	yes	—
67	C₄²	C₂×Dic₁₂₇	1	2	yes	—
68	C₂³	C₂×C₄×D₁₂₇	1	1	yes	characteristic
69	C₂×C₄	—	2	4	no	—
70	C₂×C₄	—	127	8	no	—
71	C₂×C₄	—	254	4	no	—
72	C₂×C₄	—	254	4	no	—
73	C₂×C₄	C₄×Dic₁₂₇	1	4	yes	—
74	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
75	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
76	C₂×C₄	#14523248	1	4	yes	—
77	C₈	—	2	4	no	—
78	C₈	—	254	4	no	—
79	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
80	C₂²	#16861166	1	2	yes	—
81	C₂²	M₁₆×D₁₂₇	1	4	yes	—
82	C₄	—	2	4	no	—
83	C₄	—	254	8	no	—
84	C₄	#16861166	1	2	yes	—
85	C₄	#16861168	1	2	yes	—
86	C₂	#22549722	1	1	yes	characteristic
87	C₂	M₁₆×Dic₁₂₇	1	2	yes	—
88	C₂	#22549931	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₂₇
commutator	C₂²×C₁₂₇	C₂×C₄²
Fitting	C₂×#97×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×#97	D₁₂₇
hypocenter	C₁₂₇	#1745
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
8. #16861120	C₄	semidirect
11. #97×C₁₂₇	C₄	semidirect
45. C₁₂₇	#1745	semidirect
56. #97	Dic₁₂₇	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
C₄²×Dic₁₂₇	2
#22549630	2
C₂×#97×C₁₂₇	2
#1745	127