Subgroups — #30638332

Order 16256.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550249	C₂	1	1	yes	maximal, characteristic
2	#22550226	C₂	1	2	yes	maximal, complemented
3	#22550224	C₂	1	2	yes	maximal
4	C₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523237	—	2	2	no	—
7	#16861209	C₂²	1	1	yes	characteristic
8	#16861206	C₂²	1	1	yes	characteristic
9	#16861133	—	2	4	no	—
10	#16861129	—	2	4	no	—
11	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
13	#106×C₁₂₇	—	2	2	no	—
14	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₅₄	—	2	1	no	—
16	C₂×D₅₀₈	—	2	1	no	—
17	C₂×D₅₀₈	—	2	2	no	—
18	#14523237	—	4	4	no	—
19	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
20	C₄⋊Dic₁₂₇	—	2	2	no	—
21	C₂×Q₈×C₁₂₇	—	2	2	no	—
22	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₂₇	—	2	2	no	—
24	C₄⋊C₄×C₁₂₇	—	2	2	no	—
25	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂²×D₁₂₇	—	4	1	no	—
27	Dic₂₅₄	—	4	2	no	—
28	D₅₀₈	—	4	2	no	—
29	D₅₀₈	—	4	4	no	—
30	C₂×Dic₁₂₇	—	4	1	no	—
31	C₁₂₇⋊C₈	—	2	4	no	—
32	Q₈×C₁₂₇	—	4	4	no	—
33	C₂×C₄×C₁₂₇	—	4	2	no	—
34	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
36	C₈×C₁₂₇	SD₁₆	1	4	yes	—
37	C₂²×C₁₂₇	#127	1	1	yes	characteristic
38	C₂×D₁₂₇	—	8	1	no	—
39	C₂×D₁₂₇	—	8	2	no	—
40	C₄×C₁₂₇	—	8	2	no	—
41	C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
42	C₄×C₁₂₇	#135	1	4	yes	—
43	Dic₁₂₇	—	8	1	no	—
44	C₂×C₁₂₇	#492	1	1	yes	characteristic
45	C₂×C₁₂₇	#495	1	2	yes	—
46	D₁₂₇	—	16	1	no	—
47	#1708	—	127	1	no	maximal, Sylow
48	C₁₂₇	#1708	1	1	yes	characteristic, complemented, Sylow
49	#498	D₁₂₇	1	1	yes	characteristic, complemented
50	#492	—	127	1	no	—
51	#487	—	127	2	no	—
52	#486	—	127	2	no	—
53	C₈²	—	127	1	no	—
54	C₂×SD₁₆	—	254	2	no	—
55	C₄⋊Q₈	—	127	1	no	—
56	C₄⋊Q₈	C₂×D₁₂₇	1	2	yes	—
57	#127	—	127	1	no	—
58	#106	—	2	2	no	—
59	#103	—	254	4	no	—
60	#102	—	254	4	no	—
61	C₄×C₈	—	127	2	no	—
62	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	254	1	no	—
65	C₂×D₄	—	254	2	no	—
66	C₂×D₄	—	254	1	no	—
67	SD₁₆	—	508	4	no	—
68	C₂×C₈	—	127	4	no	—
69	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	254	2	no	—
73	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
74	C₂³	—	508	1	no	—
75	Q₈	—	4	4	no	—
76	Q₈	—	508	2	no	—
77	D₄	—	508	2	no	—
78	D₄	—	508	4	no	—
79	C₂×C₄	—	4	2	no	—
80	C₂×C₄	—	508	1	no	—
81	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
83	C₈	—	254	4	no	—
84	C₈	#14523235	1	4	yes	—
85	C₂²	—	1016	1	no	—
86	C₂²	—	1016	2	no	—
87	C₂²	#16861205	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	—	1016	1	no	—
90	C₄	C₂×#14523235	1	2	yes	—
91	C₄	#16861230	1	4	yes	—
92	C₂	—	2032	1	no	—
93	C₂	#22550250	1	1	yes	characteristic
94	C₂	#22550257	1	2	yes	—

	subgroup	G/H
center	C₂²	#16861205
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#498×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#127
prime radical	#16861206	C₂²
hypercenter	#498	D₁₂₇
hypocenter	C₁₂₇	#1708
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
2. #22550226	C₂	semidirect
4. C₈×C₁₂₇⋊C₈	C₂	semidirect
5. #498×C₁₂₇	C₂	semidirect
48. C₁₂₇	#1708	semidirect
49. #498	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550249	2
#22550226	2
#22550224	2
C₈×C₁₂₇⋊C₈	2
#498×C₁₂₇	2
#1708	127