Subgroups — #30638364

Order 16256.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550272	C₂	1	2	yes	maximal
2	#22550256	C₂	1	1	yes	maximal, characteristic
3	#22550226	C₂	1	2	yes	maximal
4	#22549597	C₂	1	1	yes	maximal, characteristic
5	#498×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₀₈	—	2	2	no	—
7	#16861209	C₂²	1	1	yes	characteristic
8	#16861209	C₂²	1	1	yes	characteristic
9	#16861140	—	2	4	no	—
10	#16861133	—	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×Dic₂₅₄	—	2	1	no	—
17	Dic₅₀₈	—	4	4	no	—
18	C₂×C₁₂₇⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₁₂₇	—	2	2	no	—
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	Dic₂₅₄	—	4	2	no	—
27	Dic₂₅₄	—	4	2	no	—
28	C₂×Dic₁₂₇	—	4	1	no	—
29	C₂×Dic₁₂₇	—	4	1	no	—
30	C₁₂₇⋊C₈	—	2	4	no	—
31	Q₈×C₁₂₇	—	4	4	no	—
32	C₂×C₄×C₁₂₇	—	4	2	no	—
33	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₂₇	SD₁₆	1	4	yes	—
36	C₂²×C₁₂₇	#127	1	1	yes	characteristic
37	C₄×C₁₂₇	—	8	2	no	—
38	C₄×C₁₂₇	C₂×SD₁₆	1	2	yes	—
39	C₄×C₁₂₇	#137	1	4	yes	—
40	Dic₁₂₇	—	8	1	no	—
41	Dic₁₂₇	—	8	1	no	—
42	C₂×C₁₂₇	#492	1	1	yes	characteristic
43	C₂×C₁₂₇	#497	1	2	yes	—
44	#1716	—	127	1	no	maximal, Sylow
45	C₁₂₇	#1716	1	1	yes	characteristic, complemented, Sylow
46	#498	—	127	2	no	—
47	#498	D₁₂₇	1	1	yes	characteristic
48	#494	—	127	1	no	—
49	#487	—	127	2	no	—
50	#322	—	127	1	no	—
51	C₂×Q₁₆	—	254	2	no	—
52	C₄⋊Q₈	—	127	1	no	—
53	C₄⋊Q₈	—	127	1	no	—
54	C₄⋊Q₈	C₂×D₁₂₇	1	2	yes	—
55	#106	—	2	2	no	—
56	#106	—	254	4	no	—
57	#103	—	254	4	no	—
58	C₄×C₈	—	127	2	no	—
59	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	254	1	no	—
62	C₂×Q₈	—	254	1	no	—
63	Q₁₆	—	508	4	no	—
64	C₂×C₈	—	127	4	no	—
65	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	254	2	no	—
69	C₄⋊C₄	—	254	2	no	—
70	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
71	Q₈	—	4	4	no	—
72	Q₈	—	508	2	no	—
73	Q₈	—	508	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	508	1	no	—
76	C₂×C₄	—	508	1	no	—
77	C₂×C₄	D₄×D₁₂₇	1	2	yes	—
78	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
79	C₈	—	254	4	no	—
80	C₈	#14523235	1	4	yes	—
81	C₂²	#16861205	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	1016	1	no	—
84	C₄	—	1016	1	no	—
85	C₄	C₂×#14523235	1	2	yes	—
86	C₄	#16861238	1	4	yes	—
87	C₂	#22550250	1	1	yes	characteristic
88	C₂	#22550267	1	2	yes	—

	subgroup	G/H
center	C₂²	#16861205
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#498×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	#127
hypercenter	#498	D₁₂₇
hypocenter	C₁₂₇	#1716
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
45. C₁₂₇	#1716	semidirect

Maximal subgroups

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

subgroup	index
#22550272	2
#22550256	2
#22550226	2
#22549597	2
#498×C₁₂₇	2
#1716	127