Subgroups — #30638401

Order 16256.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550281	C₂	1	2	yes	maximal, complemented
2	#22550256	C₂	1	1	yes	maximal, characteristic, complemented
3	#22550249	C₂	1	1	yes	maximal, characteristic
4	#22550223	C₂	1	2	yes	maximal
5	#322×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₀₈	—	2	2	no	—
7	C₂×#14523237	—	2	2	no	—
8	#16861209	C₂²	1	1	yes	characteristic
9	#16861209	C₂²	1	2	yes	—
10	#16861206	C₂²	1	1	yes	characteristic
11	#16861143	—	2	4	no	—
12	#16861131	—	2	4	no	—
13	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₅₄	—	2	1	no	—
16	C₂×Dic₂₅₄	—	2	2	no	—
17	C₂×D₅₀₈	—	2	1	no	—
18	C₂×D₅₀₈	—	2	2	no	—
19	Dic₅₀₈	—	4	4	no	—
20	#14523237	—	4	4	no	—
21	C₄⋊Dic₁₂₇	—	2	2	no	—
22	C₄⋊Dic₁₂₇	—	2	2	no	—
23	C₄⋊Dic₁₂₇	—	2	2	no	—
24	C₂×C₈×C₁₂₇	D₄	1	4	yes	—
25	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
26	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂²×D₁₂₇	—	4	1	no	—
28	Dic₂₅₄	—	4	2	no	—
29	Dic₂₅₄	—	4	4	no	—
30	D₅₀₈	—	4	2	no	—
31	D₅₀₈	—	4	4	no	—
32	C₂×Dic₁₂₇	—	4	2	no	—
33	C₂×Dic₁₂₇	—	4	1	no	—
34	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
36	C₈×C₁₂₇	—	2	4	no	—
37	C₈×C₁₂₇	D₈	1	4	yes	—
38	C₂²×C₁₂₇	#127	1	1	yes	characteristic
39	C₂×D₁₂₇	—	8	1	no	—
40	C₂×D₁₂₇	—	8	2	no	—
41	C₄×C₁₂₇	C₂×D₈	1	2	yes	—
42	C₄×C₁₂₇	#137	1	4	yes	—
43	Dic₁₂₇	—	8	1	no	—
44	Dic₁₂₇	—	8	2	no	—
45	C₂×C₁₂₇	#493	1	1	yes	characteristic
46	C₂×C₁₂₇	#497	1	2	yes	—
47	D₁₂₇	—	16	1	no	—
48	#1722	—	127	1	no	maximal, Sylow
49	C₁₂₇	#1722	1	1	yes	characteristic, complemented, Sylow
50	#500	—	127	2	no	—
51	#494	—	127	1	no	—
52	#492	—	127	1	no	—
53	#486	—	127	2	no	—
54	#322	D₁₂₇	1	1	yes	characteristic, complemented
55	C₂×Q₁₆	—	254	2	no	—
56	C₂×SD₁₆	—	254	2	no	—
57	C₄⋊Q₈	—	127	2	no	—
58	C₄⋊Q₈	—	127	1	no	—
59	#127	—	127	1	no	—
60	#107	—	254	4	no	—
61	#102	—	254	4	no	—
62	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
63	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
64	C₂×Q₈	—	254	1	no	—
65	C₂×Q₈	—	254	2	no	—
66	C₂×D₄	—	254	2	no	—
67	C₂×D₄	—	254	1	no	—
68	Q₁₆	—	508	4	no	—
69	SD₁₆	—	508	4	no	—
70	C₂×C₈	D₅₀₈	1	4	yes	—
71	C₂×C₈	D₅₀₈	1	2	yes	—
72	C₄⋊C₄	—	254	2	no	—
73	C₄⋊C₄	—	254	2	no	—
74	C₄⋊C₄	—	254	2	no	—
75	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
76	C₂³	—	508	1	no	—
77	Q₈	—	508	2	no	—
78	Q₈	—	508	4	no	—
79	D₄	—	508	2	no	—
80	D₄	—	508	4	no	—
81	C₂×C₄	—	508	2	no	—
82	C₂×C₄	—	508	1	no	—
83	C₂×C₄	C₂×D₅₀₈	1	1	yes	characteristic
84	C₂×C₄	C₂×D₅₀₈	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	D₁₀₁₆	1	4	yes	—
87	C₂²	—	1016	1	no	—
88	C₂²	—	1016	2	no	—
89	C₂²	#16861206	1	1	yes	characteristic
90	C₄	—	1016	1	no	—
91	C₄	—	1016	2	no	—
92	C₄	C₂×D₁₀₁₆	1	2	yes	—
93	C₄	#16861242	1	4	yes	—
94	C₂	—	2032	1	no	—
95	C₂	#22550253	1	1	yes	characteristic
96	C₂	#22550270	1	2	yes	—

	subgroup	G/H
center	C₂²	#16861206
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	#322×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#127
prime radical	#16861206	C₂²
hypercenter	#322	D₁₂₇
hypocenter	C₁₂₇	#1722
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
1. #22550281	C₂	semidirect
2. #22550256	C₂	semidirect
5. #322×C₁₂₇	C₂	semidirect
49. C₁₂₇	#1722	semidirect
54. #322	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550281	2
#22550256	2
#22550249	2
#22550223	2
#322×C₁₂₇	2
#1722	127