Subgroups — #30639835

Order 16256.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550676	C₂	1	3	yes	maximal, complemented
2	#22549844	C₂	1	3	yes	maximal, complemented
3	#338×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14523248	C₂²	1	3	yes	—
5	#16861242	—	2	6	no	—
6	#16861237	—	2	6	no	—
7	#16861231	—	2	6	no	—
8	C₂²×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
9	(C₂×D₁₂₇)⋊C₈	—	2	6	no	—
10	C₂×M₁₆×C₁₂₇	C₂²	1	3	yes	—
11	#14523248	—	2	6	no	—
12	#14523248	—	2	6	no	—
13	C₂×Dic₂₅₄	D₄	1	3	yes	—
14	C₂×D₅₀₈	D₄	1	3	yes	—
15	C₂×C₂²⋊D₁₂₇	—	2	3	no	—
16	C₂×C₄×D₁₂₇	—	2	3	no	—
17	Dic₅₀₈	—	2	6	no	—
18	#14523237	—	2	12	no	—
19	D₁₀₁₆	—	2	6	no	—
20	C₂×C₁₂₇⋊C₈	—	2	3	no	—
21	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₂₇	—	2	6	no	—
23	C₂×C₈×C₁₂₇	D₄	1	6	yes	—
24	C₂²×D₁₂₇	—	2	3	no	—
25	Dic₂₅₄	—	2	6	no	—
26	D₅₀₈	—	2	6	no	—
27	C₂²⋊D₁₂₇	—	4	6	no	—
28	C₂×Dic₁₂₇	—	2	3	no	—
29	C₄×D₁₂₇	—	4	3	no	—
30	C₄×D₁₂₇	—	4	3	no	—
31	C₁₂₇⋊C₈	—	4	1	no	—
32	C₂³×C₁₂₇	#41	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
35	C₈×C₁₂₇	—	2	6	no	—
36	C₂²×C₁₂₇	—	4	1	no	—
37	C₂²×C₁₂₇	#121	1	3	yes	—
38	C₂×D₁₂₇	—	4	6	no	—
39	C₂×D₁₂₇	—	4	3	no	—
40	C₄×C₁₂₇	#120	1	3	yes	—
41	C₄×C₁₂₇	#127	1	1	yes	characteristic
42	Dic₁₂₇	—	4	3	no	—
43	C₂×C₁₂₇	—	2	3	no	—
44	C₂×C₁₂₇	#392	1	1	yes	characteristic
45	D₁₂₇	—	8	3	no	—
46	#2017	—	127	1	no	maximal, Sylow
47	C₁₂₇	#2017	1	1	yes	characteristic, complemented, Sylow
48	#575	—	127	3	no	—
49	#408	—	127	3	no	—
50	#338	D₁₂₇	1	1	yes	characteristic, complemented
51	C₂×#41	—	127	3	no	—
52	#137	—	254	6	no	—
53	AGL(1,ℤ/8)	—	254	6	no	—
54	#135	—	254	6	no	—
55	C₂×M₁₆	C₂×D₁₂₇	1	3	yes	—
56	C₂²×C₈	—	127	1	no	—
57	C₂²⋊C₈	—	254	6	no	—
58	#41	—	254	6	no	—
59	#41	—	254	6	no	—
60	C₂×Q₈	—	127	3	no	—
61	C₂×D₄	—	127	3	no	—
62	C₂×D₄	—	254	3	no	—
63	C₂²×C₄	—	254	3	no	—
64	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
65	Q₁₆	—	254	6	no	—
66	SD₁₆	—	254	12	no	—
67	D₈	—	254	6	no	—
68	M₁₆	—	2	6	no	—
69	C₂×C₈	—	254	3	no	—
70	C₂×C₈	D₅₀₈	1	6	yes	—
71	C₂³	—	254	3	no	—
72	C₂³	#14523249	1	1	yes	characteristic
73	Q₈	—	254	6	no	—
74	D₄	—	508	6	no	—
75	D₄	—	254	6	no	—
76	C₂×C₄	—	508	3	no	—
77	C₂×C₄	—	508	3	no	—
78	C₂×C₄	—	254	3	no	—
79	C₂×C₄	D₄×D₁₂₇	1	3	yes	—
80	C₂×C₄	C₂×D₅₀₈	1	3	yes	—
81	C₈	—	2	6	no	—
82	C₈	—	508	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	508	6	no	—
85	C₂²	—	508	3	no	—
86	C₂²	#16861184	1	3	yes	—
87	C₄	—	508	3	no	—
88	C₄	#16861179	1	3	yes	—
89	C₄	#16861206	1	1	yes	characteristic
90	C₂	—	2	3	no	—
91	C₂	—	1016	3	no	—
92	C₂	#22549786	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16861206
commutator, cosocle	C₂²×C₄×C₁₂₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#338×C₁₂₇	C₂
socle	C₂×C₁₂₇	#392
hypercenter	#338	D₁₂₇
hypocenter	C₁₂₇	#2017
LCS	G ▷ C₂²×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #22550676	C₂	semidirect
2. #22549844	C₂	semidirect
3. #338×C₁₂₇	C₂	semidirect
47. C₁₂₇	#2017	semidirect
50. #338	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550676	2
#22549844	2
#338×C₁₂₇	2
#2017	127