Subgroups — #30636762

Order 16256.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550241	C₂	1	1	yes	maximal, characteristic
2	#22549654	C₂	1	1	yes	maximal, characteristic, complemented
3	#347×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#16861206	C₄	1	1	yes	characteristic
5	#16861201	C₄	1	1	yes	characteristic
6	#16861149	—	2	1	no	—
7	#16861131	—	2	1	no	—
8	#16861131	—	2	1	no	—
9	M₃₂×C₁₂₇	—	2	1	no	—
10	#97×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×D₅₀₈	—	2	1	no	—
12	C₂×D₅₀₈	—	4	1	no	—
13	C₄⋊Dic₁₂₇	—	2	1	no	—
14	Dic₁₂₇⋊C₄	—	4	1	no	—
15	C₁₂₇⋊C₁₆	—	2	2	no	—
16	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
17	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₂₇	—	2	2	no	—
20	C₂²×D₁₂₇	—	4	1	no	—
21	D₅₀₈	—	4	1	no	—
22	D₅₀₈	—	4	1	no	—
23	D₅₀₈	—	8	2	no	—
24	C₂×Dic₁₂₇	—	4	1	no	—
25	C₂×C₄×C₁₂₇	—	2	1	no	—
26	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₈×C₁₂₇	—	2	1	no	—
28	C₈×C₁₂₇	—	2	1	no	—
29	C₂²×C₁₂₇	#100	1	1	yes	characteristic
30	C₂×D₁₂₇	—	8	2	no	—
31	C₂×D₁₂₇	—	8	1	no	—
32	C₄×C₁₂₇	—	4	1	no	—
33	C₄×C₁₂₇	#104	1	1	yes	characteristic
34	C₄×C₁₂₇	#104	1	1	yes	characteristic
35	Dic₁₂₇	—	8	1	no	—
36	C₂×C₁₂₇	—	2	1	no	—
37	C₂×C₁₂₇	#329	1	1	yes	characteristic
38	D₁₂₇	—	16	1	no	—
39	#1355	—	127	1	no	maximal, Sylow
40	C₁₂₇	#1355	1	1	yes	characteristic, complemented, Sylow
41	#490	—	127	1	no	—
42	#347	—	127	1	no	—
43	#347	D₁₂₇	1	1	yes	characteristic, complemented
44	#127	—	127	1	no	—
45	#125	—	127	1	no	—
46	M₃₂	—	2	1	no	—
47	M₃₂	—	254	1	no	—
48	#102	—	254	1	no	—
49	#102	—	254	1	no	—
50	#97	C₂×D₁₂₇	1	1	yes	characteristic
51	C₂×D₄	—	508	1	no	—
52	C₂×D₄	—	254	1	no	—
53	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
54	C₂×C₈	D₅₀₈	1	1	yes	characteristic
55	C₄⋊C₄	—	508	1	no	—
56	C₄⋊C₄	—	254	1	no	—
57	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
58	C₁₆	—	2	2	no	—
59	C₁₆	—	254	2	no	—
60	C₂³	—	508	1	no	—
61	D₄	—	1016	2	no	—
62	D₄	—	508	1	no	—
63	D₄	—	508	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	508	1	no	—
66	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₂²	—	1016	2	no	—
70	C₂²	—	1016	1	no	—
71	C₂²	#16861125	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	1016	1	no	—
74	C₄	#16861135	1	1	yes	characteristic
75	C₄	#16861137	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	2032	1	no	—
78	C₂	#22549615	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549615
commutator	C₄²×C₁₂₇	C₂×C₄
Frattini	#97	C₂×D₁₂₇
Fitting	#347×C₁₂₇	C₂
socle	C₂×C₁₂₇	#329
prime radical	#16861206	C₄
cosocle	#97×C₁₂₇	C₂²
hypercenter	#347	D₁₂₇
hypocenter	C₁₂₇	#1355
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
2. #22549654	C₂	semidirect
3. #347×C₁₂₇	C₂	semidirect
40. C₁₂₇	#1355	semidirect
43. #347	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550241	2
#22549654	2
#347×C₁₂₇	2
#1355	127