Subgroups — #30636671

Order 16256.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549902	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₁₂₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523232	C₄	1	1	yes	characteristic
5	#16861168	C₄	1	1	yes	characteristic
6	C₂×C₁₂₇⋊C₁₆	—	2	1	no	—
7	Dic₁₂₇⋊C₈	—	2	1	no	—
8	Dic₁₂₇⋊C₈	—	2	1	no	—
9	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₁₂₇	—	2	1	no	—
11	#14523232	—	4	1	no	—
12	C₂×C₁₂₇⋊C₈	—	2	1	no	—
13	C₄⋊Dic₁₂₇	C₈	1	1	yes	characteristic
14	C₄⋊Dic₁₂₇	C₈	1	1	yes	characteristic
15	C₂²⋊Dic₁₂₇	—	2	1	no	—
16	C₄×Dic₁₂₇	—	2	1	no	—
17	C₁₂₇⋊C₁₆	—	4	1	no	—
18	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	—	4	1	no	—
20	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
22	C₁₆×C₁₂₇	—	4	1	no	—
23	C₂×Dic₁₂₇	—	2	1	no	—
24	C₂×Dic₁₂₇	—	2	1	no	—
25	C₁₂₇⋊C₈	—	4	1	no	—
26	C₂³×C₁₂₇	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	—	2	1	no	—
28	C₂×C₄×C₁₂₇	—	2	1	no	—
29	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
31	C₈×C₁₂₇	—	2	1	no	—
32	C₈×C₁₂₇	—	2	1	no	—
33	C₂²×C₁₂₇	—	2	1	no	—
34	C₂²×C₁₂₇	—	2	1	no	—
35	C₂²×C₁₂₇	—	2	1	no	—
36	C₂²×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₁₂₇	—	2	1	no	—
38	C₄×C₁₂₇	#99	1	1	yes	characteristic
39	C₄×C₁₂₇	#101	1	1	yes	characteristic
40	Dic₁₂₇	—	4	1	no	—
41	Dic₁₂₇	—	4	1	no	—
42	C₂×C₁₂₇	—	4	1	no	—
43	C₂×C₁₂₇	#324	1	1	yes	characteristic
44	C₂×C₁₂₇	#350	1	1	yes	characteristic
45	C₂×C₁₂₇	#350	1	1	yes	characteristic
46	#1323	—	127	1	no	maximal, Sylow
47	C₁₂₇	#1323	1	1	yes	characteristic, complemented, Sylow
48	#424	—	127	1	no	—
49	C₂²⋊C₁₆	—	127	1	no	—
50	C₂²⋊C₁₆	D₁₂₇	1	1	yes	characteristic
51	C₂×M₁₆	—	127	1	no	—
52	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
53	#117	—	127	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	254	1	no	—
56	C₄⋊C₈	—	254	1	no	—
57	C₄⋊C₈	—	254	1	no	—
58	C₂²×C₄	C₄×D₁₂₇	1	1	yes	characteristic
59	M₁₆	—	508	1	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	254	1	no	—
62	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
63	C₂×C₈	D₅₀₈	1	1	yes	characteristic
64	C₄⋊C₄	—	127	1	no	—
65	C₄⋊C₄	—	127	1	no	—
66	C₂²⋊C₄	—	254	1	no	—
67	C₄²	—	254	1	no	—
68	C₁₆	—	4	1	no	—
69	C₁₆	—	508	1	no	—
70	C₂³	#14523231	1	1	yes	characteristic
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	254	1	no	—
74	C₂×C₄	—	254	1	no	—
75	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	1	no	—
79	C₈	—	508	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	(C₂×D₁₂₇)⋊C₈	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	508	1	no	—
86	C₄	—	508	1	no	—
87	C₄	#16861123	1	1	yes	characteristic
88	C₄	#16861127	1	1	yes	characteristic
89	C₂	—	4	1	no	—
90	C₂	#22549601	1	1	yes	characteristic
91	C₂	#22549659	1	1	yes	characteristic
92	C₂	#22549661	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₁₂₇)⋊C₄
commutator	C₂×C₄×C₁₂₇	C₂×C₈
Frattini	C₂²×C₈	C₂×D₁₂₇
Fitting	C₂²⋊C₁₆×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₈
prime radical	C₂³×C₁₂₇	M₁₆
cosocle	C₂²×C₈×C₁₂₇	C₂²
hypercenter	C₂²⋊C₁₆	D₁₂₇
hypocenter	C₁₂₇	#1323
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
47. C₁₂₇	#1323	semidirect

Maximal subgroups

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

subgroup	index
#22549902	2
C₂²⋊(C₁₂₇⋊C₁₆)	2
C₂²⋊C₁₆×C₁₂₇	2
#1323	127