Subgroups — #30636896

Order 16256.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550250	C₂	1	1	yes	maximal, characteristic
2	#22549675	C₂	1	1	yes	maximal, characteristic, complemented
3	#356×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14523235	—	2	1	no	—
5	C₂×#14523235	—	2	1	no	—
6	#16861206	C₄	1	1	yes	characteristic
7	#16861128	—	2	1	no	—
8	C₄×C₁₂₇⋊C₈	C₄	1	1	yes	characteristic
9	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	#101×C₁₂₇	—	2	1	no	—
11	C₂×D₅₀₈	—	2	1	no	—
12	C₂×D₅₀₈	—	4	1	no	—
13	#14523235	—	4	2	no	—
14	#14523235	—	4	2	no	—
15	#14523232	—	4	1	no	—
16	C₂×C₁₂₇⋊C₈	—	2	1	no	—
17	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
19	M₁₆×C₁₂₇	—	4	1	no	—
20	C₄⋊C₄×C₁₂₇	—	4	1	no	—
21	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₂₇	—	4	1	no	—
23	D₅₀₈	—	4	1	no	—
24	D₅₀₈	—	4	1	no	—
25	D₅₀₈	—	8	2	no	—
26	C₁₂₇⋊C₈	—	2	2	no	—
27	C₁₂₇⋊C₈	—	8	1	no	—
28	Q₈×C₁₂₇	—	4	1	no	—
29	Q₈×C₁₂₇	—	4	1	no	—
30	C₂×C₄×C₁₂₇	—	2	1	no	—
31	C₂×C₄×C₁₂₇	—	2	1	no	—
32	C₂×C₄×C₁₂₇	—	2	1	no	—
33	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₈×C₁₂₇	—	8	1	no	—
35	C₂²×C₁₂₇	#99	1	1	yes	characteristic
36	C₂×D₁₂₇	—	8	2	no	—
37	C₂×D₁₂₇	—	8	1	no	—
38	C₄×C₁₂₇	—	2	2	no	—
39	C₄×C₁₂₇	—	2	1	no	—
40	C₄×C₁₂₇	—	4	1	no	—
41	C₄×C₁₂₇	—	4	1	no	—
42	C₂×C₁₂₇	—	2	1	no	—
43	C₂×C₁₂₇	#353	1	1	yes	characteristic
44	D₁₂₇	—	16	1	no	—
45	#1410	—	127	1	no	maximal, Sylow
46	C₁₂₇	#1410	1	1	yes	characteristic, complemented, Sylow
47	#492	—	127	1	no	—
48	#356	—	127	1	no	—
49	#356	D₁₂₇	1	1	yes	characteristic, complemented
50	C₂×SD₁₆	—	254	1	no	—
51	C₂×SD₁₆	—	254	1	no	—
52	C₄⋊Q₈	C₂×D₁₂₇	1	1	yes	characteristic
53	#127	—	127	1	no	—
54	#101	—	2	1	no	—
55	#101	—	254	1	no	—
56	C₄×C₈	—	127	1	no	—
57	C₂×Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
58	C₂×Q₈	D₅₀₈	1	1	yes	characteristic
59	C₂×D₄	—	508	1	no	—
60	C₂×D₄	—	254	1	no	—
61	SD₁₆	—	508	2	no	—
62	SD₁₆	—	508	2	no	—
63	M₁₆	—	4	1	no	—
64	M₁₆	—	508	1	no	—
65	C₂×C₈	—	254	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
68	C₂³	—	508	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	D₄	—	1016	2	no	—
72	D₄	—	508	1	no	—
73	D₄	—	508	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
78	C₈	—	8	1	no	—
79	C₈	—	1016	1	no	—
80	C₈	—	254	2	no	—
81	C₂²	—	1016	2	no	—
82	C₂²	—	1016	1	no	—
83	C₂²	#16861123	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	2	1	no	—
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	2032	1	no	—
90	C₂	#22549668	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549668
commutator	C₄²×C₁₂₇	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₂₇
Fitting	#356×C₁₂₇	C₂
socle	C₂×C₁₂₇	#353
prime radical	#16861206	C₄
cosocle	C₄⋊Q₈×C₁₂₇	C₂²
hypercenter	#356	D₁₂₇
hypocenter	C₁₂₇	#1410
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
2. #22549675	C₂	semidirect
3. #356×C₁₂₇	C₂	semidirect
46. C₁₂₇	#1410	semidirect
49. #356	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550250	2
#22549675	2
#356×C₁₂₇	2
#1410	127