Subgroups — #30647194

Order 16256.

716 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550354	C₂	1	2	yes	maximal, complemented
2	#22549850	C₂	1	12	yes	maximal, complemented
3	C₂×2₋¹⁺⁴×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	Q₈×Dic₁₂₇	—	2	8	no	—
5	#16861168	C₂²	1	12	yes	—
6	#16861124	C₂²	1	16	yes	—
7	2₋¹⁺⁴×C₁₂₇	—	2	8	no	—
8	C₂×#41×C₁₂₇	C₄	1	4	yes	complemented
9	C₂×#41×C₁₂₇	C₂²	1	6	yes	—
10	C₂²×Q₈×C₁₂₇	C₄	1	4	yes	complemented
11	C₂²×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₂₇	—	2	12	no	—
13	C₂²⋊Dic₁₂₇	C₂³	1	8	yes	—
14	C₄×Dic₁₂₇	—	2	12	no	—
15	#41×C₁₂₇	—	2	16	no	—
16	#41×C₁₂₇	—	2	24	no	—
17	C₂×Q₈×C₁₂₇	—	2	16	no	—
18	C₂×Q₈×C₁₂₇	C₂×C₄	1	6	yes	—
19	C₂×Q₈×C₁₂₇	C₂×C₄	1	4	yes	—
20	C₂×Q₈×C₁₂₇	D₄	1	8	yes	—
21	C₂×D₄×C₁₂₇	C₂×C₄	1	6	yes	—
22	C₂×D₄×C₁₂₇	C₂³	1	4	yes	—
23	C₂²×C₄×C₁₂₇	C₂×C₄	1	12	yes	—
24	C₂²×C₄×C₁₂₇	C₂³	1	3	yes	—
25	C₂×Dic₁₂₇	—	2	8	no	—
26	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
27	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
28	Q₈×C₁₂₇	—	2	12	no	—
29	Q₈×C₁₂₇	C₂²⋊C₄	1	16	yes	—
30	D₄×C₁₂₇	—	2	8	no	—
31	D₄×C₁₂₇	—	2	12	no	—
32	C₂×C₄×C₁₂₇	—	2	24	no	—
33	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
34	C₂×C₄×C₁₂₇	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₁₂₇	C₂×D₄	1	12	yes	—
36	C₂²×C₁₂₇	—	2	4	no	—
37	C₂²×C₁₂₇	—	4	1	no	—
38	C₂²×C₁₂₇	—	4	4	no	—
39	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
40	C₂²×C₁₂₇	C₂²×D₄	1	2	yes	—
41	C₄×C₁₂₇	—	2	4	no	—
42	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	12	yes	—
43	Dic₁₂₇	—	4	8	no	—
44	C₂×C₁₂₇	—	2	2	no	—
45	C₂×C₁₂₇	—	2	1	no	—
46	C₂×C₁₂₇	—	4	4	no	—
47	C₂×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
48	#2884	—	127	1	no	maximal, Sylow
49	C₁₂₇	#2884	1	1	yes	characteristic, complemented, Sylow
50	C₂×2₋¹⁺⁴	D₁₂₇	1	1	yes	characteristic
51	#519	—	127	2	no	—
52	#410	—	127	12	no	—
53	2₋¹⁺⁴	—	2	8	no	—
54	C₂×#41	Dic₁₂₇	1	4	yes	complemented
55	C₂×#41	C₂×D₁₂₇	1	6	yes	—
56	C₂²×Q₈	Dic₁₂₇	1	4	yes	complemented
57	C₂²×Q₈	C₂×D₁₂₇	1	1	yes	characteristic
58	C₄×Q₈	—	254	8	no	—
59	#117	—	127	12	no	—
60	#99	—	127	16	no	—
61	#41	—	2	16	no	—
62	#41	—	2	24	no	—
63	C₂×Q₈	—	2	16	no	—
64	C₂×Q₈	C₂×Dic₁₂₇	1	6	yes	—
65	C₂×Q₈	C₂×Dic₁₂₇	1	4	yes	—
66	C₂×Q₈	C₂²⋊D₁₂₇	1	8	yes	—
67	C₂×D₄	C₂×Dic₁₂₇	1	6	yes	—
68	C₂×D₄	C₂²×D₁₂₇	1	4	yes	—
69	C₂²×C₄	C₂×Dic₁₂₇	1	12	yes	—
70	C₂²×C₄	C₂²×D₁₂₇	1	3	yes	—
71	C₄⋊C₄	—	254	12	no	—
72	C₂²⋊C₄	—	127	8	no	—
73	C₄²	—	254	12	no	—
74	C₂³	C₂²×Dic₁₂₇	1	4	yes	—
75	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
76	Q₈	—	2	12	no	—
77	Q₈	C₂²⋊Dic₁₂₇	1	16	yes	—
78	D₄	—	2	8	no	—
79	D₄	—	2	12	no	—
80	C₂×C₄	—	2	24	no	—
81	C₂×C₄	—	254	8	no	—
82	C₂×C₄	C₂²×Dic₁₂₇	1	4	yes	—
83	C₂×C₄	C₂²×Dic₁₂₇	1	6	yes	—
84	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	12	yes	—
85	C₂²	—	2	4	no	—
86	C₂²	—	4	1	no	—
87	C₂²	—	4	4	no	—
88	C₂²	C₂³×Dic₁₂₇	1	1	yes	characteristic
89	C₂²	C₂²×C₂²⋊D₁₂₇	1	2	yes	—
90	C₄	—	2	4	no	—
91	C₄	—	508	8	no	—
92	C₄	C₂×C₂²⋊Dic₁₂₇	1	12	yes	—
93	C₂	—	2	2	no	—
94	C₂	—	2	1	no	—
95	C₂	—	4	4	no	—
96	C₂	C₂²×C₂²⋊Dic₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊Dic₁₂₇
commutator	C₂²×C₁₂₇	C₂³×C₄
Frattini	C₂³	C₂³×D₁₂₇
Fitting, prime radical	C₂×2₋¹⁺⁴×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂²×C₂²⋊C₄
cosocle	C₂³×C₁₂₇	C₂⁴
hypercenter	C₂×2₋¹⁺⁴	D₁₂₇
hypocenter	C₁₂₇	#2884
LCS	G ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ C₂×2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #22550354	C₂	semidirect
2. #22549850	C₂	semidirect
8. C₂×#41×C₁₂₇	C₄	semidirect
10. C₂²×Q₈×C₁₂₇	C₄	semidirect
49. C₁₂₇	#2884	semidirect
54. C₂×#41	Dic₁₂₇	semidirect
56. C₂²×Q₈	Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550354	2
#22549850	2
C₂×2₋¹⁺⁴×C₁₂₇	2
#2884	127