Subgroups — #367×D₁₂₇ #30640508

Order 16256.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×D₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#107×D₁₂₇	C₂	1	2	yes	maximal
3	#22549704	C₂	1	1	yes	maximal, characteristic, complemented
4	#22549703	C₂	1	2	yes	maximal, complemented
5	#367×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₂₇	C₂²	1	1	yes	characteristic
7	C₄⋊C₄×D₁₂₇	—	2	2	no	—
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₁₆×D₁₂₇	C₄	1	4	yes	complemented
10	#16861143	C₂²	1	2	yes	—
11	#16861142	—	2	2	no	—
12	C₂×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	#107×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄×D₁₂₇	—	4	2	no	—
15	C₂×C₄×D₁₂₇	D₄	1	1	yes	characteristic
16	C₂×C₁₂₇⋊C₈	D₄	1	1	yes	characteristic
17	C₈×D₁₂₇	C₂×C₄	1	2	yes	—
18	C₈×D₁₂₇	Q₈	1	2	yes	—
19	C₄⋊Dic₁₂₇	—	2	2	no	—
20	Dic₁₂₇⋊C₄	—	4	2	no	—
21	C₁₂₇⋊C₁₆	C₂×C₄	1	2	yes	complemented
22	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₂₇	—	2	2	no	—
24	C₁₆×C₁₂₇	C₂×C₄	1	2	yes	complemented
25	C₂²×D₁₂₇	D₈	1	1	yes	characteristic
26	C₂×Dic₁₂₇	—	4	2	no	—
27	C₂×Dic₁₂₇	D₈	1	1	yes	characteristic
28	C₄×D₁₂₇	—	8	4	no	—
29	C₄×D₁₂₇	C₄⋊C₄	1	2	yes	—
30	C₄×D₁₂₇	Q₁₆	1	2	yes	—
31	C₁₂₇⋊C₈	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₂₇	—	4	2	no	—
33	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
37	C₂×D₁₂₇	#107	1	2	yes	—
38	C₂×D₁₂₇	SD₃₂	1	4	yes	—
39	C₄×C₁₂₇	—	8	2	no	—
40	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
42	Dic₁₂₇	—	8	2	no	—
43	Dic₁₂₇	#107	1	2	yes	—
44	C₂×C₁₂₇	C₂×#107	1	1	yes	characteristic
45	C₂×C₁₂₇	C₂×SD₃₂	1	2	yes	—
46	D₁₂₇	#367	1	4	yes	direct product factor
47	C₂×#367	—	127	1	no	maximal, Sylow
48	C₁₂₇	C₂×#367	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₁₆	—	127	1	no	—
50	C₂×#107	—	127	2	no	—
51	#367	—	127	1	no	—
52	#367	—	127	2	no	—
53	#367	D₁₂₇	1	1	yes	characteristic, direct product factor
54	C₂²×C₈	—	127	1	no	—
55	C₂×C₄⋊C₄	—	254	2	no	—
56	C₂×C₁₆	—	127	1	no	—
57	C₂×C₁₆	—	127	4	no	—
58	C₂×C₁₆	C₂×D₁₂₇	1	1	yes	characteristic
59	#107	—	127	2	no	—
60	#107	—	254	2	no	—
61	#107	C₂×D₁₂₇	1	2	yes	—
62	C₂²×C₄	—	127	1	no	—
63	C₂²×C₄	—	508	2	no	—
64	C₂×C₈	—	127	1	no	—
65	C₂×C₈	—	127	2	no	—
66	C₂×C₈	—	127	2	no	—
67	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	508	2	no	—
70	C₄⋊C₄	—	254	2	no	—
71	C₁₆	—	127	2	no	—
72	C₁₆	C₄×D₁₂₇	1	2	yes	complemented
73	C₂³	—	127	1	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	127	1	no	—
76	C₂×C₄	—	127	2	no	—
77	C₂×C₄	—	127	2	no	—
78	C₂×C₄	—	1016	4	no	—
79	C₂×C₄	—	508	2	no	—
80	C₂×C₄	D₄×D₁₂₇	1	1	yes	characteristic
81	C₈	—	127	2	no	—
82	C₈	C₂×C₄×D₁₂₇	1	1	yes	characteristic
83	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
84	C₂²	—	127	4	no	—
85	C₂²	—	127	2	no	—
86	C₂²	D₈×D₁₂₇	1	1	yes	characteristic
87	C₄	—	8	2	no	—
88	C₄	—	127	2	no	—
89	C₄	—	1016	2	no	—
90	C₄	C₄⋊C₄×D₁₂₇	1	1	yes	characteristic
91	C₄	Q₁₆×D₁₂₇	1	1	yes	characteristic
92	C₂	—	127	4	no	—
93	C₂	#107×D₁₂₇	1	1	yes	characteristic
94	C₂	SD₃₂×D₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₂₇
commutator	C₈×C₁₂₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#367×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×D₈
prime radical	C₂²×D₁₂₇	D₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#367	D₁₂₇
hypocenter	C₁₂₇	C₂×#367
LCS	G ▷ C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
3. #22549704	C₂	semidirect
4. #22549703	C₂	semidirect
5. #367×C₁₂₇	C₂	semidirect
9. C₁₆×D₁₂₇	C₄	semidirect
21. C₁₂₇⋊C₁₆	C₂×C₄	semidirect
24. C₁₆×C₁₂₇	C₂×C₄	semidirect
46. D₁₂₇	#367	direct (N × H)
48. C₁₂₇	C₂×#367	semidirect
53. #367	D₁₂₇	direct (N × H)
72. C₁₆	C₄×D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆×D₁₂₇	2
#107×D₁₂₇	2
#22549704	2
#22549703	2
#367×C₁₂₇	2
C₂×#367	127

Subgroups — #367×D127 #30640508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #367×D₁₂₇ #30640508