Subgroups — #30640632

Order 16256.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₂₅₄	C₂	1	1	yes	maximal, characteristic
2	Dic₁₂₇⋊C₁₆	C₂	1	2	yes	maximal
3	C₄×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₁₂₇	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₂₅₄	C₄	1	1	yes	characteristic
7	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₂₇⋊C₈)	C₄	1	1	yes	characteristic
10	Dic₁₂₇⋊C₈	C₄	1	2	yes	—
11	C₈×Dic₁₂₇	C₂²	1	2	yes	—
12	C₂×C₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₅₄	C₈	1	1	yes	characteristic
15	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₂₇	C₈	1	1	yes	characteristic
17	Dic₁₂₇⋊C₄	C₈	1	2	yes	—
18	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
19	C₁₂₇⋊C₁₆	—	2	1	no	—
20	C₁₂₇⋊C₁₆	Q₈	1	2	yes	complemented
21	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₁₆×C₁₂₇	—	2	2	no	—
25	Dic₂₅₄	C₁₆	1	4	yes	complemented
26	C₂×Dic₁₂₇	C₂×C₈	1	2	yes	—
27	C₂×Dic₁₂₇	C₂×C₈	1	2	yes	—
28	C₁₂₇⋊C₈	—	2	2	no	—
29	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₂₇	—	2	1	no	—
33	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₁₂₇	#41	1	1	yes	characteristic
35	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₁₂₇	—	2	1	no	—
37	C₄×C₁₂₇	C₂×C₁₆	1	2	yes	—
38	C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₁₂₇	#131	1	1	yes	characteristic
40	Dic₁₂₇	—	2	2	no	—
41	Dic₁₂₇	C₂×C₁₆	1	4	yes	—
42	C₂×C₁₂₇	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
44	C₂×C₁₂₇	#504	1	1	yes	characteristic
45	Q₈×C₁₆	—	127	1	no	maximal, Sylow
46	C₁₂₇	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
47	C₈×Q₈	—	127	1	no	—
48	C₄⋊C₁₆	—	127	2	no	—
49	C₄⋊C₁₆	D₁₂₇	1	1	yes	characteristic
50	C₄×C₁₆	—	127	2	no	—
51	C₄×C₁₆	—	127	1	no	—
52	C₄×Q₈	—	127	1	no	—
53	C₂×C₁₆	—	127	1	no	—
54	C₂×C₁₆	—	127	1	no	—
55	C₂×C₁₆	C₂×D₁₂₇	1	2	yes	—
56	C₄⋊C₈	—	127	2	no	—
57	C₄⋊C₈	—	127	1	no	—
58	C₄×C₈	—	127	2	no	—
59	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
60	C₂×Q₈	—	127	1	no	—
61	C₂×C₈	—	127	2	no	—
62	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
64	C₄⋊C₄	—	127	2	no	—
65	C₄⋊C₄	—	127	1	no	—
66	C₄²	—	127	2	no	—
67	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
68	C₁₆	—	2	2	no	—
69	C₁₆	—	127	2	no	—
70	C₁₆	—	254	1	no	—
71	Q₈	—	127	4	no	—
72	C₂×C₄	—	127	2	no	—
73	C₂×C₄	—	127	2	no	—
74	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
75	C₂×C₄	C₈×D₁₂₇	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	254	2	no	—
79	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
80	C₈	#14523247	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₁₂₇	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	127	4	no	—
84	C₄	—	254	2	no	—
85	C₄	C₁₆×D₁₂₇	1	2	yes	—
86	C₄	#16861174	1	1	yes	characteristic
87	C₄	#16861217	1	1	yes	characteristic
88	C₂	#22550000	1	1	yes	characteristic
89	C₂	C₂×C₁₆×D₁₂₇	1	1	yes	characteristic
90	C₂	#22550294	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₂₇
commutator	C₂×C₁₂₇	C₂²×C₁₆
Fitting	C₄⋊C₁₆×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂²×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₄⋊C₁₆	D₁₂₇
hypocenter	C₁₂₇	Q₈×C₁₆
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
20. C₁₂₇⋊C₁₆	Q₈	semidirect
25. Dic₂₅₄	C₁₆	semidirect
46. C₁₂₇	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₂₅₄	2
Dic₁₂₇⋊C₁₆	2
C₄×C₁₂₇⋊C₁₆	2
C₁₆×Dic₁₂₇	2
C₄⋊C₁₆×C₁₂₇	2
Q₈×C₁₆	127