Subgroups — #30640639

Order 16256.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₁₂₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₁₂₇⋊C₁₆)	C₂	1	2	yes	maximal
3	#22549652	C₂	1	2	yes	maximal
4	C₄×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₈×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₁₂₇	C₄	1	1	yes	characteristic
10	C₄⋊C₈×C₁₂₇	C₄	1	2	yes	—
11	C₄⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
12	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
14	C₁₂₇⋊C₁₆	—	2	1	no	—
15	C₁₂₇⋊C₁₆	—	2	2	no	—
16	C₁₂₇⋊C₁₆	Q₈	1	2	yes	complemented
17	C₂×Q₈×C₁₂₇	C₈	1	1	yes	characteristic
18	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₂₇	C₈	1	2	yes	—
22	C₄⋊C₄×C₁₂₇	C₈	1	1	yes	characteristic
23	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	Q₈×C₁₂₇	—	2	2	no	—
26	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₁₂₇	—	2	1	no	—
32	C₈×C₁₂₇	—	2	2	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	2	no	—
37	C₄×C₁₂₇	—	2	2	no	—
38	C₄×C₁₂₇	—	2	1	no	—
39	C₄×C₁₂₇	M₃₂	1	2	yes	—
40	C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
41	C₄×C₁₂₇	#131	1	1	yes	characteristic
42	C₂×C₁₂₇	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
44	C₂×C₁₂₇	#504	1	1	yes	characteristic
45	#2183	—	127	1	no	maximal, Sylow
46	C₁₂₇	#2183	1	1	yes	characteristic, complemented, Sylow
47	C₈×Q₈	D₁₂₇	1	1	yes	characteristic
48	C₄⋊C₁₆	—	127	1	no	—
49	C₄⋊C₁₆	—	127	2	no	—
50	#346	—	127	2	no	—
51	C₄×C₁₆	—	127	1	no	—
52	C₄×Q₈	Dic₁₂₇	1	1	yes	characteristic
53	C₂×C₁₆	—	127	1	no	—
54	C₂×C₁₆	—	127	1	no	—
55	C₂×C₁₆	—	127	2	no	—
56	C₄⋊C₈	Dic₁₂₇	1	2	yes	—
57	C₄⋊C₈	Dic₁₂₇	1	1	yes	characteristic
58	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
59	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
60	C₂×Q₈	C₁₂₇⋊C₈	1	1	yes	characteristic
61	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
62	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
63	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
64	C₄⋊C₄	C₁₂₇⋊C₈	1	2	yes	—
65	C₄⋊C₄	C₁₂₇⋊C₈	1	1	yes	characteristic
66	C₄²	C₂×Dic₁₂₇	1	2	yes	—
67	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
68	C₁₆	—	127	2	no	—
69	C₁₆	—	254	1	no	—
70	C₁₆	—	254	2	no	—
71	Q₈	—	2	2	no	—
72	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
73	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
74	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
75	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
76	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	2	no	—
79	C₈	Q₈×D₁₂₇	1	1	yes	characteristic
80	C₈	#14523249	1	1	yes	characteristic
81	C₂²	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	2	2	no	—
84	C₄	—	2	1	no	—
85	C₄	#16861149	1	2	yes	—
86	C₄	Q₈×Dic₁₂₇	1	1	yes	characteristic
87	C₄	#16861216	1	1	yes	characteristic
88	C₂	Q₈×C₁₂₇⋊C₈	1	1	yes	characteristic
89	C₂	C₂×#16861149	1	1	yes	characteristic
90	C₂	#22550293	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₂₇
commutator, socle, prime radical	C₂²×C₁₂₇	C₂²×C₈
Fitting	C₈×Q₈×C₁₂₇	C₂
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₈×Q₈	D₁₂₇
hypocenter	C₁₂₇	#2183
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
16. C₁₂₇⋊C₁₆	Q₈	semidirect
46. C₁₂₇	#2183	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₁₂₇⋊C₁₆)	2
#22549652	2
C₄×C₁₂₇⋊C₁₆	2
C₈×Q₈×C₁₂₇	2
#2183	127