Subgroups — #30636670

Order 16256.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₁₂₇⋊C₁₆)	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₂₇⋊C₁₆	—	2	2	no	—
4	C₂²×C₈×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₁₂₇	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₁₂₇	—	2	2	no	—
8	C₁₂₇⋊C₁₆	—	4	2	no	—
9	C₂²×C₄×C₁₂₇	C₈	1	1	yes	characteristic
10	C₂²×C₄×C₁₂₇	C₈	1	1	yes	characteristic
11	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₁₂₇	—	2	1	no	—
13	C₂×C₈×C₁₂₇	—	4	1	no	—
14	C₂×C₈×C₁₂₇	—	4	1	no	—
15	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
16	C₄²×C₁₂₇	—	2	1	no	—
17	C₄²×C₁₂₇	—	2	1	no	—
18	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
19	C₂×C₄×C₁₂₇	—	2	1	no	—
20	C₂×C₄×C₁₂₇	—	2	1	no	—
21	C₂×C₄×C₁₂₇	—	2	1	no	—
22	C₂×C₄×C₁₂₇	—	2	1	no	—
23	C₂×C₄×C₁₂₇	—	2	1	no	—
24	C₂×C₄×C₁₂₇	—	4	1	no	—
25	C₂×C₄×C₁₂₇	C₁₆	1	2	yes	complemented
26	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
28	C₈×C₁₂₇	—	2	2	no	—
29	C₈×C₁₂₇	—	4	1	no	—
30	C₂²×C₁₂₇	—	2	1	no	—
31	C₂²×C₁₂₇	—	2	1	no	—
32	C₂²×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₂²×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
34	C₂²×C₁₂₇	M₃₂	1	1	yes	characteristic
35	C₄×C₁₂₇	—	2	1	no	—
36	C₄×C₁₂₇	—	2	2	no	—
37	C₄×C₁₂₇	—	4	1	no	—
38	C₄×C₁₂₇	#99	1	1	yes	characteristic
39	C₄×C₁₂₇	#101	1	1	yes	characteristic
40	C₂×C₁₂₇	—	2	1	no	—
41	C₂×C₁₂₇	—	2	1	no	—
42	C₂×C₁₂₇	#324	1	1	yes	characteristic
43	C₂×C₁₂₇	C₂²⋊C₁₆	1	1	yes	characteristic
44	C₂×C₁₂₇	#349	1	1	yes	characteristic
45	#1322	—	127	1	no	maximal, Sylow
46	C₁₂₇	#1322	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₄⋊C₈	D₁₂₇	1	1	yes	characteristic
48	C₂²⋊C₁₆	—	127	2	no	—
49	C₂²×C₈	Dic₁₂₇	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
51	C₂×C₄²	Dic₁₂₇	1	1	yes	characteristic
52	C₂×C₁₆	—	254	2	no	—
53	C₄⋊C₈	—	2	2	no	—
54	C₂²×C₄	C₁₂₇⋊C₈	1	1	yes	characteristic
55	C₂²×C₄	C₁₂₇⋊C₈	1	1	yes	characteristic
56	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	4	1	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	C₂²⋊D₁₂₇	1	2	yes	—
61	C₄²	—	2	1	no	—
62	C₄²	—	2	1	no	—
63	C₁₆	—	508	2	no	—
64	C₂³	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	C₁₂₇⋊C₁₆	1	2	yes	complemented
72	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
73	C₂×C₄	#14523232	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	—	4	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	C₂²⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
79	C₂²	C₂×C₁₂₇⋊C₁₆	1	1	yes	characteristic
80	C₂²	#16861149	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	#16861124	1	1	yes	characteristic
85	C₄	#16861128	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#22549602	1	1	yes	characteristic
89	C₂	C₂²⋊(C₁₂₇⋊C₁₆)	1	1	yes	characteristic
90	C₂	#22549658	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₂₇
commutator	C₂²×C₁₂₇	C₂×C₁₆
Frattini	C₂²×C₈	C₂×D₁₂₇
Fitting	C₂×C₄⋊C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂²⋊C₈
prime radical	C₂³×C₁₂₇	C₂×C₈
cosocle	C₂²×C₈×C₁₂₇	C₂²
hypercenter	C₂×C₄⋊C₈	D₁₂₇
hypocenter	C₁₂₇	#1322
LCS	G ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
25. C₂×C₄×C₁₂₇	C₁₆	semidirect
46. C₁₂₇	#1322	semidirect
71. C₂×C₄	C₁₂₇⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊(C₁₂₇⋊C₁₆)	2
C₂×C₄⋊C₈×C₁₂₇	2
#1322	127