Subgroups — #30636674

Order 16256.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549602	C₂	1	2	yes	maximal
2	#382×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊(C₁₂₇⋊C₈)	—	2	2	no	—
4	C₂×C₄⋊C₄×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₁₂₇	C₄	1	1	yes	characteristic
7	#95×C₁₂₇	—	2	2	no	—
8	C₂×C₁₂₇⋊C₈	—	4	2	no	—
9	C₂²×C₄×C₁₂₇	—	2	1	no	—
10	C₂²×C₄×C₁₂₇	—	2	1	no	—
11	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
13	C₄⋊C₄×C₁₂₇	—	4	1	no	—
14	C₄⋊C₄×C₁₂₇	—	4	1	no	—
15	C₄²×C₁₂₇	—	2	1	no	—
16	C₄²×C₁₂₇	C₈	1	2	yes	complemented
17	C₁₂₇⋊C₈	—	8	2	no	—
18	C₂³×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₂₇	—	2	1	no	—
20	C₂×C₄×C₁₂₇	—	2	1	no	—
21	C₂×C₄×C₁₂₇	—	2	2	no	—
22	C₂×C₄×C₁₂₇	—	2	2	no	—
23	C₂×C₄×C₁₂₇	—	4	1	no	—
24	C₂×C₄×C₁₂₇	—	4	1	no	—
25	C₂×C₄×C₁₂₇	—	4	2	no	—
26	C₂×C₄×C₁₂₇	—	4	2	no	—
27	C₂×C₄×C₁₂₇	—	4	1	no	—
28	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
30	C₂²×C₁₂₇	—	2	1	no	—
31	C₂²×C₁₂₇	—	2	1	no	—
32	C₂²×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₂²×C₁₂₇	#99	1	1	yes	characteristic
34	C₂²×C₁₂₇	#101	1	1	yes	characteristic
35	C₄×C₁₂₇	—	2	1	no	—
36	C₄×C₁₂₇	—	2	1	no	—
37	C₄×C₁₂₇	—	4	2	no	—
38	C₄×C₁₂₇	—	4	2	no	—
39	C₄×C₁₂₇	—	8	1	no	—
40	C₂×C₁₂₇	—	2	1	no	—
41	C₂×C₁₂₇	—	2	1	no	—
42	C₂×C₁₂₇	#324	1	1	yes	characteristic
43	C₂×C₁₂₇	#354	1	1	yes	characteristic
44	C₂×C₁₂₇	#355	1	1	yes	characteristic
45	#1324	—	127	1	no	maximal, Sylow
46	C₁₂₇	#1324	1	1	yes	characteristic, complemented, Sylow
47	#382	D₁₂₇	1	1	yes	characteristic
48	#324	—	127	2	no	—
49	C₂×C₄⋊C₄	Dic₁₂₇	1	1	yes	characteristic
50	C₂×C₄⋊C₄	C₂×D₁₂₇	1	1	yes	characteristic
51	C₂×C₄²	Dic₁₂₇	1	1	yes	characteristic
52	C₂²⋊C₈	—	254	2	no	—
53	#95	—	2	2	no	—
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
57	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
58	C₂×C₈	—	508	2	no	—
59	C₄⋊C₄	—	4	1	no	—
60	C₄⋊C₄	—	4	1	no	—
61	C₄²	—	2	1	no	—
62	C₄²	C₁₂₇⋊C₈	1	2	yes	complemented
63	C₂³	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	4	1	no	—
69	C₂×C₄	—	4	1	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	4	2	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
74	C₂×C₄	#14523232	1	1	yes	characteristic
75	C₈	—	1016	2	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	C₂²⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
79	C₂²	#16861124	1	1	yes	characteristic
80	C₂²	#16861128	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	2	1	no	—
83	C₄	—	4	2	no	—
84	C₄	—	4	2	no	—
85	C₄	—	8	1	no	—
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#22549602	1	1	yes	characteristic
89	C₂	#22549671	1	1	yes	characteristic
90	C₂	#22549673	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16861124
commutator	C₂×C₄×C₁₂₇	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₁₂₇
Fitting	#382×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#99
prime radical	C₂³×C₁₂₇	C₂²⋊C₄
cosocle	C₂×C₄⋊C₄×C₁₂₇	C₂²
hypercenter	#382	D₁₂₇
hypocenter	C₁₂₇	#1324
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #382

Complements

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

N	H ≅ G/N	type
16. C₄²×C₁₂₇	C₈	semidirect
46. C₁₂₇	#1324	semidirect
62. C₄²	C₁₂₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22549602	2
#382×C₁₂₇	2
#1324	127