Subgroups — #30640486

Order 16256.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861149	C₂	1	2	yes	maximal
2	C₄⋊(C₁₂₇⋊C₁₆)	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861149	—	2	4	no	—
5	C₂×C₁₂₇⋊C₁₆	C₂²	1	4	yes	—
6	C₂²×C₈×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₂₇	C₄	1	1	yes	characteristic
9	C₄×C₈×C₁₂₇	C₄	1	2	yes	—
10	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
11	C₁₂₇⋊C₁₆	—	2	4	no	—
12	C₂²×C₄×C₁₂₇	C₈	1	2	yes	—
13	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₂₇	—	2	2	no	—
15	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
16	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
18	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
19	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₄²×C₁₂₇	C₈	1	1	yes	characteristic
21	C₄²×C₁₂₇	C₈	1	1	yes	characteristic
22	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	—	2	4	no	—
25	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
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	M₁₆	1	2	yes	—
30	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₁₂₇	—	2	2	no	—
32	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
33	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
34	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₁₂₇	—	2	2	no	—
36	C₂²×C₁₂₇	C₄⋊C₈	1	2	yes	—
37	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
38	C₄×C₁₂₇	—	2	2	no	—
39	C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
40	C₄×C₁₂₇	M₃₂	1	4	yes	complemented
41	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₁₂₇	—	2	2	no	—
44	C₂×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₂₇	C₂×M₃₂	1	2	yes	—
46	#2150	—	127	1	no	maximal, Sylow
47	C₁₂₇	#2150	1	1	yes	characteristic, complemented, Sylow
48	C₂×M₃₂	—	127	2	no	—
49	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
50	C₄⋊C₁₆	—	127	4	no	—
51	C₂²×C₈	Dic₁₂₇	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
53	C₂×C₄²	Dic₁₂₇	1	1	yes	characteristic
54	M₃₂	—	254	4	no	—
55	C₂×C₁₆	—	127	4	no	—
56	C₄×C₈	Dic₁₂₇	1	2	yes	—
57	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
58	C₂²×C₄	C₁₂₇⋊C₈	1	2	yes	—
59	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
62	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
63	C₂×C₈	D₅₀₈	1	2	yes	—
64	C₂×C₈	Dic₂₅₄	1	2	yes	—
65	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
66	C₄²	C₁₂₇⋊C₈	1	1	yes	characteristic
67	C₄²	C₁₂₇⋊C₈	1	1	yes	characteristic
68	C₄²	C₂×Dic₁₂₇	1	2	yes	—
69	C₁₆	—	254	4	no	—
70	C₂³	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	C₄⋊Dic₁₂₇	1	2	yes	—
73	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
74	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
75	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
76	C₂×C₄	#14523232	1	2	yes	—
77	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₈	C₄⋊Dic₁₂₇	1	2	yes	—
80	C₈	C₂×D₅₀₈	1	1	yes	characteristic
81	C₈	C₂×Dic₂₅₄	1	1	yes	characteristic
82	C₂²	—	2	2	no	—
83	C₂²	C₄⋊(C₁₂₇⋊C₈)	1	2	yes	—
84	C₂²	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	C₄⋊(C₁₂₇⋊C₈)	1	2	yes	—
87	C₄	#16861149	1	4	yes	complemented
88	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
89	C₄	C₂×#14523232	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic
92	C₂	C₂×#16861149	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₈
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
prime radical	C₂³×C₁₂₇	C₂×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#2150
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊(C₁₂₇⋊C₁₆)	C₂	semidirect
40. C₄×C₁₂₇	M₃₂	semidirect
47. C₁₂₇	#2150	semidirect
87. C₄	#16861149	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861149	2
C₄⋊(C₁₂₇⋊C₁₆)	2
C₂×C₄×C₈×C₁₂₇	2
#2150	127