Subgroups — #30640489

Order 16256.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861149	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₂₇⋊C₁₆)	C₂	1	4	yes	maximal, complemented
4	#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#16861149	—	2	2	no	—
6	C₂×C₁₂₇⋊C₁₆	—	2	2	no	—
7	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
9	C₂×M₁₆×C₁₂₇	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	#117×C₁₂₇	C₄	1	1	yes	characteristic
12	#97×C₁₂₇	C₄	1	2	yes	—
13	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
14	C₁₂₇⋊C₁₆	—	2	2	no	—
15	C₁₂₇⋊C₁₆	—	2	2	no	—
16	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	M₁₆×C₁₂₇	—	2	2	no	—
18	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
21	C₂×C₈×C₁₂₇	Q₈	1	2	yes	—
22	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₂₇	C₈	1	2	yes	—
24	C₂²⋊C₄×C₁₂₇	C₈	1	2	yes	—
25	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
26	C₂³×C₁₂₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₂₇	C₂×C₈	1	4	yes	—
29	C₂×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
31	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₂₇	—	2	2	no	—
33	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
34	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₂₇	—	2	2	no	—
37	C₂²×C₁₂₇	C₄⋊C₈	1	2	yes	—
38	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₂₇	—	2	4	no	—
40	C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
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	#504	1	2	yes	—
46	#2151	—	127	1	no	maximal, Sylow
47	C₁₂₇	#2151	1	1	yes	characteristic, complemented, Sylow
48	C₂×M₃₂	—	127	1	no	—
49	C₂²×C₁₆	—	127	1	no	—
50	#405	D₁₂₇	1	1	yes	characteristic
51	C₄⋊C₁₆	—	127	4	no	—
52	C₂×M₁₆	Dic₁₂₇	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
54	#117	Dic₁₂₇	1	1	yes	characteristic
55	M₃₂	—	254	2	no	—
56	C₂×C₁₆	—	127	2	no	—
57	C₂×C₁₆	—	127	2	no	—
58	C₂×C₁₆	—	254	2	no	—
59	#97	Dic₁₂₇	1	2	yes	—
60	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
61	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
64	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
65	C₂×C₈	D₅₀₈	1	2	yes	—
66	C₂×C₈	Dic₂₅₄	1	2	yes	—
67	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
68	C₄⋊C₄	C₁₂₇⋊C₈	1	2	yes	—
69	C₂²⋊C₄	C₁₂₇⋊C₈	1	2	yes	—
70	C₄²	C₂×Dic₁₂₇	1	2	yes	—
71	C₁₆	—	254	2	no	—
72	C₁₆	—	254	2	no	—
73	C₂³	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
74	C₂×C₄	C₄⋊Dic₁₂₇	1	2	yes	—
75	C₂×C₄	C₂×C₁₂₇⋊C₈	1	4	yes	—
76	C₂×C₄	C₂×C₁₂₇⋊C₈	1	1	yes	characteristic
77	C₂×C₄	#14523232	1	2	yes	—
78	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	C₄⋊Dic₁₂₇	1	2	yes	—
81	C₈	C₂×D₅₀₈	1	1	yes	characteristic
82	C₈	C₂×Dic₂₅₄	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	C₄⋊(C₁₂₇⋊C₈)	1	2	yes	—
85	C₂²	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	C₄⋊(C₁₂₇⋊C₈)	1	2	yes	—
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	#22550293	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂²×C₈
Fitting	#405×C₁₂₇	C₂
prime radical	C₂³×C₁₂₇	C₂×C₈
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#405	D₁₂₇
hypocenter	C₁₂₇	#2151
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₁₂₇⋊C₁₆)	C₂	semidirect
47. C₁₂₇	#2151	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
C₂²×C₁₂₇⋊C₁₆	2
C₄⋊(C₁₂₇⋊C₁₆)	2
#405×C₁₂₇	2
#2151	127