Subgroups — #30640496

Order 16256.

152 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	#22549697	C₂	1	4	yes	maximal, complemented
3	#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#16861149	—	2	2	no	—
5	#16861149	C₂²	1	4	yes	—
6	C₂×C₁₂₇⋊C₁₆	—	2	2	no	—
7	C₂×M₁₆×C₁₂₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
9	#117×C₁₂₇	C₄	1	1	yes	characteristic
10	#97×C₁₂₇	C₄	1	2	yes	—
11	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
12	C₁₂₇⋊C₁₆	—	2	4	no	—
13	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	M₁₆×C₁₂₇	C₈	1	4	yes	—
15	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
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	C₄⋊C₄×C₁₂₇	—	2	1	no	—
21	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
22	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂³×C₁₂₇	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	—	2	2	no	—
25	C₂×C₄×C₁₂₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₂₇	M₁₆	1	1	yes	characteristic
28	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
29	C₈×C₁₂₇	C₄⋊C₄	1	2	yes	—
30	C₈×C₁₂₇	C₂×C₈	1	4	yes	—
31	C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
32	C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₁₂₇	—	4	1	no	—
34	C₂²×C₁₂₇	C₄⋊C₈	1	2	yes	—
35	C₂²×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
36	C₄×C₁₂₇	—	4	2	no	—
37	C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
38	C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
40	C₂×C₁₂₇	—	2	1	no	—
41	C₂×C₁₂₇	—	2	2	no	—
42	C₂×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
43	#2153	—	127	1	no	maximal, Sylow
44	C₁₂₇	#2153	1	1	yes	characteristic, complemented, Sylow
45	C₂×M₃₂	—	127	2	no	—
46	#405	D₁₂₇	1	1	yes	characteristic
47	#364	—	127	4	no	—
48	C₂×M₁₆	Dic₁₂₇	1	1	yes	characteristic
49	C₂²×C₈	C₂×D₁₂₇	1	1	yes	characteristic
50	#117	Dic₁₂₇	1	1	yes	characteristic
51	M₃₂	—	127	4	no	—
52	M₃₂	—	254	2	no	—
53	C₂×C₁₆	—	254	2	no	—
54	#97	Dic₁₂₇	1	2	yes	—
55	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
56	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
57	M₁₆	C₁₂₇⋊C₈	1	4	yes	—
58	C₂×C₈	C₂×Dic₁₂₇	1	1	yes	characteristic
59	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
60	C₂×C₈	D₅₀₈	1	2	yes	—
61	C₂×C₈	Dic₂₅₄	1	2	yes	—
62	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
63	C₄⋊C₄	—	2	1	no	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	C₂×Dic₁₂₇	1	2	yes	—
66	C₁₆	—	254	4	no	—
67	C₂³	#14523232	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	C₄⋊Dic₁₂₇	1	2	yes	—
70	C₂×C₄	C₂×C₁₂₇⋊C₈	1	2	yes	—
71	C₂×C₄	#14523232	1	1	yes	characteristic
72	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
73	C₈	C₄⋊Dic₁₂₇	1	2	yes	—
74	C₈	C₂×C₁₂₇⋊C₈	1	4	yes	—
75	C₈	C₂×D₅₀₈	1	1	yes	characteristic
76	C₈	C₂×Dic₂₅₄	1	1	yes	characteristic
77	C₂²	—	4	1	no	—
78	C₂²	C₄⋊(C₁₂₇⋊C₈)	1	2	yes	—
79	C₂²	C₂×#14523232	1	1	yes	characteristic
80	C₄	—	4	2	no	—
81	C₄	C₄⋊(C₁₂₇⋊C₈)	1	2	yes	—
82	C₄	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
83	C₄	C₂²×C₁₂₇⋊C₈	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×C₄⋊(C₁₂₇⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₅₀₈
commutator	C₄×C₁₂₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#405×C₁₂₇	C₂
socle	C₂×C₁₂₇	C₂×C₄⋊C₈
prime radical	C₂³×C₁₂₇	M₁₆
cosocle	C₂×C₈×C₁₂₇	C₂³
hypercenter	#405	D₁₂₇
hypocenter	C₁₂₇	#2153
LCS	G ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #22549697	C₂	semidirect
44. C₁₂₇	#2153	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
#22549697	2
#405×C₁₂₇	2
#2153	127