Subgroups — #30638861

Order 16256.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549742	C₂	1	2	yes	maximal
2	#22549639	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
5	C₁₂₇⋊#95	—	2	4	no	—
6	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₂₇	C₄	1	4	yes	complemented
9	C₂²×Dic₁₂₇	—	2	4	no	—
10	C₄⋊Dic₁₂₇	—	2	8	no	—
11	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
12	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
13	C₂×C₈×C₁₂₇	—	2	4	no	—
14	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
15	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
16	C₄²×C₁₂₇	Q₈	1	2	yes	—
17	C₂×Dic₁₂₇	—	4	4	no	—
18	C₂×Dic₁₂₇	—	4	8	no	—
19	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₂₇	—	2	4	no	—
21	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	#41	1	4	yes	—
25	C₈×C₁₂₇	—	2	4	no	—
26	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₂₇	#124	1	2	yes	—
28	C₂²×C₁₂₇	#135	1	4	yes	—
29	C₄×C₁₂₇	—	2	4	no	—
30	C₄×C₁₂₇	#117	1	2	yes	—
31	C₄×C₁₂₇	#125	1	2	yes	—
32	Dic₁₂₇	—	8	4	no	—
33	C₂×C₁₂₇	#382	1	1	yes	characteristic
34	C₂×C₁₂₇	#427	1	2	yes	—
35	C₂×C₁₂₇	#488	1	4	yes	—
36	#1835	—	127	1	no	maximal, Sylow
37	C₁₂₇	#1835	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
39	#382	—	127	2	no	—
40	#340	—	127	4	no	—
41	C₂²×C₈	C₂×D₁₂₇	1	2	yes	—
42	C₂×C₄⋊C₄	—	127	4	no	—
43	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
44	C₄×C₈	Dic₁₂₇	1	4	yes	complemented
45	#95	—	254	4	no	—
46	C₂²×C₄	—	254	4	no	—
47	C₂²×C₄	D₅₀₈	1	2	yes	—
48	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	C₂×Dic₁₂₇	1	4	yes	—
51	C₄⋊C₄	—	254	8	no	—
52	C₄²	C₂×Dic₁₂₇	1	2	yes	—
53	C₄²	Dic₂₅₄	1	2	yes	—
54	C₂³	C₂×D₅₀₈	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	508	4	no	—
57	C₂×C₄	—	508	8	no	—
58	C₂×C₄	C₄⋊Dic₁₂₇	1	4	yes	—
59	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
61	C₂×C₄	#14523248	1	4	yes	—
62	C₈	—	2	4	no	—
63	C₂²	C₂×C₄⋊Dic₁₂₇	1	1	yes	characteristic
64	C₂²	#16861197	1	2	yes	—
65	C₂²	#16861231	1	4	yes	—
66	C₄	—	2	4	no	—
67	C₄	—	1016	4	no	—
68	C₄	#16861168	1	2	yes	—
69	C₄	#16861201	1	2	yes	—
70	C₂	#22549742	1	1	yes	characteristic
71	C₂	#22549914	1	2	yes	—
72	C₂	#22550231	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₀₈
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#1835
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₄×C₈×C₁₂₇	C₄	semidirect
37. C₁₂₇	#1835	semidirect
44. C₄×C₈	Dic₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22549742	2
#22549639	2
C₂×C₄×C₈×C₁₂₇	2
#1835	127