Subgroups — #30637013

Order 16256.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549929	C₂	1	2	yes	maximal
2	#22549597	C₂	1	2	yes	maximal, complemented
3	C₈×C₁₂₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₁₂₇⋊C₈)	C₄	1	4	yes	—
6	C₂²⋊(C₁₂₇⋊C₈)	C₄	1	4	yes	—
7	C₄×C₁₂₇⋊C₈	C₂²	1	4	yes	—
8	C₂²×C₈×C₁₂₇	C₄	1	2	yes	—
9	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₁₂₇	C₄	1	2	yes	—
11	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₁₂₇⋊C₈	C₂×C₄	1	8	yes	—
14	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₂₇	C₈	1	8	yes	complemented
17	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
19	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₁₂₇⋊C₈	—	2	8	no	—
23	C₂³×C₁₂₇	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	—	2	4	no	—
25	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
26	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	C₂×C₈	1	4	yes	—
28	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
29	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
30	C₈×C₁₂₇	C₂×C₈	1	4	yes	—
31	C₈×C₁₂₇	C₂×C₈	1	4	yes	—
32	C₂²×C₁₂₇	—	2	2	no	—
33	C₂²×C₁₂₇	C₄×C₈	1	2	yes	—
34	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
35	C₄×C₁₂₇	—	2	2	no	—
36	C₄×C₁₂₇	C₄×C₈	1	2	yes	—
37	C₄×C₁₂₇	C₂²×C₈	1	2	yes	—
38	C₄×C₁₂₇	#131	1	4	yes	—
39	C₂×C₁₂₇	—	2	2	no	—
40	C₂×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
41	C₂×C₁₂₇	#405	1	2	yes	—
42	#1450	—	127	1	no	maximal, Sylow
43	C₁₂₇	#1450	1	1	yes	characteristic, complemented, Sylow
44	#431	—	127	2	no	—
45	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
46	#322	—	127	2	no	—
47	C₈²	—	127	2	no	—
48	C₂²×C₈	Dic₁₂₇	1	2	yes	—
49	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
50	C₄⋊C₈	—	127	4	no	—
51	C₂²⋊C₈	—	127	4	no	—
52	C₄×C₈	—	127	4	no	—
53	C₄×C₈	Dic₁₂₇	1	2	yes	—
54	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
55	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
56	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
57	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
58	C₂×C₈	—	127	8	no	—
59	C₂×C₈	C₁₂₇⋊C₈	1	8	yes	complemented
60	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
61	C₂×C₈	C₂×Dic₁₂₇	1	2	yes	—
62	C₄²	C₄×D₁₂₇	1	2	yes	—
63	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
64	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
65	C₂³	C₄×Dic₁₂₇	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
68	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
69	C₂×C₄	C₂×C₁₂₇⋊C₈	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
71	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
72	C₈	—	254	8	no	—
73	C₈	C₂×C₁₂₇⋊C₈	1	4	yes	—
74	C₈	C₂×C₁₂₇⋊C₈	1	4	yes	—
75	C₂²	—	2	2	no	—
76	C₂²	C₄×C₁₂₇⋊C₈	1	2	yes	—
77	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	C₄×C₁₂₇⋊C₈	1	2	yes	—
80	C₄	C₂²×C₁₂₇⋊C₈	1	2	yes	—
81	C₄	#16861215	1	4	yes	—
82	C₂	—	2	2	no	—
83	C₂	C₂×C₄×C₁₂₇⋊C₈	1	1	yes	characteristic
84	C₂	#22549831	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₁₂₇
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
socle	C₂²×C₁₂₇	C₂×C₄²
prime radical	C₂³×C₁₂₇	C₄²
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#1450
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22549597	C₂	semidirect
3. C₈×C₁₂₇⋊C₈	C₂	semidirect
16. C₂×C₈×C₁₂₇	C₈	semidirect
43. C₁₂₇	#1450	semidirect
59. C₂×C₈	C₁₂₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22549929	2
#22549597	2
C₈×C₁₂₇⋊C₈	2
C₂×C₄×C₈×C₁₂₇	2
#1450	127