Subgroups — #30638476

Order 16256.

296 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₂₇⋊C₈	C₂	1	2	yes	maximal
2	#22549631	C₂	1	4	yes	maximal
3	C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₂₇⋊C₈	C₂²	1	4	yes	—
5	C₄×C₁₂₇⋊C₈	C₄	1	8	yes	complemented
6	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
7	C₂×C₄²×C₁₂₇	C₄	1	2	yes	—
8	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
9	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×C₁₂₇⋊C₈	—	2	8	no	—
11	C₂×C₁₂₇⋊C₈	C₂×C₄	1	8	yes	—
12	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
13	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
16	C₄²×C₁₂₇	—	2	4	no	—
17	C₄²×C₁₂₇	—	2	4	no	—
18	C₄²×C₁₂₇	C₈	1	8	yes	complemented
19	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
20	C₁₂₇⋊C₈	—	2	8	no	—
21	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₄×C₁₂₇	—	2	4	no	—
23	C₂×C₄×C₁₂₇	—	2	4	no	—
24	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
25	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
26	C₂×C₄×C₁₂₇	C₂×C₈	1	8	yes	—
27	C₂×C₄×C₁₂₇	C₂×C₈	1	4	yes	—
28	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	—
29	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
30	C₂×C₄×C₁₂₇	#41	1	4	yes	—
31	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
32	C₂²×C₁₂₇	#117	1	2	yes	—
33	C₂²×C₁₂₇	C₂²×C₈	1	2	yes	—
34	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
35	C₄×C₁₂₇	—	2	8	no	—
36	C₄×C₁₂₇	C₄×C₈	1	4	yes	—
37	C₄×C₁₂₇	#97	1	4	yes	—
38	C₄×C₁₂₇	#117	1	4	yes	—
39	C₂×C₁₂₇	#376	1	1	yes	characteristic
40	C₂×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
41	C₂×C₁₂₇	C₂×#97	1	1	yes	characteristic
42	C₂×C₁₂₇	#431	1	4	yes	—
43	#1744	—	127	1	no	maximal, Sylow
44	C₁₂₇	#1744	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄×C₈	—	127	2	no	—
46	C₄³	D₁₂₇	1	1	yes	characteristic
47	#336	—	127	4	no	—
48	C₂²×C₈	—	127	4	no	—
49	C₂×C₄²	Dic₁₂₇	1	2	yes	—
50	C₂×C₄²	Dic₁₂₇	1	2	yes	—
51	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
52	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
53	C₄×C₈	—	127	8	no	—
54	C₂²×C₄	C₂×Dic₁₂₇	1	4	yes	—
55	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
56	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
57	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
58	C₂×C₈	—	127	8	no	—
59	C₂×C₈	—	254	8	no	—
60	C₄²	—	2	4	no	—
61	C₄²	—	2	4	no	—
62	C₄²	C₁₂₇⋊C₈	1	8	yes	complemented
63	C₄²	C₄×D₁₂₇	1	4	yes	—
64	C₂³	C₂²×Dic₁₂₇	1	1	yes	characteristic
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
68	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
69	C₂×C₄	C₂×C₁₂₇⋊C₈	1	8	yes	—
70	C₂×C₄	C₂×C₁₂₇⋊C₈	1	4	yes	—
71	C₂×C₄	#14523232	1	4	yes	—
72	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
73	C₂×C₄	#14523248	1	4	yes	—
74	C₈	—	254	8	no	—
75	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
76	C₂²	#16861168	1	2	yes	—
77	C₂²	C₂²×C₁₂₇⋊C₈	1	2	yes	—
78	C₂²	C₂×#14523232	1	2	yes	—
79	C₄	—	2	8	no	—
80	C₄	C₄×C₁₂₇⋊C₈	1	4	yes	—
81	C₄	#16861120	1	4	yes	—
82	C₄	#16861166	1	4	yes	—
83	C₂	#22549722	1	1	yes	characteristic
84	C₂	C₂×C₄×C₁₂₇⋊C₈	1	1	yes	characteristic
85	C₂	C₂×#16861120	1	1	yes	characteristic
86	C₂	#22549929	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₂₇
commutator	C₂×C₁₂₇	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	C₄³×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	C₄³	D₁₂₇
hypocenter	C₁₂₇	#1744
LCS	G ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
5. C₄×C₁₂₇⋊C₈	C₄	semidirect
18. C₄²×C₁₂₇	C₈	semidirect
44. C₁₂₇	#1744	semidirect
62. C₄²	C₁₂₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₂₇⋊C₈	2
#22549631	2
C₄³×C₁₂₇	2
#1744	127