Subgroups — #30637034

Order 16256.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523231	C₂	1	1	yes	maximal, characteristic
2	C₄×#14523231	C₂	1	2	yes	maximal
3	#22549598	C₂	1	2	yes	maximal, complemented
4	C₈×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#322×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523231	C₄	1	4	yes	—
7	C₂×#14523231	C₄	1	2	yes	—
8	C₄²×D₁₂₇	C₂²	1	1	yes	characteristic
9	#16861119	C₄	1	4	yes	—
10	#16861119	C₄	1	2	yes	—
11	C₄×C₁₂₇⋊C₈	C₂²	1	2	yes	—
12	C₄×C₁₂₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
15	C₂×C₄×D₁₂₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×D₁₂₇	C₂×C₄	1	1	yes	characteristic
17	#14523231	—	2	8	no	—
18	#14523231	—	2	4	no	—
19	C₂×C₁₂₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₂₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₂₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
24	C₂×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
25	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
26	C₂²×D₁₂₇	C₄²	1	1	yes	characteristic
27	C₂×Dic₁₂₇	C₄²	1	1	yes	characteristic
28	C₂×Dic₁₂₇	C₄²	1	2	yes	—
29	C₄×D₁₂₇	—	2	4	no	—
30	C₄×D₁₂₇	—	2	2	no	—
31	C₁₂₇⋊C₈	—	2	2	no	—
32	C₁₂₇⋊C₈	M₁₆	1	8	yes	complemented
33	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₂₇	—	2	4	no	—
36	C₈×C₁₂₇	M₁₆	1	4	yes	complemented
37	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
38	C₂×D₁₂₇	—	2	1	no	—
39	C₂×D₁₂₇	—	2	2	no	—
40	C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
41	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
42	Dic₁₂₇	—	2	1	no	—
43	Dic₁₂₇	—	2	2	no	—
44	C₂×C₁₂₇	C₄×M₁₆	1	1	yes	characteristic
45	C₂×C₁₂₇	C₄×M₁₆	1	2	yes	—
46	D₁₂₇	—	4	1	no	—
47	#1455	—	127	1	no	maximal, Sylow
48	C₁₂₇	#1455	1	1	yes	characteristic, complemented, Sylow
49	C₄×M₁₆	—	127	2	no	—
50	C₄×M₁₆	—	127	1	no	—
51	#322	—	127	2	no	—
52	#322	D₁₂₇	1	1	yes	characteristic, complemented
53	C₈²	—	127	1	no	—
54	C₂×M₁₆	—	127	4	no	—
55	C₂×M₁₆	—	127	2	no	—
56	C₂×C₄²	—	127	1	no	—
57	#97	—	127	4	no	—
58	#97	—	127	2	no	—
59	C₄×C₈	—	127	2	no	—
60	C₄×C₈	—	127	1	no	—
61	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
62	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
63	C₂²×C₄	—	127	2	no	—
64	C₂²×C₄	—	127	1	no	—
65	M₁₆	—	254	8	no	—
66	M₁₆	—	254	4	no	—
67	C₂×C₈	—	127	4	no	—
68	C₂×C₈	—	127	2	no	—
69	C₂×C₈	C₄×D₁₂₇	1	4	yes	—
70	C₂×C₈	C₄×D₁₂₇	1	2	yes	—
71	C₄²	—	127	2	no	—
72	C₄²	—	127	1	no	—
73	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
74	C₂³	—	127	1	no	—
75	C₂×C₄	—	127	1	no	—
76	C₂×C₄	—	127	2	no	—
77	C₂×C₄	—	254	4	no	—
78	C₂×C₄	—	254	2	no	—
79	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
80	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	127	8	no	—
83	C₈	—	254	2	no	—
84	C₈	#14523231	1	4	yes	complemented
85	C₂²	—	254	1	no	—
86	C₂²	—	254	2	no	—
87	C₂²	C₄²×D₁₂₇	1	1	yes	characteristic
88	C₄	—	254	1	no	—
89	C₄	—	254	2	no	—
90	C₄	M₁₆×D₁₂₇	1	4	yes	—
91	C₄	C₂×#14523231	1	2	yes	—
92	C₂	—	508	1	no	—
93	C₂	C₄×#14523231	1	1	yes	characteristic
94	C₂	#22549828	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂×C₄²
Fitting	#322×C₁₂₇	C₂
prime radical	C₂²×D₁₂₇	C₄²
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	#322	D₁₂₇
hypocenter	C₁₂₇	#1455
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. #22549598	C₂	semidirect
4. C₈×C₁₂₇⋊C₈	C₂	semidirect
5. #322×C₁₂₇	C₂	semidirect
32. C₁₂₇⋊C₈	M₁₆	semidirect
36. C₈×C₁₂₇	M₁₆	semidirect
48. C₁₂₇	#1455	semidirect
52. #322	D₁₂₇	semidirect
84. C₈	#14523231	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523231	2
#22549598	2
C₈×C₁₂₇⋊C₈	2
#322×C₁₂₇	2
#1455	127