Subgroups — C₁₂₇×#2873 #30635825

Order 16256.

564 conjugacy classes of subgroups.

The subgroup lattice has 124 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#405×C₁₂₇	C₂	1	8	yes	maximal, complemented
4	C₄×M₁₆×C₁₂₇	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₁₂₇	C₄	1	16	yes	complemented
7	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	—
8	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	—
9	C₂²×C₈×C₁₂₇	C₂²	1	4	yes	—
10	#117×C₁₂₇	C₄	1	4	yes	—
11	#117×C₁₂₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
13	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
14	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
15	#97×C₁₂₇	C₂²	1	8	yes	complemented
16	C₄×C₈×C₁₂₇	C₂²	1	8	yes	complemented
17	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₁₂₇	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
24	M₁₆×C₁₂₇	C₂×C₄	1	16	yes	—
25	M₁₆×C₁₂₇	C₂×C₄	1	16	yes	complemented
26	C₂×C₈×C₁₂₇	C₂×C₄	1	16	yes	complemented
27	C₂×C₈×C₁₂₇	C₂³	1	8	yes	—
28	C₄⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
29	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
30	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₄²×C₁₂₇	C₂³	1	4	yes	—
32	C₂³×C₁₂₇	—	4	2	no	—
33	C₂³×C₁₂₇	C₄²	1	4	yes	—
34	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
35	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	—	4	4	no	—
37	C₂×C₄×C₁₂₇	C₄²	1	4	yes	—
38	C₂×C₄×C₁₂₇	C₄²	1	8	yes	—
39	C₂×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
40	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
43	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
46	C₈×C₁₂₇	C₂²×C₄	1	16	yes	—
47	C₂²×C₁₂₇	—	2	2	no	—
48	C₂²×C₁₂₇	—	2	4	no	—
49	C₂²×C₁₂₇	—	4	4	no	—
50	C₂²×C₁₂₇	C₂×C₄²	1	4	yes	—
51	C₂²×C₁₂₇	C₂×C₄²	1	2	yes	—
52	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
53	C₄×C₁₂₇	—	2	8	no	—
54	C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
55	C₄×C₁₂₇	C₂×C₄²	1	4	yes	—
56	C₄×C₁₂₇	C₂³×C₄	1	2	yes	—
57	C₂×C₁₂₇	—	2	2	no	—
58	C₂×C₁₂₇	—	2	4	no	—
59	C₂×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
60	C₂×C₁₂₇	#568	1	2	yes	—
61	#2873	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₁₂₇	#2873	1	1	yes	characteristic, direct product factor, Sylow
63	C₂²×M₁₆	C₂×C₁₂₇	1	2	yes	—
64	C₂×#117	C₂×C₁₂₇	1	1	yes	characteristic
65	#405	C₂×C₁₂₇	1	8	yes	complemented
66	C₄×M₁₆	C₂×C₁₂₇	1	4	yes	complemented
67	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
68	C₂×M₁₆	C₄×C₁₂₇	1	16	yes	complemented
69	C₂×M₁₆	C₂²×C₁₂₇	1	4	yes	—
70	C₂×M₁₆	C₂²×C₁₂₇	1	4	yes	—
71	C₂²×C₈	C₂²×C₁₂₇	1	4	yes	—
72	#117	C₄×C₁₂₇	1	4	yes	—
73	#117	C₂²×C₁₂₇	1	4	yes	—
74	C₂×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
75	C₂×C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
76	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
77	#97	C₂²×C₁₂₇	1	8	yes	complemented
78	C₄×C₈	C₂²×C₁₂₇	1	8	yes	complemented
79	C₂⁴	C₂×C₄×C₁₂₇	1	1	yes	characteristic
80	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
81	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
82	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
83	C₂²×C₄	C₂×C₄×C₁₂₇	1	8	yes	—
84	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
85	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
86	M₁₆	C₂×C₄×C₁₂₇	1	16	yes	—
87	M₁₆	C₂×C₄×C₁₂₇	1	16	yes	complemented
88	C₂×C₈	C₂×C₄×C₁₂₇	1	16	yes	complemented
89	C₂×C₈	C₂³×C₁₂₇	1	8	yes	—
90	C₄⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
91	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
92	C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
93	C₄²	C₂³×C₁₂₇	1	4	yes	—
94	C₂³	—	4	2	no	—
95	C₂³	C₄²×C₁₂₇	1	4	yes	—
96	C₂³	C₂²×C₄×C₁₂₇	1	2	yes	—
97	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	C₄²×C₁₂₇	1	4	yes	—
100	C₂×C₄	C₄²×C₁₂₇	1	8	yes	—
101	C₂×C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
102	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
103	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
104	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
105	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
106	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
107	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
108	C₈	C₂²×C₄×C₁₂₇	1	16	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	—	2	4	no	—
111	C₂²	—	4	4	no	—
112	C₂²	C₂×C₄²×C₁₂₇	1	4	yes	—
113	C₂²	C₂×C₄²×C₁₂₇	1	2	yes	—
114	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
115	C₄	—	2	8	no	—
116	C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
117	C₄	C₂×C₄²×C₁₂₇	1	4	yes	—
118	C₄	C₂³×C₄×C₁₂₇	1	2	yes	—
119	C₂	—	2	2	no	—
120	C₂	—	2	4	no	—
121	C₂	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
122	C₂	#568×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂⁴
commutator	C₂	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂³×C₄
prime radical	C₂⁴×C₁₂₇	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #405×C₁₂₇	C₂	semidirect
4. C₄×M₁₆×C₁₂₇	C₂	semidirect
6. C₂×M₁₆×C₁₂₇	C₄	semidirect
15. #97×C₁₂₇	C₂²	semidirect
16. C₄×C₈×C₁₂₇	C₂²	semidirect
25. M₁₆×C₁₂₇	C₂×C₄	semidirect
26. C₂×C₈×C₁₂₇	C₂×C₄	semidirect
61. #2873	C₁₂₇	direct (N × H)
62. C₁₂₇	#2873	direct (N × H)
65. #405	C₂×C₁₂₇	semidirect
66. C₄×M₁₆	C₂×C₁₂₇	semidirect
68. C₂×M₁₆	C₄×C₁₂₇	semidirect
77. #97	C₂²×C₁₂₇	semidirect
78. C₄×C₈	C₂²×C₁₂₇	semidirect
87. M₁₆	C₂×C₄×C₁₂₇	semidirect
88. C₂×C₈	C₂×C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₁₂₇	2
C₂×#117×C₁₂₇	2
#405×C₁₂₇	2
C₄×M₁₆×C₁₂₇	2
#2873	127

Subgroups — C127×#2873 #30635825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2873 #30635825