Subgroups — C₁₂₇×#13549 #44577222

Order 32512.

238 conjugacy classes of subgroups.

The subgroup lattice has 130 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2108	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1397	C₂	1	2	yes	maximal, complemented
5	C₄×C₃₂×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
8	#405×C₁₂₇	C₄	1	1	yes	characteristic
9	M₆₄×C₁₂₇	C₄	1	4	yes	complemented
10	C₂×C₃₂×C₁₂₇	C₄	1	4	yes	complemented
11	C₂×C₃₂×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
14	#346×C₁₂₇	C₄	1	2	yes	—
15	C₄×C₁₆×C₁₂₇	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₂₇	C₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	#117×C₁₂₇	C₈	1	1	yes	characteristic
19	M₃₂×C₁₂₇	C₈	1	4	yes	—
20	C₂×C₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₁₂₇	C₂³	1	1	yes	characteristic
25	#97×C₁₂₇	C₈	1	2	yes	—
26	C₄×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
27	C₃₂×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₃₂×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₃₂×C₁₂₇	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
31	M₁₆×C₁₂₇	C₁₆	1	4	yes	—
32	C₂×C₈×C₁₂₇	C₄²	1	2	yes	—
33	C₂×C₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₁₂₇	C₁₆	1	2	yes	—
38	C₂²⋊C₄×C₁₂₇	C₁₆	1	2	yes	—
39	C₄²×C₁₂₇	C₂×C₈	1	2	yes	—
40	C₁₆×C₁₂₇	C₄²	1	2	yes	—
41	C₁₆×C₁₂₇	C₂×C₈	1	4	yes	—
42	C₁₆×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
43	C₁₆×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₂³×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₁₂₇	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₁₂₇	C₂×C₁₆	1	4	yes	—
47	C₂×C₄×C₁₂₇	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄×C₁₂₇	C₂×C₁₆	1	2	yes	—
49	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₁₂₇	C₄×C₈	1	2	yes	—
51	C₈×C₁₂₇	C₂×C₁₆	1	4	yes	—
52	C₈×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
53	C₈×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
54	C₂²×C₁₂₇	—	2	2	no	—
55	C₂²×C₁₂₇	C₄×C₁₆	1	2	yes	—
56	C₂²×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
57	C₄×C₁₂₇	—	2	4	no	—
58	C₄×C₁₂₇	C₄×C₁₆	1	2	yes	—
59	C₄×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
61	#13549	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₁₂₇	—	2	2	no	—
63	C₂×C₁₂₇	C₂×C₄×C₁₆	1	1	yes	characteristic
64	C₂×C₁₂₇	#2258	1	2	yes	—
65	C₂×M₆₄	C₂×C₁₂₇	1	1	yes	characteristic
66	C₂²×C₃₂	C₂×C₁₂₇	1	1	yes	characteristic
67	#2108	C₂×C₁₂₇	1	1	yes	characteristic
68	#1397	C₂×C₁₂₇	1	2	yes	complemented
69	C₄×C₃₂	C₂×C₁₂₇	1	2	yes	complemented
70	C₁₂₇	#13549	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×M₃₂	C₄×C₁₂₇	1	1	yes	characteristic
72	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
73	#405	C₄×C₁₂₇	1	1	yes	characteristic
74	M₆₄	C₄×C₁₂₇	1	4	yes	complemented
75	C₂×C₃₂	C₄×C₁₂₇	1	4	yes	complemented
76	C₂×C₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
77	C₂×C₃₂	C₂²×C₁₂₇	1	2	yes	—
78	C₂×C₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
79	#346	C₄×C₁₂₇	1	2	yes	—
80	C₄×C₁₆	C₂²×C₁₂₇	1	2	yes	—
81	C₂×M₁₆	C₈×C₁₂₇	1	1	yes	characteristic
82	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
83	#117	C₈×C₁₂₇	1	1	yes	characteristic
84	M₃₂	C₈×C₁₂₇	1	4	yes	—
85	C₂×C₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
86	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
87	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
88	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
89	C₂×C₁₆	C₂³×C₁₂₇	1	1	yes	characteristic
90	#97	C₈×C₁₂₇	1	2	yes	—
91	C₄×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
92	C₃₂	C₂×C₄×C₁₂₇	1	2	yes	—
93	C₃₂	C₂×C₄×C₁₂₇	1	2	yes	—
94	C₃₂	C₂×C₄×C₁₂₇	1	4	yes	—
95	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
96	M₁₆	C₁₆×C₁₂₇	1	4	yes	—
97	C₂×C₈	C₄²×C₁₂₇	1	2	yes	—
98	C₂×C₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
99	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
100	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
101	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
102	C₄⋊C₄	C₁₆×C₁₂₇	1	2	yes	—
103	C₂²⋊C₄	C₁₆×C₁₂₇	1	2	yes	—
104	C₄²	C₂×C₈×C₁₂₇	1	2	yes	—
105	C₁₆	C₄²×C₁₂₇	1	2	yes	—
106	C₁₆	C₂×C₈×C₁₂₇	1	4	yes	—
107	C₁₆	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
108	C₁₆	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
109	C₂³	C₂×C₁₆×C₁₂₇	1	1	yes	characteristic
110	C₂×C₄	C₄×C₈×C₁₂₇	1	2	yes	—
111	C₂×C₄	C₂×C₁₆×C₁₂₇	1	4	yes	—
112	C₂×C₄	C₂×C₁₆×C₁₂₇	1	2	yes	—
113	C₂×C₄	C₂×C₁₆×C₁₂₇	1	1	yes	characteristic
114	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
115	C₈	C₄×C₈×C₁₂₇	1	2	yes	—
116	C₈	C₂×C₁₆×C₁₂₇	1	4	yes	—
117	C₈	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
118	C₈	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	C₄×C₁₆×C₁₂₇	1	2	yes	—
121	C₂²	C₂²×C₁₆×C₁₂₇	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	C₄×C₁₆×C₁₂₇	1	2	yes	—
124	C₄	C₂×C₄×C₈×C₁₂₇	1	1	yes	characteristic
125	C₄	C₂²×C₁₆×C₁₂₇	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×C₄×C₁₆×C₁₂₇	1	1	yes	characteristic
128	C₂	C₁₂₇×#2258	1	2	yes	—

	subgroup	G/H
center	C₂×C₃₂×C₁₂₇	C₂²
commutator	C₂	C₂×C₄×C₁₆×C₁₂₇
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂²×C₁₆
prime radical	C₂³×C₁₂₇	C₂×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1397	C₂	semidirect
5. C₄×C₃₂×C₁₂₇	C₂	semidirect
9. M₆₄×C₁₂₇	C₄	semidirect
10. C₂×C₃₂×C₁₂₇	C₄	semidirect
61. #13549	C₁₂₇	direct (N × H)
68. #1397	C₂×C₁₂₇	semidirect
69. C₄×C₃₂	C₂×C₁₂₇	semidirect
70. C₁₂₇	#13549	direct (N × H)
74. M₆₄	C₄×C₁₂₇	semidirect
75. C₂×C₃₂	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₆₄×C₁₂₇	2
C₂²×C₃₂×C₁₂₇	2
C₁₂₇×#2108	2
C₁₂₇×#1397	2
C₄×C₃₂×C₁₂₇	2
#13549	127

Subgroups — C127×#13549 #44577222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13549 #44577222