Subgroups — C₁₂₇×#11487 #44575160

Order 32512.

430 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2162	C₂	1	2	yes	maximal, complemented
3	C₁₂₇×#1314	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₁₂₇	C₄	1	2	yes	—
5	C₂×C₄×D₄×C₁₂₇	C₄	1	1	yes	characteristic
6	C₈×D₄×C₁₂₇	—	2	2	no	—
7	C₈×D₄×C₁₂₇	—	2	2	no	—
8	C₂×C₄⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
9	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
12	C₂²⋊C₁₆×C₁₂₇	C₂²	1	4	yes	—
13	C₄×C₁₆×C₁₂₇	—	2	2	no	—
14	C₂²×D₄×C₁₂₇	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₁₂₇	—	2	2	no	—
17	C₂²×C₈×C₁₂₇	—	2	2	no	—
18	C₂²×C₈×C₁₂₇	—	4	4	no	—
19	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₄×D₄×C₁₂₇	—	2	2	no	—
23	C₄×D₄×C₁₂₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₂₇	C₈	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₂₇	C₈	1	2	yes	—
26	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₁₂₇	—	2	4	no	—
28	C₄⋊C₈×C₁₂₇	—	2	2	no	—
29	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
30	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
31	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
32	C₂⁴×C₁₂₇	C₂×C₈	1	2	yes	—
33	C₂×D₄×C₁₂₇	—	2	2	no	—
34	C₂×D₄×C₁₂₇	—	2	2	no	—
35	C₂×D₄×C₁₂₇	C₁₆	1	4	yes	complemented
36	C₂²×C₄×C₁₂₇	—	2	2	no	—
37	C₂²×C₄×C₁₂₇	—	2	2	no	—
38	C₂²×C₄×C₁₂₇	—	4	2	no	—
39	C₂²×C₄×C₁₂₇	—	4	2	no	—
40	C₂²×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₂₇	—	2	2	no	—
45	C₂×C₈×C₁₂₇	—	2	2	no	—
46	C₂×C₈×C₁₂₇	—	4	1	no	—
47	C₂×C₈×C₁₂₇	—	4	4	no	—
48	C₂×C₈×C₁₂₇	—	4	4	no	—
49	C₂×C₈×C₁₂₇	—	4	2	no	—
50	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
53	C₄⋊C₄×C₁₂₇	—	2	2	no	—
54	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
55	C₄²×C₁₂₇	C₂²⋊C₄	1	2	yes	—
56	C₄²×C₁₂₇	M₁₆	1	2	yes	—
57	C₁₆×C₁₂₇	—	4	4	no	—
58	C₂³×C₁₂₇	—	2	2	no	—
59	C₂³×C₁₂₇	—	2	2	no	—
60	C₂³×C₁₂₇	—	4	2	no	—
61	C₂³×C₁₂₇	—	4	2	no	—
62	C₂³×C₁₂₇	C₂×C₁₆	1	4	yes	—
63	C₂³×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
64	D₄×C₁₂₇	—	2	8	no	—
65	C₂×C₄×C₁₂₇	—	2	1	no	—
66	C₂×C₄×C₁₂₇	—	2	1	no	—
67	C₂×C₄×C₁₂₇	—	2	2	no	—
68	C₂×C₄×C₁₂₇	—	4	1	no	—
69	C₂×C₄×C₁₂₇	—	4	4	no	—
70	C₂×C₄×C₁₂₇	—	4	2	no	—
71	C₂×C₄×C₁₂₇	—	4	2	no	—
72	C₂×C₄×C₁₂₇	—	4	2	no	—
73	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
74	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₁₂₇	C₂×C₁₆	1	2	yes	—
76	C₂×C₄×C₁₂₇	M₃₂	1	2	yes	—
77	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
79	C₈×C₁₂₇	—	2	2	no	—
80	C₈×C₁₂₇	—	2	1	no	—
81	C₈×C₁₂₇	—	4	2	no	—
82	C₈×C₁₂₇	#100	1	2	yes	—
83	C₂²×C₁₂₇	—	2	1	no	—
84	C₂²×C₁₂₇	—	2	1	no	—
85	C₂²×C₁₂₇	—	2	4	no	—
86	C₂²×C₁₂₇	—	4	4	no	—
87	C₂²×C₁₂₇	—	4	4	no	—
88	C₂²×C₁₂₇	—	4	2	no	—
89	C₂²×C₁₂₇	—	4	2	no	—
90	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
91	C₂²×C₁₂₇	C₂²×C₁₆	1	1	yes	characteristic
92	C₂²×C₁₂₇	C₂×M₃₂	1	1	yes	characteristic
93	C₄×C₁₂₇	—	2	2	no	—
94	C₄×C₁₂₇	—	2	1	no	—
95	C₄×C₁₂₇	—	4	2	no	—
96	C₄×C₁₂₇	C₂²⋊C₁₆	1	4	yes	—
97	C₄×C₁₂₇	#410	1	1	yes	characteristic
98	C₄×C₁₂₇	C₂×#100	1	1	yes	characteristic
99	#11487	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₁₂₇	—	2	1	no	—
101	C₂×C₁₂₇	—	2	1	no	—
102	C₂×C₁₂₇	—	4	4	no	—
103	C₂×C₁₂₇	#1460	1	1	yes	characteristic
104	C₂×C₁₂₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
105	C₂×C₁₂₇	#2115	1	1	yes	characteristic
106	C₂×C₈×D₄	C₂×C₁₂₇	1	1	yes	characteristic
107	#2162	C₂×C₁₂₇	1	2	yes	complemented
108	#1314	C₂×C₁₂₇	1	4	yes	complemented
109	C₁₂₇	#11487	1	1	yes	characteristic, direct product factor, Sylow
110	C₂³×C₈	C₄×C₁₂₇	1	2	yes	—
111	C₂×C₄×D₄	C₄×C₁₂₇	1	1	yes	characteristic
112	C₈×D₄	—	2	2	no	—
113	C₈×D₄	—	2	2	no	—
114	C₂×C₄⋊C₈	C₄×C₁₂₇	1	1	yes	characteristic
115	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
117	C₄⋊C₁₆	—	2	2	no	—
118	C₂²⋊C₁₆	C₂²×C₁₂₇	1	4	yes	—
119	C₄×C₁₆	—	2	2	no	—
120	C₂²×D₄	C₈×C₁₂₇	1	1	yes	characteristic
121	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	4	4	no	—
125	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
126	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
127	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
128	C₄×D₄	—	2	2	no	—
129	C₄×D₄	—	2	2	no	—
130	C₂×C₄⋊C₄	C₈×C₁₂₇	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	C₈×C₁₂₇	1	2	yes	—
132	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
133	C₂×C₁₆	—	2	4	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₂²⋊C₈	—	2	4	no	—
136	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
137	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
138	C₂⁴	C₂×C₈×C₁₂₇	1	2	yes	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	C₁₆×C₁₂₇	1	4	yes	complemented
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	—	2	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	—	4	2	no	—
146	C₂²×C₄	C₂×C₈×C₁₂₇	1	2	yes	—
147	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
149	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
158	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
159	C₄⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	4	no	—
161	C₄²	C₂²⋊C₄×C₁₂₇	1	2	yes	—
162	C₄²	M₁₆×C₁₂₇	1	2	yes	—
163	C₁₆	—	4	4	no	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	2	no	—
166	C₂³	—	4	2	no	—
167	C₂³	—	4	2	no	—
168	C₂³	C₂×C₁₆×C₁₂₇	1	4	yes	—
169	C₂³	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
170	D₄	—	2	8	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
180	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
181	C₂×C₄	C₂×C₁₆×C₁₂₇	1	2	yes	—
182	C₂×C₄	M₃₂×C₁₂₇	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
184	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
185	C₈	—	2	2	no	—
186	C₈	—	2	1	no	—
187	C₈	—	4	2	no	—
188	C₈	#100×C₁₂₇	1	2	yes	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	—	4	4	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
197	C₂²	C₂²×C₁₆×C₁₂₇	1	1	yes	characteristic
198	C₂²	C₂×M₃₂×C₁₂₇	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	1	no	—
201	C₄	—	4	2	no	—
202	C₄	C₂²⋊C₁₆×C₁₂₇	1	4	yes	—
203	C₄	#410×C₁₂₇	1	1	yes	characteristic
204	C₄	C₂×#100×C₁₂₇	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	4	4	no	—
208	C₂	C₁₂₇×#1460	1	1	yes	characteristic
209	C₂	C₂×C₂²⋊C₁₆×C₁₂₇	1	1	yes	characteristic
210	C₂	C₁₂₇×#2115	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₁₆×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₁₂₇	C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₂₇×#2162	C₂	semidirect
3. C₁₂₇×#1314	C₂	semidirect
35. C₂×D₄×C₁₂₇	C₁₆	semidirect
99. #11487	C₁₂₇	direct (N × H)
107. #2162	C₂×C₁₂₇	semidirect
108. #1314	C₂×C₁₂₇	semidirect
109. C₁₂₇	#11487	direct (N × H)
141. C₂×D₄	C₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₂₇	2
C₁₂₇×#2162	2
C₁₂₇×#1314	2
#11487	127

Subgroups — C127×#11487 #44575160

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#11487 #44575160