Subgroups — C₁₂₇×#20834 #44584507

Order 32512.

1646 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#424×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₂×#407×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#2280	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1850	C₂	1	8	yes	maximal, complemented
6	C₁₂₇×#1729	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₂₇	—	2	4	no	—
11	C₂³×C₈×C₁₂₇	—	2	8	no	—
12	C₂³×C₈×C₁₂₇	C₂²	1	4	yes	—
13	C₂³×C₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
15	#424×C₁₂₇	—	2	8	no	—
16	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	8	yes	complemented
17	#407×C₁₂₇	—	2	4	no	—
18	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	—
19	#381×C₁₂₇	C₄	1	2	yes	—
20	#380×C₁₂₇	C₄	1	4	yes	—
21	#379×C₁₂₇	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	—
23	#336×C₁₂₇	C₂²	1	8	yes	complemented
24	C₂⁵×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₂₇	—	2	4	no	—
26	C₂³×C₄×C₁₂₇	—	2	4	no	—
27	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₂₇	D₄	1	4	yes	—
30	C₂³×C₄×C₁₂₇	Q₈	1	4	yes	—
31	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
32	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₁₂₇	—	2	2	no	—
34	C₂×M₁₆×C₁₂₇	—	2	2	no	—
35	C₂×M₁₆×C₁₂₇	—	2	4	no	—
36	C₂×M₁₆×C₁₂₇	—	2	4	no	—
37	C₂²×C₈×C₁₂₇	—	2	32	no	—
38	C₂²×C₈×C₁₂₇	—	2	8	no	—
39	C₂²×C₈×C₁₂₇	—	2	4	no	—
40	C₂²×C₈×C₁₂₇	—	2	4	no	—
41	C₂²×C₈×C₁₂₇	—	2	16	no	—
42	C₂²×C₈×C₁₂₇	D₄	1	8	yes	—
43	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
46	#117×C₁₂₇	—	2	4	no	—
47	#117×C₁₂₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
49	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
50	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
51	C₂×C₄²×C₁₂₇	C₂³	1	4	yes	—
52	C₄⋊C₈×C₁₂₇	—	2	16	no	—
53	C₂²⋊C₈×C₁₂₇	—	2	8	no	—
54	#95×C₁₂₇	C₂×C₄	1	8	yes	—
55	C₂⁴×C₁₂₇	—	2	4	no	—
56	C₂⁴×C₁₂₇	—	2	8	no	—
57	C₂⁴×C₁₂₇	C₄⋊C₄	1	4	yes	—
58	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
59	C₂⁴×C₁₂₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₂₇	—	2	4	no	—
61	C₂²×C₄×C₁₂₇	—	2	8	no	—
62	C₂²×C₄×C₁₂₇	—	2	8	no	—
63	C₂²×C₄×C₁₂₇	—	2	16	no	—
64	C₂²×C₄×C₁₂₇	—	2	8	no	—
65	C₂²×C₄×C₁₂₇	—	2	4	no	—
66	C₂²×C₄×C₁₂₇	—	2	4	no	—
67	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
69	C₂²×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
76	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₁₂₇	#41	1	4	yes	—
78	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₁₂₇	—	4	4	no	—
80	M₁₆×C₁₂₇	—	4	4	no	—
81	C₂×C₈×C₁₂₇	—	2	32	no	—
82	C₂×C₈×C₁₂₇	—	2	16	no	—
83	C₂×C₈×C₁₂₇	—	2	4	no	—
84	C₂×C₈×C₁₂₇	—	2	4	no	—
85	C₂×C₈×C₁₂₇	—	2	2	no	—
86	C₂×C₈×C₁₂₇	—	2	2	no	—
87	C₂×C₈×C₁₂₇	—	2	4	no	—
88	C₂×C₈×C₁₂₇	—	2	4	no	—
89	C₂×C₈×C₁₂₇	C₂×D₄	1	8	yes	—
90	C₄⋊C₄×C₁₂₇	—	2	8	no	—
91	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
92	C₂²⋊C₄×C₁₂₇	—	2	16	no	—
93	C₄²×C₁₂₇	—	2	4	no	—
94	C₄²×C₁₂₇	—	2	4	no	—
95	C₂³×C₁₂₇	—	2	32	no	—
96	C₂³×C₁₂₇	—	2	4	no	—
97	C₂³×C₁₂₇	—	2	4	no	—
98	C₂³×C₁₂₇	—	2	4	no	—
99	C₂³×C₁₂₇	—	2	8	no	—
100	C₂³×C₁₂₇	—	2	16	no	—
101	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
102	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
103	C₂³×C₁₂₇	C₄×D₄	1	4	yes	—
104	C₂³×C₁₂₇	#131	1	8	yes	—
105	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	—	2	4	no	—
107	C₂×C₄×C₁₂₇	—	2	8	no	—
108	C₂×C₄×C₁₂₇	—	2	4	no	—
109	C₂×C₄×C₁₂₇	—	2	8	no	—
110	C₂×C₄×C₁₂₇	—	2	4	no	—
111	C₂×C₄×C₁₂₇	—	2	16	no	—
112	C₂×C₄×C₁₂₇	—	2	8	no	—
113	C₂×C₄×C₁₂₇	—	2	16	no	—
114	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
115	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
117	C₂×C₄×C₁₂₇	#120	1	4	yes	—
118	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	8	yes	—
119	C₂×C₄×C₁₂₇	#123	1	4	yes	—
120	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
121	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
124	C₈×C₁₂₇	—	2	8	no	—
125	C₈×C₁₂₇	—	4	2	no	—
126	C₈×C₁₂₇	—	4	2	no	—
127	C₂²×C₁₂₇	—	2	32	no	—
128	C₂²×C₁₂₇	—	2	4	no	—
129	C₂²×C₁₂₇	—	2	4	no	—
130	C₂²×C₁₂₇	—	2	8	no	—
131	C₂²×C₁₂₇	—	2	16	no	—
132	C₂²×C₁₂₇	—	2	4	no	—
133	C₂²×C₁₂₇	#385	1	4	yes	—
134	C₂²×C₁₂₇	#424	1	8	yes	complemented
135	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
137	C₂²×C₁₂₇	C₂×#131	1	4	yes	—
138	C₄×C₁₂₇	—	2	4	no	—
139	C₄×C₁₂₇	—	4	8	no	—
140	C₄×C₁₂₇	#385	1	4	yes	—
141	C₄×C₁₂₇	C₂×#120	1	1	yes	characteristic
142	C₄×C₁₂₇	C₂×C₂²⋊Q₈	1	2	yes	—
143	C₄×C₁₂₇	C₂×#123	1	1	yes	characteristic
144	#20834	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₁₂₇	—	2	4	no	—
146	C₂×C₁₂₇	—	2	8	no	—
147	C₂×C₁₂₇	C₂×#385	1	1	yes	characteristic
148	C₂×C₁₂₇	C₂×#424	1	2	yes	—
149	C₂×C₁₂₇	#2929	1	4	yes	—
150	C₂⁴×C₈	C₂×C₁₂₇	1	1	yes	characteristic
151	C₂×#424	C₂×C₁₂₇	1	2	yes	complemented
152	C₂×#407	C₂×C₁₂₇	1	1	yes	characteristic
153	#2280	C₂×C₁₂₇	1	1	yes	characteristic
154	#1850	C₂×C₁₂₇	1	8	yes	complemented
155	#1729	C₂×C₁₂₇	1	2	yes	complemented
156	C₁₂₇	#20834	1	1	yes	characteristic, direct product factor, Sylow
157	C₂⁴×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
158	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
159	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
160	C₂³×C₈	—	2	4	no	—
161	C₂³×C₈	—	2	8	no	—
162	C₂³×C₈	C₂²×C₁₂₇	1	4	yes	—
163	C₂³×C₈	C₂²×C₁₂₇	1	2	yes	—
164	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
165	#424	—	2	8	no	—
166	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	8	yes	complemented
167	#407	—	2	4	no	—
168	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	4	yes	—
169	#381	C₄×C₁₂₇	1	2	yes	—
170	#380	C₄×C₁₂₇	1	4	yes	—
171	#379	C₄×C₁₂₇	1	2	yes	—
172	C₄×C₂²⋊C₄	C₂²×C₁₂₇	1	4	yes	—
173	#336	C₂²×C₁₂₇	1	8	yes	complemented
174	C₂⁵	C₂×C₄×C₁₂₇	1	1	yes	characteristic
175	C₂³×C₄	—	2	4	no	—
176	C₂³×C₄	—	2	4	no	—
177	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
178	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
179	C₂³×C₄	D₄×C₁₂₇	1	4	yes	—
180	C₂³×C₄	Q₈×C₁₂₇	1	4	yes	—
181	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
182	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	4	no	—
186	C₂×M₁₆	—	2	4	no	—
187	C₂²×C₈	—	2	32	no	—
188	C₂²×C₈	—	2	8	no	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	—	2	16	no	—
192	C₂²×C₈	D₄×C₁₂₇	1	8	yes	—
193	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
194	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
195	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
196	#117	—	2	4	no	—
197	#117	—	2	4	no	—
198	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
199	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
200	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
201	C₂×C₄²	C₂³×C₁₂₇	1	4	yes	—
202	C₄⋊C₈	—	2	16	no	—
203	C₂²⋊C₈	—	2	8	no	—
204	#95	C₂×C₄×C₁₂₇	1	8	yes	—
205	C₂⁴	—	2	4	no	—
206	C₂⁴	—	2	8	no	—
207	C₂⁴	C₄⋊C₄×C₁₂₇	1	4	yes	—
208	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
209	C₂⁴	C₂²×C₄×C₁₂₇	1	2	yes	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	2	16	no	—
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
218	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
219	C₂²×C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
220	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
223	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
225	C₂²×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
226	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
227	C₂²×C₄	#41×C₁₂₇	1	4	yes	—
228	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
229	M₁₆	—	4	4	no	—
230	M₁₆	—	4	4	no	—
231	C₂×C₈	—	2	32	no	—
232	C₂×C₈	—	2	16	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂×D₄×C₁₂₇	1	8	yes	—
240	C₄⋊C₄	—	2	8	no	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	2	16	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₂³	—	2	32	no	—
246	C₂³	—	2	4	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	8	no	—
250	C₂³	—	2	16	no	—
251	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
252	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
253	C₂³	C₄×D₄×C₁₂₇	1	4	yes	—
254	C₂³	#131×C₁₂₇	1	8	yes	—
255	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	16	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	16	no	—
264	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
265	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
266	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
267	C₂×C₄	#120×C₁₂₇	1	4	yes	—
268	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	8	yes	—
269	C₂×C₄	#123×C₁₂₇	1	4	yes	—
270	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
271	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
272	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
273	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
274	C₈	—	2	8	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	2	no	—
277	C₂²	—	2	32	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	8	no	—
281	C₂²	—	2	16	no	—
282	C₂²	—	2	4	no	—
283	C₂²	#385×C₁₂₇	1	4	yes	—
284	C₂²	#424×C₁₂₇	1	8	yes	complemented
285	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
286	C₂²	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
287	C₂²	C₂×#131×C₁₂₇	1	4	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	4	8	no	—
290	C₄	#385×C₁₂₇	1	4	yes	—
291	C₄	C₂×#120×C₁₂₇	1	1	yes	characteristic
292	C₄	C₂×C₂²⋊Q₈×C₁₂₇	1	2	yes	—
293	C₄	C₂×#123×C₁₂₇	1	1	yes	characteristic
294	C₂	—	2	4	no	—
295	C₂	—	2	8	no	—
296	C₂	C₂×#385×C₁₂₇	1	1	yes	characteristic
297	C₂	C₂×#424×C₁₂₇	1	2	yes	—
298	C₂	C₁₂₇×#2929	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂³×C₄
prime radical	C₂⁵×C₁₂₇	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#424×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1850	C₂	semidirect
6. C₁₂₇×#1729	C₂	semidirect
16. C₂×C₄⋊C₈×C₁₂₇	C₂²	semidirect
23. #336×C₁₂₇	C₂²	semidirect
134. C₂²×C₁₂₇	#424	semidirect
144. #20834	C₁₂₇	direct (N × H)
151. C₂×#424	C₂×C₁₂₇	semidirect
154. #1850	C₂×C₁₂₇	semidirect
155. #1729	C₂×C₁₂₇	semidirect
156. C₁₂₇	#20834	direct (N × H)
166. C₂×C₄⋊C₈	C₂²×C₁₂₇	semidirect
173. #336	C₂²×C₁₂₇	semidirect
284. C₂²	#424×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₈×C₁₂₇	2
C₂×#424×C₁₂₇	2
C₂×#407×C₁₂₇	2
C₁₂₇×#2280	2
C₁₂₇×#1850	2
C₁₂₇×#1729	2
#20834	127

Subgroups — C127×#20834 #44584507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20834 #44584507