Subgroups — C₁₂₇×#19648 #44583321

Order 32512.

664 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3231	C₂	1	2	yes	maximal
2	C₁₂₇×#3185	C₂	1	2	yes	maximal
3	C₁₂₇×#3170	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#3160	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1718	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1716	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1713	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₂₇×#1710	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₂₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₂₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
12	#533×C₁₂₇	C₂²	1	2	yes	—
13	#533×C₁₂₇	C₂²	1	2	yes	—
14	#500×C₁₂₇	C₂²	1	1	yes	characteristic
15	#500×C₁₂₇	C₂²	1	1	yes	characteristic
16	#500×C₁₂₇	C₂²	1	1	yes	characteristic
17	#500×C₁₂₇	C₂²	1	2	yes	—
18	#499×C₁₂₇	—	2	2	no	—
19	#498×C₁₂₇	C₂²	1	2	yes	—
20	#498×C₁₂₇	C₂²	1	2	yes	—
21	#498×C₁₂₇	C₂²	1	2	yes	—
22	#494×C₁₂₇	C₂²	1	1	yes	characteristic
23	#494×C₁₂₇	C₂²	1	2	yes	—
24	#487×C₁₂₇	C₂²	1	2	yes	—
25	#487×C₁₂₇	C₂²	1	2	yes	—
26	#487×C₁₂₇	C₂²	1	2	yes	—
27	#487×C₁₂₇	C₂²	1	2	yes	—
28	#487×C₁₂₇	C₂²	1	2	yes	—
29	#485×C₁₂₇	—	2	4	no	—
30	#479×C₁₂₇	—	2	4	no	—
31	#470×C₁₂₇	—	2	4	no	—
32	#467×C₁₂₇	—	2	2	no	—
33	#441×C₁₂₇	—	2	4	no	—
34	C₄×Q₁₆×C₁₂₇	—	2	2	no	—
35	#431×C₁₂₇	C₂²	1	1	yes	characteristic
36	#427×C₁₂₇	—	2	2	no	—
37	C₄×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
39	#322×C₁₂₇	C₂²	1	1	yes	characteristic
40	#322×C₁₂₇	C₂²	1	1	yes	characteristic
41	C₈²×C₁₂₇	C₂²	1	1	yes	characteristic
42	C₈²×C₁₂₇	C₂²	1	1	yes	characteristic
43	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
44	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
45	C₂×M₁₆×C₁₂₇	D₄	1	2	yes	—
46	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
47	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
51	#125×C₁₂₇	—	2	2	no	—
52	#125×C₁₂₇	—	2	2	no	—
53	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
54	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
55	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
56	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
57	C₄×Q₈×C₁₂₇	—	2	2	no	—
58	C₄×Q₈×C₁₂₇	—	2	2	no	—
59	C₄×Q₈×C₁₂₇	—	2	2	no	—
60	C₄×Q₈×C₁₂₇	—	2	2	no	—
61	#117×C₁₂₇	—	2	2	no	—
62	#117×C₁₂₇	—	2	2	no	—
63	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
64	#107×C₁₂₇	—	2	2	no	—
65	#107×C₁₂₇	—	2	2	no	—
66	#107×C₁₂₇	—	2	2	no	—
67	#107×C₁₂₇	—	2	4	no	—
68	#106×C₁₂₇	—	2	4	no	—
69	#106×C₁₂₇	—	2	4	no	—
70	#106×C₁₂₇	—	2	4	no	—
71	C₄⋊C₈×C₁₂₇	D₄	1	2	yes	—
72	#103×C₁₂₇	—	2	4	no	—
73	#103×C₁₂₇	—	2	4	no	—
74	#103×C₁₂₇	—	2	4	no	—
75	#103×C₁₂₇	—	2	4	no	—
76	#103×C₁₂₇	—	2	4	no	—
77	C₂²⋊C₈×C₁₂₇	D₄	1	2	yes	—
78	#97×C₁₂₇	D₄	1	2	yes	—
79	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
80	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₁₂₇	—	2	2	no	—
87	C₂×Q₈×C₁₂₇	—	2	2	no	—
88	C₂×Q₈×C₁₂₇	—	2	2	no	—
89	C₂×Q₈×C₁₂₇	—	2	2	no	—
90	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
93	Q₁₆×C₁₂₇	—	4	8	no	—
94	Q₁₆×C₁₂₇	—	4	4	no	—
95	M₁₆×C₁₂₇	—	2	4	no	—
96	C₂×C₈×C₁₂₇	—	2	4	no	—
97	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
103	C₄⋊C₄×C₁₂₇	—	2	2	no	—
104	C₄⋊C₄×C₁₂₇	—	2	2	no	—
105	C₄⋊C₄×C₁₂₇	—	2	2	no	—
106	C₄⋊C₄×C₁₂₇	—	2	2	no	—
107	C₄⋊C₄×C₁₂₇	—	2	2	no	—
108	C₄⋊C₄×C₁₂₇	—	2	2	no	—
109	C₄⋊C₄×C₁₂₇	—	2	2	no	—
110	C₄⋊C₄×C₁₂₇	—	2	2	no	—
111	C₄⋊C₄×C₁₂₇	—	4	2	no	—
112	C₄⋊C₄×C₁₂₇	—	4	2	no	—
113	C₄⋊C₄×C₁₂₇	—	4	2	no	—
114	C₄⋊C₄×C₁₂₇	—	4	2	no	—
115	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
116	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
117	C₄²×C₁₂₇	—	4	2	no	—
118	C₄²×C₁₂₇	—	4	2	no	—
119	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
121	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
123	C₂³×C₁₂₇	#127	1	1	yes	characteristic
124	Q₈×C₁₂₇	—	4	4	no	—
125	Q₈×C₁₂₇	—	4	4	no	—
126	Q₈×C₁₂₇	—	4	4	no	—
127	Q₈×C₁₂₇	—	4	4	no	—
128	C₂×C₄×C₁₂₇	—	2	2	no	—
129	C₂×C₄×C₁₂₇	—	2	2	no	—
130	C₂×C₄×C₁₂₇	—	2	2	no	—
131	C₂×C₄×C₁₂₇	—	4	2	no	—
132	C₂×C₄×C₁₂₇	—	4	2	no	—
133	C₂×C₄×C₁₂₇	—	4	2	no	—
134	C₂×C₄×C₁₂₇	—	4	2	no	—
135	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
136	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
137	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
138	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₁₂₇	—	2	2	no	—
142	C₈×C₁₂₇	—	2	2	no	—
143	C₈×C₁₂₇	—	2	2	no	—
144	C₈×C₁₂₇	—	2	2	no	—
145	C₈×C₁₂₇	#135	1	4	yes	—
146	C₈×C₁₂₇	#137	1	4	yes	—
147	C₂²×C₁₂₇	—	2	1	no	—
148	C₂²×C₁₂₇	—	2	1	no	—
149	C₂²×C₁₂₇	—	2	1	no	—
150	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
151	C₄×C₁₂₇	—	2	1	no	—
152	C₄×C₁₂₇	—	2	1	no	—
153	C₄×C₁₂₇	—	2	1	no	—
154	C₄×C₁₂₇	—	8	2	no	—
155	C₄×C₁₂₇	—	8	2	no	—
156	C₄×C₁₂₇	—	8	2	no	—
157	C₄×C₁₂₇	—	8	2	no	—
158	C₄×C₁₂₇	C₂×#135	1	2	yes	—
159	C₄×C₁₂₇	C₂×#137	1	2	yes	—
160	C₄×C₁₂₇	#578	1	2	yes	—
161	#19648	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂×C₁₂₇	—	4	1	no	—
163	C₂×C₁₂₇	#3147	1	1	yes	characteristic
164	C₂×C₁₂₇	#3152	1	1	yes	characteristic
165	C₂×C₁₂₇	#3156	1	1	yes	characteristic
166	#3231	C₂×C₁₂₇	1	2	yes	—
167	#3185	C₂×C₁₂₇	1	2	yes	—
168	#3170	C₂×C₁₂₇	1	1	yes	characteristic
169	#3160	C₂×C₁₂₇	1	1	yes	characteristic
170	#1718	C₂×C₁₂₇	1	2	yes	complemented
171	#1716	C₂×C₁₂₇	1	2	yes	complemented
172	#1713	C₂×C₁₂₇	1	1	yes	characteristic, complemented
173	#1710	C₂×C₁₂₇	1	1	yes	characteristic, complemented
174	#1707	C₂×C₁₂₇	1	1	yes	characteristic, complemented
175	#1707	C₂×C₁₂₇	1	1	yes	characteristic, complemented
176	C₈×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
177	C₁₂₇	#19648	1	1	yes	characteristic, direct product factor, Sylow
178	#533	C₂²×C₁₂₇	1	2	yes	—
179	#533	C₂²×C₁₂₇	1	2	yes	—
180	#500	C₂²×C₁₂₇	1	1	yes	characteristic
181	#500	C₂²×C₁₂₇	1	1	yes	characteristic
182	#500	C₂²×C₁₂₇	1	1	yes	characteristic
183	#500	C₂²×C₁₂₇	1	2	yes	—
184	#499	—	2	2	no	—
185	#498	C₂²×C₁₂₇	1	2	yes	—
186	#498	C₂²×C₁₂₇	1	2	yes	—
187	#498	C₂²×C₁₂₇	1	2	yes	—
188	#494	C₂²×C₁₂₇	1	1	yes	characteristic
189	#494	C₂²×C₁₂₇	1	2	yes	—
190	#487	C₂²×C₁₂₇	1	2	yes	—
191	#487	C₂²×C₁₂₇	1	2	yes	—
192	#487	C₂²×C₁₂₇	1	2	yes	—
193	#487	C₂²×C₁₂₇	1	2	yes	—
194	#487	C₂²×C₁₂₇	1	2	yes	—
195	#485	—	2	4	no	—
196	#479	—	2	4	no	—
197	#470	—	2	4	no	—
198	#467	—	2	2	no	—
199	#441	—	2	4	no	—
200	C₄×Q₁₆	—	2	2	no	—
201	#431	C₂²×C₁₂₇	1	1	yes	characteristic
202	#427	—	2	2	no	—
203	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
205	#322	C₂²×C₁₂₇	1	1	yes	characteristic
206	#322	C₂²×C₁₂₇	1	1	yes	characteristic
207	C₈²	C₂²×C₁₂₇	1	1	yes	characteristic
208	C₈²	C₂²×C₁₂₇	1	1	yes	characteristic
209	C₂×Q₁₆	—	2	2	no	—
210	C₂×Q₁₆	—	2	4	no	—
211	C₂×M₁₆	D₄×C₁₂₇	1	2	yes	—
212	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
214	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
215	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
216	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
217	#125	—	2	2	no	—
218	#125	—	2	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	—	2	2	no	—
221	C₂²⋊Q₈	—	2	2	no	—
222	C₂²⋊Q₈	—	2	2	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	#117	—	2	2	no	—
228	#117	—	2	2	no	—
229	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
230	#107	—	2	2	no	—
231	#107	—	2	2	no	—
232	#107	—	2	2	no	—
233	#107	—	2	4	no	—
234	#106	—	2	4	no	—
235	#106	—	2	4	no	—
236	#106	—	2	4	no	—
237	C₄⋊C₈	D₄×C₁₂₇	1	2	yes	—
238	#103	—	2	4	no	—
239	#103	—	2	4	no	—
240	#103	—	2	4	no	—
241	#103	—	2	4	no	—
242	#103	—	2	4	no	—
243	C₂²⋊C₈	D₄×C₁₂₇	1	2	yes	—
244	#97	D₄×C₁₂₇	1	2	yes	—
245	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
246	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
251	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
259	Q₁₆	—	4	8	no	—
260	Q₁₆	—	4	4	no	—
261	M₁₆	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
265	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
266	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
267	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	—	4	2	no	—
285	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
288	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
289	C₂³	#127×C₁₂₇	1	1	yes	characteristic
290	Q₈	—	4	4	no	—
291	Q₈	—	4	4	no	—
292	Q₈	—	4	4	no	—
293	Q₈	—	4	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
302	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
303	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
307	C₈	—	2	2	no	—
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	2	2	no	—
311	C₈	#135×C₁₂₇	1	4	yes	—
312	C₈	#137×C₁₂₇	1	4	yes	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	C₂×#135×C₁₂₇	1	2	yes	—
325	C₄	C₂×#137×C₁₂₇	1	2	yes	—
326	C₄	#578×C₁₂₇	1	2	yes	—
327	C₂	—	4	1	no	—
328	C₂	C₁₂₇×#3147	1	1	yes	characteristic
329	C₂	C₁₂₇×#3152	1	1	yes	characteristic
330	C₂	C₁₂₇×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₂³×C₁₂₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₂₇×#1718	C₂	semidirect
6. C₁₂₇×#1716	C₂	semidirect
7. C₁₂₇×#1713	C₂	semidirect
8. C₁₂₇×#1710	C₂	semidirect
9. C₁₂₇×#1707	C₂	semidirect
10. C₁₂₇×#1707	C₂	semidirect
161. #19648	C₁₂₇	direct (N × H)
170. #1718	C₂×C₁₂₇	semidirect
171. #1716	C₂×C₁₂₇	semidirect
172. #1713	C₂×C₁₂₇	semidirect
173. #1710	C₂×C₁₂₇	semidirect
174. #1707	C₂×C₁₂₇	semidirect
175. #1707	C₂×C₁₂₇	semidirect
177. C₁₂₇	#19648	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#3231	2
C₁₂₇×#3185	2
C₁₂₇×#3170	2
C₁₂₇×#3160	2
C₁₂₇×#1718	2
C₁₂₇×#1716	2
C₁₂₇×#1713	2
C₁₂₇×#1710	2
C₁₂₇×#1707	2
C₈×M₁₆×C₁₂₇	2
#19648	127

Subgroups — C127×#19648 #44583321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19648 #44583321