Subgroups — C₁₂₇×#19687 #44583360

Order 32512.

764 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3228	C₂	1	2	yes	maximal, complemented
2	C₁₂₇×#3182	C₂	1	2	yes	maximal, complemented
3	C₁₂₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#3147	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1723	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1722	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1717	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1714	C₂	1	2	yes	maximal, complemented
9	C₁₂₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₁₂₇	—	2	2	no	—
11	#533×C₁₂₇	C₂²	1	2	yes	—
12	#532×C₁₂₇	C₂²	1	2	yes	—
13	#500×C₁₂₇	C₂²	1	1	yes	characteristic
14	#500×C₁₂₇	C₂²	1	2	yes	—
15	#500×C₁₂₇	C₂²	1	2	yes	—
16	#498×C₁₂₇	C₂²	1	2	yes	—
17	#495×C₁₂₇	—	2	2	no	—
18	#494×C₁₂₇	C₂²	1	1	yes	characteristic
19	#494×C₁₂₇	C₂²	1	1	yes	characteristic
20	#493×C₁₂₇	C₂²	1	1	yes	characteristic
21	#492×C₁₂₇	C₂²	1	2	yes	—
22	#492×C₁₂₇	C₂²	1	2	yes	—
23	#487×C₁₂₇	C₂²	1	2	yes	—
24	#487×C₁₂₇	C₂²	1	2	yes	—
25	#486×C₁₂₇	C₂²	1	2	yes	—
26	#486×C₁₂₇	C₂²	1	2	yes	—
27	#486×C₁₂₇	C₂²	1	2	yes	—
28	#482×C₁₂₇	—	2	4	no	—
29	#478×C₁₂₇	—	2	4	no	—
30	#468×C₁₂₇	—	2	4	no	—
31	#467×C₁₂₇	—	2	2	no	—
32	#440×C₁₂₇	—	2	4	no	—
33	C₄×Q₁₆×C₁₂₇	—	2	2	no	—
34	#431×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₄×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
37	#322×C₁₂₇	C₂²	1	2	yes	—
38	#322×C₁₂₇	C₂²	1	2	yes	—
39	C₂×#41×C₁₂₇	—	2	2	no	—
40	#135×C₁₂₇	—	4	8	no	—
41	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
42	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
43	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
44	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
45	C₂×D₈×C₁₂₇	—	2	2	no	—
46	C₂×M₁₆×C₁₂₇	D₄	1	2	yes	—
47	C₂²×C₈×C₁₂₇	D₄	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	#127×C₁₂₇	C₂³	1	2	yes	—
52	#125×C₁₂₇	—	2	2	no	—
53	#124×C₁₂₇	—	2	2	no	—
54	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
55	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
56	#121×C₁₂₇	—	2	2	no	—
57	#121×C₁₂₇	—	2	2	no	—
58	C₄×Q₈×C₁₂₇	—	2	2	no	—
59	C₄×Q₈×C₁₂₇	—	2	2	no	—
60	C₄×D₄×C₁₂₇	—	2	2	no	—
61	C₄×D₄×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	4	no	—
66	#107×C₁₂₇	—	2	4	no	—
67	#106×C₁₂₇	—	2	4	no	—
68	C₄⋊C₈×C₁₂₇	D₄	1	2	yes	—
69	#103×C₁₂₇	—	2	4	no	—
70	#103×C₁₂₇	—	2	4	no	—
71	#102×C₁₂₇	—	2	4	no	—
72	#102×C₁₂₇	—	2	4	no	—
73	#102×C₁₂₇	—	2	4	no	—
74	C₂²⋊C₈×C₁₂₇	D₄	1	2	yes	—
75	#97×C₁₂₇	D₄	1	2	yes	—
76	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
77	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
78	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
80	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
81	#41×C₁₂₇	—	4	8	no	—
82	C₂×Q₈×C₁₂₇	—	2	2	no	—
83	C₂×Q₈×C₁₂₇	—	2	2	no	—
84	C₂×Q₈×C₁₂₇	—	2	2	no	—
85	C₂×D₄×C₁₂₇	—	2	2	no	—
86	C₂×D₄×C₁₂₇	—	2	2	no	—
87	C₂×D₄×C₁₂₇	—	2	2	no	—
88	C₂×D₄×C₁₂₇	—	4	2	no	—
89	C₂²×C₄×C₁₂₇	—	4	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	4	no	—
94	Q₁₆×C₁₂₇	—	4	4	no	—
95	SD₁₆×C₁₂₇	—	4	8	no	—
96	SD₁₆×C₁₂₇	—	4	8	no	—
97	D₈×C₁₂₇	—	4	4	no	—
98	M₁₆×C₁₂₇	—	4	2	no	—
99	C₂×C₈×C₁₂₇	—	2	4	no	—
100	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
102	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
103	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
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	—	4	2	no	—
111	C₄⋊C₄×C₁₂₇	—	4	2	no	—
112	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
113	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
114	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
115	C₄²×C₁₂₇	—	4	2	no	—
116	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
120	C₂³×C₁₂₇	—	4	2	no	—
121	C₂³×C₁₂₇	#127	1	1	yes	characteristic
122	Q₈×C₁₂₇	—	4	4	no	—
123	Q₈×C₁₂₇	—	4	4	no	—
124	Q₈×C₁₂₇	—	4	4	no	—
125	D₄×C₁₂₇	—	4	4	no	—
126	D₄×C₁₂₇	—	4	4	no	—
127	D₄×C₁₂₇	—	4	4	no	—
128	D₄×C₁₂₇	—	8	4	no	—
129	C₂×C₄×C₁₂₇	—	2	2	no	—
130	C₂×C₄×C₁₂₇	—	2	2	no	—
131	C₂×C₄×C₁₂₇	—	2	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	—	8	4	no	—
136	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
137	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
138	C₂×C₄×C₁₂₇	#127	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	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₁₂₇	—	2	2	no	—
143	C₈×C₁₂₇	—	2	4	no	—
144	C₈×C₁₂₇	—	2	4	no	—
145	C₈×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	—
146	C₂²×C₁₂₇	—	2	1	no	—
147	C₂²×C₁₂₇	—	2	1	no	—
148	C₂²×C₁₂₇	—	2	1	no	—
149	C₂²×C₁₂₇	—	8	2	no	—
150	C₂²×C₁₂₇	—	8	2	no	—
151	C₂²×C₁₂₇	—	8	2	no	—
152	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
153	C₄×C₁₂₇	—	2	1	no	—
154	C₄×C₁₂₇	—	2	1	no	—
155	C₄×C₁₂₇	—	2	1	no	—
156	C₄×C₁₂₇	—	8	2	no	—
157	C₄×C₁₂₇	—	8	2	no	—
158	C₄×C₁₂₇	—	8	2	no	—
159	C₄×C₁₂₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
160	C₄×C₁₂₇	#575	1	2	yes	—
161	C₄×C₁₂₇	#578	1	2	yes	—
162	#19687	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₁₂₇	—	4	1	no	—
164	C₂×C₁₂₇	—	16	2	no	—
165	C₂×C₁₂₇	#3150	1	1	yes	characteristic
166	C₂×C₁₂₇	#3151	1	1	yes	characteristic
167	C₂×C₁₂₇	#3156	1	1	yes	characteristic
168	#3228	C₂×C₁₂₇	1	2	yes	complemented
169	#3182	C₂×C₁₂₇	1	2	yes	complemented
170	#3170	C₂×C₁₂₇	1	1	yes	characteristic, complemented
171	#3147	C₂×C₁₂₇	1	1	yes	characteristic
172	#1723	C₂×C₁₂₇	1	2	yes	complemented
173	#1722	C₂×C₁₂₇	1	2	yes	complemented
174	#1717	C₂×C₁₂₇	1	2	yes	complemented
175	#1714	C₂×C₁₂₇	1	2	yes	complemented
176	#1451	C₂×C₁₂₇	1	1	yes	characteristic, complemented
177	C₁₂₇	#19687	1	1	yes	characteristic, direct product factor, Sylow
178	C₂×#135	—	2	2	no	—
179	#533	C₂²×C₁₂₇	1	2	yes	—
180	#532	C₂²×C₁₂₇	1	2	yes	—
181	#500	C₂²×C₁₂₇	1	1	yes	characteristic
182	#500	C₂²×C₁₂₇	1	2	yes	—
183	#500	C₂²×C₁₂₇	1	2	yes	—
184	#498	C₂²×C₁₂₇	1	2	yes	—
185	#495	—	2	2	no	—
186	#494	C₂²×C₁₂₇	1	1	yes	characteristic
187	#494	C₂²×C₁₂₇	1	1	yes	characteristic
188	#493	C₂²×C₁₂₇	1	1	yes	characteristic
189	#492	C₂²×C₁₂₇	1	2	yes	—
190	#492	C₂²×C₁₂₇	1	2	yes	—
191	#487	C₂²×C₁₂₇	1	2	yes	—
192	#487	C₂²×C₁₂₇	1	2	yes	—
193	#486	C₂²×C₁₂₇	1	2	yes	—
194	#486	C₂²×C₁₂₇	1	2	yes	—
195	#486	C₂²×C₁₂₇	1	2	yes	—
196	#482	—	2	4	no	—
197	#478	—	2	4	no	—
198	#468	—	2	4	no	—
199	#467	—	2	2	no	—
200	#440	—	2	4	no	—
201	C₄×Q₁₆	—	2	2	no	—
202	#431	C₂²×C₁₂₇	1	1	yes	characteristic
203	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
204	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
205	#322	C₂²×C₁₂₇	1	2	yes	—
206	#322	C₂²×C₁₂₇	1	2	yes	—
207	C₂×#41	—	2	2	no	—
208	#135	—	4	8	no	—
209	C₂×Q₁₆	—	2	2	no	—
210	C₂×Q₁₆	—	2	2	no	—
211	C₂×SD₁₆	—	2	4	no	—
212	C₂×SD₁₆	—	2	4	no	—
213	C₂×D₈	—	2	2	no	—
214	C₂×M₁₆	D₄×C₁₂₇	1	2	yes	—
215	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
216	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
217	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
218	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
219	#127	C₂³×C₁₂₇	1	2	yes	—
220	#125	—	2	2	no	—
221	#124	—	2	2	no	—
222	C₂²⋊Q₈	—	2	2	no	—
223	C₂²⋊Q₈	—	2	2	no	—
224	#121	—	2	2	no	—
225	#121	—	2	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	—	2	2	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	#117	—	2	2	no	—
231	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
232	#107	—	2	2	no	—
233	#107	—	2	4	no	—
234	#107	—	2	4	no	—
235	#106	—	2	4	no	—
236	C₄⋊C₈	D₄×C₁₂₇	1	2	yes	—
237	#103	—	2	4	no	—
238	#103	—	2	4	no	—
239	#102	—	2	4	no	—
240	#102	—	2	4	no	—
241	#102	—	2	4	no	—
242	C₂²⋊C₈	D₄×C₁₂₇	1	2	yes	—
243	#97	D₄×C₁₂₇	1	2	yes	—
244	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
245	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
248	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
249	#41	—	4	8	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
261	Q₁₆	—	4	4	no	—
262	Q₁₆	—	4	4	no	—
263	SD₁₆	—	4	8	no	—
264	SD₁₆	—	4	8	no	—
265	D₈	—	4	4	no	—
266	M₁₆	—	4	2	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
269	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
270	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
271	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
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	—	2	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
287	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
288	C₂³	—	4	2	no	—
289	C₂³	#127×C₁₂₇	1	1	yes	characteristic
290	Q₈	—	4	4	no	—
291	Q₈	—	4	4	no	—
292	Q₈	—	4	4	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	4	no	—
295	D₄	—	4	4	no	—
296	D₄	—	8	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
305	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
306	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	2	4	no	—
312	C₈	—	2	4	no	—
313	C₈	AGL(1,ℤ/8)×C₁₂₇	1	4	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	—	8	2	no	—
320	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	8	2	no	—
327	C₄	C₂×AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
328	C₄	#575×C₁₂₇	1	2	yes	—
329	C₄	#578×C₁₂₇	1	2	yes	—
330	C₂	—	4	1	no	—
331	C₂	—	16	2	no	—
332	C₂	C₁₂₇×#3150	1	1	yes	characteristic
333	C₂	C₁₂₇×#3151	1	1	yes	characteristic
334	C₂	C₁₂₇×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₁₂₇×#3147	C₂
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
1. C₁₂₇×#3228	C₂	semidirect
2. C₁₂₇×#3182	C₂	semidirect
3. C₁₂₇×#3170	C₂	semidirect
5. C₁₂₇×#1723	C₂	semidirect
6. C₁₂₇×#1722	C₂	semidirect
7. C₁₂₇×#1717	C₂	semidirect
8. C₁₂₇×#1714	C₂	semidirect
9. C₁₂₇×#1451	C₂	semidirect
162. #19687	C₁₂₇	direct (N × H)
168. #3228	C₂×C₁₂₇	semidirect
169. #3182	C₂×C₁₂₇	semidirect
170. #3170	C₂×C₁₂₇	semidirect
172. #1723	C₂×C₁₂₇	semidirect
173. #1722	C₂×C₁₂₇	semidirect
174. #1717	C₂×C₁₂₇	semidirect
175. #1714	C₂×C₁₂₇	semidirect
176. #1451	C₂×C₁₂₇	semidirect
177. C₁₂₇	#19687	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×#3228	2
C₁₂₇×#3182	2
C₁₂₇×#3170	2
C₁₂₇×#3147	2
C₁₂₇×#1723	2
C₁₂₇×#1722	2
C₁₂₇×#1717	2
C₁₂₇×#1714	2
C₁₂₇×#1451	2
#19687	127

Subgroups — C127×#19687 #44583360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19687 #44583360