Subgroups — C₁₂₇×#19688 #44583361

Order 32512.

724 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×#3210	C₂	1	2	yes	maximal, complemented
2	C₁₂₇×#3182	C₂	1	2	yes	maximal, complemented
3	C₁₂₇×#3169	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1723	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1722	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1719	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1715	C₂	1	2	yes	maximal, complemented
9	C₁₂₇×#1451	C₂	1	1	yes	maximal, characteristic, complemented
10	#533×C₁₂₇	C₂²	1	2	yes	—
11	#532×C₁₂₇	C₂²	1	2	yes	—
12	#500×C₁₂₇	C₂²	1	1	yes	characteristic
13	#500×C₁₂₇	C₂²	1	1	yes	characteristic
14	#500×C₁₂₇	C₂²	1	1	yes	characteristic
15	#500×C₁₂₇	C₂²	1	2	yes	—
16	#499×C₁₂₇	—	2	2	no	—
17	#498×C₁₂₇	C₂²	1	2	yes	—
18	#494×C₁₂₇	C₂²	1	2	yes	—
19	#493×C₁₂₇	C₂²	1	1	yes	characteristic
20	#492×C₁₂₇	C₂²	1	2	yes	—
21	#492×C₁₂₇	C₂²	1	2	yes	—
22	#487×C₁₂₇	C₂²	1	2	yes	—
23	#487×C₁₂₇	C₂²	1	2	yes	—
24	#486×C₁₂₇	C₂²	1	2	yes	—
25	#486×C₁₂₇	C₂²	1	2	yes	—
26	#486×C₁₂₇	C₂²	1	2	yes	—
27	#468×C₁₂₇	—	2	4	no	—
28	#466×C₁₂₇	—	2	2	no	—
29	#464×C₁₂₇	—	2	4	no	—
30	#452×C₁₂₇	—	2	4	no	—
31	#440×C₁₂₇	—	2	4	no	—
32	C₄×D₈×C₁₂₇	—	2	2	no	—
33	#431×C₁₂₇	C₂²	1	1	yes	characteristic
34	#427×C₁₂₇	—	2	2	no	—
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	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
41	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
42	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
43	C₂×D₈×C₁₂₇	—	2	2	no	—
44	C₂×M₁₆×C₁₂₇	D₄	1	2	yes	—
45	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
46	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
49	#127×C₁₂₇	C₂³	1	2	yes	—
50	#125×C₁₂₇	—	2	2	no	—
51	#124×C₁₂₇	—	2	2	no	—
52	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
53	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
54	#121×C₁₂₇	—	2	2	no	—
55	#121×C₁₂₇	—	2	2	no	—
56	C₄×Q₈×C₁₂₇	—	2	2	no	—
57	C₄×Q₈×C₁₂₇	—	2	2	no	—
58	C₄×D₄×C₁₂₇	—	2	2	no	—
59	C₄×D₄×C₁₂₇	—	2	2	no	—
60	#117×C₁₂₇	—	2	2	no	—
61	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
62	#107×C₁₂₇	—	2	2	no	—
63	#107×C₁₂₇	—	2	2	no	—
64	#107×C₁₂₇	—	2	2	no	—
65	#107×C₁₂₇	—	2	4	no	—
66	#106×C₁₂₇	—	2	4	no	—
67	C₄⋊C₈×C₁₂₇	D₄	1	2	yes	—
68	#103×C₁₂₇	—	2	4	no	—
69	#103×C₁₂₇	—	2	4	no	—
70	#102×C₁₂₇	—	2	4	no	—
71	#102×C₁₂₇	—	2	4	no	—
72	#102×C₁₂₇	—	2	4	no	—
73	C₂²⋊C₈×C₁₂₇	D₄	1	2	yes	—
74	#97×C₁₂₇	D₄	1	2	yes	—
75	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
76	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
79	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
80	#41×C₁₂₇	—	8	4	no	—
81	C₂×Q₈×C₁₂₇	—	2	2	no	—
82	C₂×Q₈×C₁₂₇	—	2	2	no	—
83	C₂×Q₈×C₁₂₇	—	2	2	no	—
84	C₂×D₄×C₁₂₇	—	2	2	no	—
85	C₂×D₄×C₁₂₇	—	2	2	no	—
86	C₂×D₄×C₁₂₇	—	2	2	no	—
87	C₂×D₄×C₁₂₇	—	4	2	no	—
88	C₂²×C₄×C₁₂₇	—	4	2	no	—
89	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₁₂₇	—	8	4	no	—
93	SD₁₆×C₁₂₇	—	4	8	no	—
94	SD₁₆×C₁₂₇	—	8	4	no	—
95	D₈×C₁₂₇	—	4	4	no	—
96	M₁₆×C₁₂₇	—	4	2	no	—
97	C₂×C₈×C₁₂₇	—	2	4	no	—
98	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₁₂₇	—	2	2	no	—
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	—	4	2	no	—
109	C₄⋊C₄×C₁₂₇	—	4	2	no	—
110	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
111	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
112	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
113	C₄²×C₁₂₇	—	4	2	no	—
114	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
118	C₂³×C₁₂₇	—	4	2	no	—
119	C₂³×C₁₂₇	#127	1	1	yes	characteristic
120	Q₈×C₁₂₇	—	4	4	no	—
121	Q₈×C₁₂₇	—	4	4	no	—
122	Q₈×C₁₂₇	—	4	4	no	—
123	D₄×C₁₂₇	—	4	4	no	—
124	D₄×C₁₂₇	—	4	4	no	—
125	D₄×C₁₂₇	—	4	4	no	—
126	D₄×C₁₂₇	—	8	4	no	—
127	C₂×C₄×C₁₂₇	—	2	2	no	—
128	C₂×C₄×C₁₂₇	—	2	2	no	—
129	C₂×C₄×C₁₂₇	—	2	2	no	—
130	C₂×C₄×C₁₂₇	—	4	2	no	—
131	C₂×C₄×C₁₂₇	—	4	2	no	—
132	C₂×C₄×C₁₂₇	—	4	2	no	—
133	C₂×C₄×C₁₂₇	—	8	4	no	—
134	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
135	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
136	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
137	C₂×C₄×C₁₂₇	C₂²×D₄	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	C₈×C₁₂₇	—	2	2	no	—
141	C₈×C₁₂₇	—	2	4	no	—
142	C₈×C₁₂₇	—	2	4	no	—
143	C₈×C₁₂₇	#137	1	4	yes	—
144	C₂²×C₁₂₇	—	2	1	no	—
145	C₂²×C₁₂₇	—	2	1	no	—
146	C₂²×C₁₂₇	—	2	1	no	—
147	C₂²×C₁₂₇	—	8	2	no	—
148	C₂²×C₁₂₇	—	8	2	no	—
149	C₂²×C₁₂₇	—	8	2	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	C₂×#137	1	2	yes	—
158	C₄×C₁₂₇	#575	1	2	yes	—
159	C₄×C₁₂₇	#576	1	2	yes	—
160	#19688	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₁₂₇	—	4	1	no	—
162	C₂×C₁₂₇	—	16	2	no	—
163	C₂×C₁₂₇	#3150	1	1	yes	characteristic
164	C₂×C₁₂₇	#3152	1	1	yes	characteristic
165	C₂×C₁₂₇	#3155	1	1	yes	characteristic
166	#3210	C₂×C₁₂₇	1	2	yes	complemented
167	#3182	C₂×C₁₂₇	1	2	yes	complemented
168	#3169	C₂×C₁₂₇	1	1	yes	characteristic
169	#3160	C₂×C₁₂₇	1	1	yes	characteristic, complemented
170	#1723	C₂×C₁₂₇	1	2	yes	complemented
171	#1722	C₂×C₁₂₇	1	2	yes	complemented
172	#1719	C₂×C₁₂₇	1	2	yes	complemented
173	#1715	C₂×C₁₂₇	1	2	yes	complemented
174	#1451	C₂×C₁₂₇	1	1	yes	characteristic, complemented
175	C₁₂₇	#19688	1	1	yes	characteristic, direct product factor, Sylow
176	#533	C₂²×C₁₂₇	1	2	yes	—
177	#532	C₂²×C₁₂₇	1	2	yes	—
178	#500	C₂²×C₁₂₇	1	1	yes	characteristic
179	#500	C₂²×C₁₂₇	1	1	yes	characteristic
180	#500	C₂²×C₁₂₇	1	1	yes	characteristic
181	#500	C₂²×C₁₂₇	1	2	yes	—
182	#499	—	2	2	no	—
183	#498	C₂²×C₁₂₇	1	2	yes	—
184	#494	C₂²×C₁₂₇	1	2	yes	—
185	#493	C₂²×C₁₂₇	1	1	yes	characteristic
186	#492	C₂²×C₁₂₇	1	2	yes	—
187	#492	C₂²×C₁₂₇	1	2	yes	—
188	#487	C₂²×C₁₂₇	1	2	yes	—
189	#487	C₂²×C₁₂₇	1	2	yes	—
190	#486	C₂²×C₁₂₇	1	2	yes	—
191	#486	C₂²×C₁₂₇	1	2	yes	—
192	#486	C₂²×C₁₂₇	1	2	yes	—
193	#468	—	2	4	no	—
194	#466	—	2	2	no	—
195	#464	—	2	4	no	—
196	#452	—	2	4	no	—
197	#440	—	2	4	no	—
198	C₄×D₈	—	2	2	no	—
199	#431	C₂²×C₁₂₇	1	1	yes	characteristic
200	#427	—	2	2	no	—
201	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
202	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
203	#322	C₂²×C₁₂₇	1	2	yes	—
204	#322	C₂²×C₁₂₇	1	2	yes	—
205	C₂×#41	—	2	2	no	—
206	C₂×Q₁₆	—	2	4	no	—
207	C₂×SD₁₆	—	2	4	no	—
208	C₂×SD₁₆	—	2	4	no	—
209	C₂×D₈	—	2	2	no	—
210	C₂×M₁₆	D₄×C₁₂₇	1	2	yes	—
211	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
212	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
213	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
214	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
215	#127	C₂³×C₁₂₇	1	2	yes	—
216	#125	—	2	2	no	—
217	#124	—	2	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	#121	—	2	2	no	—
221	#121	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	C₄×D₄	—	2	2	no	—
226	#117	—	2	2	no	—
227	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
228	#107	—	2	2	no	—
229	#107	—	2	2	no	—
230	#107	—	2	2	no	—
231	#107	—	2	4	no	—
232	#106	—	2	4	no	—
233	C₄⋊C₈	D₄×C₁₂₇	1	2	yes	—
234	#103	—	2	4	no	—
235	#103	—	2	4	no	—
236	#102	—	2	4	no	—
237	#102	—	2	4	no	—
238	#102	—	2	4	no	—
239	C₂²⋊C₈	D₄×C₁₂₇	1	2	yes	—
240	#97	D₄×C₁₂₇	1	2	yes	—
241	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
242	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
244	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
245	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
246	#41	—	8	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
258	Q₁₆	—	8	4	no	—
259	SD₁₆	—	4	8	no	—
260	SD₁₆	—	8	4	no	—
261	D₈	—	4	4	no	—
262	M₁₆	—	4	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
265	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
266	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
267	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
268	C₄⋊C₄	—	2	2	no	—
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	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₄²	—	4	2	no	—
280	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
283	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
284	C₂³	—	4	2	no	—
285	C₂³	#127×C₁₂₇	1	1	yes	characteristic
286	Q₈	—	4	4	no	—
287	Q₈	—	4	4	no	—
288	Q₈	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	8	4	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
301	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
302	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×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	C₈	—	2	2	no	—
307	C₈	—	2	4	no	—
308	C₈	—	2	4	no	—
309	C₈	#137×C₁₂₇	1	4	yes	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	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	C₂×#137×C₁₂₇	1	2	yes	—
324	C₄	#575×C₁₂₇	1	2	yes	—
325	C₄	#576×C₁₂₇	1	2	yes	—
326	C₂	—	4	1	no	—
327	C₂	—	16	2	no	—
328	C₂	C₁₂₇×#3150	1	1	yes	characteristic
329	C₂	C₁₂₇×#3152	1	1	yes	characteristic
330	C₂	C₁₂₇×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₁₂₇×#3169	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×#3210	C₂	semidirect
2. C₁₂₇×#3182	C₂	semidirect
4. C₁₂₇×#3160	C₂	semidirect
5. C₁₂₇×#1723	C₂	semidirect
6. C₁₂₇×#1722	C₂	semidirect
7. C₁₂₇×#1719	C₂	semidirect
8. C₁₂₇×#1715	C₂	semidirect
9. C₁₂₇×#1451	C₂	semidirect
160. #19688	C₁₂₇	direct (N × H)
166. #3210	C₂×C₁₂₇	semidirect
167. #3182	C₂×C₁₂₇	semidirect
169. #3160	C₂×C₁₂₇	semidirect
170. #1723	C₂×C₁₂₇	semidirect
171. #1722	C₂×C₁₂₇	semidirect
172. #1719	C₂×C₁₂₇	semidirect
173. #1715	C₂×C₁₂₇	semidirect
174. #1451	C₂×C₁₂₇	semidirect
175. C₁₂₇	#19688	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×#3210	2
C₁₂₇×#3182	2
C₁₂₇×#3169	2
C₁₂₇×#3160	2
C₁₂₇×#1723	2
C₁₂₇×#1722	2
C₁₂₇×#1719	2
C₁₂₇×#1715	2
C₁₂₇×#1451	2
#19688	127

Subgroups — C127×#19688 #44583361

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19688 #44583361