Subgroups — C₁₂₇×#19690 #44583363

Order 32512.

656 conjugacy classes of subgroups.

The subgroup lattice has 312 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×#1723	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1723	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1718	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1716	C₂	1	2	yes	maximal, complemented
9	C₁₂₇×#1451	C₂	1	1	yes	maximal, characteristic
10	#533×C₁₂₇	C₂²	1	2	yes	—
11	#533×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	#498×C₁₂₇	C₂²	1	2	yes	—
19	#498×C₁₂₇	C₂²	1	2	yes	—
20	#494×C₁₂₇	C₂²	1	1	yes	characteristic
21	#494×C₁₂₇	C₂²	1	2	yes	—
22	#487×C₁₂₇	C₂²	1	2	yes	—
23	#487×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	#485×C₁₂₇	—	2	4	no	—
28	#479×C₁₂₇	—	2	4	no	—
29	#470×C₁₂₇	—	2	4	no	—
30	#467×C₁₂₇	—	2	2	no	—
31	#441×C₁₂₇	—	2	4	no	—
32	C₄×Q₁₆×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×Q₁₆×C₁₂₇	—	2	2	no	—
40	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
41	C₂×M₁₆×C₁₂₇	D₄	1	2	yes	—
42	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
43	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
45	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
47	#125×C₁₂₇	—	2	2	no	—
48	#125×C₁₂₇	—	2	2	no	—
49	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
50	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
51	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
52	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
53	C₄×Q₈×C₁₂₇	—	2	2	no	—
54	C₄×Q₈×C₁₂₇	—	2	2	no	—
55	C₄×Q₈×C₁₂₇	—	2	2	no	—
56	C₄×Q₈×C₁₂₇	—	2	2	no	—
57	#117×C₁₂₇	—	2	2	no	—
58	#117×C₁₂₇	—	2	2	no	—
59	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
60	#107×C₁₂₇	—	2	2	no	—
61	#107×C₁₂₇	—	2	2	no	—
62	#107×C₁₂₇	—	2	2	no	—
63	#107×C₁₂₇	—	2	4	no	—
64	#106×C₁₂₇	—	2	4	no	—
65	#106×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	#103×C₁₂₇	—	2	4	no	—
71	#103×C₁₂₇	—	2	4	no	—
72	#103×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	2	yes	—
78	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
79	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
80	C₂×Q₈×C₁₂₇	—	2	2	no	—
81	C₂×Q₈×C₁₂₇	—	2	2	no	—
82	C₂×Q₈×C₁₂₇	—	2	2	no	—
83	C₂×Q₈×C₁₂₇	—	2	2	no	—
84	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
87	Q₁₆×C₁₂₇	—	4	8	no	—
88	Q₁₆×C₁₂₇	—	4	4	no	—
89	M₁₆×C₁₂₇	—	4	2	no	—
90	C₂×C₈×C₁₂₇	—	2	4	no	—
91	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
95	C₄⋊C₄×C₁₂₇	—	2	2	no	—
96	C₄⋊C₄×C₁₂₇	—	2	2	no	—
97	C₄⋊C₄×C₁₂₇	—	2	2	no	—
98	C₄⋊C₄×C₁₂₇	—	2	2	no	—
99	C₄⋊C₄×C₁₂₇	—	2	2	no	—
100	C₄⋊C₄×C₁₂₇	—	2	2	no	—
101	C₄⋊C₄×C₁₂₇	—	2	2	no	—
102	C₄⋊C₄×C₁₂₇	—	2	2	no	—
103	C₄⋊C₄×C₁₂₇	—	4	2	no	—
104	C₄⋊C₄×C₁₂₇	—	4	2	no	—
105	C₄⋊C₄×C₁₂₇	—	4	2	no	—
106	C₄⋊C₄×C₁₂₇	—	4	2	no	—
107	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
108	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
109	C₄²×C₁₂₇	—	4	2	no	—
110	C₄²×C₁₂₇	—	4	2	no	—
111	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
115	C₂³×C₁₂₇	#127	1	1	yes	characteristic
116	Q₈×C₁₂₇	—	4	4	no	—
117	Q₈×C₁₂₇	—	4	4	no	—
118	Q₈×C₁₂₇	—	4	4	no	—
119	Q₈×C₁₂₇	—	4	4	no	—
120	C₂×C₄×C₁₂₇	—	2	2	no	—
121	C₂×C₄×C₁₂₇	—	2	2	no	—
122	C₂×C₄×C₁₂₇	—	2	2	no	—
123	C₂×C₄×C₁₂₇	—	4	2	no	—
124	C₂×C₄×C₁₂₇	—	4	2	no	—
125	C₂×C₄×C₁₂₇	—	4	2	no	—
126	C₂×C₄×C₁₂₇	—	4	2	no	—
127	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
128	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
129	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
130	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₁₂₇	—	2	2	no	—
134	C₈×C₁₂₇	—	2	4	no	—
135	C₈×C₁₂₇	—	2	4	no	—
136	C₈×C₁₂₇	#137	1	4	yes	—
137	C₂²×C₁₂₇	—	2	1	no	—
138	C₂²×C₁₂₇	—	2	1	no	—
139	C₂²×C₁₂₇	—	2	1	no	—
140	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
141	C₄×C₁₂₇	—	2	1	no	—
142	C₄×C₁₂₇	—	2	1	no	—
143	C₄×C₁₂₇	—	2	1	no	—
144	C₄×C₁₂₇	—	8	2	no	—
145	C₄×C₁₂₇	—	8	2	no	—
146	C₄×C₁₂₇	—	8	2	no	—
147	C₄×C₁₂₇	—	8	2	no	—
148	C₄×C₁₂₇	C₂×#137	1	2	yes	—
149	C₄×C₁₂₇	#575	1	2	yes	—
150	C₄×C₁₂₇	#578	1	2	yes	—
151	#19690	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂×C₁₂₇	—	4	1	no	—
153	C₂×C₁₂₇	#3150	1	1	yes	characteristic
154	C₂×C₁₂₇	#3152	1	1	yes	characteristic
155	C₂×C₁₂₇	#3156	1	1	yes	characteristic
156	#3231	C₂×C₁₂₇	1	2	yes	—
157	#3185	C₂×C₁₂₇	1	2	yes	—
158	#3170	C₂×C₁₂₇	1	1	yes	characteristic
159	#3160	C₂×C₁₂₇	1	1	yes	characteristic
160	#1723	C₂×C₁₂₇	1	2	yes	complemented
161	#1723	C₂×C₁₂₇	1	2	yes	complemented
162	#1718	C₂×C₁₂₇	1	2	yes	complemented
163	#1716	C₂×C₁₂₇	1	2	yes	complemented
164	#1451	C₂×C₁₂₇	1	1	yes	characteristic
165	C₁₂₇	#19690	1	1	yes	characteristic, direct product factor, Sylow
166	#533	C₂²×C₁₂₇	1	2	yes	—
167	#533	C₂²×C₁₂₇	1	2	yes	—
168	#500	C₂²×C₁₂₇	1	1	yes	characteristic
169	#500	C₂²×C₁₂₇	1	1	yes	characteristic
170	#500	C₂²×C₁₂₇	1	1	yes	characteristic
171	#500	C₂²×C₁₂₇	1	2	yes	—
172	#499	—	2	2	no	—
173	#498	C₂²×C₁₂₇	1	2	yes	—
174	#498	C₂²×C₁₂₇	1	2	yes	—
175	#498	C₂²×C₁₂₇	1	2	yes	—
176	#494	C₂²×C₁₂₇	1	1	yes	characteristic
177	#494	C₂²×C₁₂₇	1	2	yes	—
178	#487	C₂²×C₁₂₇	1	2	yes	—
179	#487	C₂²×C₁₂₇	1	2	yes	—
180	#487	C₂²×C₁₂₇	1	2	yes	—
181	#487	C₂²×C₁₂₇	1	2	yes	—
182	#487	C₂²×C₁₂₇	1	2	yes	—
183	#485	—	2	4	no	—
184	#479	—	2	4	no	—
185	#470	—	2	4	no	—
186	#467	—	2	2	no	—
187	#441	—	2	4	no	—
188	C₄×Q₁₆	—	2	2	no	—
189	#431	C₂²×C₁₂₇	1	1	yes	characteristic
190	#427	—	2	2	no	—
191	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
193	#322	C₂²×C₁₂₇	1	2	yes	—
194	#322	C₂²×C₁₂₇	1	2	yes	—
195	C₂×Q₁₆	—	2	2	no	—
196	C₂×Q₁₆	—	2	4	no	—
197	C₂×M₁₆	D₄×C₁₂₇	1	2	yes	—
198	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
199	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
200	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
201	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
202	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
203	#125	—	2	2	no	—
204	#125	—	2	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₂²⋊Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	#117	—	2	2	no	—
214	#117	—	2	2	no	—
215	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
216	#107	—	2	2	no	—
217	#107	—	2	2	no	—
218	#107	—	2	2	no	—
219	#107	—	2	4	no	—
220	#106	—	2	4	no	—
221	#106	—	2	4	no	—
222	#106	—	2	4	no	—
223	C₄⋊C₈	D₄×C₁₂₇	1	2	yes	—
224	#103	—	2	4	no	—
225	#103	—	2	4	no	—
226	#103	—	2	4	no	—
227	#103	—	2	4	no	—
228	#103	—	2	4	no	—
229	C₂²⋊C₈	D₄×C₁₂₇	1	2	yes	—
230	#97	D₄×C₁₂₇	1	2	yes	—
231	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
232	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
233	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
234	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
235	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
243	Q₁₆	—	4	8	no	—
244	Q₁₆	—	4	4	no	—
245	M₁₆	—	4	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
248	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
249	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
250	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₄²	—	4	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
269	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
270	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
271	C₂³	#127×C₁₂₇	1	1	yes	characteristic
272	Q₈	—	4	4	no	—
273	Q₈	—	4	4	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	4	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
284	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
285	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
286	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
288	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	4	no	—
291	C₈	—	2	4	no	—
292	C₈	#137×C₁₂₇	1	4	yes	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	C₂×#137×C₁₂₇	1	2	yes	—
305	C₄	#575×C₁₂₇	1	2	yes	—
306	C₄	#578×C₁₂₇	1	2	yes	—
307	C₂	—	4	1	no	—
308	C₂	C₁₂₇×#3150	1	1	yes	characteristic
309	C₂	C₁₂₇×#3152	1	1	yes	characteristic
310	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×#1723	C₂	semidirect
6. C₁₂₇×#1723	C₂	semidirect
7. C₁₂₇×#1718	C₂	semidirect
8. C₁₂₇×#1716	C₂	semidirect
151. #19690	C₁₂₇	direct (N × H)
160. #1723	C₂×C₁₂₇	semidirect
161. #1723	C₂×C₁₂₇	semidirect
162. #1718	C₂×C₁₂₇	semidirect
163. #1716	C₂×C₁₂₇	semidirect
165. C₁₂₇	#19690	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×#1723	2
C₁₂₇×#1718	2
C₁₂₇×#1716	2
C₁₂₇×#1451	2
#19690	127

Subgroups — C127×#19690 #44583363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19690 #44583363