Subgroups — C₁₂₇×#19717 #44583390

Order 32512.

720 conjugacy classes of subgroups.

The subgroup lattice has 320 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, complemented
2	C₁₂₇×#3208	C₂	1	2	yes	maximal
3	C₁₂₇×#3185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#3138	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1720	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1719	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1714	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1708	C₂	1	2	yes	maximal, complemented
9	C₁₂₇×#1453	C₂	1	1	yes	maximal, characteristic, complemented
10	#533×C₁₂₇	C₂²	1	1	yes	characteristic
11	#533×C₁₂₇	C₂²	1	2	yes	—
12	C₂×#127×C₁₂₇	C₂²	1	1	yes	characteristic
13	#500×C₁₂₇	C₂²	1	2	yes	complemented
14	#498×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
15	#498×C₁₂₇	C₂²	1	2	yes	complemented
16	#494×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
17	#492×C₁₂₇	C₂²	1	2	yes	—
18	#492×C₁₂₇	C₂²	1	2	yes	—
19	#490×C₁₂₇	—	2	2	no	—
20	#487×C₁₂₇	C₂²	1	2	yes	complemented
21	#487×C₁₂₇	C₂²	1	2	yes	complemented
22	#487×C₁₂₇	C₂²	1	2	yes	complemented
23	#486×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	#485×C₁₂₇	—	2	4	no	—
28	#479×C₁₂₇	—	2	4	no	—
29	#470×C₁₂₇	—	2	2	no	—
30	#460×C₁₂₇	—	2	4	no	—
31	#450×C₁₂₇	—	2	4	no	—
32	#441×C₁₂₇	—	2	2	no	—
33	#431×C₁₂₇	C₂²	1	2	yes	—
34	#418×C₁₂₇	—	2	2	no	—
35	C₄×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
36	#322×C₁₂₇	C₂²	1	2	yes	complemented
37	#322×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
38	C₈²×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₁₂₇	—	2	1	no	—
40	C₂²×D₄×C₁₂₇	—	2	2	no	—
41	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
42	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
43	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
44	C₂×M₁₆×C₁₂₇	D₄	1	2	yes	—
45	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
46	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
48	#127×C₁₂₇	—	2	2	no	—
49	#127×C₁₂₇	—	2	1	no	—
50	#127×C₁₂₇	C₂³	1	2	yes	—
51	#125×C₁₂₇	—	2	2	no	—
52	#125×C₁₂₇	—	2	1	no	—
53	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
54	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
55	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
56	C₄×Q₈×C₁₂₇	—	2	2	no	—
57	C₄×Q₈×C₁₂₇	—	2	2	no	—
58	C₄×Q₈×C₁₂₇	—	2	2	no	—
59	#117×C₁₂₇	—	2	1	no	—
60	#117×C₁₂₇	—	2	2	no	—
61	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
62	#107×C₁₂₇	—	2	4	no	—
63	#106×C₁₂₇	—	2	2	no	—
64	#106×C₁₂₇	—	2	4	no	—
65	C₄⋊C₈×C₁₂₇	D₄	1	4	yes	—
66	#103×C₁₂₇	—	2	4	no	—
67	#103×C₁₂₇	—	2	4	no	—
68	#103×C₁₂₇	—	2	4	no	—
69	#102×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	4	yes	—
74	#97×C₁₂₇	D₄	1	2	yes	—
75	C₄×C₈×C₁₂₇	C₂³	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	1	yes	characteristic
79	C₂⁴×C₁₂₇	—	4	1	no	—
80	C₂×Q₈×C₁₂₇	—	2	2	no	—
81	C₂×Q₈×C₁₂₇	—	2	2	no	—
82	C₂×Q₈×C₁₂₇	—	2	2	no	—
83	C₂×D₄×C₁₂₇	—	2	2	no	—
84	C₂×D₄×C₁₂₇	—	2	2	no	—
85	C₂×D₄×C₁₂₇	—	2	2	no	—
86	C₂×D₄×C₁₂₇	—	4	4	no	—
87	C₂×D₄×C₁₂₇	—	4	1	no	—
88	C₂×D₄×C₁₂₇	—	4	2	no	—
89	C₂×D₄×C₁₂₇	—	8	4	no	—
90	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₁₂₇	—	4	4	no	—
93	SD₁₆×C₁₂₇	—	8	4	no	—
94	SD₁₆×C₁₂₇	—	8	4	no	—
95	M₁₆×C₁₂₇	—	4	2	no	—
96	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
99	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
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	—	2	2	no	—
104	C₄⋊C₄×C₁₂₇	—	2	2	no	—
105	C₄⋊C₄×C₁₂₇	—	2	2	no	—
106	C₄⋊C₄×C₁₂₇	—	4	2	no	—
107	C₄⋊C₄×C₁₂₇	—	4	2	no	—
108	C₄⋊C₄×C₁₂₇	—	4	2	no	—
109	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
110	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
111	C₄²×C₁₂₇	—	4	1	no	—
112	C₄²×C₁₂₇	—	4	2	no	—
113	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
115	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
116	C₂³×C₁₂₇	—	4	2	no	—
117	C₂³×C₁₂₇	—	8	2	no	—
118	C₂³×C₁₂₇	—	8	4	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	2	no	—
127	D₄×C₁₂₇	—	8	4	no	—
128	C₂×C₄×C₁₂₇	—	2	2	no	—
129	C₂×C₄×C₁₂₇	—	2	4	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	#127	1	1	yes	characteristic
134	C₂×C₄×C₁₂₇	#127	1	2	yes	—
135	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
137	C₈×C₁₂₇	—	2	2	no	—
138	C₈×C₁₂₇	—	2	2	no	—
139	C₈×C₁₂₇	—	2	4	no	—
140	C₈×C₁₂₇	—	2	4	no	—
141	C₂²×C₁₂₇	—	2	1	no	—
142	C₂²×C₁₂₇	—	2	2	no	—
143	C₂²×C₁₂₇	—	8	2	no	—
144	C₂²×C₁₂₇	—	8	2	no	—
145	C₂²×C₁₂₇	—	8	2	no	—
146	C₂²×C₁₂₇	—	16	4	no	—
147	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
148	C₄×C₁₂₇	—	2	1	no	—
149	C₄×C₁₂₇	—	2	2	no	—
150	C₄×C₁₂₇	—	8	2	no	—
151	C₄×C₁₂₇	—	8	2	no	—
152	C₄×C₁₂₇	—	8	2	no	—
153	C₄×C₁₂₇	#575	1	2	yes	—
154	C₄×C₁₂₇	#577	1	4	yes	—
155	#19717	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₁₂₇	—	4	1	no	—
157	C₂×C₁₂₇	—	16	2	no	—
158	C₂×C₁₂₇	#3153	1	1	yes	characteristic
159	C₂×C₁₂₇	#3154	1	2	yes	—
160	#3231	C₂×C₁₂₇	1	2	yes	complemented
161	#3208	C₂×C₁₂₇	1	2	yes	—
162	#3185	C₂×C₁₂₇	1	1	yes	characteristic, complemented
163	#3138	C₂×C₁₂₇	1	1	yes	characteristic
164	#1720	C₂×C₁₂₇	1	2	yes	complemented
165	#1719	C₂×C₁₂₇	1	2	yes	complemented
166	#1714	C₂×C₁₂₇	1	2	yes	complemented
167	#1708	C₂×C₁₂₇	1	2	yes	complemented
168	#1453	C₂×C₁₂₇	1	1	yes	characteristic, complemented
169	C₁₂₇	#19717	1	1	yes	characteristic, direct product factor, Sylow
170	#533	C₂²×C₁₂₇	1	1	yes	characteristic
171	#533	C₂²×C₁₂₇	1	2	yes	—
172	C₂×#127	C₂²×C₁₂₇	1	1	yes	characteristic
173	#500	C₂²×C₁₂₇	1	2	yes	complemented
174	#498	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
175	#498	C₂²×C₁₂₇	1	2	yes	complemented
176	#494	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
177	#492	C₂²×C₁₂₇	1	2	yes	—
178	#492	C₂²×C₁₂₇	1	2	yes	—
179	#490	—	2	2	no	—
180	#487	C₂²×C₁₂₇	1	2	yes	complemented
181	#487	C₂²×C₁₂₇	1	2	yes	complemented
182	#487	C₂²×C₁₂₇	1	2	yes	complemented
183	#486	C₂²×C₁₂₇	1	2	yes	—
184	#486	C₂²×C₁₂₇	1	2	yes	—
185	#486	C₂²×C₁₂₇	1	2	yes	—
186	#486	C₂²×C₁₂₇	1	2	yes	—
187	#485	—	2	4	no	—
188	#479	—	2	4	no	—
189	#470	—	2	2	no	—
190	#460	—	2	4	no	—
191	#450	—	2	4	no	—
192	#441	—	2	2	no	—
193	#431	C₂²×C₁₂₇	1	2	yes	—
194	#418	—	2	2	no	—
195	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
196	#322	C₂²×C₁₂₇	1	2	yes	complemented
197	#322	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
198	C₈²	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
199	C₂²×D₄	—	2	1	no	—
200	C₂²×D₄	—	2	2	no	—
201	C₂×Q₁₆	—	2	2	no	—
202	C₂×SD₁₆	—	2	4	no	—
203	C₂×SD₁₆	—	2	4	no	—
204	C₂×M₁₆	D₄×C₁₂₇	1	2	yes	—
205	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
206	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
207	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
208	#127	—	2	2	no	—
209	#127	—	2	1	no	—
210	#127	C₂³×C₁₂₇	1	2	yes	—
211	#125	—	2	2	no	—
212	#125	—	2	1	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	#117	—	2	1	no	—
220	#117	—	2	2	no	—
221	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
222	#107	—	2	4	no	—
223	#106	—	2	2	no	—
224	#106	—	2	4	no	—
225	C₄⋊C₈	D₄×C₁₂₇	1	4	yes	—
226	#103	—	2	4	no	—
227	#103	—	2	4	no	—
228	#103	—	2	4	no	—
229	#102	—	2	4	no	—
230	#102	—	2	4	no	—
231	#102	—	2	4	no	—
232	#102	—	2	4	no	—
233	C₂²⋊C₈	D₄×C₁₂₇	1	4	yes	—
234	#97	D₄×C₁₂₇	1	2	yes	—
235	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
237	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
239	C₂⁴	—	4	1	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	8	4	no	—
250	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
252	Q₁₆	—	4	4	no	—
253	SD₁₆	—	8	4	no	—
254	SD₁₆	—	8	4	no	—
255	M₁₆	—	4	2	no	—
256	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
257	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
259	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	2	no	—
273	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
275	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
276	C₂³	—	4	2	no	—
277	C₂³	—	8	2	no	—
278	C₂³	—	8	4	no	—
279	C₂³	#127×C₁₂₇	1	1	yes	characteristic
280	Q₈	—	4	4	no	—
281	Q₈	—	4	4	no	—
282	Q₈	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	4	no	—
286	D₄	—	8	2	no	—
287	D₄	—	8	4	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
294	C₂×C₄	#127×C₁₂₇	1	2	yes	—
295	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	—	2	4	no	—
300	C₈	—	2	4	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	16	4	no	—
307	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	#575×C₁₂₇	1	2	yes	—
314	C₄	#577×C₁₂₇	1	4	yes	—
315	C₂	—	4	1	no	—
316	C₂	—	16	2	no	—
317	C₂	C₁₂₇×#3153	1	1	yes	characteristic
318	C₂	C₁₂₇×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₂×#127×C₁₂₇	C₂²
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×#3231	C₂	semidirect
3. C₁₂₇×#3185	C₂	semidirect
5. C₁₂₇×#1720	C₂	semidirect
6. C₁₂₇×#1719	C₂	semidirect
7. C₁₂₇×#1714	C₂	semidirect
8. C₁₂₇×#1708	C₂	semidirect
9. C₁₂₇×#1453	C₂	semidirect
13. #500×C₁₂₇	C₂²	semidirect
14. #498×C₁₂₇	C₂²	semidirect
15. #498×C₁₂₇	C₂²	semidirect
16. #494×C₁₂₇	C₂²	semidirect
20. #487×C₁₂₇	C₂²	semidirect
21. #487×C₁₂₇	C₂²	semidirect
22. #487×C₁₂₇	C₂²	semidirect
36. #322×C₁₂₇	C₂²	semidirect
37. #322×C₁₂₇	C₂²	semidirect
38. C₈²×C₁₂₇	C₂²	semidirect
155. #19717	C₁₂₇	direct (N × H)
160. #3231	C₂×C₁₂₇	semidirect
162. #3185	C₂×C₁₂₇	semidirect
164. #1720	C₂×C₁₂₇	semidirect
165. #1719	C₂×C₁₂₇	semidirect
166. #1714	C₂×C₁₂₇	semidirect
167. #1708	C₂×C₁₂₇	semidirect
168. #1453	C₂×C₁₂₇	semidirect
169. C₁₂₇	#19717	direct (N × H)
173. #500	C₂²×C₁₂₇	semidirect
174. #498	C₂²×C₁₂₇	semidirect
175. #498	C₂²×C₁₂₇	semidirect
176. #494	C₂²×C₁₂₇	semidirect
180. #487	C₂²×C₁₂₇	semidirect
181. #487	C₂²×C₁₂₇	semidirect
182. #487	C₂²×C₁₂₇	semidirect
196. #322	C₂²×C₁₂₇	semidirect
197. #322	C₂²×C₁₂₇	semidirect
198. C₈²	C₂²×C₁₂₇	semidirect

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×#3208	2
C₁₂₇×#3185	2
C₁₂₇×#3138	2
C₁₂₇×#1720	2
C₁₂₇×#1719	2
C₁₂₇×#1714	2
C₁₂₇×#1708	2
C₁₂₇×#1453	2
#19717	127

Subgroups — C127×#19717 #44583390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19717 #44583390