Subgroups — C₁₂₇×#19654 #44583327

Order 32512.

798 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3227	C₂	1	2	yes	maximal, complemented
2	C₁₂₇×#3206	C₂	1	2	yes	maximal, complemented
3	C₂×#498×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#3147	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1716	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1715	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1708	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1705	C₂	1	2	yes	maximal, complemented
9	C₁₂₇×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₁₂₇	—	2	2	no	—
11	#532×C₁₂₇	C₂²	1	1	yes	characteristic
12	#532×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
14	#498×C₁₂₇	—	2	2	no	—
15	#498×C₁₂₇	C₂²	1	2	yes	—
16	#498×C₁₂₇	C₂²	1	2	yes	—
17	#498×C₁₂₇	C₂²	1	4	yes	—
18	#495×C₁₂₇	—	2	2	no	—
19	#494×C₁₂₇	C₂²	1	1	yes	characteristic
20	#493×C₁₂₇	C₂²	1	1	yes	characteristic
21	#492×C₁₂₇	C₂²	1	1	yes	characteristic
22	#492×C₁₂₇	C₂²	1	1	yes	characteristic
23	#492×C₁₂₇	C₂²	1	4	yes	—
24	#487×C₁₂₇	C₂²	1	4	yes	—
25	#486×C₁₂₇	C₂²	1	4	yes	—
26	#481×C₁₂₇	—	2	4	no	—
27	#476×C₁₂₇	—	2	4	no	—
28	#461×C₁₂₇	—	2	4	no	—
29	#448×C₁₂₇	—	2	4	no	—
30	#431×C₁₂₇	C₂²	1	2	yes	—
31	C₂×#106×C₁₂₇	—	2	2	no	—
32	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
33	#322×C₁₂₇	C₂²	1	2	yes	—
34	C₈²×C₁₂₇	C₂²	1	2	yes	—
35	C₂×#41×C₁₂₇	—	2	1	no	—
36	C₂×#41×C₁₂₇	—	2	1	no	—
37	C₂²×Q₈×C₁₂₇	—	2	2	no	—
38	#135×C₁₂₇	—	4	8	no	—
39	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
40	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
41	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
42	C₂×SD₁₆×C₁₂₇	—	2	8	no	—
43	C₂×D₈×C₁₂₇	—	2	2	no	—
44	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
45	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
46	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
47	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
48	C₄⋊Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₁₂₇	C₂³	1	4	yes	—
50	C₄⋊Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
51	#127×C₁₂₇	C₂³	1	1	yes	characteristic
52	#127×C₁₂₇	C₂³	1	1	yes	characteristic
53	#124×C₁₂₇	—	2	1	no	—
54	#124×C₁₂₇	—	2	1	no	—
55	#121×C₁₂₇	—	2	2	no	—
56	#121×C₁₂₇	—	2	2	no	—
57	C₄×D₄×C₁₂₇	—	2	2	no	—
58	C₄×D₄×C₁₂₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
60	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
61	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
62	#106×C₁₂₇	—	2	4	no	—
63	#106×C₁₂₇	—	2	4	no	—
64	#106×C₁₂₇	—	2	8	no	—
65	C₄⋊C₈×C₁₂₇	D₄	1	4	yes	—
66	#103×C₁₂₇	—	2	8	no	—
67	#102×C₁₂₇	—	2	8	no	—
68	C₂²⋊C₈×C₁₂₇	D₄	1	4	yes	—
69	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
70	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₁₂₇	C₂³	1	4	yes	—
73	#41×C₁₂₇	—	4	4	no	—
74	#41×C₁₂₇	—	4	4	no	—
75	C₂×Q₈×C₁₂₇	—	2	4	no	—
76	C₂×Q₈×C₁₂₇	—	2	1	no	—
77	C₂×Q₈×C₁₂₇	—	2	1	no	—
78	C₂×Q₈×C₁₂₇	—	4	2	no	—
79	C₂×Q₈×C₁₂₇	—	8	4	no	—
80	C₂×D₄×C₁₂₇	—	2	2	no	—
81	C₂×D₄×C₁₂₇	—	2	2	no	—
82	C₂×D₄×C₁₂₇	—	2	1	no	—
83	C₂×D₄×C₁₂₇	—	2	1	no	—
84	C₂×D₄×C₁₂₇	—	4	1	no	—
85	C₂×D₄×C₁₂₇	—	4	1	no	—
86	C₂²×C₄×C₁₂₇	—	4	2	no	—
87	C₂²×C₄×C₁₂₇	—	4	1	no	—
88	C₂²×C₄×C₁₂₇	—	4	1	no	—
89	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₁₂₇	—	4	4	no	—
92	SD₁₆×C₁₂₇	—	4	4	no	—
93	SD₁₆×C₁₂₇	—	4	4	no	—
94	SD₁₆×C₁₂₇	—	8	8	no	—
95	D₈×C₁₂₇	—	4	4	no	—
96	C₂×C₈×C₁₂₇	SD₁₆	1	8	yes	complemented
97	C₂×C₈×C₁₂₇	C₂×D₄	1	8	yes	—
98	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
100	C₄⋊C₄×C₁₂₇	—	2	4	no	—
101	C₄⋊C₄×C₁₂₇	—	2	2	no	—
102	C₄⋊C₄×C₁₂₇	—	2	4	no	—
103	C₄⋊C₄×C₁₂₇	—	2	2	no	—
104	C₄⋊C₄×C₁₂₇	—	4	2	no	—
105	C₄⋊C₄×C₁₂₇	—	4	2	no	—
106	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
107	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
108	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
110	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
111	C₂³×C₁₂₇	—	4	1	no	—
112	C₂³×C₁₂₇	—	4	1	no	—
113	C₂³×C₁₂₇	#127	1	1	yes	characteristic
114	Q₈×C₁₂₇	—	4	2	no	—
115	Q₈×C₁₂₇	—	4	2	no	—
116	Q₈×C₁₂₇	—	4	8	no	—
117	Q₈×C₁₂₇	—	8	4	no	—
118	D₄×C₁₂₇	—	4	2	no	—
119	D₄×C₁₂₇	—	4	2	no	—
120	D₄×C₁₂₇	—	4	4	no	—
121	D₄×C₁₂₇	—	4	4	no	—
122	D₄×C₁₂₇	—	8	2	no	—
123	D₄×C₁₂₇	—	8	2	no	—
124	C₂×C₄×C₁₂₇	—	2	4	no	—
125	C₂×C₄×C₁₂₇	—	4	1	no	—
126	C₂×C₄×C₁₂₇	—	4	1	no	—
127	C₂×C₄×C₁₂₇	—	4	4	no	—
128	C₂×C₄×C₁₂₇	—	8	4	no	—
129	C₂×C₄×C₁₂₇	—	8	2	no	—
130	C₂×C₄×C₁₂₇	—	8	2	no	—
131	C₂×C₄×C₁₂₇	#127	1	2	yes	—
132	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
133	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	4	yes	—
134	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
136	C₈×C₁₂₇	—	2	8	no	—
137	C₈×C₁₂₇	C₂×SD₁₆	1	4	yes	—
138	C₈×C₁₂₇	C₂×SD₁₆	1	4	yes	—
139	C₂²×C₁₂₇	—	2	2	no	—
140	C₂²×C₁₂₇	—	8	1	no	—
141	C₂²×C₁₂₇	—	8	1	no	—
142	C₂²×C₁₂₇	—	8	2	no	—
143	C₂²×C₁₂₇	—	8	2	no	—
144	C₂²×C₁₂₇	#492	1	2	yes	—
145	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
146	C₄×C₁₂₇	—	2	2	no	—
147	C₄×C₁₂₇	—	8	1	no	—
148	C₄×C₁₂₇	—	8	1	no	—
149	C₄×C₁₂₇	—	8	4	no	—
150	C₄×C₁₂₇	#492	1	2	yes	—
151	C₄×C₁₂₇	C₂²×SD₁₆	1	2	yes	—
152	C₄×C₁₂₇	#577	1	4	yes	—
153	#19654	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂×C₁₂₇	—	2	2	no	—
155	C₂×C₁₂₇	—	16	1	no	—
156	C₂×C₁₂₇	—	16	1	no	—
157	C₂×C₁₂₇	C₂×#492	1	1	yes	characteristic
158	C₂×C₁₂₇	#3154	1	2	yes	—
159	#3227	C₂×C₁₂₇	1	2	yes	complemented
160	#3206	C₂×C₁₂₇	1	2	yes	complemented
161	C₂×#498	C₂×C₁₂₇	1	1	yes	characteristic, complemented
162	#3147	C₂×C₁₂₇	1	1	yes	characteristic
163	#1716	C₂×C₁₂₇	1	2	yes	complemented
164	#1715	C₂×C₁₂₇	1	2	yes	complemented
165	#1708	C₂×C₁₂₇	1	2	yes	complemented
166	#1705	C₂×C₁₂₇	1	2	yes	complemented
167	#1450	C₂×C₁₂₇	1	1	yes	characteristic, complemented
168	C₁₂₇	#19654	1	1	yes	characteristic, direct product factor, Sylow
169	C₂×#135	—	2	2	no	—
170	#532	C₂²×C₁₂₇	1	1	yes	characteristic
171	#532	C₂²×C₁₂₇	1	1	yes	characteristic
172	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
173	#498	—	2	2	no	—
174	#498	C₂²×C₁₂₇	1	2	yes	—
175	#498	C₂²×C₁₂₇	1	2	yes	—
176	#498	C₂²×C₁₂₇	1	4	yes	—
177	#495	—	2	2	no	—
178	#494	C₂²×C₁₂₇	1	1	yes	characteristic
179	#493	C₂²×C₁₂₇	1	1	yes	characteristic
180	#492	C₂²×C₁₂₇	1	1	yes	characteristic
181	#492	C₂²×C₁₂₇	1	1	yes	characteristic
182	#492	C₂²×C₁₂₇	1	4	yes	—
183	#487	C₂²×C₁₂₇	1	4	yes	—
184	#486	C₂²×C₁₂₇	1	4	yes	—
185	#481	—	2	4	no	—
186	#476	—	2	4	no	—
187	#461	—	2	4	no	—
188	#448	—	2	4	no	—
189	#431	C₂²×C₁₂₇	1	2	yes	—
190	C₂×#106	—	2	2	no	—
191	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
192	#322	C₂²×C₁₂₇	1	2	yes	—
193	C₈²	C₂²×C₁₂₇	1	2	yes	—
194	C₂×#41	—	2	1	no	—
195	C₂×#41	—	2	1	no	—
196	C₂²×Q₈	—	2	2	no	—
197	#135	—	4	8	no	—
198	C₂×Q₁₆	—	2	2	no	—
199	C₂×SD₁₆	—	2	2	no	—
200	C₂×SD₁₆	—	2	2	no	—
201	C₂×SD₁₆	—	2	8	no	—
202	C₂×D₈	—	2	2	no	—
203	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	C₂³×C₁₂₇	1	4	yes	—
208	C₄⋊Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
209	C₄⋊Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
210	#127	C₂³×C₁₂₇	1	1	yes	characteristic
211	#127	C₂³×C₁₂₇	1	1	yes	characteristic
212	#124	—	2	1	no	—
213	#124	—	2	1	no	—
214	#121	—	2	2	no	—
215	#121	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
221	#106	—	2	4	no	—
222	#106	—	2	4	no	—
223	#106	—	2	8	no	—
224	C₄⋊C₈	D₄×C₁₂₇	1	4	yes	—
225	#103	—	2	8	no	—
226	#102	—	2	8	no	—
227	C₂²⋊C₈	D₄×C₁₂₇	1	4	yes	—
228	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
229	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₁₂₇	1	4	yes	—
231	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
232	#41	—	4	4	no	—
233	#41	—	4	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	8	4	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
250	Q₁₆	—	4	4	no	—
251	SD₁₆	—	4	4	no	—
252	SD₁₆	—	4	4	no	—
253	SD₁₆	—	8	8	no	—
254	D₈	—	4	4	no	—
255	C₂×C₈	SD₁₆×C₁₂₇	1	8	yes	complemented
256	C₂×C₈	C₂×D₄×C₁₂₇	1	8	yes	—
257	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
258	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
268	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
269	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
270	C₂³	—	4	1	no	—
271	C₂³	—	4	1	no	—
272	C₂³	#127×C₁₂₇	1	1	yes	characteristic
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	4	8	no	—
276	Q₈	—	8	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	4	no	—
280	D₄	—	4	4	no	—
281	D₄	—	8	2	no	—
282	D₄	—	8	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	#127×C₁₂₇	1	2	yes	—
291	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
292	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	4	yes	—
293	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
295	C₈	—	2	8	no	—
296	C₈	C₂×SD₁₆×C₁₂₇	1	4	yes	—
297	C₈	C₂×SD₁₆×C₁₂₇	1	4	yes	—
298	C₂²	—	2	2	no	—
299	C₂²	—	8	1	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	#492×C₁₂₇	1	2	yes	—
304	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	8	1	no	—
307	C₄	—	8	1	no	—
308	C₄	—	8	4	no	—
309	C₄	#492×C₁₂₇	1	2	yes	—
310	C₄	C₂²×SD₁₆×C₁₂₇	1	2	yes	—
311	C₄	#577×C₁₂₇	1	4	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	16	1	no	—
314	C₂	—	16	1	no	—
315	C₂	C₂×#492×C₁₂₇	1	1	yes	characteristic
316	C₂	C₁₂₇×#3154	1	2	yes	—

	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×#3227	C₂	semidirect
2. C₁₂₇×#3206	C₂	semidirect
3. C₂×#498×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1716	C₂	semidirect
6. C₁₂₇×#1715	C₂	semidirect
7. C₁₂₇×#1708	C₂	semidirect
8. C₁₂₇×#1705	C₂	semidirect
9. C₁₂₇×#1450	C₂	semidirect
96. C₂×C₈×C₁₂₇	SD₁₆	semidirect
153. #19654	C₁₂₇	direct (N × H)
159. #3227	C₂×C₁₂₇	semidirect
160. #3206	C₂×C₁₂₇	semidirect
161. C₂×#498	C₂×C₁₂₇	semidirect
163. #1716	C₂×C₁₂₇	semidirect
164. #1715	C₂×C₁₂₇	semidirect
165. #1708	C₂×C₁₂₇	semidirect
166. #1705	C₂×C₁₂₇	semidirect
167. #1450	C₂×C₁₂₇	semidirect
168. C₁₂₇	#19654	direct (N × H)
255. C₂×C₈	SD₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3227	2
C₁₂₇×#3206	2
C₂×#498×C₁₂₇	2
C₁₂₇×#3147	2
C₁₂₇×#1716	2
C₁₂₇×#1715	2
C₁₂₇×#1708	2
C₁₂₇×#1705	2
C₁₂₇×#1450	2
#19654	127

Subgroups — C127×#19654 #44583327

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19654 #44583327