Subgroups — C₁₃₉×#8963 #48003559

Order 35584.

694 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2975×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#127×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1478×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	#1460×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#1276×C₁₃₉	C₂	1	2	yes	maximal, complemented
6	C₂×#127×C₁₃₉	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃₉	—	2	2	no	—
8	C₂×C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
10	#436×C₁₃₉	—	2	4	no	—
11	#431×C₁₃₉	C₂²	1	2	yes	—
12	#431×C₁₃₉	C₂²	1	2	yes	—
13	#407×C₁₃₉	—	2	2	no	—
14	#390×C₁₃₉	—	2	2	no	—
15	#390×C₁₃₉	C₄	1	1	yes	characteristic
16	#390×C₁₃₉	C₄	1	1	yes	characteristic
17	#384×C₁₃₉	C₄	1	1	yes	characteristic
18	C₄³×C₁₃₉	C₂²	1	1	yes	characteristic
19	D₄⋊C₈×C₁₃₉	—	2	4	no	—
20	#323×C₁₃₉	—	2	2	no	—
21	C₂²×D₄×C₁₃₉	—	2	2	no	—
22	C₂²×D₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×D₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃₉	—	2	1	no	—
25	C₂³×C₄×C₁₃₉	D₄	1	2	yes	—
26	C₂×M₁₆×C₁₃₉	—	2	4	no	—
27	#127×C₁₃₉	—	2	2	no	—
28	#127×C₁₃₉	—	2	2	no	—
29	C₄×D₄×C₁₃₉	—	2	2	no	—
30	C₄×D₄×C₁₃₉	—	2	2	no	—
31	C₄×D₄×C₁₃₉	—	2	2	no	—
32	C₄×D₄×C₁₃₉	—	2	2	no	—
33	C₄×D₄×C₁₃₉	—	4	8	no	—
34	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₁₃₉	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₁₃₉	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃₉	—	2	1	no	—
42	C₂×C₄²×C₁₃₉	—	2	1	no	—
43	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₁₃₉	—	2	2	no	—
47	C₄⋊C₈×C₁₃₉	D₄	1	4	yes	complemented
48	C₂²⋊C₈×C₁₃₉	—	2	4	no	—
49	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
50	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
51	C₄×C₈×C₁₃₉	—	2	2	no	—
52	C₄×C₈×C₁₃₉	—	2	2	no	—
53	C₂⁴×C₁₃₉	—	2	1	no	—
54	C₂⁴×C₁₃₉	C₂²⋊C₄	1	2	yes	—
55	C₂×D₄×C₁₃₉	—	2	8	no	—
56	C₂×D₄×C₁₃₉	—	2	2	no	—
57	C₂×D₄×C₁₃₉	—	2	2	no	—
58	C₂×D₄×C₁₃₉	—	2	1	no	—
59	C₂×D₄×C₁₃₉	—	2	1	no	—
60	C₂×D₄×C₁₃₉	—	2	1	no	—
61	C₂×D₄×C₁₃₉	—	2	2	no	—
62	C₂×D₄×C₁₃₉	—	2	2	no	—
63	C₂×D₄×C₁₃₉	—	4	4	no	—
64	C₂×D₄×C₁₃₉	—	4	4	no	—
65	C₂×D₄×C₁₃₉	M₁₆	1	2	yes	complemented
66	C₂²×C₄×C₁₃₉	—	2	1	no	—
67	C₂²×C₄×C₁₃₉	—	2	1	no	—
68	C₂²×C₄×C₁₃₉	—	2	2	no	—
69	C₂²×C₄×C₁₃₉	—	2	2	no	—
70	C₂²×C₄×C₁₃₉	—	2	1	no	—
71	C₂²×C₄×C₁₃₉	—	4	4	no	—
72	C₂²×C₄×C₁₃₉	—	4	4	no	—
73	C₂²×C₄×C₁₃₉	—	4	1	no	—
74	C₂²×C₄×C₁₃₉	—	4	1	no	—
75	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
76	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
79	M₁₆×C₁₃₉	—	8	4	no	—
80	C₂×C₈×C₁₃₉	—	2	4	no	—
81	C₂×C₈×C₁₃₉	—	4	2	no	—
82	C₄⋊C₄×C₁₃₉	—	2	2	no	—
83	C₄⋊C₄×C₁₃₉	—	2	2	no	—
84	C₄⋊C₄×C₁₃₉	—	4	4	no	—
85	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
86	C₂²⋊C₄×C₁₃₉	—	4	4	no	—
87	C₂²⋊C₄×C₁₃₉	—	4	4	no	—
88	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
89	C₄²×C₁₃₉	—	2	4	no	—
90	C₄²×C₁₃₉	—	2	1	no	—
91	C₄²×C₁₃₉	—	2	2	no	—
92	C₄²×C₁₃₉	—	4	2	no	—
93	C₄²×C₁₃₉	M₁₆	1	2	yes	complemented
94	C₄²×C₁₃₉	C₂×D₄	1	2	yes	—
95	C₄²×C₁₃₉	#41	1	2	yes	—
96	C₂³×C₁₃₉	—	2	2	no	—
97	C₂³×C₁₃₉	—	2	2	no	—
98	C₂³×C₁₃₉	—	2	2	no	—
99	C₂³×C₁₃₉	—	2	2	no	—
100	C₂³×C₁₃₉	—	4	2	no	—
101	C₂³×C₁₃₉	—	4	2	no	—
102	C₂³×C₁₃₉	—	4	2	no	—
103	C₂³×C₁₃₉	—	4	2	no	—
104	C₂³×C₁₃₉	—	4	1	no	—
105	C₂³×C₁₃₉	—	4	1	no	—
106	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	D₄×C₁₃₉	—	2	4	no	—
108	D₄×C₁₃₉	—	4	16	no	—
109	D₄×C₁₃₉	—	4	2	no	—
110	D₄×C₁₃₉	—	4	2	no	—
111	D₄×C₁₃₉	—	4	2	no	—
112	C₂×C₄×C₁₃₉	—	2	2	no	—
113	C₂×C₄×C₁₃₉	—	2	2	no	—
114	C₂×C₄×C₁₃₉	—	2	2	no	—
115	C₂×C₄×C₁₃₉	—	2	2	no	—
116	C₂×C₄×C₁₃₉	—	2	2	no	—
117	C₂×C₄×C₁₃₉	—	2	2	no	—
118	C₂×C₄×C₁₃₉	—	4	1	no	—
119	C₂×C₄×C₁₃₉	—	4	1	no	—
120	C₂×C₄×C₁₃₉	—	4	1	no	—
121	C₂×C₄×C₁₃₉	—	4	4	no	—
122	C₂×C₄×C₁₃₉	—	4	2	no	—
123	C₂×C₄×C₁₃₉	—	4	2	no	—
124	C₂×C₄×C₁₃₉	—	4	1	no	—
125	C₂×C₄×C₁₃₉	—	8	2	no	—
126	C₂×C₄×C₁₃₉	—	8	2	no	—
127	C₂×C₄×C₁₃₉	—	8	2	no	—
128	C₂×C₄×C₁₃₉	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₁₃₉	#120	1	1	yes	characteristic
130	C₂×C₄×C₁₃₉	#121	1	2	yes	—
131	C₂×C₄×C₁₃₉	#123	1	1	yes	characteristic
132	C₂×C₄×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
133	C₂×C₄×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
134	C₂×C₄×C₁₃₉	#131	1	2	yes	—
135	C₈×C₁₃₉	—	4	4	no	—
136	C₈×C₁₃₉	—	8	2	no	—
137	C₂²×C₁₃₉	—	2	1	no	—
138	C₂²×C₁₃₉	—	2	1	no	—
139	C₂²×C₁₃₉	—	4	4	no	—
140	C₂²×C₁₃₉	—	4	4	no	—
141	C₂²×C₁₃₉	—	4	2	no	—
142	C₂²×C₁₃₉	—	4	2	no	—
143	C₂²×C₁₃₉	—	4	2	no	—
144	C₂²×C₁₃₉	—	4	2	no	—
145	C₂²×C₁₃₉	—	8	2	no	—
146	C₂²×C₁₃₉	—	8	2	no	—
147	C₂²×C₁₃₉	—	8	1	no	—
148	C₂²×C₁₃₉	—	8	1	no	—
149	C₂²×C₁₃₉	#386	1	1	yes	characteristic
150	C₂²×C₁₃₉	#407	1	1	yes	characteristic
151	C₂²×C₁₃₉	#408	1	1	yes	characteristic
152	C₄×C₁₃₉	—	2	4	no	—
153	C₄×C₁₃₉	—	2	2	no	—
154	C₄×C₁₃₉	—	2	1	no	—
155	C₄×C₁₃₉	—	4	2	no	—
156	C₄×C₁₃₉	—	4	2	no	—
157	C₄×C₁₃₉	—	8	1	no	—
158	C₄×C₁₃₉	#435	1	2	yes	—
159	C₄×C₁₃₉	#436	1	2	yes	—
160	C₄×C₁₃₉	#453	1	2	yes	—
161	C₂×C₁₃₉	—	2	1	no	—
162	C₂×C₁₃₉	—	2	1	no	—
163	C₂×C₁₃₉	—	8	2	no	—
164	C₂×C₁₃₉	—	8	2	no	—
165	C₂×C₁₃₉	#1870	1	1	yes	characteristic
166	C₂×C₁₃₉	#1884	1	1	yes	characteristic
167	C₂×C₁₃₉	#1887	1	1	yes	characteristic
168	#8963	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₁₃₉	#8963	1	1	yes	characteristic, direct product factor, Sylow
170	#2975	C₂×C₁₃₉	1	1	yes	characteristic, complemented
171	C₄×#127	C₂×C₁₃₉	1	1	yes	characteristic
172	#1478	C₂×C₁₃₉	1	2	yes	complemented
173	#1460	C₂×C₁₃₉	1	1	yes	characteristic, complemented
174	#1276	C₂×C₁₃₉	1	2	yes	complemented
175	C₂×#127	C₄×C₁₃₉	1	1	yes	characteristic
176	C₂×C₄×D₄	—	2	2	no	—
177	C₂×C₄×D₄	C₂²×C₁₃₉	1	1	yes	characteristic
178	C₂×C₄×D₄	C₂²×C₁₃₉	1	1	yes	characteristic
179	#436	—	2	4	no	—
180	#431	C₂²×C₁₃₉	1	2	yes	—
181	#431	C₂²×C₁₃₉	1	2	yes	—
182	#407	—	2	2	no	—
183	#390	—	2	2	no	—
184	#390	C₄×C₁₃₉	1	1	yes	characteristic
185	#390	C₄×C₁₃₉	1	1	yes	characteristic
186	#384	C₄×C₁₃₉	1	1	yes	characteristic
187	C₄³	C₂²×C₁₃₉	1	1	yes	characteristic
188	D₄⋊C₈	—	2	4	no	—
189	#323	—	2	2	no	—
190	C₂²×D₄	—	2	2	no	—
191	C₂²×D₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
192	C₂²×D₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
193	C₂³×C₄	—	2	1	no	—
194	C₂³×C₄	D₄×C₁₃₉	1	2	yes	—
195	C₂×M₁₆	—	2	4	no	—
196	#127	—	2	2	no	—
197	#127	—	2	2	no	—
198	C₄×D₄	—	2	2	no	—
199	C₄×D₄	—	2	2	no	—
200	C₄×D₄	—	2	2	no	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	4	8	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₂²⋊C₄	—	2	1	no	—
209	C₂×C₂²⋊C₄	D₄×C₁₃₉	1	2	yes	—
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	—	2	1	no	—
212	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
213	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
215	C₄⋊C₈	—	2	2	no	—
216	C₄⋊C₈	D₄×C₁₃₉	1	4	yes	complemented
217	C₂²⋊C₈	—	2	4	no	—
218	C₂²⋊C₈	—	2	2	no	—
219	C₂²⋊C₈	—	2	2	no	—
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	—	2	2	no	—
222	C₂⁴	—	2	1	no	—
223	C₂⁴	C₂²⋊C₄×C₁₃₉	1	2	yes	—
224	C₂×D₄	—	2	8	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	—	4	4	no	—
234	C₂×D₄	M₁₆×C₁₃₉	1	2	yes	complemented
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
245	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
248	M₁₆	—	8	4	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	4	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₂²⋊C₄	—	2	4	no	—
255	C₂²⋊C₄	—	4	4	no	—
256	C₂²⋊C₄	—	4	4	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	M₁₆×C₁₃₉	1	2	yes	complemented
263	C₄²	C₂×D₄×C₁₃₉	1	2	yes	—
264	C₄²	#41×C₁₃₉	1	2	yes	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
276	D₄	—	2	4	no	—
277	D₄	—	4	16	no	—
278	D₄	—	4	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	C₄×D₄×C₁₃₉	1	2	yes	—
298	C₂×C₄	#120×C₁₃₉	1	1	yes	characteristic
299	C₂×C₄	#121×C₁₃₉	1	2	yes	—
300	C₂×C₄	#123×C₁₃₉	1	1	yes	characteristic
301	C₂×C₄	C₂×M₁₆×C₁₃₉	1	1	yes	characteristic
302	C₂×C₄	C₂×M₁₆×C₁₃₉	1	1	yes	characteristic
303	C₂×C₄	#131×C₁₃₉	1	2	yes	—
304	C₈	—	4	4	no	—
305	C₈	—	8	2	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	#386×C₁₃₉	1	1	yes	characteristic
319	C₂²	#407×C₁₃₉	1	1	yes	characteristic
320	C₂²	#408×C₁₃₉	1	1	yes	characteristic
321	C₄	—	2	4	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	8	1	no	—
327	C₄	#435×C₁₃₉	1	2	yes	—
328	C₄	#436×C₁₃₉	1	2	yes	—
329	C₄	#453×C₁₃₉	1	2	yes	—
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	8	2	no	—
333	C₂	—	8	2	no	—
334	C₂	#1870×C₁₃₉	1	1	yes	characteristic
335	C₂	#1884×C₁₃₉	1	1	yes	characteristic
336	C₂	#1887×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₉	#120
commutator	C₂²×C₄	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
socle	C₂²×C₁₃₉	#386
prime radical	C₂×#127×C₁₃₉	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₉ ◁ C₂×C₄²×C₁₃₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #2975×C₁₃₉	C₂	semidirect
3. #1478×C₁₃₉	C₂	semidirect
4. #1460×C₁₃₉	C₂	semidirect
5. #1276×C₁₃₉	C₂	semidirect
47. C₄⋊C₈×C₁₃₉	D₄	semidirect
65. C₂×D₄×C₁₃₉	M₁₆	semidirect
93. C₄²×C₁₃₉	M₁₆	semidirect
168. #8963	C₁₃₉	direct (N × H)
169. C₁₃₉	#8963	direct (N × H)
170. #2975	C₂×C₁₃₉	semidirect
172. #1478	C₂×C₁₃₉	semidirect
173. #1460	C₂×C₁₃₉	semidirect
174. #1276	C₂×C₁₃₉	semidirect
216. C₄⋊C₈	D₄×C₁₃₉	semidirect
234. C₂×D₄	M₁₆×C₁₃₉	semidirect
262. C₄²	M₁₆×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#2975×C₁₃₉	2
C₄×#127×C₁₃₉	2
#1478×C₁₃₉	2
#1460×C₁₃₉	2
#1276×C₁₃₉	2
#8963	139

Subgroups — C139×#8963 #48003559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#8963 #48003559