Subgroups — C₁₃₉×#33743 #48028339

Order 35584.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3177×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#3176×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#2253×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#2248×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#2245×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#2244×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#2220×C₁₃₉	C₂	1	2	yes	maximal, complemented
8	#2217×C₁₃₉	C₂	1	2	yes	maximal, complemented
9	#2175×C₁₃₉	C₂	1	2	yes	maximal, complemented
10	C₄×SD₃₂×C₁₃₉	C₂	1	2	yes	maximal, complemented
11	#2108×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#541×C₁₃₉	—	2	1	no	—
13	#540×C₁₃₉	—	2	1	no	—
14	C₂×SD₃₂×C₁₃₉	C₂²	1	2	yes	—
15	C₂×SD₃₂×C₁₃₉	C₂²	1	2	yes	—
16	C₂×M₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
18	#499×C₁₃₉	C₂²	1	1	yes	characteristic
19	#499×C₁₃₉	C₂²	1	1	yes	characteristic
20	#494×C₁₃₉	C₂²	1	1	yes	characteristic
21	#493×C₁₃₉	C₂²	1	1	yes	characteristic
22	#491×C₁₃₉	—	2	1	no	—
23	#490×C₁₃₉	—	2	1	no	—
24	#470×C₁₃₉	—	2	1	no	—
25	#469×C₁₃₉	—	2	1	no	—
26	#467×C₁₃₉	C₂²	1	2	yes	—
27	#466×C₁₃₉	C₂²	1	2	yes	—
28	#442×C₁₃₉	—	2	1	no	—
29	#441×C₁₃₉	—	2	1	no	—
30	C₄×Q₁₆×C₁₃₉	C₂²	1	2	yes	—
31	C₄×D₈×C₁₃₉	C₂²	1	2	yes	—
32	#405×C₁₃₉	C₂²	1	1	yes	characteristic
33	#367×C₁₃₉	C₂²	1	2	yes	—
34	#366×C₁₃₉	C₂²	1	2	yes	—
35	#358×C₁₃₉	C₂²	1	2	yes	—
36	#358×C₁₃₉	C₂²	1	2	yes	—
37	#357×C₁₃₉	C₂²	1	2	yes	—
38	#357×C₁₃₉	C₂²	1	2	yes	—
39	#346×C₁₃₉	C₂²	1	1	yes	characteristic
40	#346×C₁₃₉	C₂²	1	1	yes	characteristic
41	C₄×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
42	C₄×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
43	C₂×Q₁₆×C₁₃₉	—	2	1	no	—
44	C₂×Q₁₆×C₁₃₉	C₂³	1	2	yes	—
45	C₂×D₈×C₁₃₉	—	2	1	no	—
46	C₂×D₈×C₁₃₉	C₂³	1	2	yes	—
47	C₂×M₁₆×C₁₃₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₁₃₉	—	2	1	no	—
50	#127×C₁₃₉	—	2	1	no	—
51	#126×C₁₃₉	—	4	2	no	—
52	#125×C₁₃₉	—	2	1	no	—
53	#125×C₁₃₉	—	2	1	no	—
54	#125×C₁₃₉	—	4	2	no	—
55	#123×C₁₃₉	—	4	2	no	—
56	C₂²⋊Q₈×C₁₃₉	—	2	2	no	—
57	#121×C₁₃₉	—	2	2	no	—
58	#121×C₁₃₉	—	4	2	no	—
59	C₄×Q₈×C₁₃₉	—	2	2	no	—
60	C₄×D₄×C₁₃₉	—	2	2	no	—
61	#117×C₁₃₉	D₄	1	1	yes	characteristic
62	SD₃₂×C₁₃₉	—	2	4	no	—
63	SD₃₂×C₁₃₉	—	2	4	no	—
64	M₃₂×C₁₃₉	—	2	2	no	—
65	C₂×C₁₆×C₁₃₉	—	2	2	no	—
66	C₂×C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₁₃₉	C₂³	1	1	yes	characteristic
70	#107×C₁₃₉	C₂³	1	2	yes	—
71	#107×C₁₃₉	C₂³	1	2	yes	—
72	#106×C₁₃₉	—	2	1	no	—
73	#106×C₁₃₉	—	2	1	no	—
74	#103×C₁₃₉	—	2	2	no	—
75	#103×C₁₃₉	—	2	2	no	—
76	#102×C₁₃₉	—	2	2	no	—
77	#102×C₁₃₉	—	2	2	no	—
78	#97×C₁₃₉	D₄	1	1	yes	characteristic
79	#97×C₁₃₉	D₄	1	1	yes	characteristic
80	C₄×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₁₃₉	—	2	2	no	—
83	C₂×D₄×C₁₃₉	—	2	2	no	—
84	C₂×D₄×C₁₃₉	—	4	2	no	—
85	C₂×D₄×C₁₃₉	—	4	2	no	—
86	C₂²×C₄×C₁₃₉	—	4	2	no	—
87	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₁₃₉	—	2	2	no	—
89	Q₁₆×C₁₃₉	—	2	2	no	—
90	Q₁₆×C₁₃₉	—	4	2	no	—
91	D₈×C₁₃₉	—	2	2	no	—
92	D₈×C₁₃₉	—	2	2	no	—
93	D₈×C₁₃₉	—	4	2	no	—
94	M₁₆×C₁₃₉	—	2	2	no	—
95	C₂×C₈×C₁₃₉	—	2	1	no	—
96	C₂×C₈×C₁₃₉	—	2	1	no	—
97	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃₉	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₉	—	2	2	no	—
102	C₄⋊C₄×C₁₃₉	—	2	2	no	—
103	C₄⋊C₄×C₁₃₉	—	4	2	no	—
104	C₄⋊C₄×C₁₃₉	—	4	2	no	—
105	C₄⋊C₄×C₁₃₉	—	4	2	no	—
106	C₄⋊C₄×C₁₃₉	—	4	2	no	—
107	C₄⋊C₄×C₁₃₉	—	4	2	no	—
108	C₄⋊C₄×C₁₃₉	D₈	1	2	yes	—
109	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
110	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
111	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
112	C₂²⋊C₄×C₁₃₉	D₈	1	2	yes	—
113	C₄²×C₁₃₉	—	4	2	no	—
114	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₁₃₉	—	2	1	no	—
117	C₁₆×C₁₃₉	—	2	1	no	—
118	C₁₆×C₁₃₉	#41	1	2	yes	—
119	C₁₆×C₁₃₉	#41	1	2	yes	—
120	C₂³×C₁₃₉	—	4	2	no	—
121	C₂³×C₁₃₉	C₂×D₈	1	1	yes	characteristic
122	Q₈×C₁₃₉	—	4	2	no	—
123	Q₈×C₁₃₉	—	4	2	no	—
124	D₄×C₁₃₉	—	4	2	no	—
125	D₄×C₁₃₉	—	4	2	no	—
126	D₄×C₁₃₉	—	16	2	no	—
127	D₄×C₁₃₉	—	16	2	no	—
128	C₂×C₄×C₁₃₉	—	2	1	no	—
129	C₂×C₄×C₁₃₉	—	2	1	no	—
130	C₂×C₄×C₁₃₉	—	4	2	no	—
131	C₂×C₄×C₁₃₉	—	4	2	no	—
132	C₂×C₄×C₁₃₉	—	4	2	no	—
133	C₂×C₄×C₁₃₉	—	16	2	no	—
134	C₂×C₄×C₁₃₉	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃₉	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₁₃₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₁₃₉	—	2	1	no	—
139	C₈×C₁₃₉	—	2	1	no	—
140	C₈×C₁₃₉	#135	1	2	yes	—
141	C₈×C₁₃₉	C₂×#41	1	1	yes	characteristic
142	C₈×C₁₃₉	C₂×#41	1	1	yes	characteristic
143	C₂²×C₁₃₉	—	2	2	no	—
144	C₂²×C₁₃₉	—	2	1	no	—
145	C₂²×C₁₃₉	—	8	2	no	—
146	C₂²×C₁₃₉	—	8	2	no	—
147	C₂²×C₁₃₉	—	8	2	no	—
148	C₂²×C₁₃₉	C₂²×D₈	1	1	yes	characteristic
149	C₄×C₁₃₉	—	2	2	no	—
150	C₄×C₁₃₉	—	2	2	no	—
151	C₄×C₁₃₉	—	2	1	no	—
152	C₄×C₁₃₉	—	8	2	no	—
153	C₄×C₁₃₉	—	8	2	no	—
154	C₄×C₁₃₉	—	8	2	no	—
155	C₄×C₁₃₉	#532	1	1	yes	characteristic
156	C₄×C₁₃₉	C₂×#135	1	1	yes	characteristic
157	C₂×C₁₃₉	—	4	1	no	—
158	C₂×C₁₃₉	—	16	2	no	—
159	C₂×C₁₃₉	#3169	1	1	yes	characteristic
160	C₂×C₁₃₉	#3416	1	2	yes	—
161	#33743	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₁₃₉	#33743	1	1	yes	characteristic, direct product factor, Sylow
163	#3177	C₂×C₁₃₉	1	1	yes	characteristic, complemented
164	#3176	C₂×C₁₃₉	1	1	yes	characteristic
165	#2253	C₂×C₁₃₉	1	1	yes	characteristic, complemented
166	#2248	C₂×C₁₃₉	1	1	yes	characteristic, complemented
167	#2245	C₂×C₁₃₉	1	1	yes	characteristic, complemented
168	#2244	C₂×C₁₃₉	1	1	yes	characteristic, complemented
169	#2220	C₂×C₁₃₉	1	2	yes	complemented
170	#2217	C₂×C₁₃₉	1	2	yes	complemented
171	#2175	C₂×C₁₃₉	1	2	yes	complemented
172	C₄×SD₃₂	C₂×C₁₃₉	1	2	yes	complemented
173	#2108	C₂×C₁₃₉	1	1	yes	characteristic, complemented
174	#541	—	2	1	no	—
175	#540	—	2	1	no	—
176	C₂×SD₃₂	C₂²×C₁₃₉	1	2	yes	—
177	C₂×SD₃₂	C₂²×C₁₃₉	1	2	yes	—
178	C₂×M₃₂	C₂²×C₁₃₉	1	1	yes	characteristic
179	C₂²×C₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
180	#499	C₂²×C₁₃₉	1	1	yes	characteristic
181	#499	C₂²×C₁₃₉	1	1	yes	characteristic
182	#494	C₂²×C₁₃₉	1	1	yes	characteristic
183	#493	C₂²×C₁₃₉	1	1	yes	characteristic
184	#491	—	2	1	no	—
185	#490	—	2	1	no	—
186	#470	—	2	1	no	—
187	#469	—	2	1	no	—
188	#467	C₂²×C₁₃₉	1	2	yes	—
189	#466	C₂²×C₁₃₉	1	2	yes	—
190	#442	—	2	1	no	—
191	#441	—	2	1	no	—
192	C₄×Q₁₆	C₂²×C₁₃₉	1	2	yes	—
193	C₄×D₈	C₂²×C₁₃₉	1	2	yes	—
194	#405	C₂²×C₁₃₉	1	1	yes	characteristic
195	#367	C₂²×C₁₃₉	1	2	yes	—
196	#366	C₂²×C₁₃₉	1	2	yes	—
197	#358	C₂²×C₁₃₉	1	2	yes	—
198	#358	C₂²×C₁₃₉	1	2	yes	—
199	#357	C₂²×C₁₃₉	1	2	yes	—
200	#357	C₂²×C₁₃₉	1	2	yes	—
201	#346	C₂²×C₁₃₉	1	1	yes	characteristic
202	#346	C₂²×C₁₃₉	1	1	yes	characteristic
203	C₄×C₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
204	C₄×C₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	C₂³×C₁₃₉	1	2	yes	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	C₂³×C₁₃₉	1	2	yes	—
209	C₂×M₁₆	D₄×C₁₃₉	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
211	C₄⋊Q₈	—	2	1	no	—
212	#127	—	2	1	no	—
213	#126	—	4	2	no	—
214	#125	—	2	1	no	—
215	#125	—	2	1	no	—
216	#125	—	4	2	no	—
217	#123	—	4	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	#121	—	2	2	no	—
220	#121	—	4	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×D₄	—	2	2	no	—
223	#117	D₄×C₁₃₉	1	1	yes	characteristic
224	SD₃₂	—	2	4	no	—
225	SD₃₂	—	2	4	no	—
226	M₃₂	—	2	2	no	—
227	C₂×C₁₆	—	2	2	no	—
228	C₂×C₁₆	C₂³×C₁₃₉	1	1	yes	characteristic
229	C₂×C₁₆	C₂³×C₁₃₉	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₁₃₉	1	1	yes	characteristic
231	C₂×C₁₆	C₂³×C₁₃₉	1	1	yes	characteristic
232	#107	C₂³×C₁₃₉	1	2	yes	—
233	#107	C₂³×C₁₃₉	1	2	yes	—
234	#106	—	2	1	no	—
235	#106	—	2	1	no	—
236	#103	—	2	2	no	—
237	#103	—	2	2	no	—
238	#102	—	2	2	no	—
239	#102	—	2	2	no	—
240	#97	D₄×C₁₃₉	1	1	yes	characteristic
241	#97	D₄×C₁₃₉	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
244	C₂×Q₈	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	4	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	2	2	no	—
255	D₈	—	4	2	no	—
256	M₁₆	—	2	2	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
261	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
262	C₂×C₈	C₂⁴×C₁₃₉	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	D₈×C₁₃₉	1	2	yes	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	D₈×C₁₃₉	1	2	yes	—
275	C₄²	—	4	2	no	—
276	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	2	1	no	—
280	C₁₆	#41×C₁₃₉	1	2	yes	—
281	C₁₆	#41×C₁₃₉	1	2	yes	—
282	C₂³	—	4	2	no	—
283	C₂³	C₂×D₈×C₁₃₉	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	16	2	no	—
289	D₄	—	16	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	16	2	no	—
296	C₂×C₄	C₂×D₈×C₁₃₉	1	2	yes	—
297	C₂×C₄	C₂×D₈×C₁₃₉	1	2	yes	—
298	C₂×C₄	C₂×D₈×C₁₃₉	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	#135×C₁₃₉	1	2	yes	—
303	C₈	C₂×#41×C₁₃₉	1	1	yes	characteristic
304	C₈	C₂×#41×C₁₃₉	1	1	yes	characteristic
305	C₂²	—	2	2	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	C₂²×D₈×C₁₃₉	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	#532×C₁₃₉	1	1	yes	characteristic
318	C₄	C₂×#135×C₁₃₉	1	1	yes	characteristic
319	C₂	—	4	1	no	—
320	C₂	—	16	2	no	—
321	C₂	#3169×C₁₃₉	1	1	yes	characteristic
322	C₂	#3416×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₉
prime radical	#3176×C₁₃₉	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ #117×C₁₃₉ ◁ #405×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #3177×C₁₃₉	C₂	semidirect
3. #2253×C₁₃₉	C₂	semidirect
4. #2248×C₁₃₉	C₂	semidirect
5. #2245×C₁₃₉	C₂	semidirect
6. #2244×C₁₃₉	C₂	semidirect
7. #2220×C₁₃₉	C₂	semidirect
8. #2217×C₁₃₉	C₂	semidirect
9. #2175×C₁₃₉	C₂	semidirect
10. C₄×SD₃₂×C₁₃₉	C₂	semidirect
11. #2108×C₁₃₉	C₂	semidirect
161. #33743	C₁₃₉	direct (N × H)
162. C₁₃₉	#33743	direct (N × H)
163. #3177	C₂×C₁₃₉	semidirect
165. #2253	C₂×C₁₃₉	semidirect
166. #2248	C₂×C₁₃₉	semidirect
167. #2245	C₂×C₁₃₉	semidirect
168. #2244	C₂×C₁₃₉	semidirect
169. #2220	C₂×C₁₃₉	semidirect
170. #2217	C₂×C₁₃₉	semidirect
171. #2175	C₂×C₁₃₉	semidirect
172. C₄×SD₃₂	C₂×C₁₃₉	semidirect
173. #2108	C₂×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#3177×C₁₃₉	2
#3176×C₁₃₉	2
#2253×C₁₃₉	2
#2248×C₁₃₉	2
#2245×C₁₃₉	2
#2244×C₁₃₉	2
#2220×C₁₃₉	2
#2217×C₁₃₉	2
#2175×C₁₃₉	2
C₄×SD₃₂×C₁₃₉	2
#2108×C₁₃₉	2
#33743	139

Subgroups — C139×#33743 #48028339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#33743 #48028339