Subgroups — C₁₃₉×#19192 #48013788

Order 35584.

674 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	#3206×C₁₃₉	C₂	1	2	yes	maximal
2	C₂×#500×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#2931×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#1693×C₁₃₉	C₂	1	4	yes	maximal, complemented
5	#1673×C₁₃₉	C₂	1	4	yes	maximal, complemented
6	#1564×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#1549×C₁₃₉	C₂	1	2	yes	maximal, complemented
8	C₂×#131×C₁₃₉	—	2	1	no	—
9	#532×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	2	yes	—
11	#500×C₁₃₉	—	2	1	no	—
12	#500×C₁₃₉	—	2	1	no	—
13	#500×C₁₃₉	C₂²	1	2	yes	—
14	#500×C₁₃₉	C₂²	1	2	yes	—
15	#492×C₁₃₉	—	2	2	no	—
16	#487×C₁₃₉	—	2	2	no	—
17	#461×C₁₃₉	C₂²	1	8	yes	—
18	#448×C₁₃₉	—	2	4	no	—
19	#436×C₁₃₉	—	2	1	no	—
20	#436×C₁₃₉	C₂²	1	2	yes	—
21	C₈×D₄×C₁₃₉	—	2	1	no	—
22	C₈×D₄×C₁₃₉	C₂²	1	2	yes	—
23	#431×C₁₃₉	C₂²	1	2	yes	—
24	C₂×#107×C₁₃₉	—	2	1	no	—
25	C₂×#107×C₁₃₉	—	2	1	no	—
26	#423×C₁₃₉	C₂²	1	1	yes	characteristic
27	#408×C₁₃₉	—	2	1	no	—
28	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
29	#335×C₁₃₉	C₂²	1	2	yes	—
30	#334×C₁₃₉	C₂²	1	2	yes	—
31	#332×C₁₃₉	C₂²	1	8	yes	—
32	C₂×#41×C₁₃₉	—	2	1	no	—
33	C₂²×Q₈×C₁₃₉	—	2	2	no	—
34	C₂×SD₁₆×C₁₃₉	—	2	8	no	—
35	#131×C₁₃₉	—	4	4	no	—
36	C₂×M₁₆×C₁₃₉	—	2	1	no	—
37	C₂×M₁₆×C₁₃₉	—	2	2	no	—
38	C₂²×C₈×C₁₃₉	—	2	2	no	—
39	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
41	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
42	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
43	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
44	C₄⋊Q₈×C₁₃₉	D₄	1	1	yes	characteristic
45	C₄⋊Q₈×C₁₃₉	C₂³	1	4	yes	—
46	#127×C₁₃₉	D₄	1	1	yes	characteristic
47	#124×C₁₃₉	—	2	1	no	—
48	#121×C₁₃₉	—	2	1	no	—
49	#121×C₁₃₉	D₄	1	2	yes	—
50	C₄×D₄×C₁₃₉	—	2	1	no	—
51	C₄×D₄×C₁₃₉	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
53	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
54	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
55	#107×C₁₃₉	—	2	2	no	—
56	#107×C₁₃₉	—	2	2	no	—
57	#107×C₁₃₉	—	2	2	no	—
58	#107×C₁₃₉	—	2	2	no	—
59	C₄⋊C₈×C₁₃₉	—	2	1	no	—
60	C₄⋊C₈×C₁₃₉	C₂³	1	2	yes	—
61	C₄⋊C₈×C₁₃₉	C₂³	1	4	yes	—
62	#103×C₁₃₉	—	2	8	no	—
63	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
64	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
65	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
66	C₄×C₈×C₁₃₉	—	2	2	no	—
67	C₄×C₈×C₁₃₉	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
71	#41×C₁₃₉	—	4	2	no	—
72	#41×C₁₃₉	—	4	2	no	—
73	C₂×Q₈×C₁₃₉	—	2	1	no	—
74	C₂×Q₈×C₁₃₉	—	2	4	no	—
75	C₂×Q₈×C₁₃₉	—	4	2	no	—
76	C₂×Q₈×C₁₃₉	—	8	4	no	—
77	C₂×D₄×C₁₃₉	—	2	2	no	—
78	C₂×D₄×C₁₃₉	—	2	1	no	—
79	C₂×D₄×C₁₃₉	—	2	1	no	—
80	C₂×D₄×C₁₃₉	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃₉	—	2	2	no	—
82	C₂²×C₄×C₁₃₉	—	4	2	no	—
83	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃₉	#41	1	1	yes	characteristic
86	SD₁₆×C₁₃₉	—	8	8	no	—
87	M₁₆×C₁₃₉	—	4	2	no	—
88	M₁₆×C₁₃₉	—	4	4	no	—
89	C₂×C₈×C₁₃₉	—	2	2	no	—
90	C₂×C₈×C₁₃₉	—	2	4	no	—
91	C₂×C₈×C₁₃₉	—	2	2	no	—
92	C₂×C₈×C₁₃₉	—	4	4	no	—
93	C₂×C₈×C₁₃₉	SD₁₆	1	4	yes	complemented
94	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
98	C₄⋊C₄×C₁₃₉	—	2	1	no	—
99	C₄⋊C₄×C₁₃₉	—	2	4	no	—
100	C₄⋊C₄×C₁₃₉	—	2	4	no	—
101	C₄⋊C₄×C₁₃₉	—	4	2	no	—
102	C₄⋊C₄×C₁₃₉	—	4	2	no	—
103	C₄⋊C₄×C₁₃₉	C₂×D₄	1	2	yes	—
104	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
105	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
106	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃₉	#41	1	1	yes	characteristic
109	C₄²×C₁₃₉	C₂⁴	1	1	yes	characteristic
110	C₂³×C₁₃₉	—	2	2	no	—
111	C₂³×C₁₃₉	#121	1	1	yes	characteristic
112	Q₈×C₁₃₉	—	4	2	no	—
113	Q₈×C₁₃₉	—	4	8	no	—
114	Q₈×C₁₃₉	—	8	4	no	—
115	D₄×C₁₃₉	—	2	4	no	—
116	D₄×C₁₃₉	—	4	4	no	—
117	D₄×C₁₃₉	—	4	2	no	—
118	D₄×C₁₃₉	—	4	2	no	—
119	C₂×C₄×C₁₃₉	—	2	2	no	—
120	C₂×C₄×C₁₃₉	—	2	2	no	—
121	C₂×C₄×C₁₃₉	—	2	2	no	—
122	C₂×C₄×C₁₃₉	—	4	2	no	—
123	C₂×C₄×C₁₃₉	—	4	2	no	—
124	C₂×C₄×C₁₃₉	—	4	4	no	—
125	C₂×C₄×C₁₃₉	—	8	4	no	—
126	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
127	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
128	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
129	C₂×C₄×C₁₃₉	C₂×SD₁₆	1	2	yes	—
130	C₂×C₄×C₁₃₉	#137	1	2	yes	—
131	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
134	C₈×C₁₃₉	—	2	1	no	—
135	C₈×C₁₃₉	—	2	1	no	—
136	C₈×C₁₃₉	—	4	2	no	—
137	C₈×C₁₃₉	—	4	4	no	—
138	C₈×C₁₃₉	C₂×SD₁₆	1	2	yes	—
139	C₈×C₁₃₉	C₂×SD₁₆	1	2	yes	—
140	C₂²×C₁₃₉	—	2	1	no	—
141	C₂²×C₁₃₉	—	2	1	no	—
142	C₂²×C₁₃₉	—	4	2	no	—
143	C₂²×C₁₃₉	—	4	2	no	—
144	C₂²×C₁₃₉	—	4	2	no	—
145	C₂²×C₁₃₉	#461	1	2	yes	—
146	C₂²×C₁₃₉	C₂×#121	1	1	yes	characteristic
147	C₄×C₁₃₉	—	2	1	no	—
148	C₄×C₁₃₉	—	2	1	no	—
149	C₄×C₁₃₉	—	4	2	no	—
150	C₄×C₁₃₉	—	8	4	no	—
151	C₄×C₁₃₉	#461	1	2	yes	—
152	C₄×C₁₃₉	#473	1	2	yes	—
153	C₄×C₁₃₉	C₂²×SD₁₆	1	1	yes	characteristic
154	C₄×C₁₃₉	C₂×#137	1	1	yes	characteristic
155	C₄×C₁₃₉	#576	1	2	yes	—
156	C₂×C₁₃₉	—	2	2	no	—
157	C₂×C₁₃₉	—	8	2	no	—
158	C₂×C₁₃₉	C₂×#461	1	1	yes	characteristic
159	C₂×C₁₃₉	#3058	1	1	yes	characteristic
160	C₂×C₁₃₉	C₂×#473	1	1	yes	characteristic
161	#19192	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₁₃₉	#19192	1	1	yes	characteristic, direct product factor, Sylow
163	#3206	C₂×C₁₃₉	1	2	yes	—
164	C₂×#500	C₂×C₁₃₉	1	1	yes	characteristic, complemented
165	#2931	C₂×C₁₃₉	1	1	yes	characteristic
166	#1693	C₂×C₁₃₉	1	4	yes	complemented
167	#1673	C₂×C₁₃₉	1	4	yes	complemented
168	#1564	C₂×C₁₃₉	1	1	yes	characteristic, complemented
169	#1549	C₂×C₁₃₉	1	2	yes	complemented
170	C₂×#131	—	2	1	no	—
171	#532	C₂²×C₁₃₉	1	1	yes	characteristic
172	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	2	yes	—
173	#500	—	2	1	no	—
174	#500	—	2	1	no	—
175	#500	C₂²×C₁₃₉	1	2	yes	—
176	#500	C₂²×C₁₃₉	1	2	yes	—
177	#492	—	2	2	no	—
178	#487	—	2	2	no	—
179	#461	C₂²×C₁₃₉	1	8	yes	—
180	#448	—	2	4	no	—
181	#436	—	2	1	no	—
182	#436	C₂²×C₁₃₉	1	2	yes	—
183	C₈×D₄	—	2	1	no	—
184	C₈×D₄	C₂²×C₁₃₉	1	2	yes	—
185	#431	C₂²×C₁₃₉	1	2	yes	—
186	C₂×#107	—	2	1	no	—
187	C₂×#107	—	2	1	no	—
188	#423	C₂²×C₁₃₉	1	1	yes	characteristic
189	#408	—	2	1	no	—
190	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
191	#335	C₂²×C₁₃₉	1	2	yes	—
192	#334	C₂²×C₁₃₉	1	2	yes	—
193	#332	C₂²×C₁₃₉	1	8	yes	—
194	C₂×#41	—	2	1	no	—
195	C₂²×Q₈	—	2	2	no	—
196	C₂×SD₁₆	—	2	8	no	—
197	#131	—	4	4	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
202	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	D₄×C₁₃₉	1	1	yes	characteristic
207	C₄⋊Q₈	C₂³×C₁₃₉	1	4	yes	—
208	#127	D₄×C₁₃₉	1	1	yes	characteristic
209	#124	—	2	1	no	—
210	#121	—	2	1	no	—
211	#121	D₄×C₁₃₉	1	2	yes	—
212	C₄×D₄	—	2	1	no	—
213	C₄×D₄	C₂³×C₁₃₉	1	2	yes	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
217	#107	—	2	2	no	—
218	#107	—	2	2	no	—
219	#107	—	2	2	no	—
220	#107	—	2	2	no	—
221	C₄⋊C₈	—	2	1	no	—
222	C₄⋊C₈	C₂³×C₁₃₉	1	2	yes	—
223	C₄⋊C₈	C₂³×C₁₃₉	1	4	yes	—
224	#103	—	2	8	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₂²⋊C₈	—	2	2	no	—
228	C₄×C₈	—	2	2	no	—
229	C₄×C₈	D₄×C₁₃₉	1	1	yes	characteristic
230	C₄×C₈	D₄×C₁₃₉	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
233	#41	—	4	2	no	—
234	#41	—	4	2	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	8	4	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₂×D₄×C₁₃₉	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
247	C₂²×C₄	#41×C₁₃₉	1	1	yes	characteristic
248	SD₁₆	—	8	8	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	4	no	—
255	C₂×C₈	SD₁₆×C₁₃₉	1	4	yes	complemented
256	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	C₂×D₄×C₁₃₉	1	2	yes	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
269	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
270	C₄²	#41×C₁₃₉	1	1	yes	characteristic
271	C₄²	C₂⁴×C₁₃₉	1	1	yes	characteristic
272	C₂³	—	2	2	no	—
273	C₂³	#121×C₁₃₉	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	4	8	no	—
276	Q₈	—	8	4	no	—
277	D₄	—	2	4	no	—
278	D₄	—	4	4	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	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
289	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
290	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
291	C₂×C₄	C₂×SD₁₆×C₁₃₉	1	2	yes	—
292	C₂×C₄	#137×C₁₃₉	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₁₃₉	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	4	no	—
300	C₈	C₂×SD₁₆×C₁₃₉	1	2	yes	—
301	C₈	C₂×SD₁₆×C₁₃₉	1	2	yes	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	#461×C₁₃₉	1	2	yes	—
308	C₂²	C₂×#121×C₁₃₉	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	4	no	—
313	C₄	#461×C₁₃₉	1	2	yes	—
314	C₄	#473×C₁₃₉	1	2	yes	—
315	C₄	C₂²×SD₁₆×C₁₃₉	1	1	yes	characteristic
316	C₄	C₂×#137×C₁₃₉	1	1	yes	characteristic
317	C₄	#576×C₁₃₉	1	2	yes	—
318	C₂	—	2	2	no	—
319	C₂	—	8	2	no	—
320	C₂	C₂×#461×C₁₃₉	1	1	yes	characteristic
321	C₂	#3058×C₁₃₉	1	1	yes	characteristic
322	C₂	C₂×#473×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₉
prime radical	#532×C₁₃₉	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₂×C₄²×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#500×C₁₃₉	C₂	semidirect
4. #1693×C₁₃₉	C₂	semidirect
5. #1673×C₁₃₉	C₂	semidirect
6. #1564×C₁₃₉	C₂	semidirect
7. #1549×C₁₃₉	C₂	semidirect
93. C₂×C₈×C₁₃₉	SD₁₆	semidirect
161. #19192	C₁₃₉	direct (N × H)
162. C₁₃₉	#19192	direct (N × H)
164. C₂×#500	C₂×C₁₃₉	semidirect
166. #1693	C₂×C₁₃₉	semidirect
167. #1673	C₂×C₁₃₉	semidirect
168. #1564	C₂×C₁₃₉	semidirect
169. #1549	C₂×C₁₃₉	semidirect
255. C₂×C₈	SD₁₆×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#3206×C₁₃₉	2
C₂×#500×C₁₃₉	2
#2931×C₁₃₉	2
#1693×C₁₃₉	2
#1673×C₁₃₉	2
#1564×C₁₃₉	2
#1549×C₁₃₉	2
#19192	139

Subgroups — C139×#19192 #48013788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#19192 #48013788