Subgroups — C₁₃₉×#18696 #48013292

Order 35584.

946 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3548×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#3253×C₁₃₉	C₂	1	2	yes	maximal, complemented
3	#3131×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#1635×C₁₃₉	C₂	1	4	yes	maximal, complemented
5	#1625×C₁₃₉	C₂	1	4	yes	maximal, complemented
6	#1561×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#1492×C₁₃₉	C₂	1	2	yes	maximal, complemented
8	#562×C₁₃₉	—	2	2	no	—
9	#554×C₁₃₉	—	2	2	no	—
10	#550×C₁₃₉	—	2	2	no	—
11	#547×C₁₃₉	C₂²	1	4	yes	—
12	#540×C₁₃₉	—	2	2	no	—
13	#537×C₁₃₉	—	2	4	no	—
14	#533×C₁₃₉	C₂²	1	1	yes	characteristic
15	#532×C₁₃₉	C₂²	1	1	yes	characteristic
16	#532×C₁₃₉	C₂²	1	2	yes	—
17	#488×C₁₃₉	—	2	2	no	—
18	#487×C₁₃₉	C₂²	1	2	yes	—
19	#486×C₁₃₉	C₂²	1	2	yes	—
20	#476×C₁₃₉	—	2	2	no	—
21	#476×C₁₃₉	C₂²	1	4	yes	—
22	#469×C₁₃₉	—	2	4	no	—
23	#459×C₁₃₉	—	2	2	no	—
24	#459×C₁₃₉	C₂²	1	4	yes	—
25	#423×C₁₃₉	C₂²	1	2	yes	—
26	#416×C₁₃₉	—	2	2	no	—
27	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
28	#332×C₁₃₉	C₂²	1	4	yes	—
29	D₄⋊C₈×C₁₃₉	C₂²	1	8	yes	—
30	C₂×#41×C₁₃₉	—	2	4	no	—
31	C₂×#41×C₁₃₉	—	2	2	no	—
32	C₂×#41×C₁₃₉	—	2	1	no	—
33	C₂×D₈×C₁₃₉	—	2	4	no	—
34	C₂×M₁₆×C₁₃₉	—	2	2	no	—
35	C₂²×C₈×C₁₃₉	—	2	1	no	—
36	C₄⋊Q₈×C₁₃₉	D₄	1	2	yes	—
37	C₄⋊Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₁₃₉	C₂³	1	2	yes	—
39	#127×C₁₃₉	—	2	2	no	—
40	#127×C₁₃₉	D₄	1	2	yes	—
41	#127×C₁₃₉	C₂³	1	1	yes	characteristic
42	#125×C₁₃₉	—	2	2	no	—
43	#125×C₁₃₉	D₄	1	2	yes	—
44	#124×C₁₃₉	—	2	2	no	—
45	#124×C₁₃₉	—	2	1	no	—
46	#123×C₁₃₉	—	2	8	no	—
47	C₂²⋊Q₈×C₁₃₉	—	2	8	no	—
48	C₂²⋊Q₈×C₁₃₉	—	2	2	no	—
49	C₂²⋊Q₈×C₁₃₉	—	2	4	no	—
50	#121×C₁₃₉	—	2	8	no	—
51	#121×C₁₃₉	—	2	8	no	—
52	#121×C₁₃₉	—	2	4	no	—
53	#121×C₁₃₉	—	2	2	no	—
54	#121×C₁₃₉	—	2	2	no	—
55	#121×C₁₃₉	D₄	1	4	yes	—
56	C₄×Q₈×C₁₃₉	—	2	2	no	—
57	C₄×D₄×C₁₃₉	—	2	4	no	—
58	C₄×D₄×C₁₃₉	—	2	2	no	—
59	C₄×D₄×C₁₃₉	—	2	2	no	—
60	C₄×D₄×C₁₃₉	C₂³	1	4	yes	—
61	#117×C₁₃₉	D₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
63	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
64	#106×C₁₃₉	—	2	4	no	—
65	C₄⋊C₈×C₁₃₉	—	2	2	no	—
66	C₄⋊C₈×C₁₃₉	C₂³	1	4	yes	—
67	#103×C₁₃₉	—	2	4	no	—
68	#102×C₁₃₉	—	2	4	no	—
69	#102×C₁₃₉	—	2	4	no	—
70	#102×C₁₃₉	—	2	4	no	—
71	C₄×C₈×C₁₃₉	—	2	1	no	—
72	C₄×C₈×C₁₃₉	C₂³	1	2	yes	—
73	#41×C₁₃₉	—	4	4	no	—
74	#41×C₁₃₉	—	8	2	no	—
75	#41×C₁₃₉	—	8	2	no	—
76	#41×C₁₃₉	—	8	4	no	—
77	#41×C₁₃₉	—	8	4	no	—
78	C₂×Q₈×C₁₃₉	—	2	2	no	—
79	C₂×Q₈×C₁₃₉	—	2	2	no	—
80	C₂×Q₈×C₁₃₉	—	2	1	no	—
81	C₂×D₄×C₁₃₉	—	2	8	no	—
82	C₂×D₄×C₁₃₉	—	2	4	no	—
83	C₂×D₄×C₁₃₉	—	2	2	no	—
84	C₂×D₄×C₁₃₉	—	2	2	no	—
85	C₂×D₄×C₁₃₉	—	2	1	no	—
86	C₂×D₄×C₁₃₉	—	2	2	no	—
87	C₂×D₄×C₁₃₉	—	4	1	no	—
88	C₂×D₄×C₁₃₉	C₂×D₄	1	4	yes	—
89	C₂²×C₄×C₁₃₉	—	2	8	no	—
90	C₂²×C₄×C₁₃₉	—	2	4	no	—
91	C₂²×C₄×C₁₃₉	—	4	1	no	—
92	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
94	D₈×C₁₃₉	—	8	4	no	—
95	M₁₆×C₁₃₉	—	8	2	no	—
96	C₂×C₈×C₁₃₉	—	2	2	no	—
97	C₂×C₈×C₁₃₉	—	2	4	no	—
98	C₂×C₈×C₁₃₉	—	4	2	no	—
99	C₄⋊C₄×C₁₃₉	—	2	8	no	—
100	C₄⋊C₄×C₁₃₉	—	2	2	no	—
101	C₄⋊C₄×C₁₃₉	—	2	4	no	—
102	C₄⋊C₄×C₁₃₉	—	2	2	no	—
103	C₄⋊C₄×C₁₃₉	—	2	2	no	—
104	C₄⋊C₄×C₁₃₉	C₂×D₄	1	4	yes	—
105	C₄⋊C₄×C₁₃₉	C₂×D₄	1	4	yes	—
106	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
107	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
108	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
109	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
110	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
111	C₄²×C₁₃₉	—	2	2	no	—
112	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₃₉	C₂×D₄	1	2	yes	—
114	C₄²×C₁₃₉	C₂⁴	1	1	yes	characteristic
115	C₂³×C₁₃₉	—	2	4	no	—
116	C₂³×C₁₃₉	—	4	1	no	—
117	C₂³×C₁₃₉	#120	1	1	yes	characteristic
118	Q₈×C₁₃₉	—	4	2	no	—
119	Q₈×C₁₃₉	—	4	2	no	—
120	Q₈×C₁₃₉	—	4	2	no	—
121	Q₈×C₁₃₉	—	8	2	no	—
122	D₄×C₁₃₉	—	2	4	no	—
123	D₄×C₁₃₉	—	2	4	no	—
124	D₄×C₁₃₉	—	4	2	no	—
125	D₄×C₁₃₉	—	4	2	no	—
126	D₄×C₁₃₉	—	4	2	no	—
127	D₄×C₁₃₉	—	4	4	no	—
128	D₄×C₁₃₉	—	8	2	no	—
129	D₄×C₁₃₉	—	8	4	no	—
130	D₄×C₁₃₉	—	8	8	no	—
131	D₄×C₁₃₉	—	8	2	no	—
132	C₂×C₄×C₁₃₉	—	2	2	no	—
133	C₂×C₄×C₁₃₉	—	2	2	no	—
134	C₂×C₄×C₁₃₉	—	2	2	no	—
135	C₂×C₄×C₁₃₉	—	2	4	no	—
136	C₂×C₄×C₁₃₉	—	2	4	no	—
137	C₂×C₄×C₁₃₉	—	4	4	no	—
138	C₂×C₄×C₁₃₉	—	4	4	no	—
139	C₂×C₄×C₁₃₉	—	4	1	no	—
140	C₂×C₄×C₁₃₉	—	8	8	no	—
141	C₂×C₄×C₁₃₉	—	8	2	no	—
142	C₂×C₄×C₁₃₉	#120	1	2	yes	—
143	C₂×C₄×C₁₃₉	#120	1	1	yes	characteristic
144	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₁₃₉	C₂²×D₄	1	2	yes	—
146	C₈×C₁₃₉	—	4	2	no	—
147	C₈×C₁₃₉	—	4	4	no	—
148	C₂²×C₁₃₉	—	2	2	no	—
149	C₂²×C₁₃₉	—	2	1	no	—
150	C₂²×C₁₃₉	—	4	4	no	—
151	C₂²×C₁₃₉	—	4	4	no	—
152	C₂²×C₁₃₉	—	4	4	no	—
153	C₂²×C₁₃₉	—	8	2	no	—
154	C₂²×C₁₃₉	—	8	1	no	—
155	C₂²×C₁₃₉	C₂×#120	1	1	yes	characteristic
156	C₄×C₁₃₉	—	2	2	no	—
157	C₄×C₁₃₉	—	2	1	no	—
158	C₄×C₁₃₉	—	4	4	no	—
159	C₄×C₁₃₉	—	4	4	no	—
160	C₄×C₁₃₉	—	8	1	no	—
161	C₄×C₁₃₉	#454	1	2	yes	—
162	C₄×C₁₃₉	#576	1	2	yes	—
163	C₄×C₁₃₉	#577	1	2	yes	—
164	C₂×C₁₃₉	—	4	1	no	—
165	C₂×C₁₃₉	—	8	4	no	—
166	C₂×C₁₃₉	—	16	1	no	—
167	C₂×C₁₃₉	#3012	1	2	yes	—
168	C₂×C₁₃₉	C₂×#454	1	1	yes	characteristic
169	#18696	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₁₃₉	#18696	1	1	yes	characteristic, direct product factor, Sylow
171	#3548	C₂×C₁₃₉	1	1	yes	characteristic, complemented
172	#3253	C₂×C₁₃₉	1	2	yes	complemented
173	#3131	C₂×C₁₃₉	1	1	yes	characteristic, complemented
174	#1635	C₂×C₁₃₉	1	4	yes	complemented
175	#1625	C₂×C₁₃₉	1	4	yes	complemented
176	#1561	C₂×C₁₃₉	1	1	yes	characteristic, complemented
177	#1492	C₂×C₁₃₉	1	2	yes	complemented
178	#562	—	2	2	no	—
179	#554	—	2	2	no	—
180	#550	—	2	2	no	—
181	#547	C₂²×C₁₃₉	1	4	yes	—
182	#540	—	2	2	no	—
183	#537	—	2	4	no	—
184	#533	C₂²×C₁₃₉	1	1	yes	characteristic
185	#532	C₂²×C₁₃₉	1	1	yes	characteristic
186	#532	C₂²×C₁₃₉	1	2	yes	—
187	#488	—	2	2	no	—
188	#487	C₂²×C₁₃₉	1	2	yes	—
189	#486	C₂²×C₁₃₉	1	2	yes	—
190	#476	—	2	2	no	—
191	#476	C₂²×C₁₃₉	1	4	yes	—
192	#469	—	2	4	no	—
193	#459	—	2	2	no	—
194	#459	C₂²×C₁₃₉	1	4	yes	—
195	#423	C₂²×C₁₃₉	1	2	yes	—
196	#416	—	2	2	no	—
197	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
198	#332	C₂²×C₁₃₉	1	4	yes	—
199	D₄⋊C₈	C₂²×C₁₃₉	1	8	yes	—
200	C₂×#41	—	2	4	no	—
201	C₂×#41	—	2	2	no	—
202	C₂×#41	—	2	1	no	—
203	C₂×D₈	—	2	4	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₄⋊Q₈	D₄×C₁₃₉	1	2	yes	—
207	C₄⋊Q₈	C₂³×C₁₃₉	1	1	yes	characteristic
208	C₄⋊Q₈	C₂³×C₁₃₉	1	2	yes	—
209	#127	—	2	2	no	—
210	#127	D₄×C₁₃₉	1	2	yes	—
211	#127	C₂³×C₁₃₉	1	1	yes	characteristic
212	#125	—	2	2	no	—
213	#125	D₄×C₁₃₉	1	2	yes	—
214	#124	—	2	2	no	—
215	#124	—	2	1	no	—
216	#123	—	2	8	no	—
217	C₂²⋊Q₈	—	2	8	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	2	4	no	—
220	#121	—	2	8	no	—
221	#121	—	2	8	no	—
222	#121	—	2	4	no	—
223	#121	—	2	2	no	—
224	#121	—	2	2	no	—
225	#121	D₄×C₁₃₉	1	4	yes	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×D₄	—	2	4	no	—
228	C₄×D₄	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	C₂³×C₁₃₉	1	4	yes	—
231	#117	D₄×C₁₃₉	1	2	yes	—
232	C₂×C₄⋊C₄	—	2	4	no	—
233	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
234	#106	—	2	4	no	—
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	C₂³×C₁₃₉	1	4	yes	—
237	#103	—	2	4	no	—
238	#102	—	2	4	no	—
239	#102	—	2	4	no	—
240	#102	—	2	4	no	—
241	C₄×C₈	—	2	1	no	—
242	C₄×C₈	C₂³×C₁₃₉	1	2	yes	—
243	#41	—	4	4	no	—
244	#41	—	8	2	no	—
245	#41	—	8	2	no	—
246	#41	—	8	4	no	—
247	#41	—	8	4	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	1	no	—
251	C₂×D₄	—	2	8	no	—
252	C₂×D₄	—	2	4	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	C₂×D₄×C₁₃₉	1	4	yes	—
259	C₂²×C₄	—	2	8	no	—
260	C₂²×C₄	—	2	4	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
264	D₈	—	8	4	no	—
265	M₁₆	—	8	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	4	2	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	C₂×D₄×C₁₃₉	1	4	yes	—
275	C₄⋊C₄	C₂×D₄×C₁₃₉	1	4	yes	—
276	C₂²⋊C₄	—	2	8	no	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	4	no	—
279	C₂²⋊C₄	—	2	4	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₁₃₉	1	2	yes	—
284	C₄²	C₂⁴×C₁₃₉	1	1	yes	characteristic
285	C₂³	—	2	4	no	—
286	C₂³	—	4	1	no	—
287	C₂³	#120×C₁₃₉	1	1	yes	characteristic
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	2	4	no	—
293	D₄	—	2	4	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	4	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	2	no	—
299	D₄	—	8	4	no	—
300	D₄	—	8	8	no	—
301	D₄	—	8	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	8	8	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	#120×C₁₃₉	1	2	yes	—
313	C₂×C₄	#120×C₁₃₉	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₄×C₁₃₉	1	2	yes	—
316	C₈	—	4	2	no	—
317	C₈	—	4	4	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	4	4	no	—
321	C₂²	—	4	4	no	—
322	C₂²	—	4	4	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	1	no	—
325	C₂²	C₂×#120×C₁₃₉	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	4	no	—
329	C₄	—	4	4	no	—
330	C₄	—	8	1	no	—
331	C₄	#454×C₁₃₉	1	2	yes	—
332	C₄	#576×C₁₃₉	1	2	yes	—
333	C₄	#577×C₁₃₉	1	2	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	8	4	no	—
336	C₂	—	16	1	no	—
337	C₂	#3012×C₁₃₉	1	2	yes	—
338	C₂	C₂×#454×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₉
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
1. #3548×C₁₃₉	C₂	semidirect
2. #3253×C₁₃₉	C₂	semidirect
3. #3131×C₁₃₉	C₂	semidirect
4. #1635×C₁₃₉	C₂	semidirect
5. #1625×C₁₃₉	C₂	semidirect
6. #1561×C₁₃₉	C₂	semidirect
7. #1492×C₁₃₉	C₂	semidirect
169. #18696	C₁₃₉	direct (N × H)
170. C₁₃₉	#18696	direct (N × H)
171. #3548	C₂×C₁₃₉	semidirect
172. #3253	C₂×C₁₃₉	semidirect
173. #3131	C₂×C₁₃₉	semidirect
174. #1635	C₂×C₁₃₉	semidirect
175. #1625	C₂×C₁₃₉	semidirect
176. #1561	C₂×C₁₃₉	semidirect
177. #1492	C₂×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#3548×C₁₃₉	2
#3253×C₁₃₉	2
#3131×C₁₃₉	2
#1635×C₁₃₉	2
#1625×C₁₃₉	2
#1561×C₁₃₉	2
#1492×C₁₃₉	2
#18696	139

Subgroups — C139×#18696 #48013292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#18696 #48013292