Subgroups — C₁₃₉×#12585 #48007181

Order 35584.

822 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#413×C₁₃₉	C₂	1	2	yes	maximal
2	#2297×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1394×C₁₃₉	C₂	1	4	yes	maximal, complemented
4	C₂²×#41×C₁₃₉	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₁₃₉	—	2	2	no	—
6	C₂×#117×C₁₃₉	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃₉	C₄	1	1	yes	characteristic
8	#413×C₁₃₉	—	2	4	no	—
9	#413×C₁₃₉	C₄	1	8	yes	complemented
10	C₂×#101×C₁₃₉	C₂²	1	4	yes	—
11	C₂×#100×C₁₃₉	—	2	2	no	—
12	#391×C₁₃₉	C₂²	1	2	yes	—
13	#390×C₁₃₉	C₄	1	2	yes	—
14	#386×C₁₃₉	—	2	2	no	—
15	C₄×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
16	C₂×#41×C₁₃₉	—	4	2	no	—
17	C₂×#41×C₁₃₉	—	4	1	no	—
18	C₂×#41×C₁₃₉	—	4	1	no	—
19	C₂×#41×C₁₃₉	C₂×C₄	1	4	yes	—
20	C₂×#41×C₁₃₉	D₄	1	2	yes	—
21	C₂×#41×C₁₃₉	Q₈	1	2	yes	—
22	C₂²×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
23	C₂²×D₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×D₄×C₁₃₉	D₄	1	2	yes	—
25	C₂³×C₄×C₁₃₉	—	2	1	no	—
26	C₂³×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃₉	D₄	1	2	yes	—
28	C₂×M₁₆×C₁₃₉	—	2	8	no	—
29	C₂×M₁₆×C₁₃₉	—	2	4	no	—
30	C₂×M₁₆×C₁₃₉	—	4	2	no	—
31	C₂×M₁₆×C₁₃₉	—	4	4	no	—
32	C₂²×C₈×C₁₃₉	—	4	2	no	—
33	#117×C₁₃₉	—	4	1	no	—
34	#117×C₁₃₉	—	4	1	no	—
35	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
36	C₂×C₂²⋊C₄×C₁₃₉	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₁₃₉	—	2	4	no	—
38	C₂×C₄²×C₁₃₉	—	2	1	no	—
39	C₂×C₄²×C₁₃₉	—	2	1	no	—
40	C₂×C₄²×C₁₃₉	—	4	2	no	—
41	C₂×C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
42	#101×C₁₃₉	—	2	4	no	—
43	#101×C₁₃₉	C₂×C₄	1	8	yes	complemented
44	#100×C₁₃₉	—	2	8	no	—
45	#95×C₁₃₉	—	2	4	no	—
46	C₂⁴×C₁₃₉	C₂²⋊C₄	1	2	yes	—
47	C₂⁴×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
48	#41×C₁₃₉	—	4	4	no	—
49	#41×C₁₃₉	—	4	4	no	—
50	#41×C₁₃₉	—	4	2	no	—
51	#41×C₁₃₉	—	4	2	no	—
52	#41×C₁₃₉	—	4	2	no	—
53	#41×C₁₃₉	—	4	2	no	—
54	C₂×Q₈×C₁₃₉	—	4	1	no	—
55	C₂×Q₈×C₁₃₉	—	4	1	no	—
56	C₂×Q₈×C₁₃₉	C₂²×C₄	1	2	yes	—
57	C₂×Q₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
58	C₂×Q₈×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×D₄×C₁₃₉	—	4	1	no	—
60	C₂×D₄×C₁₃₉	—	4	1	no	—
61	C₂×D₄×C₁₃₉	—	4	2	no	—
62	C₂×D₄×C₁₃₉	—	4	2	no	—
63	C₂×D₄×C₁₃₉	C₄²	1	2	yes	—
64	C₂×D₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
65	C₂×D₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
66	C₂×D₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₁₃₉	—	2	2	no	—
68	C₂²×C₄×C₁₃₉	—	2	1	no	—
69	C₂²×C₄×C₁₃₉	—	2	2	no	—
70	C₂²×C₄×C₁₃₉	—	2	4	no	—
71	C₂²×C₄×C₁₃₉	—	2	1	no	—
72	C₂²×C₄×C₁₃₉	—	2	2	no	—
73	C₂²×C₄×C₁₃₉	—	2	2	no	—
74	C₂²×C₄×C₁₃₉	—	2	2	no	—
75	C₂²×C₄×C₁₃₉	—	4	4	no	—
76	C₂²×C₄×C₁₃₉	—	4	2	no	—
77	C₂²×C₄×C₁₃₉	C₄²	1	2	yes	—
78	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
79	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
81	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
82	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
84	M₁₆×C₁₃₉	—	2	8	no	—
85	M₁₆×C₁₃₉	—	4	4	no	—
86	M₁₆×C₁₃₉	—	4	8	no	—
87	C₂×C₈×C₁₃₉	—	4	4	no	—
88	C₂×C₈×C₁₃₉	—	4	8	no	—
89	C₄⋊C₄×C₁₃₉	—	4	2	no	—
90	C₂²⋊C₄×C₁₃₉	—	4	8	no	—
91	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
92	C₄²×C₁₃₉	—	4	2	no	—
93	C₄²×C₁₃₉	—	4	2	no	—
94	C₄²×C₁₃₉	—	4	1	no	—
95	C₄²×C₁₃₉	—	4	1	no	—
96	C₂³×C₁₃₉	—	2	1	no	—
97	C₂³×C₁₃₉	—	2	1	no	—
98	C₂³×C₁₃₉	—	2	2	no	—
99	C₂³×C₁₃₉	—	2	2	no	—
100	C₂³×C₁₃₉	—	4	2	no	—
101	C₂³×C₁₃₉	—	4	4	no	—
102	C₂³×C₁₃₉	#95	1	2	yes	—
103	C₂³×C₁₃₉	#95	1	4	yes	—
104	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	Q₈×C₁₃₉	—	4	2	no	—
106	Q₈×C₁₃₉	—	4	1	no	—
107	Q₈×C₁₃₉	—	4	1	no	—
108	D₄×C₁₃₉	—	4	2	no	—
109	D₄×C₁₃₉	—	4	4	no	—
110	D₄×C₁₃₉	—	4	2	no	—
111	D₄×C₁₃₉	—	4	4	no	—
112	C₂×C₄×C₁₃₉	—	2	1	no	—
113	C₂×C₄×C₁₃₉	—	2	1	no	—
114	C₂×C₄×C₁₃₉	—	2	2	no	—
115	C₂×C₄×C₁₃₉	—	2	2	no	—
116	C₂×C₄×C₁₃₉	—	2	4	no	—
117	C₂×C₄×C₁₃₉	—	2	4	no	—
118	C₂×C₄×C₁₃₉	—	2	2	no	—
119	C₂×C₄×C₁₃₉	—	2	2	no	—
120	C₂×C₄×C₁₃₉	—	2	2	no	—
121	C₂×C₄×C₁₃₉	—	2	4	no	—
122	C₂×C₄×C₁₃₉	—	2	2	no	—
123	C₂×C₄×C₁₃₉	—	2	4	no	—
124	C₂×C₄×C₁₃₉	—	2	4	no	—
125	C₂×C₄×C₁₃₉	—	4	2	no	—
126	C₂×C₄×C₁₃₉	—	4	8	no	—
127	C₂×C₄×C₁₃₉	—	4	2	no	—
128	C₂×C₄×C₁₃₉	—	4	4	no	—
129	C₂×C₄×C₁₃₉	—	4	2	no	—
130	C₂×C₄×C₁₃₉	#95	1	2	yes	—
131	C₂×C₄×C₁₃₉	#99	1	1	yes	characteristic
132	C₂×C₄×C₁₃₉	#99	1	2	yes	—
133	C₂×C₄×C₁₃₉	#99	1	1	yes	characteristic
134	C₂×C₄×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
136	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
138	C₈×C₁₃₉	—	4	8	no	—
139	C₂²×C₁₃₉	—	2	2	no	—
140	C₂²×C₁₃₉	—	2	2	no	—
141	C₂²×C₁₃₉	—	2	4	no	—
142	C₂²×C₁₃₉	—	4	2	no	—
143	C₂²×C₁₃₉	—	4	4	no	—
144	C₂²×C₁₃₉	—	4	2	no	—
145	C₂²×C₁₃₉	—	4	2	no	—
146	C₂²×C₁₃₉	—	4	4	no	—
147	C₂²×C₁₃₉	—	8	2	no	—
148	C₂²×C₁₃₉	C₂×#95	1	1	yes	characteristic
149	C₂²×C₁₃₉	C₂×#99	1	1	yes	characteristic
150	C₂²×C₁₃₉	C₂×#99	1	1	yes	characteristic
151	C₄×C₁₃₉	—	2	2	no	—
152	C₄×C₁₃₉	—	2	1	no	—
153	C₄×C₁₃₉	—	2	1	no	—
154	C₄×C₁₃₉	—	2	2	no	—
155	C₄×C₁₃₉	—	4	4	no	—
156	C₄×C₁₃₉	—	8	2	no	—
157	C₄×C₁₃₉	#342	1	2	yes	—
158	C₄×C₁₃₉	#342	1	2	yes	—
159	C₂×C₁₃₉	—	2	1	no	—
160	C₂×C₁₃₉	—	2	1	no	—
161	C₂×C₁₃₉	—	4	2	no	—
162	C₂×C₁₃₉	—	4	4	no	—
163	C₂×C₁₃₉	C₂×#342	1	1	yes	characteristic
164	C₂×C₁₃₉	#2132	1	2	yes	—
165	#12585	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₉	#12585	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×#413	C₂×C₁₃₉	1	2	yes	—
168	#2297	C₂×C₁₃₉	1	1	yes	characteristic
169	#1394	C₂×C₁₃₉	1	4	yes	complemented
170	C₂²×#41	C₂²×C₁₃₉	1	1	yes	characteristic
171	C₂²×M₁₆	—	2	2	no	—
172	C₂×#117	C₄×C₁₃₉	1	1	yes	characteristic
173	C₂²×C₄²	C₄×C₁₃₉	1	1	yes	characteristic
174	#413	—	2	4	no	—
175	#413	C₄×C₁₃₉	1	8	yes	complemented
176	C₂×#101	C₂²×C₁₃₉	1	4	yes	—
177	C₂×#100	—	2	2	no	—
178	#391	C₂²×C₁₃₉	1	2	yes	—
179	#390	C₄×C₁₃₉	1	2	yes	—
180	#386	—	2	2	no	—
181	C₄×C₂²⋊C₄	—	2	2	no	—
182	C₂×#41	—	4	2	no	—
183	C₂×#41	—	4	1	no	—
184	C₂×#41	—	4	1	no	—
185	C₂×#41	C₂×C₄×C₁₃₉	1	4	yes	—
186	C₂×#41	D₄×C₁₃₉	1	2	yes	—
187	C₂×#41	Q₈×C₁₃₉	1	2	yes	—
188	C₂²×Q₈	C₂³×C₁₃₉	1	1	yes	characteristic
189	C₂²×D₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
190	C₂²×D₄	D₄×C₁₃₉	1	2	yes	—
191	C₂³×C₄	—	2	1	no	—
192	C₂³×C₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
193	C₂³×C₄	D₄×C₁₃₉	1	2	yes	—
194	C₂×M₁₆	—	2	8	no	—
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	4	2	no	—
197	C₂×M₁₆	—	4	4	no	—
198	C₂²×C₈	—	4	2	no	—
199	#117	—	4	1	no	—
200	#117	—	4	1	no	—
201	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	1	no	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₄²	—	2	1	no	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	4	2	no	—
207	C₂×C₄²	C₂×C₄×C₁₃₉	1	2	yes	—
208	#101	—	2	4	no	—
209	#101	C₂×C₄×C₁₃₉	1	8	yes	complemented
210	#100	—	2	8	no	—
211	#95	—	2	4	no	—
212	C₂⁴	C₂²⋊C₄×C₁₃₉	1	2	yes	—
213	C₂⁴	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
214	#41	—	4	4	no	—
215	#41	—	4	4	no	—
216	#41	—	4	2	no	—
217	#41	—	4	2	no	—
218	#41	—	4	2	no	—
219	#41	—	4	2	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	1	no	—
222	C₂×Q₈	C₂²×C₄×C₁₃₉	1	2	yes	—
223	C₂×Q₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
224	C₂×Q₈	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
225	C₂×D₄	—	4	1	no	—
226	C₂×D₄	—	4	1	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	C₄²×C₁₃₉	1	2	yes	—
230	C₂×D₄	C₂²⋊C₄×C₁₃₉	1	4	yes	—
231	C₂×D₄	C₄⋊C₄×C₁₃₉	1	2	yes	—
232	C₂×D₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	C₄²×C₁₃₉	1	2	yes	—
244	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	4	yes	—
245	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
246	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	2	yes	—
247	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
248	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
250	M₁₆	—	2	8	no	—
251	M₁₆	—	4	4	no	—
252	M₁₆	—	4	8	no	—
253	C₂×C₈	—	4	4	no	—
254	C₂×C₈	—	4	8	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	8	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	2	no	—
260	C₄²	—	4	1	no	—
261	C₄²	—	4	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	4	no	—
268	C₂³	#95×C₁₃₉	1	2	yes	—
269	C₂³	#95×C₁₃₉	1	4	yes	—
270	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	4	1	no	—
273	Q₈	—	4	1	no	—
274	D₄	—	4	2	no	—
275	D₄	—	4	4	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	4	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	8	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	#95×C₁₃₉	1	2	yes	—
297	C₂×C₄	#99×C₁₃₉	1	2	yes	—
298	C₂×C₄	#99×C₁₃₉	1	1	yes	characteristic
299	C₂×C₄	#99×C₁₃₉	1	1	yes	characteristic
300	C₂×C₄	C₂×C₄²×C₁₃₉	1	1	yes	characteristic
301	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	2	yes	—
302	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
303	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
304	C₈	—	4	8	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	8	2	no	—
314	C₂²	C₂×#95×C₁₃₉	1	1	yes	characteristic
315	C₂²	C₂×#99×C₁₃₉	1	1	yes	characteristic
316	C₂²	C₂×#99×C₁₃₉	1	1	yes	characteristic
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	4	no	—
322	C₄	—	8	2	no	—
323	C₄	#342×C₁₃₉	1	2	yes	—
324	C₄	#342×C₁₃₉	1	2	yes	—
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	4	2	no	—
328	C₂	—	4	4	no	—
329	C₂	C₂×#342×C₁₃₉	1	1	yes	characteristic
330	C₂	#2132×C₁₃₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₉	#99
commutator	C₂×C₄	C₂×C₄²×C₁₃₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₃₉
socle	C₂²×C₁₃₉	C₂×#99
prime radical	C₂²×#41×C₁₃₉	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₉ ◁ C₂²×C₄×C₁₃₉ ◁ C₂²×#41×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1394×C₁₃₉	C₂	semidirect
9. #413×C₁₃₉	C₄	semidirect
43. #101×C₁₃₉	C₂×C₄	semidirect
165. #12585	C₁₃₉	direct (N × H)
166. C₁₃₉	#12585	direct (N × H)
169. #1394	C₂×C₁₃₉	semidirect
175. #413	C₄×C₁₃₉	semidirect
209. #101	C₂×C₄×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#413×C₁₃₉	2
#2297×C₁₃₉	2
#1394×C₁₃₉	2
#12585	139

Subgroups — C139×#12585 #48007181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#12585 #48007181