Subgroups — C₁₃₉×#8612 #48003208

Order 35584.

486 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#1810×C₁₃₉	C₂	1	2	yes	maximal, complemented
3	#1465×C₁₃₉	C₂	1	2	yes	maximal
4	#1293×C₁₃₉	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₁₃₉	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₁₃₉	—	2	2	no	—
9	C₈×D₄×C₁₃₉	—	2	2	no	—
10	C₂×#108×C₁₃₉	—	2	2	no	—
11	C₂×C₄⋊C₈×C₁₃₉	C₄	1	1	yes	characteristic
12	#407×C₁₃₉	C₂²	1	4	yes	—
13	C₂×C₂²⋊C₈×C₁₃₉	C₄	1	1	yes	characteristic
14	C₂×C₂²⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃₉	C₄	1	1	yes	characteristic
16	#341×C₁₃₉	—	2	2	no	—
17	#324×C₁₃₉	—	2	2	no	—
18	#323×C₁₃₉	—	2	2	no	—
19	C₂²×D₄×C₁₃₉	Q₈	1	1	yes	characteristic
20	C₂³×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₁₃₉	—	2	4	no	—
23	C₂²×C₈×C₁₃₉	—	2	1	no	—
24	C₂²×C₈×C₁₃₉	—	2	1	no	—
25	C₂²×C₈×C₁₃₉	—	4	2	no	—
26	C₂²×C₈×C₁₃₉	—	4	2	no	—
27	C₂²×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₉	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃₉	D₄	1	2	yes	—
33	C₄×D₄×C₁₃₉	—	2	2	no	—
34	C₄×D₄×C₁₃₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₃₉	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
39	#108×C₁₃₉	—	4	4	no	—
40	C₄⋊C₈×C₁₃₉	—	2	2	no	—
41	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
42	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
43	C₂²⋊C₈×C₁₃₉	—	2	4	no	—
44	C₂²⋊C₈×C₁₃₉	—	2	4	no	—
45	C₄×C₈×C₁₃₉	—	2	1	no	—
46	C₄×C₈×C₁₃₉	—	2	1	no	—
47	C₂⁴×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
49	C₂×D₄×C₁₃₉	—	2	2	no	—
50	C₂×D₄×C₁₃₉	—	2	2	no	—
51	C₂×D₄×C₁₃₉	—	2	2	no	—
52	C₂²×C₄×C₁₃₉	—	2	1	no	—
53	C₂²×C₄×C₁₃₉	—	2	1	no	—
54	C₂²×C₄×C₁₃₉	—	4	1	no	—
55	C₂²×C₄×C₁₃₉	—	4	1	no	—
56	C₂²×C₄×C₁₃₉	—	4	1	no	—
57	C₂²×C₄×C₁₃₉	—	4	1	no	—
58	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₉	#41	1	2	yes	—
65	M₁₆×C₁₃₉	—	4	4	no	—
66	M₁₆×C₁₃₉	—	4	4	no	—
67	C₂×C₈×C₁₃₉	—	2	4	no	—
68	C₂×C₈×C₁₃₉	—	2	2	no	—
69	C₂×C₈×C₁₃₉	—	2	1	no	—
70	C₂×C₈×C₁₃₉	—	2	1	no	—
71	C₂×C₈×C₁₃₉	—	4	1	no	—
72	C₂×C₈×C₁₃₉	—	4	2	no	—
73	C₂×C₈×C₁₃₉	—	4	2	no	—
74	C₂×C₈×C₁₃₉	—	4	2	no	—
75	C₂×C₈×C₁₃₉	—	4	1	no	—
76	C₂×C₈×C₁₃₉	—	4	1	no	—
77	C₂×C₈×C₁₃₉	—	4	4	no	—
78	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₁₃₉	—	2	2	no	—
82	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
83	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
84	C₄²×C₁₃₉	—	2	1	no	—
85	C₄²×C₁₃₉	—	2	1	no	—
86	C₂³×C₁₃₉	—	2	1	no	—
87	C₂³×C₁₃₉	—	2	1	no	—
88	C₂³×C₁₃₉	—	2	1	no	—
89	C₂³×C₁₃₉	—	2	1	no	—
90	C₂³×C₁₃₉	—	4	1	no	—
91	C₂³×C₁₃₉	—	4	1	no	—
92	C₂³×C₁₃₉	—	4	1	no	—
93	C₂³×C₁₃₉	—	4	1	no	—
94	C₂³×C₁₃₉	#108	1	2	yes	complemented
95	C₂³×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
97	D₄×C₁₃₉	—	2	8	no	—
98	C₂×C₄×C₁₃₉	—	2	1	no	—
99	C₂×C₄×C₁₃₉	—	2	1	no	—
100	C₂×C₄×C₁₃₉	—	2	1	no	—
101	C₂×C₄×C₁₃₉	—	2	2	no	—
102	C₂×C₄×C₁₃₉	—	2	1	no	—
103	C₂×C₄×C₁₃₉	—	2	2	no	—
104	C₂×C₄×C₁₃₉	—	2	1	no	—
105	C₂×C₄×C₁₃₉	—	2	1	no	—
106	C₂×C₄×C₁₃₉	—	4	1	no	—
107	C₂×C₄×C₁₃₉	—	4	2	no	—
108	C₂×C₄×C₁₃₉	—	4	2	no	—
109	C₂×C₄×C₁₃₉	—	4	1	no	—
110	C₂×C₄×C₁₃₉	—	4	1	no	—
111	C₂×C₄×C₁₃₉	—	4	1	no	—
112	C₂×C₄×C₁₃₉	—	4	1	no	—
113	C₂×C₄×C₁₃₉	#108	1	2	yes	complemented
114	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₉	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₃₉	#121	1	2	yes	—
117	C₈×C₁₃₉	—	2	1	no	—
118	C₈×C₁₃₉	—	2	2	no	—
119	C₈×C₁₃₉	—	4	1	no	—
120	C₈×C₁₃₉	—	4	1	no	—
121	C₈×C₁₃₉	—	8	4	no	—
122	C₈×C₁₃₉	#99	1	2	yes	—
123	C₂²×C₁₃₉	—	2	1	no	—
124	C₂²×C₁₃₉	—	2	1	no	—
125	C₂²×C₁₃₉	—	2	2	no	—
126	C₂²×C₁₃₉	—	2	2	no	—
127	C₂²×C₁₃₉	—	2	2	no	—
128	C₂²×C₁₃₉	—	2	2	no	—
129	C₂²×C₁₃₉	—	4	2	no	—
130	C₂²×C₁₃₉	—	4	2	no	—
131	C₂²×C₁₃₉	—	4	2	no	—
132	C₂²×C₁₃₉	—	4	2	no	—
133	C₂²×C₁₃₉	—	4	1	no	—
134	C₂²×C₁₃₉	—	4	1	no	—
135	C₂²×C₁₃₉	#380	1	1	yes	characteristic
136	C₂²×C₁₃₉	C₂×#108	1	1	yes	characteristic
137	C₂²×C₁₃₉	C₂×#108	1	1	yes	characteristic
138	C₄×C₁₃₉	—	2	2	no	—
139	C₄×C₁₃₉	—	2	1	no	—
140	C₄×C₁₃₉	—	2	2	no	—
141	C₄×C₁₃₉	—	4	1	no	—
142	C₄×C₁₃₉	—	4	1	no	—
143	C₄×C₁₃₉	C₂×#99	1	1	yes	characteristic
144	C₄×C₁₃₉	#410	1	1	yes	characteristic
145	C₂×C₁₃₉	—	2	1	no	—
146	C₂×C₁₃₉	—	2	1	no	—
147	C₂×C₁₃₉	—	4	2	no	—
148	C₂×C₁₃₉	—	4	2	no	—
149	C₂×C₁₃₉	#1799	1	1	yes	characteristic
150	C₂×C₁₃₉	#1810	1	1	yes	characteristic
151	C₂×C₁₃₉	#1813	1	1	yes	characteristic
152	#8612	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₁₃₉	#8612	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×C₈×D₄	C₂×C₁₃₉	1	1	yes	characteristic
155	#1810	C₂×C₁₃₉	1	2	yes	complemented
156	#1465	C₂×C₁₃₉	1	2	yes	—
157	#1293	C₂×C₁₃₉	1	2	yes	complemented
158	C₂³×C₈	C₄×C₁₃₉	1	1	yes	characteristic
159	C₂³×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
160	C₂×C₄×D₄	C₂²×C₁₃₉	1	1	yes	characteristic
161	C₈×D₄	—	2	2	no	—
162	C₈×D₄	—	2	2	no	—
163	C₂×#108	—	2	2	no	—
164	C₂×C₄⋊C₈	C₄×C₁₃₉	1	1	yes	characteristic
165	#407	C₂²×C₁₃₉	1	4	yes	—
166	C₂×C₂²⋊C₈	C₄×C₁₃₉	1	1	yes	characteristic
167	C₂×C₂²⋊C₈	C₂²×C₁₃₉	1	1	yes	characteristic
168	C₂×C₄×C₈	C₄×C₁₃₉	1	1	yes	characteristic
169	#341	—	2	2	no	—
170	#324	—	2	2	no	—
171	#323	—	2	2	no	—
172	C₂²×D₄	Q₈×C₁₃₉	1	1	yes	characteristic
173	C₂³×C₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
174	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
175	C₂×M₁₆	—	2	4	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	4	2	no	—
179	C₂²×C₈	—	4	2	no	—
180	C₂²×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
181	C₂²×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
182	C₂²×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
183	C₂²×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
184	C₂²×C₈	D₄×C₁₃₉	1	2	yes	—
185	C₂²×C₈	D₄×C₁₃₉	1	2	yes	—
186	C₄×D₄	—	2	2	no	—
187	C₄×D₄	—	2	2	no	—
188	C₂×C₄⋊C₄	D₄×C₁₃₉	1	1	yes	characteristic
189	C₂×C₂²⋊C₄	D₄×C₁₃₉	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	Q₈×C₁₃₉	1	1	yes	characteristic
191	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
192	#108	—	4	4	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₂²⋊C₈	—	2	4	no	—
197	C₂²⋊C₈	—	2	4	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	C₂⁴	C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
201	C₂⁴	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	4	1	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
212	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
213	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
217	C₂²×C₄	#41×C₁₃₉	1	2	yes	—
218	M₁₆	—	4	4	no	—
219	M₁₆	—	4	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	2	yes	—
233	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	2	2	no	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	4	1	no	—
244	C₂³	—	4	1	no	—
245	C₂³	—	4	1	no	—
246	C₂³	—	4	1	no	—
247	C₂³	#108×C₁₃₉	1	2	yes	complemented
248	C₂³	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
249	C₂³	C₂²⋊Q₈×C₁₃₉	1	2	yes	—
250	D₄	—	2	8	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	#108×C₁₃₉	1	2	yes	complemented
267	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
268	C₂×C₄	#117×C₁₃₉	1	1	yes	characteristic
269	C₂×C₄	#121×C₁₃₉	1	2	yes	—
270	C₈	—	2	1	no	—
271	C₈	—	2	2	no	—
272	C₈	—	4	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	8	4	no	—
275	C₈	#99×C₁₃₉	1	2	yes	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	1	no	—
287	C₂²	—	4	1	no	—
288	C₂²	#380×C₁₃₉	1	1	yes	characteristic
289	C₂²	C₂×#108×C₁₃₉	1	1	yes	characteristic
290	C₂²	C₂×#108×C₁₃₉	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	C₂×#99×C₁₃₉	1	1	yes	characteristic
297	C₄	#410×C₁₃₉	1	1	yes	characteristic
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	—	4	2	no	—
301	C₂	—	4	2	no	—
302	C₂	#1799×C₁₃₉	1	1	yes	characteristic
303	C₂	#1810×C₁₃₉	1	1	yes	characteristic
304	C₂	#1813×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	#380
commutator	C₂²×C₄	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₉
prime radical	C₂²×D₄×C₁₃₉	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₂³×C₄×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1810×C₁₃₉	C₂	semidirect
4. #1293×C₁₃₉	C₂	semidirect
94. C₂³×C₁₃₉	#108	semidirect
113. C₂×C₄×C₁₃₉	#108	semidirect
152. #8612	C₁₃₉	direct (N × H)
153. C₁₃₉	#8612	direct (N × H)
155. #1810	C₂×C₁₃₉	semidirect
157. #1293	C₂×C₁₃₉	semidirect
247. C₂³	#108×C₁₃₉	semidirect
266. C₂×C₄	#108×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₁₃₉	2
#1810×C₁₃₉	2
#1465×C₁₃₉	2
#1293×C₁₃₉	2
#8612	139

Subgroups — C139×#8612 #48003208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#8612 #48003208