Subgroups — C₁₃₉×#8398 #48002994

Order 35584.

782 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#1801×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#1549×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	#1486×C₁₃₉	C₂	1	2	yes	maximal
5	#1461×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#554×C₁₃₉	—	2	4	no	—
7	#554×C₁₃₉	—	2	4	no	—
8	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂×#125×C₁₃₉	C₄	1	2	yes	—
10	C₂×C₂²⋊Q₈×C₁₃₉	—	2	1	no	—
11	C₂×C₂²⋊Q₈×C₁₃₉	—	2	2	no	—
12	C₂×C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₁₃₉	C₄	1	2	yes	—
14	C₂×C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₁₃₉	—	2	1	no	—
16	#431×C₁₃₉	C₂²	1	2	yes	—
17	#423×C₁₃₉	C₂²	1	2	yes	—
18	C₂×#101×C₁₃₉	—	2	1	no	—
19	#343×C₁₃₉	—	2	1	no	—
20	#332×C₁₃₉	—	2	4	no	—
21	D₄⋊C₈×C₁₃₉	—	2	4	no	—
22	#324×C₁₃₉	—	2	2	no	—
23	C₂²×Q₈×C₁₃₉	—	2	1	no	—
24	C₂²×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
25	C₂²×D₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₃₉	—	2	1	no	—
27	C₂³×C₄×C₁₃₉	—	2	2	no	—
28	C₂×M₁₆×C₁₃₉	—	2	2	no	—
29	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
30	C₄⋊Q₈×C₁₃₉	D₄	1	4	yes	—
31	#125×C₁₃₉	—	2	4	no	—
32	#125×C₁₃₉	—	2	4	no	—
33	C₂²⋊Q₈×C₁₃₉	—	2	8	no	—
34	C₂²⋊Q₈×C₁₃₉	—	2	16	no	—
35	C₄×Q₈×C₁₃₉	—	2	2	no	—
36	C₄×Q₈×C₁₃₉	—	2	2	no	—
37	C₄×D₄×C₁₃₉	—	2	4	no	—
38	C₄×D₄×C₁₃₉	—	2	4	no	—
39	C₄×D₄×C₁₃₉	—	2	2	no	—
40	C₄×D₄×C₁₃₉	D₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₃₉	—	2	8	no	—
42	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₁₃₉	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₁₃₉	—	2	1	no	—
51	C₂×C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₁₃₉	—	2	2	no	—
54	C₄⋊C₈×C₁₃₉	—	2	2	no	—
55	C₄⋊C₈×C₁₃₉	—	2	2	no	—
56	#101×C₁₃₉	—	4	2	no	—
57	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
58	C₄×C₈×C₁₃₉	—	2	2	no	—
59	C₂⁴×C₁₃₉	—	2	1	no	—
60	C₂×Q₈×C₁₃₉	—	2	2	no	—
61	C₂×Q₈×C₁₃₉	—	2	4	no	—
62	C₂×Q₈×C₁₃₉	—	2	2	no	—
63	C₂×Q₈×C₁₃₉	—	2	1	no	—
64	C₂×Q₈×C₁₃₉	—	2	1	no	—
65	C₂×Q₈×C₁₃₉	—	8	2	no	—
66	C₂×D₄×C₁₃₉	—	2	4	no	—
67	C₂×D₄×C₁₃₉	—	2	1	no	—
68	C₂×D₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₃₉	—	2	1	no	—
70	C₂²×C₄×C₁₃₉	—	2	2	no	—
71	C₂²×C₄×C₁₃₉	—	2	8	no	—
72	C₂²×C₄×C₁₃₉	—	2	4	no	—
73	C₂²×C₄×C₁₃₉	—	4	2	no	—
74	C₂²×C₄×C₁₃₉	—	4	1	no	—
75	C₂²×C₄×C₁₃₉	—	4	1	no	—
76	C₂²×C₄×C₁₃₉	—	8	4	no	—
77	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
82	M₁₆×C₁₃₉	—	4	2	no	—
83	M₁₆×C₁₃₉	—	4	2	no	—
84	C₂×C₈×C₁₃₉	—	4	2	no	—
85	C₂×C₈×C₁₃₉	—	4	2	no	—
86	C₄⋊C₄×C₁₃₉	—	2	8	no	—
87	C₄⋊C₄×C₁₃₉	—	2	8	no	—
88	C₄⋊C₄×C₁₃₉	—	2	1	no	—
89	C₄⋊C₄×C₁₃₉	—	2	2	no	—
90	C₄⋊C₄×C₁₃₉	—	2	2	no	—
91	C₄⋊C₄×C₁₃₉	—	2	4	no	—
92	C₄⋊C₄×C₁₃₉	—	2	4	no	—
93	C₄⋊C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
94	C₄⋊C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
95	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
96	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
97	C₄²×C₁₃₉	—	2	2	no	—
98	C₄²×C₁₃₉	—	2	2	no	—
99	C₄²×C₁₃₉	C₂×D₄	1	2	yes	—
100	C₄²×C₁₃₉	C₂×D₄	1	2	yes	—
101	C₂³×C₁₃₉	—	2	4	no	—
102	C₂³×C₁₃₉	—	2	2	no	—
103	C₂³×C₁₃₉	—	4	2	no	—
104	C₂³×C₁₃₉	—	4	1	no	—
105	C₂³×C₁₃₉	—	4	1	no	—
106	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	Q₈×C₁₃₉	—	4	2	no	—
108	Q₈×C₁₃₉	—	4	2	no	—
109	Q₈×C₁₃₉	—	8	4	no	—
110	D₄×C₁₃₉	—	2	4	no	—
111	D₄×C₁₃₉	—	2	2	no	—
112	D₄×C₁₃₉	#101	1	4	yes	complemented
113	C₂×C₄×C₁₃₉	—	2	2	no	—
114	C₂×C₄×C₁₃₉	—	2	2	no	—
115	C₂×C₄×C₁₃₉	—	2	2	no	—
116	C₂×C₄×C₁₃₉	—	2	2	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	—	4	1	no	—
121	C₂×C₄×C₁₃₉	—	4	4	no	—
122	C₂×C₄×C₁₃₉	—	4	1	no	—
123	C₂×C₄×C₁₃₉	—	4	1	no	—
124	C₂×C₄×C₁₃₉	—	4	4	no	—
125	C₂×C₄×C₁₃₉	—	4	4	no	—
126	C₂×C₄×C₁₃₉	—	4	2	no	—
127	C₂×C₄×C₁₃₉	—	8	8	no	—
128	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃₉	#120	1	2	yes	—
131	C₂×C₄×C₁₃₉	#120	1	2	yes	—
132	C₂×C₄×C₁₃₉	#135	1	2	yes	—
133	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₁₃₉	#137	1	1	yes	characteristic
135	C₈×C₁₃₉	—	8	2	no	—
136	C₈×C₁₃₉	—	8	2	no	—
137	C₂²×C₁₃₉	—	2	1	no	—
138	C₂²×C₁₃₉	—	2	1	no	—
139	C₂²×C₁₃₉	—	2	4	no	—
140	C₂²×C₁₃₉	—	4	4	no	—
141	C₂²×C₁₃₉	—	4	4	no	—
142	C₂²×C₁₃₉	—	4	2	no	—
143	C₂²×C₁₃₉	—	4	2	no	—
144	C₂²×C₁₃₉	#379	1	1	yes	characteristic
145	C₂²×C₁₃₉	#416	1	1	yes	characteristic
146	C₂²×C₁₃₉	#417	1	1	yes	characteristic
147	C₄×C₁₃₉	—	2	2	no	—
148	C₄×C₁₃₉	—	2	1	no	—
149	C₄×C₁₃₉	—	4	4	no	—
150	C₄×C₁₃₉	—	4	2	no	—
151	C₄×C₁₃₉	—	4	2	no	—
152	C₄×C₁₃₉	—	4	2	no	—
153	C₄×C₁₃₉	C₂×#101	1	2	yes	—
154	C₄×C₁₃₉	#449	1	2	yes	—
155	C₄×C₁₃₉	#452	1	2	yes	—
156	C₂×C₁₃₉	—	2	1	no	—
157	C₂×C₁₃₉	—	2	1	no	—
158	C₂×C₁₃₉	—	4	4	no	—
159	C₂×C₁₃₉	#1784	1	1	yes	characteristic
160	C₂×C₁₃₉	#1788	1	1	yes	characteristic
161	C₂×C₁₃₉	#1795	1	1	yes	characteristic
162	#8398	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₁₃₉	#8398	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×#554	C₂×C₁₃₉	1	1	yes	characteristic
165	#1801	C₂×C₁₃₉	1	1	yes	characteristic, complemented
166	#1549	C₂×C₁₃₉	1	2	yes	complemented
167	#1486	C₂×C₁₃₉	1	2	yes	—
168	#1461	C₂×C₁₃₉	1	1	yes	characteristic, complemented
169	#554	—	2	4	no	—
170	#554	—	2	4	no	—
171	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
172	C₂×#125	C₄×C₁₃₉	1	2	yes	—
173	C₂×C₂²⋊Q₈	—	2	1	no	—
174	C₂×C₂²⋊Q₈	—	2	2	no	—
175	C₂×C₄×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
176	C₂×C₄×D₄	C₄×C₁₃₉	1	2	yes	—
177	C₂×C₄×D₄	C₂²×C₁₃₉	1	1	yes	characteristic
178	C₂²×C₄⋊C₄	—	2	1	no	—
179	#431	C₂²×C₁₃₉	1	2	yes	—
180	#423	C₂²×C₁₃₉	1	2	yes	—
181	C₂×#101	—	2	1	no	—
182	#343	—	2	1	no	—
183	#332	—	2	4	no	—
184	D₄⋊C₈	—	2	4	no	—
185	#324	—	2	2	no	—
186	C₂²×Q₈	—	2	1	no	—
187	C₂²×Q₈	D₄×C₁₃₉	1	1	yes	characteristic
188	C₂²×D₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
189	C₂³×C₄	—	2	1	no	—
190	C₂³×C₄	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	D₄×C₁₃₉	1	4	yes	—
194	#125	—	2	4	no	—
195	#125	—	2	4	no	—
196	C₂²⋊Q₈	—	2	8	no	—
197	C₂²⋊Q₈	—	2	16	no	—
198	C₄×Q₈	—	2	2	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×D₄	—	2	4	no	—
201	C₄×D₄	—	2	4	no	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	D₄×C₁₃₉	1	4	yes	—
204	C₂×C₄⋊C₄	—	2	8	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
210	C₂×C₄⋊C₄	D₄×C₁₃₉	1	1	yes	characteristic
211	C₂×C₄⋊C₄	D₄×C₁₃₉	1	2	yes	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	2	1	no	—
214	C₂×C₄²	C₂×C₄×C₁₃₉	1	2	yes	—
215	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	2	2	no	—
219	#101	—	4	2	no	—
220	C₂²⋊C₈	—	2	2	no	—
221	C₄×C₈	—	2	2	no	—
222	C₂⁴	—	2	1	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	1	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	8	2	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	8	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	8	4	no	—
240	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
241	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
242	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	C₂²⋊C₄×C₁₃₉	1	4	yes	—
257	C₄⋊C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
258	C₂²⋊C₄	—	2	8	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	C₂×D₄×C₁₃₉	1	2	yes	—
263	C₄²	C₂×D₄×C₁₃₉	1	2	yes	—
264	C₂³	—	2	4	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	4	2	no	—
267	C₂³	—	4	1	no	—
268	C₂³	—	4	1	no	—
269	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
270	Q₈	—	4	2	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	8	4	no	—
273	D₄	—	2	4	no	—
274	D₄	—	2	2	no	—
275	D₄	#101×C₁₃₉	1	4	yes	complemented
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	8	8	no	—
291	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
292	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
293	C₂×C₄	#120×C₁₃₉	1	2	yes	—
294	C₂×C₄	#120×C₁₃₉	1	2	yes	—
295	C₂×C₄	#135×C₁₃₉	1	2	yes	—
296	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	1	yes	characteristic
297	C₂×C₄	#137×C₁₃₉	1	1	yes	characteristic
298	C₈	—	8	2	no	—
299	C₈	—	8	2	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	#379×C₁₃₉	1	1	yes	characteristic
308	C₂²	#416×C₁₃₉	1	1	yes	characteristic
309	C₂²	#417×C₁₃₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	4	no	—
315	C₄	—	4	2	no	—
316	C₄	C₂×#101×C₁₃₉	1	2	yes	—
317	C₄	#449×C₁₃₉	1	2	yes	—
318	C₄	#452×C₁₃₉	1	2	yes	—
319	C₂	—	2	1	no	—
320	C₂	—	2	1	no	—
321	C₂	—	4	4	no	—
322	C₂	#1784×C₁₃₉	1	1	yes	characteristic
323	C₂	#1788×C₁₃₉	1	1	yes	characteristic
324	C₂	#1795×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	#379
commutator	C₂²×C₄	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
prime radical	C₂²×D₄×C₁₃₉	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ 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. #1801×C₁₃₉	C₂	semidirect
3. #1549×C₁₃₉	C₂	semidirect
5. #1461×C₁₃₉	C₂	semidirect
112. D₄×C₁₃₉	#101	semidirect
162. #8398	C₁₃₉	direct (N × H)
163. C₁₃₉	#8398	direct (N × H)
165. #1801	C₂×C₁₃₉	semidirect
166. #1549	C₂×C₁₃₉	semidirect
168. #1461	C₂×C₁₃₉	semidirect
275. D₄	#101×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#554×C₁₃₉	2
#1801×C₁₃₉	2
#1549×C₁₃₉	2
#1486×C₁₃₉	2
#1461×C₁₃₉	2
#8398	139

Subgroups — C139×#8398 #48002994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#8398 #48002994