Subgroups — C₁₃₉×#7909 #48002505

Order 35584.

1318 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×#407×C₁₃₉	C₂	1	2	yes	maximal
2	C₂²×#95×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1280×C₁₃₉	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₁₃₉	C₄	1	2	yes	—
5	C₂⁴×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
6	C₂²×M₁₆×C₁₃₉	—	2	2	no	—
7	#407×C₁₃₉	—	2	4	no	—
8	#407×C₁₃₉	C₄	1	8	yes	complemented
9	C₂×C₂²⋊C₈×C₁₃₉	—	2	2	no	—
10	C₂×C₂²⋊C₈×C₁₃₉	C₂²	1	4	yes	—
11	C₂×#95×C₁₃₉	—	2	2	no	—
12	C₂×#95×C₁₃₉	—	2	2	no	—
13	C₂×#95×C₁₃₉	C₄	1	2	yes	—
14	C₂×#95×C₁₃₉	C₂²	1	2	yes	—
15	C₂⁵×C₁₃₉	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₃₉	—	2	2	no	—
17	C₂³×C₄×C₁₃₉	—	2	2	no	—
18	C₂³×C₄×C₁₃₉	—	2	8	no	—
19	C₂³×C₄×C₁₃₉	—	4	4	no	—
20	C₂³×C₄×C₁₃₉	—	4	4	no	—
21	C₂³×C₄×C₁₃₉	—	4	4	no	—
22	C₂³×C₄×C₁₃₉	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃₉	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₁₃₉	D₄	1	2	yes	—
26	C₂³×C₄×C₁₃₉	D₄	1	2	yes	—
27	C₂³×C₄×C₁₃₉	D₄	1	2	yes	—
28	C₂³×C₄×C₁₃₉	Q₈	1	2	yes	—
29	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
30	C₂×M₁₆×C₁₃₉	—	2	8	no	—
31	C₂×M₁₆×C₁₃₉	—	4	8	no	—
32	C₂²×C₈×C₁₃₉	—	2	4	no	—
33	C₂²⋊C₈×C₁₃₉	—	2	8	no	—
34	C₂²⋊C₈×C₁₃₉	—	2	4	no	—
35	C₂²⋊C₈×C₁₃₉	C₂×C₄	1	8	yes	complemented
36	#95×C₁₃₉	—	2	4	no	—
37	#95×C₁₃₉	—	2	4	no	—
38	C₂⁴×C₁₃₉	—	2	4	no	—
39	C₂⁴×C₁₃₉	—	2	2	no	—
40	C₂⁴×C₁₃₉	—	2	2	no	—
41	C₂⁴×C₁₃₉	—	2	4	no	—
42	C₂⁴×C₁₃₉	C₄²	1	2	yes	—
43	C₂⁴×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂⁴×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂⁴×C₁₃₉	C₄⋊C₄	1	2	yes	—
46	C₂⁴×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₉	—	2	8	no	—
48	C₂²×C₄×C₁₃₉	—	2	8	no	—
49	C₂²×C₄×C₁₃₉	—	2	8	no	—
50	C₂²×C₄×C₁₃₉	—	2	4	no	—
51	C₂²×C₄×C₁₃₉	—	2	4	no	—
52	C₂²×C₄×C₁₃₉	—	4	8	no	—
53	C₂²×C₄×C₁₃₉	—	4	2	no	—
54	C₂²×C₄×C₁₃₉	—	4	4	no	—
55	C₂²×C₄×C₁₃₉	—	4	4	no	—
56	C₂²×C₄×C₁₃₉	—	4	4	no	—
57	C₂²×C₄×C₁₃₉	—	4	2	no	—
58	C₂²×C₄×C₁₃₉	—	4	32	no	—
59	C₂²×C₄×C₁₃₉	—	4	4	no	—
60	C₂²×C₄×C₁₃₉	—	4	4	no	—
61	C₂²×C₄×C₁₃₉	—	4	4	no	—
62	C₂²×C₄×C₁₃₉	—	4	4	no	—
63	C₂²×C₄×C₁₃₉	—	4	4	no	—
64	C₂²×C₄×C₁₃₉	—	4	4	no	—
65	C₂²×C₄×C₁₃₉	C₄²	1	2	yes	—
66	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
68	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
72	C₂²×C₄×C₁₃₉	M₁₆	1	8	yes	complemented
73	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
77	M₁₆×C₁₃₉	—	4	16	no	—
78	C₂×C₈×C₁₃₉	—	2	8	no	—
79	C₂×C₈×C₁₃₉	—	4	8	no	—
80	C₂³×C₁₃₉	—	2	16	no	—
81	C₂³×C₁₃₉	—	2	16	no	—
82	C₂³×C₁₃₉	—	2	4	no	—
83	C₂³×C₁₃₉	—	2	4	no	—
84	C₂³×C₁₃₉	—	2	4	no	—
85	C₂³×C₁₃₉	—	2	4	no	—
86	C₂³×C₁₃₉	—	2	4	no	—
87	C₂³×C₁₃₉	—	2	2	no	—
88	C₂³×C₁₃₉	—	2	2	no	—
89	C₂³×C₁₃₉	—	2	2	no	—
90	C₂³×C₁₃₉	—	2	2	no	—
91	C₂³×C₁₃₉	—	2	1	no	—
92	C₂³×C₁₃₉	—	2	1	no	—
93	C₂³×C₁₃₉	—	2	1	no	—
94	C₂³×C₁₃₉	—	2	1	no	—
95	C₂³×C₁₃₉	—	2	4	no	—
96	C₂³×C₁₃₉	#95	1	2	yes	—
97	C₂³×C₁₃₉	#95	1	2	yes	—
98	C₂³×C₁₃₉	#99	1	4	yes	—
99	C₂³×C₁₃₉	#100	1	2	yes	—
100	C₂³×C₁₃₉	#101	1	2	yes	—
101	C₂³×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
102	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂³×C₁₃₉	C₂×M₁₆	1	4	yes	—
105	C₂×C₄×C₁₃₉	—	2	8	no	—
106	C₂×C₄×C₁₃₉	—	2	8	no	—
107	C₂×C₄×C₁₃₉	—	2	4	no	—
108	C₂×C₄×C₁₃₉	—	2	4	no	—
109	C₂×C₄×C₁₃₉	—	2	8	no	—
110	C₂×C₄×C₁₃₉	—	4	2	no	—
111	C₂×C₄×C₁₃₉	—	4	4	no	—
112	C₂×C₄×C₁₃₉	—	4	2	no	—
113	C₂×C₄×C₁₃₉	—	4	4	no	—
114	C₂×C₄×C₁₃₉	—	4	2	no	—
115	C₂×C₄×C₁₃₉	—	4	1	no	—
116	C₂×C₄×C₁₃₉	—	4	1	no	—
117	C₂×C₄×C₁₃₉	—	4	16	no	—
118	C₂×C₄×C₁₃₉	—	4	16	no	—
119	C₂×C₄×C₁₃₉	—	4	8	no	—
120	C₂×C₄×C₁₃₉	—	4	4	no	—
121	C₂×C₄×C₁₃₉	—	4	4	no	—
122	C₂×C₄×C₁₃₉	—	4	8	no	—
123	C₂×C₄×C₁₃₉	#95	1	4	yes	—
124	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
125	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
126	C₈×C₁₃₉	—	4	8	no	—
127	C₂²×C₁₃₉	—	2	16	no	—
128	C₂²×C₁₃₉	—	2	16	no	—
129	C₂²×C₁₃₉	—	2	4	no	—
130	C₂²×C₁₃₉	—	2	4	no	—
131	C₂²×C₁₃₉	—	2	4	no	—
132	C₂²×C₁₃₉	—	2	4	no	—
133	C₂²×C₁₃₉	—	2	4	no	—
134	C₂²×C₁₃₉	—	2	4	no	—
135	C₂²×C₁₃₉	—	2	2	no	—
136	C₂²×C₁₃₉	—	2	2	no	—
137	C₂²×C₁₃₉	—	2	2	no	—
138	C₂²×C₁₃₉	—	2	1	no	—
139	C₂²×C₁₃₉	—	2	1	no	—
140	C₂²×C₁₃₉	—	2	2	no	—
141	C₂²×C₁₃₉	—	2	1	no	—
142	C₂²×C₁₃₉	—	2	1	no	—
143	C₂²×C₁₃₉	#342	1	4	yes	—
144	C₂²×C₁₃₉	#343	1	4	yes	—
145	C₂²×C₁₃₉	C₂×#95	1	1	yes	characteristic
146	C₂²×C₁₃₉	C₄×M₁₆	1	2	yes	—
147	C₂²×C₁₃₉	#407	1	2	yes	—
148	C₂²×C₁₃₉	C₂×#99	1	2	yes	—
149	C₂²×C₁₃₉	C₂×#100	1	1	yes	characteristic
150	C₂²×C₁₃₉	C₂×#101	1	1	yes	characteristic
151	C₂²×C₁₃₉	#423	1	2	yes	—
152	C₄×C₁₃₉	—	2	2	no	—
153	C₄×C₁₃₉	—	2	4	no	—
154	C₄×C₁₃₉	—	2	2	no	—
155	C₄×C₁₃₉	—	4	8	no	—
156	C₂×C₁₃₉	—	2	4	no	—
157	C₂×C₁₃₉	—	2	4	no	—
158	C₂×C₁₃₉	—	2	2	no	—
159	C₂×C₁₃₉	—	2	2	no	—
160	C₂×C₁₃₉	#1462	1	2	yes	—
161	C₂×C₁₃₉	#1463	1	2	yes	—
162	C₂×C₁₃₉	#1728	1	1	yes	characteristic
163	C₂×C₁₃₉	C₂×#342	1	1	yes	characteristic
164	C₂×C₁₃₉	C₂×#343	1	1	yes	characteristic
165	#7909	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₉	#7909	1	1	yes	characteristic, direct product factor, Sylow
167	C₂×#407	C₂×C₁₃₉	1	2	yes	—
168	C₂²×#95	C₂×C₁₃₉	1	1	yes	characteristic
169	#1280	C₂×C₁₃₉	1	4	yes	complemented
170	C₂⁴×C₄	C₄×C₁₃₉	1	2	yes	—
171	C₂⁴×C₄	C₂²×C₁₃₉	1	1	yes	characteristic
172	C₂²×M₁₆	—	2	2	no	—
173	#407	—	2	4	no	—
174	#407	C₄×C₁₃₉	1	8	yes	complemented
175	C₂×C₂²⋊C₈	—	2	2	no	—
176	C₂×C₂²⋊C₈	C₂²×C₁₃₉	1	4	yes	—
177	C₂×#95	—	2	2	no	—
178	C₂×#95	—	2	2	no	—
179	C₂×#95	C₄×C₁₃₉	1	2	yes	—
180	C₂×#95	C₂²×C₁₃₉	1	2	yes	—
181	C₂⁵	C₂×C₄×C₁₃₉	1	1	yes	characteristic
182	C₂³×C₄	—	2	2	no	—
183	C₂³×C₄	—	2	2	no	—
184	C₂³×C₄	—	2	8	no	—
185	C₂³×C₄	—	4	4	no	—
186	C₂³×C₄	—	4	4	no	—
187	C₂³×C₄	—	4	4	no	—
188	C₂³×C₄	C₂×C₄×C₁₃₉	1	4	yes	—
189	C₂³×C₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
190	C₂³×C₄	C₂×C₄×C₁₃₉	1	4	yes	—
191	C₂³×C₄	D₄×C₁₃₉	1	2	yes	—
192	C₂³×C₄	D₄×C₁₃₉	1	2	yes	—
193	C₂³×C₄	D₄×C₁₃₉	1	2	yes	—
194	C₂³×C₄	Q₈×C₁₃₉	1	2	yes	—
195	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
196	C₂×M₁₆	—	2	8	no	—
197	C₂×M₁₆	—	4	8	no	—
198	C₂²×C₈	—	2	4	no	—
199	C₂²⋊C₈	—	2	8	no	—
200	C₂²⋊C₈	—	2	4	no	—
201	C₂²⋊C₈	C₂×C₄×C₁₃₉	1	8	yes	complemented
202	#95	—	2	4	no	—
203	#95	—	2	4	no	—
204	C₂⁴	—	2	4	no	—
205	C₂⁴	—	2	2	no	—
206	C₂⁴	—	2	2	no	—
207	C₂⁴	—	2	4	no	—
208	C₂⁴	C₄²×C₁₃₉	1	2	yes	—
209	C₂⁴	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
210	C₂⁴	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
211	C₂⁴	C₄⋊C₄×C₁₃₉	1	2	yes	—
212	C₂⁴	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	2	8	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	8	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	32	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	4	4	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	C₄²×C₁₃₉	1	2	yes	—
232	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
233	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	4	yes	—
234	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	4	yes	—
235	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
236	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	2	yes	—
237	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
238	C₂²×C₄	M₁₆×C₁₃₉	1	8	yes	complemented
239	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
243	M₁₆	—	4	16	no	—
244	C₂×C₈	—	2	8	no	—
245	C₂×C₈	—	4	8	no	—
246	C₂³	—	2	16	no	—
247	C₂³	—	2	16	no	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	1	no	—
254	C₂³	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	2	4	no	—
261	C₂³	—	2	4	no	—
262	C₂³	#95×C₁₃₉	1	2	yes	—
263	C₂³	#95×C₁₃₉	1	2	yes	—
264	C₂³	#99×C₁₃₉	1	4	yes	—
265	C₂³	#100×C₁₃₉	1	2	yes	—
266	C₂³	#101×C₁₃₉	1	2	yes	—
267	C₂³	C₂×C₄²×C₁₃₉	1	1	yes	characteristic
268	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
269	C₂³	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
270	C₂³	C₂×M₁₆×C₁₃₉	1	4	yes	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	16	no	—
284	C₂×C₄	—	4	16	no	—
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	8	no	—
289	C₂×C₄	#95×C₁₃₉	1	4	yes	—
290	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	2	yes	—
291	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
292	C₈	—	4	8	no	—
293	C₂²	—	2	16	no	—
294	C₂²	—	2	16	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	#342×C₁₃₉	1	4	yes	—
310	C₂²	#343×C₁₃₉	1	4	yes	—
311	C₂²	C₂×#95×C₁₃₉	1	1	yes	characteristic
312	C₂²	C₄×M₁₆×C₁₃₉	1	2	yes	—
313	C₂²	#407×C₁₃₉	1	2	yes	—
314	C₂²	C₂×#99×C₁₃₉	1	2	yes	—
315	C₂²	C₂×#100×C₁₃₉	1	1	yes	characteristic
316	C₂²	C₂×#101×C₁₃₉	1	1	yes	characteristic
317	C₂²	#423×C₁₃₉	1	2	yes	—
318	C₄	—	2	2	no	—
319	C₄	—	2	4	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	8	no	—
322	C₂	—	2	2	no	—
323	C₂	—	2	2	no	—
324	C₂	—	2	4	no	—
325	C₂	—	2	4	no	—
326	C₂	#1462×C₁₃₉	1	2	yes	—
327	C₂	#1463×C₁₃₉	1	2	yes	—
328	C₂	#1728×C₁₃₉	1	1	yes	characteristic
329	C₂	C₂×#342×C₁₃₉	1	1	yes	characteristic
330	C₂	C₂×#343×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₁₃₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₉
prime radical	C₂⁵×C₁₃₉	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ C₂⁴×C₄×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1280×C₁₃₉	C₂	semidirect
8. #407×C₁₃₉	C₄	semidirect
35. C₂²⋊C₈×C₁₃₉	C₂×C₄	semidirect
72. C₂²×C₄×C₁₃₉	M₁₆	semidirect
165. #7909	C₁₃₉	direct (N × H)
166. C₁₃₉	#7909	direct (N × H)
169. #1280	C₂×C₁₃₉	semidirect
174. #407	C₄×C₁₃₉	semidirect
201. C₂²⋊C₈	C₂×C₄×C₁₃₉	semidirect
238. C₂²×C₄	M₁₆×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#407×C₁₃₉	2
C₂²×#95×C₁₃₉	2
#1280×C₁₃₉	2
#7909	139

Subgroups — C139×#7909 #48002505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#7909 #48002505