Subgroups — C₁₃₉×#9061 #48003657

Order 35584.

822 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	C₂×#493×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#2366×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#1982×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	C₂×#331×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#1279×C₁₃₉	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₁₃₉	—	2	2	no	—
7	C₂×#127×C₁₃₉	C₂²	1	1	yes	characteristic
8	C₂×#127×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₁₃₉	—	2	2	no	—
10	#493×C₁₃₉	—	2	2	no	—
11	#493×C₁₃₉	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
13	C₂×#102×C₁₃₉	—	2	2	no	—
14	C₂×#102×C₁₃₉	—	2	2	no	—
15	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	#390×C₁₃₉	—	2	2	no	—
17	#386×C₁₃₉	—	2	2	no	—
18	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
19	#331×C₁₃₉	—	2	2	no	—
20	C₂²×D₄×C₁₃₉	—	2	2	no	—
21	C₂²×D₄×C₁₃₉	—	2	2	no	—
22	C₂²×D₄×C₁₃₉	—	2	1	no	—
23	C₂²×D₄×C₁₃₉	D₄	1	2	yes	—
24	C₂³×C₄×C₁₃₉	—	2	2	no	—
25	C₂×D₈×C₁₃₉	—	2	8	no	—
26	C₂×D₈×C₁₃₉	—	4	4	no	—
27	C₂×D₈×C₁₃₉	—	4	4	no	—
28	C₂²×C₈×C₁₃₉	—	2	1	no	—
29	C₂²×C₈×C₁₃₉	—	2	2	no	—
30	#127×C₁₃₉	—	2	1	no	—
31	#127×C₁₃₉	—	2	2	no	—
32	#127×C₁₃₉	—	2	2	no	—
33	#127×C₁₃₉	C₂×C₄	1	2	yes	—
34	#127×C₁₃₉	C₂×C₄	1	2	yes	complemented
35	#127×C₁₃₉	D₄	1	2	yes	—
36	C₄×D₄×C₁₃₉	—	4	4	no	—
37	C₄×D₄×C₁₃₉	—	4	4	no	—
38	C₂×C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₁₃₉	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
42	C₂×C₄²×C₁₃₉	—	2	2	no	—
43	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₁₃₉	—	2	2	no	—
45	C₄⋊C₈×C₁₃₉	—	2	2	no	—
46	#102×C₁₃₉	—	4	4	no	—
47	#102×C₁₃₉	—	4	4	no	—
48	C₄×C₈×C₁₃₉	—	2	1	no	—
49	C₄×C₈×C₁₃₉	C₂×C₄	1	2	yes	complemented
50	#95×C₁₃₉	—	2	2	no	—
51	C₂⁴×C₁₃₉	—	2	2	no	—
52	C₂⁴×C₁₃₉	—	4	1	no	—
53	C₂×D₄×C₁₃₉	—	2	2	no	—
54	C₂×D₄×C₁₃₉	—	2	4	no	—
55	C₂×D₄×C₁₃₉	—	2	4	no	—
56	C₂×D₄×C₁₃₉	—	2	4	no	—
57	C₂×D₄×C₁₃₉	—	2	2	no	—
58	C₂×D₄×C₁₃₉	—	4	2	no	—
59	C₂×D₄×C₁₃₉	—	4	2	no	—
60	C₂×D₄×C₁₃₉	—	4	2	no	—
61	C₂×D₄×C₁₃₉	—	4	2	no	—
62	C₂×D₄×C₁₃₉	—	4	2	no	—
63	C₂×D₄×C₁₃₉	—	4	1	no	—
64	C₂×D₄×C₁₃₉	—	4	2	no	—
65	C₂×D₄×C₁₃₉	—	4	2	no	—
66	C₂×D₄×C₁₃₉	—	4	2	no	—
67	C₂×D₄×C₁₃₉	—	4	2	no	—
68	C₂×D₄×C₁₃₉	—	4	2	no	—
69	C₂×D₄×C₁₃₉	—	4	2	no	—
70	C₂×D₄×C₁₃₉	—	4	2	no	—
71	C₂×D₄×C₁₃₉	—	4	2	no	—
72	C₂×D₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
73	C₂²×C₄×C₁₃₉	—	2	2	no	—
74	C₂²×C₄×C₁₃₉	—	2	2	no	—
75	C₂²×C₄×C₁₃₉	—	4	8	no	—
76	C₂²×C₄×C₁₃₉	—	8	2	no	—
77	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
79	D₈×C₁₃₉	—	4	16	no	—
80	C₂×C₈×C₁₃₉	—	2	2	no	—
81	C₂×C₈×C₁₃₉	—	2	2	no	—
82	C₂×C₈×C₁₃₉	—	2	2	no	—
83	C₂×C₈×C₁₃₉	—	4	2	no	—
84	C₂×C₈×C₁₃₉	—	4	2	no	—
85	C₂×C₈×C₁₃₉	—	4	2	no	—
86	C₄⋊C₄×C₁₃₉	—	2	2	no	—
87	C₄⋊C₄×C₁₃₉	—	2	2	no	—
88	C₄⋊C₄×C₁₃₉	—	2	2	no	—
89	C₄⋊C₄×C₁₃₉	D₈	1	4	yes	complemented
90	C₂²⋊C₄×C₁₃₉	—	4	8	no	—
91	C₂²⋊C₄×C₁₃₉	—	8	2	no	—
92	C₄²×C₁₃₉	—	4	2	no	—
93	C₄²×C₁₃₉	—	4	2	no	—
94	C₄²×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₃₉	#41	1	2	yes	—
97	C₂³×C₁₃₉	—	2	4	no	—
98	C₂³×C₁₃₉	—	4	4	no	—
99	C₂³×C₁₃₉	—	4	4	no	—
100	C₂³×C₁₃₉	—	4	2	no	—
101	C₂³×C₁₃₉	—	4	2	no	—
102	C₂³×C₁₃₉	—	4	2	no	—
103	C₂³×C₁₃₉	—	8	2	no	—
104	C₂³×C₁₃₉	—	8	2	no	—
105	C₂³×C₁₃₉	—	8	1	no	—
106	C₂³×C₁₃₉	—	8	1	no	—
107	C₂³×C₁₃₉	#120	1	1	yes	characteristic
108	D₄×C₁₃₉	—	4	4	no	—
109	D₄×C₁₃₉	—	4	4	no	—
110	D₄×C₁₃₉	—	4	4	no	—
111	D₄×C₁₃₉	—	4	4	no	—
112	D₄×C₁₃₉	—	4	4	no	—
113	D₄×C₁₃₉	—	4	4	no	—
114	D₄×C₁₃₉	—	4	4	no	—
115	D₄×C₁₃₉	—	4	4	no	—
116	D₄×C₁₃₉	—	8	2	no	—
117	D₄×C₁₃₉	—	8	2	no	—
118	D₄×C₁₃₉	—	8	2	no	—
119	D₄×C₁₃₉	—	8	2	no	—
120	C₂×C₄×C₁₃₉	—	2	1	no	—
121	C₂×C₄×C₁₃₉	—	2	1	no	—
122	C₂×C₄×C₁₃₉	—	2	4	no	—
123	C₂×C₄×C₁₃₉	—	4	2	no	—
124	C₂×C₄×C₁₃₉	—	4	2	no	—
125	C₂×C₄×C₁₃₉	—	4	2	no	—
126	C₂×C₄×C₁₃₉	—	4	2	no	—
127	C₂×C₄×C₁₃₉	—	4	2	no	—
128	C₂×C₄×C₁₃₉	—	8	8	no	—
129	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃₉	C₄×D₄	1	2	yes	—
131	C₂×C₄×C₁₃₉	#121	1	2	yes	—
132	C₂×C₄×C₁₃₉	#123	1	1	yes	characteristic
133	C₂×C₄×C₁₃₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₁₃₉	#135	1	2	yes	—
135	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
136	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
137	C₈×C₁₃₉	—	2	4	no	—
138	C₈×C₁₃₉	—	8	2	no	—
139	C₂²×C₁₃₉	—	4	8	no	—
140	C₂²×C₁₃₉	—	4	4	no	—
141	C₂²×C₁₃₉	—	8	2	no	—
142	C₂²×C₁₃₉	—	8	2	no	—
143	C₂²×C₁₃₉	—	8	4	no	—
144	C₂²×C₁₃₉	—	8	4	no	—
145	C₂²×C₁₃₉	—	8	2	no	—
146	C₂²×C₁₃₉	—	16	1	no	—
147	C₂²×C₁₃₉	—	16	1	no	—
148	C₂²×C₁₃₉	—	16	2	no	—
149	C₂²×C₁₃₉	#386	1	1	yes	characteristic
150	C₂²×C₁₃₉	#447	1	2	yes	—
151	C₂²×C₁₃₉	#449	1	2	yes	—
152	C₂²×C₁₃₉	#453	1	1	yes	characteristic
153	C₂²×C₁₃₉	#453	1	1	yes	characteristic
154	C₄×C₁₃₉	—	2	1	no	—
155	C₄×C₁₃₉	—	2	1	no	—
156	C₄×C₁₃₉	—	4	4	no	—
157	C₄×C₁₃₉	—	8	2	no	—
158	C₄×C₁₃₉	C₄×D₈	1	2	yes	—
159	C₄×C₁₃₉	#442	1	2	yes	—
160	C₄×C₁₃₉	#459	1	2	yes	—
161	C₄×C₁₃₉	#463	1	2	yes	—
162	C₂×C₁₃₉	—	8	4	no	—
163	C₂×C₁₃₉	—	16	2	no	—
164	C₂×C₁₃₉	#1619	1	1	yes	characteristic
165	C₂×C₁₃₉	#1620	1	1	yes	characteristic
166	C₂×C₁₃₉	#1621	1	2	yes	—
167	C₂×C₁₃₉	#1879	1	2	yes	—
168	C₂×C₁₃₉	#1887	1	1	yes	characteristic
169	#9061	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₁₃₉	#9061	1	1	yes	characteristic, direct product factor, Sylow
171	C₂×#493	C₂×C₁₃₉	1	1	yes	characteristic
172	#2366	C₂×C₁₃₉	1	1	yes	characteristic, complemented
173	#1982	C₂×C₁₃₉	1	2	yes	complemented
174	C₂×#331	C₂×C₁₃₉	1	1	yes	characteristic, complemented
175	#1279	C₂×C₁₃₉	1	2	yes	complemented
176	C₂²×D₈	—	2	2	no	—
177	C₂×#127	C₂²×C₁₃₉	1	1	yes	characteristic
178	C₂×#127	C₂²×C₁₃₉	1	1	yes	characteristic
179	C₂×C₄×D₄	—	2	2	no	—
180	#493	—	2	2	no	—
181	#493	C₄×C₁₃₉	1	4	yes	complemented
182	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
183	C₂×#102	—	2	2	no	—
184	C₂×#102	—	2	2	no	—
185	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
186	#390	—	2	2	no	—
187	#386	—	2	2	no	—
188	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
189	#331	—	2	2	no	—
190	C₂²×D₄	—	2	2	no	—
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	—	2	2	no	—
193	C₂²×D₄	D₄×C₁₃₉	1	2	yes	—
194	C₂³×C₄	—	2	2	no	—
195	C₂×D₈	—	2	8	no	—
196	C₂×D₈	—	4	4	no	—
197	C₂×D₈	—	4	4	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	2	2	no	—
200	#127	—	2	1	no	—
201	#127	—	2	2	no	—
202	#127	—	2	2	no	—
203	#127	C₂×C₄×C₁₃₉	1	2	yes	—
204	#127	C₂×C₄×C₁₃₉	1	2	yes	complemented
205	#127	D₄×C₁₃₉	1	2	yes	—
206	C₄×D₄	—	4	4	no	—
207	C₄×D₄	—	4	4	no	—
208	C₂×C₄⋊C₄	D₄×C₁₃₉	1	2	yes	—
209	C₂×C₄⋊C₄	D₄×C₁₃₉	1	2	yes	—
210	C₂×C₂²⋊C₄	—	2	4	no	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	—	2	2	no	—
216	#102	—	4	4	no	—
217	#102	—	4	4	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	C₂×C₄×C₁₃₉	1	2	yes	complemented
220	#95	—	2	2	no	—
221	C₂⁴	—	2	2	no	—
222	C₂⁴	—	4	1	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	C₂²⋊C₄×C₁₃₉	1	4	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	4	8	no	—
246	C₂²×C₄	—	8	2	no	—
247	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
249	D₈	—	4	16	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	D₈×C₁₃₉	1	4	yes	complemented
260	C₂²⋊C₄	—	4	8	no	—
261	C₂²⋊C₄	—	8	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	4	2	no	—
264	C₄²	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
265	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
266	C₄²	#41×C₁₃₉	1	2	yes	—
267	C₂³	—	2	4	no	—
268	C₂³	—	4	4	no	—
269	C₂³	—	4	4	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	8	2	no	—
274	C₂³	—	8	2	no	—
275	C₂³	—	8	1	no	—
276	C₂³	—	8	1	no	—
277	C₂³	#120×C₁₃₉	1	1	yes	characteristic
278	D₄	—	4	4	no	—
279	D₄	—	4	4	no	—
280	D₄	—	4	4	no	—
281	D₄	—	4	4	no	—
282	D₄	—	4	4	no	—
283	D₄	—	4	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	4	no	—
286	D₄	—	8	2	no	—
287	D₄	—	8	2	no	—
288	D₄	—	8	2	no	—
289	D₄	—	8	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	8	no	—
299	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
300	C₂×C₄	C₄×D₄×C₁₃₉	1	2	yes	—
301	C₂×C₄	#121×C₁₃₉	1	2	yes	—
302	C₂×C₄	#123×C₁₃₉	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₁₃₉	1	2	yes	—
304	C₂×C₄	#135×C₁₃₉	1	2	yes	—
305	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	2	yes	—
306	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	2	yes	—
307	C₈	—	2	4	no	—
308	C₈	—	8	2	no	—
309	C₂²	—	4	8	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	4	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	16	1	no	—
317	C₂²	—	16	1	no	—
318	C₂²	—	16	2	no	—
319	C₂²	#386×C₁₃₉	1	1	yes	characteristic
320	C₂²	#447×C₁₃₉	1	2	yes	—
321	C₂²	#449×C₁₃₉	1	2	yes	—
322	C₂²	#453×C₁₃₉	1	1	yes	characteristic
323	C₂²	#453×C₁₃₉	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	4	no	—
327	C₄	—	8	2	no	—
328	C₄	C₄×D₈×C₁₃₉	1	2	yes	—
329	C₄	#442×C₁₃₉	1	2	yes	—
330	C₄	#459×C₁₃₉	1	2	yes	—
331	C₄	#463×C₁₃₉	1	2	yes	—
332	C₂	—	8	4	no	—
333	C₂	—	16	2	no	—
334	C₂	#1619×C₁₃₉	1	1	yes	characteristic
335	C₂	#1620×C₁₃₉	1	1	yes	characteristic
336	C₂	#1621×C₁₃₉	1	2	yes	—
337	C₂	#1879×C₁₃₉	1	2	yes	—
338	C₂	#1887×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₉	#120
commutator	C₄²	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
prime radical	C₂×#493×C₁₃₉	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ C₂×C₄²×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2366×C₁₃₉	C₂	semidirect
3. #1982×C₁₃₉	C₂	semidirect
4. C₂×#331×C₁₃₉	C₂	semidirect
5. #1279×C₁₃₉	C₂	semidirect
11. #493×C₁₃₉	C₄	semidirect
34. #127×C₁₃₉	C₂×C₄	semidirect
49. C₄×C₈×C₁₃₉	C₂×C₄	semidirect
89. C₄⋊C₄×C₁₃₉	D₈	semidirect
169. #9061	C₁₃₉	direct (N × H)
170. C₁₃₉	#9061	direct (N × H)
172. #2366	C₂×C₁₃₉	semidirect
173. #1982	C₂×C₁₃₉	semidirect
174. C₂×#331	C₂×C₁₃₉	semidirect
175. #1279	C₂×C₁₃₉	semidirect
181. #493	C₄×C₁₃₉	semidirect
204. #127	C₂×C₄×C₁₃₉	semidirect
219. C₄×C₈	C₂×C₄×C₁₃₉	semidirect
259. C₄⋊C₄	D₈×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₁₃₉	2
#2366×C₁₃₉	2
#1982×C₁₃₉	2
C₂×#331×C₁₃₉	2
#1279×C₁₃₉	2
#9061	139

Subgroups — C139×#9061 #48003657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#9061 #48003657