Subgroups — C₁₃₉×#8445 #48003041

Order 35584.

868 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#559×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#1957×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1803×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	C₂×#331×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈⋊C₈×C₁₃₉	C₂	1	2	yes	maximal, complemented
6	#559×C₁₃₉	—	2	2	no	—
7	#559×C₁₃₉	—	2	4	no	—
8	#559×C₁₃₉	C₄	1	4	yes	complemented
9	C₂×#127×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×#124×C₁₃₉	—	2	2	no	—
11	C₂×#123×C₁₃₉	—	2	2	no	—
12	C₂×C₄×Q₈×C₁₃₉	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₁₃₉	—	2	2	no	—
14	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
15	C₂×#103×C₁₃₉	—	2	2	no	—
16	C₂×#102×C₁₃₉	—	2	2	no	—
17	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
18	#384×C₁₃₉	C₂²	1	1	yes	characteristic
19	#340×C₁₃₉	—	2	2	no	—
20	#331×C₁₃₉	—	2	2	no	—
21	Q₈⋊C₈×C₁₃₉	—	2	4	no	—
22	C₂²×Q₈×C₁₃₉	D₄	1	2	yes	—
23	C₂²×D₄×C₁₃₉	—	2	2	no	—
24	C₂²×D₄×C₁₃₉	D₄	1	2	yes	—
25	C₂³×C₄×C₁₃₉	—	2	2	no	—
26	C₂²×C₈×C₁₃₉	—	2	1	no	—
27	C₂²×C₈×C₁₃₉	—	2	2	no	—
28	#127×C₁₃₉	—	2	2	no	—
29	#127×C₁₃₉	C₂×C₄	1	2	yes	—
30	#127×C₁₃₉	D₄	1	2	yes	—
31	#124×C₁₃₉	—	2	8	no	—
32	#124×C₁₃₉	—	2	8	no	—
33	#123×C₁₃₉	—	2	8	no	—
34	#123×C₁₃₉	—	2	8	no	—
35	C₄×Q₈×C₁₃₉	—	2	2	no	—
36	C₄×Q₈×C₁₃₉	—	2	2	no	—
37	C₄×Q₈×C₁₃₉	C₂×C₄	1	4	yes	—
38	C₄×Q₈×C₁₃₉	D₄	1	4	yes	—
39	C₄×D₄×C₁₃₉	—	2	8	no	—
40	C₄×D₄×C₁₃₉	—	2	8	no	—
41	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₁₃₉	—	2	4	no	—
46	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
47	C₂×C₄²×C₁₃₉	—	2	2	no	—
48	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₁₃₉	—	2	2	no	—
50	C₄⋊C₈×C₁₃₉	—	2	2	no	—
51	#103×C₁₃₉	—	4	4	no	—
52	#102×C₁₃₉	—	4	4	no	—
53	C₄×C₈×C₁₃₉	—	2	2	no	—
54	C₂⁴×C₁₃₉	—	2	2	no	—
55	C₂×Q₈×C₁₃₉	—	2	2	no	—
56	C₂×Q₈×C₁₃₉	—	2	2	no	—
57	C₂×Q₈×C₁₃₉	—	2	2	no	—
58	C₂×Q₈×C₁₃₉	—	2	2	no	—
59	C₂×Q₈×C₁₃₉	—	2	2	no	—
60	C₂×Q₈×C₁₃₉	C₂²⋊C₄	1	4	yes	—
61	C₂×D₄×C₁₃₉	—	2	4	no	—
62	C₂×D₄×C₁₃₉	—	2	4	no	—
63	C₂×D₄×C₁₃₉	—	2	2	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	C₂²⋊C₄	1	4	yes	—
71	C₂²×C₄×C₁₃₉	—	2	2	no	—
72	C₂²×C₄×C₁₃₉	—	2	2	no	—
73	C₂²×C₄×C₁₃₉	—	2	8	no	—
74	C₂²×C₄×C₁₃₉	—	4	1	no	—
75	C₂²×C₄×C₁₃₉	—	8	2	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	C₂×C₈×C₁₃₉	—	2	2	no	—
80	C₂×C₈×C₁₃₉	—	4	1	no	—
81	C₂×C₈×C₁₃₉	—	4	1	no	—
82	C₂×C₈×C₁₃₉	—	4	2	no	—
83	C₂×C₈×C₁₃₉	—	4	2	no	—
84	C₂×C₈×C₁₃₉	—	4	2	no	—
85	C₄⋊C₄×C₁₃₉	—	2	4	no	—
86	C₄⋊C₄×C₁₃₉	—	2	4	no	—
87	C₄⋊C₄×C₁₃₉	—	2	2	no	—
88	C₄⋊C₄×C₁₃₉	—	4	2	no	—
89	C₄⋊C₄×C₁₃₉	—	4	2	no	—
90	C₄⋊C₄×C₁₃₉	—	4	2	no	—
91	C₄⋊C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
92	C₂²⋊C₄×C₁₃₉	—	2	16	no	—
93	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
94	C₄²×C₁₃₉	—	2	4	no	—
95	C₄²×C₁₃₉	—	2	4	no	—
96	C₄²×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₃₉	C₂×D₄	1	2	yes	—
98	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₁₃₉	—	2	4	no	—
100	C₂³×C₁₃₉	—	4	4	no	—
101	C₂³×C₁₃₉	—	4	4	no	—
102	C₂³×C₁₃₉	—	4	2	no	—
103	C₂³×C₁₃₉	—	4	2	no	—
104	C₂³×C₁₃₉	#120	1	1	yes	characteristic
105	Q₈×C₁₃₉	—	2	4	no	—
106	Q₈×C₁₃₉	—	2	4	no	—
107	Q₈×C₁₃₉	—	4	2	no	—
108	Q₈×C₁₃₉	—	4	2	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	—	8	2	no	—
115	C₂×C₄×C₁₃₉	—	2	1	no	—
116	C₂×C₄×C₁₃₉	—	2	1	no	—
117	C₂×C₄×C₁₃₉	—	2	4	no	—
118	C₂×C₄×C₁₃₉	—	2	4	no	—
119	C₂×C₄×C₁₃₉	—	4	2	no	—
120	C₂×C₄×C₁₃₉	—	4	2	no	—
121	C₂×C₄×C₁₃₉	—	4	2	no	—
122	C₂×C₄×C₁₃₉	—	4	2	no	—
123	C₂×C₄×C₁₃₉	—	8	8	no	—
124	C₂×C₄×C₁₃₉	—	8	2	no	—
125	C₂×C₄×C₁₃₉	—	8	1	no	—
126	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₁₃₉	#120	1	1	yes	characteristic
129	C₂×C₄×C₁₃₉	#120	1	2	yes	—
130	C₂×C₄×C₁₃₉	#135	1	2	yes	—
131	C₂×C₄×C₁₃₉	#135	1	2	yes	—
132	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₄×C₁₃₉	#137	1	2	yes	—
134	C₈×C₁₃₉	—	4	2	no	—
135	C₈×C₁₃₉	—	8	2	no	—
136	C₂²×C₁₃₉	—	4	4	no	—
137	C₂²×C₁₃₉	—	4	8	no	—
138	C₂²×C₁₃₉	—	8	2	no	—
139	C₂²×C₁₃₉	—	8	2	no	—
140	C₂²×C₁₃₉	—	8	4	no	—
141	C₂²×C₁₃₉	#379	1	1	yes	characteristic
142	C₂²×C₁₃₉	#449	1	2	yes	—
143	C₂²×C₁₃₉	#452	1	2	yes	—
144	C₂²×C₁₃₉	#453	1	1	yes	characteristic
145	C₂²×C₁₃₉	#455	1	1	yes	characteristic
146	C₄×C₁₃₉	—	2	1	no	—
147	C₄×C₁₃₉	—	2	1	no	—
148	C₄×C₁₃₉	—	4	4	no	—
149	C₄×C₁₃₉	—	4	4	no	—
150	C₄×C₁₃₉	—	16	1	no	—
151	C₄×C₁₃₉	#416	1	2	yes	—
152	C₄×C₁₃₉	#417	1	2	yes	—
153	C₄×C₁₃₉	#449	1	2	yes	—
154	C₄×C₁₃₉	#452	1	2	yes	—
155	C₂×C₁₃₉	—	8	4	no	—
156	C₂×C₁₃₉	#1620	1	1	yes	characteristic
157	C₂×C₁₃₉	#1630	1	1	yes	characteristic
158	C₂×C₁₃₉	#1640	1	2	yes	—
159	C₂×C₁₃₉	#1788	1	2	yes	—
160	C₂×C₁₃₉	#1792	1	1	yes	characteristic
161	#8445	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₁₃₉	#8445	1	1	yes	characteristic, direct product factor, Sylow
163	C₂×#559	C₂×C₁₃₉	1	1	yes	characteristic
164	#1957	C₂×C₁₃₉	1	1	yes	characteristic
165	#1803	C₂×C₁₃₉	1	2	yes	complemented
166	C₂×#331	C₂×C₁₃₉	1	1	yes	characteristic
167	C₂×Q₈⋊C₈	C₂×C₁₃₉	1	2	yes	complemented
168	#559	—	2	2	no	—
169	#559	—	2	4	no	—
170	#559	C₄×C₁₃₉	1	4	yes	complemented
171	C₂×#127	C₂²×C₁₃₉	1	1	yes	characteristic
172	C₂×#124	—	2	2	no	—
173	C₂×#123	—	2	2	no	—
174	C₂×C₄×Q₈	C₂²×C₁₃₉	1	2	yes	—
175	C₂×C₄×D₄	—	2	2	no	—
176	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
177	C₂×#103	—	2	2	no	—
178	C₂×#102	—	2	2	no	—
179	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
180	#384	C₂²×C₁₃₉	1	1	yes	characteristic
181	#340	—	2	2	no	—
182	#331	—	2	2	no	—
183	Q₈⋊C₈	—	2	4	no	—
184	C₂²×Q₈	D₄×C₁₃₉	1	2	yes	—
185	C₂²×D₄	—	2	2	no	—
186	C₂²×D₄	D₄×C₁₃₉	1	2	yes	—
187	C₂³×C₄	—	2	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	2	no	—
190	#127	—	2	2	no	—
191	#127	C₂×C₄×C₁₃₉	1	2	yes	—
192	#127	D₄×C₁₃₉	1	2	yes	—
193	#124	—	2	8	no	—
194	#124	—	2	8	no	—
195	#123	—	2	8	no	—
196	#123	—	2	8	no	—
197	C₄×Q₈	—	2	2	no	—
198	C₄×Q₈	—	2	2	no	—
199	C₄×Q₈	C₂×C₄×C₁₃₉	1	4	yes	—
200	C₄×Q₈	D₄×C₁₃₉	1	4	yes	—
201	C₄×D₄	—	2	8	no	—
202	C₄×D₄	—	2	8	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	D₄×C₁₃₉	1	2	yes	—
207	C₂×C₂²⋊C₄	—	2	4	no	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	2	2	no	—
213	#103	—	4	4	no	—
214	#102	—	4	4	no	—
215	C₄×C₈	—	2	2	no	—
216	C₂⁴	—	2	2	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	C₂²⋊C₄×C₁₃₉	1	4	yes	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	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	C₂²⋊C₄×C₁₃₉	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	8	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	8	2	no	—
238	C₂²×C₄	—	8	2	no	—
239	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	C₂²⋊C₄×C₁₃₉	1	4	yes	—
254	C₂²⋊C₄	—	2	16	no	—
255	C₂²⋊C₄	—	2	8	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	4	no	—
258	C₄²	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₁₃₉	1	2	yes	—
260	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
261	C₂³	—	2	4	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	4	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	2	no	—
266	C₂³	#120×C₁₃₉	1	1	yes	characteristic
267	Q₈	—	2	4	no	—
268	Q₈	—	2	4	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	D₄	—	4	4	no	—
272	D₄	—	4	4	no	—
273	D₄	—	4	4	no	—
274	D₄	—	4	4	no	—
275	D₄	—	4	4	no	—
276	D₄	—	8	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	8	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
289	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	2	yes	—
290	C₂×C₄	#120×C₁₃₉	1	1	yes	characteristic
291	C₂×C₄	#120×C₁₃₉	1	2	yes	—
292	C₂×C₄	#135×C₁₃₉	1	2	yes	—
293	C₂×C₄	#135×C₁₃₉	1	2	yes	—
294	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	2	yes	—
295	C₂×C₄	#137×C₁₃₉	1	2	yes	—
296	C₈	—	4	2	no	—
297	C₈	—	8	2	no	—
298	C₂²	—	4	8	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	4	no	—
303	C₂²	#379×C₁₃₉	1	1	yes	characteristic
304	C₂²	#449×C₁₃₉	1	2	yes	—
305	C₂²	#452×C₁₃₉	1	2	yes	—
306	C₂²	#453×C₁₃₉	1	1	yes	characteristic
307	C₂²	#455×C₁₃₉	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	4	4	no	—
311	C₄	—	4	4	no	—
312	C₄	—	16	1	no	—
313	C₄	#416×C₁₃₉	1	2	yes	—
314	C₄	#417×C₁₃₉	1	2	yes	—
315	C₄	#449×C₁₃₉	1	2	yes	—
316	C₄	#452×C₁₃₉	1	2	yes	—
317	C₂	—	8	4	no	—
318	C₂	#1620×C₁₃₉	1	1	yes	characteristic
319	C₂	#1630×C₁₃₉	1	1	yes	characteristic
320	C₂	#1640×C₁₃₉	1	2	yes	—
321	C₂	#1788×C₁₃₉	1	2	yes	—
322	C₂	#1792×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×#127×C₁₃₉	C₂²
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
3. #1803×C₁₃₉	C₂	semidirect
5. C₂×Q₈⋊C₈×C₁₃₉	C₂	semidirect
8. #559×C₁₃₉	C₄	semidirect
161. #8445	C₁₃₉	direct (N × H)
162. C₁₃₉	#8445	direct (N × H)
165. #1803	C₂×C₁₃₉	semidirect
167. C₂×Q₈⋊C₈	C₂×C₁₃₉	semidirect
170. #559	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×#559×C₁₃₉	2
#1957×C₁₃₉	2
#1803×C₁₃₉	2
C₂×#331×C₁₃₉	2
C₂×Q₈⋊C₈×C₁₃₉	2
#8445	139

Subgroups — C139×#8445 #48003041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#8445 #48003041