Subgroups — C₁₃₉×#19751 #48014347

Order 35584.

684 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3185×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#3185×C₁₃₉	C₂	1	2	yes	maximal
3	#3165×C₁₃₉	C₂	1	2	yes	maximal
4	#3139×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	#1723×C₁₃₉	C₂	1	2	yes	maximal, complemented
6	#1718×C₁₃₉	C₂	1	2	yes	maximal, complemented
7	#1716×C₁₃₉	C₂	1	2	yes	maximal, complemented
8	#1707×C₁₃₉	C₂	1	2	yes	maximal, complemented
9	#1455×C₁₃₉	C₂	1	1	yes	maximal, characteristic
10	#533×C₁₃₉	C₂²	1	1	yes	characteristic
11	#533×C₁₃₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	#501×C₁₃₉	—	2	4	no	—
14	#500×C₁₃₉	C₂²	1	2	yes	—
15	#500×C₁₃₉	C₂²	1	2	yes	—
16	#498×C₁₃₉	C₂²	1	2	yes	—
17	#498×C₁₃₉	C₂²	1	2	yes	—
18	#498×C₁₃₉	C₂²	1	2	yes	—
19	#498×C₁₃₉	C₂²	1	1	yes	characteristic
20	#494×C₁₃₉	C₂²	1	2	yes	—
21	#494×C₁₃₉	C₂²	1	1	yes	characteristic
22	#491×C₁₃₉	—	2	2	no	—
23	#487×C₁₃₉	C₂²	1	2	yes	—
24	#487×C₁₃₉	C₂²	1	2	yes	—
25	#487×C₁₃₉	C₂²	1	2	yes	—
26	#487×C₁₃₉	C₂²	1	2	yes	—
27	#487×C₁₃₉	C₂²	1	2	yes	—
28	#470×C₁₃₉	—	2	4	no	—
29	#470×C₁₃₉	—	2	2	no	—
30	#441×C₁₃₉	—	2	2	no	—
31	#441×C₁₃₉	—	2	4	no	—
32	#428×C₁₃₉	—	2	4	no	—
33	#419×C₁₃₉	—	2	2	no	—
34	C₄×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
35	C₄×M₁₆×C₁₃₉	C₂²	1	2	yes	—
36	#322×C₁₃₉	C₂²	1	2	yes	—
37	#322×C₁₃₉	C₂²	1	1	yes	characteristic
38	C₈²×C₁₃₉	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₁₃₉	—	2	1	no	—
40	C₂×Q₁₆×C₁₃₉	—	2	2	no	—
41	C₂×Q₁₆×C₁₃₉	—	2	4	no	—
42	C₂×M₁₆×C₁₃₉	D₄	1	4	yes	—
43	C₂×M₁₆×C₁₃₉	D₄	1	2	yes	—
44	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
45	C₄⋊Q₈×C₁₃₉	—	2	1	no	—
46	C₄⋊Q₈×C₁₃₉	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₁₃₉	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₁₃₉	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₁₃₉	C₂³	1	2	yes	—
50	#125×C₁₃₉	—	2	1	no	—
51	#125×C₁₃₉	—	2	2	no	—
52	C₂²⋊Q₈×C₁₃₉	—	2	2	no	—
53	C₂²⋊Q₈×C₁₃₉	—	2	2	no	—
54	C₂²⋊Q₈×C₁₃₉	—	2	2	no	—
55	C₄×Q₈×C₁₃₉	—	2	2	no	—
56	C₄×Q₈×C₁₃₉	—	2	2	no	—
57	C₄×Q₈×C₁₃₉	—	2	2	no	—
58	#117×C₁₃₉	—	2	1	no	—
59	#117×C₁₃₉	—	2	2	no	—
60	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
61	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
62	#107×C₁₃₉	—	2	4	no	—
63	#107×C₁₃₉	—	2	4	no	—
64	#106×C₁₃₉	—	2	4	no	—
65	#106×C₁₃₉	—	2	4	no	—
66	#106×C₁₃₉	—	2	2	no	—
67	#106×C₁₃₉	—	2	4	no	—
68	#103×C₁₃₉	—	2	4	no	—
69	#103×C₁₃₉	—	2	4	no	—
70	#103×C₁₃₉	—	2	4	no	—
71	#103×C₁₃₉	—	2	4	no	—
72	#103×C₁₃₉	—	2	4	no	—
73	#97×C₁₃₉	D₄	1	2	yes	—
74	#97×C₁₃₉	D₄	1	4	yes	—
75	C₄×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₁₃₉	C₂³	1	2	yes	—
78	C₄×C₈×C₁₃₉	C₂³	1	2	yes	—
79	C₂×Q₈×C₁₃₉	—	2	2	no	—
80	C₂×Q₈×C₁₃₉	—	2	2	no	—
81	C₂×Q₈×C₁₃₉	—	2	2	no	—
82	C₂×Q₈×C₁₃₉	—	2	2	no	—
83	C₂×Q₈×C₁₃₉	—	4	4	no	—
84	C₂×Q₈×C₁₃₉	—	4	1	no	—
85	C₂²×C₄×C₁₃₉	—	4	1	no	—
86	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₁₃₉	—	4	4	no	—
89	Q₁₆×C₁₃₉	—	4	8	no	—
90	M₁₆×C₁₃₉	—	2	8	no	—
91	M₁₆×C₁₃₉	—	4	2	no	—
92	C₂×C₈×C₁₃₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃₉	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₁₃₉	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₁₃₉	C₂×D₄	1	4	yes	—
96	C₄⋊C₄×C₁₃₉	—	2	2	no	—
97	C₄⋊C₄×C₁₃₉	—	2	2	no	—
98	C₄⋊C₄×C₁₃₉	—	2	2	no	—
99	C₄⋊C₄×C₁₃₉	—	2	2	no	—
100	C₄⋊C₄×C₁₃₉	—	2	2	no	—
101	C₄⋊C₄×C₁₃₉	—	2	2	no	—
102	C₄⋊C₄×C₁₃₉	—	2	2	no	—
103	C₄⋊C₄×C₁₃₉	—	2	2	no	—
104	C₄⋊C₄×C₁₃₉	—	4	2	no	—
105	C₄⋊C₄×C₁₃₉	—	4	2	no	—
106	C₄⋊C₄×C₁₃₉	—	4	2	no	—
107	C₄⋊C₄×C₁₃₉	—	4	2	no	—
108	C₂²⋊C₄×C₁₃₉	—	4	1	no	—
109	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
110	C₄²×C₁₃₉	—	4	2	no	—
111	C₄²×C₁₃₉	—	4	1	no	—
112	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₃₉	C₂×D₄	1	2	yes	—
114	C₄²×C₁₃₉	C₂⁴	1	1	yes	characteristic
115	C₂³×C₁₃₉	#127	1	1	yes	characteristic
116	Q₈×C₁₃₉	—	4	4	no	—
117	Q₈×C₁₃₉	—	4	4	no	—
118	Q₈×C₁₃₉	—	4	4	no	—
119	Q₈×C₁₃₉	—	4	4	no	—
120	Q₈×C₁₃₉	—	8	2	no	—
121	C₂×C₄×C₁₃₉	—	2	4	no	—
122	C₂×C₄×C₁₃₉	—	2	2	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	—	8	2	no	—
128	C₂×C₄×C₁₃₉	#127	1	2	yes	—
129	C₂×C₄×C₁₃₉	#127	1	1	yes	characteristic
130	C₂×C₄×C₁₃₉	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₁₃₉	—	2	4	no	—
133	C₈×C₁₃₉	—	2	2	no	—
134	C₈×C₁₃₉	—	2	2	no	—
135	C₈×C₁₃₉	#137	1	8	yes	—
136	C₂²×C₁₃₉	—	2	2	no	—
137	C₂²×C₁₃₉	—	2	1	no	—
138	C₂²×C₁₃₉	C₂×#127	1	1	yes	characteristic
139	C₄×C₁₃₉	—	2	2	no	—
140	C₄×C₁₃₉	—	2	1	no	—
141	C₄×C₁₃₉	—	8	2	no	—
142	C₄×C₁₃₉	—	8	2	no	—
143	C₄×C₁₃₉	—	8	2	no	—
144	C₄×C₁₃₉	—	8	2	no	—
145	C₄×C₁₃₉	C₂×#137	1	4	yes	—
146	C₄×C₁₃₉	#575	1	2	yes	—
147	C₂×C₁₃₉	—	4	1	no	—
148	C₂×C₁₃₉	#3152	1	2	yes	—
149	C₂×C₁₃₉	#3153	1	1	yes	characteristic
150	#19751	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₁₃₉	#19751	1	1	yes	characteristic, direct product factor, Sylow
152	#3185	C₂×C₁₃₉	1	2	yes	—
153	#3185	C₂×C₁₃₉	1	1	yes	characteristic
154	#3165	C₂×C₁₃₉	1	2	yes	—
155	#3139	C₂×C₁₃₉	1	1	yes	characteristic
156	#1723	C₂×C₁₃₉	1	2	yes	complemented
157	#1718	C₂×C₁₃₉	1	2	yes	complemented
158	#1716	C₂×C₁₃₉	1	2	yes	complemented
159	#1707	C₂×C₁₃₉	1	2	yes	complemented
160	#1455	C₂×C₁₃₉	1	1	yes	characteristic
161	#533	C₂²×C₁₃₉	1	1	yes	characteristic
162	#533	C₂²×C₁₃₉	1	2	yes	—
163	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
164	#501	—	2	4	no	—
165	#500	C₂²×C₁₃₉	1	2	yes	—
166	#500	C₂²×C₁₃₉	1	2	yes	—
167	#498	C₂²×C₁₃₉	1	2	yes	—
168	#498	C₂²×C₁₃₉	1	2	yes	—
169	#498	C₂²×C₁₃₉	1	2	yes	—
170	#498	C₂²×C₁₃₉	1	1	yes	characteristic
171	#494	C₂²×C₁₃₉	1	2	yes	—
172	#494	C₂²×C₁₃₉	1	1	yes	characteristic
173	#491	—	2	2	no	—
174	#487	C₂²×C₁₃₉	1	2	yes	—
175	#487	C₂²×C₁₃₉	1	2	yes	—
176	#487	C₂²×C₁₃₉	1	2	yes	—
177	#487	C₂²×C₁₃₉	1	2	yes	—
178	#487	C₂²×C₁₃₉	1	2	yes	—
179	#470	—	2	4	no	—
180	#470	—	2	2	no	—
181	#441	—	2	2	no	—
182	#441	—	2	4	no	—
183	#428	—	2	4	no	—
184	#419	—	2	2	no	—
185	C₄×M₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
186	C₄×M₁₆	C₂²×C₁₃₉	1	2	yes	—
187	#322	C₂²×C₁₃₉	1	2	yes	—
188	#322	C₂²×C₁₃₉	1	1	yes	characteristic
189	C₈²	C₂²×C₁₃₉	1	1	yes	characteristic
190	C₂²×Q₈	—	2	1	no	—
191	C₂×Q₁₆	—	2	2	no	—
192	C₂×Q₁₆	—	2	4	no	—
193	C₂×M₁₆	D₄×C₁₃₉	1	4	yes	—
194	C₂×M₁₆	D₄×C₁₃₉	1	2	yes	—
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	1	no	—
197	C₄⋊Q₈	C₂³×C₁₃₉	1	2	yes	—
198	C₄⋊Q₈	C₂³×C₁₃₉	1	2	yes	—
199	C₄⋊Q₈	C₂³×C₁₃₉	1	2	yes	—
200	C₄⋊Q₈	C₂³×C₁₃₉	1	2	yes	—
201	#125	—	2	1	no	—
202	#125	—	2	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	2	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	#117	—	2	1	no	—
210	#117	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
213	#107	—	2	4	no	—
214	#107	—	2	4	no	—
215	#106	—	2	4	no	—
216	#106	—	2	4	no	—
217	#106	—	2	2	no	—
218	#106	—	2	4	no	—
219	#103	—	2	4	no	—
220	#103	—	2	4	no	—
221	#103	—	2	4	no	—
222	#103	—	2	4	no	—
223	#103	—	2	4	no	—
224	#97	D₄×C₁₃₉	1	2	yes	—
225	#97	D₄×C₁₃₉	1	4	yes	—
226	C₄×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
227	C₄×C₈	C₂³×C₁₃₉	1	2	yes	—
228	C₄×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₁₃₉	1	2	yes	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	4	4	no	—
235	C₂×Q₈	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
239	Q₁₆	—	4	4	no	—
240	Q₁₆	—	4	8	no	—
241	M₁₆	—	2	8	no	—
242	M₁₆	—	4	2	no	—
243	C₂×C₈	C₂×D₄×C₁₃₉	1	2	yes	—
244	C₂×C₈	C₂×D₄×C₁₃₉	1	4	yes	—
245	C₂×C₈	C₂×D₄×C₁₃₉	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₁₃₉	1	4	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	4	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₁₃₉	1	2	yes	—
265	C₄²	C₂⁴×C₁₃₉	1	1	yes	characteristic
266	C₂³	#127×C₁₃₉	1	1	yes	characteristic
267	Q₈	—	4	4	no	—
268	Q₈	—	4	4	no	—
269	Q₈	—	4	4	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	8	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	#127×C₁₃₉	1	2	yes	—
280	C₂×C₄	#127×C₁₃₉	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₁₃₉	1	2	yes	—
282	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	#137×C₁₃₉	1	8	yes	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	1	no	—
289	C₂²	C₂×#127×C₁₃₉	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	8	2	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	C₂×#137×C₁₃₉	1	4	yes	—
297	C₄	#575×C₁₃₉	1	2	yes	—
298	C₂	—	4	1	no	—
299	C₂	#3152×C₁₃₉	1	2	yes	—
300	C₂	#3153×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₉
prime radical	C₂³×C₁₃₉	#127
LCS	G ▷ C₄² ▷ 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
5. #1723×C₁₃₉	C₂	semidirect
6. #1718×C₁₃₉	C₂	semidirect
7. #1716×C₁₃₉	C₂	semidirect
8. #1707×C₁₃₉	C₂	semidirect
150. #19751	C₁₃₉	direct (N × H)
151. C₁₃₉	#19751	direct (N × H)
156. #1723	C₂×C₁₃₉	semidirect
157. #1718	C₂×C₁₃₉	semidirect
158. #1716	C₂×C₁₃₉	semidirect
159. #1707	C₂×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#3185×C₁₃₉	2
#3165×C₁₃₉	2
#3139×C₁₃₉	2
#1723×C₁₃₉	2
#1718×C₁₃₉	2
#1716×C₁₃₉	2
#1707×C₁₃₉	2
#1455×C₁₃₉	2
#19751	139

Subgroups — C139×#19751 #48014347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#19751 #48014347