Subgroups — C₁₃₉×#12586 #48007182

Order 35584.

746 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#101×C₁₃₉	C₂	1	2	yes	maximal
2	#2310×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1394×C₁₃₉	C₂	1	4	yes	maximal, complemented
4	C₂³×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₁₃₉	—	2	2	no	—
6	C₂×#117×C₁₃₉	C₄	1	1	yes	characteristic
7	C₂×#117×C₁₃₉	C₄	1	1	yes	characteristic
8	C₂×#101×C₁₃₉	—	2	2	no	—
9	C₂×#101×C₁₃₉	—	2	4	no	—
10	C₂×#101×C₁₃₉	C₄	1	8	yes	complemented
11	C₂×#101×C₁₃₉	C₂²	1	4	yes	—
12	#391×C₁₃₉	—	2	2	no	—
13	#391×C₁₃₉	C₄	1	2	yes	—
14	#391×C₁₃₉	C₂²	1	2	yes	—
15	#381×C₁₃₉	—	2	2	no	—
16	C₂²×Q₈×C₁₃₉	—	4	2	no	—
17	C₂²×Q₈×C₁₃₉	—	4	2	no	—
18	C₂²×Q₈×C₁₃₉	C₂×C₄	1	4	yes	—
19	C₂²×Q₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×Q₈×C₁₃₉	D₄	1	2	yes	—
21	C₂²×Q₈×C₁₃₉	D₄	1	2	yes	—
22	C₂²×Q₈×C₁₃₉	Q₈	1	2	yes	—
23	C₂²×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
24	C₂³×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₃₉	D₄	1	2	yes	—
26	C₂×M₁₆×C₁₃₉	—	2	8	no	—
27	C₂×M₁₆×C₁₃₉	—	2	4	no	—
28	C₂×M₁₆×C₁₃₉	—	4	4	no	—
29	C₂×M₁₆×C₁₃₉	—	4	2	no	—
30	C₂²×C₈×C₁₃₉	—	4	2	no	—
31	#117×C₁₃₉	—	4	2	no	—
32	#117×C₁₃₉	—	4	2	no	—
33	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₁₃₉	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₁₃₉	—	2	1	no	—
37	C₂×C₄²×C₁₃₉	—	2	1	no	—
38	C₂×C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
39	#101×C₁₃₉	—	2	8	no	—
40	#101×C₁₃₉	—	2	4	no	—
41	#101×C₁₃₉	C₂×C₄	1	8	yes	—
42	#95×C₁₃₉	—	2	4	no	—
43	#95×C₁₃₉	—	2	4	no	—
44	C₂⁴×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×Q₈×C₁₃₉	—	4	4	no	—
46	C₂×Q₈×C₁₃₉	—	4	4	no	—
47	C₂×Q₈×C₁₃₉	—	4	1	no	—
48	C₂×Q₈×C₁₃₉	—	4	1	no	—
49	C₂×Q₈×C₁₃₉	—	4	4	no	—
50	C₂×Q₈×C₁₃₉	—	4	2	no	—
51	C₂×Q₈×C₁₃₉	—	4	2	no	—
52	C₂×Q₈×C₁₃₉	—	4	4	no	—
53	C₂×Q₈×C₁₃₉	—	4	1	no	—
54	C₂×Q₈×C₁₃₉	—	4	1	no	—
55	C₂×Q₈×C₁₃₉	C₄²	1	2	yes	—
56	C₂×Q₈×C₁₃₉	C₂²⋊C₄	1	4	yes	—
57	C₂×Q₈×C₁₃₉	C₄⋊C₄	1	4	yes	—
58	C₂×Q₈×C₁₃₉	C₄⋊C₄	1	2	yes	—
59	C₂×Q₈×C₁₃₉	C₂²×C₄	1	2	yes	—
60	C₂×Q₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
61	C₂×Q₈×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₉	—	2	2	no	—
63	C₂²×C₄×C₁₃₉	—	2	2	no	—
64	C₂²×C₄×C₁₃₉	—	2	2	no	—
65	C₂²×C₄×C₁₃₉	—	2	4	no	—
66	C₂²×C₄×C₁₃₉	—	2	2	no	—
67	C₂²×C₄×C₁₃₉	—	2	2	no	—
68	C₂²×C₄×C₁₃₉	—	2	2	no	—
69	C₂²×C₄×C₁₃₉	C₄²	1	2	yes	—
70	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
73	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
77	M₁₆×C₁₃₉	—	2	8	no	—
78	M₁₆×C₁₃₉	—	4	8	no	—
79	M₁₆×C₁₃₉	—	4	4	no	—
80	C₂×C₈×C₁₃₉	—	4	8	no	—
81	C₂×C₈×C₁₃₉	—	4	4	no	—
82	C₄⋊C₄×C₁₃₉	—	4	2	no	—
83	C₄⋊C₄×C₁₃₉	—	4	2	no	—
84	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
85	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
86	C₄²×C₁₃₉	—	4	2	no	—
87	C₄²×C₁₃₉	—	4	2	no	—
88	C₂³×C₁₃₉	—	4	2	no	—
89	C₂³×C₁₃₉	#95	1	2	yes	—
90	C₂³×C₁₃₉	#99	1	2	yes	—
91	C₂³×C₁₃₉	#99	1	2	yes	—
92	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	Q₈×C₁₃₉	—	4	2	no	—
94	Q₈×C₁₃₉	—	4	4	no	—
95	Q₈×C₁₃₉	—	4	2	no	—
96	Q₈×C₁₃₉	—	4	4	no	—
97	Q₈×C₁₃₉	—	4	2	no	—
98	Q₈×C₁₃₉	—	4	1	no	—
99	Q₈×C₁₃₉	—	4	1	no	—
100	C₂×C₄×C₁₃₉	—	2	4	no	—
101	C₂×C₄×C₁₃₉	—	2	4	no	—
102	C₂×C₄×C₁₃₉	—	2	4	no	—
103	C₂×C₄×C₁₃₉	—	2	4	no	—
104	C₂×C₄×C₁₃₉	—	2	4	no	—
105	C₂×C₄×C₁₃₉	—	2	4	no	—
106	C₂×C₄×C₁₃₉	—	2	1	no	—
107	C₂×C₄×C₁₃₉	—	2	1	no	—
108	C₂×C₄×C₁₃₉	—	2	2	no	—
109	C₂×C₄×C₁₃₉	—	2	2	no	—
110	C₂×C₄×C₁₃₉	—	2	1	no	—
111	C₂×C₄×C₁₃₉	—	2	1	no	—
112	C₂×C₄×C₁₃₉	—	2	2	no	—
113	C₂×C₄×C₁₃₉	—	2	2	no	—
114	C₂×C₄×C₁₃₉	—	4	4	no	—
115	C₂×C₄×C₁₃₉	—	4	4	no	—
116	C₂×C₄×C₁₃₉	—	4	2	no	—
117	C₂×C₄×C₁₃₉	—	4	2	no	—
118	C₂×C₄×C₁₃₉	#95	1	2	yes	—
119	C₂×C₄×C₁₃₉	#95	1	4	yes	—
120	C₂×C₄×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
121	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
122	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
124	C₈×C₁₃₉	—	4	8	no	—
125	C₂²×C₁₃₉	—	2	2	no	—
126	C₂²×C₁₃₉	—	2	4	no	—
127	C₂²×C₁₃₉	—	4	2	no	—
128	C₂²×C₁₃₉	—	4	2	no	—
129	C₂²×C₁₃₉	#342	1	4	yes	—
130	C₂²×C₁₃₉	C₂×#95	1	1	yes	characteristic
131	C₂²×C₁₃₉	C₂×#99	1	1	yes	characteristic
132	C₂²×C₁₃₉	C₂×#99	1	1	yes	characteristic
133	C₄×C₁₃₉	—	2	2	no	—
134	C₄×C₁₃₉	—	2	4	no	—
135	C₄×C₁₃₉	—	2	2	no	—
136	C₄×C₁₃₉	—	2	1	no	—
137	C₄×C₁₃₉	—	2	1	no	—
138	C₄×C₁₃₉	—	2	2	no	—
139	C₄×C₁₃₉	—	8	2	no	—
140	C₄×C₁₃₉	—	8	2	no	—
141	C₂×C₁₃₉	—	2	1	no	—
142	C₂×C₁₃₉	—	2	4	no	—
143	C₂×C₁₃₉	—	2	1	no	—
144	C₂×C₁₃₉	C₂×#342	1	1	yes	characteristic
145	C₂×C₁₃₉	#2133	1	2	yes	—
146	#12586	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₁₃₉	#12586	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×#101	C₂×C₁₃₉	1	2	yes	—
149	#2310	C₂×C₁₃₉	1	1	yes	characteristic
150	#1394	C₂×C₁₃₉	1	4	yes	complemented
151	C₂³×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
152	C₂²×M₁₆	—	2	2	no	—
153	C₂×#117	C₄×C₁₃₉	1	1	yes	characteristic
154	C₂×#117	C₄×C₁₃₉	1	1	yes	characteristic
155	C₂×#101	—	2	2	no	—
156	C₂×#101	—	2	4	no	—
157	C₂×#101	C₄×C₁₃₉	1	8	yes	complemented
158	C₂×#101	C₂²×C₁₃₉	1	4	yes	—
159	#391	—	2	2	no	—
160	#391	C₄×C₁₃₉	1	2	yes	—
161	#391	C₂²×C₁₃₉	1	2	yes	—
162	#381	—	2	2	no	—
163	C₂²×Q₈	—	4	2	no	—
164	C₂²×Q₈	—	4	2	no	—
165	C₂²×Q₈	C₂×C₄×C₁₃₉	1	4	yes	—
166	C₂²×Q₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
167	C₂²×Q₈	D₄×C₁₃₉	1	2	yes	—
168	C₂²×Q₈	D₄×C₁₃₉	1	2	yes	—
169	C₂²×Q₈	Q₈×C₁₃₉	1	2	yes	—
170	C₂²×Q₈	C₂³×C₁₃₉	1	1	yes	characteristic
171	C₂³×C₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
172	C₂³×C₄	D₄×C₁₃₉	1	2	yes	—
173	C₂×M₁₆	—	2	8	no	—
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	—	4	2	no	—
176	C₂×M₁₆	—	4	4	no	—
177	C₂²×C₈	—	4	2	no	—
178	#117	—	4	2	no	—
179	#117	—	4	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
182	C₂×C₂²⋊C₄	—	2	1	no	—
183	C₂×C₂²⋊C₄	—	2	1	no	—
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	C₂×C₄×C₁₃₉	1	2	yes	—
186	#101	—	2	8	no	—
187	#101	—	2	4	no	—
188	#101	C₂×C₄×C₁₃₉	1	8	yes	—
189	#95	—	2	4	no	—
190	#95	—	2	4	no	—
191	C₂⁴	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
192	C₂×Q₈	—	4	1	no	—
193	C₂×Q₈	—	4	1	no	—
194	C₂×Q₈	—	4	4	no	—
195	C₂×Q₈	—	4	4	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×Q₈	—	4	2	no	—
198	C₂×Q₈	—	4	4	no	—
199	C₂×Q₈	—	4	4	no	—
200	C₂×Q₈	—	4	1	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	C₄²×C₁₃₉	1	2	yes	—
203	C₂×Q₈	C₂²⋊C₄×C₁₃₉	1	4	yes	—
204	C₂×Q₈	C₄⋊C₄×C₁₃₉	1	4	yes	—
205	C₂×Q₈	C₄⋊C₄×C₁₃₉	1	2	yes	—
206	C₂×Q₈	C₂²×C₄×C₁₃₉	1	2	yes	—
207	C₂×Q₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
208	C₂×Q₈	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₄²×C₁₃₉	1	2	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
218	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	4	yes	—
220	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	2	yes	—
221	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
224	M₁₆	—	2	8	no	—
225	M₁₆	—	4	4	no	—
226	M₁₆	—	4	8	no	—
227	C₂×C₈	—	4	4	no	—
228	C₂×C₈	—	4	8	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	4	2	no	—
233	C₄²	—	4	2	no	—
234	C₄²	—	4	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	#95×C₁₃₉	1	2	yes	—
237	C₂³	#99×C₁₃₉	1	2	yes	—
238	C₂³	#99×C₁₃₉	1	2	yes	—
239	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
240	Q₈	—	4	2	no	—
241	Q₈	—	4	4	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	4	1	no	—
246	Q₈	—	4	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	#95×C₁₃₉	1	2	yes	—
266	C₂×C₄	#95×C₁₃₉	1	4	yes	—
267	C₂×C₄	C₂×C₄²×C₁₃₉	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	2	yes	—
269	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
270	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
271	C₈	—	4	8	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	#342×C₁₃₉	1	4	yes	—
277	C₂²	C₂×#95×C₁₃₉	1	1	yes	characteristic
278	C₂²	C₂×#99×C₁₃₉	1	1	yes	characteristic
279	C₂²	C₂×#99×C₁₃₉	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	4	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	8	2	no	—
287	C₄	—	8	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	4	no	—
290	C₂	—	2	1	no	—
291	C₂	C₂×#342×C₁₃₉	1	1	yes	characteristic
292	C₂	#2133×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂×#99
commutator	C₂×C₄	C₂×C₄²×C₁₃₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₃₉
prime radical	C₂⁴×C₁₃₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₂⁴×C₁₃₉ ◁ C₂³×Q₈×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. #1394×C₁₃₉	C₂	semidirect
10. C₂×#101×C₁₃₉	C₄	semidirect
146. #12586	C₁₃₉	direct (N × H)
147. C₁₃₉	#12586	direct (N × H)
150. #1394	C₂×C₁₃₉	semidirect
157. C₂×#101	C₄×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#101×C₁₃₉	2
#2310×C₁₃₉	2
#1394×C₁₃₉	2
#12586	139

Subgroups — C139×#12586 #48007182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#12586 #48007182