Subgroups — C₁₃₉×#8691 #48003287

Order 35584.

358 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2966×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₄×#108×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#1568×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#1529×C₁₃₉	C₂	1	2	yes	maximal
5	#1291×C₁₃₉	C₂	1	2	yes	maximal, complemented
6	#529×C₁₃₉	C₂²	1	1	yes	characteristic
7	#446×C₁₃₉	C₄	1	1	yes	characteristic
8	C₈×Q₈×C₁₃₉	C₄	1	1	yes	characteristic
9	#436×C₁₃₉	C₄	1	1	yes	characteristic
10	#435×C₁₃₉	—	2	1	no	—
11	C₈×D₄×C₁₃₉	—	2	1	no	—
12	C₈×D₄×C₁₃₉	C₄	1	1	yes	characteristic
13	#433×C₁₃₉	—	2	1	no	—
14	#432×C₁₃₉	C₂²	1	2	yes	—
15	#432×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×#108×C₁₃₉	—	2	1	no	—
17	#405×C₁₃₉	—	2	1	no	—
18	C₄×M₁₆×C₁₃₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
20	#341×C₁₃₉	—	2	1	no	—
21	#335×C₁₃₉	—	2	1	no	—
22	#334×C₁₃₉	—	2	1	no	—
23	#330×C₁₃₉	—	2	2	no	—
24	#329×C₁₃₉	—	2	2	no	—
25	C₂×M₁₆×C₁₃₉	—	2	1	no	—
26	C₂×M₁₆×C₁₃₉	—	2	2	no	—
27	C₂²×C₈×C₁₃₉	—	2	1	no	—
28	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
30	#126×C₁₃₉	—	2	1	no	—
31	#125×C₁₃₉	D₄	1	1	yes	characteristic
32	#124×C₁₃₉	D₄	1	1	yes	characteristic
33	#123×C₁₃₉	—	2	1	no	—
34	C₂²⋊Q₈×C₁₃₉	Q₈	1	1	yes	characteristic
35	#121×C₁₃₉	Q₈	1	1	yes	characteristic
36	C₄×Q₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₁₃₉	—	2	1	no	—
38	C₄×D₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
39	#117×C₁₃₉	—	2	1	no	—
40	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
41	#108×C₁₃₉	—	2	4	no	—
42	C₄⋊C₈×C₁₃₉	—	2	1	no	—
43	C₄⋊C₈×C₁₃₉	—	2	1	no	—
44	C₄⋊C₈×C₁₃₉	—	2	2	no	—
45	C₄⋊C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
46	C₄⋊C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
47	C₂²⋊C₈×C₁₃₉	—	2	1	no	—
48	C₂²⋊C₈×C₁₃₉	—	2	1	no	—
49	C₂²⋊C₈×C₁₃₉	—	2	1	no	—
50	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
51	#97×C₁₃₉	—	2	1	no	—
52	#97×C₁₃₉	—	2	1	no	—
53	#97×C₁₃₉	—	2	2	no	—
54	#97×C₁₃₉	—	2	2	no	—
55	C₄×C₈×C₁₃₉	—	2	1	no	—
56	C₄×C₈×C₁₃₉	—	2	2	no	—
57	C₄×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃₉	D₄	1	1	yes	characteristic
61	C₂×Q₈×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₄×C₁₃₉	—	2	1	no	—
63	C₂×D₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₉	—	2	1	no	—
65	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₉	#41	1	1	yes	characteristic
68	M₁₆×C₁₃₉	—	2	4	no	—
69	M₁₆×C₁₃₉	—	4	2	no	—
70	M₁₆×C₁₃₉	—	4	2	no	—
71	C₂×C₈×C₁₃₉	—	2	2	no	—
72	C₂×C₈×C₁₃₉	—	2	1	no	—
73	C₂×C₈×C₁₃₉	—	2	1	no	—
74	C₂×C₈×C₁₃₉	—	4	1	no	—
75	C₂×C₈×C₁₃₉	—	4	2	no	—
76	C₂×C₈×C₁₃₉	—	4	2	no	—
77	C₂×C₈×C₁₃₉	—	4	2	no	—
78	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₃₉	—	2	1	no	—
83	C₄⋊C₄×C₁₃₉	—	2	1	no	—
84	C₄⋊C₄×C₁₃₉	—	2	1	no	—
85	C₄⋊C₄×C₁₃₉	—	2	1	no	—
86	C₄⋊C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
89	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
90	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
91	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
92	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
93	C₄²×C₁₃₉	—	2	1	no	—
94	C₄²×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₃₉	#41	1	1	yes	characteristic
98	C₂³×C₁₃₉	—	2	1	no	—
99	C₂³×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
100	Q₈×C₁₃₉	—	4	1	no	—
101	D₄×C₁₃₉	—	4	1	no	—
102	D₄×C₁₃₉	—	4	2	no	—
103	C₂×C₄×C₁₃₉	—	2	1	no	—
104	C₂×C₄×C₁₃₉	—	2	1	no	—
105	C₂×C₄×C₁₃₉	—	2	1	no	—
106	C₂×C₄×C₁₃₉	—	2	1	no	—
107	C₂×C₄×C₁₃₉	—	2	1	no	—
108	C₂×C₄×C₁₃₉	—	4	1	no	—
109	C₂×C₄×C₁₃₉	—	4	1	no	—
110	C₂×C₄×C₁₃₉	—	4	1	no	—
111	C₂×C₄×C₁₃₉	—	4	1	no	—
112	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₈×C₁₃₉	—	2	1	no	—
119	C₈×C₁₃₉	—	2	1	no	—
120	C₈×C₁₃₉	—	2	1	no	—
121	C₈×C₁₃₉	—	4	4	no	—
122	C₈×C₁₃₉	—	4	1	no	—
123	C₈×C₁₃₉	—	4	1	no	—
124	C₈×C₁₃₉	—	8	2	no	—
125	C₈×C₁₃₉	#104	1	2	yes	—
126	C₂²×C₁₃₉	—	2	1	no	—
127	C₂²×C₁₃₉	—	2	1	no	—
128	C₂²×C₁₃₉	—	2	1	no	—
129	C₂²×C₁₃₉	—	4	1	no	—
130	C₂²×C₁₃₉	—	4	1	no	—
131	C₂²×C₁₃₉	—	4	1	no	—
132	C₂²×C₁₃₉	#380	1	1	yes	characteristic
133	C₄×C₁₃₉	—	2	1	no	—
134	C₄×C₁₃₉	—	2	1	no	—
135	C₄×C₁₃₉	—	2	1	no	—
136	C₄×C₁₃₉	—	2	1	no	—
137	C₄×C₁₃₉	—	2	1	no	—
138	C₄×C₁₃₉	—	4	1	no	—
139	C₄×C₁₃₉	—	4	1	no	—
140	C₄×C₁₃₉	—	4	1	no	—
141	C₄×C₁₃₉	C₂×#104	1	1	yes	characteristic
142	C₄×C₁₃₉	#421	1	1	yes	characteristic
143	C₂×C₁₃₉	—	4	1	no	—
144	C₂×C₁₃₉	—	8	1	no	—
145	C₂×C₁₃₉	#1806	1	1	yes	characteristic
146	C₂×C₁₃₉	#1813	1	1	yes	characteristic
147	C₂×C₁₃₉	#1814	1	1	yes	characteristic
148	#8691	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₁₃₉	#8691	1	1	yes	characteristic, direct product factor, Sylow
150	#2966	C₂×C₁₃₉	1	1	yes	characteristic
151	C₄×#108	C₂×C₁₃₉	1	1	yes	characteristic, complemented
152	#1568	C₂×C₁₃₉	1	1	yes	characteristic, complemented
153	#1529	C₂×C₁₃₉	1	2	yes	—
154	#1291	C₂×C₁₃₉	1	2	yes	complemented
155	#529	C₂²×C₁₃₉	1	1	yes	characteristic
156	#446	C₄×C₁₃₉	1	1	yes	characteristic
157	C₈×Q₈	C₄×C₁₃₉	1	1	yes	characteristic
158	#436	C₄×C₁₃₉	1	1	yes	characteristic
159	#435	—	2	1	no	—
160	C₈×D₄	—	2	1	no	—
161	C₈×D₄	C₄×C₁₃₉	1	1	yes	characteristic
162	#433	—	2	1	no	—
163	#432	C₂²×C₁₃₉	1	2	yes	—
164	#432	C₂²×C₁₃₉	1	1	yes	characteristic
165	C₂×#108	—	2	1	no	—
166	#405	—	2	1	no	—
167	C₄×M₁₆	C₂²×C₁₃₉	1	2	yes	—
168	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
169	#341	—	2	1	no	—
170	#335	—	2	1	no	—
171	#334	—	2	1	no	—
172	#330	—	2	2	no	—
173	#329	—	2	2	no	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
178	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
179	#126	—	2	1	no	—
180	#125	D₄×C₁₃₉	1	1	yes	characteristic
181	#124	D₄×C₁₃₉	1	1	yes	characteristic
182	#123	—	2	1	no	—
183	C₂²⋊Q₈	Q₈×C₁₃₉	1	1	yes	characteristic
184	#121	Q₈×C₁₃₉	1	1	yes	characteristic
185	C₄×Q₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
188	#117	—	2	1	no	—
189	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
190	#108	—	2	4	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
195	C₄⋊C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
196	C₂²⋊C₈	—	2	1	no	—
197	C₂²⋊C₈	—	2	1	no	—
198	C₂²⋊C₈	—	2	1	no	—
199	C₂²⋊C₈	—	2	2	no	—
200	#97	—	2	1	no	—
201	#97	—	2	1	no	—
202	#97	—	2	2	no	—
203	#97	—	2	2	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	2	no	—
206	C₄×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
207	C₄×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
208	C₄×C₈	D₄×C₁₃₉	1	1	yes	characteristic
209	C₄×C₈	D₄×C₁₃₉	1	1	yes	characteristic
210	C₂×Q₈	C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
216	C₂²×C₄	#41×C₁₃₉	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	4	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
236	C₄⋊C₄	C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
237	C₂²⋊C₄	—	2	1	no	—
238	C₂²⋊C₄	—	2	1	no	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	1	no	—
241	C₂²⋊C₄	—	2	1	no	—
242	C₄²	—	2	1	no	—
243	C₄²	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
244	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
246	C₄²	#41×C₁₃₉	1	1	yes	characteristic
247	C₂³	—	2	1	no	—
248	C₂³	C₂²⋊Q₈×C₁₃₉	1	1	yes	characteristic
249	Q₈	—	4	1	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
263	C₂×C₄	#117×C₁₃₉	1	1	yes	characteristic
264	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
265	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
266	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	4	4	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	—	8	2	no	—
274	C₈	#104×C₁₃₉	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	1	no	—
281	C₂²	#380×C₁₃₉	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	4	1	no	—
290	C₄	C₂×#104×C₁₃₉	1	1	yes	characteristic
291	C₄	#421×C₁₃₉	1	1	yes	characteristic
292	C₂	—	4	1	no	—
293	C₂	—	8	1	no	—
294	C₂	#1806×C₁₃₉	1	1	yes	characteristic
295	C₂	#1813×C₁₃₉	1	1	yes	characteristic
296	C₂	#1814×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	#380
commutator	C₄²	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
prime radical	#121×C₁₃₉	Q₈
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
2. C₄×#108×C₁₃₉	C₂	semidirect
3. #1568×C₁₃₉	C₂	semidirect
5. #1291×C₁₃₉	C₂	semidirect
148. #8691	C₁₃₉	direct (N × H)
149. C₁₃₉	#8691	direct (N × H)
151. C₄×#108	C₂×C₁₃₉	semidirect
152. #1568	C₂×C₁₃₉	semidirect
154. #1291	C₂×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#2966×C₁₃₉	2
C₄×#108×C₁₃₉	2
#1568×C₁₃₉	2
#1529×C₁₃₉	2
#1291×C₁₃₉	2
#8691	139

Subgroups — C139×#8691 #48003287

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#8691 #48003287