Subgroups — C₁₃₉×#9103 #48003699

Order 35584.

538 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2975×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#2326×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1479×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	#1460×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	#1274×C₁₃₉	C₂	1	2	yes	maximal, complemented
6	C₂×#124×C₁₃₉	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
9	#436×C₁₃₉	—	2	4	no	—
10	#431×C₁₃₉	C₂²	1	2	yes	—
11	#431×C₁₃₉	C₂²	1	2	yes	—
12	#407×C₁₃₉	—	2	2	no	—
13	#391×C₁₃₉	C₄	1	1	yes	characteristic
14	#390×C₁₃₉	C₄	1	1	yes	characteristic
15	#388×C₁₃₉	—	2	2	no	—
16	#385×C₁₃₉	—	2	2	no	—
17	#384×C₁₃₉	C₄	1	1	yes	characteristic
18	#376×C₁₃₉	C₂²	1	1	yes	characteristic
19	Q₈⋊C₈×C₁₃₉	—	2	4	no	—
20	#323×C₁₃₉	—	2	2	no	—
21	C₂²×Q₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃₉	D₄	1	2	yes	—
24	C₂×M₁₆×C₁₃₉	—	2	4	no	—
25	#124×C₁₃₉	—	2	2	no	—
26	#124×C₁₃₉	—	2	2	no	—
27	C₄×Q₈×C₁₃₉	—	2	2	no	—
28	C₄×Q₈×C₁₃₉	—	2	2	no	—
29	C₄×D₄×C₁₃₉	—	2	2	no	—
30	C₄×D₄×C₁₃₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
37	C₂×C₂²⋊C₄×C₁₃₉	D₄	1	2	yes	—
38	C₂×C₄²×C₁₃₉	—	2	1	no	—
39	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₁₃₉	—	2	2	no	—
43	C₄⋊C₈×C₁₃₉	—	2	2	no	—
44	C₂²⋊C₈×C₁₃₉	—	2	4	no	—
45	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
46	C₂²⋊C₈×C₁₃₉	—	2	2	no	—
47	C₄×C₈×C₁₃₉	—	2	2	no	—
48	C₄×C₈×C₁₃₉	D₄	1	4	yes	—
49	#95×C₁₃₉	—	2	2	no	—
50	#95×C₁₃₉	—	2	1	no	—
51	#95×C₁₃₉	—	2	1	no	—
52	C₂⁴×C₁₃₉	C₂²⋊C₄	1	2	yes	—
53	C₂×Q₈×C₁₃₉	—	2	1	no	—
54	C₂×Q₈×C₁₃₉	—	2	2	no	—
55	C₂×Q₈×C₁₃₉	—	2	2	no	—
56	C₂×Q₈×C₁₃₉	M₁₆	1	2	yes	complemented
57	C₂×D₄×C₁₃₉	—	2	2	no	—
58	C₂×D₄×C₁₃₉	—	2	1	no	—
59	C₂×D₄×C₁₃₉	—	2	1	no	—
60	C₂×D₄×C₁₃₉	—	2	2	no	—
61	C₂²×C₄×C₁₃₉	—	2	1	no	—
62	C₂²×C₄×C₁₃₉	—	2	1	no	—
63	C₂²×C₄×C₁₃₉	—	2	2	no	—
64	C₂²×C₄×C₁₃₉	—	2	2	no	—
65	C₂²×C₄×C₁₃₉	—	2	1	no	—
66	C₂²×C₄×C₁₃₉	—	2	1	no	—
67	C₂²×C₄×C₁₃₉	—	8	2	no	—
68	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
72	M₁₆×C₁₃₉	—	8	4	no	—
73	C₂×C₈×C₁₃₉	—	2	4	no	—
74	C₂×C₈×C₁₃₉	—	4	2	no	—
75	C₄⋊C₄×C₁₃₉	—	2	2	no	—
76	C₄⋊C₄×C₁₃₉	—	2	2	no	—
77	C₄⋊C₄×C₁₃₉	—	4	1	no	—
78	C₄⋊C₄×C₁₃₉	—	4	1	no	—
79	C₄⋊C₄×C₁₃₉	—	8	2	no	—
80	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
81	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
82	C₂²⋊C₄×C₁₃₉	—	8	2	no	—
83	C₄²×C₁₃₉	—	2	1	no	—
84	C₄²×C₁₃₉	—	2	2	no	—
85	C₄²×C₁₃₉	—	4	2	no	—
86	C₄²×C₁₃₉	M₁₆	1	2	yes	complemented
87	C₄²×C₁₃₉	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃₉	#41	1	2	yes	—
89	C₂³×C₁₃₉	—	2	2	no	—
90	C₂³×C₁₃₉	—	2	2	no	—
91	C₂³×C₁₃₉	—	2	2	no	—
92	C₂³×C₁₃₉	—	4	2	no	—
93	C₂³×C₁₃₉	—	4	2	no	—
94	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	Q₈×C₁₃₉	—	2	4	no	—
96	Q₈×C₁₃₉	—	4	2	no	—
97	D₄×C₁₃₉	—	4	2	no	—
98	D₄×C₁₃₉	—	4	2	no	—
99	C₂×C₄×C₁₃₉	—	2	2	no	—
100	C₂×C₄×C₁₃₉	—	2	2	no	—
101	C₂×C₄×C₁₃₉	—	2	2	no	—
102	C₂×C₄×C₁₃₉	—	4	1	no	—
103	C₂×C₄×C₁₃₉	—	4	1	no	—
104	C₂×C₄×C₁₃₉	—	4	4	no	—
105	C₂×C₄×C₁₃₉	—	4	1	no	—
106	C₂×C₄×C₁₃₉	—	4	1	no	—
107	C₂×C₄×C₁₃₉	—	4	1	no	—
108	C₂×C₄×C₁₃₉	—	4	1	no	—
109	C₂×C₄×C₁₃₉	—	4	2	no	—
110	C₂×C₄×C₁₃₉	—	4	1	no	—
111	C₂×C₄×C₁₃₉	—	4	2	no	—
112	C₂×C₄×C₁₃₉	—	4	1	no	—
113	C₂×C₄×C₁₃₉	—	4	1	no	—
114	C₂×C₄×C₁₃₉	—	4	1	no	—
115	C₂×C₄×C₁₃₉	—	8	2	no	—
116	C₂×C₄×C₁₃₉	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₃₉	#120	1	1	yes	characteristic
118	C₂×C₄×C₁₃₉	#121	1	2	yes	—
119	C₂×C₄×C₁₃₉	#123	1	1	yes	characteristic
120	C₂×C₄×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₃₉	#131	1	2	yes	—
123	C₈×C₁₃₉	—	4	4	no	—
124	C₈×C₁₃₉	—	8	2	no	—
125	C₂²×C₁₃₉	—	2	1	no	—
126	C₂²×C₁₃₉	—	2	1	no	—
127	C₂²×C₁₃₉	—	4	4	no	—
128	C₂²×C₁₃₉	—	4	2	no	—
129	C₂²×C₁₃₉	—	4	2	no	—
130	C₂²×C₁₃₉	—	4	2	no	—
131	C₂²×C₁₃₉	—	8	2	no	—
132	C₂²×C₁₃₉	#386	1	1	yes	characteristic
133	C₂²×C₁₃₉	#407	1	1	yes	characteristic
134	C₂²×C₁₃₉	#408	1	1	yes	characteristic
135	C₄×C₁₃₉	—	2	2	no	—
136	C₄×C₁₃₉	—	2	1	no	—
137	C₄×C₁₃₉	—	2	1	no	—
138	C₄×C₁₃₉	—	4	2	no	—
139	C₄×C₁₃₉	—	4	2	no	—
140	C₄×C₁₃₉	—	4	2	no	—
141	C₄×C₁₃₉	—	8	1	no	—
142	C₄×C₁₃₉	—	8	1	no	—
143	C₄×C₁₃₉	#435	1	2	yes	—
144	C₄×C₁₃₉	#436	1	2	yes	—
145	C₂×C₁₃₉	—	2	1	no	—
146	C₂×C₁₃₉	—	2	1	no	—
147	C₂×C₁₃₉	—	8	2	no	—
148	C₂×C₁₃₉	#1870	1	1	yes	characteristic
149	C₂×C₁₃₉	#1909	1	1	yes	characteristic
150	C₂×C₁₃₉	#1911	1	1	yes	characteristic
151	#9103	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₁₃₉	#9103	1	1	yes	characteristic, direct product factor, Sylow
153	#2975	C₂×C₁₃₉	1	1	yes	characteristic
154	#2326	C₂×C₁₃₉	1	1	yes	characteristic
155	#1479	C₂×C₁₃₉	1	2	yes	complemented
156	#1460	C₂×C₁₃₉	1	1	yes	characteristic
157	#1274	C₂×C₁₃₉	1	2	yes	complemented
158	C₂×#124	C₄×C₁₃₉	1	1	yes	characteristic
159	C₂×C₄×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
160	C₂×C₄×D₄	C₂²×C₁₃₉	1	1	yes	characteristic
161	#436	—	2	4	no	—
162	#431	C₂²×C₁₃₉	1	2	yes	—
163	#431	C₂²×C₁₃₉	1	2	yes	—
164	#407	—	2	2	no	—
165	#391	C₄×C₁₃₉	1	1	yes	characteristic
166	#390	C₄×C₁₃₉	1	1	yes	characteristic
167	#388	—	2	2	no	—
168	#385	—	2	2	no	—
169	#384	C₄×C₁₃₉	1	1	yes	characteristic
170	#376	C₂²×C₁₃₉	1	1	yes	characteristic
171	Q₈⋊C₈	—	2	4	no	—
172	#323	—	2	2	no	—
173	C₂²×Q₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
174	C₂²×D₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
175	C₂³×C₄	D₄×C₁₃₉	1	2	yes	—
176	C₂×M₁₆	—	2	4	no	—
177	#124	—	2	2	no	—
178	#124	—	2	2	no	—
179	C₄×Q₈	—	2	2	no	—
180	C₄×Q₈	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
186	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	D₄×C₁₃₉	1	2	yes	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
192	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₂²⋊C₈	—	2	4	no	—
197	C₂²⋊C₈	—	2	2	no	—
198	C₂²⋊C₈	—	2	2	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	D₄×C₁₃₉	1	4	yes	—
201	#95	—	2	2	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	C₂⁴	C₂²⋊C₄×C₁₃₉	1	2	yes	—
205	C₂×Q₈	—	2	1	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	M₁₆×C₁₃₉	1	2	yes	complemented
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	8	2	no	—
220	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
224	M₁₆	—	8	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	4	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	4	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	8	2	no	—
232	C₂²⋊C₄	—	2	8	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₂²⋊C₄	—	8	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	M₁₆×C₁₃₉	1	2	yes	complemented
239	C₄²	C₂×D₄×C₁₃₉	1	2	yes	—
240	C₄²	#41×C₁₃₉	1	2	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	4	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	—	4	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	C₄×D₄×C₁₃₉	1	2	yes	—
269	C₂×C₄	#120×C₁₃₉	1	1	yes	characteristic
270	C₂×C₄	#121×C₁₃₉	1	2	yes	—
271	C₂×C₄	#123×C₁₃₉	1	1	yes	characteristic
272	C₂×C₄	C₂×M₁₆×C₁₃₉	1	1	yes	characteristic
273	C₂×C₄	C₂×M₁₆×C₁₃₉	1	1	yes	characteristic
274	C₂×C₄	#131×C₁₃₉	1	2	yes	—
275	C₈	—	4	4	no	—
276	C₈	—	8	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	#386×C₁₃₉	1	1	yes	characteristic
285	C₂²	#407×C₁₃₉	1	1	yes	characteristic
286	C₂²	#408×C₁₃₉	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	2	no	—
291	C₄	—	4	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	8	1	no	—
294	C₄	—	8	1	no	—
295	C₄	#435×C₁₃₉	1	2	yes	—
296	C₄	#436×C₁₃₉	1	2	yes	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	8	2	no	—
300	C₂	#1870×C₁₃₉	1	1	yes	characteristic
301	C₂	#1909×C₁₃₉	1	1	yes	characteristic
302	C₂	#1911×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
prime radical	C₂²×D₄×C₁₃₉	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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
3. #1479×C₁₃₉	C₂	semidirect
5. #1274×C₁₃₉	C₂	semidirect
56. C₂×Q₈×C₁₃₉	M₁₆	semidirect
86. C₄²×C₁₃₉	M₁₆	semidirect
151. #9103	C₁₃₉	direct (N × H)
152. C₁₃₉	#9103	direct (N × H)
155. #1479	C₂×C₁₃₉	semidirect
157. #1274	C₂×C₁₃₉	semidirect
208. C₂×Q₈	M₁₆×C₁₃₉	semidirect
238. C₄²	M₁₆×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#2975×C₁₃₉	2
#2326×C₁₃₉	2
#1479×C₁₃₉	2
#1460×C₁₃₉	2
#1274×C₁₃₉	2
#9103	139

Subgroups — C139×#9103 #48003699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#9103 #48003699