Subgroups — C₁₃₉×#10892 #48005488

Order 35584.

818 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#382×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#2274×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#343×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#1736×C₁₃₉	C₂	1	2	yes	maximal
5	C₂×#339×C₁₃₉	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₁₃₉	C₂²	1	2	yes	—
7	C₂×#117×C₁₃₉	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
11	#382×C₁₃₉	—	2	2	no	—
12	#382×C₁₃₉	C₄	1	4	yes	complemented
13	C₄×C₄⋊C₄×C₁₃₉	—	2	4	no	—
14	C₄×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
15	C₂×#95×C₁₃₉	—	2	2	no	—
16	#343×C₁₃₉	—	2	2	no	—
17	#340×C₁₃₉	—	2	4	no	—
18	#339×C₁₃₉	C₄	1	8	yes	complemented
19	C₂³×C₄×C₁₃₉	—	2	1	no	—
20	C₂³×C₄×C₁₃₉	—	2	1	no	—
21	C₂³×C₄×C₁₃₉	D₄	1	2	yes	—
22	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₁₃₉	—	2	2	no	—
24	C₂×M₁₆×C₁₃₉	—	2	2	no	—
25	C₂×M₁₆×C₁₃₉	—	2	2	no	—
26	C₂×M₁₆×C₁₃₉	—	2	2	no	—
27	C₂×M₁₆×C₁₃₉	—	2	2	no	—
28	C₂×M₁₆×C₁₃₉	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₃₉	—	2	2	no	—
30	#117×C₁₃₉	—	2	2	no	—
31	#117×C₁₃₉	—	2	2	no	—
32	#117×C₁₃₉	C₂×C₄	1	4	yes	—
33	#117×C₁₃₉	Q₈	1	4	yes	—
34	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₃₉	D₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₃₉	D₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
44	C₂×C₄²×C₁₃₉	—	2	1	no	—
45	C₂×C₄²×C₁₃₉	—	2	2	no	—
46	C₂×C₄²×C₁₃₉	—	2	2	no	—
47	C₂×C₄²×C₁₃₉	—	2	4	no	—
48	C₂×C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
50	#95×C₁₃₉	—	2	4	no	—
51	#95×C₁₃₉	—	2	8	no	—
52	C₂⁴×C₁₃₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₉	—	2	2	no	—
54	C₂²×C₄×C₁₃₉	—	2	4	no	—
55	C₂²×C₄×C₁₃₉	—	2	4	no	—
56	C₂²×C₄×C₁₃₉	—	2	2	no	—
57	C₂²×C₄×C₁₃₉	—	2	2	no	—
58	C₂²×C₄×C₁₃₉	—	2	2	no	—
59	C₂²×C₄×C₁₃₉	—	4	2	no	—
60	C₂²×C₄×C₁₃₉	—	4	2	no	—
61	C₂²×C₄×C₁₃₉	—	4	4	no	—
62	C₂²×C₄×C₁₃₉	—	4	4	no	—
63	C₂²×C₄×C₁₃₉	—	4	1	no	—
64	C₂²×C₄×C₁₃₉	—	4	1	no	—
65	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₃₉	#41	1	2	yes	—
72	C₂²×C₄×C₁₃₉	#41	1	2	yes	—
73	C₂²×C₄×C₁₃₉	#41	1	2	yes	—
74	M₁₆×C₁₃₉	—	2	4	no	—
75	M₁₆×C₁₃₉	—	2	4	no	—
76	M₁₆×C₁₃₉	—	4	2	no	—
77	M₁₆×C₁₃₉	—	4	2	no	—
78	C₂×C₈×C₁₃₉	—	2	4	no	—
79	C₂×C₈×C₁₃₉	—	4	2	no	—
80	C₂×C₈×C₁₃₉	—	4	2	no	—
81	C₄⋊C₄×C₁₃₉	—	2	4	no	—
82	C₄⋊C₄×C₁₃₉	—	2	4	no	—
83	C₄⋊C₄×C₁₃₉	—	2	4	no	—
84	C₄⋊C₄×C₁₃₉	—	2	4	no	—
85	C₄⋊C₄×C₁₃₉	—	4	2	no	—
86	C₄⋊C₄×C₁₃₉	C₄⋊C₄	1	8	yes	—
87	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
88	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
89	C₄²×C₁₃₉	—	2	2	no	—
90	C₄²×C₁₃₉	—	2	4	no	—
91	C₄²×C₁₃₉	—	2	4	no	—
92	C₄²×C₁₃₉	—	2	4	no	—
93	C₄²×C₁₃₉	—	8	4	no	—
94	C₄²×C₁₃₉	C₄²	1	4	yes	complemented
95	C₂³×C₁₃₉	—	2	2	no	—
96	C₂³×C₁₃₉	—	2	2	no	—
97	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂³×C₁₃₉	C₄×D₄	1	2	yes	—
99	C₂³×C₁₃₉	#120	1	1	yes	characteristic
100	C₂³×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₁₃₉	#123	1	1	yes	characteristic
102	C₂×C₄×C₁₃₉	—	2	4	no	—
103	C₂×C₄×C₁₃₉	—	2	8	no	—
104	C₂×C₄×C₁₃₉	—	2	4	no	—
105	C₂×C₄×C₁₃₉	—	4	8	no	—
106	C₂×C₄×C₁₃₉	—	4	2	no	—
107	C₂×C₄×C₁₃₉	—	4	2	no	—
108	C₂×C₄×C₁₃₉	—	4	4	no	—
109	C₂×C₄×C₁₃₉	—	4	4	no	—
110	C₂×C₄×C₁₃₉	—	4	4	no	—
111	C₂×C₄×C₁₃₉	—	4	4	no	—
112	C₂×C₄×C₁₃₉	—	4	4	no	—
113	C₂×C₄×C₁₃₉	—	4	2	no	—
114	C₂×C₄×C₁₃₉	—	8	2	no	—
115	C₂×C₄×C₁₃₉	—	8	2	no	—
116	C₂×C₄×C₁₃₉	—	8	2	no	—
117	C₂×C₄×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₃₉	#117	1	2	yes	—
120	C₂×C₄×C₁₃₉	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₃₉	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃₉	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₁₃₉	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₁₃₉	#121	1	2	yes	—
127	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
128	C₂×C₄×C₁₃₉	#123	1	2	yes	—
129	C₂×C₄×C₁₃₉	#124	1	1	yes	characteristic
130	C₂×C₄×C₁₃₉	#125	1	2	yes	—
131	C₂×C₄×C₁₃₉	#126	1	2	yes	—
132	C₂×C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
133	C₈×C₁₃₉	—	4	4	no	—
134	C₂²×C₁₃₉	—	2	4	no	—
135	C₂²×C₁₃₉	—	2	4	no	—
136	C₂²×C₁₃₉	—	2	1	no	—
137	C₂²×C₁₃₉	—	2	1	no	—
138	C₂²×C₁₃₉	—	2	1	no	—
139	C₂²×C₁₃₉	—	2	1	no	—
140	C₂²×C₁₃₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₁₃₉	#385	1	2	yes	—
142	C₂²×C₁₃₉	#386	1	1	yes	characteristic
143	C₂²×C₁₃₉	#387	1	2	yes	—
144	C₂²×C₁₃₉	#391	1	1	yes	characteristic
145	C₂²×C₁₃₉	#454	1	2	yes	—
146	C₂²×C₁₃₉	#455	1	2	yes	—
147	C₄×C₁₃₉	—	4	4	no	—
148	C₄×C₁₃₉	—	4	8	no	—
149	C₄×C₁₃₉	—	4	4	no	—
150	C₄×C₁₃₉	—	8	2	no	—
151	C₄×C₁₃₉	C₄×C₄⋊C₄	1	2	yes	—
152	C₄×C₁₃₉	#387	1	2	yes	—
153	C₄×C₁₃₉	#388	1	2	yes	—
154	C₄×C₁₃₉	#389	1	2	yes	—
155	C₂×C₁₃₉	—	2	1	no	—
156	C₂×C₁₃₉	—	2	2	no	—
157	C₂×C₁₃₉	—	2	1	no	—
158	C₂×C₁₃₉	#1436	1	1	yes	characteristic
159	C₂×C₁₃₉	#1860	1	4	yes	—
160	C₂×C₁₃₉	#1885	1	1	yes	characteristic
161	C₂×C₁₃₉	#1888	1	1	yes	characteristic
162	#10892	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₁₃₉	#10892	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×#382	C₂×C₁₃₉	1	1	yes	characteristic
165	#2274	C₂×C₁₃₉	1	1	yes	characteristic
166	C₂×#343	C₂×C₁₃₉	1	1	yes	characteristic
167	#1736	C₂×C₁₃₉	1	2	yes	—
168	C₂×#339	C₂×C₁₃₉	1	2	yes	—
169	C₂²×M₁₆	C₂²×C₁₃₉	1	2	yes	—
170	C₂×#117	C₂²×C₁₃₉	1	2	yes	—
171	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	1	yes	characteristic
172	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	1	yes	characteristic
173	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
174	#382	—	2	2	no	—
175	#382	C₄×C₁₃₉	1	4	yes	complemented
176	C₄×C₄⋊C₄	—	2	4	no	—
177	C₄×C₂²⋊C₄	—	2	2	no	—
178	C₂×#95	—	2	2	no	—
179	#343	—	2	2	no	—
180	#340	—	2	4	no	—
181	#339	C₄×C₁₃₉	1	8	yes	complemented
182	C₂³×C₄	—	2	1	no	—
183	C₂³×C₄	—	2	1	no	—
184	C₂³×C₄	D₄×C₁₃₉	1	2	yes	—
185	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	C₂×C₄×C₁₃₉	1	4	yes	—
192	C₂²×C₈	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	C₂×C₄×C₁₃₉	1	4	yes	—
196	#117	Q₈×C₁₃₉	1	4	yes	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
204	C₂×C₄⋊C₄	D₄×C₁₃₉	1	2	yes	—
205	C₂×C₄⋊C₄	D₄×C₁₃₉	1	4	yes	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	—	2	4	no	—
211	C₂×C₄²	C₂×C₄×C₁₃₉	1	4	yes	—
212	C₂×C₄²	C₂×C₄×C₁₃₉	1	2	yes	—
213	#95	—	2	4	no	—
214	#95	—	2	8	no	—
215	C₂⁴	C₂×D₄×C₁₃₉	1	1	yes	characteristic
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	4	yes	—
229	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
233	C₂²×C₄	C₂×Q₈×C₁₃₉	1	2	yes	—
234	C₂²×C₄	#41×C₁₃₉	1	2	yes	—
235	C₂²×C₄	#41×C₁₃₉	1	2	yes	—
236	C₂²×C₄	#41×C₁₃₉	1	2	yes	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	4	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	4	2	no	—
243	C₂×C₈	—	4	2	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	C₄⋊C₄×C₁₃₉	1	8	yes	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	8	4	no	—
257	C₄²	C₄²×C₁₃₉	1	4	yes	complemented
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
261	C₂³	C₄×D₄×C₁₃₉	1	2	yes	—
262	C₂³	#120×C₁₃₉	1	1	yes	characteristic
263	C₂³	C₂²⋊Q₈×C₁₃₉	1	2	yes	—
264	C₂³	#123×C₁₃₉	1	1	yes	characteristic
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	4	8	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	8	2	no	—
279	C₂×C₄	—	8	2	no	—
280	C₂×C₄	C₂×C₄²×C₁₃₉	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
282	C₂×C₄	#117×C₁₃₉	1	2	yes	—
283	C₂×C₄	#117×C₁₃₉	1	1	yes	characteristic
284	C₂×C₄	C₄×D₄×C₁₃₉	1	2	yes	—
285	C₂×C₄	C₄×D₄×C₁₃₉	1	2	yes	—
286	C₂×C₄	C₄×D₄×C₁₃₉	1	2	yes	—
287	C₂×C₄	C₄×Q₈×C₁₃₉	1	2	yes	—
288	C₂×C₄	C₄×Q₈×C₁₃₉	1	2	yes	—
289	C₂×C₄	#121×C₁₃₉	1	2	yes	—
290	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	2	yes	—
291	C₂×C₄	#123×C₁₃₉	1	2	yes	—
292	C₂×C₄	#124×C₁₃₉	1	1	yes	characteristic
293	C₂×C₄	#125×C₁₃₉	1	2	yes	—
294	C₂×C₄	#126×C₁₃₉	1	2	yes	—
295	C₂×C₄	C₄⋊Q₈×C₁₃₉	1	1	yes	characteristic
296	C₈	—	4	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	C₄×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
304	C₂²	#385×C₁₃₉	1	2	yes	—
305	C₂²	#386×C₁₃₉	1	1	yes	characteristic
306	C₂²	#387×C₁₃₉	1	2	yes	—
307	C₂²	#391×C₁₃₉	1	1	yes	characteristic
308	C₂²	#454×C₁₃₉	1	2	yes	—
309	C₂²	#455×C₁₃₉	1	2	yes	—
310	C₄	—	4	4	no	—
311	C₄	—	4	8	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	2	no	—
314	C₄	C₄×C₄⋊C₄×C₁₃₉	1	2	yes	—
315	C₄	#387×C₁₃₉	1	2	yes	—
316	C₄	#388×C₁₃₉	1	2	yes	—
317	C₄	#389×C₁₃₉	1	2	yes	—
318	C₂	—	2	1	no	—
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	#1436×C₁₃₉	1	1	yes	characteristic
322	C₂	#1860×C₁₃₉	1	4	yes	—
323	C₂	#1885×C₁₃₉	1	1	yes	characteristic
324	C₂	#1888×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃₉	#120
commutator	C₂×C₄	C₂×C₄²×C₁₃₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₉
prime radical	C₂⁴×C₁₃₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ C₂³×C₄×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
12. #382×C₁₃₉	C₄	semidirect
18. #339×C₁₃₉	C₄	semidirect
94. C₄²×C₁₃₉	C₄²	semidirect
162. #10892	C₁₃₉	direct (N × H)
163. C₁₃₉	#10892	direct (N × H)
175. #382	C₄×C₁₃₉	semidirect
181. #339	C₄×C₁₃₉	semidirect
257. C₄²	C₄²×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#382×C₁₃₉	2
#2274×C₁₃₉	2
C₂×#343×C₁₃₉	2
#1736×C₁₃₉	2
C₂×#339×C₁₃₉	2
#10892	139

Subgroups — C139×#10892 #48005488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#10892 #48005488