Subgroups — C₁₃₉×#18701 #48013297

Order 35584.

924 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	#3548×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#3254×C₁₃₉	C₂	1	2	yes	maximal, complemented
3	#3160×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#1654×C₁₃₉	C₂	1	4	yes	maximal, complemented
5	#1647×C₁₃₉	C₂	1	4	yes	maximal, complemented
6	#1561×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#1492×C₁₃₉	C₂	1	2	yes	maximal, complemented
8	#562×C₁₃₉	—	2	2	no	—
9	#554×C₁₃₉	—	2	2	no	—
10	#550×C₁₃₉	—	2	2	no	—
11	#547×C₁₃₉	C₂²	1	4	yes	—
12	#540×C₁₃₉	—	2	2	no	—
13	#537×C₁₃₉	—	2	4	no	—
14	#533×C₁₃₉	C₂²	1	1	yes	characteristic
15	#533×C₁₃₉	C₂²	1	1	yes	characteristic
16	#532×C₁₃₉	C₂²	1	2	yes	—
17	#500×C₁₃₉	C₂²	1	2	yes	—
18	#499×C₁₃₉	—	2	2	no	—
19	#498×C₁₃₉	C₂²	1	2	yes	—
20	#477×C₁₃₉	—	2	2	no	—
21	#474×C₁₃₉	C₂²	1	4	yes	—
22	#468×C₁₃₉	—	2	4	no	—
23	#461×C₁₃₉	C₂²	1	4	yes	—
24	#460×C₁₃₉	—	2	2	no	—
25	#427×C₁₃₉	—	2	2	no	—
26	#423×C₁₃₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
28	#332×C₁₃₉	C₂²	1	4	yes	—
29	D₄⋊C₈×C₁₃₉	C₂²	1	4	yes	—
30	D₄⋊C₈×C₁₃₉	C₂²	1	4	yes	—
31	C₂×#41×C₁₃₉	—	2	4	no	—
32	C₂×#41×C₁₃₉	—	2	2	no	—
33	C₂×SD₁₆×C₁₃₉	—	2	4	no	—
34	C₂×M₁₆×C₁₃₉	—	2	2	no	—
35	C₂²×C₈×C₁₃₉	—	2	1	no	—
36	C₄⋊Q₈×C₁₃₉	D₄	1	2	yes	—
37	C₄⋊Q₈×C₁₃₉	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₁₃₉	C₂³	1	2	yes	—
39	#127×C₁₃₉	—	2	2	no	—
40	#127×C₁₃₉	D₄	1	2	yes	—
41	#125×C₁₃₉	—	2	2	no	—
42	#125×C₁₃₉	—	2	1	no	—
43	#125×C₁₃₉	D₄	1	2	yes	—
44	#124×C₁₃₉	—	2	2	no	—
45	#123×C₁₃₉	—	2	8	no	—
46	C₂²⋊Q₈×C₁₃₉	—	2	8	no	—
47	C₂²⋊Q₈×C₁₃₉	—	2	2	no	—
48	C₂²⋊Q₈×C₁₃₉	—	2	4	no	—
49	C₂²⋊Q₈×C₁₃₉	—	2	2	no	—
50	#121×C₁₃₉	—	2	8	no	—
51	#121×C₁₃₉	—	2	8	no	—
52	#121×C₁₃₉	—	2	4	no	—
53	#121×C₁₃₉	—	2	2	no	—
54	#121×C₁₃₉	D₄	1	4	yes	—
55	C₄×Q₈×C₁₃₉	—	2	2	no	—
56	C₄×Q₈×C₁₃₉	—	2	2	no	—
57	C₄×D₄×C₁₃₉	—	2	4	no	—
58	C₄×D₄×C₁₃₉	—	2	2	no	—
59	C₄×D₄×C₁₃₉	C₂³	1	4	yes	—
60	#117×C₁₃₉	—	2	1	no	—
61	#117×C₁₃₉	D₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
63	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
64	#107×C₁₃₉	—	2	4	no	—
65	#107×C₁₃₉	—	2	4	no	—
66	#106×C₁₃₉	—	2	4	no	—
67	C₄⋊C₈×C₁₃₉	—	2	2	no	—
68	C₄⋊C₈×C₁₃₉	C₂³	1	4	yes	—
69	#103×C₁₃₉	—	2	4	no	—
70	#102×C₁₃₉	—	2	4	no	—
71	C₄×C₈×C₁₃₉	—	2	1	no	—
72	C₄×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
74	#41×C₁₃₉	—	8	2	no	—
75	#41×C₁₃₉	—	8	2	no	—
76	#41×C₁₃₉	—	8	4	no	—
77	#41×C₁₃₉	—	8	4	no	—
78	C₂×Q₈×C₁₃₉	—	2	2	no	—
79	C₂×Q₈×C₁₃₉	—	2	2	no	—
80	C₂×Q₈×C₁₃₉	—	2	2	no	—
81	C₂×D₄×C₁₃₉	—	2	8	no	—
82	C₂×D₄×C₁₃₉	—	2	4	no	—
83	C₂×D₄×C₁₃₉	—	2	2	no	—
84	C₂×D₄×C₁₃₉	—	2	2	no	—
85	C₂×D₄×C₁₃₉	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₁₃₉	—	2	8	no	—
87	C₂²×C₄×C₁₃₉	—	2	4	no	—
88	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
90	SD₁₆×C₁₃₉	—	8	4	no	—
91	M₁₆×C₁₃₉	—	8	2	no	—
92	C₂×C₈×C₁₃₉	—	2	4	no	—
93	C₂×C₈×C₁₃₉	—	2	1	no	—
94	C₂×C₈×C₁₃₉	—	2	1	no	—
95	C₂×C₈×C₁₃₉	—	2	4	no	—
96	C₄⋊C₄×C₁₃₉	—	2	8	no	—
97	C₄⋊C₄×C₁₃₉	—	2	2	no	—
98	C₄⋊C₄×C₁₃₉	—	2	4	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	—	4	2	no	—
103	C₄⋊C₄×C₁₃₉	C₂×D₄	1	4	yes	—
104	C₄⋊C₄×C₁₃₉	C₂×D₄	1	4	yes	—
105	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
106	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
107	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
108	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
109	C₂²⋊C₄×C₁₃₉	—	4	1	no	—
110	C₄²×C₁₃₉	—	2	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	—	2	4	no	—
116	C₂³×C₁₃₉	#120	1	1	yes	characteristic
117	Q₈×C₁₃₉	—	4	2	no	—
118	Q₈×C₁₃₉	—	4	2	no	—
119	Q₈×C₁₃₉	—	4	2	no	—
120	Q₈×C₁₃₉	—	4	2	no	—
121	Q₈×C₁₃₉	—	8	2	no	—
122	D₄×C₁₃₉	—	2	4	no	—
123	D₄×C₁₃₉	—	2	4	no	—
124	D₄×C₁₃₉	—	4	4	no	—
125	D₄×C₁₃₉	—	8	2	no	—
126	D₄×C₁₃₉	—	8	4	no	—
127	D₄×C₁₃₉	—	8	8	no	—
128	C₂×C₄×C₁₃₉	—	2	2	no	—
129	C₂×C₄×C₁₃₉	—	2	2	no	—
130	C₂×C₄×C₁₃₉	—	2	2	no	—
131	C₂×C₄×C₁₃₉	—	2	4	no	—
132	C₂×C₄×C₁₃₉	—	2	4	no	—
133	C₂×C₄×C₁₃₉	—	4	4	no	—
134	C₂×C₄×C₁₃₉	—	4	4	no	—
135	C₂×C₄×C₁₃₉	—	4	2	no	—
136	C₂×C₄×C₁₃₉	—	8	8	no	—
137	C₂×C₄×C₁₃₉	#120	1	2	yes	—
138	C₂×C₄×C₁₃₉	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃₉	C₂²×D₄	1	2	yes	—
141	C₈×C₁₃₉	—	2	2	no	—
142	C₈×C₁₃₉	—	2	2	no	—
143	C₈×C₁₃₉	—	4	4	no	—
144	C₂²×C₁₃₉	—	2	2	no	—
145	C₂²×C₁₃₉	—	2	1	no	—
146	C₂²×C₁₃₉	—	4	4	no	—
147	C₂²×C₁₃₉	—	4	4	no	—
148	C₂²×C₁₃₉	—	4	4	no	—
149	C₂²×C₁₃₉	C₂×#120	1	1	yes	characteristic
150	C₄×C₁₃₉	—	2	2	no	—
151	C₄×C₁₃₉	—	2	1	no	—
152	C₄×C₁₃₉	—	4	4	no	—
153	C₄×C₁₃₉	—	4	4	no	—
154	C₄×C₁₃₉	—	8	2	no	—
155	C₄×C₁₃₉	#456	1	2	yes	—
156	C₄×C₁₃₉	#576	1	2	yes	—
157	C₄×C₁₃₉	#577	1	2	yes	—
158	C₂×C₁₃₉	—	4	1	no	—
159	C₂×C₁₃₉	—	8	4	no	—
160	C₂×C₁₃₉	#3012	1	2	yes	—
161	C₂×C₁₃₉	C₂×#456	1	1	yes	characteristic
162	#18701	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₁₃₉	#18701	1	1	yes	characteristic, direct product factor, Sylow
164	#3548	C₂×C₁₃₉	1	1	yes	characteristic
165	#3254	C₂×C₁₃₉	1	2	yes	complemented
166	#3160	C₂×C₁₃₉	1	1	yes	characteristic, complemented
167	#1654	C₂×C₁₃₉	1	4	yes	complemented
168	#1647	C₂×C₁₃₉	1	4	yes	complemented
169	#1561	C₂×C₁₃₉	1	1	yes	characteristic, complemented
170	#1492	C₂×C₁₃₉	1	2	yes	complemented
171	#562	—	2	2	no	—
172	#554	—	2	2	no	—
173	#550	—	2	2	no	—
174	#547	C₂²×C₁₃₉	1	4	yes	—
175	#540	—	2	2	no	—
176	#537	—	2	4	no	—
177	#533	C₂²×C₁₃₉	1	1	yes	characteristic
178	#533	C₂²×C₁₃₉	1	1	yes	characteristic
179	#532	C₂²×C₁₃₉	1	2	yes	—
180	#500	C₂²×C₁₃₉	1	2	yes	—
181	#499	—	2	2	no	—
182	#498	C₂²×C₁₃₉	1	2	yes	—
183	#477	—	2	2	no	—
184	#474	C₂²×C₁₃₉	1	4	yes	—
185	#468	—	2	4	no	—
186	#461	C₂²×C₁₃₉	1	4	yes	—
187	#460	—	2	2	no	—
188	#427	—	2	2	no	—
189	#423	C₂²×C₁₃₉	1	2	yes	—
190	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
191	#332	C₂²×C₁₃₉	1	4	yes	—
192	D₄⋊C₈	C₂²×C₁₃₉	1	4	yes	—
193	D₄⋊C₈	C₂²×C₁₃₉	1	4	yes	—
194	C₂×#41	—	2	4	no	—
195	C₂×#41	—	2	2	no	—
196	C₂×SD₁₆	—	2	4	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₄⋊Q₈	D₄×C₁₃₉	1	2	yes	—
200	C₄⋊Q₈	C₂³×C₁₃₉	1	2	yes	—
201	C₄⋊Q₈	C₂³×C₁₃₉	1	2	yes	—
202	#127	—	2	2	no	—
203	#127	D₄×C₁₃₉	1	2	yes	—
204	#125	—	2	2	no	—
205	#125	—	2	1	no	—
206	#125	D₄×C₁₃₉	1	2	yes	—
207	#124	—	2	2	no	—
208	#123	—	2	8	no	—
209	C₂²⋊Q₈	—	2	8	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	2	4	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	#121	—	2	8	no	—
214	#121	—	2	8	no	—
215	#121	—	2	4	no	—
216	#121	—	2	2	no	—
217	#121	D₄×C₁₃₉	1	4	yes	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×D₄	—	2	4	no	—
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	C₂³×C₁₃₉	1	4	yes	—
223	#117	—	2	1	no	—
224	#117	D₄×C₁₃₉	1	2	yes	—
225	C₂×C₄⋊C₄	—	2	4	no	—
226	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
227	#107	—	2	4	no	—
228	#107	—	2	4	no	—
229	#106	—	2	4	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	C₂³×C₁₃₉	1	4	yes	—
232	#103	—	2	4	no	—
233	#102	—	2	4	no	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
237	#41	—	8	2	no	—
238	#41	—	8	2	no	—
239	#41	—	8	4	no	—
240	#41	—	8	4	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×D₄	—	2	8	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	C₂×D₄×C₁₃₉	1	4	yes	—
249	C₂²×C₄	—	2	8	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
253	SD₁₆	—	8	4	no	—
254	M₁₆	—	8	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	4	no	—
259	C₄⋊C₄	—	2	8	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	C₂×D₄×C₁₃₉	1	4	yes	—
267	C₄⋊C₄	C₂×D₄×C₁₃₉	1	4	yes	—
268	C₂²⋊C₄	—	2	8	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₂²⋊C₄	—	2	4	no	—
272	C₂²⋊C₄	—	4	1	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₁₃₉	1	2	yes	—
277	C₄²	C₂⁴×C₁₃₉	1	1	yes	characteristic
278	C₂³	—	2	4	no	—
279	C₂³	#120×C₁₃₉	1	1	yes	characteristic
280	Q₈	—	4	2	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	2	no	—
285	D₄	—	2	4	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	8	2	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	8	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	8	no	—
300	C₂×C₄	#120×C₁₃₉	1	2	yes	—
301	C₂×C₄	#120×C₁₃₉	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₁₃₉	1	2	yes	—
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	4	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	C₂×#120×C₁₃₉	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	8	2	no	—
318	C₄	#456×C₁₃₉	1	2	yes	—
319	C₄	#576×C₁₃₉	1	2	yes	—
320	C₄	#577×C₁₃₉	1	2	yes	—
321	C₂	—	4	1	no	—
322	C₂	—	8	4	no	—
323	C₂	#3012×C₁₃₉	1	2	yes	—
324	C₂	C₂×#456×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃₉
prime radical	#3548×C₁₃₉	C₂
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. #3254×C₁₃₉	C₂	semidirect
3. #3160×C₁₃₉	C₂	semidirect
4. #1654×C₁₃₉	C₂	semidirect
5. #1647×C₁₃₉	C₂	semidirect
6. #1561×C₁₃₉	C₂	semidirect
7. #1492×C₁₃₉	C₂	semidirect
162. #18701	C₁₃₉	direct (N × H)
163. C₁₃₉	#18701	direct (N × H)
165. #3254	C₂×C₁₃₉	semidirect
166. #3160	C₂×C₁₃₉	semidirect
167. #1654	C₂×C₁₃₉	semidirect
168. #1647	C₂×C₁₃₉	semidirect
169. #1561	C₂×C₁₃₉	semidirect
170. #1492	C₂×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#3548×C₁₃₉	2
#3254×C₁₃₉	2
#3160×C₁₃₉	2
#1654×C₁₃₉	2
#1647×C₁₃₉	2
#1561×C₁₃₉	2
#1492×C₁₃₉	2
#18701	139

Subgroups — C139×#18701 #48013297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#18701 #48013297