Subgroups — C₁₃₉×#10170 #48004766

Order 35584.

574 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3206×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#3152×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#487×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#1997×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#1546×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#1492×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	#1278×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#137×C₁₃₉	—	2	2	no	—
9	#532×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
12	#497×C₁₃₉	—	2	2	no	—
13	#494×C₁₃₉	—	2	1	no	—
14	#492×C₁₃₉	—	2	1	no	—
15	#492×C₁₃₉	—	2	1	no	—
16	#487×C₁₃₉	—	2	2	no	—
17	#487×C₁₃₉	—	2	1	no	—
18	#487×C₁₃₉	—	2	2	no	—
19	#461×C₁₃₉	—	2	2	no	—
20	#448×C₁₃₉	—	2	2	no	—
21	#431×C₁₃₉	C₂²	1	1	yes	characteristic
22	#423×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₂×#103×C₁₃₉	—	2	2	no	—
24	C₄×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
26	#343×C₁₃₉	—	2	2	no	—
27	#332×C₁₃₉	—	2	2	no	—
28	D₄⋊C₈×C₁₃₉	—	2	2	no	—
29	C₂×#41×C₁₃₉	—	2	1	no	—
30	C₂²×Q₈×C₁₃₉	—	2	1	no	—
31	C₂²×Q₈×C₁₃₉	D₄	1	2	yes	—
32	#137×C₁₃₉	—	4	4	no	—
33	#137×C₁₃₉	—	4	4	no	—
34	C₂×Q₁₆×C₁₃₉	—	2	4	no	—
35	C₂×SD₁₆×C₁₃₉	—	2	4	no	—
36	C₂×SD₁₆×C₁₃₉	—	2	4	no	—
37	C₂×M₁₆×C₁₃₉	—	2	1	no	—
38	C₂×M₁₆×C₁₃₉	D₄	1	2	yes	—
39	C₂²×C₈×C₁₃₉	—	2	1	no	—
40	C₄⋊Q₈×C₁₃₉	—	2	1	no	—
41	C₄⋊Q₈×C₁₃₉	—	2	1	no	—
42	C₄⋊Q₈×C₁₃₉	—	2	1	no	—
43	C₄⋊Q₈×C₁₃₉	—	2	1	no	—
44	C₄⋊Q₈×C₁₃₉	—	2	1	no	—
45	C₄⋊Q₈×C₁₃₉	—	2	1	no	—
46	C₄⋊Q₈×C₁₃₉	—	2	1	no	—
47	C₄⋊Q₈×C₁₃₉	D₄	1	1	yes	characteristic
48	C₄⋊Q₈×C₁₃₉	D₄	1	2	yes	—
49	#127×C₁₃₉	D₄	1	1	yes	characteristic
50	#124×C₁₃₉	—	2	1	no	—
51	#121×C₁₃₉	—	2	1	no	—
52	#121×C₁₃₉	—	2	1	no	—
53	C₄×D₄×C₁₃₉	—	2	1	no	—
54	C₄×D₄×C₁₃₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
59	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₁₃₉	—	2	1	no	—
61	C₄⋊C₈×C₁₃₉	—	2	1	no	—
62	C₄⋊C₈×C₁₃₉	D₄	1	2	yes	—
63	#103×C₁₃₉	—	2	8	no	—
64	#103×C₁₃₉	—	2	4	no	—
65	C₂²⋊C₈×C₁₃₉	—	2	1	no	—
66	#97×C₁₃₉	—	2	1	no	—
67	C₄×C₈×C₁₃₉	—	2	1	no	—
68	C₄×C₈×C₁₃₉	—	2	1	no	—
69	C₄×C₈×C₁₃₉	—	2	1	no	—
70	C₄×C₈×C₁₃₉	D₄	1	2	yes	—
71	#41×C₁₃₉	—	4	2	no	—
72	#41×C₁₃₉	—	4	2	no	—
73	C₂×Q₈×C₁₃₉	—	2	2	no	—
74	C₂×Q₈×C₁₃₉	—	2	2	no	—
75	C₂×Q₈×C₁₃₉	—	2	1	no	—
76	C₂×Q₈×C₁₃₉	—	2	2	no	—
77	C₂×Q₈×C₁₃₉	—	4	4	no	—
78	C₂×Q₈×C₁₃₉	—	4	1	no	—
79	C₂×Q₈×C₁₃₉	—	4	2	no	—
80	C₂×Q₈×C₁₃₉	—	4	2	no	—
81	C₂×D₄×C₁₃₉	—	2	1	no	—
82	C₂×D₄×C₁₃₉	—	2	1	no	—
83	C₂×D₄×C₁₃₉	—	2	1	no	—
84	C₂×D₄×C₁₃₉	—	4	1	no	—
85	C₂²×C₄×C₁₃₉	—	2	2	no	—
86	C₂²×C₄×C₁₃₉	—	4	1	no	—
87	C₂²×C₄×C₁₃₉	—	4	1	no	—
88	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃₉	#41	1	1	yes	characteristic
91	Q₁₆×C₁₃₉	—	4	8	no	—
92	SD₁₆×C₁₃₉	—	4	8	no	—
93	SD₁₆×C₁₃₉	—	8	4	no	—
94	M₁₆×C₁₃₉	—	2	2	no	—
95	M₁₆×C₁₃₉	—	4	2	no	—
96	M₁₆×C₁₃₉	SD₁₆	1	4	yes	complemented
97	C₂×C₈×C₁₃₉	—	2	2	no	—
98	C₂×C₈×C₁₃₉	—	2	2	no	—
99	C₂×C₈×C₁₃₉	—	2	2	no	—
100	C₂×C₈×C₁₃₉	—	4	2	no	—
101	C₂×C₈×C₁₃₉	—	4	1	no	—
102	C₄⋊C₄×C₁₃₉	—	2	2	no	—
103	C₄⋊C₄×C₁₃₉	—	2	1	no	—
104	C₄⋊C₄×C₁₃₉	—	2	2	no	—
105	C₄⋊C₄×C₁₃₉	—	2	1	no	—
106	C₄⋊C₄×C₁₃₉	—	4	1	no	—
107	C₄⋊C₄×C₁₃₉	—	4	1	no	—
108	C₄⋊C₄×C₁₃₉	—	4	1	no	—
109	C₄⋊C₄×C₁₃₉	—	4	1	no	—
110	C₄⋊C₄×C₁₃₉	—	4	1	no	—
111	C₄⋊C₄×C₁₃₉	—	4	1	no	—
112	C₂²⋊C₄×C₁₃₉	—	4	1	no	—
113	C₂²⋊C₄×C₁₃₉	—	4	1	no	—
114	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
118	C₂³×C₁₃₉	—	4	1	no	—
119	C₂³×C₁₃₉	#121	1	1	yes	characteristic
120	Q₈×C₁₃₉	—	4	4	no	—
121	Q₈×C₁₃₉	—	4	4	no	—
122	Q₈×C₁₃₉	—	4	2	no	—
123	Q₈×C₁₃₉	—	4	4	no	—
124	Q₈×C₁₃₉	—	8	1	no	—
125	Q₈×C₁₃₉	—	8	1	no	—
126	D₄×C₁₃₉	—	4	2	no	—
127	D₄×C₁₃₉	—	4	2	no	—
128	D₄×C₁₃₉	—	4	2	no	—
129	D₄×C₁₃₉	—	8	2	no	—
130	C₂×C₄×C₁₃₉	—	2	2	no	—
131	C₂×C₄×C₁₃₉	—	2	2	no	—
132	C₂×C₄×C₁₃₉	—	4	1	no	—
133	C₂×C₄×C₁₃₉	—	4	2	no	—
134	C₂×C₄×C₁₃₉	—	4	2	no	—
135	C₂×C₄×C₁₃₉	—	8	2	no	—
136	C₂×C₄×C₁₃₉	—	8	1	no	—
137	C₂×C₄×C₁₃₉	—	8	1	no	—
138	C₂×C₄×C₁₃₉	—	8	2	no	—
139	C₂×C₄×C₁₃₉	#120	1	1	yes	characteristic
140	C₂×C₄×C₁₃₉	#120	1	1	yes	characteristic
141	C₂×C₄×C₁₃₉	#120	1	1	yes	characteristic
142	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₃₉	#127	1	1	yes	characteristic
145	C₂×C₄×C₁₃₉	C₂×SD₁₆	1	2	yes	—
146	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
147	C₈×C₁₃₉	—	2	4	no	—
148	C₈×C₁₃₉	—	4	2	no	—
149	C₈×C₁₃₉	—	4	2	no	—
150	C₈×C₁₃₉	—	8	1	no	—
151	C₂²×C₁₃₉	—	2	1	no	—
152	C₂²×C₁₃₉	—	2	1	no	—
153	C₂²×C₁₃₉	—	8	1	no	—
154	C₂²×C₁₃₉	—	8	1	no	—
155	C₂²×C₁₃₉	—	8	1	no	—
156	C₂²×C₁₃₉	#392	1	1	yes	characteristic
157	C₂²×C₁₃₉	#460	1	2	yes	—
158	C₄×C₁₃₉	—	2	1	no	—
159	C₄×C₁₃₉	—	2	1	no	—
160	C₄×C₁₃₉	—	8	2	no	—
161	C₄×C₁₃₉	—	8	1	no	—
162	C₄×C₁₃₉	—	8	2	no	—
163	C₄×C₁₃₉	#450	1	2	yes	—
164	C₄×C₁₃₉	#455	1	2	yes	—
165	C₄×C₁₃₉	#473	1	2	yes	—
166	C₄×C₁₃₉	#492	1	1	yes	characteristic
167	C₄×C₁₃₉	#496	1	1	yes	characteristic
168	C₂×C₁₃₉	—	2	2	no	—
169	C₂×C₁₃₉	—	16	1	no	—
170	C₂×C₁₃₉	#2004	1	1	yes	characteristic
171	C₂×C₁₃₉	#2014	1	1	yes	characteristic
172	C₂×C₁₃₉	#2020	1	1	yes	characteristic
173	#10170	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
174	C₁₃₉	#10170	1	1	yes	characteristic, direct product factor, Sylow
175	#3206	C₂×C₁₃₉	1	1	yes	characteristic
176	#3152	C₂×C₁₃₉	1	1	yes	characteristic
177	C₂×#487	C₂×C₁₃₉	1	1	yes	characteristic, complemented
178	#1997	C₂×C₁₃₉	1	1	yes	characteristic, complemented
179	#1546	C₂×C₁₃₉	1	1	yes	characteristic, complemented
180	#1492	C₂×C₁₃₉	1	1	yes	characteristic
181	#1278	C₂×C₁₃₉	1	1	yes	characteristic, complemented
182	C₂×#137	—	2	2	no	—
183	#532	C₂²×C₁₃₉	1	1	yes	characteristic
184	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
185	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
186	#497	—	2	2	no	—
187	#494	—	2	1	no	—
188	#492	—	2	1	no	—
189	#492	—	2	1	no	—
190	#487	—	2	2	no	—
191	#487	—	2	1	no	—
192	#487	—	2	2	no	—
193	#461	—	2	2	no	—
194	#448	—	2	2	no	—
195	#431	C₂²×C₁₃₉	1	1	yes	characteristic
196	#423	C₂²×C₁₃₉	1	1	yes	characteristic
197	C₂×#103	—	2	2	no	—
198	C₄×M₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
199	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
200	#343	—	2	2	no	—
201	#332	—	2	2	no	—
202	D₄⋊C₈	—	2	2	no	—
203	C₂×#41	—	2	1	no	—
204	C₂²×Q₈	—	2	1	no	—
205	C₂²×Q₈	D₄×C₁₃₉	1	2	yes	—
206	#137	—	4	4	no	—
207	#137	—	4	4	no	—
208	C₂×Q₁₆	—	2	4	no	—
209	C₂×SD₁₆	—	2	4	no	—
210	C₂×SD₁₆	—	2	4	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	D₄×C₁₃₉	1	2	yes	—
213	C₂²×C₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	D₄×C₁₃₉	1	2	yes	—
222	C₄⋊Q₈	D₄×C₁₃₉	1	1	yes	characteristic
223	#127	D₄×C₁₃₉	1	1	yes	characteristic
224	#124	—	2	1	no	—
225	#121	—	2	1	no	—
226	#121	—	2	1	no	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	D₄×C₁₃₉	1	2	yes	—
237	#103	—	2	8	no	—
238	#103	—	2	4	no	—
239	C₂²⋊C₈	—	2	1	no	—
240	#97	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	C₄×C₈	—	2	1	no	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	D₄×C₁₃₉	1	2	yes	—
245	#41	—	4	2	no	—
246	#41	—	4	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	4	1	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
264	C₂²×C₄	#41×C₁₃₉	1	1	yes	characteristic
265	Q₁₆	—	4	8	no	—
266	SD₁₆	—	4	8	no	—
267	SD₁₆	—	8	4	no	—
268	M₁₆	—	2	2	no	—
269	M₁₆	—	4	2	no	—
270	M₁₆	SD₁₆×C₁₃₉	1	4	yes	complemented
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	4	1	no	—
288	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
289	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
290	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
292	C₂³	—	4	1	no	—
293	C₂³	#121×C₁₃₉	1	1	yes	characteristic
294	Q₈	—	4	4	no	—
295	Q₈	—	4	4	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	4	4	no	—
298	Q₈	—	8	1	no	—
299	Q₈	—	8	1	no	—
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	8	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	—	8	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	#120×C₁₃₉	1	1	yes	characteristic
314	C₂×C₄	#120×C₁₃₉	1	1	yes	characteristic
315	C₂×C₄	#120×C₁₃₉	1	1	yes	characteristic
316	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
317	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
318	C₂×C₄	#127×C₁₃₉	1	1	yes	characteristic
319	C₂×C₄	C₂×SD₁₆×C₁₃₉	1	2	yes	—
320	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	2	yes	—
321	C₈	—	2	4	no	—
322	C₈	—	4	2	no	—
323	C₈	—	4	2	no	—
324	C₈	—	8	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	#392×C₁₃₉	1	1	yes	characteristic
331	C₂²	#460×C₁₃₉	1	2	yes	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	2	no	—
337	C₄	#450×C₁₃₉	1	2	yes	—
338	C₄	#455×C₁₃₉	1	2	yes	—
339	C₄	#473×C₁₃₉	1	2	yes	—
340	C₄	#492×C₁₃₉	1	1	yes	characteristic
341	C₄	#496×C₁₃₉	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	16	1	no	—
344	C₂	#2004×C₁₃₉	1	1	yes	characteristic
345	C₂	#2014×C₁₃₉	1	1	yes	characteristic
346	C₂	#2020×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
prime radical	#532×C₁₃₉	C₂²
LCS	G ▷ C₂×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
3. C₂×#487×C₁₃₉	C₂	semidirect
4. #1997×C₁₃₉	C₂	semidirect
5. #1546×C₁₃₉	C₂	semidirect
7. #1278×C₁₃₉	C₂	semidirect
96. M₁₆×C₁₃₉	SD₁₆	semidirect
173. #10170	C₁₃₉	direct (N × H)
174. C₁₃₉	#10170	direct (N × H)
177. C₂×#487	C₂×C₁₃₉	semidirect
178. #1997	C₂×C₁₃₉	semidirect
179. #1546	C₂×C₁₃₉	semidirect
181. #1278	C₂×C₁₃₉	semidirect
270. M₁₆	SD₁₆×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#3206×C₁₃₉	2
#3152×C₁₃₉	2
C₂×#487×C₁₃₉	2
#1997×C₁₃₉	2
#1546×C₁₃₉	2
#1492×C₁₃₉	2
#1278×C₁₃₉	2
#10170	139

Subgroups — C139×#10170 #48004766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#10170 #48004766