Subgroups — C₁₃₉×#29264 #48023860

Order 35584.

686 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2331×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#2327×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#2097×C₁₃₉	C₂	1	2	yes	maximal
4	#2024×C₁₃₉	C₂	1	2	yes	maximal
5	#1981×C₁₃₉	C₂	1	2	yes	maximal
6	#1949×C₁₃₉	C₂	1	2	yes	maximal
7	#1865×C₁₃₉	C₂	1	2	yes	maximal
8	#1863×C₁₃₉	C₂	1	2	yes	maximal
9	#1832×C₁₃₉	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₃₉	C₂²	1	2	yes	—
13	C₂×#107×C₁₃₉	C₂²	1	2	yes	—
14	C₂×#106×C₁₃₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
16	C₂×#103×C₁₃₉	C₂²	1	2	yes	—
17	C₂×#103×C₁₃₉	C₂²	1	2	yes	—
18	C₂×#103×C₁₃₉	C₂²	1	2	yes	—
19	C₂×#103×C₁₃₉	C₂²	1	2	yes	—
20	#391×C₁₃₉	—	2	2	no	—
21	#391×C₁₃₉	C₂²	1	2	yes	—
22	#389×C₁₃₉	C₂²	1	2	yes	—
23	#387×C₁₃₉	—	2	2	no	—
24	#387×C₁₃₉	—	2	2	no	—
25	#387×C₁₃₉	—	2	2	no	—
26	#382×C₁₃₉	C₂²	1	1	yes	characteristic
27	#382×C₁₃₉	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
29	#376×C₁₃₉	C₂²	1	1	yes	characteristic
30	#340×C₁₃₉	C₂²	1	2	yes	—
31	#340×C₁₃₉	C₂²	1	2	yes	—
32	#336×C₁₃₉	C₂²	1	2	yes	—
33	#336×C₁₃₉	C₂²	1	1	yes	characteristic
34	#336×C₁₃₉	C₂²	1	1	yes	characteristic
35	C₂²×Q₈×C₁₃₉	—	2	1	no	—
36	C₂²×Q₈×C₁₃₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃₉	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃₉	C₂³	1	2	yes	—
39	C₄⋊Q₈×C₁₃₉	—	2	4	no	—
40	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
41	C₄⋊Q₈×C₁₃₉	Q₈	1	4	yes	complemented
42	C₄×Q₈×C₁₃₉	—	4	2	no	—
43	C₄×Q₈×C₁₃₉	—	4	2	no	—
44	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₃₉	—	2	2	no	—
52	C₂×C₄²×C₁₃₉	—	2	2	no	—
53	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
56	#107×C₁₃₉	—	4	2	no	—
57	#106×C₁₃₉	—	4	2	no	—
58	C₄⋊C₈×C₁₃₉	—	2	4	no	—
59	#103×C₁₃₉	—	2	4	no	—
60	#103×C₁₃₉	—	2	4	no	—
61	#103×C₁₃₉	—	4	2	no	—
62	#103×C₁₃₉	—	4	2	no	—
63	#95×C₁₃₉	—	2	2	no	—
64	#95×C₁₃₉	—	2	1	no	—
65	#95×C₁₃₉	—	2	2	no	—
66	#95×C₁₃₉	—	2	1	no	—
67	#95×C₁₃₉	—	2	2	no	—
68	#95×C₁₃₉	—	2	2	no	—
69	#95×C₁₃₉	—	2	2	no	—
70	#95×C₁₃₉	D₄	1	2	yes	—
71	#95×C₁₃₉	D₄	1	2	yes	—
72	C₂×Q₈×C₁₃₉	—	2	4	no	—
73	C₂×Q₈×C₁₃₉	—	2	2	no	—
74	C₂×Q₈×C₁₃₉	—	2	2	no	—
75	C₂×Q₈×C₁₃₉	—	2	2	no	—
76	C₂×Q₈×C₁₃₉	—	2	2	no	—
77	C₂×Q₈×C₁₃₉	—	2	2	no	—
78	C₂×Q₈×C₁₃₉	—	8	2	no	—
79	C₂×Q₈×C₁₃₉	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₁₃₉	—	2	2	no	—
81	C₂²×C₄×C₁₃₉	—	2	2	no	—
82	C₂²×C₄×C₁₃₉	—	2	2	no	—
83	C₂²×C₄×C₁₃₉	—	2	2	no	—
84	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₁₃₉	—	2	2	no	—
90	C₂×C₈×C₁₃₉	—	2	2	no	—
91	C₂×C₈×C₁₃₉	—	2	2	no	—
92	C₂×C₈×C₁₃₉	—	2	2	no	—
93	C₂×C₈×C₁₃₉	—	2	2	no	—
94	C₂×C₈×C₁₃₉	—	2	2	no	—
95	C₄⋊C₄×C₁₃₉	—	2	4	no	—
96	C₄⋊C₄×C₁₃₉	—	2	8	no	—
97	C₄⋊C₄×C₁₃₉	—	2	2	no	—
98	C₄⋊C₄×C₁₃₉	—	2	2	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	4	no	—
103	C₄⋊C₄×C₁₃₉	—	8	2	no	—
104	C₄⋊C₄×C₁₃₉	#41	1	4	yes	—
105	C₄²×C₁₃₉	—	2	2	no	—
106	C₄²×C₁₃₉	—	2	2	no	—
107	C₄²×C₁₃₉	—	4	4	no	—
108	C₄²×C₁₃₉	—	8	2	no	—
109	C₄²×C₁₃₉	C₂×Q₈	1	2	yes	—
110	C₄²×C₁₃₉	#41	1	2	yes	—
111	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₁₃₉	—	2	4	no	—
113	Q₈×C₁₃₉	—	2	4	no	—
114	Q₈×C₁₃₉	—	4	2	no	—
115	Q₈×C₁₃₉	—	4	2	no	—
116	Q₈×C₁₃₉	—	8	4	no	—
117	C₂×C₄×C₁₃₉	—	2	2	no	—
118	C₂×C₄×C₁₃₉	—	2	2	no	—
119	C₂×C₄×C₁₃₉	—	2	4	no	—
120	C₂×C₄×C₁₃₉	—	2	1	no	—
121	C₂×C₄×C₁₃₉	—	2	1	no	—
122	C₂×C₄×C₁₃₉	—	2	4	no	—
123	C₂×C₄×C₁₃₉	—	2	2	no	—
124	C₂×C₄×C₁₃₉	—	2	2	no	—
125	C₂×C₄×C₁₃₉	—	2	2	no	—
126	C₂×C₄×C₁₃₉	—	2	2	no	—
127	C₂×C₄×C₁₃₉	—	2	2	no	—
128	C₂×C₄×C₁₃₉	—	4	2	no	—
129	C₂×C₄×C₁₃₉	—	4	2	no	—
130	C₂×C₄×C₁₃₉	—	4	2	no	—
131	C₂×C₄×C₁₃₉	—	4	2	no	—
132	C₂×C₄×C₁₃₉	—	4	2	no	—
133	C₂×C₄×C₁₃₉	—	4	2	no	—
134	C₂×C₄×C₁₃₉	—	4	2	no	—
135	C₂×C₄×C₁₃₉	—	4	2	no	—
136	C₂×C₄×C₁₃₉	—	4	2	no	—
137	C₂×C₄×C₁₃₉	—	4	2	no	—
138	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	4	yes	—
139	C₂×C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
141	C₂×C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
143	C₈×C₁₃₉	—	4	2	no	—
144	C₈×C₁₃₉	—	4	2	no	—
145	C₂²×C₁₃₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂²×C₁₃₉	#525	1	1	yes	characteristic
147	C₂²×C₁₃₉	#535	1	1	yes	characteristic
148	C₂²×C₁₃₉	#577	1	2	yes	—
149	C₂²×C₁₃₉	#578	1	2	yes	—
150	C₄×C₁₃₉	—	2	4	no	—
151	C₄×C₁₃₉	—	4	4	no	—
152	C₄×C₁₃₉	—	4	1	no	—
153	C₄×C₁₃₉	—	4	4	no	—
154	C₄×C₁₃₉	—	4	1	no	—
155	C₄×C₁₃₉	—	4	2	no	—
156	C₄×C₁₃₉	—	8	2	no	—
157	C₄×C₁₃₉	—	8	2	no	—
158	C₄×C₁₃₉	#554	1	2	yes	—
159	C₄×C₁₃₉	#559	1	2	yes	—
160	C₂×C₁₃₉	#2580	1	1	yes	characteristic
161	C₂×C₁₃₉	#3082	1	2	yes	—
162	C₂×C₁₃₉	#3123	1	1	yes	characteristic
163	C₂×C₁₃₉	#3127	1	1	yes	characteristic
164	C₂×C₁₃₉	#3225	1	2	yes	—
165	#29264	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₁₃₉	#29264	1	1	yes	characteristic, direct product factor, Sylow
167	#2331	C₂×C₁₃₉	1	1	yes	characteristic
168	#2327	C₂×C₁₃₉	1	1	yes	characteristic
169	#2097	C₂×C₁₃₉	1	2	yes	—
170	#2024	C₂×C₁₃₉	1	2	yes	—
171	#1981	C₂×C₁₃₉	1	2	yes	—
172	#1949	C₂×C₁₃₉	1	2	yes	—
173	#1865	C₂×C₁₃₉	1	2	yes	—
174	#1863	C₂×C₁₃₉	1	2	yes	—
175	#1832	C₂×C₁₃₉	1	1	yes	characteristic
176	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
178	C₂×C₄×Q₈	C₂²×C₁₃₉	1	2	yes	—
179	C₂×#107	C₂²×C₁₃₉	1	2	yes	—
180	C₂×#106	C₂²×C₁₃₉	1	2	yes	—
181	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
182	C₂×#103	C₂²×C₁₃₉	1	2	yes	—
183	C₂×#103	C₂²×C₁₃₉	1	2	yes	—
184	C₂×#103	C₂²×C₁₃₉	1	2	yes	—
185	C₂×#103	C₂²×C₁₃₉	1	2	yes	—
186	#391	—	2	2	no	—
187	#391	C₂²×C₁₃₉	1	2	yes	—
188	#389	C₂²×C₁₃₉	1	2	yes	—
189	#387	—	2	2	no	—
190	#387	—	2	2	no	—
191	#387	—	2	2	no	—
192	#382	C₂²×C₁₃₉	1	1	yes	characteristic
193	#382	C₂²×C₁₃₉	1	1	yes	characteristic
194	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
195	#376	C₂²×C₁₃₉	1	1	yes	characteristic
196	#340	C₂²×C₁₃₉	1	2	yes	—
197	#340	C₂²×C₁₃₉	1	2	yes	—
198	#336	C₂²×C₁₃₉	1	2	yes	—
199	#336	C₂²×C₁₃₉	1	1	yes	characteristic
200	#336	C₂²×C₁₃₉	1	1	yes	characteristic
201	C₂²×Q₈	—	2	1	no	—
202	C₂²×Q₈	C₂³×C₁₃₉	1	2	yes	—
203	C₂²×C₈	C₂³×C₁₃₉	1	2	yes	—
204	C₂²×C₈	C₂³×C₁₃₉	1	2	yes	—
205	C₄⋊Q₈	—	2	4	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	Q₈×C₁₃₉	1	4	yes	complemented
208	C₄×Q₈	—	4	2	no	—
209	C₄×Q₈	—	4	2	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
222	#107	—	4	2	no	—
223	#106	—	4	2	no	—
224	C₄⋊C₈	—	2	4	no	—
225	#103	—	2	4	no	—
226	#103	—	2	4	no	—
227	#103	—	4	2	no	—
228	#103	—	4	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	1	no	—
231	#95	—	2	2	no	—
232	#95	—	2	1	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	D₄×C₁₃₉	1	2	yes	—
237	#95	D₄×C₁₃₉	1	2	yes	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	8	2	no	—
245	C₂×Q₈	C₂×Q₈×C₁₃₉	1	4	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
254	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	8	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	8	2	no	—
270	C₄⋊C₄	#41×C₁₃₉	1	4	yes	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	4	no	—
274	C₄²	—	8	2	no	—
275	C₄²	C₂×Q₈×C₁₃₉	1	2	yes	—
276	C₄²	#41×C₁₃₉	1	2	yes	—
277	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
278	Q₈	—	2	4	no	—
279	Q₈	—	2	4	no	—
280	Q₈	—	4	2	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	8	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	4	yes	—
305	C₂×C₄	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
307	C₂×C₄	C₂×#41×C₁₃₉	1	1	yes	characteristic
308	C₂×C₄	H(5,𝔽₂)×C₁₃₉	1	2	yes	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₂²	C₂×C₂²⋊Q₈×C₁₃₉	1	1	yes	characteristic
312	C₂²	#525×C₁₃₉	1	1	yes	characteristic
313	C₂²	#535×C₁₃₉	1	1	yes	characteristic
314	C₂²	#577×C₁₃₉	1	2	yes	—
315	C₂²	#578×C₁₃₉	1	2	yes	—
316	C₄	—	2	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	4	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	#554×C₁₃₉	1	2	yes	—
325	C₄	#559×C₁₃₉	1	2	yes	—
326	C₂	#2580×C₁₃₉	1	1	yes	characteristic
327	C₂	#3082×C₁₃₉	1	2	yes	—
328	C₂	#3123×C₁₃₉	1	1	yes	characteristic
329	C₂	#3127×C₁₃₉	1	1	yes	characteristic
330	C₂	#3225×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ #376×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
41. C₄⋊Q₈×C₁₃₉	Q₈	semidirect
165. #29264	C₁₃₉	direct (N × H)
166. C₁₃₉	#29264	direct (N × H)
207. C₄⋊Q₈	Q₈×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#2331×C₁₃₉	2
#2327×C₁₃₉	2
#2097×C₁₃₉	2
#2024×C₁₃₉	2
#1981×C₁₃₉	2
#1949×C₁₃₉	2
#1865×C₁₃₉	2
#1863×C₁₃₉	2
#1832×C₁₃₉	2
#29264	139

Subgroups — C139×#29264 #48023860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#29264 #48023860