Subgroups — C₁₃₉×#29291 #48023887

Order 35584.

652 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#125×C₁₃₉	C₂	1	2	yes	maximal
2	#2088×C₁₃₉	C₂	1	2	yes	maximal
3	#2087×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#2086×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	#1989×C₁₃₉	C₂	1	2	yes	maximal
6	#1989×C₁₃₉	C₂	1	2	yes	maximal
7	#1949×C₁₃₉	C₂	1	2	yes	maximal
8	#1949×C₁₃₉	C₂	1	2	yes	maximal
9	#1842×C₁₃₉	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₁₃₉	C₂²	1	2	yes	—
11	C₂×#107×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×#107×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂×#106×C₁₃₉	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
18	#389×C₁₃₉	C₂²	1	2	yes	—
19	#389×C₁₃₉	C₂²	1	2	yes	—
20	#387×C₁₃₉	—	2	2	no	—
21	#387×C₁₃₉	—	2	2	no	—
22	#382×C₁₃₉	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₃₉	—	2	2	no	—
24	C₄×C₄⋊C₄×C₁₃₉	—	2	2	no	—
25	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
27	C₄³×C₁₃₉	C₂²	1	1	yes	characteristic
28	#340×C₁₃₉	C₂²	1	2	yes	—
29	#340×C₁₃₉	C₂²	1	1	yes	characteristic
30	#340×C₁₃₉	C₂²	1	2	yes	—
31	#340×C₁₃₉	C₂²	1	1	yes	characteristic
32	#340×C₁₃₉	C₂²	1	2	yes	—
33	#340×C₁₃₉	C₂²	1	2	yes	—
34	#340×C₁₃₉	C₂²	1	2	yes	—
35	C₂²×C₈×C₁₃₉	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
38	#125×C₁₃₉	—	2	4	no	—
39	#125×C₁₃₉	Q₈	1	8	yes	—
40	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
48	C₂×C₄²×C₁₃₉	—	2	2	no	—
49	C₂×C₄²×C₁₃₉	—	2	2	no	—
50	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
53	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃₉	C₂³	1	2	yes	—
55	#107×C₁₃₉	—	4	1	no	—
56	#107×C₁₃₉	—	4	1	no	—
57	#106×C₁₃₉	—	4	2	no	—
58	C₄⋊C₈×C₁₃₉	—	2	4	no	—
59	C₄⋊C₈×C₁₃₉	—	2	4	no	—
60	C₄⋊C₈×C₁₃₉	—	4	1	no	—
61	C₄⋊C₈×C₁₃₉	—	4	1	no	—
62	#95×C₁₃₉	—	2	2	no	—
63	#95×C₁₃₉	—	2	2	no	—
64	#95×C₁₃₉	—	2	2	no	—
65	#95×C₁₃₉	—	2	2	no	—
66	#95×C₁₃₉	—	2	2	no	—
67	#95×C₁₃₉	—	2	2	no	—
68	C₂²×C₄×C₁₃₉	—	2	2	no	—
69	C₂²×C₄×C₁₃₉	—	2	2	no	—
70	C₂²×C₄×C₁₃₉	—	2	2	no	—
71	C₂²×C₄×C₁₃₉	—	2	2	no	—
72	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₃₉	—	2	2	no	—
78	C₂×C₈×C₁₃₉	—	2	2	no	—
79	C₂×C₈×C₁₃₉	—	2	2	no	—
80	C₂×C₈×C₁₃₉	—	2	2	no	—
81	C₂×C₈×C₁₃₉	—	2	2	no	—
82	C₂×C₈×C₁₃₉	—	2	1	no	—
83	C₂×C₈×C₁₃₉	—	2	1	no	—
84	C₄⋊C₄×C₁₃₉	—	2	8	no	—
85	C₄⋊C₄×C₁₃₉	—	2	8	no	—
86	C₄⋊C₄×C₁₃₉	—	2	2	no	—
87	C₄⋊C₄×C₁₃₉	—	2	2	no	—
88	C₄⋊C₄×C₁₃₉	—	2	2	no	—
89	C₄⋊C₄×C₁₃₉	—	2	2	no	—
90	C₄⋊C₄×C₁₃₉	—	2	2	no	—
91	C₄⋊C₄×C₁₃₉	—	2	2	no	—
92	C₄⋊C₄×C₁₃₉	—	4	4	no	—
93	C₄⋊C₄×C₁₃₉	—	4	4	no	—
94	C₄⋊C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
95	C₄⋊C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
96	C₄²×C₁₃₉	—	2	2	no	—
97	C₄²×C₁₃₉	—	2	1	no	—
98	C₄²×C₁₃₉	—	2	1	no	—
99	C₄²×C₁₃₉	—	4	1	no	—
100	C₄²×C₁₃₉	—	4	2	no	—
101	C₄²×C₁₃₉	—	4	4	no	—
102	C₄²×C₁₃₉	—	4	4	no	—
103	C₄²×C₁₃₉	C₂×Q₈	1	4	yes	—
104	C₄²×C₁₃₉	#41	1	4	yes	—
105	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃₉	—	2	1	no	—
107	C₂×C₄×C₁₃₉	—	2	2	no	—
108	C₂×C₄×C₁₃₉	—	2	2	no	—
109	C₂×C₄×C₁₃₉	—	2	2	no	—
110	C₂×C₄×C₁₃₉	—	2	2	no	—
111	C₂×C₄×C₁₃₉	—	2	1	no	—
112	C₂×C₄×C₁₃₉	—	2	4	no	—
113	C₂×C₄×C₁₃₉	—	2	4	no	—
114	C₂×C₄×C₁₃₉	—	2	1	no	—
115	C₂×C₄×C₁₃₉	—	2	2	no	—
116	C₂×C₄×C₁₃₉	—	4	2	no	—
117	C₂×C₄×C₁₃₉	—	4	2	no	—
118	C₂×C₄×C₁₃₉	—	4	2	no	—
119	C₂×C₄×C₁₃₉	—	4	2	no	—
120	C₂×C₄×C₁₃₉	—	4	2	no	—
121	C₂×C₄×C₁₃₉	—	4	2	no	—
122	C₂×C₄×C₁₃₉	—	4	2	no	—
123	C₂×C₄×C₁₃₉	—	4	2	no	—
124	C₂×C₄×C₁₃₉	—	4	2	no	—
125	C₂×C₄×C₁₃₉	—	4	2	no	—
126	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	4	yes	—
127	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	4	yes	—
128	C₂×C₄×C₁₃₉	#135	1	2	yes	—
129	C₂×C₄×C₁₃₉	C₂²×Q₈	1	2	yes	—
130	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
131	C₂×C₄×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
133	C₈×C₁₃₉	—	4	2	no	—
134	C₈×C₁₃₉	—	4	1	no	—
135	C₈×C₁₃₉	—	4	1	no	—
136	C₂²×C₁₃₉	C₂×C₂²⋊Q₈	1	2	yes	—
137	C₂²×C₁₃₉	#537	1	1	yes	characteristic
138	C₂²×C₁₃₉	C₂×#135	1	1	yes	characteristic
139	C₂²×C₁₃₉	#575	1	2	yes	—
140	C₂²×C₁₃₉	#575	1	1	yes	characteristic
141	C₄×C₁₃₉	—	2	4	no	—
142	C₄×C₁₃₉	—	2	4	no	—
143	C₄×C₁₃₉	—	2	2	no	—
144	C₄×C₁₃₉	—	4	4	no	—
145	C₄×C₁₃₉	—	4	4	no	—
146	C₄×C₁₃₉	—	4	1	no	—
147	C₄×C₁₃₉	—	8	2	no	—
148	C₄×C₁₃₉	—	8	2	no	—
149	C₄×C₁₃₉	#557	1	2	yes	—
150	C₄×C₁₃₉	Q₈²	1	1	yes	characteristic
151	C₄×C₁₃₉	#559	1	1	yes	characteristic
152	C₂×C₁₃₉	#2582	1	1	yes	characteristic
153	C₂×C₁₃₉	#3076	1	2	yes	—
154	C₂×C₁₃₉	#3080	1	2	yes	—
155	C₂×C₁₃₉	#3250	1	1	yes	characteristic
156	C₂×C₁₃₉	#3256	1	1	yes	characteristic
157	#29291	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₁₃₉	#29291	1	1	yes	characteristic, direct product factor, Sylow
159	C₄×#125	C₂×C₁₃₉	1	2	yes	—
160	#2088	C₂×C₁₃₉	1	2	yes	—
161	#2087	C₂×C₁₃₉	1	1	yes	characteristic
162	#2086	C₂×C₁₃₉	1	1	yes	characteristic
163	#1989	C₂×C₁₃₉	1	2	yes	—
164	#1989	C₂×C₁₃₉	1	2	yes	—
165	#1949	C₂×C₁₃₉	1	2	yes	—
166	#1949	C₂×C₁₃₉	1	2	yes	—
167	#1842	C₂×C₁₃₉	1	1	yes	characteristic
168	C₂×#125	C₂²×C₁₃₉	1	2	yes	—
169	C₂×#107	C₂²×C₁₃₉	1	1	yes	characteristic
170	C₂×#107	C₂²×C₁₃₉	1	1	yes	characteristic
171	C₂×#106	C₂²×C₁₃₉	1	2	yes	—
172	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
173	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
174	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	1	yes	characteristic
176	#389	C₂²×C₁₃₉	1	2	yes	—
177	#389	C₂²×C₁₃₉	1	2	yes	—
178	#387	—	2	2	no	—
179	#387	—	2	2	no	—
180	#382	C₂²×C₁₃₉	1	2	yes	—
181	C₄×C₄⋊C₄	—	2	2	no	—
182	C₄×C₄⋊C₄	—	2	2	no	—
183	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
184	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
185	C₄³	C₂²×C₁₃₉	1	1	yes	characteristic
186	#340	C₂²×C₁₃₉	1	2	yes	—
187	#340	C₂²×C₁₃₉	1	1	yes	characteristic
188	#340	C₂²×C₁₃₉	1	2	yes	—
189	#340	C₂²×C₁₃₉	1	1	yes	characteristic
190	#340	C₂²×C₁₃₉	1	2	yes	—
191	#340	C₂²×C₁₃₉	1	2	yes	—
192	#340	C₂²×C₁₃₉	1	2	yes	—
193	C₂²×C₈	C₂³×C₁₃₉	1	2	yes	—
194	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
196	#125	—	2	4	no	—
197	#125	Q₈×C₁₃₉	1	8	yes	—
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	2	no	—
202	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
209	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
211	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₁₃₉	1	2	yes	—
213	#107	—	4	1	no	—
214	#107	—	4	1	no	—
215	#106	—	4	2	no	—
216	C₄⋊C₈	—	2	4	no	—
217	C₄⋊C₈	—	2	4	no	—
218	C₄⋊C₈	—	4	1	no	—
219	C₄⋊C₈	—	4	1	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
234	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₄⋊C₄	—	2	8	no	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	C₂×Q₈×C₁₃₉	1	4	yes	—
253	C₄⋊C₄	C₂×Q₈×C₁₃₉	1	4	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	4	1	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	4	no	—
260	C₄²	—	4	4	no	—
261	C₄²	C₂×Q₈×C₁₃₉	1	4	yes	—
262	C₄²	#41×C₁₃₉	1	4	yes	—
263	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	4	yes	—
285	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	4	yes	—
286	C₂×C₄	#135×C₁₃₉	1	2	yes	—
287	C₂×C₄	C₂²×Q₈×C₁₃₉	1	2	yes	—
288	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
289	C₂×C₄	H(5,𝔽₂)×C₁₃₉	1	1	yes	characteristic
290	C₂×C₄	2₋¹⁺⁴×C₁₃₉	1	1	yes	characteristic
291	C₈	—	4	2	no	—
292	C₈	—	4	1	no	—
293	C₈	—	4	1	no	—
294	C₂²	C₂×C₂²⋊Q₈×C₁₃₉	1	2	yes	—
295	C₂²	#537×C₁₃₉	1	1	yes	characteristic
296	C₂²	C₂×#135×C₁₃₉	1	1	yes	characteristic
297	C₂²	#575×C₁₃₉	1	2	yes	—
298	C₂²	#575×C₁₃₉	1	1	yes	characteristic
299	C₄	—	2	4	no	—
300	C₄	—	2	4	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	4	no	—
303	C₄	—	4	4	no	—
304	C₄	—	4	1	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	#557×C₁₃₉	1	2	yes	—
308	C₄	Q₈²×C₁₃₉	1	1	yes	characteristic
309	C₄	#559×C₁₃₉	1	1	yes	characteristic
310	C₂	#2582×C₁₃₉	1	1	yes	characteristic
311	C₂	#3076×C₁₃₉	1	2	yes	—
312	C₂	#3080×C₁₃₉	1	2	yes	—
313	C₂	#3250×C₁₃₉	1	1	yes	characteristic
314	C₂	#3256×C₁₃₉	1	1	yes	characteristic

	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 ◁ C₄³×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
157. #29291	C₁₃₉	direct (N × H)
158. C₁₃₉	#29291	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#125×C₁₃₉	2
#2088×C₁₃₉	2
#2087×C₁₃₉	2
#2086×C₁₃₉	2
#1989×C₁₃₉	2
#1949×C₁₃₉	2
#1842×C₁₃₉	2
#29291	139

Subgroups — C139×#29291 #48023887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#29291 #48023887