Subgroups — C₁₃₉×#29146 #48023742

Order 35584.

742 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2859×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#124×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#2026×C₁₃₉	C₂	1	2	yes	maximal, complemented
4	#2025×C₁₃₉	C₂	1	2	yes	maximal
5	#1959×C₁₃₉	C₂	1	2	yes	maximal
6	#1836×C₁₃₉	C₂	1	2	yes	maximal, complemented
7	#1805×C₁₃₉	C₂	1	2	yes	maximal, complemented
8	#1804×C₁₃₉	C₂	1	2	yes	maximal
9	#1767×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#125×C₁₃₉	C₂²	1	2	yes	—
11	C₂×#124×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
15	C₂×#103×C₁₃₉	C₂²	1	4	yes	—
16	C₂×#102×C₁₃₉	C₂²	1	4	yes	—
17	C₂×#97×C₁₃₉	C₂²	1	2	yes	—
18	#391×C₁₃₉	C₂²	1	1	yes	characteristic
19	#390×C₁₃₉	C₂²	1	1	yes	characteristic
20	#389×C₁₃₉	—	2	2	no	—
21	#389×C₁₃₉	—	2	2	no	—
22	#388×C₁₃₉	—	2	2	no	—
23	#384×C₁₃₉	C₂²	1	2	yes	—
24	#382×C₁₃₉	C₂²	1	1	yes	characteristic
25	#382×C₁₃₉	C₂²	1	2	yes	—
26	C₄×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
27	C₄³×C₁₃₉	C₂²	1	1	yes	characteristic
28	#340×C₁₃₉	C₂²	1	4	yes	—
29	#340×C₁₃₉	C₂²	1	4	yes	—
30	#336×C₁₃₉	C₂²	1	2	yes	—
31	C₂²×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
32	C₂²×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₁₃₉	—	2	1	no	—
34	C₂²×C₈×C₁₃₉	C₂³	1	4	yes	—
35	#125×C₁₃₉	—	4	4	no	—
36	#124×C₁₃₉	—	2	2	no	—
37	#124×C₁₃₉	Q₈	1	4	yes	—
38	C₄×Q₈×C₁₃₉	—	2	4	no	—
39	C₄×D₄×C₁₃₉	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
42	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
43	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₁₃₉	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
50	C₂×C₄²×C₁₃₉	—	2	1	no	—
51	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
52	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
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	C₄⋊C₈×C₁₃₉	—	2	4	no	—
57	#103×C₁₃₉	—	4	4	no	—
58	#102×C₁₃₉	—	4	4	no	—
59	#97×C₁₃₉	—	2	4	no	—
60	#95×C₁₃₉	—	2	2	no	—
61	#95×C₁₃₉	—	2	2	no	—
62	#95×C₁₃₉	—	2	2	no	—
63	#95×C₁₃₉	—	2	2	no	—
64	C₂⁴×C₁₃₉	—	2	1	no	—
65	C₂×Q₈×C₁₃₉	—	2	1	no	—
66	C₂×Q₈×C₁₃₉	—	2	2	no	—
67	C₂×Q₈×C₁₃₉	—	2	2	no	—
68	C₂×Q₈×C₁₃₉	C₂×Q₈	1	2	yes	—
69	C₂×D₄×C₁₃₉	—	2	2	no	—
70	C₂×D₄×C₁₃₉	—	2	1	no	—
71	C₂×D₄×C₁₃₉	—	2	2	no	—
72	C₂×D₄×C₁₃₉	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₁₃₉	—	2	1	no	—
74	C₂²×C₄×C₁₃₉	—	2	1	no	—
75	C₂²×C₄×C₁₃₉	—	2	1	no	—
76	C₂²×C₄×C₁₃₉	—	2	4	no	—
77	C₂²×C₄×C₁₃₉	—	4	4	no	—
78	C₂²×C₄×C₁₃₉	—	4	2	no	—
79	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₁₃₉	—	2	4	no	—
85	C₂×C₈×C₁₃₉	—	2	4	no	—
86	C₂×C₈×C₁₃₉	—	2	4	no	—
87	C₄⋊C₄×C₁₃₉	—	2	2	no	—
88	C₄⋊C₄×C₁₃₉	—	2	2	no	—
89	C₄⋊C₄×C₁₃₉	—	2	4	no	—
90	C₄⋊C₄×C₁₃₉	—	2	4	no	—
91	C₄⋊C₄×C₁₃₉	—	4	4	no	—
92	C₄⋊C₄×C₁₃₉	—	4	4	no	—
93	C₄⋊C₄×C₁₃₉	—	4	8	no	—
94	C₄⋊C₄×C₁₃₉	—	4	2	no	—
95	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
96	C₂²⋊C₄×C₁₃₉	—	4	4	no	—
97	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
98	C₄²×C₁₃₉	—	2	2	no	—
99	C₄²×C₁₃₉	—	4	2	no	—
100	C₄²×C₁₃₉	—	4	2	no	—
101	C₄²×C₁₃₉	—	4	2	no	—
102	C₄²×C₁₃₉	C₂×Q₈	1	2	yes	—
103	C₄²×C₁₃₉	#41	1	2	yes	—
104	C₄²×C₁₃₉	#41	1	4	yes	—
105	C₂³×C₁₃₉	—	2	2	no	—
106	C₂³×C₁₃₉	—	4	1	no	—
107	C₂³×C₁₃₉	—	4	1	no	—
108	C₂³×C₁₃₉	—	4	2	no	—
109	C₂³×C₁₃₉	—	4	2	no	—
110	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₁₃₉	—	2	4	no	—
112	Q₈×C₁₃₉	—	4	2	no	—
113	D₄×C₁₃₉	—	2	4	no	—
114	D₄×C₁₃₉	—	4	2	no	—
115	C₂×C₄×C₁₃₉	—	2	2	no	—
116	C₂×C₄×C₁₃₉	—	2	2	no	—
117	C₂×C₄×C₁₃₉	—	2	2	no	—
118	C₂×C₄×C₁₃₉	—	2	2	no	—
119	C₂×C₄×C₁₃₉	—	2	2	no	—
120	C₂×C₄×C₁₃₉	—	2	2	no	—
121	C₂×C₄×C₁₃₉	—	2	2	no	—
122	C₂×C₄×C₁₃₉	—	2	2	no	—
123	C₂×C₄×C₁₃₉	—	4	1	no	—
124	C₂×C₄×C₁₃₉	—	4	1	no	—
125	C₂×C₄×C₁₃₉	—	4	1	no	—
126	C₂×C₄×C₁₃₉	—	4	1	no	—
127	C₂×C₄×C₁₃₉	—	4	4	no	—
128	C₂×C₄×C₁₃₉	—	4	2	no	—
129	C₂×C₄×C₁₃₉	—	4	4	no	—
130	C₂×C₄×C₁₃₉	—	4	4	no	—
131	C₂×C₄×C₁₃₉	—	4	2	no	—
132	C₂×C₄×C₁₃₉	—	8	4	no	—
133	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	4	yes	—
134	C₂×C₄×C₁₃₉	#124	1	2	yes	—
135	C₂×C₄×C₁₃₉	#124	1	2	yes	—
136	C₂×C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
138	C₂×C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
140	C₈×C₁₃₉	—	4	4	no	—
141	C₂²×C₁₃₉	—	4	2	no	—
142	C₂²×C₁₃₉	—	4	4	no	—
143	C₂²×C₁₃₉	—	8	1	no	—
144	C₂²×C₁₃₉	—	8	1	no	—
145	C₂²×C₁₃₉	—	8	2	no	—
146	C₂²×C₁₃₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
147	C₂²×C₁₃₉	C₂×#124	1	1	yes	characteristic
148	C₂²×C₁₃₉	#535	1	1	yes	characteristic
149	C₂²×C₁₃₉	#575	1	2	yes	—
150	C₂²×C₁₃₉	#575	1	2	yes	—
151	C₄×C₁₃₉	—	2	4	no	—
152	C₄×C₁₃₉	—	2	2	no	—
153	C₄×C₁₃₉	—	2	2	no	—
154	C₄×C₁₃₉	—	4	2	no	—
155	C₄×C₁₃₉	—	4	2	no	—
156	C₄×C₁₃₉	—	8	1	no	—
157	C₄×C₁₃₉	—	8	1	no	—
158	C₄×C₁₃₉	—	8	4	no	—
159	C₄×C₁₃₉	#554	1	2	yes	—
160	C₄×C₁₃₉	#555	1	2	yes	—
161	C₂×C₁₃₉	—	8	2	no	—
162	C₂×C₁₃₉	#2563	1	1	yes	characteristic
163	C₂×C₁₃₉	#3080	1	2	yes	—
164	C₂×C₁₃₉	#3135	1	1	yes	characteristic
165	C₂×C₁₃₉	#3140	1	1	yes	characteristic
166	C₂×C₁₃₉	#3219	1	2	yes	—
167	#29146	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₁₃₉	#29146	1	1	yes	characteristic, direct product factor, Sylow
169	#2859	C₂×C₁₃₉	1	1	yes	characteristic, complemented
170	C₄×#124	C₂×C₁₃₉	1	1	yes	characteristic
171	#2026	C₂×C₁₃₉	1	2	yes	complemented
172	#2025	C₂×C₁₃₉	1	2	yes	—
173	#1959	C₂×C₁₃₉	1	2	yes	—
174	#1836	C₂×C₁₃₉	1	2	yes	complemented
175	#1805	C₂×C₁₃₉	1	2	yes	complemented
176	#1804	C₂×C₁₃₉	1	2	yes	—
177	#1767	C₂×C₁₃₉	1	1	yes	characteristic, complemented
178	C₂×#125	C₂²×C₁₃₉	1	2	yes	—
179	C₂×#124	C₂²×C₁₃₉	1	1	yes	characteristic
180	C₂×C₄×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
181	C₂×C₄×D₄	C₂²×C₁₃₉	1	1	yes	characteristic
182	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
183	C₂×#103	C₂²×C₁₃₉	1	4	yes	—
184	C₂×#102	C₂²×C₁₃₉	1	4	yes	—
185	C₂×#97	C₂²×C₁₃₉	1	2	yes	—
186	#391	C₂²×C₁₃₉	1	1	yes	characteristic
187	#390	C₂²×C₁₃₉	1	1	yes	characteristic
188	#389	—	2	2	no	—
189	#389	—	2	2	no	—
190	#388	—	2	2	no	—
191	#384	C₂²×C₁₃₉	1	2	yes	—
192	#382	C₂²×C₁₃₉	1	2	yes	—
193	#382	C₂²×C₁₃₉	1	1	yes	characteristic
194	C₄×C₂²⋊C₄	—	2	2	no	—
195	C₄³	C₂²×C₁₃₉	1	1	yes	characteristic
196	#340	C₂²×C₁₃₉	1	4	yes	—
197	#340	C₂²×C₁₃₉	1	4	yes	—
198	#336	C₂²×C₁₃₉	1	2	yes	—
199	C₂²×Q₈	C₂³×C₁₃₉	1	1	yes	characteristic
200	C₂²×D₄	C₂³×C₁₃₉	1	1	yes	characteristic
201	C₂³×C₄	—	2	1	no	—
202	C₂²×C₈	C₂³×C₁₃₉	1	4	yes	—
203	#125	—	4	4	no	—
204	#124	—	2	2	no	—
205	#124	Q₈×C₁₃₉	1	4	yes	—
206	C₄×Q₈	—	2	4	no	—
207	C₄×D₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	4	yes	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₂²⋊C₄	—	2	1	no	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
220	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
221	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
224	C₄⋊C₈	—	2	4	no	—
225	#103	—	4	4	no	—
226	#102	—	4	4	no	—
227	#97	—	2	4	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	C₂⁴	—	2	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	C₂×Q₈×C₁₃₉	1	2	yes	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	C₂×Q₈×C₁₃₉	1	2	yes	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
251	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	4	8	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	2	8	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	C₂×Q₈×C₁₃₉	1	2	yes	—
271	C₄²	#41×C₁₃₉	1	2	yes	—
272	C₄²	#41×C₁₃₉	1	4	yes	—
273	C₂³	—	2	2	no	—
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
279	Q₈	—	2	4	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	2	4	no	—
282	D₄	—	4	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	4	yes	—
302	C₂×C₄	#124×C₁₃₉	1	2	yes	—
303	C₂×C₄	#124×C₁₃₉	1	2	yes	—
304	C₂×C₄	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
305	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
306	C₂×C₄	C₂×#41×C₁₃₉	1	1	yes	characteristic
307	C₂×C₄	H(5,𝔽₂)×C₁₃₉	1	2	yes	—
308	C₈	—	4	4	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	—	8	2	no	—
314	C₂²	C₂×C₂²⋊Q₈×C₁₃₉	1	1	yes	characteristic
315	C₂²	C₂×#124×C₁₃₉	1	1	yes	characteristic
316	C₂²	#535×C₁₃₉	1	1	yes	characteristic
317	C₂²	#575×C₁₃₉	1	2	yes	—
318	C₂²	#575×C₁₃₉	1	2	yes	—
319	C₄	—	2	4	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	4	no	—
327	C₄	#554×C₁₃₉	1	2	yes	—
328	C₄	#555×C₁₃₉	1	2	yes	—
329	C₂	—	8	2	no	—
330	C₂	#2563×C₁₃₉	1	1	yes	characteristic
331	C₂	#3080×C₁₃₉	1	2	yes	—
332	C₂	#3135×C₁₃₉	1	1	yes	characteristic
333	C₂	#3140×C₁₃₉	1	1	yes	characteristic
334	C₂	#3219×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₉
prime radical	C₂²×D₄×C₁₃₉	C₂³
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
1. #2859×C₁₃₉	C₂	semidirect
3. #2026×C₁₃₉	C₂	semidirect
6. #1836×C₁₃₉	C₂	semidirect
7. #1805×C₁₃₉	C₂	semidirect
9. #1767×C₁₃₉	C₂	semidirect
167. #29146	C₁₃₉	direct (N × H)
168. C₁₃₉	#29146	direct (N × H)
169. #2859	C₂×C₁₃₉	semidirect
171. #2026	C₂×C₁₃₉	semidirect
174. #1836	C₂×C₁₃₉	semidirect
175. #1805	C₂×C₁₃₉	semidirect
177. #1767	C₂×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#2859×C₁₃₉	2
C₄×#124×C₁₃₉	2
#2026×C₁₃₉	2
#2025×C₁₃₉	2
#1959×C₁₃₉	2
#1836×C₁₃₉	2
#1805×C₁₃₉	2
#1804×C₁₃₉	2
#1767×C₁₃₉	2
#29146	139

Subgroups — C139×#29146 #48023742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#29146 #48023742