Subgroups — C₁₃₉×#10717 #48005313

Order 35584.

1374 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×M₁₆×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#2280×C₁₃₉	C₂	1	2	yes	maximal
3	C₂×H(3,ℤ/4)×C₁₃₉	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
5	C₂²×M₁₆×C₁₃₉	—	2	4	no	—
6	C₂²×M₁₆×C₁₃₉	—	2	8	no	—
7	C₂²×M₁₆×C₁₃₉	C₄	1	2	yes	complemented
8	C₂²×M₁₆×C₁₃₉	C₂²	1	2	yes	—
9	C₂³×C₈×C₁₃₉	C₄	1	2	yes	complemented
10	C₂×#117×C₁₃₉	C₂²	1	4	yes	—
11	#381×C₁₃₉	—	2	2	no	—
12	#380×C₁₃₉	—	2	4	no	—
13	#379×C₁₃₉	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₁₃₉	C₄	1	8	yes	complemented
15	H(3,ℤ/4)×C₁₃₉	—	2	8	no	—
16	C₂⁵×C₁₃₉	D₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₃₉	—	2	4	no	—
18	C₂³×C₄×C₁₃₉	—	2	4	no	—
19	C₂³×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₃₉	C₂×C₄	1	4	yes	—
22	C₂³×C₄×C₁₃₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃₉	D₄	1	4	yes	—
24	C₂³×C₄×C₁₃₉	Q₈	1	1	yes	characteristic
25	C₂³×C₄×C₁₃₉	Q₈	1	1	yes	characteristic
26	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₁₃₉	—	2	32	no	—
28	C₂×M₁₆×C₁₃₉	—	2	4	no	—
29	C₂×M₁₆×C₁₃₉	—	2	4	no	—
30	C₂×M₁₆×C₁₃₉	—	2	4	no	—
31	C₂×M₁₆×C₁₃₉	—	2	8	no	—
32	C₂×M₁₆×C₁₃₉	D₄	1	8	yes	—
33	C₂²×C₈×C₁₃₉	—	2	4	no	—
34	C₂²×C₈×C₁₃₉	—	2	8	no	—
35	C₂²×C₈×C₁₃₉	C₂×C₄	1	4	yes	—
36	#117×C₁₃₉	—	2	8	no	—
37	#117×C₁₃₉	—	2	8	no	—
38	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
39	C₂×C₂²⋊C₄×C₁₃₉	—	2	8	no	—
40	C₂×C₂²⋊C₄×C₁₃₉	—	2	4	no	—
41	C₂×C₄²×C₁₃₉	C₂×C₄	1	8	yes	—
42	#95×C₁₃₉	—	2	8	no	—
43	C₂⁴×C₁₃₉	—	2	4	no	—
44	C₂⁴×C₁₃₉	—	4	4	no	—
45	C₂⁴×C₁₃₉	C₂²⋊C₄	1	4	yes	—
46	C₂⁴×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
47	C₂⁴×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂⁴×C₁₃₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₉	—	2	4	no	—
50	C₂²×C₄×C₁₃₉	—	2	2	no	—
51	C₂²×C₄×C₁₃₉	—	2	2	no	—
52	C₂²×C₄×C₁₃₉	—	2	2	no	—
53	C₂²×C₄×C₁₃₉	—	2	2	no	—
54	C₂²×C₄×C₁₃₉	—	2	16	no	—
55	C₂²×C₄×C₁₃₉	—	2	8	no	—
56	C₂²×C₄×C₁₃₉	—	2	4	no	—
57	C₂²×C₄×C₁₃₉	—	2	4	no	—
58	C₂²×C₄×C₁₃₉	—	2	8	no	—
59	C₂²×C₄×C₁₃₉	—	2	8	no	—
60	C₂²×C₄×C₁₃₉	C₄²	1	2	yes	—
61	C₂²×C₄×C₁₃₉	C₄²	1	2	yes	—
62	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
63	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₃₉	#41	1	4	yes	—
75	M₁₆×C₁₃₉	—	2	16	no	—
76	M₁₆×C₁₃₉	—	2	16	no	—
77	C₂×C₈×C₁₃₉	—	2	16	no	—
78	C₂×C₈×C₁₃₉	—	2	4	no	—
79	C₂×C₈×C₁₃₉	—	2	4	no	—
80	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	8	yes	complemented
81	C₄⋊C₄×C₁₃₉	—	4	4	no	—
82	C₄⋊C₄×C₁₃₉	—	4	4	no	—
83	C₂²⋊C₄×C₁₃₉	—	4	16	no	—
84	C₂²⋊C₄×C₁₃₉	—	4	8	no	—
85	C₄²×C₁₃₉	—	2	16	no	—
86	C₂³×C₁₃₉	—	2	4	no	—
87	C₂³×C₁₃₉	—	2	2	no	—
88	C₂³×C₁₃₉	—	2	8	no	—
89	C₂³×C₁₃₉	—	2	2	no	—
90	C₂³×C₁₃₉	—	2	8	no	—
91	C₂³×C₁₃₉	—	4	4	no	—
92	C₂³×C₁₃₉	—	4	2	no	—
93	C₂³×C₁₃₉	—	4	2	no	—
94	C₂³×C₁₃₉	—	4	16	no	—
95	C₂³×C₁₃₉	#95	1	2	yes	—
96	C₂³×C₁₃₉	#95	1	2	yes	—
97	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂³×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₁₃₉	#121	1	4	yes	—
100	C₂×C₄×C₁₃₉	—	2	2	no	—
101	C₂×C₄×C₁₃₉	—	2	2	no	—
102	C₂×C₄×C₁₃₉	—	2	8	no	—
103	C₂×C₄×C₁₃₉	—	2	4	no	—
104	C₂×C₄×C₁₃₉	—	2	4	no	—
105	C₂×C₄×C₁₃₉	—	2	2	no	—
106	C₂×C₄×C₁₃₉	—	2	2	no	—
107	C₂×C₄×C₁₃₉	—	2	2	no	—
108	C₂×C₄×C₁₃₉	—	2	2	no	—
109	C₂×C₄×C₁₃₉	—	2	8	no	—
110	C₂×C₄×C₁₃₉	—	2	8	no	—
111	C₂×C₄×C₁₃₉	—	4	16	no	—
112	C₂×C₄×C₁₃₉	—	4	8	no	—
113	C₂×C₄×C₁₃₉	#95	1	2	yes	—
114	C₂×C₄×C₁₃₉	#95	1	2	yes	—
115	C₂×C₄×C₁₃₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₃₉	#117	1	2	yes	—
120	C₂×C₄×C₁₃₉	C₄×D₄	1	8	yes	—
121	C₂×C₄×C₁₃₉	#120	1	4	yes	—
122	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	4	yes	—
123	C₂×C₄×C₁₃₉	#123	1	4	yes	—
124	C₈×C₁₃₉	—	2	8	no	—
125	C₂²×C₁₃₉	—	2	8	no	—
126	C₂²×C₁₃₉	—	2	2	no	—
127	C₂²×C₁₃₉	—	2	4	no	—
128	C₂²×C₁₃₉	—	2	2	no	—
129	C₂²×C₁₃₉	—	2	2	no	—
130	C₂²×C₁₃₉	—	2	2	no	—
131	C₂²×C₁₃₉	—	2	4	no	—
132	C₂²×C₁₃₉	—	4	4	no	—
133	C₂²×C₁₃₉	—	4	16	no	—
134	C₂²×C₁₃₉	—	4	4	no	—
135	C₂²×C₁₃₉	H(3,ℤ/4)	1	4	yes	complemented
136	C₂²×C₁₃₉	C₂×#95	1	1	yes	characteristic
137	C₂²×C₁₃₉	#380	1	2	yes	—
138	C₂²×C₁₃₉	#386	1	4	yes	—
139	C₄×C₁₃₉	—	2	2	no	—
140	C₄×C₁₃₉	—	2	2	no	—
141	C₄×C₁₃₉	—	8	8	no	—
142	C₄×C₁₃₉	C₄×C₂²⋊C₄	1	2	yes	—
143	C₄×C₁₃₉	#379	1	1	yes	characteristic
144	C₄×C₁₃₉	#381	1	1	yes	characteristic
145	C₄×C₁₃₉	#385	1	4	yes	—
146	C₂×C₁₃₉	—	2	4	no	—
147	C₂×C₁₃₉	—	2	4	no	—
148	C₂×C₁₃₉	—	2	2	no	—
149	C₂×C₁₃₉	—	4	2	no	—
150	C₂×C₁₃₉	C₂²⋊#95	1	1	yes	characteristic
151	C₂×C₁₃₉	C₂×H(3,ℤ/4)	1	2	yes	—
152	#10717	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₁₃₉	#10717	1	1	yes	characteristic, direct product factor, Sylow
154	C₂³×M₁₆	C₂×C₁₃₉	1	1	yes	characteristic
155	#2280	C₂×C₁₃₉	1	2	yes	—
156	C₂×H(3,ℤ/4)	C₂×C₁₃₉	1	4	yes	complemented
157	C₂⁴×C₄	C₂²×C₁₃₉	1	1	yes	characteristic
158	C₂²×M₁₆	—	2	4	no	—
159	C₂²×M₁₆	—	2	8	no	—
160	C₂²×M₁₆	C₄×C₁₃₉	1	2	yes	complemented
161	C₂²×M₁₆	C₂²×C₁₃₉	1	2	yes	—
162	C₂³×C₈	C₄×C₁₃₉	1	2	yes	complemented
163	C₂×#117	C₂²×C₁₃₉	1	4	yes	—
164	#381	—	2	2	no	—
165	#380	—	2	4	no	—
166	#379	—	2	2	no	—
167	C₄×C₂²⋊C₄	C₄×C₁₃₉	1	8	yes	complemented
168	H(3,ℤ/4)	—	2	8	no	—
169	C₂⁵	D₄×C₁₃₉	1	1	yes	characteristic
170	C₂³×C₄	—	2	4	no	—
171	C₂³×C₄	—	2	4	no	—
172	C₂³×C₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
173	C₂³×C₄	C₂×C₄×C₁₃₉	1	1	yes	characteristic
174	C₂³×C₄	C₂×C₄×C₁₃₉	1	4	yes	—
175	C₂³×C₄	D₄×C₁₃₉	1	1	yes	characteristic
176	C₂³×C₄	D₄×C₁₃₉	1	4	yes	—
177	C₂³×C₄	Q₈×C₁₃₉	1	1	yes	characteristic
178	C₂³×C₄	Q₈×C₁₃₉	1	1	yes	characteristic
179	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
180	C₂×M₁₆	—	2	32	no	—
181	C₂×M₁₆	—	2	4	no	—
182	C₂×M₁₆	—	2	4	no	—
183	C₂×M₁₆	—	2	4	no	—
184	C₂×M₁₆	—	2	8	no	—
185	C₂×M₁₆	D₄×C₁₃₉	1	8	yes	—
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	2	8	no	—
188	C₂²×C₈	C₂×C₄×C₁₃₉	1	4	yes	—
189	#117	—	2	8	no	—
190	#117	—	2	8	no	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₂²⋊C₄	—	2	8	no	—
193	C₂×C₂²⋊C₄	—	2	4	no	—
194	C₂×C₄²	C₂×C₄×C₁₃₉	1	8	yes	—
195	#95	—	2	8	no	—
196	C₂⁴	—	2	4	no	—
197	C₂⁴	—	4	4	no	—
198	C₂⁴	C₂²⋊C₄×C₁₃₉	1	4	yes	—
199	C₂⁴	C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
200	C₂⁴	C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
201	C₂⁴	C₂×D₄×C₁₃₉	1	1	yes	characteristic
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	16	no	—
208	C₂²×C₄	—	2	8	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	8	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	C₄²×C₁₃₉	1	2	yes	—
214	C₂²×C₄	C₄²×C₁₃₉	1	2	yes	—
215	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	4	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	2	yes	—
218	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
219	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
220	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
221	C₂²×C₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₁₃₉	1	4	yes	—
226	C₂²×C₄	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
227	C₂²×C₄	#41×C₁₃₉	1	4	yes	—
228	M₁₆	—	2	16	no	—
229	M₁₆	—	2	16	no	—
230	C₂×C₈	—	2	16	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	8	yes	complemented
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	—	4	4	no	—
236	C₂²⋊C₄	—	4	16	no	—
237	C₂²⋊C₄	—	4	8	no	—
238	C₄²	—	2	16	no	—
239	C₂³	—	2	4	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	8	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	8	no	—
244	C₂³	—	4	4	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	2	no	—
247	C₂³	—	4	16	no	—
248	C₂³	#95×C₁₃₉	1	2	yes	—
249	C₂³	#95×C₁₃₉	1	2	yes	—
250	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	2	yes	—
251	C₂³	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
252	C₂³	#121×C₁₃₉	1	4	yes	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	4	16	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	#95×C₁₃₉	1	2	yes	—
267	C₂×C₄	#95×C₁₃₉	1	2	yes	—
268	C₂×C₄	C₂×C₄²×C₁₃₉	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	2	yes	—
270	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
271	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	2	yes	—
272	C₂×C₄	#117×C₁₃₉	1	2	yes	—
273	C₂×C₄	C₄×D₄×C₁₃₉	1	8	yes	—
274	C₂×C₄	#120×C₁₃₉	1	4	yes	—
275	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	4	yes	—
276	C₂×C₄	#123×C₁₃₉	1	4	yes	—
277	C₈	—	2	8	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	16	no	—
287	C₂²	—	4	4	no	—
288	C₂²	H(3,ℤ/4)×C₁₃₉	1	4	yes	complemented
289	C₂²	C₂×#95×C₁₃₉	1	1	yes	characteristic
290	C₂²	#380×C₁₃₉	1	2	yes	—
291	C₂²	#386×C₁₃₉	1	4	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	8	8	no	—
295	C₄	C₄×C₂²⋊C₄×C₁₃₉	1	2	yes	—
296	C₄	#379×C₁₃₉	1	1	yes	characteristic
297	C₄	#381×C₁₃₉	1	1	yes	characteristic
298	C₄	#385×C₁₃₉	1	4	yes	—
299	C₂	—	2	4	no	—
300	C₂	—	2	4	no	—
301	C₂	—	2	2	no	—
302	C₂	—	4	2	no	—
303	C₂	C₂²⋊#95×C₁₃₉	1	1	yes	characteristic
304	C₂	C₂×H(3,ℤ/4)×C₁₃₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₉	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₁₃₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃₉
socle	C₂²×C₁₃₉	C₂×#95
prime radical	C₂⁵×C₁₃₉	D₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×H(3,ℤ/4)×C₁₃₉	C₂	semidirect
7. C₂²×M₁₆×C₁₃₉	C₄	semidirect
9. C₂³×C₈×C₁₃₉	C₄	semidirect
14. C₄×C₂²⋊C₄×C₁₃₉	C₄	semidirect
80. C₂×C₈×C₁₃₉	C₂²⋊C₄	semidirect
135. C₂²×C₁₃₉	H(3,ℤ/4)	semidirect
152. #10717	C₁₃₉	direct (N × H)
153. C₁₃₉	#10717	direct (N × H)
156. C₂×H(3,ℤ/4)	C₂×C₁₃₉	semidirect
160. C₂²×M₁₆	C₄×C₁₃₉	semidirect
162. C₂³×C₈	C₄×C₁₃₉	semidirect
167. C₄×C₂²⋊C₄	C₄×C₁₃₉	semidirect
233. C₂×C₈	C₂²⋊C₄×C₁₃₉	semidirect
288. C₂²	H(3,ℤ/4)×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂³×M₁₆×C₁₃₉	2
#2280×C₁₃₉	2
C₂×H(3,ℤ/4)×C₁₃₉	2
#10717	139

Subgroups — C139×#10717 #48005313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#10717 #48005313