Subgroups — C₂×#1437×C₁₃₉ #48008356

Order 35584.

5230 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊C₄×C₁₃₉	C₂	1	3	yes	maximal
2	C₂³×C₄²×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#95×C₁₃₉	C₂	1	3	yes	maximal, complemented
4	#1437×C₁₃₉	C₂	1	8	yes	maximal, direct product factor
5	C₂⁶×C₁₃₉	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₁₃₉	C₂²	1	3	yes	—
7	C₂⁴×C₄×C₁₃₉	C₂²	1	3	yes	—
8	C₂²×C₂²⋊C₄×C₁₃₉	—	2	12	no	—
9	C₂²×C₂²⋊C₄×C₁₃₉	—	2	12	no	—
10	C₂²×C₂²⋊C₄×C₁₃₉	C₄	1	24	yes	complemented
11	C₂²×C₂²⋊C₄×C₁₃₉	C₂²	1	12	yes	—
12	C₂²×C₄²×C₁₃₉	—	2	12	no	—
13	C₂²×C₄²×C₁₃₉	C₂²	1	4	yes	complemented
14	C₂×#95×C₁₃₉	—	2	12	no	—
15	C₂×#95×C₁₃₉	C₂²	1	12	yes	complemented
16	C₂⁵×C₁₃₉	—	2	6	no	—
17	C₂⁵×C₁₃₉	—	2	6	no	—
18	C₂⁵×C₁₃₉	—	2	12	no	—
19	C₂⁵×C₁₃₉	C₂×C₄	1	12	yes	—
20	C₂⁵×C₁₃₉	C₂³	1	2	yes	—
21	C₂⁵×C₁₃₉	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₁₃₉	—	2	12	no	—
23	C₂³×C₄×C₁₃₉	—	2	12	no	—
24	C₂³×C₄×C₁₃₉	—	2	6	no	—
25	C₂³×C₄×C₁₃₉	—	2	6	no	—
26	C₂³×C₄×C₁₃₉	—	2	12	no	—
27	C₂³×C₄×C₁₃₉	—	2	12	no	—
28	C₂³×C₄×C₁₃₉	C₂×C₄	1	12	yes	—
29	C₂³×C₄×C₁₃₉	C₂×C₄	1	12	yes	—
30	C₂³×C₄×C₁₃₉	D₄	1	12	yes	—
31	C₂³×C₄×C₁₃₉	D₄	1	12	yes	—
32	C₂³×C₄×C₁₃₉	C₂³	1	6	yes	—
33	C₂³×C₄×C₁₃₉	C₂³	1	6	yes	—
34	C₂×C₂²⋊C₄×C₁₃₉	—	2	96	no	—
35	C₂×C₂²⋊C₄×C₁₃₉	—	2	24	no	—
36	C₂×C₂²⋊C₄×C₁₃₉	—	2	24	no	—
37	C₂×C₂²⋊C₄×C₁₃₉	C₂×C₄	1	48	yes	complemented
38	C₂×C₄²×C₁₃₉	—	2	32	no	—
39	C₂×C₄²×C₁₃₉	—	2	24	no	—
40	#95×C₁₃₉	—	2	24	no	—
41	C₂⁴×C₁₃₉	—	2	24	no	—
42	C₂⁴×C₁₃₉	—	2	24	no	—
43	C₂⁴×C₁₃₉	—	2	48	no	—
44	C₂⁴×C₁₃₉	—	2	48	no	—
45	C₂⁴×C₁₃₉	—	2	6	no	—
46	C₂⁴×C₁₃₉	—	2	6	no	—
47	C₂⁴×C₁₃₉	—	2	12	no	—
48	C₂⁴×C₁₃₉	—	4	4	no	—
49	C₂⁴×C₁₃₉	—	4	24	no	—
50	C₂⁴×C₁₃₉	—	4	12	no	—
51	C₂⁴×C₁₃₉	—	4	12	no	—
52	C₂⁴×C₁₃₉	—	4	8	no	—
53	C₂⁴×C₁₃₉	—	4	4	no	—
54	C₂⁴×C₁₃₉	C₄²	1	16	yes	complemented
55	C₂⁴×C₁₃₉	C₂²×C₄	1	12	yes	—
56	C₂⁴×C₁₃₉	C₂²×C₄	1	6	yes	—
57	C₂⁴×C₁₃₉	C₂×D₄	1	6	yes	—
58	C₂⁴×C₁₃₉	C₂×D₄	1	6	yes	—
59	C₂⁴×C₁₃₉	#41	1	12	yes	—
60	C₂⁴×C₁₃₉	C₂⁴	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₉	—	2	48	no	—
62	C₂²×C₄×C₁₃₉	—	2	24	no	—
63	C₂²×C₄×C₁₃₉	—	2	48	no	—
64	C₂²×C₄×C₁₃₉	—	2	24	no	—
65	C₂²×C₄×C₁₃₉	—	2	12	no	—
66	C₂²×C₄×C₁₃₉	—	2	12	no	—
67	C₂²×C₄×C₁₃₉	—	2	24	no	—
68	C₂²×C₄×C₁₃₉	—	2	48	no	—
69	C₂²×C₄×C₁₃₉	—	2	12	no	—
70	C₂²×C₄×C₁₃₉	—	2	6	no	—
71	C₂²×C₄×C₁₃₉	—	2	6	no	—
72	C₂²×C₄×C₁₃₉	—	2	12	no	—
73	C₂²×C₄×C₁₃₉	—	4	24	no	—
74	C₂²×C₄×C₁₃₉	—	4	12	no	—
75	C₂²×C₄×C₁₃₉	—	4	12	no	—
76	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	48	yes	complemented
77	C₂²×C₄×C₁₃₉	C₂²×C₄	1	12	yes	—
78	C₂²×C₄×C₁₃₉	C₂²×C₄	1	12	yes	—
79	C₂²×C₄×C₁₃₉	C₂×D₄	1	12	yes	—
80	C₂²×C₄×C₁₃₉	C₂×D₄	1	12	yes	—
81	C₂²⋊C₄×C₁₃₉	—	2	96	no	—
82	C₄²×C₁₃₉	—	2	32	no	—
83	C₂³×C₁₃₉	—	2	12	no	—
84	C₂³×C₁₃₉	—	2	12	no	—
85	C₂³×C₁₃₉	—	2	24	no	—
86	C₂³×C₁₃₉	—	2	24	no	—
87	C₂³×C₁₃₉	—	2	96	no	—
88	C₂³×C₁₃₉	—	4	2	no	—
89	C₂³×C₁₃₉	—	4	12	no	—
90	C₂³×C₁₃₉	—	4	6	no	—
91	C₂³×C₁₃₉	—	4	6	no	—
92	C₂³×C₁₃₉	—	4	4	no	—
93	C₂³×C₁₃₉	—	4	2	no	—
94	C₂³×C₁₃₉	—	4	48	no	—
95	C₂³×C₁₃₉	—	4	96	no	—
96	C₂³×C₁₃₉	—	4	48	no	—
97	C₂³×C₁₃₉	C₂×C₄²	1	4	yes	—
98	C₂³×C₁₃₉	C₂×C₄²	1	8	yes	complemented
99	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	12	yes	—
100	C₂³×C₁₃₉	#117	1	12	yes	—
101	C₂³×C₁₃₉	C₄×D₄	1	24	yes	—
102	C₂³×C₁₃₉	C₄×D₄	1	24	yes	—
103	C₂³×C₁₃₉	#120	1	4	yes	—
104	C₂³×C₁₃₉	#121	1	24	yes	—
105	C₂³×C₁₃₉	#123	1	12	yes	—
106	C₂³×C₁₃₉	#124	1	12	yes	—
107	C₂³×C₁₃₉	#126	1	8	yes	—
108	C₂³×C₁₃₉	#127	1	4	yes	—
109	C₂³×C₁₃₉	C₂³×C₄	1	3	yes	—
110	C₂³×C₁₃₉	C₂²×D₄	1	3	yes	—
111	C₂³×C₁₃₉	C₂²×D₄	1	3	yes	—
112	C₂³×C₁₃₉	C₂×#41	1	6	yes	—
113	C₂×C₄×C₁₃₉	—	2	24	no	—
114	C₂×C₄×C₁₃₉	—	2	12	no	—
115	C₂×C₄×C₁₃₉	—	2	24	no	—
116	C₂×C₄×C₁₃₉	—	2	12	no	—
117	C₂×C₄×C₁₃₉	—	2	96	no	—
118	C₂×C₄×C₁₃₉	—	2	12	no	—
119	C₂×C₄×C₁₃₉	—	2	24	no	—
120	C₂×C₄×C₁₃₉	—	4	12	no	—
121	C₂×C₄×C₁₃₉	—	4	6	no	—
122	C₂×C₄×C₁₃₉	—	4	6	no	—
123	C₂×C₄×C₁₃₉	—	4	48	no	—
124	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	24	yes	complemented
125	C₂²×C₁₃₉	—	2	24	no	—
126	C₂²×C₁₃₉	—	4	64	no	—
127	C₂²×C₁₃₉	—	4	12	no	—
128	C₂²×C₁₃₉	—	4	24	no	—
129	C₂²×C₁₃₉	—	4	12	no	—
130	C₂²×C₁₃₉	C₄×C₂²⋊C₄	1	24	yes	—
131	C₂²×C₁₃₉	#386	1	24	yes	—
132	C₂²×C₁₃₉	#388	1	48	yes	—
133	C₂²×C₁₃₉	#390	1	24	yes	—
134	C₂²×C₁₃₉	C₂²×C₄²	1	1	yes	characteristic
135	C₂²×C₁₃₉	C₂²×C₂²⋊C₄	1	3	yes	—
136	C₂²×C₁₃₉	C₂×#117	1	3	yes	—
137	C₂²×C₁₃₉	C₂×C₄×D₄	1	6	yes	—
138	C₂²×C₁₃₉	C₂×C₄×D₄	1	6	yes	—
139	C₂²×C₁₃₉	C₂×#120	1	1	yes	characteristic
140	C₂²×C₁₃₉	C₂×#121	1	6	yes	—
141	C₂²×C₁₃₉	C₂×#123	1	3	yes	—
142	C₂²×C₁₃₉	C₂×#124	1	3	yes	—
143	C₂²×C₁₃₉	C₂×#126	1	2	yes	—
144	C₂²×C₁₃₉	C₂×#127	1	1	yes	characteristic
145	C₄×C₁₃₉	—	2	24	no	—
146	C₄×C₁₃₉	—	4	12	no	—
147	C₂×C₁₃₉	—	4	8	no	—
148	C₂×C₁₃₉	#1437	1	16	yes	direct product factor
149	C₂×C₁₃₉	C₂×C₄×C₂²⋊C₄	1	3	yes	—
150	C₂×C₁₃₉	C₂×#386	1	3	yes	—
151	C₂×C₁₃₉	C₂×#388	1	6	yes	—
152	C₂×C₁₃₉	C₂×#390	1	3	yes	—
153	C₂×#1437	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₁₃₉	C₂×#1437	1	1	yes	characteristic, direct product factor, Sylow
155	C₂³×C₂²⋊C₄	C₂×C₁₃₉	1	3	yes	—
156	C₂³×C₄²	C₂×C₁₃₉	1	1	yes	characteristic, complemented
157	C₂²×#95	C₂×C₁₃₉	1	3	yes	complemented
158	#1437	C₂×C₁₃₉	1	8	yes	direct product factor
159	C₂⁶	C₂²×C₁₃₉	1	1	yes	characteristic
160	C₂⁴×C₄	C₂²×C₁₃₉	1	3	yes	—
161	C₂⁴×C₄	C₂²×C₁₃₉	1	3	yes	—
162	C₂²×C₂²⋊C₄	—	2	12	no	—
163	C₂²×C₂²⋊C₄	—	2	12	no	—
164	C₂²×C₂²⋊C₄	C₄×C₁₃₉	1	24	yes	complemented
165	C₂²×C₂²⋊C₄	C₂²×C₁₃₉	1	12	yes	—
166	C₂²×C₄²	—	2	12	no	—
167	C₂²×C₄²	C₂²×C₁₃₉	1	4	yes	complemented
168	C₂×#95	—	2	12	no	—
169	C₂×#95	C₂²×C₁₃₉	1	12	yes	complemented
170	C₂⁵	—	2	6	no	—
171	C₂⁵	—	2	6	no	—
172	C₂⁵	—	2	12	no	—
173	C₂⁵	C₂×C₄×C₁₃₉	1	12	yes	—
174	C₂⁵	C₂³×C₁₃₉	1	2	yes	—
175	C₂⁵	C₂³×C₁₃₉	1	1	yes	characteristic
176	C₂³×C₄	—	2	12	no	—
177	C₂³×C₄	—	2	12	no	—
178	C₂³×C₄	—	2	6	no	—
179	C₂³×C₄	—	2	6	no	—
180	C₂³×C₄	—	2	12	no	—
181	C₂³×C₄	—	2	12	no	—
182	C₂³×C₄	C₂×C₄×C₁₃₉	1	12	yes	—
183	C₂³×C₄	C₂×C₄×C₁₃₉	1	12	yes	—
184	C₂³×C₄	D₄×C₁₃₉	1	12	yes	—
185	C₂³×C₄	D₄×C₁₃₉	1	12	yes	—
186	C₂³×C₄	C₂³×C₁₃₉	1	6	yes	—
187	C₂³×C₄	C₂³×C₁₃₉	1	6	yes	—
188	C₂×C₂²⋊C₄	—	2	96	no	—
189	C₂×C₂²⋊C₄	—	2	24	no	—
190	C₂×C₂²⋊C₄	—	2	24	no	—
191	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₉	1	48	yes	complemented
192	C₂×C₄²	—	2	32	no	—
193	C₂×C₄²	—	2	24	no	—
194	#95	—	2	24	no	—
195	C₂⁴	—	2	24	no	—
196	C₂⁴	—	2	24	no	—
197	C₂⁴	—	2	48	no	—
198	C₂⁴	—	2	48	no	—
199	C₂⁴	—	2	6	no	—
200	C₂⁴	—	2	6	no	—
201	C₂⁴	—	2	12	no	—
202	C₂⁴	—	4	4	no	—
203	C₂⁴	—	4	24	no	—
204	C₂⁴	—	4	12	no	—
205	C₂⁴	—	4	12	no	—
206	C₂⁴	—	4	8	no	—
207	C₂⁴	—	4	4	no	—
208	C₂⁴	C₄²×C₁₃₉	1	16	yes	complemented
209	C₂⁴	C₂²×C₄×C₁₃₉	1	12	yes	—
210	C₂⁴	C₂²×C₄×C₁₃₉	1	6	yes	—
211	C₂⁴	C₂×D₄×C₁₃₉	1	6	yes	—
212	C₂⁴	C₂×D₄×C₁₃₉	1	6	yes	—
213	C₂⁴	#41×C₁₃₉	1	12	yes	—
214	C₂⁴	C₂⁴×C₁₃₉	1	1	yes	characteristic
215	C₂²×C₄	—	2	48	no	—
216	C₂²×C₄	—	2	24	no	—
217	C₂²×C₄	—	2	48	no	—
218	C₂²×C₄	—	2	24	no	—
219	C₂²×C₄	—	2	12	no	—
220	C₂²×C₄	—	2	12	no	—
221	C₂²×C₄	—	2	24	no	—
222	C₂²×C₄	—	2	48	no	—
223	C₂²×C₄	—	2	12	no	—
224	C₂²×C₄	—	2	6	no	—
225	C₂²×C₄	—	2	6	no	—
226	C₂²×C₄	—	2	12	no	—
227	C₂²×C₄	—	4	24	no	—
228	C₂²×C₄	—	4	12	no	—
229	C₂²×C₄	—	4	12	no	—
230	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	48	yes	complemented
231	C₂²×C₄	C₂²×C₄×C₁₃₉	1	12	yes	—
232	C₂²×C₄	C₂²×C₄×C₁₃₉	1	12	yes	—
233	C₂²×C₄	C₂×D₄×C₁₃₉	1	12	yes	—
234	C₂²×C₄	C₂×D₄×C₁₃₉	1	12	yes	—
235	C₂²⋊C₄	—	2	96	no	—
236	C₄²	—	2	32	no	—
237	C₂³	—	2	12	no	—
238	C₂³	—	2	12	no	—
239	C₂³	—	2	24	no	—
240	C₂³	—	2	24	no	—
241	C₂³	—	2	96	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	12	no	—
244	C₂³	—	4	6	no	—
245	C₂³	—	4	6	no	—
246	C₂³	—	4	4	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	48	no	—
249	C₂³	—	4	96	no	—
250	C₂³	—	4	48	no	—
251	C₂³	C₂×C₄²×C₁₃₉	1	4	yes	—
252	C₂³	C₂×C₄²×C₁₃₉	1	8	yes	complemented
253	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	12	yes	—
254	C₂³	#117×C₁₃₉	1	12	yes	—
255	C₂³	C₄×D₄×C₁₃₉	1	24	yes	—
256	C₂³	C₄×D₄×C₁₃₉	1	24	yes	—
257	C₂³	#120×C₁₃₉	1	4	yes	—
258	C₂³	#121×C₁₃₉	1	24	yes	—
259	C₂³	#123×C₁₃₉	1	12	yes	—
260	C₂³	#124×C₁₃₉	1	12	yes	—
261	C₂³	#126×C₁₃₉	1	8	yes	—
262	C₂³	#127×C₁₃₉	1	4	yes	—
263	C₂³	C₂³×C₄×C₁₃₉	1	3	yes	—
264	C₂³	C₂²×D₄×C₁₃₉	1	3	yes	—
265	C₂³	C₂²×D₄×C₁₃₉	1	3	yes	—
266	C₂³	C₂×#41×C₁₃₉	1	6	yes	—
267	C₂×C₄	—	2	24	no	—
268	C₂×C₄	—	2	12	no	—
269	C₂×C₄	—	2	24	no	—
270	C₂×C₄	—	2	12	no	—
271	C₂×C₄	—	2	96	no	—
272	C₂×C₄	—	2	12	no	—
273	C₂×C₄	—	2	24	no	—
274	C₂×C₄	—	4	12	no	—
275	C₂×C₄	—	4	6	no	—
276	C₂×C₄	—	4	6	no	—
277	C₂×C₄	—	4	48	no	—
278	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	24	yes	complemented
279	C₂²	—	2	24	no	—
280	C₂²	—	4	12	no	—
281	C₂²	—	4	24	no	—
282	C₂²	—	4	12	no	—
283	C₂²	—	4	64	no	—
284	C₂²	C₄×C₂²⋊C₄×C₁₃₉	1	24	yes	—
285	C₂²	#386×C₁₃₉	1	24	yes	—
286	C₂²	#388×C₁₃₉	1	48	yes	—
287	C₂²	#390×C₁₃₉	1	24	yes	—
288	C₂²	C₂²×C₄²×C₁₃₉	1	1	yes	characteristic
289	C₂²	C₂²×C₂²⋊C₄×C₁₃₉	1	3	yes	—
290	C₂²	C₂×#117×C₁₃₉	1	3	yes	—
291	C₂²	C₂×C₄×D₄×C₁₃₉	1	6	yes	—
292	C₂²	C₂×C₄×D₄×C₁₃₉	1	6	yes	—
293	C₂²	C₂×#120×C₁₃₉	1	1	yes	characteristic
294	C₂²	C₂×#121×C₁₃₉	1	6	yes	—
295	C₂²	C₂×#123×C₁₃₉	1	3	yes	—
296	C₂²	C₂×#124×C₁₃₉	1	3	yes	—
297	C₂²	C₂×#126×C₁₃₉	1	2	yes	—
298	C₂²	C₂×#127×C₁₃₉	1	1	yes	characteristic
299	C₄	—	2	24	no	—
300	C₄	—	4	12	no	—
301	C₂	—	4	8	no	—
302	C₂	#1437×C₁₃₉	1	16	yes	direct product factor
303	C₂	C₂×C₄×C₂²⋊C₄×C₁₃₉	1	3	yes	—
304	C₂	C₂×#386×C₁₃₉	1	3	yes	—
305	C₂	C₂×#388×C₁₃₉	1	6	yes	—
306	C₂	C₂×#390×C₁₃₉	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₁₃₉	C₂³
commutator	C₂²	C₂²×C₄²×C₁₃₉
Frattini, cosocle	C₂⁴	C₂⁴×C₁₃₉
prime radical	C₂⁶×C₁₃₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄²×C₁₃₉	C₂	semidirect
3. C₂²×#95×C₁₃₉	C₂	semidirect
4. #1437×C₁₃₉	C₂	direct (N × H)
10. C₂²×C₂²⋊C₄×C₁₃₉	C₄	semidirect
13. C₂²×C₄²×C₁₃₉	C₂²	semidirect
15. C₂×#95×C₁₃₉	C₂²	semidirect
37. C₂×C₂²⋊C₄×C₁₃₉	C₂×C₄	semidirect
54. C₂⁴×C₁₃₉	C₄²	semidirect
76. C₂²×C₄×C₁₃₉	C₂²⋊C₄	semidirect
98. C₂³×C₁₃₉	C₂×C₄²	semidirect
124. C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	semidirect
148. C₂×C₁₃₉	#1437	direct (N × H)
153. C₂×#1437	C₁₃₉	direct (N × H)
154. C₁₃₉	C₂×#1437	direct (N × H)
156. C₂³×C₄²	C₂×C₁₃₉	semidirect
157. C₂²×#95	C₂×C₁₃₉	semidirect
158. #1437	C₂×C₁₃₉	direct (N × H)
164. C₂²×C₂²⋊C₄	C₄×C₁₃₉	semidirect
167. C₂²×C₄²	C₂²×C₁₃₉	semidirect
169. C₂×#95	C₂²×C₁₃₉	semidirect
191. C₂×C₂²⋊C₄	C₂×C₄×C₁₃₉	semidirect
208. C₂⁴	C₄²×C₁₃₉	semidirect
230. C₂²×C₄	C₂²⋊C₄×C₁₃₉	semidirect
252. C₂³	C₂×C₄²×C₁₃₉	semidirect
278. C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	semidirect
302. C₂	#1437×C₁₃₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊C₄×C₁₃₉	2
C₂³×C₄²×C₁₃₉	2
C₂²×#95×C₁₃₉	2
#1437×C₁₃₉	2
C₂×#1437	139

Subgroups — C2×#1437×C139 #48008356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1437×C₁₃₉ #48008356