Subgroups — C₁₃₉×#35626 #48030222

Order 35584.

2526 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#124×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#518×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#2483×C₁₃₉	C₂	1	8	yes	maximal, complemented
4	#2447×C₁₃₉	C₂	1	16	yes	maximal, complemented
5	#2310×C₁₃₉	C₂	1	2	yes	maximal, complemented
6	#2309×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	#2296×C₁₃₉	C₂	1	2	yes	maximal, complemented
8	C₂³×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×#124×C₁₃₉	—	2	8	no	—
11	C₂×#124×C₁₃₉	C₂²	1	8	yes	—
12	#518×C₁₃₉	—	2	8	no	—
13	C₂×C₄×D₄×C₁₃₉	C₂²	1	8	yes	—
14	C₂×#117×C₁₃₉	C₂²	1	2	yes	—
15	C₂×#117×C₁₃₉	C₂²	1	2	yes	—
16	C₂²×C₂²⋊C₄×C₁₃₉	C₂²	1	4	yes	—
17	C₂²×C₂²⋊C₄×C₁₃₉	C₂²	1	4	yes	—
18	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
19	#397×C₁₃₉	C₂²	1	32	yes	complemented
20	#391×C₁₃₉	C₂²	1	8	yes	complemented
21	#391×C₁₃₉	C₂²	1	8	yes	complemented
22	#390×C₁₃₉	C₂²	1	8	yes	—
23	#389×C₁₃₉	C₂²	1	8	yes	complemented
24	#385×C₁₃₉	C₂²	1	8	yes	—
25	#385×C₁₃₉	C₂²	1	16	yes	complemented
26	#381×C₁₃₉	C₂²	1	8	yes	—
27	#381×C₁₃₉	C₂²	1	16	yes	complemented
28	#379×C₁₃₉	C₂²	1	4	yes	—
29	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	8	yes	complemented
30	C₂⁵×C₁₃₉	C₂³	1	2	yes	—
31	C₂²×Q₈×C₁₃₉	—	2	4	no	—
32	C₂²×Q₈×C₁₃₉	—	2	8	no	—
33	C₂²×Q₈×C₁₃₉	C₂³	1	4	yes	—
34	C₂²×D₄×C₁₃₉	—	2	8	no	—
35	C₂²×D₄×C₁₃₉	D₄	1	8	yes	—
36	C₂²×D₄×C₁₃₉	C₂³	1	4	yes	—
37	C₂³×C₄×C₁₃₉	C₂³	1	2	yes	—
38	C₂³×C₄×C₁₃₉	C₂³	1	2	yes	—
39	C₂³×C₄×C₁₃₉	C₂³	1	8	yes	—
40	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
41	#124×C₁₃₉	—	2	32	no	—
42	C₄×D₄×C₁₃₉	—	2	32	no	—
43	#117×C₁₃₉	—	2	8	no	—
44	#117×C₁₃₉	—	2	8	no	—
45	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	16	yes	—
48	C₂×C₂²⋊C₄×C₁₃₉	—	2	16	no	—
49	C₂×C₂²⋊C₄×C₁₃₉	—	2	16	no	—
50	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	8	yes	—
51	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	16	yes	—
52	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	4	yes	—
53	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	4	yes	—
54	C₂×C₂²⋊C₄×C₁₃₉	C₂³	1	16	yes	—
55	C₂×C₄²×C₁₃₉	—	2	4	no	—
56	C₂×C₄²×C₁₃₉	C₂³	1	4	yes	—
57	C₂×C₄²×C₁₃₉	C₂³	1	4	yes	—
58	C₂×C₄²×C₁₃₉	C₂³	1	4	yes	—
59	#95×C₁₃₉	C₂³	1	16	yes	—
60	#95×C₁₃₉	C₂³	1	32	yes	—
61	C₂⁴×C₁₃₉	—	2	8	no	—
62	C₂⁴×C₁₃₉	—	4	8	no	—
63	C₂⁴×C₁₃₉	C₂×D₄	1	8	yes	—
64	C₂⁴×C₁₃₉	C₂⁴	1	4	yes	—
65	C₂⁴×C₁₃₉	C₂⁴	1	1	yes	characteristic
66	C₂×Q₈×C₁₃₉	—	2	8	no	—
67	C₂×Q₈×C₁₃₉	—	2	8	no	—
68	C₂×Q₈×C₁₃₉	—	2	8	no	—
69	C₂×Q₈×C₁₃₉	—	4	16	no	—
70	C₂×D₄×C₁₃₉	—	2	32	no	—
71	C₂×D₄×C₁₃₉	—	2	8	no	—
72	C₂×D₄×C₁₃₉	—	2	8	no	—
73	C₂×D₄×C₁₃₉	C₂×D₄	1	16	yes	—
74	C₂²×C₄×C₁₃₉	—	2	4	no	—
75	C₂²×C₄×C₁₃₉	—	2	4	no	—
76	C₂²×C₄×C₁₃₉	—	2	8	no	—
77	C₂²×C₄×C₁₃₉	—	2	16	no	—
78	C₂²×C₄×C₁₃₉	—	4	4	no	—
79	C₂²×C₄×C₁₃₉	—	4	16	no	—
80	C₂²×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₁₃₉	#41	1	8	yes	—
82	C₂²×C₄×C₁₃₉	C₂⁴	1	4	yes	—
83	C₂²×C₄×C₁₃₉	C₂⁴	1	8	yes	—
84	C₂²×C₄×C₁₃₉	C₂⁴	1	8	yes	—
85	C₂²×C₄×C₁₃₉	C₂⁴	1	4	yes	—
86	C₂²×C₄×C₁₃₉	C₂⁴	1	2	yes	—
87	C₄⋊C₄×C₁₃₉	—	2	8	no	—
88	C₄⋊C₄×C₁₃₉	—	2	8	no	—
89	C₄⋊C₄×C₁₃₉	—	4	16	no	—
90	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
91	C₂²⋊C₄×C₁₃₉	—	2	16	no	—
92	C₂²⋊C₄×C₁₃₉	—	2	32	no	—
93	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
94	C₂²⋊C₄×C₁₃₉	—	4	16	no	—
95	C₄²×C₁₃₉	—	2	8	no	—
96	C₄²×C₁₃₉	—	2	8	no	—
97	C₄²×C₁₃₉	—	2	8	no	—
98	C₂³×C₁₃₉	—	2	4	no	—
99	C₂³×C₁₃₉	—	2	8	no	—
100	C₂³×C₁₃₉	—	2	16	no	—
101	C₂³×C₁₃₉	—	2	16	no	—
102	C₂³×C₁₃₉	—	4	16	no	—
103	C₂³×C₁₃₉	—	8	16	no	—
104	C₂³×C₁₃₉	C₂²×D₄	1	8	yes	—
105	C₂³×C₁₃₉	C₂²×D₄	1	2	yes	—
106	C₂³×C₁₃₉	C₂×#41	1	4	yes	—
107	C₂³×C₁₃₉	C₂⁵	1	1	yes	characteristic
108	Q₈×C₁₃₉	—	4	16	no	—
109	D₄×C₁₃₉	—	2	32	no	—
110	C₂×C₄×C₁₃₉	—	2	4	no	—
111	C₂×C₄×C₁₃₉	—	2	8	no	—
112	C₂×C₄×C₁₃₉	—	2	4	no	—
113	C₂×C₄×C₁₃₉	—	2	8	no	—
114	C₂×C₄×C₁₃₉	—	2	8	no	—
115	C₂×C₄×C₁₃₉	—	2	16	no	—
116	C₂×C₄×C₁₃₉	—	2	8	no	—
117	C₂×C₄×C₁₃₉	—	2	16	no	—
118	C₂×C₄×C₁₃₉	—	4	8	no	—
119	C₂×C₄×C₁₃₉	—	4	8	no	—
120	C₂×C₄×C₁₃₉	—	8	16	no	—
121	C₂×C₄×C₁₃₉	#123	1	16	yes	—
122	C₂×C₄×C₁₃₉	C₂²×D₄	1	4	yes	—
123	C₂×C₄×C₁₃₉	C₂×#41	1	4	yes	—
124	C₂×C₄×C₁₃₉	C₂×#41	1	4	yes	—
125	C₂²×C₁₃₉	—	2	4	no	—
126	C₂²×C₁₃₉	—	2	8	no	—
127	C₂²×C₁₃₉	—	2	8	no	—
128	C₂²×C₁₃₉	—	2	16	no	—
129	C₂²×C₁₃₉	—	4	16	no	—
130	C₂²×C₁₃₉	—	8	16	no	—
131	C₂²×C₁₃₉	C₂×#123	1	4	yes	—
132	C₂²×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
133	C₂²×C₁₃₉	C₂²×#41	1	2	yes	—
134	C₂²×C₁₃₉	#585	1	4	yes	—
135	C₄×C₁₃₉	—	4	4	no	—
136	C₄×C₁₃₉	—	4	4	no	—
137	C₄×C₁₃₉	—	4	8	no	—
138	C₄×C₁₃₉	—	4	8	no	—
139	C₄×C₁₃₉	C₂×#123	1	8	yes	—
140	C₂×C₁₃₉	—	2	4	no	—
141	C₂×C₁₃₉	—	4	8	no	—
142	C₂×C₁₃₉	C₂²×#123	1	1	yes	characteristic
143	C₂×C₁₃₉	#3485	1	2	yes	—
144	C₂×C₁₃₉	#3510	1	4	yes	—
145	#35626	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₁₃₉	#35626	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²×#124	C₂×C₁₃₉	1	1	yes	characteristic
148	C₂×#518	C₂×C₁₃₉	1	1	yes	characteristic
149	#2483	C₂×C₁₃₉	1	8	yes	complemented
150	#2447	C₂×C₁₃₉	1	16	yes	complemented
151	#2310	C₂×C₁₃₉	1	2	yes	complemented
152	#2309	C₂×C₁₃₉	1	1	yes	characteristic
153	#2296	C₂×C₁₃₉	1	2	yes	complemented
154	C₂³×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
155	C₂³×D₄	C₂²×C₁₃₉	1	1	yes	characteristic
156	C₂×#124	—	2	8	no	—
157	C₂×#124	C₂²×C₁₃₉	1	8	yes	—
158	#518	—	2	8	no	—
159	C₂×C₄×D₄	C₂²×C₁₃₉	1	8	yes	—
160	C₂×#117	C₂²×C₁₃₉	1	2	yes	—
161	C₂×#117	C₂²×C₁₃₉	1	2	yes	—
162	C₂²×C₂²⋊C₄	C₂²×C₁₃₉	1	4	yes	—
163	C₂²×C₂²⋊C₄	C₂²×C₁₃₉	1	4	yes	—
164	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
165	#397	C₂²×C₁₃₉	1	32	yes	complemented
166	#391	C₂²×C₁₃₉	1	8	yes	complemented
167	#391	C₂²×C₁₃₉	1	8	yes	complemented
168	#390	C₂²×C₁₃₉	1	8	yes	—
169	#389	C₂²×C₁₃₉	1	8	yes	complemented
170	#385	C₂²×C₁₃₉	1	8	yes	—
171	#385	C₂²×C₁₃₉	1	16	yes	complemented
172	#381	C₂²×C₁₃₉	1	8	yes	—
173	#381	C₂²×C₁₃₉	1	16	yes	complemented
174	#379	C₂²×C₁₃₉	1	4	yes	—
175	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	8	yes	complemented
176	C₂⁵	C₂³×C₁₃₉	1	2	yes	—
177	C₂²×Q₈	—	2	4	no	—
178	C₂²×Q₈	—	2	8	no	—
179	C₂²×Q₈	C₂³×C₁₃₉	1	4	yes	—
180	C₂²×D₄	—	2	8	no	—
181	C₂²×D₄	D₄×C₁₃₉	1	8	yes	—
182	C₂²×D₄	C₂³×C₁₃₉	1	4	yes	—
183	C₂³×C₄	C₂³×C₁₃₉	1	2	yes	—
184	C₂³×C₄	C₂³×C₁₃₉	1	2	yes	—
185	C₂³×C₄	C₂³×C₁₃₉	1	8	yes	—
186	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
187	#124	—	2	32	no	—
188	C₄×D₄	—	2	32	no	—
189	#117	—	2	8	no	—
190	#117	—	2	8	no	—
191	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	4	yes	—
192	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	4	yes	—
193	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	16	yes	—
194	C₂×C₂²⋊C₄	—	2	16	no	—
195	C₂×C₂²⋊C₄	—	2	16	no	—
196	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	4	yes	—
197	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	8	yes	—
198	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	16	yes	—
199	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	4	yes	—
200	C₂×C₂²⋊C₄	C₂³×C₁₃₉	1	16	yes	—
201	C₂×C₄²	—	2	4	no	—
202	C₂×C₄²	C₂³×C₁₃₉	1	4	yes	—
203	C₂×C₄²	C₂³×C₁₃₉	1	4	yes	—
204	C₂×C₄²	C₂³×C₁₃₉	1	4	yes	—
205	#95	C₂³×C₁₃₉	1	16	yes	—
206	#95	C₂³×C₁₃₉	1	32	yes	—
207	C₂⁴	—	2	8	no	—
208	C₂⁴	—	4	8	no	—
209	C₂⁴	C₂×D₄×C₁₃₉	1	8	yes	—
210	C₂⁴	C₂⁴×C₁₃₉	1	4	yes	—
211	C₂⁴	C₂⁴×C₁₃₉	1	1	yes	characteristic
212	C₂×Q₈	—	2	8	no	—
213	C₂×Q₈	—	2	8	no	—
214	C₂×Q₈	—	2	8	no	—
215	C₂×Q₈	—	4	16	no	—
216	C₂×D₄	—	2	32	no	—
217	C₂×D₄	—	2	8	no	—
218	C₂×D₄	—	2	8	no	—
219	C₂×D₄	C₂×D₄×C₁₃₉	1	16	yes	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	16	no	—
224	C₂²×C₄	—	4	4	no	—
225	C₂²×C₄	—	4	16	no	—
226	C₂²×C₄	C₂×D₄×C₁₃₉	1	4	yes	—
227	C₂²×C₄	#41×C₁₃₉	1	8	yes	—
228	C₂²×C₄	C₂⁴×C₁₃₉	1	4	yes	—
229	C₂²×C₄	C₂⁴×C₁₃₉	1	8	yes	—
230	C₂²×C₄	C₂⁴×C₁₃₉	1	8	yes	—
231	C₂²×C₄	C₂⁴×C₁₃₉	1	4	yes	—
232	C₂²×C₄	C₂⁴×C₁₃₉	1	2	yes	—
233	C₄⋊C₄	—	2	8	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	4	16	no	—
236	C₂²⋊C₄	—	2	8	no	—
237	C₂²⋊C₄	—	2	16	no	—
238	C₂²⋊C₄	—	2	32	no	—
239	C₂²⋊C₄	—	2	8	no	—
240	C₂²⋊C₄	—	4	16	no	—
241	C₄²	—	2	8	no	—
242	C₄²	—	2	8	no	—
243	C₄²	—	2	8	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	2	8	no	—
246	C₂³	—	2	16	no	—
247	C₂³	—	2	16	no	—
248	C₂³	—	4	16	no	—
249	C₂³	—	8	16	no	—
250	C₂³	C₂²×D₄×C₁₃₉	1	8	yes	—
251	C₂³	C₂²×D₄×C₁₃₉	1	2	yes	—
252	C₂³	C₂×#41×C₁₃₉	1	4	yes	—
253	C₂³	C₂⁵×C₁₃₉	1	1	yes	characteristic
254	Q₈	—	4	16	no	—
255	D₄	—	2	32	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	8	no	—
261	C₂×C₄	—	2	16	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	16	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	—	8	16	no	—
267	C₂×C₄	#123×C₁₃₉	1	16	yes	—
268	C₂×C₄	C₂²×D₄×C₁₃₉	1	4	yes	—
269	C₂×C₄	C₂×#41×C₁₃₉	1	4	yes	—
270	C₂×C₄	C₂×#41×C₁₃₉	1	4	yes	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	8	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	16	no	—
275	C₂²	—	4	16	no	—
276	C₂²	—	8	16	no	—
277	C₂²	C₂×#123×C₁₃₉	1	4	yes	—
278	C₂²	C₂³×D₄×C₁₃₉	1	1	yes	characteristic
279	C₂²	C₂²×#41×C₁₃₉	1	2	yes	—
280	C₂²	#585×C₁₃₉	1	4	yes	—
281	C₄	—	4	4	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	8	no	—
284	C₄	—	4	8	no	—
285	C₄	C₂×#123×C₁₃₉	1	8	yes	—
286	C₂	—	2	4	no	—
287	C₂	—	4	8	no	—
288	C₂	C₂²×#123×C₁₃₉	1	1	yes	characteristic
289	C₂	#3485×C₁₃₉	1	2	yes	—
290	C₂	#3510×C₁₃₉	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #2483×C₁₃₉	C₂	semidirect
4. #2447×C₁₃₉	C₂	semidirect
5. #2310×C₁₃₉	C₂	semidirect
7. #2296×C₁₃₉	C₂	semidirect
19. #397×C₁₃₉	C₂²	semidirect
20. #391×C₁₃₉	C₂²	semidirect
21. #391×C₁₃₉	C₂²	semidirect
23. #389×C₁₃₉	C₂²	semidirect
25. #385×C₁₃₉	C₂²	semidirect
27. #381×C₁₃₉	C₂²	semidirect
29. C₄×C₄⋊C₄×C₁₃₉	C₂²	semidirect
145. #35626	C₁₃₉	direct (N × H)
146. C₁₃₉	#35626	direct (N × H)
149. #2483	C₂×C₁₃₉	semidirect
150. #2447	C₂×C₁₃₉	semidirect
151. #2310	C₂×C₁₃₉	semidirect
153. #2296	C₂×C₁₃₉	semidirect
165. #397	C₂²×C₁₃₉	semidirect
166. #391	C₂²×C₁₃₉	semidirect
167. #391	C₂²×C₁₃₉	semidirect
169. #389	C₂²×C₁₃₉	semidirect
171. #385	C₂²×C₁₃₉	semidirect
173. #381	C₂²×C₁₃₉	semidirect
175. C₄×C₄⋊C₄	C₂²×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#124×C₁₃₉	2
C₂×#518×C₁₃₉	2
#2483×C₁₃₉	2
#2447×C₁₃₉	2
#2310×C₁₃₉	2
#2309×C₁₃₉	2
#2296×C₁₃₉	2
#35626	139

Subgroups — C139×#35626 #48030222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#35626 #48030222