Subgroups — C₁₃₉×#13123 #48007719

Order 35584.

312 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2153×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#2117×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#350×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#2108×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#1757×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	#1379×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#1378×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂×M₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
13	#413×C₁₃₉	C₄	1	1	yes	characteristic
14	#410×C₁₃₉	C₄	1	1	yes	characteristic
15	#405×C₁₃₉	C₂²	1	1	yes	characteristic
16	#405×C₁₃₉	C₂²	1	1	yes	characteristic
17	#364×C₁₃₉	—	2	2	no	—
18	#350×C₁₃₉	—	2	2	no	—
19	#350×C₁₃₉	—	2	1	no	—
20	#350×C₁₃₉	—	2	1	no	—
21	#346×C₁₃₉	—	2	1	no	—
22	C₄×C₁₆×C₁₃₉	—	2	1	no	—
23	#344×C₁₃₉	C₄	1	1	yes	characteristic
24	#344×C₁₃₉	C₄	1	1	yes	characteristic
25	C₂×#41×C₁₃₉	C₂×C₄	1	1	yes	characteristic
26	#131×C₁₃₉	—	2	1	no	—
27	#131×C₁₃₉	—	2	1	no	—
28	#131×C₁₃₉	—	2	1	no	—
29	#131×C₁₃₉	—	2	1	no	—
30	C₂×M₁₆×C₁₃₉	—	2	1	no	—
31	C₂×M₁₆×C₁₃₉	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₁₃₉	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₁₃₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₉	—	2	1	no	—
35	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
36	#117×C₁₃₉	C₂×C₄	1	1	yes	characteristic
37	#117×C₁₃₉	C₂×C₄	1	1	yes	characteristic
38	M₃₂×C₁₃₉	—	2	1	no	—
39	M₃₂×C₁₃₉	—	2	1	no	—
40	M₃₂×C₁₃₉	—	2	1	no	—
41	M₃₂×C₁₃₉	—	2	1	no	—
42	M₃₂×C₁₃₉	—	2	1	no	—
43	M₃₂×C₁₃₉	D₄	1	2	yes	—
44	C₂×C₁₆×C₁₃₉	—	2	1	no	—
45	C₂×C₁₆×C₁₃₉	—	2	1	no	—
46	C₂×C₁₆×C₁₃₉	—	2	1	no	—
47	C₂×C₁₆×C₁₃₉	—	2	1	no	—
48	C₂×C₁₆×C₁₃₉	—	2	1	no	—
49	C₂×C₁₆×C₁₃₉	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₁₃₉	D₄	1	1	yes	characteristic
51	#101×C₁₃₉	C₈	1	2	yes	—
52	#100×C₁₃₉	C₈	1	2	yes	—
53	#99×C₁₃₉	—	2	2	no	—
54	#97×C₁₃₉	—	2	1	no	—
55	#97×C₁₃₉	—	2	1	no	—
56	C₄×C₈×C₁₃₉	—	2	1	no	—
57	C₄×C₈×C₁₃₉	—	2	1	no	—
58	#41×C₁₃₉	—	2	1	no	—
59	#41×C₁₃₉	—	2	1	no	—
60	#41×C₁₃₉	—	2	1	no	—
61	#41×C₁₃₉	—	2	1	no	—
62	C₂×Q₈×C₁₃₉	C₂×C₈	1	1	yes	characteristic
63	C₂×D₄×C₁₃₉	—	2	1	no	—
64	C₂×D₄×C₁₃₉	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃₉	—	2	1	no	—
66	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
67	M₁₆×C₁₃₉	—	2	1	no	—
68	M₁₆×C₁₃₉	—	2	1	no	—
69	M₁₆×C₁₃₉	—	2	1	no	—
70	M₁₆×C₁₃₉	—	2	1	no	—
71	M₁₆×C₁₃₉	—	4	1	no	—
72	M₁₆×C₁₃₉	—	4	1	no	—
73	M₁₆×C₁₃₉	C₂×C₈	1	2	yes	—
74	M₁₆×C₁₃₉	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₁₃₉	—	2	1	no	—
76	C₂×C₈×C₁₃₉	—	2	1	no	—
77	C₂×C₈×C₁₃₉	—	2	1	no	—
78	C₂×C₈×C₁₃₉	—	4	1	no	—
79	C₂×C₈×C₁₃₉	—	4	1	no	—
80	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₃₉	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₃₉	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₃₉	#41	1	1	yes	characteristic
86	C₄⋊C₄×C₁₃₉	—	2	1	no	—
87	C₄⋊C₄×C₁₃₉	—	2	1	no	—
88	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
89	C₂²⋊C₄×C₁₃₉	M₁₆	1	2	yes	—
90	C₄²×C₁₃₉	—	2	1	no	—
91	C₄²×C₁₃₉	—	2	1	no	—
92	C₁₆×C₁₃₉	—	2	2	no	—
93	C₁₆×C₁₃₉	—	2	1	no	—
94	C₁₆×C₁₃₉	—	2	1	no	—
95	C₁₆×C₁₃₉	—	4	1	no	—
96	C₁₆×C₁₃₉	—	4	1	no	—
97	C₂³×C₁₃₉	—	2	1	no	—
98	C₂³×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
99	Q₈×C₁₃₉	—	2	1	no	—
100	Q₈×C₁₃₉	—	2	1	no	—
101	D₄×C₁₃₉	—	2	1	no	—
102	D₄×C₁₃₉	—	2	1	no	—
103	D₄×C₁₃₉	—	4	1	no	—
104	D₄×C₁₃₉	—	4	1	no	—
105	C₂×C₄×C₁₃₉	—	2	1	no	—
106	C₂×C₄×C₁₃₉	—	2	2	no	—
107	C₂×C₄×C₁₃₉	—	4	1	no	—
108	C₂×C₄×C₁₃₉	—	4	1	no	—
109	C₂×C₄×C₁₃₉	—	4	1	no	—
110	C₂×C₄×C₁₃₉	#117	1	1	yes	characteristic
111	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃₉	#131	1	1	yes	characteristic
115	C₂×C₄×C₁₃₉	#131	1	1	yes	characteristic
116	C₈×C₁₃₉	—	2	2	no	—
117	C₈×C₁₃₉	—	2	2	no	—
118	C₈×C₁₃₉	—	4	1	no	—
119	C₈×C₁₃₉	#121	1	1	yes	characteristic
120	C₈×C₁₃₉	#123	1	1	yes	characteristic
121	C₈×C₁₃₉	#124	1	1	yes	characteristic
122	C₈×C₁₃₉	#126	1	1	yes	characteristic
123	C₂²×C₁₃₉	—	4	1	no	—
124	C₂²×C₁₃₉	—	4	1	no	—
125	C₂²×C₁₃₉	—	8	1	no	—
126	C₂²×C₁₃₉	#431	1	1	yes	characteristic
127	C₂²×C₁₃₉	#435	1	1	yes	characteristic
128	C₂²×C₁₃₉	#436	1	1	yes	characteristic
129	C₄×C₁₃₉	—	4	2	no	—
130	C₄×C₁₃₉	—	4	1	no	—
131	C₄×C₁₃₉	—	8	1	no	—
132	C₄×C₁₃₉	#388	1	1	yes	characteristic
133	C₄×C₁₃₉	#433	1	1	yes	characteristic
134	C₄×C₁₃₉	C₈×D₄	1	1	yes	characteristic
135	C₄×C₁₃₉	C₈×D₄	1	1	yes	characteristic
136	C₂×C₁₃₉	—	2	1	no	—
137	C₂×C₁₃₉	—	2	1	no	—
138	C₂×C₁₃₉	—	2	1	no	—
139	C₂×C₁₃₉	—	8	1	no	—
140	C₂×C₁₃₉	#1923	1	1	yes	characteristic
141	#13123	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₁₃₉	#13123	1	1	yes	characteristic, direct product factor, Sylow
143	#2153	C₂×C₁₃₉	1	1	yes	characteristic, complemented
144	#2117	C₂×C₁₃₉	1	1	yes	characteristic
145	C₂×#350	C₂×C₁₃₉	1	1	yes	characteristic
146	#2108	C₂×C₁₃₉	1	1	yes	characteristic, complemented
147	#1757	C₂×C₁₃₉	1	1	yes	characteristic
148	#1379	C₂×C₁₃₉	1	1	yes	characteristic, complemented
149	#1378	C₂×C₁₃₉	1	1	yes	characteristic, complemented
150	C₂×#131	C₂²×C₁₃₉	1	1	yes	characteristic
151	C₂×M₃₂	C₂²×C₁₃₉	1	1	yes	characteristic
152	C₂×M₃₂	C₂²×C₁₃₉	1	1	yes	characteristic
153	C₂×M₃₂	C₂²×C₁₃₉	1	1	yes	characteristic
154	C₂²×C₁₆	C₂²×C₁₃₉	1	1	yes	characteristic
155	#413	C₄×C₁₃₉	1	1	yes	characteristic
156	#410	C₄×C₁₃₉	1	1	yes	characteristic
157	#405	C₂²×C₁₃₉	1	1	yes	characteristic
158	#405	C₂²×C₁₃₉	1	1	yes	characteristic
159	#364	—	2	2	no	—
160	#350	—	2	2	no	—
161	#350	—	2	1	no	—
162	#350	—	2	1	no	—
163	#346	—	2	1	no	—
164	C₄×C₁₆	—	2	1	no	—
165	#344	C₄×C₁₃₉	1	1	yes	characteristic
166	#344	C₄×C₁₃₉	1	1	yes	characteristic
167	C₂×#41	C₂×C₄×C₁₃₉	1	1	yes	characteristic
168	#131	—	2	1	no	—
169	#131	—	2	1	no	—
170	#131	—	2	1	no	—
171	#131	—	2	1	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	C₂×C₄×C₁₃₉	1	1	yes	characteristic
174	C₂×M₁₆	C₂×C₄×C₁₃₉	1	1	yes	characteristic
175	C₂×M₁₆	C₂×C₄×C₁₃₉	1	1	yes	characteristic
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
178	#117	C₂×C₄×C₁₃₉	1	1	yes	characteristic
179	#117	C₂×C₄×C₁₃₉	1	1	yes	characteristic
180	M₃₂	—	2	1	no	—
181	M₃₂	—	2	1	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	2	1	no	—
184	M₃₂	—	2	1	no	—
185	M₃₂	D₄×C₁₃₉	1	2	yes	—
186	C₂×C₁₆	—	2	1	no	—
187	C₂×C₁₆	—	2	1	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	D₄×C₁₃₉	1	1	yes	characteristic
192	C₂×C₁₆	D₄×C₁₃₉	1	1	yes	characteristic
193	#101	C₈×C₁₃₉	1	2	yes	—
194	#100	C₈×C₁₃₉	1	2	yes	—
195	#99	—	2	2	no	—
196	#97	—	2	1	no	—
197	#97	—	2	1	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	2	1	no	—
200	#41	—	2	1	no	—
201	#41	—	2	1	no	—
202	#41	—	2	1	no	—
203	#41	—	2	1	no	—
204	C₂×Q₈	C₂×C₈×C₁₃₉	1	1	yes	characteristic
205	C₂×D₄	—	2	1	no	—
206	C₂×D₄	C₂×C₈×C₁₃₉	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
209	M₁₆	—	2	1	no	—
210	M₁₆	—	2	1	no	—
211	M₁₆	—	2	1	no	—
212	M₁₆	—	2	1	no	—
213	M₁₆	—	4	1	no	—
214	M₁₆	—	4	1	no	—
215	M₁₆	C₂×C₈×C₁₃₉	1	2	yes	—
216	M₁₆	C₂×C₈×C₁₃₉	1	2	yes	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
224	C₂×C₈	#41×C₁₃₉	1	1	yes	characteristic
225	C₂×C₈	#41×C₁₃₉	1	1	yes	characteristic
226	C₂×C₈	#41×C₁₃₉	1	1	yes	characteristic
227	C₂×C₈	#41×C₁₃₉	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	2	1	no	—
231	C₂²⋊C₄	M₁₆×C₁₃₉	1	2	yes	—
232	C₄²	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₁₆	—	2	2	no	—
235	C₁₆	—	2	1	no	—
236	C₁₆	—	2	1	no	—
237	C₁₆	—	4	1	no	—
238	C₁₆	—	4	1	no	—
239	C₂³	—	2	1	no	—
240	C₂³	C₂×M₁₆×C₁₃₉	1	1	yes	characteristic
241	Q₈	—	2	1	no	—
242	Q₈	—	2	1	no	—
243	D₄	—	2	1	no	—
244	D₄	—	2	1	no	—
245	D₄	—	4	1	no	—
246	D₄	—	4	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	#117×C₁₃₉	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
254	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₈×C₁₃₉	1	1	yes	characteristic
256	C₂×C₄	#131×C₁₃₉	1	1	yes	characteristic
257	C₂×C₄	#131×C₁₃₉	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	2	no	—
260	C₈	—	4	1	no	—
261	C₈	#121×C₁₃₉	1	1	yes	characteristic
262	C₈	#123×C₁₃₉	1	1	yes	characteristic
263	C₈	#124×C₁₃₉	1	1	yes	characteristic
264	C₈	#126×C₁₃₉	1	1	yes	characteristic
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	8	1	no	—
268	C₂²	#431×C₁₃₉	1	1	yes	characteristic
269	C₂²	#435×C₁₃₉	1	1	yes	characteristic
270	C₂²	#436×C₁₃₉	1	1	yes	characteristic
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	8	1	no	—
274	C₄	#388×C₁₃₉	1	1	yes	characteristic
275	C₄	#433×C₁₃₉	1	1	yes	characteristic
276	C₄	C₈×D₄×C₁₃₉	1	1	yes	characteristic
277	C₄	C₈×D₄×C₁₃₉	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	8	1	no	—
282	C₂	#1923×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃₉	#121
commutator	C₂×C₄	C₂²×C₈×C₁₃₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₉
socle	C₂×C₁₃₉	#1923
prime radical	C₂×#41×C₁₃₉	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃₉ ◁ C₂²×C₈×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #2153×C₁₃₉	C₂	semidirect
4. #2108×C₁₃₉	C₂	semidirect
6. #1379×C₁₃₉	C₂	semidirect
7. #1378×C₁₃₉	C₂	semidirect
141. #13123	C₁₃₉	direct (N × H)
142. C₁₃₉	#13123	direct (N × H)
143. #2153	C₂×C₁₃₉	semidirect
146. #2108	C₂×C₁₃₉	semidirect
148. #1379	C₂×C₁₃₉	semidirect
149. #1378	C₂×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
#2153×C₁₃₉	2
#2117×C₁₃₉	2
C₂×#350×C₁₃₉	2
#2108×C₁₃₉	2
#1757×C₁₃₉	2
#1379×C₁₃₉	2
#1378×C₁₃₉	2
#13123	139

Subgroups — C139×#13123 #48007719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#13123 #48007719