Subgroups — C₁₃₉×#9659 #48004255

Order 35584.

590 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	C₂×C₄×D₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#1954×C₁₃₉	C₂	1	2	yes	maximal, complemented
3	C₂×#334×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊C₈×C₁₃₉	C₂	1	2	yes	maximal, complemented
5	C₂×#322×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₁₃₉	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃₉	C₂²	1	2	yes	—
8	C₄×D₈×C₁₃₉	—	2	2	no	—
9	C₄×D₈×C₁₃₉	—	2	2	no	—
10	C₂×#107×C₁₃₉	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
12	C₂×#102×C₁₃₉	C₄	1	2	yes	—
13	C₂×C₂²⋊C₈×C₁₃₉	—	2	4	no	—
14	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	#334×C₁₃₉	—	2	2	no	—
18	D₄⋊C₈×C₁₃₉	—	2	4	no	—
19	#322×C₁₃₉	—	2	2	no	—
20	C₂²×D₄×C₁₃₉	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₃₉	—	2	2	no	—
22	C₂×D₈×C₁₃₉	—	2	2	no	—
23	C₂×D₈×C₁₃₉	—	2	2	no	—
24	C₂×D₈×C₁₃₉	C₈	1	4	yes	complemented
25	C₂²×C₈×C₁₃₉	—	2	1	no	—
26	C₂²×C₈×C₁₃₉	—	2	2	no	—
27	C₂²×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₉	D₄	1	2	yes	—
30	C₄×D₄×C₁₃₉	—	2	2	no	—
31	C₄×D₄×C₁₃₉	—	2	2	no	—
32	C₄×D₄×C₁₃₉	—	2	2	no	—
33	C₄×D₄×C₁₃₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₁₃₉	—	2	2	no	—
36	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
37	#107×C₁₃₉	—	2	2	no	—
38	C₄⋊C₈×C₁₃₉	—	2	2	no	—
39	C₄⋊C₈×C₁₃₉	—	2	2	no	—
40	#102×C₁₃₉	—	2	4	no	—
41	C₂²⋊C₈×C₁₃₉	—	4	8	no	—
42	C₄×C₈×C₁₃₉	—	2	1	no	—
43	C₄×C₈×C₁₃₉	—	2	1	no	—
44	C₄×C₈×C₁₃₉	—	2	1	no	—
45	C₄×C₈×C₁₃₉	D₄	1	2	yes	—
46	C₄×C₈×C₁₃₉	D₄	1	2	yes	—
47	C₄×C₈×C₁₃₉	D₄	1	2	yes	—
48	C₂⁴×C₁₃₉	—	2	2	no	—
49	C₂×D₄×C₁₃₉	—	2	2	no	—
50	C₂×D₄×C₁₃₉	—	2	2	no	—
51	C₂×D₄×C₁₃₉	—	2	2	no	—
52	C₂×D₄×C₁₃₉	—	2	2	no	—
53	C₂×D₄×C₁₃₉	—	2	2	no	—
54	C₂×D₄×C₁₃₉	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₁₃₉	—	2	2	no	—
56	C₂²×C₄×C₁₃₉	—	4	2	no	—
57	C₂²×C₄×C₁₃₉	—	4	2	no	—
58	C₂²×C₄×C₁₃₉	—	4	4	no	—
59	C₂²×C₄×C₁₃₉	—	4	4	no	—
60	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₉	#41	1	1	yes	characteristic
63	D₈×C₁₃₉	—	2	8	no	—
64	C₂×C₈×C₁₃₉	—	2	1	no	—
65	C₂×C₈×C₁₃₉	—	2	1	no	—
66	C₂×C₈×C₁₃₉	—	2	2	no	—
67	C₂×C₈×C₁₃₉	—	2	2	no	—
68	C₂×C₈×C₁₃₉	—	2	4	no	—
69	C₂×C₈×C₁₃₉	—	4	2	no	—
70	C₂×C₈×C₁₃₉	—	4	2	no	—
71	C₂×C₈×C₁₃₉	—	4	4	no	—
72	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₃₉	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₃₉	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₁₃₉	M₁₆	1	2	yes	—
76	C₄⋊C₄×C₁₃₉	—	2	2	no	—
77	C₄⋊C₄×C₁₃₉	—	2	2	no	—
78	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
79	C₂²⋊C₄×C₁₃₉	—	4	2	no	—
80	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃₉	#41	1	1	yes	characteristic
84	C₂³×C₁₃₉	—	2	4	no	—
85	C₂³×C₁₃₉	—	4	2	no	—
86	C₂³×C₁₃₉	—	4	2	no	—
87	C₂³×C₁₃₉	—	4	4	no	—
88	C₂³×C₁₃₉	—	4	4	no	—
89	C₂³×C₁₃₉	C₄×D₄	1	1	yes	characteristic
90	D₄×C₁₃₉	—	2	4	no	—
91	D₄×C₁₃₉	—	2	4	no	—
92	D₄×C₁₃₉	—	4	2	no	—
93	D₄×C₁₃₉	—	4	2	no	—
94	C₂×C₄×C₁₃₉	—	2	2	no	—
95	C₂×C₄×C₁₃₉	—	4	2	no	—
96	C₂×C₄×C₁₃₉	—	4	4	no	—
97	C₂×C₄×C₁₃₉	—	8	4	no	—
98	C₂×C₄×C₁₃₉	—	8	2	no	—
99	C₂×C₄×C₁₃₉	—	8	2	no	—
100	C₂×C₄×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
103	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
104	C₂×C₄×C₁₃₉	#124	1	1	yes	characteristic
105	C₂×C₄×C₁₃₉	#127	1	1	yes	characteristic
106	C₂×C₄×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₃₉	#131	1	1	yes	characteristic
109	C₂×C₄×C₁₃₉	#131	1	1	yes	characteristic
110	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	4	yes	—
111	C₈×C₁₃₉	—	2	2	no	—
112	C₈×C₁₃₉	—	4	2	no	—
113	C₈×C₁₃₉	—	4	2	no	—
114	C₈×C₁₃₉	—	8	2	no	—
115	C₈×C₁₃₉	C₂²⋊C₈	1	4	yes	complemented
116	C₂²×C₁₃₉	—	4	8	no	—
117	C₂²×C₁₃₉	—	4	4	no	—
118	C₂²×C₁₃₉	—	8	4	no	—
119	C₂²×C₁₃₉	—	8	2	no	—
120	C₂²×C₁₃₉	—	8	2	no	—
121	C₂²×C₁₃₉	#390	1	1	yes	characteristic
122	C₂²×C₁₃₉	C₈×D₄	1	1	yes	characteristic
123	C₂²×C₁₃₉	#436	1	1	yes	characteristic
124	C₂²×C₁₃₉	#442	1	2	yes	—
125	C₂²×C₁₃₉	#444	1	2	yes	—
126	C₄×C₁₃₉	—	2	1	no	—
127	C₄×C₁₃₉	—	2	1	no	—
128	C₄×C₁₃₉	—	8	2	no	—
129	C₄×C₁₃₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₄×C₁₃₉	#408	1	1	yes	characteristic
131	C₄×C₁₃₉	C₈×D₄	1	1	yes	characteristic
132	C₄×C₁₃₉	#436	1	1	yes	characteristic
133	C₄×C₁₃₉	#469	1	2	yes	—
134	C₄×C₁₃₉	#496	1	2	yes	—
135	C₂×C₁₃₉	—	8	4	no	—
136	C₂×C₁₃₉	#1580	1	2	yes	—
137	C₂×C₁₃₉	#1594	1	2	yes	—
138	C₂×C₁₃₉	#1954	1	1	yes	characteristic
139	C₂×C₁₃₉	#1972	1	1	yes	characteristic
140	C₂×C₁₃₉	#1975	1	1	yes	characteristic
141	#9659	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₁₃₉	#9659	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×C₄×D₈	C₂×C₁₃₉	1	1	yes	characteristic
144	#1954	C₂×C₁₃₉	1	2	yes	complemented
145	C₂×#334	C₂×C₁₃₉	1	1	yes	characteristic, complemented
146	C₂×D₄⋊C₈	C₂×C₁₃₉	1	2	yes	complemented
147	C₂×#322	C₂×C₁₃₉	1	1	yes	characteristic, complemented
148	C₂²×D₈	C₄×C₁₃₉	1	1	yes	characteristic
149	C₂×C₄×D₄	C₂²×C₁₃₉	1	2	yes	—
150	C₄×D₈	—	2	2	no	—
151	C₄×D₈	—	2	2	no	—
152	C₂×#107	C₄×C₁₃₉	1	1	yes	characteristic
153	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
154	C₂×#102	C₄×C₁₃₉	1	2	yes	—
155	C₂×C₂²⋊C₈	—	2	4	no	—
156	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
159	#334	—	2	2	no	—
160	D₄⋊C₈	—	2	4	no	—
161	#322	—	2	2	no	—
162	C₂²×D₄	C₂×C₄×C₁₃₉	1	2	yes	—
163	C₂³×C₄	—	2	2	no	—
164	C₂×D₈	—	2	2	no	—
165	C₂×D₈	—	2	2	no	—
166	C₂×D₈	C₈×C₁₃₉	1	4	yes	complemented
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
170	C₂²×C₈	C₂×C₄×C₁₃₉	1	1	yes	characteristic
171	C₂²×C₈	D₄×C₁₃₉	1	2	yes	—
172	C₄×D₄	—	2	2	no	—
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	—	2	2	no	—
175	C₄×D₄	—	2	2	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
177	C₂×C₂²⋊C₄	—	2	2	no	—
178	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
179	#107	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	#102	—	2	4	no	—
183	C₂²⋊C₈	—	4	8	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	D₄×C₁₃₉	1	2	yes	—
188	C₄×C₈	D₄×C₁₃₉	1	2	yes	—
189	C₄×C₈	D₄×C₁₃₉	1	2	yes	—
190	C₂⁴	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	C₂×C₈×C₁₃₉	1	4	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
204	C₂²×C₄	#41×C₁₃₉	1	1	yes	characteristic
205	D₈	—	2	8	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	2	yes	—
215	C₂×C₈	C₂²⋊C₄×C₁₃₉	1	2	yes	—
216	C₂×C₈	C₂×C₈×C₁₃₉	1	2	yes	—
217	C₂×C₈	M₁₆×C₁₃₉	1	2	yes	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₂²⋊C₄	—	4	2	no	—
222	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
224	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
225	C₄²	#41×C₁₃₉	1	1	yes	characteristic
226	C₂³	—	2	4	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	4	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	2	no	—
231	C₂³	C₄×D₄×C₁₃₉	1	1	yes	characteristic
232	D₄	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	4	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
243	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
244	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
245	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
246	C₂×C₄	#124×C₁₃₉	1	1	yes	characteristic
247	C₂×C₄	#127×C₁₃₉	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₈×C₁₃₉	1	1	yes	characteristic
249	C₂×C₄	C₂×M₁₆×C₁₃₉	1	1	yes	characteristic
250	C₂×C₄	#131×C₁₃₉	1	1	yes	characteristic
251	C₂×C₄	#131×C₁₃₉	1	1	yes	characteristic
252	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	4	yes	—
253	C₈	—	2	2	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	8	2	no	—
257	C₈	C₂²⋊C₈×C₁₃₉	1	4	yes	complemented
258	C₂²	—	4	8	no	—
259	C₂²	—	4	4	no	—
260	C₂²	—	8	4	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	2	no	—
263	C₂²	#390×C₁₃₉	1	1	yes	characteristic
264	C₂²	C₈×D₄×C₁₃₉	1	1	yes	characteristic
265	C₂²	#436×C₁₃₉	1	1	yes	characteristic
266	C₂²	#442×C₁₃₉	1	2	yes	—
267	C₂²	#444×C₁₃₉	1	2	yes	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	2	no	—
271	C₄	C₂×C₂²⋊C₈×C₁₃₉	1	1	yes	characteristic
272	C₄	#408×C₁₃₉	1	1	yes	characteristic
273	C₄	C₈×D₄×C₁₃₉	1	1	yes	characteristic
274	C₄	#436×C₁₃₉	1	1	yes	characteristic
275	C₄	#469×C₁₃₉	1	2	yes	—
276	C₄	#496×C₁₃₉	1	2	yes	—
277	C₂	—	8	4	no	—
278	C₂	#1580×C₁₃₉	1	2	yes	—
279	C₂	#1594×C₁₃₉	1	2	yes	—
280	C₂	#1954×C₁₃₉	1	1	yes	characteristic
281	C₂	#1972×C₁₃₉	1	1	yes	characteristic
282	C₂	#1975×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
socle	C₂³×C₁₃₉	C₄×D₄
prime radical	C₂²×D₈×C₁₃₉	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₉ ◁ C₂×C₄×C₈×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #1954×C₁₃₉	C₂	semidirect
3. C₂×#334×C₁₃₉	C₂	semidirect
4. C₂×D₄⋊C₈×C₁₃₉	C₂	semidirect
5. C₂×#322×C₁₃₉	C₂	semidirect
24. C₂×D₈×C₁₃₉	C₈	semidirect
115. C₈×C₁₃₉	C₂²⋊C₈	semidirect
141. #9659	C₁₃₉	direct (N × H)
142. C₁₃₉	#9659	direct (N × H)
144. #1954	C₂×C₁₃₉	semidirect
145. C₂×#334	C₂×C₁₃₉	semidirect
146. C₂×D₄⋊C₈	C₂×C₁₃₉	semidirect
147. C₂×#322	C₂×C₁₃₉	semidirect
166. C₂×D₈	C₈×C₁₃₉	semidirect
257. C₈	C₂²⋊C₈×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₁₃₉	2
#1954×C₁₃₉	2
C₂×#334×C₁₃₉	2
C₂×D₄⋊C₈×C₁₃₉	2
C₂×#322×C₁₃₉	2
#9659	139

Subgroups — C139×#9659 #48004255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#9659 #48004255