Subgroups — C₁₃₉×#12956 #48007552

Order 35584.

460 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#442×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#365×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#361×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#2140×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	#1946×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#1386×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#1375×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₁₃₉	—	2	1	no	—
10	C₂×M₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂×M₃₂×C₁₃₉	C₂²	1	1	yes	characteristic
12	#442×C₁₃₉	—	2	2	no	—
13	#442×C₁₃₉	—	2	2	no	—
14	C₂×#108×C₁₃₉	C₂²	1	1	yes	characteristic
15	#427×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂×#106×C₁₃₉	C₂²	1	1	yes	characteristic
17	#424×C₁₃₉	—	2	1	no	—
18	C₂×#102×C₁₃₉	—	2	1	no	—
19	C₂×#97×C₁₃₉	C₂²	1	1	yes	characteristic
20	#365×C₁₃₉	—	2	2	no	—
21	#361×C₁₃₉	—	2	2	no	—
22	#357×C₁₃₉	C₄	1	4	yes	complemented
23	#339×C₁₃₉	—	2	1	no	—
24	C₂²×D₄×C₁₃₉	—	2	1	no	—
25	C₂³×C₄×C₁₃₉	—	4	1	no	—
26	C₂×D₈×C₁₃₉	—	2	1	no	—
27	C₂×D₈×C₁₃₉	—	2	1	no	—
28	C₂×D₈×C₁₃₉	—	2	1	no	—
29	C₂×D₈×C₁₃₉	C₂×C₄	1	2	yes	—
30	C₂×D₈×C₁₃₉	D₄	1	2	yes	—
31	C₂×D₈×C₁₃₉	Q₈	1	2	yes	—
32	C₂×M₁₆×C₁₃₉	—	2	1	no	—
33	C₂²×C₈×C₁₃₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
35	C₄×D₄×C₁₃₉	—	2	4	no	—
36	C₄×D₄×C₁₃₉	—	4	2	no	—
37	#117×C₁₃₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₁₃₉	—	4	1	no	—
40	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
41	M₃₂×C₁₃₉	—	2	1	no	—
42	M₃₂×C₁₃₉	—	2	1	no	—
43	M₃₂×C₁₃₉	—	2	1	no	—
44	M₃₂×C₁₃₉	—	2	1	no	—
45	C₂×C₁₆×C₁₃₉	—	2	1	no	—
46	C₂×C₁₆×C₁₃₉	C₂×C₄	1	2	yes	complemented
47	#108×C₁₃₉	—	2	1	no	—
48	#108×C₁₃₉	—	2	1	no	—
49	#107×C₁₃₉	C₂×C₄	1	2	yes	complemented
50	#106×C₁₃₉	—	2	1	no	—
51	#106×C₁₃₉	—	2	1	no	—
52	#106×C₁₃₉	D₄	1	2	yes	—
53	C₄⋊C₈×C₁₃₉	—	4	2	no	—
54	#102×C₁₃₉	—	2	4	no	—
55	#97×C₁₃₉	—	2	1	no	—
56	#97×C₁₃₉	—	2	1	no	—
57	C₂⁴×C₁₃₉	—	4	1	no	—
58	C₂×D₄×C₁₃₉	—	2	2	no	—
59	C₂×D₄×C₁₃₉	—	2	2	no	—
60	C₂×D₄×C₁₃₉	—	2	2	no	—
61	C₂×D₄×C₁₃₉	—	4	1	no	—
62	C₂×D₄×C₁₃₉	—	4	1	no	—
63	C₂×D₄×C₁₃₉	—	4	1	no	—
64	C₂²×C₄×C₁₃₉	—	2	1	no	—
65	C₂²×C₄×C₁₃₉	—	4	4	no	—
66	C₂²×C₄×C₁₃₉	—	4	1	no	—
67	C₂²×C₄×C₁₃₉	—	8	4	no	—
68	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
69	D₈×C₁₃₉	—	2	2	no	—
70	D₈×C₁₃₉	—	2	2	no	—
71	D₈×C₁₃₉	—	2	2	no	—
72	D₈×C₁₃₉	C₄⋊C₄	1	4	yes	—
73	M₁₆×C₁₃₉	—	4	1	no	—
74	M₁₆×C₁₃₉	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₈×C₁₃₉	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₃₉	#41	1	1	yes	characteristic
85	C₄⋊C₄×C₁₃₉	—	2	2	no	—
86	C₄⋊C₄×C₁₃₉	—	2	2	no	—
87	C₄⋊C₄×C₁₃₉	—	4	1	no	—
88	C₂²⋊C₄×C₁₃₉	—	4	4	no	—
89	C₂²⋊C₄×C₁₃₉	—	4	1	no	—
90	C₄²×C₁₃₉	—	2	1	no	—
91	C₄²×C₁₃₉	—	4	1	no	—
92	C₄²×C₁₃₉	SD₁₆	1	2	yes	—
93	C₁₆×C₁₃₉	—	2	2	no	—
94	C₁₆×C₁₃₉	—	4	1	no	—
95	C₂³×C₁₃₉	—	4	2	no	—
96	C₂³×C₁₃₉	—	4	2	no	—
97	C₂³×C₁₃₉	—	4	2	no	—
98	C₂³×C₁₃₉	—	8	2	no	—
99	C₂³×C₁₃₉	—	8	2	no	—
100	C₂³×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
101	D₄×C₁₃₉	—	2	4	no	—
102	D₄×C₁₃₉	—	4	2	no	—
103	D₄×C₁₃₉	—	4	2	no	—
104	D₄×C₁₃₉	—	4	2	no	—
105	C₂×C₄×C₁₃₉	—	2	2	no	—
106	C₂×C₄×C₁₃₉	—	4	2	no	—
107	C₂×C₄×C₁₃₉	—	4	2	no	—
108	C₂×C₄×C₁₃₉	—	4	2	no	—
109	C₂×C₄×C₁₃₉	—	8	8	no	—
110	C₂×C₄×C₁₃₉	—	8	2	no	—
111	C₂×C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₉	#120	1	1	yes	characteristic
113	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃₉	C₂×SD₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₃₉	#135	1	1	yes	characteristic
116	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₈×C₁₃₉	—	4	1	no	—
118	C₈×C₁₃₉	—	8	1	no	—
119	C₈×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₈×C₁₃₉	C₄×D₄	1	1	yes	characteristic
121	C₈×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₈×C₁₃₉	#123	1	1	yes	characteristic
123	C₂²×C₁₃₉	—	2	2	no	—
124	C₂²×C₁₃₉	—	4	4	no	—
125	C₂²×C₁₃₉	—	8	2	no	—
126	C₂²×C₁₃₉	—	8	2	no	—
127	C₂²×C₁₃₉	—	8	4	no	—
128	C₂²×C₁₃₉	—	8	4	no	—
129	C₂²×C₁₃₉	#442	1	1	yes	characteristic
130	C₂²×C₁₃₉	#450	1	1	yes	characteristic
131	C₂²×C₁₃₉	#478	1	1	yes	characteristic
132	C₄×C₁₃₉	—	4	2	no	—
133	C₄×C₁₃₉	—	8	2	no	—
134	C₄×C₁₃₉	—	8	2	no	—
135	C₄×C₁₃₉	#385	1	1	yes	characteristic
136	C₄×C₁₃₉	C₄×SD₁₆	1	1	yes	characteristic
137	C₄×C₁₃₉	#449	1	1	yes	characteristic
138	C₄×C₁₃₉	#476	1	1	yes	characteristic
139	C₂×C₁₃₉	—	2	1	no	—
140	C₂×C₁₃₉	—	2	1	no	—
141	C₂×C₁₃₉	—	8	4	no	—
142	C₂×C₁₃₉	#1864	1	1	yes	characteristic
143	C₂×C₁₃₉	AGL(1,ℤ/16)	1	2	yes	—
144	#12956	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₁₃₉	#12956	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×#442	C₂×C₁₃₉	1	1	yes	characteristic
147	C₂×#365	C₂×C₁₃₉	1	1	yes	characteristic, complemented
148	C₂×#361	C₂×C₁₃₉	1	1	yes	characteristic
149	#2140	C₂×C₁₃₉	1	1	yes	characteristic
150	#1946	C₂×C₁₃₉	1	1	yes	characteristic, complemented
151	#1386	C₂×C₁₃₉	1	1	yes	characteristic, complemented
152	#1375	C₂×C₁₃₉	1	1	yes	characteristic, complemented
153	C₂²×D₈	C₂²×C₁₃₉	1	1	yes	characteristic
154	C₂×C₄×D₄	—	2	1	no	—
155	C₂×M₃₂	C₂²×C₁₃₉	1	1	yes	characteristic
156	C₂×M₃₂	C₂²×C₁₃₉	1	1	yes	characteristic
157	#442	—	2	2	no	—
158	#442	—	2	2	no	—
159	C₂×#108	C₂²×C₁₃₉	1	1	yes	characteristic
160	#427	C₂²×C₁₃₉	1	1	yes	characteristic
161	C₂×#106	C₂²×C₁₃₉	1	1	yes	characteristic
162	#424	—	2	1	no	—
163	C₂×#102	—	2	1	no	—
164	C₂×#97	C₂²×C₁₃₉	1	1	yes	characteristic
165	#365	—	2	2	no	—
166	#361	—	2	2	no	—
167	#357	C₄×C₁₃₉	1	4	yes	complemented
168	#339	—	2	1	no	—
169	C₂²×D₄	—	2	1	no	—
170	C₂³×C₄	—	4	1	no	—
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	C₂×C₄×C₁₃₉	1	2	yes	—
175	C₂×D₈	D₄×C₁₃₉	1	2	yes	—
176	C₂×D₈	Q₈×C₁₃₉	1	2	yes	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂²×C₈	D₄×C₁₃₉	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₁₃₉	1	1	yes	characteristic
180	C₄×D₄	—	2	4	no	—
181	C₄×D₄	—	4	2	no	—
182	#117	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₂²⋊C₄	—	4	1	no	—
185	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	C₂×C₄×C₁₃₉	1	2	yes	complemented
192	#108	—	2	1	no	—
193	#108	—	2	1	no	—
194	#107	C₂×C₄×C₁₃₉	1	2	yes	complemented
195	#106	—	2	1	no	—
196	#106	—	2	1	no	—
197	#106	D₄×C₁₃₉	1	2	yes	—
198	C₄⋊C₈	—	4	2	no	—
199	#102	—	2	4	no	—
200	#97	—	2	1	no	—
201	#97	—	2	1	no	—
202	C₂⁴	—	4	1	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂×D₄	—	4	1	no	—
208	C₂×D₄	—	4	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	—	8	4	no	—
213	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
214	D₈	—	2	2	no	—
215	D₈	—	2	2	no	—
216	D₈	—	2	2	no	—
217	D₈	C₄⋊C₄×C₁₃₉	1	4	yes	—
218	M₁₆	—	4	1	no	—
219	M₁₆	—	4	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
226	C₂×C₈	C₂×D₄×C₁₃₉	1	1	yes	characteristic
227	C₂×C₈	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
228	C₂×C₈	#41×C₁₃₉	1	1	yes	characteristic
229	C₂×C₈	#41×C₁₃₉	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₂²⋊C₄	—	4	4	no	—
234	C₂²⋊C₄	—	4	1	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	4	1	no	—
237	C₄²	SD₁₆×C₁₃₉	1	2	yes	—
238	C₁₆	—	2	2	no	—
239	C₁₆	—	4	1	no	—
240	C₂³	—	4	2	no	—
241	C₂³	—	4	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	8	2	no	—
244	C₂³	—	8	2	no	—
245	C₂³	AGL(1,ℤ/8)×C₁₃₉	1	1	yes	characteristic
246	D₄	—	2	4	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	8	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	C₄×D₄×C₁₃₉	1	1	yes	characteristic
257	C₂×C₄	#120×C₁₃₉	1	1	yes	characteristic
258	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	1	yes	characteristic
259	C₂×C₄	C₂×SD₁₆×C₁₃₉	1	1	yes	characteristic
260	C₂×C₄	#135×C₁₃₉	1	1	yes	characteristic
261	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	1	yes	characteristic
262	C₈	—	4	1	no	—
263	C₈	—	8	1	no	—
264	C₈	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
265	C₈	C₄×D₄×C₁₃₉	1	1	yes	characteristic
266	C₈	C₂²⋊Q₈×C₁₃₉	1	1	yes	characteristic
267	C₈	#123×C₁₃₉	1	1	yes	characteristic
268	C₂²	—	2	2	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	8	2	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	4	no	—
273	C₂²	—	8	4	no	—
274	C₂²	#442×C₁₃₉	1	1	yes	characteristic
275	C₂²	#450×C₁₃₉	1	1	yes	characteristic
276	C₂²	#478×C₁₃₉	1	1	yes	characteristic
277	C₄	—	4	2	no	—
278	C₄	—	8	2	no	—
279	C₄	—	8	2	no	—
280	C₄	#385×C₁₃₉	1	1	yes	characteristic
281	C₄	C₄×SD₁₆×C₁₃₉	1	1	yes	characteristic
282	C₄	#449×C₁₃₉	1	1	yes	characteristic
283	C₄	#476×C₁₃₉	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	8	4	no	—
287	C₂	#1864×C₁₃₉	1	1	yes	characteristic
288	C₂	AGL(1,ℤ/16)×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₉	#442
commutator	C₂×C₈	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₉
prime radical	C₂²×D₈×C₁₃₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ C₂²×C₄×C₁₃₉ ◁ C₂×#97×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#365×C₁₃₉	C₂	semidirect
5. #1946×C₁₃₉	C₂	semidirect
6. #1386×C₁₃₉	C₂	semidirect
7. #1375×C₁₃₉	C₂	semidirect
22. #357×C₁₃₉	C₄	semidirect
46. C₂×C₁₆×C₁₃₉	C₂×C₄	semidirect
49. #107×C₁₃₉	C₂×C₄	semidirect
144. #12956	C₁₃₉	direct (N × H)
145. C₁₃₉	#12956	direct (N × H)
147. C₂×#365	C₂×C₁₃₉	semidirect
150. #1946	C₂×C₁₃₉	semidirect
151. #1386	C₂×C₁₃₉	semidirect
152. #1375	C₂×C₁₃₉	semidirect
167. #357	C₄×C₁₃₉	semidirect
191. C₂×C₁₆	C₂×C₄×C₁₃₉	semidirect
194. #107	C₂×C₄×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#442×C₁₃₉	2
C₂×#365×C₁₃₉	2
C₂×#361×C₁₃₉	2
#2140×C₁₃₉	2
#1946×C₁₃₉	2
#1386×C₁₃₉	2
#1375×C₁₃₉	2
#12956	139

Subgroups — C139×#12956 #48007552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#12956 #48007552