Subgroups — C₁₃₉×#8593 #48003189

Order 35584.

476 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	#1831×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	#1831×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#332×C₁₃₉	C₂	1	2	yes	maximal
5	#1296×C₁₃₉	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃₉	—	2	1	no	—
8	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
10	#391×C₁₃₉	—	2	1	no	—
11	#389×C₁₃₉	—	2	1	no	—
12	#389×C₁₃₉	C₄	1	2	yes	—
13	#384×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₁₃₉	—	2	1	no	—
15	C₄×C₄⋊C₄×C₁₃₉	C₄	1	2	yes	—
16	C₄³×C₁₃₉	C₂²	1	1	yes	characteristic
17	#336×C₁₃₉	—	2	1	no	—
18	#336×C₁₃₉	—	2	1	no	—
19	#336×C₁₃₉	—	2	1	no	—
20	#336×C₁₃₉	—	2	1	no	—
21	#332×C₁₃₉	—	2	4	no	—
22	C₂²×Q₈×C₁₃₉	—	2	1	no	—
23	C₂²×C₈×C₁₃₉	—	2	2	no	—
24	C₂²×C₈×C₁₃₉	—	2	2	no	—
25	C₄⋊Q₈×C₁₃₉	—	2	1	no	—
26	C₄⋊Q₈×C₁₃₉	—	2	1	no	—
27	C₄⋊Q₈×C₁₃₉	D₄	1	2	yes	—
28	C₄⋊Q₈×C₁₃₉	Q₈	1	2	yes	—
29	C₄×Q₈×C₁₃₉	—	4	4	no	—
30	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃₉	—	2	2	no	—
37	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₁₃₉	—	2	2	no	—
45	C₄⋊C₈×C₁₃₉	—	2	2	no	—
46	C₄⋊C₈×C₁₃₉	—	2	2	no	—
47	C₄⋊C₈×C₁₃₉	—	2	2	no	—
48	#95×C₁₃₉	—	2	2	no	—
49	#95×C₁₃₉	—	2	2	no	—
50	C₂×Q₈×C₁₃₉	—	2	2	no	—
51	C₂×Q₈×C₁₃₉	—	2	2	no	—
52	C₂×Q₈×C₁₃₉	—	4	2	no	—
53	C₂×Q₈×C₁₃₉	—	4	2	no	—
54	C₂²×C₄×C₁₃₉	—	2	2	no	—
55	C₂²×C₄×C₁₃₉	—	2	2	no	—
56	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃₉	—	4	2	no	—
64	C₂×C₈×C₁₃₉	—	4	2	no	—
65	C₂×C₈×C₁₃₉	—	4	2	no	—
66	C₂×C₈×C₁₃₉	—	4	2	no	—
67	C₂×C₈×C₁₃₉	—	4	2	no	—
68	C₂×C₈×C₁₃₉	—	4	2	no	—
69	C₄⋊C₄×C₁₃₉	—	2	2	no	—
70	C₄⋊C₄×C₁₃₉	—	2	2	no	—
71	C₄⋊C₄×C₁₃₉	—	2	2	no	—
72	C₄⋊C₄×C₁₃₉	—	4	2	no	—
73	C₄⋊C₄×C₁₃₉	—	4	4	no	—
74	C₄⋊C₄×C₁₃₉	—	4	2	no	—
75	C₄⋊C₄×C₁₃₉	—	4	2	no	—
76	C₄⋊C₄×C₁₃₉	C₄⋊C₄	1	4	yes	—
77	C₄²×C₁₃₉	—	2	2	no	—
78	C₄²×C₁₃₉	—	2	2	no	—
79	C₄²×C₁₃₉	—	2	2	no	—
80	C₄²×C₁₃₉	—	2	2	no	—
81	C₄²×C₁₃₉	—	2	2	no	—
82	C₄²×C₁₃₉	—	4	1	no	—
83	C₄²×C₁₃₉	—	4	4	no	—
84	C₄²×C₁₃₉	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
86	C₄²×C₁₃₉	#41	1	1	yes	characteristic
87	C₄²×C₁₃₉	#41	1	1	yes	characteristic
88	C₂³×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	Q₈×C₁₃₉	—	4	4	no	—
90	Q₈×C₁₃₉	—	4	4	no	—
91	C₂×C₄×C₁₃₉	—	2	1	no	—
92	C₂×C₄×C₁₃₉	—	2	1	no	—
93	C₂×C₄×C₁₃₉	—	2	1	no	—
94	C₂×C₄×C₁₃₉	—	2	1	no	—
95	C₂×C₄×C₁₃₉	—	2	4	no	—
96	C₂×C₄×C₁₃₉	—	2	1	no	—
97	C₂×C₄×C₁₃₉	—	2	1	no	—
98	C₂×C₄×C₁₃₉	—	2	1	no	—
99	C₂×C₄×C₁₃₉	—	2	1	no	—
100	C₂×C₄×C₁₃₉	—	2	1	no	—
101	C₂×C₄×C₁₃₉	—	2	1	no	—
102	C₂×C₄×C₁₃₉	—	2	1	no	—
103	C₂×C₄×C₁₃₉	—	2	1	no	—
104	C₂×C₄×C₁₃₉	—	2	1	no	—
105	C₂×C₄×C₁₃₉	—	4	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	—	4	2	no	—
110	C₂×C₄×C₁₃₉	#101	1	2	yes	—
111	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃₉	#117	1	1	yes	characteristic
113	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₃₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃₉	#135	1	2	yes	—
118	C₂×C₄×C₁₃₉	#135	1	2	yes	—
119	C₂×C₄×C₁₃₉	AGL(1,ℤ/8)	1	2	yes	—
120	C₂×C₄×C₁₃₉	#137	1	2	yes	—
121	C₈×C₁₃₉	—	8	2	no	—
122	C₈×C₁₃₉	—	8	2	no	—
123	C₂²×C₁₃₉	#380	1	1	yes	characteristic
124	C₂²×C₁₃₉	C₂×#101	1	1	yes	characteristic
125	C₂²×C₁₃₉	#413	1	1	yes	characteristic
126	C₂²×C₁₃₉	#416	1	1	yes	characteristic
127	C₂²×C₁₃₉	#416	1	1	yes	characteristic
128	C₂²×C₁₃₉	#418	1	1	yes	characteristic
129	C₂²×C₁₃₉	#419	1	1	yes	characteristic
130	C₄×C₁₃₉	—	2	1	no	—
131	C₄×C₁₃₉	—	2	1	no	—
132	C₄×C₁₃₉	—	2	2	no	—
133	C₄×C₁₃₉	—	4	1	no	—
134	C₄×C₁₃₉	—	4	4	no	—
135	C₄×C₁₃₉	—	4	1	no	—
136	C₄×C₁₃₉	—	4	1	no	—
137	C₄×C₁₃₉	—	8	2	no	—
138	C₄×C₁₃₉	#463	1	2	yes	—
139	C₄×C₁₃₉	#464	1	2	yes	—
140	C₄×C₁₃₉	#478	1	2	yes	—
141	C₄×C₁₃₉	#479	1	2	yes	—
142	C₂×C₁₃₉	#1544	1	1	yes	characteristic
143	C₂×C₁₃₉	#1553	1	1	yes	characteristic
144	C₂×C₁₃₉	#1554	1	1	yes	characteristic
145	C₂×C₁₃₉	#1556	1	1	yes	characteristic
146	C₂×C₁₃₉	#1801	1	1	yes	characteristic
147	C₂×C₁₃₉	#1804	1	1	yes	characteristic
148	C₂×C₁₃₉	#1805	1	1	yes	characteristic
149	#8593	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₁₃₉	#8593	1	1	yes	characteristic, direct product factor, Sylow
151	C₄×C₄⋊Q₈	C₂×C₁₃₉	1	1	yes	characteristic
152	#1831	C₂×C₁₃₉	1	1	yes	characteristic
153	#1831	C₂×C₁₃₉	1	1	yes	characteristic
154	C₂×#332	C₂×C₁₃₉	1	2	yes	—
155	#1296	C₂×C₁₃₉	1	2	yes	—
156	C₂×C₄⋊Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
157	C₂×C₄×Q₈	—	2	1	no	—
158	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
159	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	2	yes	—
160	#391	—	2	1	no	—
161	#389	—	2	1	no	—
162	#389	C₄×C₁₃₉	1	2	yes	—
163	#384	C₂²×C₁₃₉	1	1	yes	characteristic
164	C₄×C₄⋊C₄	—	2	1	no	—
165	C₄×C₄⋊C₄	C₄×C₁₃₉	1	2	yes	—
166	C₄³	C₂²×C₁₃₉	1	1	yes	characteristic
167	#336	—	2	1	no	—
168	#336	—	2	1	no	—
169	#336	—	2	1	no	—
170	#336	—	2	1	no	—
171	#332	—	2	4	no	—
172	C₂²×Q₈	—	2	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₄⋊Q₈	—	2	1	no	—
176	C₄⋊Q₈	—	2	1	no	—
177	C₄⋊Q₈	D₄×C₁₃₉	1	2	yes	—
178	C₄⋊Q₈	Q₈×C₁₃₉	1	2	yes	—
179	C₄×Q₈	—	4	4	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
185	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
189	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
191	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
192	C₂×C₄²	D₄×C₁₃₉	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₁₃₉	1	1	yes	characteristic
194	C₄⋊C₈	—	2	2	no	—
195	C₄⋊C₈	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	#95	—	2	2	no	—
199	#95	—	2	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	—	4	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	C₄⋊C₄×C₁₃₉	1	4	yes	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	4	4	no	—
234	C₄²	C₂×D₄×C₁₃₉	1	1	yes	characteristic
235	C₄²	C₂×Q₈×C₁₃₉	1	1	yes	characteristic
236	C₄²	#41×C₁₃₉	1	1	yes	characteristic
237	C₄²	#41×C₁₃₉	1	1	yes	characteristic
238	C₂³	C₂×C₂²⋊C₄×C₁₃₉	1	1	yes	characteristic
239	Q₈	—	4	4	no	—
240	Q₈	—	4	4	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	#101×C₁₃₉	1	2	yes	—
261	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
262	C₂×C₄	#117×C₁₃₉	1	1	yes	characteristic
263	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
264	C₂×C₄	#121×C₁₃₉	1	1	yes	characteristic
265	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	1	yes	characteristic
266	C₂×C₄	C₂²⋊Q₈×C₁₃₉	1	1	yes	characteristic
267	C₂×C₄	#135×C₁₃₉	1	2	yes	—
268	C₂×C₄	#135×C₁₃₉	1	2	yes	—
269	C₂×C₄	AGL(1,ℤ/8)×C₁₃₉	1	2	yes	—
270	C₂×C₄	#137×C₁₃₉	1	2	yes	—
271	C₈	—	8	2	no	—
272	C₈	—	8	2	no	—
273	C₂²	#380×C₁₃₉	1	1	yes	characteristic
274	C₂²	C₂×#101×C₁₃₉	1	1	yes	characteristic
275	C₂²	#413×C₁₃₉	1	1	yes	characteristic
276	C₂²	#416×C₁₃₉	1	1	yes	characteristic
277	C₂²	#416×C₁₃₉	1	1	yes	characteristic
278	C₂²	#418×C₁₃₉	1	1	yes	characteristic
279	C₂²	#419×C₁₃₉	1	1	yes	characteristic
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	2	2	no	—
283	C₄	—	4	1	no	—
284	C₄	—	4	4	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	8	2	no	—
288	C₄	#463×C₁₃₉	1	2	yes	—
289	C₄	#464×C₁₃₉	1	2	yes	—
290	C₄	#478×C₁₃₉	1	2	yes	—
291	C₄	#479×C₁₃₉	1	2	yes	—
292	C₂	#1544×C₁₃₉	1	1	yes	characteristic
293	C₂	#1553×C₁₃₉	1	1	yes	characteristic
294	C₂	#1554×C₁₃₉	1	1	yes	characteristic
295	C₂	#1556×C₁₃₉	1	1	yes	characteristic
296	C₂	#1801×C₁₃₉	1	1	yes	characteristic
297	C₂	#1804×C₁₃₉	1	1	yes	characteristic
298	C₂	#1805×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₉	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₉
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₉ ◁ C₄³×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
149. #8593	C₁₃₉	direct (N × H)
150. C₁₃₉	#8593	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₁₃₉	2
#1831×C₁₃₉	2
C₂×#332×C₁₃₉	2
#1296×C₁₃₉	2
#8593	139

Subgroups — C139×#8593 #48003189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#8593 #48003189