Subgroups — C₁₃₉×#13759 #48008355

Order 35584.

1774 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₁₃₉	C₂	1	2	yes	maximal
2	C₂×#376×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#376×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄³×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	#1436×C₁₃₉	C₂	1	8	yes	maximal
6	C₄×#95×C₁₃₉	C₂	1	2	yes	maximal
7	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
8	C₂²×C₄²×C₁₃₉	C₂²	1	4	yes	—
9	C₂²×C₄²×C₁₃₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₁₃₉	C₄	1	16	yes	complemented
15	#376×C₁₃₉	—	2	4	no	—
16	#376×C₁₃₉	C₄	1	8	yes	complemented
17	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
18	C₂×#95×C₁₃₉	C₂²	1	8	yes	—
19	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
20	C₂×#95×C₁₃₉	C₂²	1	4	yes	—
21	C₄³×C₁₃₉	—	2	4	no	—
22	C₂³×C₄×C₁₃₉	C₂³	1	4	yes	—
23	C₂³×C₄×C₁₃₉	C₂³	1	2	yes	—
24	C₂³×C₄×C₁₃₉	C₂³	1	2	yes	—
25	C₂³×C₄×C₁₃₉	C₂³	1	2	yes	—
26	C₂³×C₄×C₁₃₉	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₃₉	—	2	16	no	—
29	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	16	yes	—
30	C₂×C₄²×C₁₃₉	—	2	8	no	—
31	C₂×C₄²×C₁₃₉	—	2	4	no	—
32	C₂×C₄²×C₁₃₉	—	2	4	no	—
33	C₂×C₄²×C₁₃₉	—	2	16	no	—
34	C₂×C₄²×C₁₃₉	—	2	4	no	—
35	C₂×C₄²×C₁₃₉	—	2	4	no	—
36	C₂×C₄²×C₁₃₉	—	2	4	no	—
37	C₂×C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₁₃₉	C₂×C₄	1	8	yes	—
39	C₂×C₄²×C₁₃₉	C₂×C₄	1	8	yes	—
40	C₂×C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₁₃₉	C₂×C₄	1	8	yes	—
42	C₂×C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃₉	D₄	1	4	yes	—
44	C₂×C₄²×C₁₃₉	Q₈	1	4	yes	—
45	#95×C₁₃₉	—	2	8	no	—
46	#95×C₁₃₉	—	2	16	no	—
47	#95×C₁₃₉	C₂×C₄	1	16	yes	—
48	#95×C₁₃₉	C₂×C₄	1	16	yes	—
49	C₂⁴×C₁₃₉	C₂⁴	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₉	—	2	4	no	—
51	C₂²×C₄×C₁₃₉	—	2	4	no	—
52	C₂²×C₄×C₁₃₉	—	2	8	no	—
53	C₂²×C₄×C₁₃₉	—	2	8	no	—
54	C₂²×C₄×C₁₃₉	—	2	8	no	—
55	C₂²×C₄×C₁₃₉	—	2	4	no	—
56	C₂²×C₄×C₁₃₉	—	2	8	no	—
57	C₂²×C₄×C₁₃₉	—	2	8	no	—
58	C₂²×C₄×C₁₃₉	—	2	4	no	—
59	C₂²×C₄×C₁₃₉	—	2	4	no	—
60	C₂²×C₄×C₁₃₉	—	2	8	no	—
61	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₁₃₉	C₂²×C₄	1	8	yes	—
63	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₁₃₉	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
73	C₂²×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₁₃₉	#41	1	16	yes	—
75	C₂²×C₄×C₁₃₉	#41	1	4	yes	—
76	C₄⋊C₄×C₁₃₉	—	2	32	no	—
77	C₄²×C₁₃₉	—	2	32	no	—
78	C₄²×C₁₃₉	—	2	8	no	—
79	C₄²×C₁₃₉	—	2	16	no	—
80	C₄²×C₁₃₉	—	2	16	no	—
81	C₄²×C₁₃₉	C₄²	1	16	yes	complemented
82	C₄²×C₁₃₉	C₄⋊C₄	1	16	yes	complemented
83	C₂³×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
84	C₂³×C₁₃₉	C₂³×C₄	1	2	yes	—
85	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
86	C₂³×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
87	C₂³×C₁₃₉	C₂×#41	1	2	yes	—
88	C₂³×C₁₃₉	C₂×#41	1	8	yes	—
89	C₂×C₄×C₁₃₉	—	2	8	no	—
90	C₂×C₄×C₁₃₉	—	2	8	no	—
91	C₂×C₄×C₁₃₉	—	2	8	no	—
92	C₂×C₄×C₁₃₉	—	2	8	no	—
93	C₂×C₄×C₁₃₉	—	2	8	no	—
94	C₂×C₄×C₁₃₉	—	2	8	no	—
95	C₂×C₄×C₁₃₉	—	2	8	no	—
96	C₂×C₄×C₁₃₉	—	2	16	no	—
97	C₂×C₄×C₁₃₉	—	2	16	no	—
98	C₂×C₄×C₁₃₉	—	2	8	no	—
99	C₂×C₄×C₁₃₉	—	2	4	no	—
100	C₂×C₄×C₁₃₉	—	2	8	no	—
101	C₂×C₄×C₁₃₉	—	2	8	no	—
102	C₂×C₄×C₁₃₉	—	2	8	no	—
103	C₂×C₄×C₁₃₉	—	2	8	no	—
104	C₂×C₄×C₁₃₉	—	4	8	no	—
105	C₂×C₄×C₁₃₉	—	4	8	no	—
106	C₂×C₄×C₁₃₉	—	4	8	no	—
107	C₂×C₄×C₁₃₉	C₂×C₄²	1	8	yes	—
108	C₂×C₄×C₁₃₉	C₂×C₄²	1	4	yes	—
109	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	8	yes	—
110	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	4	yes	—
111	C₂×C₄×C₁₃₉	#117	1	16	yes	—
112	C₂×C₄×C₁₃₉	#117	1	4	yes	—
113	C₂×C₄×C₁₃₉	C₄×D₄	1	8	yes	—
114	C₂×C₄×C₁₃₉	C₄×Q₈	1	8	yes	—
115	C₂²×C₁₃₉	C₂²×C₄²	1	1	yes	characteristic
116	C₂²×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₂²×C₁₃₉	C₂×#117	1	1	yes	characteristic
118	C₂²×C₁₃₉	C₂×#117	1	4	yes	—
119	C₂²×C₁₃₉	C₂×C₄×D₄	1	2	yes	—
120	C₂²×C₁₃₉	C₂×C₄×Q₈	1	2	yes	—
121	C₂²×C₁₃₉	C₄×#41	1	4	yes	—
122	C₂²×C₁₃₉	C₄×#41	1	4	yes	—
123	C₂²×C₁₃₉	#525	1	4	yes	—
124	C₂²×C₁₃₉	#529	1	8	yes	—
125	C₂²×C₁₃₉	#533	1	4	yes	—
126	C₄×C₁₃₉	—	2	8	no	—
127	C₄×C₁₃₉	—	2	16	no	—
128	C₄×C₁₃₉	—	2	8	no	—
129	C₄×C₁₃₉	—	4	8	no	—
130	C₄×C₁₃₉	—	4	4	no	—
131	C₄×C₁₃₉	C₄×C₄⋊C₄	1	8	yes	—
132	C₂×C₁₃₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₁₃₉	C₄×#117	1	2	yes	—
134	C₂×C₁₃₉	#2290	1	4	yes	—
135	C₂×C₁₃₉	#2295	1	4	yes	—
136	C₂×C₁₃₉	#2296	1	2	yes	—
137	C₂×C₁₃₉	#2298	1	2	yes	—
138	#13759	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₁₃₉	#13759	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×C₄×C₄⋊C₄	C₂×C₁₃₉	1	2	yes	—
141	C₂×#376	C₂×C₁₃₉	1	1	yes	characteristic
142	C₂×#376	C₂×C₁₃₉	1	1	yes	characteristic
143	C₂×C₄³	C₂×C₁₃₉	1	1	yes	characteristic
144	#1436	C₂×C₁₃₉	1	8	yes	—
145	C₄×#95	C₂×C₁₃₉	1	2	yes	—
146	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	—
147	C₂²×C₄²	C₂²×C₁₃₉	1	4	yes	—
148	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
149	C₂²×C₄²	C₂²×C₁₃₉	1	2	yes	—
150	C₂²×C₄²	C₂²×C₁₃₉	1	2	yes	—
151	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
152	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
153	C₄×C₄⋊C₄	C₄×C₁₃₉	1	16	yes	complemented
154	#376	—	2	4	no	—
155	#376	C₄×C₁₃₉	1	8	yes	complemented
156	C₂×#95	C₂²×C₁₃₉	1	8	yes	—
157	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
158	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
159	C₂×#95	C₂²×C₁₃₉	1	4	yes	—
160	C₄³	—	2	4	no	—
161	C₂³×C₄	C₂³×C₁₃₉	1	4	yes	—
162	C₂³×C₄	C₂³×C₁₃₉	1	2	yes	—
163	C₂³×C₄	C₂³×C₁₃₉	1	2	yes	—
164	C₂³×C₄	C₂³×C₁₃₉	1	2	yes	—
165	C₂³×C₄	C₂³×C₁₃₉	1	4	yes	—
166	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
167	C₂×C₄⋊C₄	—	2	16	no	—
168	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	16	yes	—
169	C₂×C₄²	—	2	8	no	—
170	C₂×C₄²	—	2	4	no	—
171	C₂×C₄²	—	2	4	no	—
172	C₂×C₄²	—	2	16	no	—
173	C₂×C₄²	—	2	4	no	—
174	C₂×C₄²	—	2	4	no	—
175	C₂×C₄²	—	2	4	no	—
176	C₂×C₄²	C₂×C₄×C₁₃₉	1	4	yes	—
177	C₂×C₄²	C₂×C₄×C₁₃₉	1	8	yes	—
178	C₂×C₄²	C₂×C₄×C₁₃₉	1	8	yes	—
179	C₂×C₄²	C₂×C₄×C₁₃₉	1	4	yes	—
180	C₂×C₄²	C₂×C₄×C₁₃₉	1	8	yes	—
181	C₂×C₄²	C₂×C₄×C₁₃₉	1	4	yes	—
182	C₂×C₄²	D₄×C₁₃₉	1	4	yes	—
183	C₂×C₄²	Q₈×C₁₃₉	1	4	yes	—
184	#95	—	2	8	no	—
185	#95	—	2	16	no	—
186	#95	C₂×C₄×C₁₃₉	1	16	yes	—
187	#95	C₂×C₄×C₁₃₉	1	16	yes	—
188	C₂⁴	C₂⁴×C₁₃₉	1	1	yes	characteristic
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	8	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
201	C₂²×C₄	C₂²×C₄×C₁₃₉	1	8	yes	—
202	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
204	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
206	C₂²×C₄	C₂²×C₄×C₁₃₉	1	8	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₁₃₉	1	4	yes	—
209	C₂²×C₄	C₂×D₄×C₁₃₉	1	4	yes	—
210	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
211	C₂²×C₄	C₂×Q₈×C₁₃₉	1	4	yes	—
212	C₂²×C₄	C₂×Q₈×C₁₃₉	1	2	yes	—
213	C₂²×C₄	#41×C₁₃₉	1	16	yes	—
214	C₂²×C₄	#41×C₁₃₉	1	4	yes	—
215	C₄⋊C₄	—	2	32	no	—
216	C₄²	—	2	32	no	—
217	C₄²	—	2	8	no	—
218	C₄²	—	2	16	no	—
219	C₄²	—	2	16	no	—
220	C₄²	C₄²×C₁₃₉	1	16	yes	complemented
221	C₄²	C₄⋊C₄×C₁₃₉	1	16	yes	complemented
222	C₂³	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
223	C₂³	C₂³×C₄×C₁₃₉	1	2	yes	—
224	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
225	C₂³	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
226	C₂³	C₂×#41×C₁₃₉	1	2	yes	—
227	C₂³	C₂×#41×C₁₃₉	1	8	yes	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	8	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	16	no	—
236	C₂×C₄	—	2	16	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	8	no	—
240	C₂×C₄	—	2	8	no	—
241	C₂×C₄	—	2	8	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	—	4	8	no	—
245	C₂×C₄	—	4	8	no	—
246	C₂×C₄	C₂×C₄²×C₁₃₉	1	8	yes	—
247	C₂×C₄	C₂×C₄²×C₁₃₉	1	4	yes	—
248	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	8	yes	—
249	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	4	yes	—
250	C₂×C₄	#117×C₁₃₉	1	16	yes	—
251	C₂×C₄	#117×C₁₃₉	1	4	yes	—
252	C₂×C₄	C₄×D₄×C₁₃₉	1	8	yes	—
253	C₂×C₄	C₄×Q₈×C₁₃₉	1	8	yes	—
254	C₂²	C₂²×C₄²×C₁₃₉	1	1	yes	characteristic
255	C₂²	C₂²×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
256	C₂²	C₂×#117×C₁₃₉	1	1	yes	characteristic
257	C₂²	C₂×#117×C₁₃₉	1	4	yes	—
258	C₂²	C₂×C₄×D₄×C₁₃₉	1	2	yes	—
259	C₂²	C₂×C₄×Q₈×C₁₃₉	1	2	yes	—
260	C₂²	C₄×#41×C₁₃₉	1	4	yes	—
261	C₂²	C₄×#41×C₁₃₉	1	4	yes	—
262	C₂²	#525×C₁₃₉	1	4	yes	—
263	C₂²	#529×C₁₃₉	1	8	yes	—
264	C₂²	#533×C₁₃₉	1	4	yes	—
265	C₄	—	2	8	no	—
266	C₄	—	2	16	no	—
267	C₄	—	2	8	no	—
268	C₄	—	4	8	no	—
269	C₄	—	4	4	no	—
270	C₄	C₄×C₄⋊C₄×C₁₃₉	1	8	yes	—
271	C₂	C₂×C₄×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
272	C₂	C₄×#117×C₁₃₉	1	2	yes	—
273	C₂	#2290×C₁₃₉	1	4	yes	—
274	C₂	#2295×C₁₃₉	1	4	yes	—
275	C₂	#2296×C₁₃₉	1	2	yes	—
276	C₂	#2298×C₁₃₉	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₃₉	C₂³
commutator	C₂²	C₂²×C₄²×C₁₃₉
Frattini, cosocle	C₂⁴	C₂⁴×C₁₃₉
socle, prime radical	C₂⁴×C₁₃₉	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₄×C₄⋊C₄×C₁₃₉	C₄	semidirect
16. #376×C₁₃₉	C₄	semidirect
81. C₄²×C₁₃₉	C₄²	semidirect
82. C₄²×C₁₃₉	C₄⋊C₄	semidirect
138. #13759	C₁₃₉	direct (N × H)
139. C₁₃₉	#13759	direct (N × H)
153. C₄×C₄⋊C₄	C₄×C₁₃₉	semidirect
155. #376	C₄×C₁₃₉	semidirect
220. C₄²	C₄²×C₁₃₉	semidirect
221. C₄²	C₄⋊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×C₄×C₄⋊C₄×C₁₃₉	2
C₂×#376×C₁₃₉	2
C₂×C₄³×C₁₃₉	2
#1436×C₁₃₉	2
C₄×#95×C₁₃₉	2
#13759	139

Subgroups — C139×#13759 #48008355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#13759 #48008355