Subgroups — C₁₃₉×#20438 #48015034

Order 35584.

1110 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×#423×C₁₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#405×C₁₃₉	C₂	1	2	yes	maximal
3	C₂²×#97×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#2285×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	#1771×C₁₃₉	C₂	1	8	yes	maximal, complemented
6	#1729×C₁₃₉	C₂	1	2	yes	maximal
7	C₂²×M₁₆×C₁₃₉	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
10	C₂×#117×C₁₃₉	C₂²	1	2	yes	—
11	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
12	#423×C₁₃₉	C₄	1	8	yes	complemented
13	C₂×C₄⋊C₈×C₁₃₉	C₂²	1	4	yes	—
14	#405×C₁₃₉	—	2	8	no	—
15	C₂×#97×C₁₃₉	C₄	1	8	yes	complemented
16	C₂×#97×C₁₃₉	C₂²	1	4	yes	—
17	C₂×#97×C₁₃₉	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₁₃₉	C₂²	1	4	yes	—
19	#384×C₁₃₉	C₄	1	2	yes	—
20	#382×C₁₃₉	C₄	1	2	yes	—
21	#380×C₁₃₉	C₄	1	4	yes	—
22	C₄×C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	—
23	#336×C₁₃₉	C₂²	1	8	yes	—
24	C₂³×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₁₃₉	—	2	4	no	—
27	C₂×M₁₆×C₁₃₉	—	2	4	no	—
28	C₂×M₁₆×C₁₃₉	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₁₃₉	—	2	2	no	—
30	C₂²×C₈×C₁₃₉	—	2	2	no	—
31	C₂²×C₈×C₁₃₉	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₁₃₉	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃₉	D₄	1	4	yes	—
34	C₂²×C₈×C₁₃₉	Q₈	1	4	yes	—
35	C₂²×C₈×C₁₃₉	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₃₉	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃₉	C₂³	1	2	yes	—
38	#117×C₁₃₉	—	2	4	no	—
39	#117×C₁₃₉	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₁₃₉	C₂×C₄	1	8	yes	—
42	C₂×C₂²⋊C₄×C₁₃₉	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₁₃₉	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₁₃₉	C₂³	1	4	yes	—
48	C₂×C₄²×C₁₃₉	C₂³	1	2	yes	—
49	C₄⋊C₈×C₁₃₉	C₂×C₄	1	16	yes	—
50	#97×C₁₃₉	—	2	8	no	—
51	#97×C₁₃₉	C₂×C₄	1	16	yes	—
52	C₄×C₈×C₁₃₉	—	2	8	no	—
53	#95×C₁₃₉	C₂×C₄	1	8	yes	—
54	C₂⁴×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₉	—	4	4	no	—
56	C₂²×C₄×C₁₃₉	C₄²	1	8	yes	—
57	C₂²×C₄×C₁₃₉	C₂²×C₄	1	8	yes	—
58	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₁₃₉	#41	1	2	yes	—
66	C₂²×C₄×C₁₃₉	#41	1	2	yes	—
67	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
68	M₁₆×C₁₃₉	—	2	16	no	—
69	C₂×C₈×C₁₃₉	—	2	8	no	—
70	C₂×C₈×C₁₃₉	—	2	4	no	—
71	C₂×C₈×C₁₃₉	—	2	2	no	—
72	C₂×C₈×C₁₃₉	—	2	2	no	—
73	C₂×C₈×C₁₃₉	—	2	4	no	—
74	C₂×C₈×C₁₃₉	C₄⋊C₄	1	16	yes	complemented
75	C₂×C₈×C₁₃₉	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₁₃₉	C₂²×C₄	1	8	yes	—
77	C₂×C₈×C₁₃₉	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₁₃₉	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₁₃₉	C₂×Q₈	1	4	yes	—
80	C₄⋊C₄×C₁₃₉	—	2	8	no	—
81	C₄⋊C₄×C₁₃₉	—	4	8	no	—
82	C₂²⋊C₄×C₁₃₉	—	2	8	no	—
83	C₄²×C₁₃₉	—	2	4	no	—
84	C₄²×C₁₃₉	—	2	4	no	—
85	C₄²×C₁₃₉	C₄²	1	4	yes	—
86	C₄²×C₁₃₉	C₄²	1	4	yes	—
87	C₄²×C₁₃₉	C₂²×C₄	1	4	yes	—
88	C₄²×C₁₃₉	C₂²×C₄	1	4	yes	—
89	C₂³×C₁₃₉	—	2	4	no	—
90	C₂³×C₁₃₉	C₂×C₄²	1	2	yes	—
91	C₂³×C₁₃₉	C₄×D₄	1	2	yes	—
92	C₂³×C₁₃₉	C₄×Q₈	1	2	yes	—
93	C₂³×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₉	—	2	4	no	—
95	C₂×C₄×C₁₃₉	—	2	4	no	—
96	C₂×C₄×C₁₃₉	—	2	8	no	—
97	C₂×C₄×C₁₃₉	—	2	16	no	—
98	C₂×C₄×C₁₃₉	—	4	8	no	—
99	C₂×C₄×C₁₃₉	C₂×C₄²	1	4	yes	—
100	C₂×C₄×C₁₃₉	C₂×C₄²	1	4	yes	—
101	C₂×C₄×C₁₃₉	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₁₃₉	C₂×C₄⋊C₄	1	4	yes	—
103	C₂×C₄×C₁₃₉	#117	1	4	yes	—
104	C₂×C₄×C₁₃₉	C₄×D₄	1	4	yes	—
105	C₂×C₄×C₁₃₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₃₉	C₄×Q₈	1	4	yes	—
107	C₂×C₄×C₁₃₉	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
114	C₈×C₁₃₉	—	2	4	no	—
115	C₈×C₁₃₉	—	2	8	no	—
116	C₈×C₁₃₉	C₂×C₄⋊C₄	1	8	yes	—
117	C₂²×C₁₃₉	—	2	4	no	—
118	C₂²×C₁₃₉	—	2	4	no	—
119	C₂²×C₁₃₉	—	2	4	no	—
120	C₂²×C₁₃₉	C₄×C₄⋊C₄	1	4	yes	—
121	C₂²×C₁₃₉	C₂²×C₄²	1	1	yes	characteristic
122	C₂²×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂²×C₁₃₉	C₂×C₄×Q₈	1	1	yes	characteristic
124	C₂²×C₁₃₉	#568	1	4	yes	—
125	C₄×C₁₃₉	—	2	4	no	—
126	C₄×C₁₃₉	—	2	4	no	—
127	C₄×C₁₃₉	—	4	8	no	—
128	C₄×C₁₃₉	C₄×C₄⋊C₄	1	4	yes	—
129	C₄×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₁₃₉	C₂×#117	1	1	yes	characteristic
131	C₄×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
132	C₄×C₁₃₉	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₂×C₁₃₉	—	2	4	no	—
134	C₂×C₁₃₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₁₃₉	#2873	1	2	yes	—
136	C₂×C₁₃₉	#2932	1	2	yes	—
137	C₂×C₁₃₉	#2961	1	2	yes	—
138	#20438	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₁₃₉	#20438	1	1	yes	characteristic, direct product factor, Sylow
140	C₂×#423	C₂×C₁₃₉	1	1	yes	characteristic
141	C₂×#405	C₂×C₁₃₉	1	2	yes	—
142	C₂²×#97	C₂×C₁₃₉	1	1	yes	characteristic
143	#2285	C₂×C₁₃₉	1	1	yes	characteristic
144	#1771	C₂×C₁₃₉	1	8	yes	complemented
145	#1729	C₂×C₁₃₉	1	2	yes	—
146	C₂²×M₁₆	C₂²×C₁₃₉	1	2	yes	—
147	C₂³×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
148	C₂³×C₈	C₂²×C₁₃₉	1	1	yes	characteristic
149	C₂×#117	C₂²×C₁₃₉	1	2	yes	—
150	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
151	#423	C₄×C₁₃₉	1	8	yes	complemented
152	C₂×C₄⋊C₈	C₂²×C₁₃₉	1	4	yes	—
153	#405	—	2	8	no	—
154	C₂×#97	C₄×C₁₃₉	1	8	yes	complemented
155	C₂×#97	C₂²×C₁₃₉	1	4	yes	—
156	C₂×#97	C₂²×C₁₃₉	1	4	yes	—
157	C₂×C₄×C₈	C₂²×C₁₃₉	1	4	yes	—
158	#384	C₄×C₁₃₉	1	2	yes	—
159	#382	C₄×C₁₃₉	1	2	yes	—
160	#380	C₄×C₁₃₉	1	4	yes	—
161	C₄×C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	—
162	#336	C₂²×C₁₃₉	1	8	yes	—
163	C₂³×C₄	C₂×C₄×C₁₃₉	1	2	yes	—
164	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
165	C₂×M₁₆	—	2	4	no	—
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	C₂×C₄×C₁₃₉	1	8	yes	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	C₂×C₄×C₁₃₉	1	4	yes	—
171	C₂²×C₈	C₂×C₄×C₁₃₉	1	4	yes	—
172	C₂²×C₈	D₄×C₁₃₉	1	4	yes	—
173	C₂²×C₈	Q₈×C₁₃₉	1	4	yes	—
174	C₂²×C₈	C₂³×C₁₃₉	1	2	yes	—
175	C₂²×C₈	C₂³×C₁₃₉	1	4	yes	—
176	C₂²×C₈	C₂³×C₁₃₉	1	2	yes	—
177	#117	—	2	4	no	—
178	#117	—	2	4	no	—
179	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	4	yes	—
180	C₂×C₄⋊C₄	C₂×C₄×C₁₃₉	1	8	yes	—
181	C₂×C₂²⋊C₄	C₂×C₄×C₁₃₉	1	4	yes	—
182	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
183	C₂×C₄²	C₂×C₄×C₁₃₉	1	1	yes	characteristic
184	C₂×C₄²	C₂×C₄×C₁₃₉	1	4	yes	—
185	C₂×C₄²	C₂×C₄×C₁₃₉	1	4	yes	—
186	C₂×C₄²	C₂³×C₁₃₉	1	4	yes	—
187	C₂×C₄²	C₂³×C₁₃₉	1	2	yes	—
188	C₄⋊C₈	C₂×C₄×C₁₃₉	1	16	yes	—
189	#97	—	2	8	no	—
190	#97	C₂×C₄×C₁₃₉	1	16	yes	—
191	C₄×C₈	—	2	8	no	—
192	#95	C₂×C₄×C₁₃₉	1	8	yes	—
193	C₂⁴	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
194	C₂²×C₄	—	4	4	no	—
195	C₂²×C₄	C₄²×C₁₃₉	1	8	yes	—
196	C₂²×C₄	C₂²×C₄×C₁₃₉	1	8	yes	—
197	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
199	C₂²×C₄	C₂²×C₄×C₁₃₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₁₃₉	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
203	C₂²×C₄	C₂×Q₈×C₁₃₉	1	2	yes	—
204	C₂²×C₄	#41×C₁₃₉	1	2	yes	—
205	C₂²×C₄	#41×C₁₃₉	1	2	yes	—
206	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
207	M₁₆	—	2	16	no	—
208	C₂×C₈	—	2	8	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	C₄⋊C₄×C₁₃₉	1	16	yes	complemented
214	C₂×C₈	C₂²×C₄×C₁₃₉	1	4	yes	—
215	C₂×C₈	C₂²×C₄×C₁₃₉	1	8	yes	—
216	C₂×C₈	C₂²×C₄×C₁₃₉	1	4	yes	—
217	C₂×C₈	C₂×D₄×C₁₃₉	1	4	yes	—
218	C₂×C₈	C₂×Q₈×C₁₃₉	1	4	yes	—
219	C₄⋊C₄	—	2	8	no	—
220	C₄⋊C₄	—	4	8	no	—
221	C₂²⋊C₄	—	2	8	no	—
222	C₄²	—	2	4	no	—
223	C₄²	—	2	4	no	—
224	C₄²	C₄²×C₁₃₉	1	4	yes	—
225	C₄²	C₄²×C₁₃₉	1	4	yes	—
226	C₄²	C₂²×C₄×C₁₃₉	1	4	yes	—
227	C₄²	C₂²×C₄×C₁₃₉	1	4	yes	—
228	C₂³	—	2	4	no	—
229	C₂³	C₂×C₄²×C₁₃₉	1	2	yes	—
230	C₂³	C₄×D₄×C₁₃₉	1	2	yes	—
231	C₂³	C₄×Q₈×C₁₃₉	1	2	yes	—
232	C₂³	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	16	no	—
237	C₂×C₄	—	4	8	no	—
238	C₂×C₄	C₂×C₄²×C₁₃₉	1	4	yes	—
239	C₂×C₄	C₂×C₄²×C₁₃₉	1	4	yes	—
240	C₂×C₄	C₂×C₄²×C₁₃₉	1	2	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₁₃₉	1	4	yes	—
242	C₂×C₄	#117×C₁₃₉	1	4	yes	—
243	C₂×C₄	C₄×D₄×C₁₃₉	1	4	yes	—
244	C₂×C₄	C₄×D₄×C₁₃₉	1	2	yes	—
245	C₂×C₄	C₄×Q₈×C₁₃₉	1	4	yes	—
246	C₂×C₄	C₄×Q₈×C₁₃₉	1	2	yes	—
247	C₂×C₄	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
248	C₂×C₄	C₂³×C₄×C₁₃₉	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
250	C₂×C₄	C₂²×Q₈×C₁₃₉	1	1	yes	characteristic
251	C₂×C₄	C₂×#41×C₁₃₉	1	1	yes	characteristic
252	C₂×C₄	C₂×#41×C₁₃₉	1	1	yes	characteristic
253	C₈	—	2	4	no	—
254	C₈	—	2	8	no	—
255	C₈	C₂×C₄⋊C₄×C₁₃₉	1	8	yes	—
256	C₂²	—	2	4	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	2	4	no	—
259	C₂²	C₄×C₄⋊C₄×C₁₃₉	1	4	yes	—
260	C₂²	C₂²×C₄²×C₁₃₉	1	1	yes	characteristic
261	C₂²	C₂×C₄×D₄×C₁₃₉	1	1	yes	characteristic
262	C₂²	C₂×C₄×Q₈×C₁₃₉	1	1	yes	characteristic
263	C₂²	#568×C₁₃₉	1	4	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	2	4	no	—
266	C₄	—	4	8	no	—
267	C₄	C₄×C₄⋊C₄×C₁₃₉	1	4	yes	—
268	C₄	C₂²×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
269	C₄	C₂×#117×C₁₃₉	1	1	yes	characteristic
270	C₄	C₂×C₄×D₄×C₁₃₉	1	1	yes	characteristic
271	C₄	C₂×C₄×Q₈×C₁₃₉	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	C₂×C₄×C₄⋊C₄×C₁₃₉	1	1	yes	characteristic
274	C₂	#2873×C₁₃₉	1	2	yes	—
275	C₂	#2932×C₁₃₉	1	2	yes	—
276	C₂	#2961×C₁₃₉	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
5. #1771×C₁₃₉	C₂	semidirect
12. #423×C₁₃₉	C₄	semidirect
15. C₂×#97×C₁₃₉	C₄	semidirect
74. C₂×C₈×C₁₃₉	C₄⋊C₄	semidirect
138. #20438	C₁₃₉	direct (N × H)
139. C₁₃₉	#20438	direct (N × H)
144. #1771	C₂×C₁₃₉	semidirect
151. #423	C₄×C₁₃₉	semidirect
154. C₂×#97	C₄×C₁₃₉	semidirect
213. C₂×C₈	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×#423×C₁₃₉	2
C₂×#405×C₁₃₉	2
C₂²×#97×C₁₃₉	2
#2285×C₁₃₉	2
#1771×C₁₃₉	2
#1729×C₁₃₉	2
#20438	139

Subgroups — C139×#20438 #48015034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₉×#20438 #48015034