Subgroups — C₂×#2074×C₁₃₉ #48018222

Order 35584.

1062 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	C₂²×#103×C₁₃₉	C₂	1	2	yes	maximal
2	C₂×#391×C₁₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#382×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	#2074×C₁₃₉	C₂	1	8	yes	maximal, direct product factor
5	C₂×#340×C₁₃₉	C₂	1	2	yes	maximal
6	C₂×#336×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃₉	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
12	C₂×#103×C₁₃₉	—	2	8	no	—
13	C₂×#103×C₁₃₉	C₂²	1	8	yes	—
14	#391×C₁₃₉	—	2	2	no	—
15	#391×C₁₃₉	C₂²	1	4	yes	—
16	#382×C₁₃₉	—	2	2	no	—
17	#382×C₁₃₉	C₂²	1	4	yes	—
18	C₂×#95×C₁₃₉	—	2	2	no	—
19	C₂×#95×C₁₃₉	—	2	2	no	—
20	#340×C₁₃₉	C₂²	1	8	yes	—
21	#336×C₁₃₉	C₂²	1	4	yes	—
22	C₂²×Q₈×C₁₃₉	—	2	1	no	—
23	C₂²×Q₈×C₁₃₉	—	2	2	no	—
24	C₂²×Q₈×C₁₃₉	—	2	4	no	—
25	C₂²×Q₈×C₁₃₉	—	2	4	no	—
26	C₂²×Q₈×C₁₃₉	D₄	1	4	yes	—
27	C₂²×Q₈×C₁₃₉	C₂³	1	2	yes	—
28	C₂³×C₄×C₁₃₉	—	2	1	no	—
29	C₂³×C₄×C₁₃₉	—	2	2	no	—
30	C₂³×C₄×C₁₃₉	—	2	1	no	—
31	C₂³×C₄×C₁₃₉	D₄	1	2	yes	—
32	C₂³×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₃₉	—	2	4	no	—
34	C₂²×C₈×C₁₃₉	—	2	4	no	—
35	C₂²×C₈×C₁₃₉	—	2	4	no	—
36	C₂²×C₈×C₁₃₉	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
39	C₂×C₄⋊C₄×C₁₃₉	—	2	4	no	—
40	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₃₉	—	2	1	no	—
42	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
43	C₂×C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃₉	—	2	4	no	—
45	C₂×C₄²×C₁₃₉	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃₉	C₂³	1	2	yes	—
47	#103×C₁₃₉	—	2	16	no	—
48	#95×C₁₃₉	—	2	8	no	—
49	#95×C₁₃₉	—	2	8	no	—
50	C₂⁴×C₁₃₉	C₂×D₄	1	1	yes	characteristic
51	C₂×Q₈×C₁₃₉	—	2	4	no	—
52	C₂×Q₈×C₁₃₉	—	2	2	no	—
53	C₂×Q₈×C₁₃₉	—	2	16	no	—
54	C₂×Q₈×C₁₃₉	—	2	4	no	—
55	C₂×Q₈×C₁₃₉	—	2	4	no	—
56	C₂×Q₈×C₁₃₉	—	4	2	no	—
57	C₂×Q₈×C₁₃₉	—	4	2	no	—
58	C₂×Q₈×C₁₃₉	—	4	8	no	—
59	C₂×Q₈×C₁₃₉	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₁₃₉	—	2	4	no	—
61	C₂²×C₄×C₁₃₉	—	2	4	no	—
62	C₂²×C₄×C₁₃₉	—	2	4	no	—
63	C₂²×C₄×C₁₃₉	—	2	4	no	—
64	C₂²×C₄×C₁₃₉	—	2	2	no	—
65	C₂²×C₄×C₁₃₉	—	2	4	no	—
66	C₂²×C₄×C₁₃₉	—	2	2	no	—
67	C₂²×C₄×C₁₃₉	—	4	2	no	—
68	C₂²×C₄×C₁₃₉	—	4	2	no	—
69	C₂²×C₄×C₁₃₉	—	4	4	no	—
70	C₂²×C₄×C₁₃₉	—	4	2	no	—
71	C₂²×C₄×C₁₃₉	—	4	4	no	—
72	C₂²×C₄×C₁₃₉	—	4	4	no	—
73	C₂²×C₄×C₁₃₉	—	4	4	no	—
74	C₂²×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃₉	#41	1	2	yes	—
78	C₂²×C₄×C₁₃₉	#41	1	4	yes	—
79	C₂²×C₄×C₁₃₉	#41	1	4	yes	—
80	C₂²×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₁₃₉	—	2	4	no	—
82	C₂×C₈×C₁₃₉	—	2	4	no	—
83	C₂×C₈×C₁₃₉	—	2	4	no	—
84	C₂×C₈×C₁₃₉	—	4	8	no	—
85	C₄⋊C₄×C₁₃₉	—	2	4	no	—
86	C₄⋊C₄×C₁₃₉	—	2	4	no	—
87	C₄⋊C₄×C₁₃₉	—	2	4	no	—
88	C₄⋊C₄×C₁₃₉	—	4	4	no	—
89	C₄⋊C₄×C₁₃₉	—	4	2	no	—
90	C₄²×C₁₃₉	—	2	4	no	—
91	C₄²×C₁₃₉	—	2	4	no	—
92	C₂³×C₁₃₉	#121	1	2	yes	—
93	C₂³×C₁₃₉	#123	1	2	yes	—
94	C₂³×C₁₃₉	#124	1	2	yes	—
95	C₂³×C₁₃₉	#135	1	4	yes	—
96	C₂³×C₁₃₉	#137	1	4	yes	—
97	C₂³×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₁₃₉	—	2	8	no	—
99	Q₈×C₁₃₉	—	4	4	no	—
100	Q₈×C₁₃₉	—	4	8	no	—
101	C₂×C₄×C₁₃₉	—	2	4	no	—
102	C₂×C₄×C₁₃₉	—	2	4	no	—
103	C₂×C₄×C₁₃₉	—	2	4	no	—
104	C₂×C₄×C₁₃₉	—	2	4	no	—
105	C₂×C₄×C₁₃₉	—	4	2	no	—
106	C₂×C₄×C₁₃₉	—	4	2	no	—
107	C₂×C₄×C₁₃₉	—	4	8	no	—
108	C₂×C₄×C₁₃₉	—	4	8	no	—
109	C₂×C₄×C₁₃₉	—	4	4	no	—
110	C₂×C₄×C₁₃₉	—	4	2	no	—
111	C₂×C₄×C₁₃₉	—	4	4	no	—
112	C₂×C₄×C₁₃₉	—	4	4	no	—
113	C₂×C₄×C₁₃₉	—	4	4	no	—
114	C₂×C₄×C₁₃₉	—	8	8	no	—
115	C₂×C₄×C₁₃₉	—	8	4	no	—
116	C₂×C₄×C₁₃₉	#121	1	2	yes	—
117	C₂×C₄×C₁₃₉	#123	1	8	yes	—
118	C₂×C₄×C₁₃₉	#123	1	2	yes	—
119	C₂×C₄×C₁₃₉	#124	1	2	yes	—
120	C₂×C₄×C₁₃₉	#126	1	8	yes	—
121	C₂×C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
123	C₂×C₄×C₁₃₉	C₂×#41	1	2	yes	—
124	C₂×C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
125	C₈×C₁₃₉	—	4	4	no	—
126	C₂²×C₁₃₉	#397	1	4	yes	—
127	C₂²×C₁₃₉	#464	1	8	yes	—
128	C₂²×C₁₃₉	#485	1	8	yes	—
129	C₂²×C₁₃₉	#488	1	4	yes	—
130	C₂²×C₁₃₉	#491	1	4	yes	—
131	C₂²×C₁₃₉	C₂×#121	1	1	yes	characteristic
132	C₂²×C₁₃₉	C₂×#123	1	1	yes	characteristic
133	C₂²×C₁₃₉	C₂×#124	1	1	yes	characteristic
134	C₂²×C₁₃₉	C₂×#135	1	2	yes	—
135	C₂²×C₁₃₉	C₂×#137	1	2	yes	—
136	C₄×C₁₃₉	—	4	4	no	—
137	C₄×C₁₃₉	—	4	4	no	—
138	C₄×C₁₃₉	—	8	2	no	—
139	C₄×C₁₃₉	—	8	4	no	—
140	C₄×C₁₃₉	#397	1	4	yes	—
141	C₄×C₁₃₉	C₂×#123	1	2	yes	—
142	C₄×C₁₃₉	C₂×#126	1	2	yes	—
143	C₂×C₁₃₉	#2074	1	8	yes	direct product factor
144	C₂×C₁₃₉	C₂×#397	1	1	yes	characteristic
145	C₂×C₁₃₉	C₂×#464	1	2	yes	—
146	C₂×C₁₃₉	C₂×#485	1	2	yes	—
147	C₂×C₁₃₉	C₂×#488	1	1	yes	characteristic
148	C₂×C₁₃₉	C₂×#491	1	1	yes	characteristic
149	C₂×#2074	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₁₃₉	C₂×#2074	1	1	yes	characteristic, direct product factor, Sylow
151	C₂²×#103	C₂×C₁₃₉	1	2	yes	—
152	C₂×#391	C₂×C₁₃₉	1	1	yes	characteristic
153	C₂×#382	C₂×C₁₃₉	1	1	yes	characteristic
154	#2074	C₂×C₁₃₉	1	8	yes	direct product factor
155	C₂×#340	C₂×C₁₃₉	1	2	yes	—
156	C₂×#336	C₂×C₁₃₉	1	1	yes	characteristic
157	C₂³×Q₈	C₂²×C₁₃₉	1	1	yes	characteristic
158	C₂³×C₈	C₂²×C₁₃₉	1	2	yes	—
159	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	2	yes	—
160	C₂²×C₄⋊C₄	C₂²×C₁₃₉	1	1	yes	characteristic
161	C₂²×C₄²	C₂²×C₁₃₉	1	1	yes	characteristic
162	C₂×#103	—	2	8	no	—
163	C₂×#103	C₂²×C₁₃₉	1	8	yes	—
164	#391	—	2	2	no	—
165	#391	C₂²×C₁₃₉	1	4	yes	—
166	#382	—	2	2	no	—
167	#382	C₂²×C₁₃₉	1	4	yes	—
168	C₂×#95	—	2	2	no	—
169	C₂×#95	—	2	2	no	—
170	#340	C₂²×C₁₃₉	1	8	yes	—
171	#336	C₂²×C₁₃₉	1	4	yes	—
172	C₂²×Q₈	—	2	1	no	—
173	C₂²×Q₈	—	2	2	no	—
174	C₂²×Q₈	—	2	4	no	—
175	C₂²×Q₈	—	2	4	no	—
176	C₂²×Q₈	D₄×C₁₃₉	1	4	yes	—
177	C₂²×Q₈	C₂³×C₁₃₉	1	2	yes	—
178	C₂³×C₄	—	2	1	no	—
179	C₂³×C₄	—	2	2	no	—
180	C₂³×C₄	—	2	1	no	—
181	C₂³×C₄	D₄×C₁₃₉	1	2	yes	—
182	C₂³×C₄	C₂³×C₁₃₉	1	1	yes	characteristic
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	—	2	4	no	—
185	C₂²×C₈	—	2	4	no	—
186	C₂²×C₈	C₂³×C₁₃₉	1	4	yes	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	2	yes	—
193	C₂×C₄⋊C₄	C₂³×C₁₃₉	1	4	yes	—
194	C₂×C₄²	—	2	4	no	—
195	C₂×C₄²	D₄×C₁₃₉	1	2	yes	—
196	C₂×C₄²	C₂³×C₁₃₉	1	2	yes	—
197	#103	—	2	16	no	—
198	#95	—	2	8	no	—
199	#95	—	2	8	no	—
200	C₂⁴	C₂×D₄×C₁₃₉	1	1	yes	characteristic
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	2	16	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×Q₈	—	4	8	no	—
209	C₂×Q₈	C₂×D₄×C₁₃₉	1	4	yes	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	C₂×D₄×C₁₃₉	1	4	yes	—
225	C₂²×C₄	C₂×D₄×C₁₃₉	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₃₉	1	2	yes	—
227	C₂²×C₄	#41×C₁₃₉	1	2	yes	—
228	C₂²×C₄	#41×C₁₃₉	1	4	yes	—
229	C₂²×C₄	#41×C₁₃₉	1	4	yes	—
230	C₂²×C₄	C₂⁴×C₁₃₉	1	1	yes	characteristic
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	8	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	4	no	—
242	C₂³	#121×C₁₃₉	1	2	yes	—
243	C₂³	#123×C₁₃₉	1	2	yes	—
244	C₂³	#124×C₁₃₉	1	2	yes	—
245	C₂³	#135×C₁₃₉	1	4	yes	—
246	C₂³	#137×C₁₃₉	1	4	yes	—
247	C₂³	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
248	Q₈	—	2	8	no	—
249	Q₈	—	4	8	no	—
250	Q₈	—	4	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	8	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	8	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	#121×C₁₃₉	1	2	yes	—
267	C₂×C₄	#123×C₁₃₉	1	2	yes	—
268	C₂×C₄	#123×C₁₃₉	1	8	yes	—
269	C₂×C₄	#124×C₁₃₉	1	2	yes	—
270	C₂×C₄	#126×C₁₃₉	1	8	yes	—
271	C₂×C₄	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
272	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
273	C₂×C₄	C₂×#41×C₁₃₉	1	2	yes	—
274	C₂×C₄	C₂×#41×C₁₃₉	1	1	yes	characteristic
275	C₈	—	4	4	no	—
276	C₂²	#397×C₁₃₉	1	4	yes	—
277	C₂²	#464×C₁₃₉	1	8	yes	—
278	C₂²	#485×C₁₃₉	1	8	yes	—
279	C₂²	#488×C₁₃₉	1	4	yes	—
280	C₂²	#491×C₁₃₉	1	4	yes	—
281	C₂²	C₂×#121×C₁₃₉	1	1	yes	characteristic
282	C₂²	C₂×#123×C₁₃₉	1	1	yes	characteristic
283	C₂²	C₂×#124×C₁₃₉	1	1	yes	characteristic
284	C₂²	C₂×#135×C₁₃₉	1	2	yes	—
285	C₂²	C₂×#137×C₁₃₉	1	2	yes	—
286	C₄	—	4	4	no	—
287	C₄	—	4	4	no	—
288	C₄	—	8	4	no	—
289	C₄	—	8	2	no	—
290	C₄	#397×C₁₃₉	1	4	yes	—
291	C₄	C₂×#123×C₁₃₉	1	2	yes	—
292	C₄	C₂×#126×C₁₃₉	1	2	yes	—
293	C₂	#2074×C₁₃₉	1	8	yes	direct product factor
294	C₂	C₂×#397×C₁₃₉	1	1	yes	characteristic
295	C₂	C₂×#464×C₁₃₉	1	2	yes	—
296	C₂	C₂×#485×C₁₃₉	1	2	yes	—
297	C₂	C₂×#488×C₁₃₉	1	1	yes	characteristic
298	C₂	C₂×#491×C₁₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₃₉	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃₉ ◁ C₂²×C₄²×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. #2074×C₁₃₉	C₂	direct (N × H)
143. C₂×C₁₃₉	#2074	direct (N × H)
149. C₂×#2074	C₁₃₉	direct (N × H)
150. C₁₃₉	C₂×#2074	direct (N × H)
154. #2074	C₂×C₁₃₉	direct (N × H)
293. C₂	#2074×C₁₃₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#103×C₁₃₉	2
C₂×#391×C₁₃₉	2
C₂×#382×C₁₃₉	2
#2074×C₁₃₉	2
C₂×#340×C₁₃₉	2
C₂×#336×C₁₃₉	2
C₂×#2074	139

Subgroups — C2×#2074×C139 #48018222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2074×C₁₃₉ #48018222