Subgroups — C₂×C₃₁×#2075 #21480394

Order 7936.

1270 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#380	C₂	1	2	yes	maximal
4	C₃₁×#2075	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃₁×#340	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₂²×C₂²⋊C₄×C₃₁	—	2	2	no	—
13	C₂×C₃₁×#107	—	2	4	no	—
14	C₂×C₃₁×#107	C₂²	1	4	yes	complemented
15	C₂×C₃₁×C₂²⋊C₈	—	2	4	no	—
16	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
17	C₃₁×#380	—	2	4	no	—
18	C₃₁×#380	C₂²	1	8	yes	—
19	C₂×C₃₁×#95	—	2	2	no	—
20	C₃₁×#340	C₂²	1	8	yes	complemented
21	C₃₁×#340	C₂²	1	4	yes	complemented
22	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₃₁	—	2	2	no	—
24	C₂³×C₄×C₃₁	—	2	2	no	—
25	C₂³×C₄×C₃₁	—	2	4	no	—
26	C₂³×C₄×C₃₁	—	2	4	no	—
27	C₂³×C₄×C₃₁	—	2	2	no	—
28	C₂³×C₄×C₃₁	—	2	2	no	—
29	C₂³×C₄×C₃₁	D₄	1	2	yes	—
30	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
31	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
32	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	—	2	4	no	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	—	2	4	no	—
36	C₂²×C₈×C₃₁	D₄	1	4	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
48	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
49	C₃₁×#107	—	2	8	no	—
50	C₃₁×C₂²⋊C₈	—	2	8	no	—
51	C₃₁×#95	—	2	8	no	—
52	C₂⁴×C₃₁	—	2	2	no	—
53	C₂⁴×C₃₁	—	2	4	no	—
54	C₂⁴×C₃₁	—	2	2	no	—
55	C₂⁴×C₃₁	—	2	4	no	—
56	C₂⁴×C₃₁	D₈	1	4	yes	—
57	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
58	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	2	8	no	—
63	C₂²×C₄×C₃₁	—	2	2	no	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	—	2	4	no	—
66	C₂²×C₄×C₃₁	—	2	4	no	—
67	C₂²×C₄×C₃₁	—	2	4	no	—
68	C₂²×C₄×C₃₁	—	2	4	no	—
69	C₂²×C₄×C₃₁	—	4	4	no	—
70	C₂²×C₄×C₃₁	—	4	4	no	—
71	C₂²×C₄×C₃₁	—	4	8	no	—
72	C₂²×C₄×C₃₁	—	4	4	no	—
73	C₂²×C₄×C₃₁	—	4	4	no	—
74	C₂²×C₄×C₃₁	—	4	4	no	—
75	C₂²×C₄×C₃₁	—	4	4	no	—
76	C₂²×C₄×C₃₁	—	4	4	no	—
77	C₂²×C₄×C₃₁	—	4	4	no	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₃₁	#41	1	2	yes	—
82	C₂²×C₄×C₃₁	#41	1	4	yes	—
83	C₂²×C₄×C₃₁	#41	1	4	yes	—
84	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₃₁	—	2	8	no	—
86	C₂×C₈×C₃₁	—	2	4	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₂×C₈×C₃₁	—	4	4	no	—
90	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
91	C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	—	4	4	no	—
96	C₄⋊C₄×C₃₁	—	4	4	no	—
97	C₂²⋊C₄×C₃₁	—	4	16	no	—
98	C₂×#2075	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₃₁	—	2	2	no	—
100	C₂³×C₃₁	—	2	4	no	—
101	C₂³×C₃₁	—	2	2	no	—
102	C₂³×C₃₁	—	2	4	no	—
103	C₂³×C₃₁	—	2	8	no	—
104	C₂³×C₃₁	—	2	16	no	—
105	C₂³×C₃₁	—	4	4	no	—
106	C₂³×C₃₁	—	4	4	no	—
107	C₂³×C₃₁	—	4	8	no	—
108	C₂³×C₃₁	#121	1	2	yes	—
109	C₂³×C₃₁	#123	1	4	yes	—
110	C₂³×C₃₁	C₂×D₈	1	4	yes	—
111	C₂³×C₃₁	C₂×D₈	1	2	yes	—
112	C₂³×C₃₁	#135	1	2	yes	—
113	C₂³×C₃₁	#137	1	4	yes	—
114	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	4	8	no	—
122	C₂×C₄×C₃₁	—	4	4	no	—
123	C₂×C₄×C₃₁	—	4	4	no	—
124	C₂×C₄×C₃₁	—	4	4	no	—
125	C₂×C₄×C₃₁	—	4	4	no	—
126	C₂×C₄×C₃₁	—	4	4	no	—
127	C₂×C₄×C₃₁	—	4	4	no	—
128	C₂×C₄×C₃₁	—	8	8	no	—
129	C₂×C₄×C₃₁	—	8	8	no	—
130	C₂×C₄×C₃₁	#121	1	2	yes	—
131	C₂×C₄×C₃₁	#123	1	4	yes	—
132	C₂×C₄×C₃₁	#123	1	4	yes	—
133	C₂×C₄×C₃₁	#124	1	4	yes	—
134	C₂×C₄×C₃₁	#126	1	8	yes	—
135	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
137	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
138	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
139	C₈×C₃₁	—	2	4	no	—
140	C₈×C₃₁	—	4	2	no	—
141	C₂²×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂²×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
143	C₂×#380	C₂×C₃₁	1	2	yes	—
144	#2075	C₂×C₃₁	1	8	yes	direct product factor
145	C₂×#340	C₂×C₃₁	1	2	yes	complemented
146	C₂×#340	C₂×C₃₁	1	1	yes	characteristic, complemented
147	C₂²×C₃₁	—	2	8	no	—
148	C₂²×C₃₁	—	2	4	no	—
149	C₂²×C₃₁	—	4	4	no	—
150	C₂²×C₃₁	—	4	16	no	—
151	C₂²×C₃₁	—	4	2	no	—
152	C₂²×C₃₁	—	4	2	no	—
153	C₂²×C₃₁	#397	1	4	yes	—
154	C₂²×C₃₁	#466	1	4	yes	—
155	C₂²×C₃₁	#470	1	4	yes	—
156	C₂²×C₃₁	#480	1	8	yes	—
157	C₂²×C₃₁	#485	1	8	yes	—
158	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
159	C₂²×C₃₁	C₂×#123	1	2	yes	—
160	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
161	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
162	C₂²×C₃₁	C₂×#137	1	2	yes	—
163	C₄×C₃₁	—	4	2	no	—
164	C₄×C₃₁	—	8	4	no	—
165	C₄×C₃₁	—	8	4	no	—
166	C₄×C₃₁	#397	1	4	yes	—
167	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
168	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
169	C₄×C₃₁	C₂×#126	1	2	yes	—
170	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
171	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
172	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
173	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
174	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
175	C₂²×C₂²⋊C₄	—	2	2	no	—
176	C₂×#107	—	2	4	no	—
177	C₂×#107	C₂²×C₃₁	1	4	yes	complemented
178	C₂×C₂²⋊C₈	—	2	4	no	—
179	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
180	#380	—	2	4	no	—
181	#380	C₂²×C₃₁	1	8	yes	—
182	C₂×#95	—	2	2	no	—
183	#340	C₂²×C₃₁	1	4	yes	complemented
184	#340	C₂²×C₃₁	1	8	yes	complemented
185	C₂×C₃₁	—	4	4	no	—
186	C₂×C₃₁	#2075	1	8	yes	direct product factor
187	C₂×C₃₁	C₂×#397	1	1	yes	characteristic
188	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
189	C₂×C₃₁	C₂×#470	1	1	yes	characteristic
190	C₂×C₃₁	C₂×#480	1	2	yes	—
191	C₂×C₃₁	C₂×#485	1	2	yes	—
192	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	—	2	2	no	—
195	C₂³×C₄	—	2	4	no	—
196	C₂³×C₄	—	2	4	no	—
197	C₂³×C₄	—	2	2	no	—
198	C₂³×C₄	—	2	2	no	—
199	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
200	C₂³×C₄	D₄×C₃₁	1	2	yes	—
201	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
202	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
203	C₂²×C₈	—	2	4	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	2	4	no	—
206	C₂²×C₈	D₄×C₃₁	1	4	yes	—
207	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
208	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
216	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
217	C₂×C₂²⋊C₄	—	2	8	no	—
218	C₂×C₂²⋊C₄	—	4	8	no	—
219	#107	—	2	8	no	—
220	C₂²⋊C₈	—	2	8	no	—
221	#95	—	2	8	no	—
222	C₃₁	C₂×#2075	1	1	yes	characteristic, direct product factor, Sylow
223	C₂⁴	—	2	2	no	—
224	C₂⁴	—	2	4	no	—
225	C₂⁴	—	2	2	no	—
226	C₂⁴	—	2	4	no	—
227	C₂⁴	D₈×C₃₁	1	4	yes	—
228	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
229	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	4	8	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
252	C₂²×C₄	#41×C₃₁	1	2	yes	—
253	C₂²×C₄	#41×C₃₁	1	4	yes	—
254	C₂²×C₄	#41×C₃₁	1	4	yes	—
255	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
256	C₂×C₈	—	2	8	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	4	4	no	—
261	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₂²⋊C₄	—	4	16	no	—
269	C₂³	—	2	8	no	—
270	C₂³	—	2	16	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	4	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	4	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	4	no	—
277	C₂³	—	4	8	no	—
278	C₂³	C₃₁×#121	1	2	yes	—
279	C₂³	C₃₁×#123	1	4	yes	—
280	C₂³	C₂×D₈×C₃₁	1	4	yes	—
281	C₂³	C₂×D₈×C₃₁	1	2	yes	—
282	C₂³	C₃₁×#135	1	2	yes	—
283	C₂³	C₃₁×#137	1	4	yes	—
284	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	8	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	8	8	no	—
299	C₂×C₄	—	8	8	no	—
300	C₂×C₄	C₃₁×#121	1	2	yes	—
301	C₂×C₄	C₃₁×#123	1	4	yes	—
302	C₂×C₄	C₃₁×#123	1	4	yes	—
303	C₂×C₄	C₃₁×#124	1	4	yes	—
304	C₂×C₄	C₃₁×#126	1	8	yes	—
305	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
307	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
308	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
309	C₈	—	2	4	no	—
310	C₈	—	4	2	no	—
311	C₂²	—	2	8	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	4	16	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	C₃₁×#397	1	4	yes	—
318	C₂²	C₃₁×#466	1	4	yes	—
319	C₂²	C₃₁×#470	1	4	yes	—
320	C₂²	C₃₁×#480	1	8	yes	—
321	C₂²	C₃₁×#485	1	8	yes	—
322	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
323	C₂²	C₂×C₃₁×#123	1	2	yes	—
324	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
325	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
326	C₂²	C₂×C₃₁×#137	1	2	yes	—
327	C₄	—	4	2	no	—
328	C₄	—	8	4	no	—
329	C₄	—	8	4	no	—
330	C₄	C₃₁×#397	1	4	yes	—
331	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
332	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
333	C₄	C₂×C₃₁×#126	1	2	yes	—
334	C₂	—	4	4	no	—
335	C₂	C₃₁×#2075	1	8	yes	direct product factor
336	C₂	C₂×C₃₁×#397	1	1	yes	characteristic
337	C₂	C₂×C₃₁×#466	1	1	yes	characteristic
338	C₂	C₂×C₃₁×#470	1	1	yes	characteristic
339	C₂	C₂×C₃₁×#480	1	2	yes	—
340	C₂	C₂×C₃₁×#485	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁵×C₃₁	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×#107	C₂	semidirect
4. C₃₁×#2075	C₂	direct (N × H)
5. C₂×C₃₁×#340	C₂	semidirect
6. C₂×C₃₁×#340	C₂	semidirect
14. C₂×C₃₁×#107	C₂²	semidirect
20. C₃₁×#340	C₂²	semidirect
21. C₃₁×#340	C₂²	semidirect
98. C₂×#2075	C₃₁	direct (N × H)
141. C₂²×#107	C₂×C₃₁	semidirect
144. #2075	C₂×C₃₁	direct (N × H)
145. C₂×#340	C₂×C₃₁	semidirect
146. C₂×#340	C₂×C₃₁	semidirect
177. C₂×#107	C₂²×C₃₁	semidirect
183. #340	C₂²×C₃₁	semidirect
184. #340	C₂²×C₃₁	semidirect
186. C₂×C₃₁	#2075	direct (N × H)
222. C₃₁	C₂×#2075	direct (N × H)
335. C₂	C₃₁×#2075	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×C₃₁×#107	2
C₂²×C₃₁×C₂²⋊C₈	2
C₂×C₃₁×#380	2
C₃₁×#2075	2
C₂×C₃₁×#340	2
C₂×#2075	31

Subgroups — C2×C31×#2075 #21480394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2075 #21480394