Subgroups — C₃₁×#33132 #21489891

Order 7936.

894 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2624	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1972	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1944	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1934	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1895	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#121	C₂²	1	2	yes	—
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₃₁×#466	—	2	4	no	—
15	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#106	C₂²	1	2	yes	—
17	C₂×C₃₁×#102	C₂²	1	4	yes	—
18	C₂×C₃₁×#102	C₂²	1	2	yes	—
19	C₃₁×#405	—	2	4	no	—
20	C₂×C₃₁×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#399	—	2	2	no	—
23	C₃₁×#397	—	2	2	no	—
24	C₃₁×#394	—	2	4	no	—
25	C₃₁×#390	C₂²	1	4	yes	—
26	C₃₁×#389	C₂²	1	4	yes	—
27	C₃₁×#380	C₂²	1	2	yes	—
28	C₃₁×#340	C₂²	1	2	yes	—
29	C₃₁×#340	C₂²	1	4	yes	—
30	C₂²×D₄×C₃₁	—	2	2	no	—
31	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
32	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂×D₈×C₃₁	—	2	2	no	—
34	C₂×D₈×C₃₁	—	2	4	no	—
35	C₂×M₁₆×C₃₁	—	2	2	no	—
36	C₂×M₁₆×C₃₁	—	2	4	no	—
37	C₂²×C₈×C₃₁	—	2	4	no	—
38	C₂²×C₈×C₃₁	D₄	1	4	yes	—
39	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×#121	—	2	4	no	—
43	C₃₁×#121	—	2	4	no	—
44	C₃₁×#117	—	2	2	no	—
45	C₃₁×#117	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
47	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
54	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
55	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
56	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
57	C₃₁×#107	—	2	2	no	—
58	C₃₁×#106	—	2	4	no	—
59	C₃₁×#102	—	2	4	no	—
60	C₃₁×#102	—	2	8	no	—
61	C₃₁×#97	—	2	4	no	—
62	C₄×C₈×C₃₁	—	2	4	no	—
63	C₃₁×#95	—	2	2	no	—
64	C₃₁×#95	—	2	4	no	—
65	C₂⁴×C₃₁	—	2	2	no	—
66	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂×D₄×C₃₁	—	2	2	no	—
68	C₂×D₄×C₃₁	—	2	4	no	—
69	C₂×D₄×C₃₁	—	2	2	no	—
70	C₂×D₄×C₃₁	—	2	4	no	—
71	C₂×D₄×C₃₁	—	4	4	no	—
72	C₂×D₄×C₃₁	—	8	4	no	—
73	C₂²×C₄×C₃₁	—	2	2	no	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	4	no	—
76	C₂²×C₄×C₃₁	—	2	2	no	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
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	C₂⁴	1	1	yes	characteristic
83	D₈×C₃₁	—	4	4	no	—
84	M₁₆×C₃₁	—	2	8	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	8	no	—
89	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₃₁	#41	1	4	yes	—
92	C₂×C₈×C₃₁	#41	1	4	yes	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₄⋊C₄×C₃₁	—	2	4	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	8	4	no	—
100	C₂²⋊C₄×C₃₁	—	2	4	no	—
101	C₂²⋊C₄×C₃₁	—	4	4	no	—
102	C₂²⋊C₄×C₃₁	—	8	4	no	—
103	C₂²⋊C₄×C₃₁	—	8	4	no	—
104	C₂²⋊C₄×C₃₁	—	8	2	no	—
105	C₄²×C₃₁	—	2	2	no	—
106	C₄²×C₃₁	—	2	2	no	—
107	#33132	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₃₁	—	2	2	no	—
109	C₂³×C₃₁	—	2	2	no	—
110	C₂³×C₃₁	—	2	2	no	—
111	C₂³×C₃₁	—	4	2	no	—
112	C₂³×C₃₁	—	4	4	no	—
113	C₂³×C₃₁	—	4	4	no	—
114	C₂³×C₃₁	—	4	2	no	—
115	C₂³×C₃₁	—	4	2	no	—
116	C₂³×C₃₁	#121	1	2	yes	—
117	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
118	D₄×C₃₁	—	4	2	no	—
119	D₄×C₃₁	—	4	2	no	—
120	D₄×C₃₁	—	4	4	no	—
121	D₄×C₃₁	—	8	8	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	2	4	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	4	no	—
129	C₂×C₄×C₃₁	—	2	4	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	4	no	—
132	C₂×C₄×C₃₁	—	4	4	no	—
133	C₂×C₄×C₃₁	—	4	4	no	—
134	C₂×C₄×C₃₁	—	4	4	no	—
135	C₂×C₄×C₃₁	#121	1	2	yes	—
136	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
138	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
139	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
140	C₈×C₃₁	—	2	2	no	—
141	C₈×C₃₁	—	2	4	no	—
142	C₈×C₃₁	#123	1	4	yes	—
143	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
144	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#2624	C₂×C₃₁	1	2	yes	complemented
146	#1972	C₂×C₃₁	1	2	yes	complemented
147	#1944	C₂×C₃₁	1	2	yes	complemented
148	#1934	C₂×C₃₁	1	4	yes	complemented
149	#1895	C₂×C₃₁	1	2	yes	complemented
150	#1807	C₂×C₃₁	1	1	yes	characteristic, complemented
151	C₂²×C₃₁	—	2	2	no	—
152	C₂²×C₃₁	—	2	2	no	—
153	C₂²×C₃₁	—	2	2	no	—
154	C₂²×C₃₁	—	4	4	no	—
155	C₂²×C₃₁	—	8	2	no	—
156	C₂²×C₃₁	—	8	4	no	—
157	C₂²×C₃₁	—	8	4	no	—
158	C₂²×C₃₁	—	8	2	no	—
159	C₂²×C₃₁	—	8	2	no	—
160	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
161	C₂²×C₃₁	#532	1	2	yes	—
162	C₂²×C₃₁	#576	1	2	yes	—
163	C₂²×C₃₁	#577	1	2	yes	—
164	C₄×C₃₁	—	2	2	no	—
165	C₄×C₃₁	—	4	2	no	—
166	C₄×C₃₁	—	4	2	no	—
167	C₄×C₃₁	—	8	4	no	—
168	C₄×C₃₁	—	8	2	no	—
169	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
170	C₄×C₃₁	#525	1	1	yes	characteristic
171	C₄×C₃₁	#529	1	2	yes	—
172	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
173	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
174	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×#121	C₂²×C₃₁	1	2	yes	—
176	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
177	#466	—	2	4	no	—
178	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×#106	C₂²×C₃₁	1	2	yes	—
180	C₂×#102	C₂²×C₃₁	1	4	yes	—
181	C₂×#102	C₂²×C₃₁	1	2	yes	—
182	#405	—	2	4	no	—
183	C₂×#97	C₂²×C₃₁	1	2	yes	—
184	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
185	#399	—	2	2	no	—
186	#397	—	2	2	no	—
187	#394	—	2	4	no	—
188	#390	C₂²×C₃₁	1	4	yes	—
189	#389	C₂²×C₃₁	1	4	yes	—
190	#380	C₂²×C₃₁	1	2	yes	—
191	#340	C₂²×C₃₁	1	2	yes	—
192	#340	C₂²×C₃₁	1	4	yes	—
193	C₂×C₃₁	—	4	2	no	—
194	C₂×C₃₁	—	16	2	no	—
195	C₂×C₃₁	#2854	1	1	yes	characteristic
196	C₂×C₃₁	#3058	1	1	yes	characteristic
197	C₂×C₃₁	#3060	1	1	yes	characteristic
198	C₂×C₃₁	#3179	1	4	yes	—
199	C₂²×D₄	—	2	2	no	—
200	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
201	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
202	C₂×D₈	—	2	2	no	—
203	C₂×D₈	—	2	4	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	2	4	no	—
206	C₂²×C₈	—	2	4	no	—
207	C₂²×C₈	D₄×C₃₁	1	4	yes	—
208	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
210	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
211	#121	—	2	4	no	—
212	#121	—	2	4	no	—
213	#117	—	2	2	no	—
214	#117	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	4	no	—
216	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
217	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
218	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₂²⋊C₄	—	2	4	no	—
223	C₂×C₂²⋊C₄	—	2	4	no	—
224	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
225	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
226	#107	—	2	2	no	—
227	#106	—	2	4	no	—
228	#102	—	2	4	no	—
229	#102	—	2	8	no	—
230	#97	—	2	4	no	—
231	C₄×C₈	—	2	4	no	—
232	#95	—	2	2	no	—
233	#95	—	2	4	no	—
234	C₃₁	#33132	1	1	yes	characteristic, direct product factor, Sylow
235	C₂⁴	—	2	2	no	—
236	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	8	4	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
251	C₂²×C₄	#41×C₃₁	1	2	yes	—
252	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
253	D₈	—	4	4	no	—
254	M₁₆	—	2	8	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	8	no	—
259	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
260	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
261	C₂×C₈	#41×C₃₁	1	4	yes	—
262	C₂×C₈	#41×C₃₁	1	4	yes	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	8	4	no	—
270	C₂²⋊C₄	—	2	4	no	—
271	C₂²⋊C₄	—	4	4	no	—
272	C₂²⋊C₄	—	8	4	no	—
273	C₂²⋊C₄	—	8	4	no	—
274	C₂²⋊C₄	—	8	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	2	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	4	no	—
282	C₂³	—	4	4	no	—
283	C₂³	—	4	2	no	—
284	C₂³	—	4	2	no	—
285	C₂³	C₃₁×#121	1	2	yes	—
286	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	4	4	no	—
290	D₄	—	8	8	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	C₃₁×#121	1	2	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	2	no	—
310	C₈	—	2	4	no	—
311	C₈	C₃₁×#123	1	4	yes	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	4	no	—
318	C₂²	—	8	4	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	8	2	no	—
321	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
322	C₂²	C₃₁×#532	1	2	yes	—
323	C₂²	C₃₁×#576	1	2	yes	—
324	C₂²	C₃₁×#577	1	2	yes	—
325	C₄	—	2	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	4	no	—
329	C₄	—	8	2	no	—
330	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
331	C₄	C₃₁×#525	1	1	yes	characteristic
332	C₄	C₃₁×#529	1	2	yes	—
333	C₂	—	4	2	no	—
334	C₂	—	16	2	no	—
335	C₂	C₃₁×#2854	1	1	yes	characteristic
336	C₂	C₃₁×#3058	1	1	yes	characteristic
337	C₂	C₃₁×#3060	1	1	yes	characteristic
338	C₂	C₃₁×#3179	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#405	C₂	semidirect
3. C₃₁×#2624	C₂	semidirect
4. C₃₁×#1972	C₂	semidirect
5. C₃₁×#1944	C₂	semidirect
6. C₃₁×#1934	C₂	semidirect
7. C₃₁×#1895	C₂	semidirect
8. C₃₁×#1807	C₂	semidirect
107. #33132	C₃₁	direct (N × H)
144. C₂×#405	C₂×C₃₁	semidirect
145. #2624	C₂×C₃₁	semidirect
146. #1972	C₂×C₃₁	semidirect
147. #1944	C₂×C₃₁	semidirect
148. #1934	C₂×C₃₁	semidirect
149. #1895	C₂×C₃₁	semidirect
150. #1807	C₂×C₃₁	semidirect
234. C₃₁	#33132	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#466	2
C₂×C₃₁×#405	2
C₃₁×#2624	2
C₃₁×#1972	2
C₃₁×#1944	2
C₃₁×#1934	2
C₃₁×#1895	2
C₃₁×#1807	2
#33132	31

Subgroups — C31×#33132 #21489891

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33132 #21489891