Subgroups — C₃₁×#33213 #21489972

Order 7936.

810 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2628	C₂	1	2	yes	maximal
4	C₃₁×#1993	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1943	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1941	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1809	C₂	1	2	yes	maximal
9	C₃₁×#1807	C₂	1	2	yes	maximal
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₃₁×#498	—	2	4	no	—
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×#106	C₂²	1	2	yes	—
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₃₁×#405	—	2	4	no	—
20	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#400	—	2	2	no	—
25	C₃₁×#399	—	2	2	no	—
26	C₃₁×#395	—	2	2	no	—
27	C₃₁×#393	—	2	2	no	—
28	C₃₁×#389	C₂²	1	4	yes	—
29	C₃₁×#382	C₂²	1	2	yes	—
30	C₃₁×#380	C₂²	1	4	yes	—
31	C₃₁×#340	C₂²	1	4	yes	—
32	C₃₁×#340	C₂²	1	4	yes	—
33	C₂²×Q₈×C₃₁	—	2	2	no	—
34	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₃₁	—	2	2	no	—
36	C₂×M₁₆×C₃₁	—	2	2	no	—
37	C₂×M₁₆×C₃₁	Q₈	1	4	yes	complemented
38	C₂²×C₈×C₃₁	—	2	2	no	—
39	C₂²×C₈×C₃₁	—	2	2	no	—
40	C₂²×C₈×C₃₁	Q₈	1	4	yes	complemented
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×C₄⋊Q₈	—	2	4	no	—
46	C₃₁×C₄⋊Q₈	—	2	4	no	—
47	C₃₁×#117	—	2	2	no	—
48	C₃₁×#117	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
51	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
54	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
55	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×#107	—	2	4	no	—
59	C₃₁×#106	—	2	4	no	—
60	C₃₁×#106	—	2	4	no	—
61	C₃₁×#106	—	2	4	no	—
62	C₃₁×#97	—	2	2	no	—
63	C₃₁×#97	—	2	2	no	—
64	C₄×C₈×C₃₁	—	2	2	no	—
65	C₄×C₈×C₃₁	D₄	1	4	yes	—
66	C₃₁×#95	—	2	4	no	—
67	C₃₁×#95	—	2	2	no	—
68	C₃₁×#95	—	2	2	no	—
69	C₃₁×#95	—	2	4	no	—
70	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₃₁	—	4	4	no	—
72	C₂×Q₈×C₃₁	—	8	4	no	—
73	C₂²×C₄×C₃₁	—	2	1	no	—
74	C₂²×C₄×C₃₁	—	2	1	no	—
75	C₂²×C₄×C₃₁	—	2	1	no	—
76	C₂²×C₄×C₃₁	—	2	1	no	—
77	C₂²×C₄×C₃₁	—	2	4	no	—
78	C₂²×C₄×C₃₁	—	2	4	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₃₁	—	2	8	no	—
86	C₂×C₈×C₃₁	—	2	1	no	—
87	C₂×C₈×C₃₁	—	2	1	no	—
88	C₂×C₈×C₃₁	—	2	1	no	—
89	C₂×C₈×C₃₁	—	2	1	no	—
90	C₂×C₈×C₃₁	—	2	8	no	—
91	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
97	C₂×C₈×C₃₁	#41	1	2	yes	—
98	C₂×C₈×C₃₁	#41	1	2	yes	—
99	C₄⋊C₄×C₃₁	—	2	4	no	—
100	C₄⋊C₄×C₃₁	—	2	4	no	—
101	C₄⋊C₄×C₃₁	—	2	4	no	—
102	C₄⋊C₄×C₃₁	—	2	4	no	—
103	C₄⋊C₄×C₃₁	—	2	4	no	—
104	C₄⋊C₄×C₃₁	—	4	4	no	—
105	C₄⋊C₄×C₃₁	—	8	4	no	—
106	C₂²⋊C₄×C₃₁	—	2	4	no	—
107	C₂²⋊C₄×C₃₁	—	8	4	no	—
108	C₄²×C₃₁	—	2	1	no	—
109	C₄²×C₃₁	—	2	1	no	—
110	C₄²×C₃₁	C₂×D₄	1	2	yes	—
111	C₄²×C₃₁	#41	1	2	yes	—
112	#33213	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₃₁	—	2	1	no	—
114	C₂³×C₃₁	—	2	1	no	—
115	C₂³×C₃₁	—	2	4	no	—
116	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
117	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
118	Q₈×C₃₁	—	8	8	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	1	no	—
121	C₂×C₄×C₃₁	—	2	4	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	2	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	4	no	—
128	C₂×C₄×C₃₁	—	4	4	no	—
129	C₂×C₄×C₃₁	—	4	4	no	—
130	C₂×C₄×C₃₁	—	4	4	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	#121	1	4	yes	—
135	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
136	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
142	C₈×C₃₁	—	2	2	no	—
143	C₈×C₃₁	—	2	2	no	—
144	C₈×C₃₁	C₂²⋊Q₈	1	4	yes	—
145	C₈×C₃₁	C₂²⋊Q₈	1	4	yes	—
146	C₂×#498	C₂×C₃₁	1	1	yes	characteristic, complemented
147	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
148	#2628	C₂×C₃₁	1	2	yes	—
149	#1993	C₂×C₃₁	1	2	yes	complemented
150	#1943	C₂×C₃₁	1	2	yes	complemented
151	#1941	C₂×C₃₁	1	2	yes	complemented
152	#1846	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#1809	C₂×C₃₁	1	2	yes	—
154	#1807	C₂×C₃₁	1	2	yes	—
155	C₂²×C₃₁	—	2	2	no	—
156	C₂²×C₃₁	—	2	4	no	—
157	C₂²×C₃₁	—	4	2	no	—
158	C₂²×C₃₁	—	4	2	no	—
159	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
160	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
161	C₂²×C₃₁	#532	1	1	yes	characteristic
162	C₂²×C₃₁	#577	1	4	yes	—
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	2	4	no	—
165	C₄×C₃₁	—	4	2	no	—
166	C₄×C₃₁	—	8	4	no	—
167	C₄×C₃₁	—	8	4	no	—
168	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
169	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
170	C₄×C₃₁	#533	1	1	yes	characteristic
171	C₄×C₃₁	#533	1	1	yes	characteristic
172	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
173	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
175	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
176	#498	—	2	4	no	—
177	C₂×#107	C₂²×C₃₁	1	2	yes	—
178	C₂×#106	C₂²×C₃₁	1	2	yes	—
179	C₂×#106	C₂²×C₃₁	1	2	yes	—
180	C₂×#106	C₂²×C₃₁	1	2	yes	—
181	#405	—	2	4	no	—
182	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
183	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
186	#400	—	2	2	no	—
187	#399	—	2	2	no	—
188	#395	—	2	2	no	—
189	#393	—	2	2	no	—
190	#389	C₂²×C₃₁	1	4	yes	—
191	#382	C₂²×C₃₁	1	2	yes	—
192	#380	C₂²×C₃₁	1	4	yes	—
193	#340	C₂²×C₃₁	1	4	yes	—
194	#340	C₂²×C₃₁	1	4	yes	—
195	C₂×C₃₁	—	4	2	no	—
196	C₂×C₃₁	#2857	1	1	yes	characteristic
197	C₂×C₃₁	#3057	1	2	yes	—
198	C₂×C₃₁	#3163	1	2	yes	—
199	C₂×C₃₁	#3173	1	2	yes	—
200	C₂²×Q₈	—	2	2	no	—
201	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
202	C₂×M₁₆	—	2	2	no	—
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	Q₈×C₃₁	1	4	yes	complemented
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	Q₈×C₃₁	1	4	yes	complemented
208	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
210	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
211	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
212	C₄⋊Q₈	—	2	4	no	—
213	C₄⋊Q₈	—	2	4	no	—
214	#117	—	2	2	no	—
215	#117	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	4	no	—
218	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
220	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
221	C₂×C₂²⋊C₄	—	2	4	no	—
222	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
223	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
224	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
225	#107	—	2	4	no	—
226	#106	—	2	4	no	—
227	#106	—	2	4	no	—
228	#106	—	2	4	no	—
229	#97	—	2	2	no	—
230	#97	—	2	2	no	—
231	C₄×C₈	—	2	2	no	—
232	C₄×C₈	D₄×C₃₁	1	4	yes	—
233	#95	—	2	4	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	4	no	—
237	C₃₁	#33213	1	1	yes	characteristic, direct product factor, Sylow
238	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂×Q₈	—	4	4	no	—
240	C₂×Q₈	—	8	4	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	4	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×Q₈×C₃₁	1	2	yes	—
251	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
252	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
253	M₁₆	—	2	8	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	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	C₂×Q₈×C₃₁	1	2	yes	—
262	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
263	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
264	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
265	C₂×C₈	#41×C₃₁	1	2	yes	—
266	C₂×C₈	#41×C₃₁	1	2	yes	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	4	4	no	—
273	C₄⋊C₄	—	8	4	no	—
274	C₂²⋊C₄	—	2	4	no	—
275	C₂²⋊C₄	—	8	4	no	—
276	C₄²	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	C₂×D₄×C₃₁	1	2	yes	—
279	C₄²	#41×C₃₁	1	2	yes	—
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	2	4	no	—
283	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
284	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
285	Q₈	—	8	8	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	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	—	4	4	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	C₃₁×#121	1	4	yes	—
302	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
303	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	2	2	no	—
311	C₈	C₃₁×C₂²⋊Q₈	1	4	yes	—
312	C₈	C₃₁×C₂²⋊Q₈	1	4	yes	—
313	C₂²	—	2	2	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
318	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
319	C₂²	C₃₁×#532	1	1	yes	characteristic
320	C₂²	C₃₁×#577	1	4	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	2	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	8	4	no	—
325	C₄	—	8	4	no	—
326	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
327	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
328	C₄	C₃₁×#533	1	1	yes	characteristic
329	C₄	C₃₁×#533	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	C₃₁×#2857	1	1	yes	characteristic
332	C₂	C₃₁×#3057	1	2	yes	—
333	C₂	C₃₁×#3163	1	2	yes	—
334	C₂	C₃₁×#3173	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
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
1. C₂×C₃₁×#498	C₂	semidirect
4. C₃₁×#1993	C₂	semidirect
5. C₃₁×#1943	C₂	semidirect
6. C₃₁×#1941	C₂	semidirect
7. C₃₁×#1846	C₂	semidirect
37. C₂×M₁₆×C₃₁	Q₈	semidirect
40. C₂²×C₈×C₃₁	Q₈	semidirect
112. #33213	C₃₁	direct (N × H)
146. C₂×#498	C₂×C₃₁	semidirect
149. #1993	C₂×C₃₁	semidirect
150. #1943	C₂×C₃₁	semidirect
151. #1941	C₂×C₃₁	semidirect
152. #1846	C₂×C₃₁	semidirect
204. C₂×M₁₆	Q₈×C₃₁	semidirect
207. C₂²×C₈	Q₈×C₃₁	semidirect
237. C₃₁	#33213	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×#498	2
C₂×C₃₁×#405	2
C₃₁×#2628	2
C₃₁×#1993	2
C₃₁×#1943	2
C₃₁×#1941	2
C₃₁×#1846	2
C₃₁×#1809	2
C₃₁×#1807	2
#33213	31

Subgroups — C31×#33213 #21489972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33213 #21489972