Subgroups — C₃₁×#33222 #21489981

Order 7936.

738 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×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3079	C₂	1	2	yes	maximal
3	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1997	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1952	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1816	C₂	1	4	yes	maximal
8	C₂×C₃₁×#131	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#500	—	2	1	no	—
12	C₃₁×#500	—	2	1	no	—
13	C₃₁×#500	—	2	2	no	—
14	C₃₁×#477	—	2	2	no	—
15	C₃₁×#476	—	2	2	no	—
16	C₃₁×#475	—	2	2	no	—
17	C₃₁×#474	—	2	2	no	—
18	C₂×C₃₁×#108	C₂²	1	4	yes	—
19	C₃₁×#428	C₂²	1	4	yes	—
20	C₂×C₃₁×#107	C₂²	1	4	yes	—
21	C₃₁×#423	C₂²	1	2	yes	—
22	C₃₁×#417	C₂²	1	4	yes	—
23	C₃₁×#405	—	2	2	no	—
24	C₃₁×#405	C₂²	1	2	yes	—
25	C₄×M₁₆×C₃₁	—	2	1	no	—
26	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	—	2	1	no	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#343	C₂²	1	8	yes	—
30	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×Q₈×C₃₁	—	2	2	no	—
32	C₃₁×#131	—	2	4	no	—
33	C₃₁×#131	Q₈	1	8	yes	complemented
34	C₂×M₁₆×C₃₁	—	2	2	no	—
35	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
36	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	—	2	2	no	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₃₁×C₄⋊Q₈	—	2	2	no	—
40	C₃₁×C₄⋊Q₈	—	2	2	no	—
41	C₃₁×C₄⋊Q₈	—	2	2	no	—
42	C₃₁×C₄⋊Q₈	—	2	2	no	—
43	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
44	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
45	C₄×D₄×C₃₁	—	2	1	no	—
46	C₄×D₄×C₃₁	—	2	1	no	—
47	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
48	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
49	C₃₁×#117	—	2	1	no	—
50	C₃₁×#117	C₂³	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
53	C₂×C₄²×C₃₁	—	2	1	no	—
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×#108	—	2	8	no	—
56	C₃₁×#107	—	2	4	no	—
57	C₃₁×#107	—	2	4	no	—
58	C₃₁×#107	—	2	4	no	—
59	C₃₁×#106	—	2	4	no	—
60	C₃₁×C₄⋊C₈	—	2	2	no	—
61	C₃₁×C₄⋊C₈	—	2	2	no	—
62	C₃₁×#103	—	2	4	no	—
63	C₃₁×#102	—	2	4	no	—
64	C₃₁×#97	—	2	2	no	—
65	C₃₁×#97	—	2	1	no	—
66	C₄×C₈×C₃₁	—	2	2	no	—
67	C₄×C₈×C₃₁	—	2	1	no	—
68	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₃₁	D₄	1	2	yes	—
71	#41×C₃₁	—	2	2	no	—
72	#41×C₃₁	C₂×Q₈	1	4	yes	—
73	C₂×Q₈×C₃₁	—	4	2	no	—
74	C₂×Q₈×C₃₁	—	4	2	no	—
75	C₂×Q₈×C₃₁	—	8	4	no	—
76	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂×D₄×C₃₁	—	2	1	no	—
78	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	—	2	1	no	—
80	C₂²×C₄×C₃₁	—	2	1	no	—
81	C₂²×C₄×C₃₁	—	2	1	no	—
82	C₂²×C₄×C₃₁	—	2	4	no	—
83	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
86	M₁₆×C₃₁	—	2	2	no	—
87	M₁₆×C₃₁	—	2	8	no	—
88	M₁₆×C₃₁	—	2	8	no	—
89	M₁₆×C₃₁	C₂×Q₈	1	4	yes	—
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	8	no	—
92	C₂×C₈×C₃₁	—	2	4	no	—
93	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
96	C₂×C₈×C₃₁	#41	1	4	yes	—
97	C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₄⋊C₄×C₃₁	—	2	4	no	—
100	C₄⋊C₄×C₃₁	—	2	4	no	—
101	C₄⋊C₄×C₃₁	—	4	2	no	—
102	C₄⋊C₄×C₃₁	—	4	2	no	—
103	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²⋊C₄×C₃₁	—	2	1	no	—
106	C₂²⋊C₄×C₃₁	—	2	1	no	—
107	C₂²⋊C₄×C₃₁	—	2	1	no	—
108	C₄²×C₃₁	—	2	1	no	—
109	C₄²×C₃₁	—	2	1	no	—
110	C₄²×C₃₁	—	2	1	no	—
111	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
115	#33222	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂³×C₃₁	—	2	1	no	—
117	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
118	Q₈×C₃₁	—	2	1	no	—
119	Q₈×C₃₁	—	8	4	no	—
120	Q₈×C₃₁	—	8	4	no	—
121	Q₈×C₃₁	C₄⋊Q₈	1	2	yes	—
122	D₄×C₃₁	—	2	4	no	—
123	D₄×C₃₁	—	2	1	no	—
124	D₄×C₃₁	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	2	4	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	1	no	—
130	C₂×C₄×C₃₁	—	2	1	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	4	no	—
134	C₂×C₄×C₃₁	—	8	4	no	—
135	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
136	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
137	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
138	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
142	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
143	C₈×C₃₁	—	2	4	no	—
144	C₈×C₃₁	—	4	4	no	—
145	C₈×C₃₁	C₂²⋊Q₈	1	4	yes	—
146	C₈×C₃₁	C₂²⋊Q₈	1	4	yes	—
147	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#3079	C₂×C₃₁	1	2	yes	—
149	C₄×#131	C₂×C₃₁	1	1	yes	characteristic
150	#1997	C₂×C₃₁	1	2	yes	complemented
151	#1952	C₂×C₃₁	1	4	yes	complemented
152	#1848	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#1816	C₂×C₃₁	1	4	yes	—
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	2	2	no	—
157	C₂²×C₃₁	—	4	1	no	—
158	C₂²×C₃₁	—	4	1	no	—
159	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
160	C₂²×C₃₁	#533	1	2	yes	—
161	C₄×C₃₁	—	2	2	no	—
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	4	1	no	—
165	C₄×C₃₁	—	4	1	no	—
166	C₄×C₃₁	—	8	4	no	—
167	C₄×C₃₁	#472	1	2	yes	—
168	C₄×C₃₁	#473	1	2	yes	—
169	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
170	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
171	C₄×C₃₁	#532	1	1	yes	characteristic
172	C₂×#131	C₂²×C₃₁	1	2	yes	—
173	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
174	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
175	#500	—	2	1	no	—
176	#500	—	2	1	no	—
177	#500	—	2	2	no	—
178	#477	—	2	2	no	—
179	#476	—	2	2	no	—
180	#475	—	2	2	no	—
181	#474	—	2	2	no	—
182	C₂×#108	C₂²×C₃₁	1	4	yes	—
183	#428	C₂²×C₃₁	1	4	yes	—
184	C₂×#107	C₂²×C₃₁	1	4	yes	—
185	#423	C₂²×C₃₁	1	2	yes	—
186	#417	C₂²×C₃₁	1	4	yes	—
187	#405	—	2	2	no	—
188	#405	C₂²×C₃₁	1	2	yes	—
189	C₄×M₁₆	—	2	1	no	—
190	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
191	C₂×C₄×C₈	—	2	1	no	—
192	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
193	#343	C₂²×C₃₁	1	8	yes	—
194	C₂×C₃₁	—	2	2	no	—
195	C₂×C₃₁	—	4	2	no	—
196	C₂×C₃₁	#2857	1	1	yes	characteristic
197	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
198	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
199	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
200	C₂²×Q₈	—	2	2	no	—
201	#131	—	2	4	no	—
202	#131	Q₈×C₃₁	1	8	yes	complemented
203	C₂×M₁₆	—	2	2	no	—
204	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
205	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
208	C₄⋊Q₈	—	2	2	no	—
209	C₄⋊Q₈	—	2	2	no	—
210	C₄⋊Q₈	—	2	2	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
213	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
217	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
218	#117	—	2	1	no	—
219	#117	C₂³×C₃₁	1	1	yes	characteristic
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
224	#108	—	2	8	no	—
225	#107	—	2	4	no	—
226	#107	—	2	4	no	—
227	#107	—	2	4	no	—
228	#106	—	2	4	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	2	no	—
231	#103	—	2	4	no	—
232	#102	—	2	4	no	—
233	#97	—	2	2	no	—
234	#97	—	2	1	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
238	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
239	C₄×C₈	D₄×C₃₁	1	2	yes	—
240	C₃₁	#33222	1	1	yes	characteristic, direct product factor, Sylow
241	#41	—	2	2	no	—
242	#41	C₂×Q₈×C₃₁	1	4	yes	—
243	C₂×Q₈	—	4	2	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	8	4	no	—
246	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	4	no	—
253	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
254	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
256	M₁₆	—	2	2	no	—
257	M₁₆	—	2	8	no	—
258	M₁₆	—	2	8	no	—
259	M₁₆	C₂×Q₈×C₃₁	1	4	yes	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	8	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
264	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
265	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
266	C₂×C₈	#41×C₃₁	1	4	yes	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	—	8	4	no	—
289	Q₈	—	8	4	no	—
290	Q₈	C₃₁×C₄⋊Q₈	1	2	yes	—
291	D₄	—	2	4	no	—
292	D₄	—	2	1	no	—
293	D₄	C₃₁×C₄⋊Q₈	1	2	yes	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
305	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
306	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
307	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
310	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
311	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
312	C₈	—	2	4	no	—
313	C₈	—	4	4	no	—
314	C₈	C₃₁×C₂²⋊Q₈	1	4	yes	—
315	C₈	C₃₁×C₂²⋊Q₈	1	4	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
322	C₂²	C₃₁×#533	1	2	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	8	4	no	—
329	C₄	C₃₁×#472	1	2	yes	—
330	C₄	C₃₁×#473	1	2	yes	—
331	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
332	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
333	C₄	C₃₁×#532	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	4	2	no	—
336	C₂	C₃₁×#2857	1	1	yes	characteristic
337	C₂	C₂×C₃₁×#472	1	1	yes	characteristic
338	C₂	C₂×C₃₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#500	C₂	semidirect
4. C₃₁×#1997	C₂	semidirect
5. C₃₁×#1952	C₂	semidirect
6. C₃₁×#1848	C₂	semidirect
33. C₃₁×#131	Q₈	semidirect
115. #33222	C₃₁	direct (N × H)
147. C₂×#500	C₂×C₃₁	semidirect
150. #1997	C₂×C₃₁	semidirect
151. #1952	C₂×C₃₁	semidirect
152. #1848	C₂×C₃₁	semidirect
202. #131	Q₈×C₃₁	semidirect
240. C₃₁	#33222	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×#500	2
C₃₁×#3079	2
C₄×C₃₁×#131	2
C₃₁×#1997	2
C₃₁×#1952	2
C₃₁×#1848	2
C₃₁×#1816	2
#33222	31

Subgroups — C31×#33222 #21489981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33222 #21489981