Subgroups — C₃₁×#19271 #21476030

Order 7936.

702 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×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3039	C₂	1	2	yes	maximal
3	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1674	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1673	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1546	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#137	—	2	2	no	—
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	C₂²	1	4	yes	—
14	C₃₁×#462	—	2	2	no	—
15	C₃₁×#462	C₂²	1	4	yes	—
16	C₃₁×#461	—	2	2	no	—
17	C₃₁×#461	C₂²	1	4	yes	—
18	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
19	C₈×D₄×C₃₁	—	2	2	no	—
20	C₈×D₄×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#431	—	2	1	no	—
22	C₃₁×#431	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#107	—	2	2	no	—
24	C₃₁×#423	C₂²	1	2	yes	—
25	C₃₁×#419	—	2	2	no	—
26	C₂×C₄×C₈×C₃₁	—	2	1	no	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#335	C₂²	1	4	yes	—
29	C₃₁×#332	C₂²	1	8	yes	—
30	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₃₁	—	2	2	no	—
32	C₃₁×#137	—	4	8	no	—
33	C₂×Q₁₆×C₃₁	—	2	4	no	—
34	C₂×SD₁₆×C₃₁	—	2	4	no	—
35	C₂×M₁₆×C₃₁	—	2	2	no	—
36	C₂²×C₈×C₃₁	—	2	2	no	—
37	C₂²×C₈×C₃₁	D₄	1	2	yes	—
38	C₃₁×C₄⋊Q₈	—	2	2	no	—
39	C₃₁×C₄⋊Q₈	—	2	2	no	—
40	C₃₁×C₄⋊Q₈	—	2	2	no	—
41	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
42	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
43	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₄×D₄×C₃₁	—	2	1	no	—
45	C₄×D₄×C₃₁	—	2	1	no	—
46	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
47	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#117	—	2	1	no	—
49	C₃₁×#117	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	—	2	1	no	—
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#107	—	2	4	no	—
55	C₃₁×#107	—	2	4	no	—
56	C₃₁×C₄⋊C₈	—	2	2	no	—
57	C₃₁×C₄⋊C₈	—	2	2	no	—
58	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
59	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
60	C₃₁×#103	—	2	4	no	—
61	C₃₁×#103	—	2	4	no	—
62	C₃₁×C₂²⋊C₈	—	2	1	no	—
63	C₃₁×C₂²⋊C₈	—	2	2	no	—
64	C₄×C₈×C₃₁	—	2	2	no	—
65	C₄×C₈×C₃₁	—	2	1	no	—
66	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₃₁	C₂³	1	2	yes	—
69	#41×C₃₁	—	2	2	no	—
70	#41×C₃₁	D₈	1	4	yes	—
71	C₂×Q₈×C₃₁	—	2	4	no	—
72	C₂×Q₈×C₃₁	—	4	2	no	—
73	C₂×Q₈×C₃₁	—	4	4	no	—
74	C₂×Q₈×C₃₁	—	4	4	no	—
75	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂×D₄×C₃₁	—	2	1	no	—
77	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	—	2	1	no	—
79	C₂²×C₄×C₃₁	—	2	1	no	—
80	C₂²×C₄×C₃₁	—	2	1	no	—
81	C₂²×C₄×C₃₁	—	4	2	no	—
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
85	Q₁₆×C₃₁	—	4	8	no	—
86	SD₁₆×C₃₁	—	4	8	no	—
87	M₁₆×C₃₁	—	4	4	no	—
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	4	no	—
91	C₂×C₈×C₃₁	—	4	4	no	—
92	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₃₁	—	2	1	no	—
95	C₄⋊C₄×C₃₁	—	2	1	no	—
96	C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₄⋊C₄×C₃₁	—	2	4	no	—
98	C₄⋊C₄×C₃₁	—	4	2	no	—
99	C₄⋊C₄×C₃₁	—	4	2	no	—
100	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²⋊C₄×C₃₁	—	2	1	no	—
103	C₂²⋊C₄×C₃₁	—	2	1	no	—
104	C₂²⋊C₄×C₃₁	—	2	1	no	—
105	C₄²×C₃₁	—	2	1	no	—
106	C₄²×C₃₁	—	2	1	no	—
107	C₄²×C₃₁	—	2	1	no	—
108	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₃₁	#41	1	1	yes	characteristic
111	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
112	#19271	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₃₁	—	2	1	no	—
114	C₂³×C₃₁	#121	1	1	yes	characteristic
115	Q₈×C₃₁	—	2	1	no	—
116	Q₈×C₃₁	—	4	4	no	—
117	Q₈×C₃₁	—	4	4	no	—
118	Q₈×C₃₁	—	8	4	no	—
119	Q₈×C₃₁	C₂×D₈	1	2	yes	—
120	D₄×C₃₁	—	2	4	no	—
121	D₄×C₃₁	—	2	1	no	—
122	D₄×C₃₁	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₃₁	—	2	1	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	1	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	4	no	—
132	C₂×C₄×C₃₁	—	8	4	no	—
133	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₃₁	#135	1	2	yes	—
138	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
141	C₈×C₃₁	—	2	2	no	—
142	C₈×C₃₁	—	4	4	no	—
143	C₈×C₃₁	—	4	2	no	—
144	C₈×C₃₁	#137	1	4	yes	complemented
145	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#3039	C₂×C₃₁	1	2	yes	—
147	C₈×#41	C₂×C₃₁	1	1	yes	characteristic
148	#1674	C₂×C₃₁	1	4	yes	complemented
149	#1673	C₂×C₃₁	1	4	yes	complemented
150	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#1546	C₂×C₃₁	1	2	yes	complemented
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	2	2	no	—
155	C₂²×C₃₁	—	4	1	no	—
156	C₂²×C₃₁	—	4	1	no	—
157	C₂²×C₃₁	#466	1	2	yes	—
158	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
159	C₄×C₃₁	—	2	1	no	—
160	C₄×C₃₁	—	2	2	no	—
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	4	1	no	—
163	C₄×C₃₁	—	4	1	no	—
164	C₄×C₃₁	—	8	4	no	—
165	C₄×C₃₁	#466	1	2	yes	—
166	C₄×C₃₁	#473	1	2	yes	—
167	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
168	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
169	C₄×C₃₁	C₂×#137	1	2	yes	—
170	C₂×#137	—	2	2	no	—
171	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
172	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
173	#500	—	2	1	no	—
174	#500	—	2	1	no	—
175	#500	C₂²×C₃₁	1	4	yes	—
176	#462	—	2	2	no	—
177	#462	C₂²×C₃₁	1	4	yes	—
178	#461	—	2	2	no	—
179	#461	C₂²×C₃₁	1	4	yes	—
180	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
181	C₈×D₄	—	2	2	no	—
182	C₈×D₄	C₂²×C₃₁	1	2	yes	—
183	#431	—	2	1	no	—
184	#431	C₂²×C₃₁	1	1	yes	characteristic
185	C₂×#107	—	2	2	no	—
186	#423	C₂²×C₃₁	1	2	yes	—
187	#419	—	2	2	no	—
188	C₂×C₄×C₈	—	2	1	no	—
189	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
190	#335	C₂²×C₃₁	1	4	yes	—
191	#332	C₂²×C₃₁	1	8	yes	—
192	C₂×C₃₁	—	2	2	no	—
193	C₂×C₃₁	—	4	2	no	—
194	C₂×C₃₁	#3039	1	1	yes	characteristic
195	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
196	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
197	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
198	C₂²×Q₈	—	2	2	no	—
199	#137	—	4	8	no	—
200	C₂×Q₁₆	—	2	4	no	—
201	C₂×SD₁₆	—	2	4	no	—
202	C₂×M₁₆	—	2	2	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	D₄×C₃₁	1	2	yes	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	—	2	2	no	—
208	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
209	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
210	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	—	2	1	no	—
213	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
214	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
215	#117	—	2	1	no	—
216	#117	D₄×C₃₁	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
221	#107	—	2	4	no	—
222	#107	—	2	4	no	—
223	C₄⋊C₈	—	2	2	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
226	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
227	#103	—	2	4	no	—
228	#103	—	2	4	no	—
229	C₂²⋊C₈	—	2	1	no	—
230	C₂²⋊C₈	—	2	2	no	—
231	C₄×C₈	—	2	2	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
234	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₃₁	1	2	yes	—
236	C₃₁	#19271	1	1	yes	characteristic, direct product factor, Sylow
237	#41	—	2	2	no	—
238	#41	D₈×C₃₁	1	4	yes	—
239	C₂×Q₈	—	2	4	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	4	4	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
244	C₂×D₄	—	2	1	no	—
245	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
253	Q₁₆	—	4	8	no	—
254	SD₁₆	—	4	8	no	—
255	M₁₆	—	4	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	4	4	no	—
260	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
261	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₂²⋊C₄	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₄²	#41×C₃₁	1	1	yes	characteristic
279	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
280	C₂³	—	2	1	no	—
281	C₂³	C₃₁×#121	1	1	yes	characteristic
282	Q₈	—	2	1	no	—
283	Q₈	—	4	4	no	—
284	Q₈	—	4	4	no	—
285	Q₈	—	8	4	no	—
286	Q₈	C₂×D₈×C₃₁	1	2	yes	—
287	D₄	—	2	4	no	—
288	D₄	—	2	1	no	—
289	D₄	C₂×D₈×C₃₁	1	2	yes	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
301	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
302	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
304	C₂×C₄	C₃₁×#135	1	2	yes	—
305	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	4	no	—
310	C₈	—	4	2	no	—
311	C₈	C₃₁×#137	1	4	yes	complemented
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₃₁×#466	1	2	yes	—
318	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	8	4	no	—
325	C₄	C₃₁×#466	1	2	yes	—
326	C₄	C₃₁×#473	1	2	yes	—
327	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
328	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
329	C₄	C₂×C₃₁×#137	1	2	yes	—
330	C₂	—	2	2	no	—
331	C₂	—	4	2	no	—
332	C₂	C₃₁×#3039	1	1	yes	characteristic
333	C₂	C₂×C₃₁×#466	1	1	yes	characteristic
334	C₂	C₂×C₃₁×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ 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×#1674	C₂	semidirect
5. C₃₁×#1673	C₂	semidirect
6. C₃₁×#1567	C₂	semidirect
7. C₃₁×#1546	C₂	semidirect
112. #19271	C₃₁	direct (N × H)
144. C₈×C₃₁	#137	semidirect
145. C₂×#500	C₂×C₃₁	semidirect
148. #1674	C₂×C₃₁	semidirect
149. #1673	C₂×C₃₁	semidirect
150. #1567	C₂×C₃₁	semidirect
151. #1546	C₂×C₃₁	semidirect
236. C₃₁	#19271	direct (N × H)
311. C₈	C₃₁×#137	semidirect

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×#3039	2
C₈×#41×C₃₁	2
C₃₁×#1674	2
C₃₁×#1673	2
C₃₁×#1567	2
C₃₁×#1546	2
#19271	31

Subgroups — C31×#19271 #21476030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19271 #21476030