Subgroups — C₃₁×#10828 #21467587

Order 7936.

642 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#344	C₂	1	2	yes	maximal
3	C₂×C₃₁×#341	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1735	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1733	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	2	yes	—
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₃₁×#405	—	2	2	no	—
12	C₃₁×#405	C₄	1	4	yes	complemented
13	C₂×C₃₁×#97	—	2	1	no	—
14	C₂×C₄×C₈×C₃₁	—	2	1	no	—
15	C₃₁×#344	C₄	1	8	yes	complemented
16	C₃₁×#341	—	2	1	no	—
17	C₃₁×#341	—	2	1	no	—
18	C₃₁×#340	—	2	2	no	—
19	C₃₁×#339	—	2	4	no	—
20	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₃₁	—	2	1	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₃₁	—	2	1	no	—
32	C₂²×C₈×C₃₁	—	2	2	no	—
33	C₂²×C₈×C₃₁	—	2	2	no	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	D₄	1	2	yes	—
39	C₃₁×#117	—	2	1	no	—
40	C₃₁×#117	—	2	1	no	—
41	C₃₁×#117	—	2	2	no	—
42	C₃₁×#117	—	2	2	no	—
43	C₃₁×#117	—	2	2	no	—
44	C₃₁×#117	C₂×C₄	1	4	yes	—
45	C₃₁×#117	C₂×C₄	1	2	yes	—
46	C₃₁×#117	D₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	Q₈	1	2	yes	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	—	2	1	no	—
52	C₂×C₄²×C₃₁	—	2	2	no	—
53	C₃₁×#97	—	2	4	no	—
54	C₄×C₈×C₃₁	—	2	4	no	—
55	C₂⁴×C₃₁	#41	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	4	2	no	—
58	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	#41	1	2	yes	—
66	C₂²×C₄×C₃₁	#41	1	2	yes	—
67	C₂²×C₄×C₃₁	#41	1	2	yes	—
68	M₁₆×C₃₁	—	2	2	no	—
69	M₁₆×C₃₁	—	2	4	no	—
70	M₁₆×C₃₁	—	2	4	no	—
71	M₁₆×C₃₁	—	2	4	no	—
72	M₁₆×C₃₁	—	4	2	no	—
73	M₁₆×C₃₁	—	4	2	no	—
74	M₁₆×C₃₁	C₄²	1	4	yes	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	4	no	—
83	C₂×C₈×C₃₁	—	4	2	no	—
84	C₂×C₈×C₃₁	—	4	2	no	—
85	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	4	2	no	—
90	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
91	C₂²⋊C₄×C₃₁	—	2	2	no	—
92	C₂²⋊C₄×C₃₁	—	2	2	no	—
93	C₂²⋊C₄×C₃₁	—	2	4	no	—
94	C₂²⋊C₄×C₃₁	—	4	2	no	—
95	C₄²×C₃₁	—	2	2	no	—
96	C₄²×C₃₁	—	2	2	no	—
97	C₄²×C₃₁	—	2	4	no	—
98	C₄²×C₃₁	—	4	2	no	—
99	#10828	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	—	4	2	no	—
101	C₂³×C₃₁	#117	1	1	yes	characteristic
102	C₂³×C₃₁	#117	1	2	yes	—
103	C₂³×C₃₁	#121	1	1	yes	characteristic
104	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₃₁	#126	1	2	yes	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	4	4	no	—
109	C₂×C₄×C₃₁	—	8	4	no	—
110	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#117	1	2	yes	—
117	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₃₁	#123	1	2	yes	—
124	C₂×C₄×C₃₁	#123	1	2	yes	—
125	C₂×C₄×C₃₁	#124	1	2	yes	—
126	C₂×C₄×C₃₁	#125	1	2	yes	—
127	C₂×C₄×C₃₁	#126	1	2	yes	—
128	C₈×C₃₁	—	2	4	no	—
129	C₈×C₃₁	—	2	2	no	—
130	C₈×C₃₁	—	2	2	no	—
131	C₈×C₃₁	—	4	4	no	—
132	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
133	C₂×#344	C₂×C₃₁	1	2	yes	—
134	C₂×#341	C₂×C₃₁	1	1	yes	characteristic
135	#1735	C₂×C₃₁	1	1	yes	characteristic
136	#1733	C₂×C₃₁	1	2	yes	—
137	C₂²×C₃₁	—	2	2	no	—
138	C₂²×C₃₁	—	2	2	no	—
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	4	2	no	—
141	C₂²×C₃₁	—	4	2	no	—
142	C₂²×C₃₁	#376	1	1	yes	characteristic
143	C₂²×C₃₁	#380	1	1	yes	characteristic
144	C₂²×C₃₁	#383	1	1	yes	characteristic
145	C₂²×C₃₁	#387	1	2	yes	—
146	C₂²×C₃₁	#388	1	2	yes	—
147	C₄×C₃₁	—	4	4	no	—
148	C₄×C₃₁	—	8	4	no	—
149	C₄×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
151	C₄×C₃₁	#381	1	1	yes	characteristic
152	C₄×C₃₁	#382	1	1	yes	characteristic
153	C₄×C₃₁	#387	1	2	yes	—
154	C₄×C₃₁	#388	1	2	yes	—
155	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
156	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
157	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
158	C₂×#117	C₂²×C₃₁	1	2	yes	—
159	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
160	#405	—	2	2	no	—
161	#405	C₄×C₃₁	1	4	yes	complemented
162	C₂×#97	—	2	1	no	—
163	C₂×C₄×C₈	—	2	1	no	—
164	#344	C₄×C₃₁	1	8	yes	complemented
165	#341	—	2	1	no	—
166	#341	—	2	1	no	—
167	#340	—	2	2	no	—
168	#339	—	2	4	no	—
169	C₂×C₃₁	—	2	2	no	—
170	C₂×C₃₁	—	2	1	no	—
171	C₂×C₃₁	—	2	1	no	—
172	C₂×C₃₁	—	2	2	no	—
173	C₂×C₃₁	#1434	1	1	yes	characteristic
174	C₂×C₃₁	#1814	1	2	yes	—
175	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
184	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
185	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
191	C₂²×C₈	D₄×C₃₁	1	2	yes	—
192	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
193	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
194	#117	—	2	1	no	—
195	#117	—	2	1	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	C₂×C₄×C₃₁	1	4	yes	—
200	#117	C₂×C₄×C₃₁	1	2	yes	—
201	#117	D₄×C₃₁	1	2	yes	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	Q₈×C₃₁	1	2	yes	—
204	C₂×C₂²⋊C₄	—	2	1	no	—
205	C₂×C₂²⋊C₄	—	2	2	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	2	2	no	—
208	#97	—	2	4	no	—
209	C₄×C₈	—	2	4	no	—
210	C₃₁	#10828	1	1	yes	characteristic, direct product factor, Sylow
211	C₂⁴	#41×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	#41×C₃₁	1	2	yes	—
222	C₂²×C₄	#41×C₃₁	1	2	yes	—
223	C₂²×C₄	#41×C₃₁	1	2	yes	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	C₄²×C₃₁	1	4	yes	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
242	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	C₃₁×#117	1	1	yes	characteristic
257	C₂³	C₃₁×#117	1	2	yes	—
258	C₂³	C₃₁×#121	1	1	yes	characteristic
259	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
260	C₂³	C₃₁×#126	1	2	yes	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
268	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
269	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
270	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
271	C₂×C₄	C₃₁×#117	1	2	yes	—
272	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
273	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
275	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
276	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
277	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
278	C₂×C₄	C₃₁×#123	1	2	yes	—
279	C₂×C₄	C₃₁×#123	1	2	yes	—
280	C₂×C₄	C₃₁×#124	1	2	yes	—
281	C₂×C₄	C₃₁×#125	1	2	yes	—
282	C₂×C₄	C₃₁×#126	1	2	yes	—
283	C₈	—	2	4	no	—
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	—	4	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	C₃₁×#376	1	1	yes	characteristic
293	C₂²	C₃₁×#380	1	1	yes	characteristic
294	C₂²	C₃₁×#383	1	1	yes	characteristic
295	C₂²	C₃₁×#387	1	2	yes	—
296	C₂²	C₃₁×#388	1	2	yes	—
297	C₄	—	4	4	no	—
298	C₄	—	8	4	no	—
299	C₄	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
300	C₄	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
301	C₄	C₃₁×#381	1	1	yes	characteristic
302	C₄	C₃₁×#382	1	1	yes	characteristic
303	C₄	C₃₁×#387	1	2	yes	—
304	C₄	C₃₁×#388	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	C₃₁×#1434	1	1	yes	characteristic
310	C₂	C₃₁×#1814	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	#380
prime radical	C₂⁴×C₃₁	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₃₁×#405	C₄	semidirect
15. C₃₁×#344	C₄	semidirect
99. #10828	C₃₁	direct (N × H)
161. #405	C₄×C₃₁	semidirect
164. #344	C₄×C₃₁	semidirect
210. C₃₁	#10828	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×#405	2
C₂×C₃₁×#344	2
C₂×C₃₁×#341	2
C₃₁×#1735	2
C₃₁×#1733	2
#10828	31

Subgroups — C31×#10828 #21467587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10828 #21467587