Subgroups — C₃₁×#32828 #21489587

Order 7936.

570 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2647	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2642	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2097	C₂	1	2	yes	maximal
4	C₃₁×#2072	C₂	1	2	yes	maximal
5	C₃₁×#2068	C₂	1	2	yes	maximal
6	C₃₁×#2066	C₂	1	2	yes	maximal
7	C₃₁×#2059	C₂	1	2	yes	maximal
8	C₃₁×#2043	C₂	1	2	yes	maximal
9	C₃₁×#1832	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#107	C₂²	1	2	yes	—
13	C₂×C₃₁×#106	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#103	C₂²	1	2	yes	—
17	C₂×C₃₁×#103	C₂²	1	2	yes	—
18	C₂×C₃₁×#103	C₂²	1	2	yes	—
19	C₂×C₃₁×#103	C₂²	1	2	yes	—
20	C₃₁×#400	—	2	2	no	—
21	C₃₁×#400	—	2	2	no	—
22	C₃₁×#400	—	2	2	no	—
23	C₃₁×#398	—	2	2	no	—
24	C₃₁×#391	C₂²	1	2	yes	—
25	C₃₁×#391	C₂²	1	2	yes	—
26	C₃₁×#389	C₂²	1	2	yes	—
27	C₃₁×#389	C₂²	1	2	yes	—
28	C₃₁×#384	C₂²	1	1	yes	characteristic
29	C₃₁×#382	C₂²	1	1	yes	characteristic
30	C₃₁×#376	C₂²	1	1	yes	characteristic
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₃₁×C₄⋊Q₈	—	4	1	no	—
39	C₃₁×C₄⋊Q₈	—	4	1	no	—
40	C₃₁×#125	—	4	1	no	—
41	C₃₁×#125	—	4	1	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#107	—	4	2	no	—
54	C₃₁×#106	—	4	2	no	—
55	C₃₁×C₄⋊C₈	—	4	1	no	—
56	C₃₁×C₄⋊C₈	—	4	1	no	—
57	C₃₁×#103	—	4	2	no	—
58	C₃₁×#103	—	4	2	no	—
59	C₃₁×#103	—	4	2	no	—
60	C₃₁×#103	—	4	2	no	—
61	C₃₁×#95	—	2	2	no	—
62	C₃₁×#95	—	2	2	no	—
63	C₃₁×#95	—	2	2	no	—
64	C₃₁×#95	—	2	2	no	—
65	C₃₁×#95	—	2	4	no	—
66	C₃₁×#95	—	2	2	no	—
67	C₃₁×#95	D₄	1	2	yes	—
68	C₃₁×#95	D₄	1	2	yes	—
69	C₂×Q₈×C₃₁	—	2	2	no	—
70	C₂×Q₈×C₃₁	—	2	2	no	—
71	C₂×Q₈×C₃₁	—	2	2	no	—
72	C₂×Q₈×C₃₁	—	2	2	no	—
73	C₂×Q₈×C₃₁	—	2	2	no	—
74	C₂×Q₈×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	—	2	2	no	—
77	C₂²×C₄×C₃₁	—	2	2	no	—
78	C₂²×C₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₃₁	—	2	2	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	2	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
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	—	4	4	no	—
97	C₄⋊C₄×C₃₁	—	4	4	no	—
98	C₄⋊C₄×C₃₁	—	8	2	no	—
99	C₄⋊C₄×C₃₁	—	8	2	no	—
100	C₄⋊C₄×C₃₁	—	8	2	no	—
101	C₄²×C₃₁	—	2	1	no	—
102	C₄²×C₃₁	—	2	1	no	—
103	C₄²×C₃₁	—	2	1	no	—
104	C₄²×C₃₁	—	2	1	no	—
105	C₄²×C₃₁	—	2	1	no	—
106	C₄²×C₃₁	—	2	1	no	—
107	#32828	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₃₁	—	4	2	no	—
110	Q₈×C₃₁	—	4	2	no	—
111	Q₈×C₃₁	—	4	2	no	—
112	Q₈×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	2	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
139	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
141	C₈×C₃₁	—	4	2	no	—
142	C₈×C₃₁	—	4	2	no	—
143	#2647	C₂×C₃₁	1	1	yes	characteristic
144	#2642	C₂×C₃₁	1	1	yes	characteristic
145	#2097	C₂×C₃₁	1	2	yes	—
146	#2072	C₂×C₃₁	1	2	yes	—
147	#2068	C₂×C₃₁	1	2	yes	—
148	#2066	C₂×C₃₁	1	2	yes	—
149	#2059	C₂×C₃₁	1	2	yes	—
150	#2043	C₂×C₃₁	1	2	yes	—
151	#1832	C₂×C₃₁	1	1	yes	characteristic
152	C₂²×C₃₁	#535	1	1	yes	characteristic
153	C₂²×C₃₁	#536	1	1	yes	characteristic
154	C₂²×C₃₁	#537	1	1	yes	characteristic
155	C₂²×C₃₁	#577	1	2	yes	—
156	C₂²×C₃₁	#578	1	2	yes	—
157	C₄×C₃₁	—	4	2	no	—
158	C₄×C₃₁	—	4	2	no	—
159	C₄×C₃₁	—	4	1	no	—
160	C₄×C₃₁	—	4	1	no	—
161	C₄×C₃₁	—	8	2	no	—
162	C₄×C₃₁	—	8	2	no	—
163	C₄×C₃₁	—	8	2	no	—
164	C₄×C₃₁	—	8	2	no	—
165	C₄×C₃₁	#562	1	1	yes	characteristic
166	C₄×C₃₁	#562	1	1	yes	characteristic
167	C₄×C₃₁	#563	1	1	yes	characteristic
168	C₄×C₃₁	#563	1	1	yes	characteristic
169	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
171	C₂×#107	C₂²×C₃₁	1	2	yes	—
172	C₂×#106	C₂²×C₃₁	1	2	yes	—
173	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×#103	C₂²×C₃₁	1	2	yes	—
176	C₂×#103	C₂²×C₃₁	1	2	yes	—
177	C₂×#103	C₂²×C₃₁	1	2	yes	—
178	C₂×#103	C₂²×C₃₁	1	2	yes	—
179	#400	—	2	2	no	—
180	#400	—	2	2	no	—
181	#400	—	2	2	no	—
182	#398	—	2	2	no	—
183	#391	C₂²×C₃₁	1	2	yes	—
184	#391	C₂²×C₃₁	1	2	yes	—
185	#389	C₂²×C₃₁	1	2	yes	—
186	#389	C₂²×C₃₁	1	2	yes	—
187	#384	C₂²×C₃₁	1	1	yes	characteristic
188	#382	C₂²×C₃₁	1	1	yes	characteristic
189	#376	C₂²×C₃₁	1	1	yes	characteristic
190	#340	C₂²×C₃₁	1	2	yes	—
191	#340	C₂²×C₃₁	1	2	yes	—
192	#336	C₂²×C₃₁	1	2	yes	—
193	#336	C₂²×C₃₁	1	2	yes	—
194	C₂×C₃₁	#2820	1	1	yes	characteristic
195	C₂×C₃₁	#3225	1	2	yes	—
196	C₂×C₃₁	#3242	1	2	yes	—
197	C₂×C₃₁	#3260	1	1	yes	characteristic
198	C₂×C₃₁	#3261	1	1	yes	characteristic
199	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
200	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
201	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
202	C₄⋊Q₈	—	4	1	no	—
203	C₄⋊Q₈	—	4	1	no	—
204	#125	—	4	1	no	—
205	#125	—	4	1	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
214	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
217	#107	—	4	2	no	—
218	#106	—	4	2	no	—
219	C₄⋊C₈	—	4	1	no	—
220	C₄⋊C₈	—	4	1	no	—
221	#103	—	4	2	no	—
222	#103	—	4	2	no	—
223	#103	—	4	2	no	—
224	#103	—	4	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	4	no	—
230	#95	—	2	2	no	—
231	#95	D₄×C₃₁	1	2	yes	—
232	#95	D₄×C₃₁	1	2	yes	—
233	C₃₁	#32828	1	1	yes	characteristic, direct product factor, Sylow
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
248	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	8	2	no	—
264	C₄⋊C₄	—	8	2	no	—
265	C₄⋊C₄	—	8	2	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
300	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
301	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
302	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
303	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
304	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
305	C₈	—	4	2	no	—
306	C₈	—	4	2	no	—
307	C₂²	C₃₁×#535	1	1	yes	characteristic
308	C₂²	C₃₁×#536	1	1	yes	characteristic
309	C₂²	C₃₁×#537	1	1	yes	characteristic
310	C₂²	C₃₁×#577	1	2	yes	—
311	C₂²	C₃₁×#578	1	2	yes	—
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₃₁×#562	1	1	yes	characteristic
321	C₄	C₃₁×#562	1	1	yes	characteristic
322	C₄	C₃₁×#563	1	1	yes	characteristic
323	C₄	C₃₁×#563	1	1	yes	characteristic
324	C₂	C₃₁×#2820	1	1	yes	characteristic
325	C₂	C₃₁×#3225	1	2	yes	—
326	C₂	C₃₁×#3242	1	2	yes	—
327	C₂	C₃₁×#3260	1	1	yes	characteristic
328	C₂	C₃₁×#3261	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
107. #32828	C₃₁	direct (N × H)
233. C₃₁	#32828	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2647	2
C₃₁×#2642	2
C₃₁×#2097	2
C₃₁×#2072	2
C₃₁×#2068	2
C₃₁×#2066	2
C₃₁×#2059	2
C₃₁×#2043	2
C₃₁×#1832	2
#32828	31

Subgroups — C31×#32828 #21489587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32828 #21489587