Subgroups — C₃₁×#33906 #21490665

Order 7936.

494 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3339	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2234	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2233	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2228	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2215	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2158	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#366	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#553	—	2	2	no	—
13	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#499	C₂²	1	2	yes	—
17	C₃₁×#482	—	2	2	no	—
18	C₃₁×#481	—	2	2	no	—
19	C₃₁×#478	—	2	2	no	—
20	C₃₁×#466	C₂²	1	2	yes	—
21	C₃₁×#466	C₂²	1	2	yes	—
22	C₄×D₈×C₃₁	C₂²	1	2	yes	—
23	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#427	C₂²	1	2	yes	—
25	C₂×C₃₁×#107	C₂²	1	2	yes	—
26	C₃₁×#367	C₂²	1	2	yes	—
27	C₃₁×#367	C₂²	1	2	yes	—
28	C₃₁×#366	C₂²	1	1	yes	characteristic
29	C₃₁×#366	C₂²	1	1	yes	characteristic
30	C₃₁×#366	C₂²	1	2	yes	—
31	C₃₁×#366	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₃₁×#357	C₂²	1	2	yes	—
34	C₃₁×#357	C₂²	1	2	yes	—
35	C₃₁×#357	C₂²	1	2	yes	—
36	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×D₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×#126	—	4	2	no	—
43	C₃₁×#125	—	2	2	no	—
44	C₃₁×#123	—	4	2	no	—
45	C₃₁×#121	—	2	2	no	—
46	C₃₁×#121	—	2	2	no	—
47	C₃₁×#121	—	4	2	no	—
48	C₄×D₄×C₃₁	—	2	2	no	—
49	C₄×D₄×C₃₁	—	4	2	no	—
50	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
51	C₃₁×#117	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₃₁×D₁₆	—	2	2	no	—
54	C₂×C₁₆×C₃₁	—	2	2	no	—
55	C₂×C₁₆×C₃₁	—	2	2	no	—
56	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×#107	—	2	2	no	—
61	C₃₁×#107	C₂³	1	2	yes	—
62	C₃₁×#107	C₂³	1	2	yes	—
63	C₃₁×#107	C₂³	1	2	yes	—
64	C₃₁×#106	—	2	2	no	—
65	C₃₁×#106	—	2	2	no	—
66	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
67	C₃₁×#102	—	2	2	no	—
68	C₃₁×#102	—	2	2	no	—
69	C₃₁×#102	—	2	2	no	—
70	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
71	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
72	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₃₁	—	2	2	no	—
74	C₂×D₄×C₃₁	—	4	2	no	—
75	C₂×D₄×C₃₁	—	4	2	no	—
76	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	—	4	2	no	—
78	C₂²×C₄×C₃₁	—	4	2	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	D₈×C₃₁	—	2	2	no	—
82	D₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	1	no	—
84	C₂×C₈×C₃₁	—	2	1	no	—
85	C₂×C₈×C₃₁	—	2	1	no	—
86	C₂×C₈×C₃₁	—	2	1	no	—
87	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
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	—	4	2	no	—
95	C₄⋊C₄×C₃₁	—	4	2	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
98	C₂²⋊C₄×C₃₁	—	4	2	no	—
99	C₂²⋊C₄×C₃₁	—	4	2	no	—
100	C₂²⋊C₄×C₃₁	—	4	2	no	—
101	C₂²⋊C₄×C₃₁	—	4	2	no	—
102	C₂²⋊C₄×C₃₁	D₈	1	1	yes	characteristic
103	C₂²⋊C₄×C₃₁	D₈	1	1	yes	characteristic
104	C₄²×C₃₁	—	4	2	no	—
105	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₁₆×C₃₁	—	2	1	no	—
107	C₁₆×C₃₁	—	2	1	no	—
108	C₁₆×C₃₁	—	2	1	no	—
109	C₁₆×C₃₁	#41	1	2	yes	—
110	#33906	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₃₁	—	4	2	no	—
112	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
113	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
114	D₄×C₃₁	—	2	2	no	—
115	D₄×C₃₁	—	4	2	no	—
116	D₄×C₃₁	—	4	2	no	—
117	D₄×C₃₁	—	8	4	no	—
118	D₄×C₃₁	—	16	2	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	1	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	1	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	—	16	2	no	—
127	C₂×C₄×C₃₁	—	16	2	no	—
128	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₈×C₃₁	—	2	1	no	—
134	C₈×C₃₁	—	2	1	no	—
135	C₈×C₃₁	—	2	1	no	—
136	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
137	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
138	#3339	C₂×C₃₁	1	2	yes	complemented
139	#2253	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#2234	C₂×C₃₁	1	2	yes	complemented
141	#2233	C₂×C₃₁	1	2	yes	complemented
142	#2228	C₂×C₃₁	1	2	yes	complemented
143	#2215	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#2215	C₂×C₃₁	1	1	yes	characteristic, complemented
145	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
146	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#2158	C₂×C₃₁	1	1	yes	characteristic, complemented
148	C₂×#366	C₂×C₃₁	1	1	yes	characteristic, complemented
149	C₂²×C₃₁	—	2	1	no	—
150	C₂²×C₃₁	—	2	1	no	—
151	C₂²×C₃₁	—	2	1	no	—
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	8	2	no	—
155	C₂²×C₃₁	—	8	2	no	—
156	C₂²×C₃₁	—	8	2	no	—
157	C₂²×C₃₁	#508	1	2	yes	—
158	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
159	C₄×C₃₁	—	2	1	no	—
160	C₄×C₃₁	—	2	1	no	—
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	8	2	no	—
164	C₄×C₃₁	—	8	2	no	—
165	C₄×C₃₁	—	8	2	no	—
166	C₄×C₃₁	#547	1	1	yes	characteristic
167	C₄×C₃₁	#578	1	1	yes	characteristic
168	#553	—	2	2	no	—
169	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
170	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
171	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
172	#499	C₂²×C₃₁	1	2	yes	—
173	#482	—	2	2	no	—
174	#481	—	2	2	no	—
175	#478	—	2	2	no	—
176	#466	C₂²×C₃₁	1	2	yes	—
177	#466	C₂²×C₃₁	1	2	yes	—
178	C₄×D₈	C₂²×C₃₁	1	2	yes	—
179	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
180	#427	C₂²×C₃₁	1	2	yes	—
181	C₂×#107	C₂²×C₃₁	1	2	yes	—
182	#367	C₂²×C₃₁	1	2	yes	—
183	#367	C₂²×C₃₁	1	2	yes	—
184	#366	C₂²×C₃₁	1	1	yes	characteristic
185	#366	C₂²×C₃₁	1	1	yes	characteristic
186	#366	C₂²×C₃₁	1	2	yes	—
187	#366	C₂²×C₃₁	1	1	yes	characteristic
188	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
189	#357	C₂²×C₃₁	1	2	yes	—
190	#357	C₂²×C₃₁	1	2	yes	—
191	#357	C₂²×C₃₁	1	2	yes	—
192	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
193	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
194	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
195	C₂×C₃₁	—	2	2	no	—
196	C₂×C₃₁	—	4	1	no	—
197	C₂×C₃₁	—	16	2	no	—
198	C₂×C₃₁	#3334	1	1	yes	characteristic
199	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
200	C₂×C₃₁	#3415	1	1	yes	characteristic
201	C₂×D₈	C₂³×C₃₁	1	2	yes	—
202	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
204	#126	—	4	2	no	—
205	#125	—	2	2	no	—
206	#123	—	4	2	no	—
207	#121	—	2	2	no	—
208	#121	—	2	2	no	—
209	#121	—	4	2	no	—
210	C₄×D₄	—	2	2	no	—
211	C₄×D₄	—	4	2	no	—
212	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
213	#117	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	D₁₆	—	2	2	no	—
216	C₂×C₁₆	—	2	2	no	—
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
219	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
220	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
221	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
222	#107	—	2	2	no	—
223	#107	C₂³×C₃₁	1	2	yes	—
224	#107	C₂³×C₃₁	1	2	yes	—
225	#107	C₂³×C₃₁	1	2	yes	—
226	#106	—	2	2	no	—
227	#106	—	2	2	no	—
228	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
229	#102	—	2	2	no	—
230	#102	—	2	2	no	—
231	#102	—	2	2	no	—
232	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
233	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
234	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
235	C₃₁	#33906	1	1	yes	characteristic, direct product factor, Sylow
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	D₈×C₃₁	1	1	yes	characteristic
266	C₂²⋊C₄	D₈×C₃₁	1	1	yes	characteristic
267	C₄²	—	4	2	no	—
268	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₁₆	—	2	1	no	—
270	C₁₆	—	2	1	no	—
271	C₁₆	—	2	1	no	—
272	C₁₆	#41×C₃₁	1	2	yes	—
273	C₂³	—	4	2	no	—
274	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
275	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
276	D₄	—	2	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	8	4	no	—
280	D₄	—	16	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	16	2	no	—
289	C₂×C₄	—	16	2	no	—
290	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
292	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
293	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
299	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	C₃₁×#508	1	2	yes	—
309	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	C₃₁×#547	1	1	yes	characteristic
318	C₄	C₃₁×#578	1	1	yes	characteristic
319	C₂	—	2	2	no	—
320	C₂	—	4	1	no	—
321	C₂	—	16	2	no	—
322	C₂	C₃₁×#3334	1	1	yes	characteristic
323	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
324	C₂	C₃₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3339	C₂	semidirect
2. C₃₁×#2253	C₂	semidirect
3. C₃₁×#2234	C₂	semidirect
4. C₃₁×#2233	C₂	semidirect
5. C₃₁×#2228	C₂	semidirect
6. C₃₁×#2215	C₂	semidirect
7. C₃₁×#2215	C₂	semidirect
8. C₄×C₃₁×D₁₆	C₂	semidirect
9. D₄×C₁₆×C₃₁	C₂	semidirect
10. C₃₁×#2158	C₂	semidirect
11. C₂×C₃₁×#366	C₂	semidirect
110. #33906	C₃₁	direct (N × H)
138. #3339	C₂×C₃₁	semidirect
139. #2253	C₂×C₃₁	semidirect
140. #2234	C₂×C₃₁	semidirect
141. #2233	C₂×C₃₁	semidirect
142. #2228	C₂×C₃₁	semidirect
143. #2215	C₂×C₃₁	semidirect
144. #2215	C₂×C₃₁	semidirect
145. C₄×D₁₆	C₂×C₃₁	semidirect
146. D₄×C₁₆	C₂×C₃₁	semidirect
147. #2158	C₂×C₃₁	semidirect
148. C₂×#366	C₂×C₃₁	semidirect
235. C₃₁	#33906	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×#3339	2
C₃₁×#2253	2
C₃₁×#2234	2
C₃₁×#2233	2
C₃₁×#2228	2
C₃₁×#2215	2
C₄×C₃₁×D₁₆	2
D₄×C₁₆×C₃₁	2
C₃₁×#2158	2
C₂×C₃₁×#366	2
#33906	31

Subgroups — C31×#33906 #21490665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33906 #21490665