Subgroups — C₃₁×#33739 #21490498

Order 7936.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3175	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2243	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2220	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2216	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2215	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#2175	C₂	1	2	yes	maximal, complemented
9	C₄×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#542	—	2	2	no	—
13	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#500	C₂²	1	2	yes	—
19	C₃₁×#495	C₂²	1	2	yes	—
20	C₃₁×#489	—	2	2	no	—
21	C₃₁×#468	—	2	2	no	—
22	C₃₁×#467	C₂²	1	2	yes	—
23	C₃₁×#466	C₂²	1	2	yes	—
24	C₃₁×#440	—	2	2	no	—
25	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
26	C₄×D₈×C₃₁	C₂²	1	2	yes	—
27	C₃₁×#405	C₂²	1	1	yes	characteristic
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×#358	C₂²	1	2	yes	—
32	C₃₁×#358	C₂²	1	2	yes	—
33	C₃₁×#357	C₂²	1	2	yes	—
34	C₃₁×#357	C₂²	1	2	yes	—
35	C₃₁×#346	C₂²	1	1	yes	characteristic
36	C₃₁×#346	C₂²	1	1	yes	characteristic
37	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₃₁	—	2	2	no	—
41	C₂×D₈×C₃₁	C₂³	1	2	yes	—
42	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×C₄⋊Q₈	—	2	2	no	—
45	C₃₁×#126	—	4	2	no	—
46	C₃₁×#124	—	2	2	no	—
47	C₃₁×#124	—	4	2	no	—
48	C₃₁×#123	—	4	2	no	—
49	C₃₁×C₂²⋊Q₈	—	2	2	no	—
50	C₃₁×C₂²⋊Q₈	—	4	2	no	—
51	C₃₁×#121	—	2	2	no	—
52	C₄×Q₈×C₃₁	—	2	2	no	—
53	C₄×D₄×C₃₁	—	2	2	no	—
54	C₃₁×#117	D₄	1	1	yes	characteristic
55	C₃₁×Q₃₂	—	2	2	no	—
56	C₃₁×SD₃₂	—	2	4	no	—
57	C₃₁×D₁₆	—	2	2	no	—
58	C₃₁×M₃₂	—	2	2	no	—
59	C₂×C₁₆×C₃₁	—	2	2	no	—
60	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
64	C₃₁×#107	—	2	2	no	—
65	C₃₁×#107	C₂³	1	2	yes	—
66	C₃₁×#107	C₂³	1	2	yes	—
67	C₃₁×#103	—	2	2	no	—
68	C₃₁×#103	—	2	2	no	—
69	C₃₁×#102	—	2	2	no	—
70	C₃₁×#102	—	2	2	no	—
71	C₃₁×#97	D₄	1	1	yes	characteristic
72	C₃₁×#97	D₄	1	1	yes	characteristic
73	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×Q₈×C₃₁	—	4	2	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	4	2	no	—
79	C₂²×C₄×C₃₁	—	4	2	no	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	Q₁₆×C₃₁	—	2	2	no	—
82	Q₁₆×C₃₁	—	2	2	no	—
83	SD₁₆×C₃₁	—	4	4	no	—
84	D₈×C₃₁	—	2	2	no	—
85	D₈×C₃₁	—	2	2	no	—
86	M₁₆×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	1	no	—
88	C₂×C₈×C₃₁	—	2	1	no	—
89	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	4	2	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₄⋊C₄×C₃₁	—	4	2	no	—
98	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
99	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
100	C₂²⋊C₄×C₃₁	—	4	2	no	—
101	C₂²⋊C₄×C₃₁	—	4	2	no	—
102	C₂²⋊C₄×C₃₁	—	4	2	no	—
103	C₂²⋊C₄×C₃₁	—	4	2	no	—
104	C₂²⋊C₄×C₃₁	—	4	2	no	—
105	C₂²⋊C₄×C₃₁	D₈	1	1	yes	characteristic
106	C₂²⋊C₄×C₃₁	D₈	1	1	yes	characteristic
107	C₄²×C₃₁	—	4	2	no	—
108	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₃₁	—	2	1	no	—
111	C₁₆×C₃₁	—	2	1	no	—
112	C₁₆×C₃₁	#41	1	2	yes	—
113	C₁₆×C₃₁	#41	1	2	yes	—
114	#33739	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	—	4	2	no	—
116	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
117	Q₈×C₃₁	—	4	2	no	—
118	Q₈×C₃₁	—	4	2	no	—
119	Q₈×C₃₁	—	16	2	no	—
120	D₄×C₃₁	—	4	2	no	—
121	D₄×C₃₁	—	4	2	no	—
122	D₄×C₃₁	—	16	2	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	2	1	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	—	16	2	no	—
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	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₃₁	—	2	1	no	—
136	C₈×C₃₁	—	2	1	no	—
137	C₈×C₃₁	#135	1	2	yes	—
138	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
139	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
140	#3175	C₂×C₃₁	1	2	yes	complemented
141	#2252	C₂×C₃₁	1	1	yes	characteristic, complemented
142	#2249	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#2243	C₂×C₃₁	1	2	yes	complemented
144	#2220	C₂×C₃₁	1	2	yes	complemented
145	#2216	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#2215	C₂×C₃₁	1	1	yes	characteristic
147	#2175	C₂×C₃₁	1	2	yes	complemented
148	C₄×Q₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
149	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#2108	C₂×C₃₁	1	1	yes	characteristic, complemented
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	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₃₁	—	2	1	no	—
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	#532	1	1	yes	characteristic
167	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
168	#542	—	2	2	no	—
169	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
171	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
172	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
173	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
174	#500	C₂²×C₃₁	1	2	yes	—
175	#495	C₂²×C₃₁	1	2	yes	—
176	#489	—	2	2	no	—
177	#468	—	2	2	no	—
178	#467	C₂²×C₃₁	1	2	yes	—
179	#466	C₂²×C₃₁	1	2	yes	—
180	#440	—	2	2	no	—
181	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
182	C₄×D₈	C₂²×C₃₁	1	2	yes	—
183	#405	C₂²×C₃₁	1	1	yes	characteristic
184	#366	C₂²×C₃₁	1	1	yes	characteristic
185	#366	C₂²×C₃₁	1	1	yes	characteristic
186	#366	C₂²×C₃₁	1	2	yes	—
187	#358	C₂²×C₃₁	1	2	yes	—
188	#358	C₂²×C₃₁	1	2	yes	—
189	#357	C₂²×C₃₁	1	2	yes	—
190	#357	C₂²×C₃₁	1	2	yes	—
191	#346	C₂²×C₃₁	1	1	yes	characteristic
192	#346	C₂²×C₃₁	1	1	yes	characteristic
193	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
194	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
195	C₂×C₃₁	—	4	1	no	—
196	C₂×C₃₁	—	16	2	no	—
197	C₂×C₃₁	#3169	1	1	yes	characteristic
198	C₂×C₃₁	#3415	1	1	yes	characteristic
199	C₂×C₃₁	#3417	1	1	yes	characteristic
200	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
201	C₂×SD₁₆	—	2	2	no	—
202	C₂×D₈	C₂³×C₃₁	1	2	yes	—
203	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
205	C₄⋊Q₈	—	2	2	no	—
206	#126	—	4	2	no	—
207	#124	—	2	2	no	—
208	#124	—	4	2	no	—
209	#123	—	4	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	4	2	no	—
212	#121	—	2	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×D₄	—	2	2	no	—
215	#117	D₄×C₃₁	1	1	yes	characteristic
216	Q₃₂	—	2	2	no	—
217	SD₃₂	—	2	4	no	—
218	D₁₆	—	2	2	no	—
219	M₃₂	—	2	2	no	—
220	C₂×C₁₆	—	2	2	no	—
221	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
222	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
223	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
224	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
225	#107	—	2	2	no	—
226	#107	C₂³×C₃₁	1	2	yes	—
227	#107	C₂³×C₃₁	1	2	yes	—
228	#103	—	2	2	no	—
229	#103	—	2	2	no	—
230	#102	—	2	2	no	—
231	#102	—	2	2	no	—
232	#97	D₄×C₃₁	1	1	yes	characteristic
233	#97	D₄×C₃₁	1	1	yes	characteristic
234	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
236	C₃₁	#33739	1	1	yes	characteristic, direct product factor, Sylow
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	2	2	no	—
245	SD₁₆	—	4	4	no	—
246	D₈	—	2	2	no	—
247	D₈	—	2	2	no	—
248	M₁₆	—	2	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
253	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
254	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
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	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	D₈×C₃₁	1	1	yes	characteristic
268	C₂²⋊C₄	D₈×C₃₁	1	1	yes	characteristic
269	C₄²	—	4	2	no	—
270	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
272	C₁₆	—	2	1	no	—
273	C₁₆	—	2	1	no	—
274	C₁₆	#41×C₃₁	1	2	yes	—
275	C₁₆	#41×C₃₁	1	2	yes	—
276	C₂³	—	4	2	no	—
277	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	16	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	16	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	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	C₂×D₈×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	C₃₁×#135	1	2	yes	—
299	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
300	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	C₃₁×#532	1	1	yes	characteristic
317	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	16	2	no	—
320	C₂	C₃₁×#3169	1	1	yes	characteristic
321	C₂	C₃₁×#3415	1	1	yes	characteristic
322	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#2215	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3175	C₂	semidirect
2. C₃₁×#2252	C₂	semidirect
3. C₃₁×#2249	C₂	semidirect
4. C₃₁×#2243	C₂	semidirect
5. C₃₁×#2220	C₂	semidirect
6. C₃₁×#2216	C₂	semidirect
8. C₃₁×#2175	C₂	semidirect
9. C₄×C₃₁×Q₃₂	C₂	semidirect
10. C₄×C₃₁×D₁₆	C₂	semidirect
11. C₃₁×#2108	C₂	semidirect
114. #33739	C₃₁	direct (N × H)
140. #3175	C₂×C₃₁	semidirect
141. #2252	C₂×C₃₁	semidirect
142. #2249	C₂×C₃₁	semidirect
143. #2243	C₂×C₃₁	semidirect
144. #2220	C₂×C₃₁	semidirect
145. #2216	C₂×C₃₁	semidirect
147. #2175	C₂×C₃₁	semidirect
148. C₄×Q₃₂	C₂×C₃₁	semidirect
149. C₄×D₁₆	C₂×C₃₁	semidirect
150. #2108	C₂×C₃₁	semidirect
236. C₃₁	#33739	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×#3175	2
C₃₁×#2252	2
C₃₁×#2249	2
C₃₁×#2243	2
C₃₁×#2220	2
C₃₁×#2216	2
C₃₁×#2215	2
C₃₁×#2175	2
C₄×C₃₁×Q₃₂	2
C₄×C₃₁×D₁₆	2
C₃₁×#2108	2
#33739	31

Subgroups — C31×#33739 #21490498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33739 #21490498