Subgroups — C₃₁×#33920 #21490679

Order 7936.

502 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3342	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3334	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2236	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2234	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2225	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2224	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2220	C₂	1	2	yes	maximal, complemented
9	C₄×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2159	C₂	1	2	yes	maximal, complemented
12	C₃₁×#556	—	2	1	no	—
13	C₃₁×#547	—	2	1	no	—
14	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#500	C₂²	1	1	yes	characteristic
18	C₃₁×#485	—	2	2	no	—
19	C₃₁×#480	—	2	2	no	—
20	C₃₁×#475	—	2	1	no	—
21	C₃₁×#474	—	2	1	no	—
22	C₃₁×#467	C₂²	1	2	yes	—
23	C₃₁×#466	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#427	C₂²	1	2	yes	—
28	C₂×C₃₁×#107	C₂²	1	2	yes	—
29	C₃₁×#367	C₂²	1	1	yes	characteristic
30	C₃₁×#367	C₂²	1	2	yes	—
31	C₃₁×#366	C₂²	1	2	yes	—
32	C₃₁×#366	C₂²	1	2	yes	—
33	C₃₁×#366	C₂²	1	2	yes	—
34	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₃₁×#358	C₂²	1	1	yes	characteristic
36	C₃₁×#358	C₂²	1	2	yes	—
37	C₃₁×#357	C₂²	1	1	yes	characteristic
38	C₃₁×#357	C₂²	1	2	yes	—
39	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×#41×C₃₁	—	4	1	no	—
42	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₃₁×C₄⋊Q₈	—	2	1	no	—
46	C₃₁×C₄⋊Q₈	—	2	1	no	—
47	C₃₁×#126	—	4	2	no	—
48	C₃₁×#124	—	4	1	no	—
49	C₃₁×#123	—	4	2	no	—
50	C₃₁×C₂²⋊Q₈	—	2	2	no	—
51	C₃₁×C₂²⋊Q₈	—	4	1	no	—
52	C₃₁×#121	—	2	2	no	—
53	C₄×Q₈×C₃₁	—	2	1	no	—
54	C₄×Q₈×C₃₁	—	4	1	no	—
55	C₄×D₄×C₃₁	—	2	1	no	—
56	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
57	C₃₁×#117	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₃₁×SD₃₂	—	2	2	no	—
60	C₃₁×M₃₂	—	2	4	no	—
61	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
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	—	2	1	no	—
66	C₃₁×#107	—	2	1	no	—
67	C₃₁×#107	C₂³	1	1	yes	characteristic
68	C₃₁×#107	C₂³	1	2	yes	—
69	C₃₁×#107	C₂³	1	1	yes	characteristic
70	C₃₁×#107	C₂³	1	2	yes	—
71	C₃₁×#106	—	2	2	no	—
72	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
73	C₃₁×#103	—	2	1	no	—
74	C₃₁×#103	—	2	2	no	—
75	C₃₁×#102	—	2	1	no	—
76	C₃₁×#102	—	2	2	no	—
77	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	#41×C₃₁	—	8	4	no	—
80	C₂×Q₈×C₃₁	—	2	1	no	—
81	C₂×Q₈×C₃₁	—	4	1	no	—
82	C₂×Q₈×C₃₁	—	4	1	no	—
83	C₂×D₄×C₃₁	—	2	1	no	—
84	C₂×D₄×C₃₁	—	4	2	no	—
85	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	—	4	2	no	—
87	C₂²×C₄×C₃₁	—	4	1	no	—
88	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
89	Q₁₆×C₃₁	—	2	1	no	—
90	Q₁₆×C₃₁	—	2	1	no	—
91	D₈×C₃₁	—	2	1	no	—
92	D₈×C₃₁	—	2	1	no	—
93	C₂×C₈×C₃₁	—	2	2	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	2	1	no	—
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	4	2	no	—
103	C₄⋊C₄×C₃₁	—	4	1	no	—
104	C₄⋊C₄×C₃₁	—	4	2	no	—
105	C₄⋊C₄×C₃₁	—	4	1	no	—
106	C₄⋊C₄×C₃₁	—	4	1	no	—
107	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
108	C₂²⋊C₄×C₃₁	—	4	2	no	—
109	C₂²⋊C₄×C₃₁	—	4	2	no	—
110	C₂²⋊C₄×C₃₁	—	4	1	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
113	C₄²×C₃₁	—	4	2	no	—
114	C₄²×C₃₁	—	4	1	no	—
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₃₁	—	2	2	no	—
117	C₁₆×C₃₁	—	2	1	no	—
118	C₁₆×C₃₁	#41	1	2	yes	complemented
119	#33920	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₃₁	—	4	1	no	—
121	C₂³×C₃₁	C₂×D₈	1	2	yes	—
122	Q₈×C₃₁	—	4	1	no	—
123	Q₈×C₃₁	—	4	1	no	—
124	Q₈×C₃₁	—	8	2	no	—
125	Q₈×C₃₁	—	8	2	no	—
126	D₄×C₃₁	—	2	2	no	—
127	D₄×C₃₁	—	4	1	no	—
128	D₄×C₃₁	—	4	1	no	—
129	D₄×C₃₁	—	8	4	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	4	1	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	4	1	no	—
135	C₂×C₄×C₃₁	—	4	1	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	8	4	no	—
138	C₂×C₄×C₃₁	—	8	2	no	—
139	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
143	C₈×C₃₁	—	2	2	no	—
144	C₈×C₃₁	—	2	1	no	—
145	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
146	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
147	#3342	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#3334	C₂×C₃₁	1	1	yes	characteristic
149	#2252	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#2236	C₂×C₃₁	1	2	yes	complemented
151	#2234	C₂×C₃₁	1	2	yes	complemented
152	#2225	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#2224	C₂×C₃₁	1	1	yes	characteristic, complemented
154	#2220	C₂×C₃₁	1	2	yes	complemented
155	C₄×SD₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#2169	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#2159	C₂×C₃₁	1	2	yes	complemented
158	C₂²×C₃₁	—	2	2	no	—
159	C₂²×C₃₁	—	2	2	no	—
160	C₂²×C₃₁	—	2	2	no	—
161	C₂²×C₃₁	—	8	1	no	—
162	C₂²×C₃₁	—	8	1	no	—
163	C₂²×C₃₁	—	8	1	no	—
164	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₃₁	—	2	2	no	—
166	C₄×C₃₁	—	2	1	no	—
167	C₄×C₃₁	—	8	1	no	—
168	C₄×C₃₁	—	8	1	no	—
169	C₄×C₃₁	—	8	2	no	—
170	C₄×C₃₁	—	8	2	no	—
171	C₄×C₃₁	—	8	1	no	—
172	C₄×C₃₁	#509	1	2	yes	—
173	C₄×C₃₁	#547	1	1	yes	characteristic
174	C₄×C₃₁	#578	1	1	yes	characteristic
175	#556	—	2	1	no	—
176	#547	—	2	1	no	—
177	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
178	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
179	#500	C₂²×C₃₁	1	1	yes	characteristic
180	#500	C₂²×C₃₁	1	1	yes	characteristic
181	#485	—	2	2	no	—
182	#480	—	2	2	no	—
183	#475	—	2	1	no	—
184	#474	—	2	1	no	—
185	#467	C₂²×C₃₁	1	2	yes	—
186	#466	C₂²×C₃₁	1	2	yes	—
187	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
188	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
189	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
190	#427	C₂²×C₃₁	1	2	yes	—
191	C₂×#107	C₂²×C₃₁	1	2	yes	—
192	#367	C₂²×C₃₁	1	1	yes	characteristic
193	#367	C₂²×C₃₁	1	2	yes	—
194	#366	C₂²×C₃₁	1	2	yes	—
195	#366	C₂²×C₃₁	1	2	yes	—
196	#366	C₂²×C₃₁	1	2	yes	—
197	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
198	#358	C₂²×C₃₁	1	1	yes	characteristic
199	#358	C₂²×C₃₁	1	2	yes	—
200	#357	C₂²×C₃₁	1	1	yes	characteristic
201	#357	C₂²×C₃₁	1	2	yes	—
202	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
203	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
204	C₂×C₃₁	—	4	2	no	—
205	C₂×C₃₁	—	16	1	no	—
206	C₂×C₃₁	#3334	1	1	yes	characteristic
207	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
208	C₂×C₃₁	#3417	1	1	yes	characteristic
209	C₂×#41	—	4	1	no	—
210	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
211	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	#126	—	4	2	no	—
216	#124	—	4	1	no	—
217	#123	—	4	2	no	—
218	C₂²⋊Q₈	—	2	2	no	—
219	C₂²⋊Q₈	—	4	1	no	—
220	#121	—	2	2	no	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	—	4	1	no	—
223	C₄×D₄	—	2	1	no	—
224	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
225	#117	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	SD₃₂	—	2	2	no	—
228	M₃₂	—	2	4	no	—
229	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
230	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
231	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
232	#107	—	2	2	no	—
233	#107	—	2	1	no	—
234	#107	—	2	1	no	—
235	#107	C₂³×C₃₁	1	2	yes	—
236	#107	C₂³×C₃₁	1	1	yes	characteristic
237	#107	C₂³×C₃₁	1	1	yes	characteristic
238	#107	C₂³×C₃₁	1	2	yes	—
239	#106	—	2	2	no	—
240	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
241	#103	—	2	1	no	—
242	#103	—	2	2	no	—
243	#102	—	2	1	no	—
244	#102	—	2	2	no	—
245	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
246	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
247	C₃₁	#33920	1	1	yes	characteristic, direct product factor, Sylow
248	#41	—	8	4	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
258	Q₁₆	—	2	1	no	—
259	Q₁₆	—	2	1	no	—
260	D₈	—	2	1	no	—
261	D₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
266	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
282	C₄²	—	4	2	no	—
283	C₄²	—	4	1	no	—
284	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	#41×C₃₁	1	2	yes	complemented
288	C₂³	—	4	1	no	—
289	C₂³	C₂×D₈×C₃₁	1	2	yes	—
290	Q₈	—	4	1	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	8	2	no	—
293	Q₈	—	8	2	no	—
294	D₄	—	2	2	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	8	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	4	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
308	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
309	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	2	1	no	—
313	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
314	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	1	no	—
329	C₄	C₃₁×#509	1	2	yes	—
330	C₄	C₃₁×#547	1	1	yes	characteristic
331	C₄	C₃₁×#578	1	1	yes	characteristic
332	C₂	—	4	2	no	—
333	C₂	—	16	1	no	—
334	C₂	C₃₁×#3334	1	1	yes	characteristic
335	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
336	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×#3334	C₂
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×#3342	C₂	semidirect
3. C₃₁×#2252	C₂	semidirect
4. C₃₁×#2236	C₂	semidirect
5. C₃₁×#2234	C₂	semidirect
6. C₃₁×#2225	C₂	semidirect
7. C₃₁×#2224	C₂	semidirect
8. C₃₁×#2220	C₂	semidirect
9. C₄×C₃₁×SD₃₂	C₂	semidirect
10. C₃₁×#2169	C₂	semidirect
11. C₃₁×#2159	C₂	semidirect
118. C₁₆×C₃₁	#41	semidirect
119. #33920	C₃₁	direct (N × H)
147. #3342	C₂×C₃₁	semidirect
149. #2252	C₂×C₃₁	semidirect
150. #2236	C₂×C₃₁	semidirect
151. #2234	C₂×C₃₁	semidirect
152. #2225	C₂×C₃₁	semidirect
153. #2224	C₂×C₃₁	semidirect
154. #2220	C₂×C₃₁	semidirect
155. C₄×SD₃₂	C₂×C₃₁	semidirect
156. #2169	C₂×C₃₁	semidirect
157. #2159	C₂×C₃₁	semidirect
247. C₃₁	#33920	direct (N × H)
287. C₁₆	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3342	2
C₃₁×#3334	2
C₃₁×#2252	2
C₃₁×#2236	2
C₃₁×#2234	2
C₃₁×#2225	2
C₃₁×#2224	2
C₃₁×#2220	2
C₄×C₃₁×SD₃₂	2
C₃₁×#2169	2
C₃₁×#2159	2
#33920	31

Subgroups — C31×#33920 #21490679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33920 #21490679