Subgroups — C₃₁×#33951 #21490710

Order 7936.

498 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3380	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2255	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2228	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2226	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2224	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2177	C₂	1	2	yes	maximal
11	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#554	—	2	1	no	—
13	D₄×Q₈×C₃₁	—	2	1	no	—
14	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#500	C₂²	1	2	yes	—
17	C₃₁×#500	C₂²	1	1	yes	characteristic
18	C₃₁×#499	C₂²	1	2	yes	—
19	C₃₁×#478	—	2	2	no	—
20	C₃₁×#476	—	2	1	no	—
21	C₃₁×#474	—	2	1	no	—
22	C₃₁×#474	—	2	2	no	—
23	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₃₁	C₂²	1	2	yes	—
26	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#367	C₂²	1	2	yes	—
29	C₃₁×#367	C₂²	1	2	yes	—
30	C₃₁×#367	C₂²	1	2	yes	—
31	C₃₁×#366	C₂²	1	1	yes	characteristic
32	C₃₁×#366	C₂²	1	2	yes	—
33	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₃₁×#357	C₂²	1	1	yes	characteristic
36	C₃₁×#357	C₂²	1	1	yes	characteristic
37	C₃₁×#357	C₂²	1	2	yes	—
38	C₃₁×#357	C₂²	1	2	yes	—
39	C₃₁×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₃₁	—	4	1	no	—
42	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×C₄⋊Q₈	—	2	2	no	—
45	C₃₁×C₄⋊Q₈	—	2	1	no	—
46	C₃₁×C₄⋊Q₈	—	2	1	no	—
47	C₃₁×#125	—	2	2	no	—
48	C₃₁×C₂²⋊Q₈	—	4	1	no	—
49	C₃₁×C₂²⋊Q₈	—	4	1	no	—
50	C₃₁×C₂²⋊Q₈	—	4	2	no	—
51	C₃₁×C₂²⋊Q₈	—	4	2	no	—
52	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
53	C₄×D₄×C₃₁	—	2	2	no	—
54	C₄×D₄×C₃₁	—	2	1	no	—
55	C₄×D₄×C₃₁	—	2	1	no	—
56	C₄×D₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
58	C₃₁×D₁₆	Q₈	1	4	yes	complemented
59	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×#107	—	2	2	no	—
63	C₃₁×#107	—	2	1	no	—
64	C₃₁×#107	—	2	1	no	—
65	C₃₁×#107	C₂³	1	1	yes	characteristic
66	C₃₁×#107	C₂³	1	2	yes	—
67	C₃₁×#107	C₂³	1	1	yes	characteristic
68	C₃₁×#107	C₂³	1	2	yes	—
69	C₃₁×#106	—	2	2	no	—
70	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
71	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
72	C₃₁×#102	—	2	2	no	—
73	C₃₁×#102	—	2	1	no	—
74	C₃₁×#102	—	2	2	no	—
75	C₃₁×#102	—	2	1	no	—
76	C₄×C₈×C₃₁	C₂³	1	2	yes	—
77	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×Q₈×C₃₁	—	4	2	no	—
79	C₂×Q₈×C₃₁	—	4	1	no	—
80	C₂×Q₈×C₃₁	—	4	1	no	—
81	C₂×Q₈×C₃₁	—	8	4	no	—
82	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
83	C₂×D₄×C₃₁	—	2	1	no	—
84	C₂×D₄×C₃₁	—	2	1	no	—
85	C₂²×C₄×C₃₁	—	4	1	no	—
86	C₂²×C₄×C₃₁	—	4	2	no	—
87	C₂²×C₄×C₃₁	—	4	1	no	—
88	C₂²×C₄×C₃₁	—	4	2	no	—
89	D₈×C₃₁	—	2	1	no	—
90	D₈×C₃₁	—	2	1	no	—
91	D₈×C₃₁	C₂×Q₈	1	2	yes	—
92	D₈×C₃₁	C₂×Q₈	1	2	yes	complemented
93	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₃₁	—	2	1	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	2	1	no	—
100	C₄⋊C₄×C₃₁	—	4	2	no	—
101	C₄⋊C₄×C₃₁	—	4	1	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	D₈	1	1	yes	characteristic
106	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
107	C₂²⋊C₄×C₃₁	—	4	1	no	—
108	C₂²⋊C₄×C₃₁	—	4	2	no	—
109	C₂²⋊C₄×C₃₁	—	4	1	no	—
110	C₂²⋊C₄×C₃₁	—	4	2	no	—
111	C₄²×C₃₁	C₂×D₄	1	2	yes	—
112	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₁₆×C₃₁	—	2	1	no	—
114	C₁₆×C₃₁	—	2	2	no	—
115	C₁₆×C₃₁	C₂×Q₈	1	2	yes	complemented
116	#33951	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₃₁	—	4	1	no	—
118	C₂³×C₃₁	—	4	1	no	—
119	Q₈×C₃₁	—	2	2	no	—
120	Q₈×C₃₁	—	8	4	no	—
121	Q₈×C₃₁	—	8	2	no	—
122	Q₈×C₃₁	—	16	1	no	—
123	D₄×C₃₁	—	2	2	no	—
124	D₄×C₃₁	—	2	2	no	—
125	D₄×C₃₁	—	4	1	no	—
126	D₄×C₃₁	—	4	1	no	—
127	C₂×C₄×C₃₁	—	4	1	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	4	1	no	—
131	C₂×C₄×C₃₁	—	8	4	no	—
132	C₂×C₄×C₃₁	—	8	2	no	—
133	C₂×C₄×C₃₁	—	8	2	no	—
134	C₂×C₄×C₃₁	—	16	2	no	—
135	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₃₁	—	2	2	no	—
141	C₈×C₃₁	—	2	1	no	—
142	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
143	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
144	#3380	C₂×C₃₁	1	1	yes	characteristic, complemented
145	Q₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#2255	C₂×C₃₁	1	2	yes	complemented
147	#2254	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#2228	C₂×C₃₁	1	2	yes	complemented
149	#2226	C₂×C₃₁	1	2	yes	complemented
150	#2226	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#2224	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#2183	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#2177	C₂×C₃₁	1	2	yes	—
154	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic
155	C₂²×C₃₁	—	4	2	no	—
156	C₂²×C₃₁	—	4	2	no	—
157	C₂²×C₃₁	—	8	1	no	—
158	C₂²×C₃₁	—	8	1	no	—
159	C₂²×C₃₁	—	8	1	no	—
160	C₂²×C₃₁	—	8	1	no	—
161	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
162	C₄×C₃₁	—	2	2	no	—
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	2	1	no	—
165	C₄×C₃₁	—	8	1	no	—
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	—	8	1	no	—
168	C₄×C₃₁	—	8	2	no	—
169	C₄×C₃₁	#509	1	2	yes	—
170	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
171	C₄×C₃₁	#578	1	1	yes	characteristic
172	#554	—	2	1	no	—
173	D₄×Q₈	—	2	1	no	—
174	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
175	#500	C₂²×C₃₁	1	1	yes	characteristic
176	#500	C₂²×C₃₁	1	2	yes	—
177	#500	C₂²×C₃₁	1	1	yes	characteristic
178	#499	C₂²×C₃₁	1	2	yes	—
179	#478	—	2	2	no	—
180	#476	—	2	1	no	—
181	#474	—	2	1	no	—
182	#474	—	2	2	no	—
183	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
184	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
185	C₄×D₈	C₂²×C₃₁	1	2	yes	—
186	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
187	C₄×D₈	C₂²×C₃₁	1	2	yes	—
188	#367	C₂²×C₃₁	1	2	yes	—
189	#367	C₂²×C₃₁	1	2	yes	—
190	#367	C₂²×C₃₁	1	2	yes	—
191	#366	C₂²×C₃₁	1	1	yes	characteristic
192	#366	C₂²×C₃₁	1	2	yes	—
193	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
194	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
195	#357	C₂²×C₃₁	1	1	yes	characteristic
196	#357	C₂²×C₃₁	1	1	yes	characteristic
197	#357	C₂²×C₃₁	1	2	yes	—
198	#357	C₂²×C₃₁	1	2	yes	—
199	#346	C₂²×C₃₁	1	2	yes	—
200	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
201	C₂×C₃₁	—	8	2	no	—
202	C₂×C₃₁	—	8	2	no	—
203	C₂×C₃₁	Q₈×D₈	1	1	yes	characteristic
204	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
205	C₂×C₃₁	#3416	1	1	yes	characteristic
206	C₂²×Q₈	—	4	1	no	—
207	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
208	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	2	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	#125	—	2	2	no	—
213	C₂²⋊Q₈	—	4	1	no	—
214	C₂²⋊Q₈	—	4	1	no	—
215	C₂²⋊Q₈	—	4	2	no	—
216	C₂²⋊Q₈	—	4	2	no	—
217	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	—	2	1	no	—
220	C₄×D₄	—	2	1	no	—
221	C₄×D₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	4	1	no	—
223	D₁₆	Q₈×C₃₁	1	4	yes	complemented
224	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
225	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
226	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
227	#107	—	2	2	no	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#107	C₂³×C₃₁	1	1	yes	characteristic
231	#107	C₂³×C₃₁	1	1	yes	characteristic
232	#107	C₂³×C₃₁	1	2	yes	—
233	#107	C₂³×C₃₁	1	2	yes	—
234	#106	—	2	2	no	—
235	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
236	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
237	#102	—	2	2	no	—
238	#102	—	2	1	no	—
239	#102	—	2	2	no	—
240	#102	—	2	1	no	—
241	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₃₁	1	2	yes	—
243	C₃₁	#33951	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	4	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	8	4	no	—
248	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	2	no	—
255	D₈	—	2	1	no	—
256	D₈	—	2	1	no	—
257	D₈	C₂×Q₈×C₃₁	1	2	yes	—
258	D₈	C₂×Q₈×C₃₁	1	2	yes	complemented
259	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
260	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
272	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	C₂×D₄×C₃₁	1	2	yes	—
278	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
279	C₁₆	—	2	1	no	—
280	C₁₆	—	2	2	no	—
281	C₁₆	C₂×Q₈×C₃₁	1	2	yes	complemented
282	C₂³	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	8	4	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	16	1	no	—
288	D₄	—	2	2	no	—
289	D₄	—	2	2	no	—
290	D₄	—	4	1	no	—
291	D₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	8	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	16	2	no	—
300	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
302	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
303	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	2	1	no	—
307	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
308	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	—	8	1	no	—
314	C₂²	—	8	1	no	—
315	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	2	no	—
323	C₄	C₃₁×#509	1	2	yes	—
324	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
325	C₄	C₃₁×#578	1	1	yes	characteristic
326	C₂	—	8	2	no	—
327	C₂	—	8	2	no	—
328	C₂	Q₈×D₈×C₃₁	1	1	yes	characteristic
329	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
330	C₂	C₃₁×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ C₈×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3380	C₂	semidirect
2. Q₈×D₈×C₃₁	C₂	semidirect
3. C₃₁×#2255	C₂	semidirect
4. C₃₁×#2254	C₂	semidirect
5. C₃₁×#2228	C₂	semidirect
6. C₃₁×#2226	C₂	semidirect
7. C₃₁×#2226	C₂	semidirect
8. C₃₁×#2224	C₂	semidirect
9. C₃₁×#2183	C₂	semidirect
58. C₃₁×D₁₆	Q₈	semidirect
92. D₈×C₃₁	C₂×Q₈	semidirect
115. C₁₆×C₃₁	C₂×Q₈	semidirect
116. #33951	C₃₁	direct (N × H)
144. #3380	C₂×C₃₁	semidirect
145. Q₈×D₈	C₂×C₃₁	semidirect
146. #2255	C₂×C₃₁	semidirect
147. #2254	C₂×C₃₁	semidirect
148. #2228	C₂×C₃₁	semidirect
149. #2226	C₂×C₃₁	semidirect
150. #2226	C₂×C₃₁	semidirect
151. #2224	C₂×C₃₁	semidirect
152. #2183	C₂×C₃₁	semidirect
223. D₁₆	Q₈×C₃₁	semidirect
243. C₃₁	#33951	direct (N × H)
258. D₈	C₂×Q₈×C₃₁	semidirect
281. C₁₆	C₂×Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3380	2
Q₈×D₈×C₃₁	2
C₃₁×#2255	2
C₃₁×#2254	2
C₃₁×#2228	2
C₃₁×#2226	2
C₃₁×#2224	2
C₃₁×#2183	2
C₃₁×#2177	2
C₄×C₃₁×D₁₆	2
#33951	31

Subgroups — C31×#33951 #21490710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33951 #21490710