Subgroups — C₃₁×#33736 #21490495

Order 7936.

558 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×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2217	C₂	1	4	yes	maximal, complemented
8	C₄×C₃₁×SD₃₂	C₂	1	4	yes	maximal, complemented
9	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#533	—	2	1	no	—
11	C₃₁×#532	—	2	1	no	—
12	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
13	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#494	C₂²	1	1	yes	characteristic
17	C₃₁×#493	C₂²	1	1	yes	characteristic
18	C₃₁×#487	—	2	1	no	—
19	C₃₁×#486	—	2	1	no	—
20	C₃₁×#467	—	2	1	no	—
21	C₃₁×#467	C₂²	1	2	yes	—
22	C₃₁×#466	—	2	1	no	—
23	C₃₁×#466	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₃₁	—	2	1	no	—
25	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
26	C₄×D₈×C₃₁	—	2	1	no	—
27	C₄×D₈×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#367	C₂²	1	4	yes	—
30	C₃₁×#358	C₂²	1	4	yes	—
31	C₃₁×#357	C₂²	1	4	yes	—
32	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
33	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×#41×C₃₁	—	4	1	no	—
36	C₂×Q₁₆×C₃₁	—	2	1	no	—
37	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
38	C₂×D₈×C₃₁	—	2	1	no	—
39	C₂×D₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×C₄⋊Q₈	—	2	1	no	—
43	C₃₁×C₄⋊Q₈	—	2	1	no	—
44	C₃₁×C₄⋊Q₈	—	2	1	no	—
45	C₃₁×#127	—	2	1	no	—
46	C₃₁×#125	—	4	1	no	—
47	C₃₁×#124	—	4	1	no	—
48	C₃₁×C₂²⋊Q₈	—	2	2	no	—
49	C₃₁×C₂²⋊Q₈	—	4	1	no	—
50	C₃₁×#121	—	2	2	no	—
51	C₃₁×#121	—	4	1	no	—
52	C₄×Q₈×C₃₁	—	2	2	no	—
53	C₄×Q₈×C₃₁	—	4	1	no	—
54	C₄×D₄×C₃₁	—	2	2	no	—
55	C₄×D₄×C₃₁	—	4	1	no	—
56	C₃₁×#117	—	4	1	no	—
57	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
58	C₃₁×SD₃₂	—	2	8	no	—
59	C₂×C₁₆×C₃₁	—	2	4	no	—
60	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
62	C₃₁×#107	—	2	1	no	—
63	C₃₁×#107	—	2	1	no	—
64	C₃₁×#107	C₂³	1	2	yes	—
65	C₃₁×#107	C₂³	1	2	yes	—
66	C₃₁×#103	—	2	2	no	—
67	C₃₁×#103	—	2	2	no	—
68	C₃₁×#102	—	2	2	no	—
69	C₃₁×#102	—	2	2	no	—
70	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
71	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
72	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	#41×C₃₁	—	8	4	no	—
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×Q₈×C₃₁	—	4	1	no	—
77	C₂×Q₈×C₃₁	—	4	1	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	4	2	no	—
80	C₂×D₄×C₃₁	—	4	1	no	—
81	C₂×D₄×C₃₁	—	4	1	no	—
82	C₂²×C₄×C₃₁	—	4	2	no	—
83	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	Q₁₆×C₃₁	—	2	4	no	—
87	Q₁₆×C₃₁	—	4	2	no	—
88	D₈×C₃₁	—	2	4	no	—
89	D₈×C₃₁	—	4	2	no	—
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	4	1	no	—
99	C₄⋊C₄×C₃₁	—	4	1	no	—
100	C₄⋊C₄×C₃₁	—	4	1	no	—
101	C₄⋊C₄×C₃₁	—	4	2	no	—
102	C₄⋊C₄×C₃₁	—	4	2	no	—
103	C₄⋊C₄×C₃₁	—	4	1	no	—
104	C₂²⋊C₄×C₃₁	—	4	2	no	—
105	C₂²⋊C₄×C₃₁	—	4	2	no	—
106	C₂²⋊C₄×C₃₁	—	4	2	no	—
107	C₄²×C₃₁	—	4	2	no	—
108	C₄²×C₃₁	D₈	1	1	yes	characteristic
109	C₄²×C₃₁	D₈	1	1	yes	characteristic
110	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₃₁	—	2	2	no	—
113	C₁₆×C₃₁	#41	1	4	yes	complemented
114	#33736	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	—	8	2	no	—
120	Q₈×C₃₁	—	8	2	no	—
121	D₄×C₃₁	—	4	2	no	—
122	D₄×C₃₁	—	4	2	no	—
123	D₄×C₃₁	—	8	4	no	—
124	D₄×C₃₁	—	8	2	no	—
125	D₄×C₃₁	—	8	2	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	2	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	—	8	4	no	—
133	C₂×C₄×C₃₁	SD₃₂	1	4	yes	complemented
134	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₃₁	—	2	1	no	—
141	C₈×C₃₁	—	2	1	no	—
142	C₈×C₃₁	#135	1	2	yes	—
143	C₈×C₃₁	C₂×#41	1	2	yes	—
144	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
145	#3169	C₂×C₃₁	1	1	yes	characteristic
146	#2254	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#2248	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#2241	C₂×C₃₁	1	1	yes	characteristic, complemented
149	#2240	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#2217	C₂×C₃₁	1	4	yes	complemented
151	C₄×SD₃₂	C₂×C₃₁	1	4	yes	complemented
152	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	8	2	no	—
156	C₂²×C₃₁	—	8	2	no	—
157	C₂²×C₃₁	—	8	2	no	—
158	C₂²×C₃₁	C₂×SD₃₂	1	2	yes	—
159	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₃₁	—	2	1	no	—
161	C₄×C₃₁	—	2	1	no	—
162	C₄×C₃₁	—	8	2	no	—
163	C₄×C₃₁	—	8	2	no	—
164	C₄×C₃₁	—	8	2	no	—
165	C₄×C₃₁	C₂×SD₃₂	1	2	yes	—
166	C₄×C₃₁	C₂×SD₃₂	1	2	yes	—
167	C₄×C₃₁	#508	1	2	yes	—
168	C₄×C₃₁	#532	1	1	yes	characteristic
169	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
170	#533	—	2	1	no	—
171	#532	—	2	1	no	—
172	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
173	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
174	#500	C₂²×C₃₁	1	1	yes	characteristic
175	#500	C₂²×C₃₁	1	1	yes	characteristic
176	#494	C₂²×C₃₁	1	1	yes	characteristic
177	#493	C₂²×C₃₁	1	1	yes	characteristic
178	#487	—	2	1	no	—
179	#486	—	2	1	no	—
180	#467	—	2	1	no	—
181	#467	C₂²×C₃₁	1	2	yes	—
182	#466	—	2	1	no	—
183	#466	C₂²×C₃₁	1	2	yes	—
184	C₄×Q₁₆	—	2	1	no	—
185	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
186	C₄×D₈	—	2	1	no	—
187	C₄×D₈	C₂²×C₃₁	1	2	yes	—
188	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
189	#367	C₂²×C₃₁	1	4	yes	—
190	#358	C₂²×C₃₁	1	4	yes	—
191	#357	C₂²×C₃₁	1	4	yes	—
192	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
193	C₄×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	—	16	2	no	—
197	C₂×C₃₁	#3169	1	1	yes	characteristic
198	C₂×C₃₁	C₂²×SD₃₂	1	1	yes	characteristic
199	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
200	C₂×#41	—	4	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	C₂³×C₃₁	1	2	yes	—
205	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	#127	—	2	1	no	—
211	#125	—	4	1	no	—
212	#124	—	4	1	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	4	1	no	—
215	#121	—	2	2	no	—
216	#121	—	4	1	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	4	1	no	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	—	4	1	no	—
221	#117	—	4	1	no	—
222	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
223	SD₃₂	—	2	8	no	—
224	C₂×C₁₆	—	2	4	no	—
225	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
226	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
227	#107	—	2	1	no	—
228	#107	—	2	1	no	—
229	#107	C₂³×C₃₁	1	2	yes	—
230	#107	C₂³×C₃₁	1	2	yes	—
231	#103	—	2	2	no	—
232	#103	—	2	2	no	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
236	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
239	C₃₁	#33736	1	1	yes	characteristic, direct product factor, Sylow
240	#41	—	8	4	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
250	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
252	Q₁₆	—	2	4	no	—
253	Q₁₆	—	4	2	no	—
254	D₈	—	2	4	no	—
255	D₈	—	4	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	D₈×C₃₁	1	1	yes	characteristic
275	C₄²	D₈×C₃₁	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₁₆	—	2	2	no	—
279	C₁₆	#41×C₃₁	1	4	yes	complemented
280	C₂³	—	4	2	no	—
281	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
282	Q₈	—	4	2	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	2	no	—
285	Q₈	—	8	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	2	no	—
290	D₄	—	8	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₃₁×SD₃₂	1	4	yes	complemented
299	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
300	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	C₃₁×#135	1	2	yes	—
308	C₈	C₂×#41×C₃₁	1	2	yes	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	C₂×C₃₁×SD₃₂	1	2	yes	—
315	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	C₂×C₃₁×SD₃₂	1	2	yes	—
322	C₄	C₂×C₃₁×SD₃₂	1	2	yes	—
323	C₄	C₃₁×#508	1	2	yes	—
324	C₄	C₃₁×#532	1	1	yes	characteristic
325	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
326	C₂	—	2	2	no	—
327	C₂	—	16	2	no	—
328	C₂	C₃₁×#3169	1	1	yes	characteristic
329	C₂	C₂²×C₃₁×SD₃₂	1	1	yes	characteristic
330	C₂	C₂×C₃₁×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₈
prime radical	C₃₁×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3170	C₂	semidirect
3. C₃₁×#2254	C₂	semidirect
4. C₃₁×#2248	C₂	semidirect
5. C₃₁×#2241	C₂	semidirect
6. C₃₁×#2240	C₂	semidirect
7. C₃₁×#2217	C₂	semidirect
8. C₄×C₃₁×SD₃₂	C₂	semidirect
9. C₂×C₄×C₁₆×C₃₁	C₂	semidirect
113. C₁₆×C₃₁	#41	semidirect
114. #33736	C₃₁	direct (N × H)
133. C₂×C₄×C₃₁	SD₃₂	semidirect
144. #3170	C₂×C₃₁	semidirect
146. #2254	C₂×C₃₁	semidirect
147. #2248	C₂×C₃₁	semidirect
148. #2241	C₂×C₃₁	semidirect
149. #2240	C₂×C₃₁	semidirect
150. #2217	C₂×C₃₁	semidirect
151. C₄×SD₃₂	C₂×C₃₁	semidirect
152. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
239. C₃₁	#33736	direct (N × H)
279. C₁₆	#41×C₃₁	semidirect
298. C₂×C₄	C₃₁×SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3170	2
C₃₁×#3169	2
C₃₁×#2254	2
C₃₁×#2248	2
C₃₁×#2241	2
C₃₁×#2240	2
C₃₁×#2217	2
C₄×C₃₁×SD₃₂	2
C₂×C₄×C₁₆×C₃₁	2
#33736	31

Subgroups — C31×#33736 #21490495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33736 #21490495