Subgroups — C₃₁×#33776 #21490535

Order 7936.

592 conjugacy classes of subgroups.

The subgroup lattice has 348 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×#3147	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2248	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2247	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×#2211	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2189	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#135	—	2	1	no	—
11	C₂×C₃₁×#135	C₂²	1	2	yes	—
12	C₃₁×#533	—	2	1	no	—
13	C₃₁×#532	—	2	1	no	—
14	C₂×C₃₁×Q₃₂	C₂²	1	4	yes	—
15	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#495	—	2	1	no	—
18	C₃₁×#495	C₂²	1	2	yes	—
19	C₃₁×#494	C₂²	1	1	yes	characteristic
20	C₃₁×#494	C₂²	1	1	yes	characteristic
21	C₃₁×#493	C₂²	1	1	yes	characteristic
22	C₃₁×#492	—	2	1	no	—
23	C₃₁×#487	—	2	1	no	—
24	C₃₁×#467	—	2	1	no	—
25	C₃₁×#467	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₃₁	—	2	1	no	—
27	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₃₁×#358	C₂²	1	4	yes	—
31	C₃₁×#357	C₂²	1	4	yes	—
32	C₃₁×C₂²⋊C₁₆	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	—	2	2	no	—
36	C₃₁×#135	—	4	4	no	—
37	C₃₁×#135	—	4	4	no	—
38	C₂×Q₁₆×C₃₁	—	2	1	no	—
39	C₂×Q₁₆×C₃₁	—	2	1	no	—
40	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
41	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
42	C₂×SD₁₆×C₃₁	—	2	2	no	—
43	C₂×SD₁₆×C₃₁	—	2	2	no	—
44	C₂×D₈×C₃₁	—	2	1	no	—
45	C₂×D₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×C₄⋊Q₈	—	2	1	no	—
49	C₃₁×C₄⋊Q₈	—	2	1	no	—
50	C₃₁×C₄⋊Q₈	—	2	1	no	—
51	C₃₁×#127	—	2	1	no	—
52	C₃₁×#125	—	4	1	no	—
53	C₃₁×#124	—	2	2	no	—
54	C₃₁×C₂²⋊Q₈	—	2	2	no	—
55	C₃₁×C₂²⋊Q₈	—	4	1	no	—
56	C₃₁×#121	—	4	1	no	—
57	C₃₁×#121	—	4	1	no	—
58	C₄×Q₈×C₃₁	—	2	2	no	—
59	C₄×Q₈×C₃₁	—	4	1	no	—
60	C₄×D₄×C₃₁	—	4	1	no	—
61	C₄×D₄×C₃₁	—	4	1	no	—
62	C₃₁×#117	—	4	1	no	—
63	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
64	C₃₁×Q₃₂	—	4	4	no	—
65	C₃₁×SD₃₂	—	4	4	no	—
66	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
67	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
68	C₃₁×#107	—	2	1	no	—
69	C₃₁×#107	C₂³	1	2	yes	—
70	C₃₁×#103	—	2	2	no	—
71	C₃₁×#103	—	2	2	no	—
72	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
73	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
74	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	#41×C₃₁	—	4	4	no	—
77	#41×C₃₁	—	4	4	no	—
78	C₂×Q₈×C₃₁	—	2	2	no	—
79	C₂×Q₈×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	4	1	no	—
81	C₂×D₄×C₃₁	—	2	2	no	—
82	C₂×D₄×C₃₁	—	4	2	no	—
83	C₂×D₄×C₃₁	—	4	1	no	—
84	C₂×D₄×C₃₁	—	4	1	no	—
85	C₂²×C₄×C₃₁	—	4	2	no	—
86	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₃₁	—	2	4	no	—
90	Q₁₆×C₃₁	—	2	4	no	—
91	Q₁₆×C₃₁	—	4	2	no	—
92	Q₁₆×C₃₁	—	4	2	no	—
93	SD₁₆×C₃₁	—	4	4	no	—
94	SD₁₆×C₃₁	—	4	4	no	—
95	D₈×C₃₁	—	2	4	no	—
96	D₈×C₃₁	—	4	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	4	1	no	—
105	C₄⋊C₄×C₃₁	—	4	1	no	—
106	C₄⋊C₄×C₃₁	—	4	1	no	—
107	C₄⋊C₄×C₃₁	—	4	1	no	—
108	C₄⋊C₄×C₃₁	—	4	2	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₄⋊C₄×C₃₁	—	4	1	no	—
111	C₂²⋊C₄×C₃₁	—	4	2	no	—
112	C₂²⋊C₄×C₃₁	—	4	2	no	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₄²×C₃₁	—	4	2	no	—
115	C₄²×C₃₁	D₈	1	1	yes	characteristic
116	C₄²×C₃₁	D₈	1	1	yes	characteristic
117	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₃₁	—	2	2	no	—
120	C₁₆×C₃₁	—	2	2	no	—
121	#33776	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₃₁	—	4	2	no	—
123	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
124	Q₈×C₃₁	—	4	2	no	—
125	Q₈×C₃₁	—	4	2	no	—
126	Q₈×C₃₁	—	4	2	no	—
127	Q₈×C₃₁	—	4	2	no	—
128	Q₈×C₃₁	—	8	2	no	—
129	D₄×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	4	2	no	—
131	D₄×C₃₁	—	8	4	no	—
132	D₄×C₃₁	—	8	2	no	—
133	D₄×C₃₁	—	8	2	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	8	4	no	—
142	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₃₁	—	2	1	no	—
149	C₈×C₃₁	—	2	1	no	—
150	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
151	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
152	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#3147	C₂×C₃₁	1	1	yes	characteristic
154	#2248	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#2247	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#2241	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#2240	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#2211	C₂×C₃₁	1	4	yes	complemented
159	#2189	C₂×C₃₁	1	4	yes	complemented
160	#2162	C₂×C₃₁	1	1	yes	characteristic, complemented
161	C₂²×C₃₁	—	2	1	no	—
162	C₂²×C₃₁	—	2	1	no	—
163	C₂²×C₃₁	—	2	1	no	—
164	C₂²×C₃₁	—	8	2	no	—
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	8	2	no	—
167	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
168	C₄×C₃₁	—	2	1	no	—
169	C₄×C₃₁	—	2	1	no	—
170	C₄×C₃₁	—	2	1	no	—
171	C₄×C₃₁	—	8	2	no	—
172	C₄×C₃₁	—	8	2	no	—
173	C₄×C₃₁	—	8	2	no	—
174	C₄×C₃₁	#508	1	2	yes	—
175	C₄×C₃₁	#510	1	2	yes	—
176	C₄×C₃₁	#535	1	1	yes	characteristic
177	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₂×#135	—	2	1	no	—
179	C₂×#135	C₂²×C₃₁	1	2	yes	—
180	#533	—	2	1	no	—
181	#532	—	2	1	no	—
182	C₂×Q₃₂	C₂²×C₃₁	1	4	yes	—
183	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
184	#500	C₂²×C₃₁	1	1	yes	characteristic
185	#495	—	2	1	no	—
186	#495	C₂²×C₃₁	1	2	yes	—
187	#494	C₂²×C₃₁	1	1	yes	characteristic
188	#494	C₂²×C₃₁	1	1	yes	characteristic
189	#493	C₂²×C₃₁	1	1	yes	characteristic
190	#492	—	2	1	no	—
191	#487	—	2	1	no	—
192	#467	—	2	1	no	—
193	#467	C₂²×C₃₁	1	2	yes	—
194	C₄×Q₁₆	—	2	1	no	—
195	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
196	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
197	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
198	#358	C₂²×C₃₁	1	4	yes	—
199	#357	C₂²×C₃₁	1	4	yes	—
200	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
201	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
202	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
203	C₂×C₃₁	—	4	1	no	—
204	C₂×C₃₁	—	16	2	no	—
205	C₂×C₃₁	#3205	1	1	yes	characteristic
206	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
207	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
208	C₂×#41	—	2	2	no	—
209	#135	—	4	4	no	—
210	#135	—	4	4	no	—
211	C₂×Q₁₆	—	2	1	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
214	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×D₈	—	2	1	no	—
218	C₂×D₈	C₂³×C₃₁	1	2	yes	—
219	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	C₄⋊Q₈	—	2	1	no	—
224	#127	—	2	1	no	—
225	#125	—	4	1	no	—
226	#124	—	2	2	no	—
227	C₂²⋊Q₈	—	2	2	no	—
228	C₂²⋊Q₈	—	4	1	no	—
229	#121	—	4	1	no	—
230	#121	—	4	1	no	—
231	C₄×Q₈	—	2	2	no	—
232	C₄×Q₈	—	4	1	no	—
233	C₄×D₄	—	4	1	no	—
234	C₄×D₄	—	4	1	no	—
235	#117	—	4	1	no	—
236	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
237	Q₃₂	—	4	4	no	—
238	SD₃₂	—	4	4	no	—
239	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
240	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
241	#107	—	2	1	no	—
242	#107	C₂³×C₃₁	1	2	yes	—
243	#103	—	2	2	no	—
244	#103	—	2	2	no	—
245	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
246	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
247	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
249	C₃₁	#33776	1	1	yes	characteristic, direct product factor, Sylow
250	#41	—	4	4	no	—
251	#41	—	4	4	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	4	1	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
261	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
263	Q₁₆	—	2	4	no	—
264	Q₁₆	—	2	4	no	—
265	Q₁₆	—	4	2	no	—
266	Q₁₆	—	4	2	no	—
267	SD₁₆	—	4	4	no	—
268	SD₁₆	—	4	4	no	—
269	D₈	—	2	4	no	—
270	D₈	—	4	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₄²	—	4	2	no	—
289	C₄²	D₈×C₃₁	1	1	yes	characteristic
290	C₄²	D₈×C₃₁	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
292	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
293	C₁₆	—	2	2	no	—
294	C₁₆	—	2	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
297	Q₈	—	4	2	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	4	2	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	8	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	8	4	no	—
305	D₄	—	8	2	no	—
306	D₄	—	8	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	4	no	—
315	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
316	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
317	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
318	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
319	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
324	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	C₃₁×#508	1	2	yes	—
339	C₄	C₃₁×#510	1	2	yes	—
340	C₄	C₃₁×#535	1	1	yes	characteristic
341	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
342	C₂	—	4	1	no	—
343	C₂	—	16	2	no	—
344	C₂	C₃₁×#3205	1	1	yes	characteristic
345	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
346	C₂	C₂×C₃₁×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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×#2248	C₂	semidirect
4. C₃₁×#2247	C₂	semidirect
5. C₃₁×#2241	C₂	semidirect
6. C₃₁×#2240	C₂	semidirect
7. C₃₁×#2211	C₂	semidirect
8. C₃₁×#2189	C₂	semidirect
9. C₃₁×#2162	C₂	semidirect
121. #33776	C₃₁	direct (N × H)
152. #3170	C₂×C₃₁	semidirect
154. #2248	C₂×C₃₁	semidirect
155. #2247	C₂×C₃₁	semidirect
156. #2241	C₂×C₃₁	semidirect
157. #2240	C₂×C₃₁	semidirect
158. #2211	C₂×C₃₁	semidirect
159. #2189	C₂×C₃₁	semidirect
160. #2162	C₂×C₃₁	semidirect
249. C₃₁	#33776	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×#3170	2
C₃₁×#3147	2
C₃₁×#2248	2
C₃₁×#2247	2
C₃₁×#2241	2
C₃₁×#2240	2
C₃₁×#2211	2
C₃₁×#2189	2
C₃₁×#2162	2
#33776	31

Subgroups — C31×#33776 #21490535

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33776 #21490535