Subgroups — C₃₁×#33780 #21490539

Order 7936.

632 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2250	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2243	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2208	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2206	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2186	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	—	2	1	no	—
10	C₂×C₃₁×#135	C₂²	1	2	yes	—
11	C₃₁×#532	—	2	1	no	—
12	C₃₁×#532	—	2	1	no	—
13	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	—
14	C₂×C₃₁×D₁₆	C₂²	1	4	yes	—
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#495	—	2	1	no	—
17	C₃₁×#495	C₂²	1	2	yes	—
18	C₃₁×#494	C₂²	1	1	yes	characteristic
19	C₃₁×#493	C₂²	1	1	yes	characteristic
20	C₃₁×#493	C₂²	1	1	yes	characteristic
21	C₃₁×#492	—	2	1	no	—
22	C₃₁×#486	—	2	1	no	—
23	C₃₁×#466	—	2	1	no	—
24	C₃₁×#466	C₂²	1	2	yes	—
25	C₄×D₈×C₃₁	—	2	1	no	—
26	C₄×D₈×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₃₁×#358	C₂²	1	4	yes	—
30	C₃₁×#357	C₂²	1	4	yes	—
31	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₃₁×#346	C₂²	1	2	yes	—
33	C₂×#41×C₃₁	—	2	2	no	—
34	C₂×#41×C₃₁	—	4	1	no	—
35	C₃₁×#135	—	4	4	no	—
36	C₃₁×#135	—	4	4	no	—
37	C₂×Q₁₆×C₃₁	—	2	1	no	—
38	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
39	C₂×SD₁₆×C₃₁	—	2	2	no	—
40	C₂×SD₁₆×C₃₁	—	2	2	no	—
41	C₂×D₈×C₃₁	—	2	1	no	—
42	C₂×D₈×C₃₁	—	2	1	no	—
43	C₂×D₈×C₃₁	C₂³	1	2	yes	—
44	C₂×D₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₄⋊Q₈	—	2	1	no	—
48	C₃₁×C₄⋊Q₈	—	2	1	no	—
49	C₃₁×#127	—	2	1	no	—
50	C₃₁×#127	—	2	1	no	—
51	C₃₁×#124	—	2	2	no	—
52	C₃₁×#124	—	4	1	no	—
53	C₃₁×#121	—	2	2	no	—
54	C₃₁×#121	—	4	1	no	—
55	C₃₁×#121	—	4	1	no	—
56	C₃₁×#121	—	4	1	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	4	1	no	—
59	C₄×D₄×C₃₁	—	4	1	no	—
60	C₄×D₄×C₃₁	—	4	1	no	—
61	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
62	C₃₁×SD₃₂	—	4	4	no	—
63	C₃₁×D₁₆	—	4	4	no	—
64	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
65	C₃₁×#107	—	2	1	no	—
66	C₃₁×#107	C₂³	1	2	yes	—
67	C₃₁×#102	—	2	2	no	—
68	C₃₁×#102	—	2	2	no	—
69	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
71	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	#41×C₃₁	—	4	4	no	—
74	#41×C₃₁	—	4	4	no	—
75	#41×C₃₁	—	8	4	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	4	1	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	2	2	no	—
80	C₂×D₄×C₃₁	—	4	2	no	—
81	C₂×D₄×C₃₁	—	4	2	no	—
82	C₂×D₄×C₃₁	—	4	1	no	—
83	C₂×D₄×C₃₁	—	4	1	no	—
84	C₂×D₄×C₃₁	—	4	1	no	—
85	C₂×D₄×C₃₁	—	4	1	no	—
86	C₂²×C₄×C₃₁	—	4	2	no	—
87	C₂²×C₄×C₃₁	—	4	2	no	—
88	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₃₁	—	2	4	no	—
92	Q₁₆×C₃₁	—	4	2	no	—
93	SD₁₆×C₃₁	—	4	4	no	—
94	SD₁₆×C₃₁	—	4	4	no	—
95	D₈×C₃₁	—	2	2	no	—
96	D₈×C₃₁	—	2	4	no	—
97	D₈×C₃₁	—	2	2	no	—
98	D₈×C₃₁	—	4	2	no	—
99	D₈×C₃₁	—	4	2	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	4	1	no	—
108	C₄⋊C₄×C₃₁	—	4	1	no	—
109	C₄⋊C₄×C₃₁	—	4	1	no	—
110	C₂²⋊C₄×C₃₁	—	4	2	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	D₈	1	1	yes	characteristic
115	C₄²×C₃₁	D₈	1	1	yes	characteristic
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₃₁	—	2	4	no	—
119	#33780	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₃₁	—	4	2	no	—
121	C₂³×C₃₁	—	4	2	no	—
122	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
123	Q₈×C₃₁	—	4	2	no	—
124	Q₈×C₃₁	—	4	2	no	—
125	Q₈×C₃₁	—	8	2	no	—
126	D₄×C₃₁	—	4	2	no	—
127	D₄×C₃₁	—	4	2	no	—
128	D₄×C₃₁	—	4	2	no	—
129	D₄×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	8	2	no	—
131	D₄×C₃₁	—	8	2	no	—
132	D₄×C₃₁	—	8	2	no	—
133	D₄×C₃₁	—	8	4	no	—
134	D₄×C₃₁	—	8	4	no	—
135	D₄×C₃₁	—	8	2	no	—
136	C₂×C₄×C₃₁	—	2	1	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	4	1	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	—	8	4	no	—
143	C₂×C₄×C₃₁	—	8	4	no	—
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	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
150	C₈×C₃₁	—	2	1	no	—
151	C₈×C₃₁	—	2	1	no	—
152	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
153	C₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
154	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#2250	C₂×C₃₁	1	2	yes	complemented
157	#2243	C₂×C₃₁	1	2	yes	complemented
158	#2208	C₂×C₃₁	1	2	yes	complemented
159	#2206	C₂×C₃₁	1	2	yes	complemented
160	#2186	C₂×C₃₁	1	4	yes	complemented
161	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
162	C₂²×C₃₁	—	2	1	no	—
163	C₂²×C₃₁	—	2	1	no	—
164	C₂²×C₃₁	—	2	1	no	—
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	8	2	no	—
167	C₂²×C₃₁	—	8	2	no	—
168	C₂²×C₃₁	—	8	2	no	—
169	C₂²×C₃₁	—	8	2	no	—
170	C₂²×C₃₁	—	8	2	no	—
171	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
172	C₄×C₃₁	—	2	1	no	—
173	C₄×C₃₁	—	2	1	no	—
174	C₄×C₃₁	—	2	1	no	—
175	C₄×C₃₁	—	2	1	no	—
176	C₄×C₃₁	—	8	2	no	—
177	C₄×C₃₁	—	8	2	no	—
178	C₄×C₃₁	#509	1	2	yes	—
179	C₄×C₃₁	#535	1	1	yes	characteristic
180	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×#135	—	2	1	no	—
182	C₂×#135	C₂²×C₃₁	1	2	yes	—
183	#532	—	2	1	no	—
184	#532	—	2	1	no	—
185	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	—
186	C₂×D₁₆	C₂²×C₃₁	1	4	yes	—
187	#500	C₂²×C₃₁	1	1	yes	characteristic
188	#495	—	2	1	no	—
189	#495	C₂²×C₃₁	1	2	yes	—
190	#494	C₂²×C₃₁	1	1	yes	characteristic
191	#493	C₂²×C₃₁	1	1	yes	characteristic
192	#493	C₂²×C₃₁	1	1	yes	characteristic
193	#492	—	2	1	no	—
194	#486	—	2	1	no	—
195	#466	—	2	1	no	—
196	#466	C₂²×C₃₁	1	2	yes	—
197	C₄×D₈	—	2	1	no	—
198	C₄×D₈	C₂²×C₃₁	1	2	yes	—
199	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
200	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
201	#358	C₂²×C₃₁	1	4	yes	—
202	#357	C₂²×C₃₁	1	4	yes	—
203	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
204	#346	C₂²×C₃₁	1	2	yes	—
205	C₂×C₃₁	—	4	1	no	—
206	C₂×C₃₁	—	16	2	no	—
207	C₂×C₃₁	—	16	2	no	—
208	C₂×C₃₁	#3205	1	1	yes	characteristic
209	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
210	C₂×C₃₁	#3414	1	1	yes	characteristic
211	C₂×#41	—	2	2	no	—
212	C₂×#41	—	4	1	no	—
213	#135	—	4	4	no	—
214	#135	—	4	4	no	—
215	C₂×Q₁₆	—	2	1	no	—
216	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×SD₁₆	—	2	2	no	—
219	C₂×D₈	—	2	1	no	—
220	C₂×D₈	—	2	1	no	—
221	C₂×D₈	C₂³×C₃₁	1	2	yes	—
222	C₂×D₈	C₂³×C₃₁	1	2	yes	—
223	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	#127	—	2	1	no	—
228	#127	—	2	1	no	—
229	#124	—	2	2	no	—
230	#124	—	4	1	no	—
231	#121	—	2	2	no	—
232	#121	—	4	1	no	—
233	#121	—	4	1	no	—
234	#121	—	4	1	no	—
235	C₄×D₄	—	2	2	no	—
236	C₄×D₄	—	4	1	no	—
237	C₄×D₄	—	4	1	no	—
238	C₄×D₄	—	4	1	no	—
239	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
240	SD₃₂	—	4	4	no	—
241	D₁₆	—	4	4	no	—
242	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
243	#107	—	2	1	no	—
244	#107	C₂³×C₃₁	1	2	yes	—
245	#102	—	2	2	no	—
246	#102	—	2	2	no	—
247	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
248	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
251	C₃₁	#33780	1	1	yes	characteristic, direct product factor, Sylow
252	#41	—	4	4	no	—
253	#41	—	4	4	no	—
254	#41	—	8	4	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	4	1	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	4	1	no	—
264	C₂×D₄	—	4	1	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	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
270	Q₁₆	—	2	4	no	—
271	Q₁₆	—	4	2	no	—
272	SD₁₆	—	4	4	no	—
273	SD₁₆	—	4	4	no	—
274	D₈	—	2	2	no	—
275	D₈	—	2	4	no	—
276	D₈	—	2	2	no	—
277	D₈	—	4	2	no	—
278	D₈	—	4	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
283	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₄²	D₈×C₃₁	1	1	yes	characteristic
294	C₄²	D₈×C₃₁	1	1	yes	characteristic
295	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
296	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
297	C₁₆	—	2	4	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	8	2	no	—
310	D₄	—	8	2	no	—
311	D₄	—	8	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	4	no	—
321	C₂×C₄	—	8	4	no	—
322	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
323	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
324	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
325	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
326	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
331	C₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	8	2	no	—
347	C₄	—	8	2	no	—
348	C₄	C₃₁×#509	1	2	yes	—
349	C₄	C₃₁×#535	1	1	yes	characteristic
350	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	16	2	no	—
353	C₂	—	16	2	no	—
354	C₂	C₃₁×#3205	1	1	yes	characteristic
355	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
356	C₂	C₃₁×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
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×#3169	C₂	semidirect
2. C₃₁×#3147	C₂	semidirect
3. C₃₁×#2250	C₂	semidirect
4. C₃₁×#2243	C₂	semidirect
5. C₃₁×#2208	C₂	semidirect
6. C₃₁×#2206	C₂	semidirect
7. C₃₁×#2186	C₂	semidirect
8. C₃₁×#2163	C₂	semidirect
119. #33780	C₃₁	direct (N × H)
154. #3169	C₂×C₃₁	semidirect
155. #3147	C₂×C₃₁	semidirect
156. #2250	C₂×C₃₁	semidirect
157. #2243	C₂×C₃₁	semidirect
158. #2208	C₂×C₃₁	semidirect
159. #2206	C₂×C₃₁	semidirect
160. #2186	C₂×C₃₁	semidirect
161. #2163	C₂×C₃₁	semidirect
251. C₃₁	#33780	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×#3169	2
C₃₁×#3147	2
C₃₁×#2250	2
C₃₁×#2243	2
C₃₁×#2208	2
C₃₁×#2206	2
C₃₁×#2186	2
C₃₁×#2163	2
#33780	31

Subgroups — C31×#33780 #21490539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33780 #21490539