Subgroups — C₃₁×#33933 #21490692

Order 7936.

532 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3391	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3380	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2244	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2240	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×#2206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2206	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2177	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#554	—	2	1	no	—
13	C₃₁×#550	—	2	1	no	—
14	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#499	C₂²	1	2	yes	—
18	C₃₁×#493	C₂²	1	1	yes	characteristic
19	C₃₁×#493	C₂²	1	2	yes	—
20	C₃₁×#478	—	2	2	no	—
21	C₃₁×#476	—	2	1	no	—
22	C₃₁×#459	—	2	1	no	—
23	C₃₁×#459	—	2	2	no	—
24	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×D₈×C₃₁	C₂²	1	2	yes	—
27	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×D₈×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#367	C₂²	1	2	yes	—
30	C₃₁×#366	C₂²	1	1	yes	characteristic
31	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₃₁×#357	C₂²	1	1	yes	characteristic
34	C₃₁×#357	C₂²	1	1	yes	characteristic
35	C₃₁×#357	C₂²	1	2	yes	—
36	C₃₁×#357	C₂²	1	2	yes	—
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	C₂×#41×C₃₁	—	4	1	no	—
42	C₂×D₈×C₃₁	—	2	2	no	—
43	C₂×D₈×C₃₁	—	2	1	no	—
44	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×D₈×C₃₁	C₂³	1	2	yes	—
46	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₄⋊Q₈	—	2	1	no	—
48	C₃₁×#127	—	2	2	no	—
49	C₃₁×#127	—	2	1	no	—
50	C₃₁×#125	—	2	2	no	—
51	C₃₁×C₂²⋊Q₈	—	4	1	no	—
52	C₃₁×C₂²⋊Q₈	—	4	2	no	—
53	C₃₁×#121	—	4	1	no	—
54	C₃₁×#121	—	4	2	no	—
55	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
56	C₄×D₄×C₃₁	—	2	2	no	—
57	C₄×D₄×C₃₁	—	2	1	no	—
58	C₄×D₄×C₃₁	—	2	1	no	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
61	C₃₁×D₁₆	—	2	2	no	—
62	C₃₁×D₁₆	—	2	4	no	—
63	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
65	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×#107	—	2	1	no	—
67	C₃₁×#107	C₂³	1	1	yes	characteristic
68	C₃₁×#107	C₂³	1	1	yes	characteristic
69	C₃₁×#107	C₂³	1	2	yes	—
70	C₃₁×#106	—	2	2	no	—
71	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
73	C₃₁×#102	—	2	2	no	—
74	C₃₁×#102	—	2	1	no	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	—	2	1	no	—
77	C₄×C₈×C₃₁	C₂³	1	2	yes	—
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	#41×C₃₁	—	8	4	no	—
80	C₂×Q₈×C₃₁	—	4	1	no	—
81	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
82	C₂×D₄×C₃₁	—	2	1	no	—
83	C₂×D₄×C₃₁	—	2	2	no	—
84	C₂×D₄×C₃₁	—	2	1	no	—
85	C₂×D₄×C₃₁	—	4	2	no	—
86	C₂×D₄×C₃₁	—	4	1	no	—
87	C₂×D₄×C₃₁	—	4	1	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂²×C₄×C₃₁	—	4	1	no	—
90	C₂²×C₄×C₃₁	—	4	2	no	—
91	C₂²×C₄×C₃₁	—	4	1	no	—
92	C₂²×C₄×C₃₁	—	4	2	no	—
93	D₈×C₃₁	—	2	1	no	—
94	D₈×C₃₁	—	2	2	no	—
95	D₈×C₃₁	—	2	1	no	—
96	D₈×C₃₁	—	2	1	no	—
97	D₈×C₃₁	—	2	2	no	—
98	D₈×C₃₁	—	8	2	no	—
99	D₈×C₃₁	—	8	1	no	—
100	D₈×C₃₁	#41	1	2	yes	complemented
101	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₃₁	—	2	1	no	—
105	C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	4	2	no	—
108	C₄⋊C₄×C₃₁	—	4	1	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
111	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
112	C₂²⋊C₄×C₃₁	—	4	1	no	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₂²⋊C₄×C₃₁	—	4	1	no	—
115	C₂²⋊C₄×C₃₁	—	4	2	no	—
116	C₄²×C₃₁	C₂×D₄	1	2	yes	—
117	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₃₁	—	2	1	no	—
119	C₁₆×C₃₁	—	2	2	no	—
120	C₁₆×C₃₁	—	2	1	no	—
121	#33933	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₃₁	—	4	1	no	—
123	C₂³×C₃₁	—	4	2	no	—
124	C₂³×C₃₁	—	4	1	no	—
125	Q₈×C₃₁	—	2	2	no	—
126	Q₈×C₃₁	—	16	1	no	—
127	D₄×C₃₁	—	2	2	no	—
128	D₄×C₃₁	—	4	1	no	—
129	D₄×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	4	1	no	—
131	D₄×C₃₁	—	4	1	no	—
132	D₄×C₃₁	—	4	2	no	—
133	D₄×C₃₁	—	8	4	no	—
134	D₄×C₃₁	—	8	2	no	—
135	D₄×C₃₁	—	16	2	no	—
136	D₄×C₃₁	—	16	1	no	—
137	C₂×C₄×C₃₁	—	4	1	no	—
138	C₂×C₄×C₃₁	—	4	1	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	8	4	no	—
141	C₂×C₄×C₃₁	—	8	2	no	—
142	C₂×C₄×C₃₁	—	8	2	no	—
143	C₂×C₄×C₃₁	—	16	2	no	—
144	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₃₁	—	2	2	no	—
150	C₈×C₃₁	—	2	1	no	—
151	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
152	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
153	#3391	C₂×C₃₁	1	1	yes	characteristic, complemented
154	#3380	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#2244	C₂×C₃₁	1	2	yes	complemented
156	#2240	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#2228	C₂×C₃₁	1	2	yes	complemented
158	#2226	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#2206	C₂×C₃₁	1	2	yes	complemented
160	#2206	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#2183	C₂×C₃₁	1	1	yes	characteristic, complemented
162	#2177	C₂×C₃₁	1	2	yes	complemented
163	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
164	C₂²×C₃₁	—	4	2	no	—
165	C₂²×C₃₁	—	8	1	no	—
166	C₂²×C₃₁	—	8	1	no	—
167	C₂²×C₃₁	—	8	2	no	—
168	C₂²×C₃₁	—	8	1	no	—
169	C₂²×C₃₁	—	8	2	no	—
170	C₂²×C₃₁	—	8	1	no	—
171	C₂²×C₃₁	—	8	1	no	—
172	C₂²×C₃₁	—	8	2	no	—
173	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
174	C₄×C₃₁	—	2	2	no	—
175	C₄×C₃₁	—	2	2	no	—
176	C₄×C₃₁	—	2	1	no	—
177	C₄×C₃₁	—	8	1	no	—
178	C₄×C₃₁	—	8	1	no	—
179	C₄×C₃₁	—	8	2	no	—
180	C₄×C₃₁	#509	1	2	yes	—
181	C₄×C₃₁	#548	1	1	yes	characteristic
182	C₄×C₃₁	#577	1	1	yes	characteristic
183	#554	—	2	1	no	—
184	#550	—	2	1	no	—
185	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
186	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
187	#500	C₂²×C₃₁	1	1	yes	characteristic
188	#499	C₂²×C₃₁	1	2	yes	—
189	#493	C₂²×C₃₁	1	1	yes	characteristic
190	#493	C₂²×C₃₁	1	2	yes	—
191	#478	—	2	2	no	—
192	#476	—	2	1	no	—
193	#459	—	2	1	no	—
194	#459	—	2	2	no	—
195	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
196	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
197	C₄×D₈	C₂²×C₃₁	1	2	yes	—
198	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
199	C₄×D₈	C₂²×C₃₁	1	2	yes	—
200	#367	C₂²×C₃₁	1	2	yes	—
201	#366	C₂²×C₃₁	1	1	yes	characteristic
202	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
203	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
204	#357	C₂²×C₃₁	1	1	yes	characteristic
205	#357	C₂²×C₃₁	1	1	yes	characteristic
206	#357	C₂²×C₃₁	1	2	yes	—
207	#357	C₂²×C₃₁	1	2	yes	—
208	#357	C₂²×C₃₁	1	2	yes	—
209	#357	C₂²×C₃₁	1	2	yes	—
210	#346	C₂²×C₃₁	1	2	yes	—
211	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
212	C₂×C₃₁	—	8	2	no	—
213	C₂×C₃₁	—	16	2	no	—
214	C₂×C₃₁	—	16	1	no	—
215	C₂×C₃₁	#3358	1	1	yes	characteristic
216	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
217	C₂×C₃₁	#3415	1	1	yes	characteristic
218	C₂×#41	—	4	1	no	—
219	C₂×D₈	—	2	2	no	—
220	C₂×D₈	—	2	1	no	—
221	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
222	C₂×D₈	C₂³×C₃₁	1	2	yes	—
223	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
224	C₄⋊Q₈	—	2	1	no	—
225	#127	—	2	2	no	—
226	#127	—	2	1	no	—
227	#125	—	2	2	no	—
228	C₂²⋊Q₈	—	4	1	no	—
229	C₂²⋊Q₈	—	4	2	no	—
230	#121	—	4	1	no	—
231	#121	—	4	2	no	—
232	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
233	C₄×D₄	—	2	2	no	—
234	C₄×D₄	—	2	1	no	—
235	C₄×D₄	—	2	1	no	—
236	C₄×D₄	—	2	2	no	—
237	C₂×C₄⋊C₄	—	4	1	no	—
238	D₁₆	—	2	2	no	—
239	D₁₆	—	2	4	no	—
240	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
241	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
242	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
243	#107	—	2	1	no	—
244	#107	C₂³×C₃₁	1	1	yes	characteristic
245	#107	C₂³×C₃₁	1	2	yes	—
246	#107	C₂³×C₃₁	1	1	yes	characteristic
247	#106	—	2	2	no	—
248	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
249	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
250	#102	—	2	2	no	—
251	#102	—	2	1	no	—
252	#102	—	2	2	no	—
253	#102	—	2	1	no	—
254	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₃₁	1	2	yes	—
256	C₃₁	#33933	1	1	yes	characteristic, direct product factor, Sylow
257	#41	—	8	4	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	4	2	no	—
271	D₈	—	2	1	no	—
272	D₈	—	2	2	no	—
273	D₈	—	2	1	no	—
274	D₈	—	2	1	no	—
275	D₈	—	2	2	no	—
276	D₈	—	8	2	no	—
277	D₈	—	8	1	no	—
278	D₈	#41×C₃₁	1	2	yes	complemented
279	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
281	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
289	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	1	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₄²	C₂×D₄×C₃₁	1	2	yes	—
295	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
296	C₁₆	—	2	1	no	—
297	C₁₆	—	2	2	no	—
298	C₁₆	—	2	1	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	1	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	16	1	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	1	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	8	4	no	—
311	D₄	—	8	2	no	—
312	D₄	—	16	2	no	—
313	D₄	—	16	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	4	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	—	16	2	no	—
321	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
322	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
323	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
324	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	2	1	no	—
328	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
329	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
330	C₂²	—	4	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	2	no	—
339	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	2	no	—
346	C₄	C₃₁×#509	1	2	yes	—
347	C₄	C₃₁×#548	1	1	yes	characteristic
348	C₄	C₃₁×#577	1	1	yes	characteristic
349	C₂	—	8	2	no	—
350	C₂	—	16	2	no	—
351	C₂	—	16	1	no	—
352	C₂	C₃₁×#3358	1	1	yes	characteristic
353	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
354	C₂	C₃₁×#3415	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 ◁ 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×#3391	C₂	semidirect
2. C₃₁×#3380	C₂	semidirect
3. C₃₁×#2244	C₂	semidirect
4. C₃₁×#2240	C₂	semidirect
5. C₃₁×#2228	C₂	semidirect
6. C₃₁×#2226	C₂	semidirect
7. C₃₁×#2206	C₂	semidirect
8. C₃₁×#2206	C₂	semidirect
9. C₃₁×#2183	C₂	semidirect
10. C₃₁×#2177	C₂	semidirect
11. C₄×C₃₁×D₁₆	C₂	semidirect
100. D₈×C₃₁	#41	semidirect
121. #33933	C₃₁	direct (N × H)
153. #3391	C₂×C₃₁	semidirect
154. #3380	C₂×C₃₁	semidirect
155. #2244	C₂×C₃₁	semidirect
156. #2240	C₂×C₃₁	semidirect
157. #2228	C₂×C₃₁	semidirect
158. #2226	C₂×C₃₁	semidirect
159. #2206	C₂×C₃₁	semidirect
160. #2206	C₂×C₃₁	semidirect
161. #2183	C₂×C₃₁	semidirect
162. #2177	C₂×C₃₁	semidirect
163. C₄×D₁₆	C₂×C₃₁	semidirect
256. C₃₁	#33933	direct (N × H)
278. D₈	#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×#3391	2
C₃₁×#3380	2
C₃₁×#2244	2
C₃₁×#2240	2
C₃₁×#2228	2
C₃₁×#2226	2
C₃₁×#2206	2
C₃₁×#2183	2
C₃₁×#2177	2
C₄×C₃₁×D₁₆	2
#33933	31

Subgroups — C31×#33933 #21490692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33933 #21490692