Subgroups — C₃₁×#29264 #21486023

Order 7936.

686 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×#2331	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2327	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2097	C₂	1	2	yes	maximal
4	C₃₁×#2024	C₂	1	2	yes	maximal
5	C₃₁×#1981	C₂	1	2	yes	maximal
6	C₃₁×#1949	C₂	1	2	yes	maximal
7	C₃₁×#1865	C₂	1	2	yes	maximal
8	C₃₁×#1863	C₂	1	2	yes	maximal
9	C₃₁×#1832	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#107	C₂²	1	2	yes	—
14	C₂×C₃₁×#106	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#103	C₂²	1	2	yes	—
17	C₂×C₃₁×#103	C₂²	1	2	yes	—
18	C₂×C₃₁×#103	C₂²	1	2	yes	—
19	C₂×C₃₁×#103	C₂²	1	2	yes	—
20	C₃₁×#391	—	2	2	no	—
21	C₃₁×#391	C₂²	1	2	yes	—
22	C₃₁×#389	C₂²	1	2	yes	—
23	C₃₁×#387	—	2	2	no	—
24	C₃₁×#387	—	2	2	no	—
25	C₃₁×#387	—	2	2	no	—
26	C₃₁×#382	C₂²	1	1	yes	characteristic
27	C₃₁×#382	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#376	C₂²	1	1	yes	characteristic
30	C₃₁×#340	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₃₁×#336	C₂²	1	1	yes	characteristic
34	C₃₁×#336	C₂²	1	1	yes	characteristic
35	C₂²×Q₈×C₃₁	—	2	1	no	—
36	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₃₁×C₄⋊Q₈	—	2	4	no	—
40	C₃₁×C₄⋊Q₈	—	2	2	no	—
41	C₃₁×C₄⋊Q₈	Q₈	1	4	yes	complemented
42	C₄×Q₈×C₃₁	—	4	2	no	—
43	C₄×Q₈×C₃₁	—	4	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
51	C₂×C₄²×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	—	2	2	no	—
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×#107	—	4	2	no	—
57	C₃₁×#106	—	4	2	no	—
58	C₃₁×C₄⋊C₈	—	2	4	no	—
59	C₃₁×#103	—	2	4	no	—
60	C₃₁×#103	—	2	4	no	—
61	C₃₁×#103	—	4	2	no	—
62	C₃₁×#103	—	4	2	no	—
63	C₃₁×#95	—	2	2	no	—
64	C₃₁×#95	—	2	1	no	—
65	C₃₁×#95	—	2	2	no	—
66	C₃₁×#95	—	2	1	no	—
67	C₃₁×#95	—	2	2	no	—
68	C₃₁×#95	—	2	2	no	—
69	C₃₁×#95	—	2	2	no	—
70	C₃₁×#95	D₄	1	2	yes	—
71	C₃₁×#95	D₄	1	2	yes	—
72	C₂×Q₈×C₃₁	—	2	4	no	—
73	C₂×Q₈×C₃₁	—	2	2	no	—
74	C₂×Q₈×C₃₁	—	2	2	no	—
75	C₂×Q₈×C₃₁	—	2	2	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	2	2	no	—
78	C₂×Q₈×C₃₁	—	8	2	no	—
79	C₂×Q₈×C₃₁	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	2	2	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₄⋊C₄×C₃₁	—	2	8	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₄⋊C₄×C₃₁	—	4	4	no	—
103	C₄⋊C₄×C₃₁	—	8	2	no	—
104	C₄⋊C₄×C₃₁	#41	1	4	yes	—
105	C₄²×C₃₁	—	2	2	no	—
106	C₄²×C₃₁	—	2	2	no	—
107	C₄²×C₃₁	—	4	4	no	—
108	C₄²×C₃₁	—	8	2	no	—
109	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
110	C₄²×C₃₁	#41	1	2	yes	—
111	#29264	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₃₁	—	2	4	no	—
114	Q₈×C₃₁	—	2	4	no	—
115	Q₈×C₃₁	—	4	2	no	—
116	Q₈×C₃₁	—	4	2	no	—
117	Q₈×C₃₁	—	8	4	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	4	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	2	4	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	2	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	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
140	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
142	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
144	C₈×C₃₁	—	4	2	no	—
145	C₈×C₃₁	—	4	2	no	—
146	#2331	C₂×C₃₁	1	1	yes	characteristic
147	#2327	C₂×C₃₁	1	1	yes	characteristic
148	#2097	C₂×C₃₁	1	2	yes	—
149	#2024	C₂×C₃₁	1	2	yes	—
150	#1981	C₂×C₃₁	1	2	yes	—
151	#1949	C₂×C₃₁	1	2	yes	—
152	#1865	C₂×C₃₁	1	2	yes	—
153	#1863	C₂×C₃₁	1	2	yes	—
154	#1832	C₂×C₃₁	1	1	yes	characteristic
155	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
156	C₂²×C₃₁	#525	1	1	yes	characteristic
157	C₂²×C₃₁	#535	1	1	yes	characteristic
158	C₂²×C₃₁	#577	1	2	yes	—
159	C₂²×C₃₁	#578	1	2	yes	—
160	C₄×C₃₁	—	2	4	no	—
161	C₄×C₃₁	—	4	4	no	—
162	C₄×C₃₁	—	4	1	no	—
163	C₄×C₃₁	—	4	4	no	—
164	C₄×C₃₁	—	4	1	no	—
165	C₄×C₃₁	—	4	2	no	—
166	C₄×C₃₁	—	8	2	no	—
167	C₄×C₃₁	—	8	2	no	—
168	C₄×C₃₁	#554	1	2	yes	—
169	C₄×C₃₁	#559	1	2	yes	—
170	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
171	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
172	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
173	C₂×#107	C₂²×C₃₁	1	2	yes	—
174	C₂×#106	C₂²×C₃₁	1	2	yes	—
175	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
176	C₂×#103	C₂²×C₃₁	1	2	yes	—
177	C₂×#103	C₂²×C₃₁	1	2	yes	—
178	C₂×#103	C₂²×C₃₁	1	2	yes	—
179	C₂×#103	C₂²×C₃₁	1	2	yes	—
180	#391	—	2	2	no	—
181	#391	C₂²×C₃₁	1	2	yes	—
182	#389	C₂²×C₃₁	1	2	yes	—
183	#387	—	2	2	no	—
184	#387	—	2	2	no	—
185	#387	—	2	2	no	—
186	#382	C₂²×C₃₁	1	1	yes	characteristic
187	#382	C₂²×C₃₁	1	1	yes	characteristic
188	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
189	#376	C₂²×C₃₁	1	1	yes	characteristic
190	#340	C₂²×C₃₁	1	2	yes	—
191	#340	C₂²×C₃₁	1	2	yes	—
192	#336	C₂²×C₃₁	1	2	yes	—
193	#336	C₂²×C₃₁	1	1	yes	characteristic
194	#336	C₂²×C₃₁	1	1	yes	characteristic
195	C₂×C₃₁	#2580	1	1	yes	characteristic
196	C₂×C₃₁	#3082	1	2	yes	—
197	C₂×C₃₁	#3123	1	1	yes	characteristic
198	C₂×C₃₁	#3127	1	1	yes	characteristic
199	C₂×C₃₁	#3225	1	2	yes	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
202	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
203	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
204	C₄⋊Q₈	—	2	4	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	Q₈×C₃₁	1	4	yes	complemented
207	C₄×Q₈	—	4	2	no	—
208	C₄×Q₈	—	4	2	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
221	#107	—	4	2	no	—
222	#106	—	4	2	no	—
223	C₄⋊C₈	—	2	4	no	—
224	#103	—	2	4	no	—
225	#103	—	2	4	no	—
226	#103	—	4	2	no	—
227	#103	—	4	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	1	no	—
230	#95	—	2	2	no	—
231	#95	—	2	1	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	D₄×C₃₁	1	2	yes	—
236	#95	D₄×C₃₁	1	2	yes	—
237	C₃₁	#29264	1	1	yes	characteristic, direct product factor, Sylow
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	8	2	no	—
245	C₂×Q₈	C₂×Q₈×C₃₁	1	4	yes	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
254	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	8	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	8	2	no	—
270	C₄⋊C₄	#41×C₃₁	1	4	yes	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	4	no	—
274	C₄²	—	8	2	no	—
275	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
276	C₄²	#41×C₃₁	1	2	yes	—
277	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
278	Q₈	—	2	4	no	—
279	Q₈	—	2	4	no	—
280	Q₈	—	4	2	no	—
281	Q₈	—	4	2	no	—
282	Q₈	—	8	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
305	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
307	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
312	C₂²	C₃₁×#525	1	1	yes	characteristic
313	C₂²	C₃₁×#535	1	1	yes	characteristic
314	C₂²	C₃₁×#577	1	2	yes	—
315	C₂²	C₃₁×#578	1	2	yes	—
316	C₄	—	2	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	4	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	C₃₁×#554	1	2	yes	—
325	C₄	C₃₁×#559	1	2	yes	—
326	C₂	C₃₁×#2580	1	1	yes	characteristic
327	C₂	C₃₁×#3082	1	2	yes	—
328	C₂	C₃₁×#3123	1	1	yes	characteristic
329	C₂	C₃₁×#3127	1	1	yes	characteristic
330	C₂	C₃₁×#3225	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
41. C₃₁×C₄⋊Q₈	Q₈	semidirect
111. #29264	C₃₁	direct (N × H)
206. C₄⋊Q₈	Q₈×C₃₁	semidirect
237. C₃₁	#29264	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×#2331	2
C₃₁×#2327	2
C₃₁×#2097	2
C₃₁×#2024	2
C₃₁×#1981	2
C₃₁×#1949	2
C₃₁×#1865	2
C₃₁×#1863	2
C₃₁×#1832	2
#29264	31

Subgroups — C31×#29264 #21486023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29264 #21486023