Subgroups — C₃₁×#29146 #21485905

Order 7936.

742 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2026	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2025	C₂	1	2	yes	maximal
5	C₃₁×#1959	C₂	1	2	yes	maximal
6	C₃₁×#1836	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1805	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1804	C₂	1	2	yes	maximal
9	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#125	C₂²	1	2	yes	—
11	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×#103	C₂²	1	4	yes	—
16	C₂×C₃₁×#102	C₂²	1	4	yes	—
17	C₂×C₃₁×#97	C₂²	1	2	yes	—
18	C₃₁×#391	C₂²	1	1	yes	characteristic
19	C₃₁×#390	C₂²	1	1	yes	characteristic
20	C₃₁×#389	—	2	2	no	—
21	C₃₁×#389	—	2	2	no	—
22	C₃₁×#388	—	2	2	no	—
23	C₃₁×#384	C₂²	1	2	yes	—
24	C₃₁×#382	C₂²	1	2	yes	—
25	C₃₁×#382	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
27	C₄³×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#340	C₂²	1	4	yes	—
29	C₃₁×#340	C₂²	1	4	yes	—
30	C₃₁×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₃₁	—	2	1	no	—
34	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
35	C₃₁×#125	—	4	4	no	—
36	C₃₁×#124	—	2	2	no	—
37	C₃₁×#124	Q₈	1	4	yes	—
38	C₄×Q₈×C₃₁	—	2	4	no	—
39	C₄×D₄×C₃₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄²×C₃₁	—	2	1	no	—
51	C₂×C₄²×C₃₁	D₄	1	2	yes	—
52	C₂×C₄²×C₃₁	D₄	1	2	yes	—
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×C₄⋊C₈	—	2	4	no	—
57	C₃₁×#103	—	4	4	no	—
58	C₃₁×#102	—	4	4	no	—
59	C₃₁×#97	—	2	4	no	—
60	C₃₁×#95	—	2	2	no	—
61	C₃₁×#95	—	2	2	no	—
62	C₃₁×#95	—	2	2	no	—
63	C₃₁×#95	—	2	2	no	—
64	C₂⁴×C₃₁	—	2	1	no	—
65	C₂×Q₈×C₃₁	—	2	1	no	—
66	C₂×Q₈×C₃₁	—	2	2	no	—
67	C₂×Q₈×C₃₁	—	2	2	no	—
68	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
69	C₂×D₄×C₃₁	—	2	2	no	—
70	C₂×D₄×C₃₁	—	2	1	no	—
71	C₂×D₄×C₃₁	—	2	2	no	—
72	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₃₁	—	2	1	no	—
74	C₂²×C₄×C₃₁	—	2	1	no	—
75	C₂²×C₄×C₃₁	—	2	1	no	—
76	C₂²×C₄×C₃₁	—	2	4	no	—
77	C₂²×C₄×C₃₁	—	4	4	no	—
78	C₂²×C₄×C₃₁	—	4	2	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₃₁	—	2	4	no	—
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₂×C₈×C₃₁	—	2	4	no	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	—	4	4	no	—
92	C₄⋊C₄×C₃₁	—	4	4	no	—
93	C₄⋊C₄×C₃₁	—	4	8	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₂²⋊C₄×C₃₁	—	2	8	no	—
96	C₂²⋊C₄×C₃₁	—	4	4	no	—
97	C₂²⋊C₄×C₃₁	—	4	2	no	—
98	C₄²×C₃₁	—	2	2	no	—
99	C₄²×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	—	4	2	no	—
101	C₄²×C₃₁	—	4	2	no	—
102	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
103	C₄²×C₃₁	#41	1	2	yes	—
104	C₄²×C₃₁	#41	1	4	yes	—
105	#29146	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₃₁	—	2	2	no	—
107	C₂³×C₃₁	—	4	1	no	—
108	C₂³×C₃₁	—	4	1	no	—
109	C₂³×C₃₁	—	4	2	no	—
110	C₂³×C₃₁	—	4	2	no	—
111	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₃₁	—	2	4	no	—
113	Q₈×C₃₁	—	4	2	no	—
114	D₄×C₃₁	—	2	4	no	—
115	D₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	4	1	no	—
125	C₂×C₄×C₃₁	—	4	1	no	—
126	C₂×C₄×C₃₁	—	4	1	no	—
127	C₂×C₄×C₃₁	—	4	1	no	—
128	C₂×C₄×C₃₁	—	4	4	no	—
129	C₂×C₄×C₃₁	—	4	2	no	—
130	C₂×C₄×C₃₁	—	4	4	no	—
131	C₂×C₄×C₃₁	—	4	4	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	8	4	no	—
134	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
135	C₂×C₄×C₃₁	#124	1	2	yes	—
136	C₂×C₄×C₃₁	#124	1	2	yes	—
137	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
139	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
141	C₈×C₃₁	—	4	4	no	—
142	#2859	C₂×C₃₁	1	1	yes	characteristic, complemented
143	C₄×#124	C₂×C₃₁	1	1	yes	characteristic
144	#2026	C₂×C₃₁	1	2	yes	complemented
145	#2025	C₂×C₃₁	1	2	yes	—
146	#1959	C₂×C₃₁	1	2	yes	—
147	#1836	C₂×C₃₁	1	2	yes	complemented
148	#1805	C₂×C₃₁	1	2	yes	complemented
149	#1804	C₂×C₃₁	1	2	yes	—
150	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
151	C₂²×C₃₁	—	4	2	no	—
152	C₂²×C₃₁	—	4	4	no	—
153	C₂²×C₃₁	—	8	1	no	—
154	C₂²×C₃₁	—	8	1	no	—
155	C₂²×C₃₁	—	8	2	no	—
156	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
158	C₂²×C₃₁	#535	1	1	yes	characteristic
159	C₂²×C₃₁	#575	1	2	yes	—
160	C₂²×C₃₁	#575	1	2	yes	—
161	C₄×C₃₁	—	2	4	no	—
162	C₄×C₃₁	—	2	2	no	—
163	C₄×C₃₁	—	2	2	no	—
164	C₄×C₃₁	—	4	2	no	—
165	C₄×C₃₁	—	4	2	no	—
166	C₄×C₃₁	—	8	1	no	—
167	C₄×C₃₁	—	8	1	no	—
168	C₄×C₃₁	—	8	4	no	—
169	C₄×C₃₁	#554	1	2	yes	—
170	C₄×C₃₁	#555	1	2	yes	—
171	C₂×#125	C₂²×C₃₁	1	2	yes	—
172	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
173	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
176	C₂×#103	C₂²×C₃₁	1	4	yes	—
177	C₂×#102	C₂²×C₃₁	1	4	yes	—
178	C₂×#97	C₂²×C₃₁	1	2	yes	—
179	#391	C₂²×C₃₁	1	1	yes	characteristic
180	#390	C₂²×C₃₁	1	1	yes	characteristic
181	#389	—	2	2	no	—
182	#389	—	2	2	no	—
183	#388	—	2	2	no	—
184	#384	C₂²×C₃₁	1	2	yes	—
185	#382	C₂²×C₃₁	1	2	yes	—
186	#382	C₂²×C₃₁	1	1	yes	characteristic
187	C₄×C₂²⋊C₄	—	2	2	no	—
188	C₄³	C₂²×C₃₁	1	1	yes	characteristic
189	#340	C₂²×C₃₁	1	4	yes	—
190	#340	C₂²×C₃₁	1	4	yes	—
191	#336	C₂²×C₃₁	1	2	yes	—
192	C₂×C₃₁	—	8	2	no	—
193	C₂×C₃₁	#2563	1	1	yes	characteristic
194	C₂×C₃₁	#3080	1	2	yes	—
195	C₂×C₃₁	#3135	1	1	yes	characteristic
196	C₂×C₃₁	#3140	1	1	yes	characteristic
197	C₂×C₃₁	#3219	1	2	yes	—
198	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
199	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
200	C₂³×C₄	—	2	1	no	—
201	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
202	#125	—	4	4	no	—
203	#124	—	2	2	no	—
204	#124	Q₈×C₃₁	1	4	yes	—
205	C₄×Q₈	—	2	4	no	—
206	C₄×D₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₂²⋊C₄	—	2	1	no	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	D₄×C₃₁	1	2	yes	—
219	C₂×C₄²	D₄×C₃₁	1	2	yes	—
220	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
223	C₄⋊C₈	—	2	4	no	—
224	#103	—	4	4	no	—
225	#102	—	4	4	no	—
226	#97	—	2	4	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	C₃₁	#29146	1	1	yes	characteristic, direct product factor, Sylow
232	C₂⁴	—	2	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	4	8	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	2	8	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
271	C₄²	#41×C₃₁	1	2	yes	—
272	C₄²	#41×C₃₁	1	4	yes	—
273	C₂³	—	2	2	no	—
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
279	Q₈	—	2	4	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	2	4	no	—
282	D₄	—	4	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
302	C₂×C₄	C₃₁×#124	1	2	yes	—
303	C₂×C₄	C₃₁×#124	1	2	yes	—
304	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
306	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
308	C₈	—	4	4	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	—	8	2	no	—
314	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
315	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
316	C₂²	C₃₁×#535	1	1	yes	characteristic
317	C₂²	C₃₁×#575	1	2	yes	—
318	C₂²	C₃₁×#575	1	2	yes	—
319	C₄	—	2	4	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	1	no	—
326	C₄	—	8	4	no	—
327	C₄	C₃₁×#554	1	2	yes	—
328	C₄	C₃₁×#555	1	2	yes	—
329	C₂	—	8	2	no	—
330	C₂	C₃₁×#2563	1	1	yes	characteristic
331	C₂	C₃₁×#3080	1	2	yes	—
332	C₂	C₃₁×#3135	1	1	yes	characteristic
333	C₂	C₃₁×#3140	1	1	yes	characteristic
334	C₂	C₃₁×#3219	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2859	C₂	semidirect
3. C₃₁×#2026	C₂	semidirect
6. C₃₁×#1836	C₂	semidirect
7. C₃₁×#1805	C₂	semidirect
9. C₃₁×#1767	C₂	semidirect
105. #29146	C₃₁	direct (N × H)
142. #2859	C₂×C₃₁	semidirect
144. #2026	C₂×C₃₁	semidirect
147. #1836	C₂×C₃₁	semidirect
148. #1805	C₂×C₃₁	semidirect
150. #1767	C₂×C₃₁	semidirect
231. C₃₁	#29146	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×#2859	2
C₄×C₃₁×#124	2
C₃₁×#2026	2
C₃₁×#2025	2
C₃₁×#1959	2
C₃₁×#1836	2
C₃₁×#1805	2
C₃₁×#1804	2
C₃₁×#1767	2
#29146	31

Subgroups — C31×#29146 #21485905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29146 #21485905