Subgroups — C₃₁×#23646 #21480405

Order 7936.

906 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2896	C₂	1	2	yes	maximal
3	C₂×C₃₁×#408	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2079	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2078	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1742	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#341	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#117	C₂²	1	2	yes	—
13	C₃₁×#427	—	2	2	no	—
14	C₃₁×#427	—	2	2	no	—
15	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
16	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
17	C₃₁×#419	—	2	2	no	—
18	C₃₁×#418	—	2	2	no	—
19	C₃₁×#417	C₂²	1	8	yes	—
20	C₃₁×#416	—	2	4	no	—
21	C₃₁×#408	—	2	2	no	—
22	C₃₁×#408	C₂²	1	2	yes	—
23	C₃₁×#408	C₂²	1	2	yes	—
24	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
25	C₃₁×#343	C₂²	1	8	yes	complemented
26	C₃₁×#341	C₂²	1	2	yes	complemented
27	C₃₁×#341	C₂²	1	2	yes	complemented
28	C₂×#41×C₃₁	—	2	1	no	—
29	C₂×#41×C₃₁	—	2	1	no	—
30	C₂×#41×C₃₁	—	2	4	no	—
31	C₂×#41×C₃₁	—	2	4	no	—
32	C₂×#41×C₃₁	D₄	1	2	yes	—
33	C₂×#41×C₃₁	C₂³	1	2	yes	—
34	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
35	C₂²×D₄×C₃₁	—	2	1	no	—
36	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₃₁	—	2	1	no	—
38	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₃₁	—	2	1	no	—
40	C₂×M₁₆×C₃₁	—	2	2	no	—
41	C₂×M₁₆×C₃₁	—	2	4	no	—
42	C₂×M₁₆×C₃₁	—	2	4	no	—
43	C₂×M₁₆×C₃₁	—	2	4	no	—
44	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
45	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
46	C₂²×C₈×C₃₁	—	2	4	no	—
47	C₂²×C₈×C₃₁	—	2	1	no	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	D₄	1	2	yes	—
50	C₂²×C₈×C₃₁	D₄	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#117	—	2	2	no	—
54	C₃₁×#117	—	2	2	no	—
55	C₃₁×#117	—	2	2	no	—
56	C₃₁×#117	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
58	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄²×C₃₁	—	2	2	no	—
60	C₃₁×#107	—	2	4	no	—
61	C₃₁×#106	—	2	4	no	—
62	C₃₁×#103	—	2	8	no	—
63	C₃₁×#102	—	2	8	no	—
64	C₃₁×C₂²⋊C₈	—	2	4	no	—
65	C₃₁×C₂²⋊C₈	—	2	4	no	—
66	C₂⁴×C₃₁	—	2	1	no	—
67	C₂⁴×C₃₁	#41	1	1	yes	characteristic
68	#41×C₃₁	—	2	8	no	—
69	#41×C₃₁	—	2	4	no	—
70	#41×C₃₁	—	2	4	no	—
71	#41×C₃₁	—	4	2	no	—
72	#41×C₃₁	—	4	2	no	—
73	#41×C₃₁	—	4	4	no	—
74	C₂×Q₈×C₃₁	—	2	1	no	—
75	C₂×Q₈×C₃₁	—	2	4	no	—
76	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	2	1	no	—
80	C₂×D₄×C₃₁	—	2	4	no	—
81	C₂×D₄×C₃₁	—	4	4	no	—
82	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₁	—	2	1	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	—	2	4	no	—
87	C₂²×C₄×C₃₁	—	4	2	no	—
88	C₂²×C₄×C₃₁	—	4	2	no	—
89	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	#41	1	4	yes	—
95	C₂²×C₄×C₃₁	#41	1	4	yes	—
96	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
97	M₁₆×C₃₁	—	2	4	no	—
98	M₁₆×C₃₁	—	2	4	no	—
99	M₁₆×C₃₁	—	2	4	no	—
100	M₁₆×C₃₁	—	4	4	no	—
101	M₁₆×C₃₁	—	4	2	no	—
102	C₂×C₈×C₃₁	—	2	4	no	—
103	C₂×C₈×C₃₁	—	2	4	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	4	no	—
108	C₂×C₈×C₃₁	—	4	4	no	—
109	C₂×C₈×C₃₁	—	4	2	no	—
110	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₃₁	—	2	4	no	—
113	C₄⋊C₄×C₃₁	—	2	4	no	—
114	C₂²⋊C₄×C₃₁	—	4	2	no	—
115	C₂²⋊C₄×C₃₁	—	4	2	no	—
116	C₄²×C₃₁	—	4	2	no	—
117	C₄²×C₃₁	—	4	2	no	—
118	#23646	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₃₁	—	2	2	no	—
120	C₂³×C₃₁	—	4	2	no	—
121	C₂³×C₃₁	—	4	4	no	—
122	C₂³×C₃₁	—	4	2	no	—
123	C₂³×C₃₁	#121	1	1	yes	characteristic
124	C₂³×C₃₁	#121	1	1	yes	characteristic
125	C₂³×C₃₁	#126	1	4	yes	—
126	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
127	Q₈×C₃₁	—	2	4	no	—
128	Q₈×C₃₁	—	4	2	no	—
129	D₄×C₃₁	—	2	4	no	—
130	D₄×C₃₁	—	4	4	no	—
131	D₄×C₃₁	—	4	4	no	—
132	D₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	2	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	4	no	—
138	C₂×C₄×C₃₁	—	4	4	no	—
139	C₂×C₄×C₃₁	—	8	4	no	—
140	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
141	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
142	C₂×C₄×C₃₁	#123	1	2	yes	—
143	C₂×C₄×C₃₁	#123	1	2	yes	—
144	C₂×C₄×C₃₁	#123	1	4	yes	—
145	C₂×C₄×C₃₁	#123	1	4	yes	—
146	C₂×C₄×C₃₁	#124	1	2	yes	—
147	C₂×C₄×C₃₁	#124	1	2	yes	—
148	C₂×C₄×C₃₁	#126	1	4	yes	—
149	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
152	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
153	C₈×C₃₁	—	2	2	no	—
154	C₈×C₃₁	—	2	2	no	—
155	C₈×C₃₁	—	4	4	no	—
156	C₈×C₃₁	—	4	2	no	—
157	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#2896	C₂×C₃₁	1	2	yes	—
159	C₂×#408	C₂×C₃₁	1	1	yes	characteristic
160	#2079	C₂×C₃₁	1	4	yes	complemented
161	#2078	C₂×C₃₁	1	4	yes	complemented
162	#1742	C₂×C₃₁	1	2	yes	complemented
163	C₂×#341	C₂×C₃₁	1	1	yes	characteristic, complemented
164	C₂²×C₃₁	—	2	2	no	—
165	C₂²×C₃₁	—	2	2	no	—
166	C₂²×C₃₁	—	2	2	no	—
167	C₂²×C₃₁	—	4	4	no	—
168	C₂²×C₃₁	—	4	2	no	—
169	C₂²×C₃₁	—	4	4	no	—
170	C₂²×C₃₁	—	8	4	no	—
171	C₂²×C₃₁	#397	1	2	yes	—
172	C₂²×C₃₁	#397	1	2	yes	—
173	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
174	C₂²×C₃₁	C₂×#126	1	2	yes	—
175	C₄×C₃₁	—	4	2	no	—
176	C₄×C₃₁	—	8	4	no	—
177	C₄×C₃₁	#397	1	2	yes	—
178	C₄×C₃₁	#397	1	2	yes	—
179	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
180	C₄×C₃₁	C₂×#123	1	2	yes	—
181	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
182	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
183	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
184	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
185	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
186	C₂×#117	C₂²×C₃₁	1	2	yes	—
187	#427	—	2	2	no	—
188	#427	—	2	2	no	—
189	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
190	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
191	#419	—	2	2	no	—
192	#418	—	2	2	no	—
193	#417	C₂²×C₃₁	1	8	yes	—
194	#416	—	2	4	no	—
195	#408	—	2	2	no	—
196	#408	C₂²×C₃₁	1	2	yes	—
197	#408	C₂²×C₃₁	1	2	yes	—
198	C₂×C₂²⋊C₈	—	2	2	no	—
199	#343	C₂²×C₃₁	1	8	yes	complemented
200	#341	C₂²×C₃₁	1	2	yes	complemented
201	#341	C₂²×C₃₁	1	2	yes	complemented
202	C₂×C₃₁	—	2	2	no	—
203	C₂×C₃₁	—	2	2	no	—
204	C₂×C₃₁	—	2	2	no	—
205	C₂×C₃₁	—	8	2	no	—
206	C₂×C₃₁	C₂×#397	1	1	yes	characteristic
207	C₂×C₃₁	#3067	1	2	yes	—
208	C₂×#41	—	2	1	no	—
209	C₂×#41	—	2	1	no	—
210	C₂×#41	—	2	4	no	—
211	C₂×#41	—	2	4	no	—
212	C₂×#41	D₄×C₃₁	1	2	yes	—
213	C₂×#41	C₂³×C₃₁	1	2	yes	—
214	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
215	C₂²×D₄	—	2	1	no	—
216	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
217	C₂³×C₄	—	2	1	no	—
218	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
219	C₂×M₁₆	—	2	1	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	2	4	no	—
222	C₂×M₁₆	—	2	4	no	—
223	C₂×M₁₆	—	2	4	no	—
224	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
225	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
226	C₂²×C₈	—	2	4	no	—
227	C₂²×C₈	—	2	1	no	—
228	C₂²×C₈	—	2	2	no	—
229	C₂²×C₈	D₄×C₃₁	1	2	yes	—
230	C₂²×C₈	D₄×C₃₁	1	2	yes	—
231	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
232	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
233	#117	—	2	2	no	—
234	#117	—	2	2	no	—
235	#117	—	2	2	no	—
236	#117	—	2	2	no	—
237	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
238	C₂×C₂²⋊C₄	—	2	2	no	—
239	C₂×C₄²	—	2	2	no	—
240	#107	—	2	4	no	—
241	#106	—	2	4	no	—
242	#103	—	2	8	no	—
243	#102	—	2	8	no	—
244	C₂²⋊C₈	—	2	4	no	—
245	C₂²⋊C₈	—	2	4	no	—
246	C₃₁	#23646	1	1	yes	characteristic, direct product factor, Sylow
247	C₂⁴	—	2	1	no	—
248	C₂⁴	#41×C₃₁	1	1	yes	characteristic
249	#41	—	2	8	no	—
250	#41	—	2	4	no	—
251	#41	—	2	4	no	—
252	#41	—	4	2	no	—
253	#41	—	4	2	no	—
254	#41	—	4	4	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	4	4	no	—
263	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	#41×C₃₁	1	4	yes	—
276	C₂²×C₄	#41×C₃₁	1	4	yes	—
277	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
278	M₁₆	—	2	4	no	—
279	M₁₆	—	2	4	no	—
280	M₁₆	—	2	4	no	—
281	M₁₆	—	4	4	no	—
282	M₁₆	—	4	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	4	4	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
292	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₄²	—	4	2	no	—
298	C₄²	—	4	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	4	no	—
302	C₂³	—	4	2	no	—
303	C₂³	C₃₁×#121	1	1	yes	characteristic
304	C₂³	C₃₁×#121	1	1	yes	characteristic
305	C₂³	C₃₁×#126	1	4	yes	—
306	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
307	Q₈	—	2	4	no	—
308	Q₈	—	4	2	no	—
309	D₄	—	2	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
321	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
322	C₂×C₄	C₃₁×#123	1	2	yes	—
323	C₂×C₄	C₃₁×#123	1	2	yes	—
324	C₂×C₄	C₃₁×#123	1	4	yes	—
325	C₂×C₄	C₃₁×#123	1	4	yes	—
326	C₂×C₄	C₃₁×#124	1	2	yes	—
327	C₂×C₄	C₃₁×#124	1	2	yes	—
328	C₂×C₄	C₃₁×#126	1	4	yes	—
329	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
332	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
333	C₈	—	2	2	no	—
334	C₈	—	2	2	no	—
335	C₈	—	4	4	no	—
336	C₈	—	4	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	4	no	—
343	C₂²	—	8	4	no	—
344	C₂²	C₃₁×#397	1	2	yes	—
345	C₂²	C₃₁×#397	1	2	yes	—
346	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
347	C₂²	C₂×C₃₁×#126	1	2	yes	—
348	C₄	—	4	2	no	—
349	C₄	—	8	4	no	—
350	C₄	C₃₁×#397	1	2	yes	—
351	C₄	C₃₁×#397	1	2	yes	—
352	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
353	C₄	C₂×C₃₁×#123	1	2	yes	—
354	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
355	C₂	—	2	2	no	—
356	C₂	—	2	2	no	—
357	C₂	—	2	2	no	—
358	C₂	—	8	2	no	—
359	C₂	C₂×C₃₁×#397	1	1	yes	characteristic
360	C₂	C₃₁×#3067	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×#121
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#427	C₂	semidirect
4. C₃₁×#2079	C₂	semidirect
5. C₃₁×#2078	C₂	semidirect
6. C₃₁×#1742	C₂	semidirect
7. C₂×C₃₁×#341	C₂	semidirect
15. C₂×C₃₁×#107	C₂²	semidirect
16. C₂×C₃₁×#106	C₂²	semidirect
25. C₃₁×#343	C₂²	semidirect
26. C₃₁×#341	C₂²	semidirect
27. C₃₁×#341	C₂²	semidirect
118. #23646	C₃₁	direct (N × H)
157. C₂×#427	C₂×C₃₁	semidirect
160. #2079	C₂×C₃₁	semidirect
161. #2078	C₂×C₃₁	semidirect
162. #1742	C₂×C₃₁	semidirect
163. C₂×#341	C₂×C₃₁	semidirect
189. C₂×#107	C₂²×C₃₁	semidirect
190. C₂×#106	C₂²×C₃₁	semidirect
199. #343	C₂²×C₃₁	semidirect
200. #341	C₂²×C₃₁	semidirect
201. #341	C₂²×C₃₁	semidirect
246. C₃₁	#23646	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#427	2
C₃₁×#2896	2
C₂×C₃₁×#408	2
C₃₁×#2079	2
C₃₁×#2078	2
C₃₁×#1742	2
C₂×C₃₁×#341	2
#23646	31

Subgroups — C31×#23646 #21480405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23646 #21480405