Subgroups — C₃₁×#10147 #21466906

Order 7936.

604 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3209	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₃₁	—	2	2	no	—
9	C₃₁×#533	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₃₁×#494	—	2	1	no	—
13	C₃₁×#492	—	2	2	no	—
14	C₃₁×#492	—	2	2	no	—
15	C₃₁×#490	—	2	2	no	—
16	C₃₁×#487	—	2	1	no	—
17	C₃₁×#486	—	2	1	no	—
18	C₃₁×#486	—	2	1	no	—
19	C₃₁×#462	—	2	2	no	—
20	C₃₁×#451	—	2	2	no	—
21	C₃₁×#431	C₂²	1	1	yes	characteristic
22	C₃₁×#423	C₂²	1	1	yes	characteristic
23	C₃₁×#418	—	2	2	no	—
24	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#343	—	2	2	no	—
27	C₃₁×#331	—	2	2	no	—
28	C₃₁×Q₈⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₃₁	D₄	1	2	yes	—
30	C₂²×D₄×C₃₁	—	2	1	no	—
31	C₂²×D₄×C₃₁	—	2	1	no	—
32	C₂²×D₄×C₃₁	—	2	1	no	—
33	C₂×Q₁₆×C₃₁	—	2	4	no	—
34	C₂×SD₁₆×C₃₁	—	2	8	no	—
35	C₂×SD₁₆×C₃₁	—	4	8	no	—
36	C₂×M₁₆×C₃₁	—	2	1	no	—
37	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
38	C₂²×C₈×C₃₁	—	2	1	no	—
39	C₃₁×C₄⋊Q₈	—	2	1	no	—
40	C₃₁×C₄⋊Q₈	—	2	1	no	—
41	C₃₁×C₄⋊Q₈	—	2	1	no	—
42	C₃₁×C₄⋊Q₈	—	2	1	no	—
43	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
44	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
45	C₃₁×#127	—	2	1	no	—
46	C₃₁×#127	—	2	1	no	—
47	C₃₁×#127	—	2	1	no	—
48	C₃₁×#127	D₄	1	2	yes	—
49	C₃₁×#125	—	2	1	no	—
50	C₃₁×C₂²⋊Q₈	—	2	1	no	—
51	C₃₁×C₂²⋊Q₈	—	2	1	no	—
52	C₄×Q₈×C₃₁	—	2	1	no	—
53	C₄×Q₈×C₃₁	—	2	1	no	—
54	C₃₁×#117	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×C₄⋊C₈	—	2	1	no	—
59	C₃₁×C₄⋊C₈	—	2	1	no	—
60	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
61	C₃₁×#103	—	2	4	no	—
62	C₃₁×#102	—	2	4	no	—
63	C₃₁×#102	—	2	4	no	—
64	C₃₁×C₂²⋊C₈	—	2	1	no	—
65	C₃₁×#97	—	2	1	no	—
66	C₄×C₈×C₃₁	—	2	1	no	—
67	C₄×C₈×C₃₁	—	2	1	no	—
68	C₄×C₈×C₃₁	—	2	1	no	—
69	C₄×C₈×C₃₁	D₄	1	2	yes	—
70	C₂⁴×C₃₁	—	4	1	no	—
71	C₂×Q₈×C₃₁	—	2	2	no	—
72	C₂×Q₈×C₃₁	—	2	1	no	—
73	C₂×Q₈×C₃₁	—	2	2	no	—
74	C₂×Q₈×C₃₁	—	2	1	no	—
75	C₂×Q₈×C₃₁	—	4	4	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	4	4	no	—
80	C₂×D₄×C₃₁	—	4	1	no	—
81	C₂×D₄×C₃₁	—	4	1	no	—
82	C₂×D₄×C₃₁	—	4	1	no	—
83	C₂×D₄×C₃₁	—	4	2	no	—
84	C₂×D₄×C₃₁	—	4	2	no	—
85	C₂×D₄×C₃₁	—	8	2	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
90	Q₁₆×C₃₁	—	8	4	no	—
91	SD₁₆×C₃₁	—	4	16	no	—
92	M₁₆×C₃₁	—	2	2	no	—
93	M₁₆×C₃₁	—	2	2	no	—
94	M₁₆×C₃₁	—	4	2	no	—
95	C₂×C₈×C₃₁	—	2	4	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	4	1	no	—
100	C₄⋊C₄×C₃₁	—	2	1	no	—
101	C₄⋊C₄×C₃₁	—	2	1	no	—
102	C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₄⋊C₄×C₃₁	—	4	1	no	—
105	C₄⋊C₄×C₃₁	—	4	1	no	—
106	C₄⋊C₄×C₃₁	—	4	1	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	1	no	—
111	C₄²×C₃₁	—	4	1	no	—
112	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	#10147	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₃₁	—	4	2	no	—
118	C₂³×C₃₁	—	8	2	no	—
119	C₂³×C₃₁	—	8	2	no	—
120	C₂³×C₃₁	—	8	2	no	—
121	C₂³×C₃₁	#121	1	1	yes	characteristic
122	Q₈×C₃₁	—	4	4	no	—
123	Q₈×C₃₁	—	4	2	no	—
124	Q₈×C₃₁	—	4	4	no	—
125	Q₈×C₃₁	—	4	2	no	—
126	D₄×C₃₁	—	4	4	no	—
127	D₄×C₃₁	—	4	4	no	—
128	D₄×C₃₁	—	8	2	no	—
129	D₄×C₃₁	—	8	2	no	—
130	D₄×C₃₁	—	8	2	no	—
131	D₄×C₃₁	—	8	1	no	—
132	D₄×C₃₁	—	8	1	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	4	1	no	—
136	C₂×C₄×C₃₁	—	4	1	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	8	2	no	—
139	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
140	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
141	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
142	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
143	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
144	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
145	C₂×C₄×C₃₁	#135	1	2	yes	—
146	C₂×C₄×C₃₁	#137	1	2	yes	—
147	C₈×C₃₁	—	2	4	no	—
148	C₈×C₃₁	—	4	2	no	—
149	C₈×C₃₁	—	4	2	no	—
150	C₈×C₃₁	—	8	1	no	—
151	#3209	C₂×C₃₁	1	1	yes	characteristic, complemented
152	C₂×#492	C₂×C₃₁	1	1	yes	characteristic
153	#3138	C₂×C₃₁	1	1	yes	characteristic
154	#1997	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#1545	C₂×C₃₁	1	1	yes	characteristic
156	#1493	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#1278	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂²×C₃₁	—	2	1	no	—
159	C₂²×C₃₁	—	2	1	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	—	8	2	no	—
162	C₂²×C₃₁	—	8	2	no	—
163	C₂²×C₃₁	—	16	4	no	—
164	C₂²×C₃₁	#392	1	1	yes	characteristic
165	C₂²×C₃₁	#464	1	2	yes	—
166	C₄×C₃₁	—	2	1	no	—
167	C₄×C₃₁	—	2	1	no	—
168	C₄×C₃₁	—	8	2	no	—
169	C₄×C₃₁	—	8	1	no	—
170	C₄×C₃₁	—	8	1	no	—
171	C₄×C₃₁	#452	1	2	yes	—
172	C₄×C₃₁	#453	1	2	yes	—
173	C₄×C₃₁	#471	1	2	yes	—
174	C₄×C₃₁	#495	1	1	yes	characteristic
175	C₄×C₃₁	#497	1	1	yes	characteristic
176	C₂²×SD₁₆	—	2	2	no	—
177	#533	C₂²×C₃₁	1	1	yes	characteristic
178	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
180	#494	—	2	1	no	—
181	#492	—	2	2	no	—
182	#492	—	2	2	no	—
183	#490	—	2	2	no	—
184	#487	—	2	1	no	—
185	#486	—	2	1	no	—
186	#486	—	2	1	no	—
187	#462	—	2	2	no	—
188	#451	—	2	2	no	—
189	#431	C₂²×C₃₁	1	1	yes	characteristic
190	#423	C₂²×C₃₁	1	1	yes	characteristic
191	#418	—	2	2	no	—
192	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
193	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
194	#343	—	2	2	no	—
195	#331	—	2	2	no	—
196	Q₈⋊C₈	—	2	2	no	—
197	C₂×C₃₁	—	2	2	no	—
198	C₂×C₃₁	—	16	2	no	—
199	C₂×C₃₁	#2002	1	1	yes	characteristic
200	C₂×C₃₁	#2015	1	1	yes	characteristic
201	C₂×C₃₁	#2019	1	1	yes	characteristic
202	C₂²×Q₈	D₄×C₃₁	1	2	yes	—
203	C₂²×D₄	—	2	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂×Q₁₆	—	2	4	no	—
207	C₂×SD₁₆	—	2	8	no	—
208	C₂×SD₁₆	—	4	8	no	—
209	C₂×M₁₆	—	2	1	no	—
210	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
211	C₂²×C₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
217	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
218	#127	—	2	1	no	—
219	#127	—	2	1	no	—
220	#127	—	2	1	no	—
221	#127	D₄×C₃₁	1	2	yes	—
222	#125	—	2	1	no	—
223	C₂²⋊Q₈	—	2	1	no	—
224	C₂²⋊Q₈	—	2	1	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	—	2	1	no	—
227	#117	—	2	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	—	2	1	no	—
233	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
234	#103	—	2	4	no	—
235	#102	—	2	4	no	—
236	#102	—	2	4	no	—
237	C₂²⋊C₈	—	2	1	no	—
238	#97	—	2	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	2	1	no	—
242	C₄×C₈	D₄×C₃₁	1	2	yes	—
243	C₃₁	#10147	1	1	yes	characteristic, direct product factor, Sylow
244	C₂⁴	—	4	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	4	4	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	4	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	8	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
264	Q₁₆	—	8	4	no	—
265	SD₁₆	—	4	16	no	—
266	M₁₆	—	2	2	no	—
267	M₁₆	—	2	2	no	—
268	M₁₆	—	4	2	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₄²	—	4	1	no	—
286	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
289	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₂³	—	4	2	no	—
291	C₂³	—	8	2	no	—
292	C₂³	—	8	2	no	—
293	C₂³	—	8	2	no	—
294	C₂³	C₃₁×#121	1	1	yes	characteristic
295	Q₈	—	4	4	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	4	4	no	—
298	Q₈	—	4	2	no	—
299	D₄	—	4	4	no	—
300	D₄	—	4	4	no	—
301	D₄	—	8	2	no	—
302	D₄	—	8	2	no	—
303	D₄	—	8	2	no	—
304	D₄	—	8	1	no	—
305	D₄	—	8	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
313	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
314	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
315	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
316	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
317	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
318	C₂×C₄	C₃₁×#135	1	2	yes	—
319	C₂×C₄	C₃₁×#137	1	2	yes	—
320	C₈	—	2	4	no	—
321	C₈	—	4	2	no	—
322	C₈	—	4	2	no	—
323	C₈	—	8	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	16	4	no	—
330	C₂²	C₃₁×#392	1	1	yes	characteristic
331	C₂²	C₃₁×#464	1	2	yes	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	C₃₁×#452	1	2	yes	—
338	C₄	C₃₁×#453	1	2	yes	—
339	C₄	C₃₁×#471	1	2	yes	—
340	C₄	C₃₁×#495	1	1	yes	characteristic
341	C₄	C₃₁×#497	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	16	2	no	—
344	C₂	C₃₁×#2002	1	1	yes	characteristic
345	C₂	C₃₁×#2015	1	1	yes	characteristic
346	C₂	C₃₁×#2019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3209	C₂	semidirect
4. C₃₁×#1997	C₂	semidirect
6. C₃₁×#1493	C₂	semidirect
7. C₃₁×#1278	C₂	semidirect
116. #10147	C₃₁	direct (N × H)
151. #3209	C₂×C₃₁	semidirect
154. #1997	C₂×C₃₁	semidirect
156. #1493	C₂×C₃₁	semidirect
157. #1278	C₂×C₃₁	semidirect
243. C₃₁	#10147	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×#3209	2
C₂×C₃₁×#492	2
C₃₁×#3138	2
C₃₁×#1997	2
C₃₁×#1545	2
C₃₁×#1493	2
C₃₁×#1278	2
#10147	31

Subgroups — C31×#10147 #21466906

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10147 #21466906