Subgroups — C₃₁×#11212 #21467971

Order 7936.

714 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×#2905	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1740	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1733	C₂	1	2	yes	maximal
5	C₂×C₃₁×#339	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×#117	C₂²	1	2	yes	—
8	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#424	—	2	4	no	—
12	C₃₁×#423	—	2	2	no	—
13	C₃₁×#384	C₄	1	2	yes	complemented
14	C₃₁×#382	C₄	1	2	yes	complemented
15	C₃₁×#380	—	2	2	no	—
16	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
17	C₃₁×#342	—	2	2	no	—
18	C₃₁×#339	—	2	4	no	—
19	C₃₁×#339	—	2	4	no	—
20	C₂³×C₄×C₃₁	—	2	1	no	—
21	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₃₁×#117	—	2	2	no	—
30	C₃₁×#117	—	2	2	no	—
31	C₃₁×#117	—	2	2	no	—
32	C₃₁×#117	—	2	2	no	—
33	C₃₁×#117	—	2	2	no	—
34	C₃₁×#117	—	2	2	no	—
35	C₃₁×#117	—	2	2	no	—
36	C₃₁×#117	Q₈	1	4	yes	—
37	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
44	C₂×C₄²×C₃₁	—	2	1	no	—
45	C₂×C₄²×C₃₁	—	2	1	no	—
46	C₂×C₄²×C₃₁	—	2	1	no	—
47	C₂×C₄²×C₃₁	—	2	2	no	—
48	C₂×C₄²×C₃₁	—	2	1	no	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	D₄	1	2	yes	—
52	C₂×C₄²×C₃₁	D₄	1	2	yes	—
53	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
54	C₃₁×C₄⋊C₈	—	2	8	no	—
55	C₃₁×#95	—	2	4	no	—
56	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	—	2	1	no	—
58	C₂²×C₄×C₃₁	—	2	1	no	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	2	2	no	—
61	C₂²×C₄×C₃₁	—	2	2	no	—
62	C₂²×C₄×C₃₁	—	4	4	no	—
63	C₂²×C₄×C₃₁	—	4	2	no	—
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	#41	1	2	yes	—
74	C₂²×C₄×C₃₁	#41	1	2	yes	—
75	M₁₆×C₃₁	—	2	4	no	—
76	M₁₆×C₃₁	—	2	4	no	—
77	M₁₆×C₃₁	—	4	2	no	—
78	M₁₆×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	—	4	2	no	—
81	C₂×C₈×C₃₁	—	4	2	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	—	2	4	no	—
85	C₄⋊C₄×C₃₁	—	4	8	no	—
86	C₄⋊C₄×C₃₁	—	4	2	no	—
87	C₂²⋊C₄×C₃₁	—	2	4	no	—
88	C₂²⋊C₄×C₃₁	—	2	4	no	—
89	C₂²⋊C₄×C₃₁	—	4	2	no	—
90	C₂²⋊C₄×C₃₁	—	4	2	no	—
91	C₄²×C₃₁	—	2	1	no	—
92	C₄²×C₃₁	—	2	1	no	—
93	C₄²×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	—	2	2	no	—
95	C₄²×C₃₁	—	2	2	no	—
96	C₄²×C₃₁	—	2	2	no	—
97	C₄²×C₃₁	—	2	4	no	—
98	C₄²×C₃₁	—	4	1	no	—
99	C₄²×C₃₁	—	4	2	no	—
100	C₄²×C₃₁	—	4	1	no	—
101	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
102	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
103	#11212	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₃₁	—	4	2	no	—
105	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₃₁	#121	1	2	yes	—
107	C₂³×C₃₁	#121	1	1	yes	characteristic
108	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	4	no	—
114	C₂×C₄×C₃₁	—	4	4	no	—
115	C₂×C₄×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	4	4	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	8	4	no	—
122	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
127	C₂×C₄×C₃₁	#121	1	2	yes	—
128	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₃₁	#123	1	2	yes	—
133	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
134	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#125	1	2	yes	—
136	C₂×C₄×C₃₁	#125	1	2	yes	—
137	C₂×C₄×C₃₁	#126	1	2	yes	—
138	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
139	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
140	C₈×C₃₁	—	4	4	no	—
141	#2905	C₂×C₃₁	1	1	yes	characteristic
142	#2285	C₂×C₃₁	1	1	yes	characteristic
143	#1740	C₂×C₃₁	1	1	yes	characteristic
144	#1733	C₂×C₃₁	1	2	yes	—
145	C₂×#339	C₂×C₃₁	1	2	yes	—
146	C₂²×C₃₁	—	2	2	no	—
147	C₂²×C₃₁	—	2	4	no	—
148	C₂²×C₃₁	—	4	1	no	—
149	C₂²×C₃₁	—	4	1	no	—
150	C₂²×C₃₁	—	4	2	no	—
151	C₂²×C₃₁	#380	1	1	yes	characteristic
152	C₂²×C₃₁	#386	1	1	yes	characteristic
153	C₂²×C₃₁	#391	1	1	yes	characteristic
154	C₂²×C₃₁	#396	1	2	yes	—
155	C₂²×C₃₁	#398	1	2	yes	—
156	C₄×C₃₁	—	2	4	no	—
157	C₄×C₃₁	—	4	2	no	—
158	C₄×C₃₁	—	4	4	no	—
159	C₄×C₃₁	—	8	4	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	#387	1	2	yes	—
162	C₄×C₃₁	#389	1	2	yes	—
163	C₄×C₃₁	#394	1	1	yes	characteristic
164	C₄×C₃₁	#395	1	1	yes	characteristic
165	C₄×C₃₁	#398	1	1	yes	characteristic
166	C₄×C₃₁	#400	1	1	yes	characteristic
167	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
168	C₂×#117	C₂²×C₃₁	1	2	yes	—
169	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
171	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
172	#424	—	2	4	no	—
173	#423	—	2	2	no	—
174	#384	C₄×C₃₁	1	2	yes	complemented
175	#382	C₄×C₃₁	1	2	yes	complemented
176	#380	—	2	2	no	—
177	C₄×C₄⋊C₄	—	2	2	no	—
178	#342	—	2	2	no	—
179	#339	—	2	4	no	—
180	#339	—	2	4	no	—
181	C₂×C₃₁	—	2	1	no	—
182	C₂×C₃₁	—	2	2	no	—
183	C₂×C₃₁	—	2	1	no	—
184	C₂×C₃₁	—	2	2	no	—
185	C₂×C₃₁	#1441	1	1	yes	characteristic
186	C₂×C₃₁	#1884	1	2	yes	—
187	C₂³×C₄	—	2	1	no	—
188	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
195	C₂²×C₈	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	Q₈×C₃₁	1	4	yes	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	2	1	no	—
210	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
211	C₂×C₄²	—	2	1	no	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	2	1	no	—
216	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
217	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
218	C₂×C₄²	D₄×C₃₁	1	2	yes	—
219	C₂×C₄²	D₄×C₃₁	1	2	yes	—
220	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
221	C₄⋊C₈	—	2	8	no	—
222	#95	—	2	4	no	—
223	C₃₁	#11212	1	1	yes	characteristic, direct product factor, Sylow
224	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
233	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
239	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
241	C₂²×C₄	#41×C₃₁	1	2	yes	—
242	C₂²×C₄	#41×C₃₁	1	2	yes	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	2	4	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	4	8	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	2	4	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	4	1	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	1	no	—
269	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
270	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
271	C₂³	—	4	2	no	—
272	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
273	C₂³	C₃₁×#121	1	2	yes	—
274	C₂³	C₃₁×#121	1	1	yes	characteristic
275	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
276	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
290	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
291	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
292	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
293	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
294	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
295	C₂×C₄	C₃₁×#121	1	2	yes	—
296	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
297	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
298	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
299	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
300	C₂×C₄	C₃₁×#123	1	2	yes	—
301	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
302	C₂×C₄	C₃₁×#125	1	2	yes	—
303	C₂×C₄	C₃₁×#125	1	2	yes	—
304	C₂×C₄	C₃₁×#126	1	2	yes	—
305	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
306	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
307	C₈	—	4	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₃₁×#380	1	1	yes	characteristic
314	C₂²	C₃₁×#386	1	1	yes	characteristic
315	C₂²	C₃₁×#391	1	1	yes	characteristic
316	C₂²	C₃₁×#396	1	2	yes	—
317	C₂²	C₃₁×#398	1	2	yes	—
318	C₄	—	2	4	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	4	no	—
321	C₄	—	8	4	no	—
322	C₄	—	8	2	no	—
323	C₄	C₃₁×#387	1	2	yes	—
324	C₄	C₃₁×#389	1	2	yes	—
325	C₄	C₃₁×#394	1	1	yes	characteristic
326	C₄	C₃₁×#395	1	1	yes	characteristic
327	C₄	C₃₁×#398	1	1	yes	characteristic
328	C₄	C₃₁×#400	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	2	no	—
331	C₂	—	2	2	no	—
332	C₂	—	2	1	no	—
333	C₂	C₃₁×#1441	1	1	yes	characteristic
334	C₂	C₃₁×#1884	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#120
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	#386
prime radical	C₂⁴×C₃₁	C₂×D₄
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
13. C₃₁×#384	C₄	semidirect
14. C₃₁×#382	C₄	semidirect
103. #11212	C₃₁	direct (N × H)
174. #384	C₄×C₃₁	semidirect
175. #382	C₄×C₃₁	semidirect
223. C₃₁	#11212	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×#2905	2
C₃₁×#2285	2
C₃₁×#1740	2
C₃₁×#1733	2
C₂×C₃₁×#339	2
#11212	31

Subgroups — C31×#11212 #21467971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11212 #21467971