Subgroups — C₃₁×#29276 #21486035

Order 7936.

736 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2100	C₂	1	2	yes	maximal
4	C₃₁×#2058	C₂	1	2	yes	maximal
5	C₃₁×#1988	C₂	1	2	yes	maximal
6	C₃₁×#1948	C₂	1	2	yes	maximal
7	C₃₁×#1920	C₂	1	4	yes	maximal
8	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
12	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
13	C₂×C₃₁×#106	C₂²	1	4	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₃₁×#391	—	2	1	no	—
16	C₃₁×#391	—	2	1	no	—
17	C₃₁×#389	—	2	1	no	—
18	C₃₁×#389	—	2	1	no	—
19	C₃₁×#389	C₂²	1	2	yes	—
20	C₃₁×#389	C₂²	1	2	yes	—
21	C₃₁×#384	C₂²	1	1	yes	characteristic
22	C₃₁×#384	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
24	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
25	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₄³×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#340	C₂²	1	4	yes	—
29	C₃₁×#340	C₂²	1	4	yes	—
30	C₃₁×#340	C₂²	1	4	yes	—
31	C₃₁×#336	C₂²	1	2	yes	—
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₃₁	—	2	1	no	—
34	C₂²×Q₈×C₃₁	—	2	1	no	—
35	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
36	C₃₁×C₄⋊Q₈	—	2	4	no	—
37	C₃₁×C₄⋊Q₈	—	2	2	no	—
38	C₃₁×C₄⋊Q₈	Q₈	1	4	yes	complemented
39	C₄×Q₈×C₃₁	—	4	4	no	—
40	C₄×Q₈×C₃₁	—	4	4	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	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	1	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
53	C₂×C₄²×C₃₁	—	2	2	no	—
54	C₂×C₄²×C₃₁	—	2	2	no	—
55	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
62	C₃₁×#106	—	2	8	no	—
63	C₃₁×C₄⋊C₈	—	2	4	no	—
64	C₃₁×#95	—	2	2	no	—
65	C₃₁×#95	—	2	2	no	—
66	C₃₁×#95	—	2	2	no	—
67	C₃₁×#95	—	2	2	no	—
68	C₂×Q₈×C₃₁	—	2	4	no	—
69	C₂×Q₈×C₃₁	—	2	4	no	—
70	C₂×Q₈×C₃₁	—	4	2	no	—
71	C₂×Q₈×C₃₁	—	4	2	no	—
72	C₂×Q₈×C₃₁	—	4	2	no	—
73	C₂×Q₈×C₃₁	—	4	2	no	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	—	2	2	no	—
77	C₂²×C₄×C₃₁	—	2	2	no	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₂×C₈×C₃₁	—	2	4	no	—
87	C₂×C₈×C₃₁	—	2	4	no	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	4	2	no	—
95	C₄⋊C₄×C₃₁	—	4	2	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₄⋊C₄×C₃₁	—	4	4	no	—
98	C₄⋊C₄×C₃₁	—	4	4	no	—
99	C₄⋊C₄×C₃₁	—	4	2	no	—
100	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
101	C₄⋊C₄×C₃₁	#41	1	4	yes	—
102	C₄²×C₃₁	—	2	2	no	—
103	C₄²×C₃₁	—	2	2	no	—
104	C₄²×C₃₁	—	2	2	no	—
105	C₄²×C₃₁	—	2	2	no	—
106	C₄²×C₃₁	—	4	1	no	—
107	C₄²×C₃₁	—	4	4	no	—
108	C₄²×C₃₁	—	4	4	no	—
109	C₄²×C₃₁	SD₁₆	1	4	yes	—
110	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
111	C₄²×C₃₁	#41	1	2	yes	—
112	#29276	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₃₁	—	4	8	no	—
115	Q₈×C₃₁	—	4	8	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	2	1	no	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	1	no	—
120	C₂×C₄×C₃₁	—	2	1	no	—
121	C₂×C₄×C₃₁	—	2	1	no	—
122	C₂×C₄×C₃₁	—	2	1	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	2	4	no	—
125	C₂×C₄×C₃₁	—	2	4	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	2	1	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	2	yes	—
140	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
142	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
143	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
144	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
145	C₂×C₄×C₃₁	#137	1	2	yes	—
146	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
149	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
150	C₈×C₃₁	—	4	4	no	—
151	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
152	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic
153	#2100	C₂×C₃₁	1	2	yes	—
154	#2058	C₂×C₃₁	1	2	yes	—
155	#1988	C₂×C₃₁	1	2	yes	—
156	#1948	C₂×C₃₁	1	2	yes	—
157	#1920	C₂×C₃₁	1	4	yes	—
158	#1831	C₂×C₃₁	1	1	yes	characteristic
159	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
160	C₂²×C₃₁	#525	1	1	yes	characteristic
161	C₂²×C₃₁	#536	1	1	yes	characteristic
162	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
163	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
165	C₂²×C₃₁	#575	1	1	yes	characteristic
166	C₄×C₃₁	—	2	2	no	—
167	C₄×C₃₁	—	2	2	no	—
168	C₄×C₃₁	—	2	2	no	—
169	C₄×C₃₁	—	2	2	no	—
170	C₄×C₃₁	—	4	4	no	—
171	C₄×C₃₁	—	4	1	no	—
172	C₄×C₃₁	—	4	4	no	—
173	C₄×C₃₁	—	8	2	no	—
174	C₄×C₃₁	—	8	2	no	—
175	C₄×C₃₁	#476	1	4	yes	—
176	C₄×C₃₁	#556	1	2	yes	—
177	C₄×C₃₁	#557	1	2	yes	—
178	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
180	C₂×C₄×Q₈	—	2	1	no	—
181	C₂×C₄×Q₈	—	2	1	no	—
182	C₂×#106	C₂²×C₃₁	1	4	yes	—
183	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
184	#391	—	2	1	no	—
185	#391	—	2	1	no	—
186	#389	—	2	1	no	—
187	#389	—	2	1	no	—
188	#389	C₂²×C₃₁	1	2	yes	—
189	#389	C₂²×C₃₁	1	2	yes	—
190	#384	C₂²×C₃₁	1	1	yes	characteristic
191	#384	C₂²×C₃₁	1	1	yes	characteristic
192	C₄×C₄⋊C₄	—	2	1	no	—
193	C₄×C₄⋊C₄	—	2	1	no	—
194	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
195	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
196	C₄³	C₂²×C₃₁	1	1	yes	characteristic
197	#340	C₂²×C₃₁	1	4	yes	—
198	#340	C₂²×C₃₁	1	4	yes	—
199	#340	C₂²×C₃₁	1	4	yes	—
200	#336	C₂²×C₃₁	1	2	yes	—
201	#336	C₂²×C₃₁	1	2	yes	—
202	C₂×C₃₁	#2581	1	1	yes	characteristic
203	C₂×C₃₁	C₂×#476	1	1	yes	characteristic
204	C₂×C₃₁	#3078	1	1	yes	characteristic
205	C₂×C₃₁	#3103	1	1	yes	characteristic
206	C₂×C₃₁	#3110	1	1	yes	characteristic
207	C₂×C₃₁	#3229	1	1	yes	characteristic
208	C₂×C₃₁	#3235	1	1	yes	characteristic
209	C₂²×Q₈	—	2	1	no	—
210	C₂²×Q₈	—	2	1	no	—
211	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
212	C₄⋊Q₈	—	2	4	no	—
213	C₄⋊Q₈	—	2	2	no	—
214	C₄⋊Q₈	Q₈×C₃₁	1	4	yes	complemented
215	C₄×Q₈	—	4	4	no	—
216	C₄×Q₈	—	4	4	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
229	C₂×C₄²	—	2	2	no	—
230	C₂×C₄²	—	2	2	no	—
231	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
233	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
234	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
237	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
238	#106	—	2	8	no	—
239	C₄⋊C₈	—	2	4	no	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	C₃₁	#29276	1	1	yes	characteristic, direct product factor, Sylow
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	—	4	2	no	—
250	C₂×Q₈	—	4	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	4	4	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
278	C₄⋊C₄	#41×C₃₁	1	4	yes	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	4	1	no	—
284	C₄²	—	4	4	no	—
285	C₄²	—	4	4	no	—
286	C₄²	SD₁₆×C₃₁	1	4	yes	—
287	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
288	C₄²	#41×C₃₁	1	2	yes	—
289	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
290	Q₈	—	4	8	no	—
291	Q₈	—	4	8	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
316	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
317	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
318	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
319	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
320	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
321	C₂×C₄	C₃₁×#137	1	2	yes	—
322	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
323	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
324	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
325	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
326	C₈	—	4	4	no	—
327	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
328	C₂²	C₃₁×#525	1	1	yes	characteristic
329	C₂²	C₃₁×#536	1	1	yes	characteristic
330	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
331	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
332	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
333	C₂²	C₃₁×#575	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	4	4	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	4	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₃₁×#476	1	4	yes	—
344	C₄	C₃₁×#556	1	2	yes	—
345	C₄	C₃₁×#557	1	2	yes	—
346	C₂	C₃₁×#2581	1	1	yes	characteristic
347	C₂	C₂×C₃₁×#476	1	1	yes	characteristic
348	C₂	C₃₁×#3078	1	1	yes	characteristic
349	C₂	C₃₁×#3103	1	1	yes	characteristic
350	C₂	C₃₁×#3110	1	1	yes	characteristic
351	C₂	C₃₁×#3229	1	1	yes	characteristic
352	C₂	C₃₁×#3235	1	1	yes	characteristic

	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 ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
38. C₃₁×C₄⋊Q₈	Q₈	semidirect
112. #29276	C₃₁	direct (N × H)
214. C₄⋊Q₈	Q₈×C₃₁	semidirect
244. C₃₁	#29276	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×C₄⋊Q₈	2
C₃₁×#2100	2
C₃₁×#2058	2
C₃₁×#1988	2
C₃₁×#1948	2
C₃₁×#1920	2
C₃₁×#1831	2
#29276	31

Subgroups — C31×#29276 #21486035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29276 #21486035