Subgroups — C₃₁×#22440 #21479199

Order 7936.

1386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#441	C₂	1	2	yes	maximal, complemented
3	Q₈×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal
4	C₃₁×#1971	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1878	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1807	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1787	C₂	1	2	yes	maximal
8	C₃₁×#1752	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₃₁	—	2	8	no	—
11	C₂²×Q₁₆×C₃₁	C₄	1	8	yes	complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	—
17	C₃₁×#441	—	2	8	no	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×#103	C₂²	1	4	yes	—
20	C₂×C₃₁×#103	C₂²	1	4	yes	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#97	C₂²	1	2	yes	—
23	C₃₁×#391	—	2	4	no	—
24	C₃₁×#391	C₂²	1	4	yes	—
25	C₃₁×#380	—	2	2	no	—
26	C₃₁×#380	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
28	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#340	C₂²	1	2	yes	—
30	C₃₁×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₁	—	2	2	no	—
32	C₂²×Q₈×C₃₁	—	2	4	no	—
33	C₂²×Q₈×C₃₁	—	2	8	no	—
34	C₂²×Q₈×C₃₁	—	4	4	no	—
35	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	—
36	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₃₁	—	2	2	no	—
39	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×Q₁₆×C₃₁	—	2	8	no	—
41	C₂×Q₁₆×C₃₁	—	2	32	no	—
42	C₂×Q₁₆×C₃₁	—	2	4	no	—
43	C₂×Q₁₆×C₃₁	C₂×C₄	1	8	yes	—
44	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
45	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
46	C₂×Q₁₆×C₃₁	D₄	1	8	yes	—
47	C₂²×C₈×C₃₁	—	2	4	no	—
48	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₄×Q₈×C₃₁	—	2	8	no	—
53	C₄×Q₈×C₃₁	—	4	4	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
55	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
56	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
58	C₂×C₄²×C₃₁	—	2	4	no	—
59	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
60	C₃₁×#106	—	2	4	no	—
61	C₃₁×#103	—	2	8	no	—
62	C₃₁×#103	—	4	4	no	—
63	C₃₁×C₂²⋊C₈	—	2	2	no	—
64	C₃₁×#97	—	2	4	no	—
65	C₃₁×#95	—	2	4	no	—
66	C₃₁×#95	—	2	4	no	—
67	C₃₁×#95	—	2	2	no	—
68	C₃₁×#95	D₄	1	2	yes	—
69	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂×Q₈×C₃₁	—	2	8	no	—
71	C₂×Q₈×C₃₁	—	2	2	no	—
72	C₂×Q₈×C₃₁	—	2	2	no	—
73	C₂×Q₈×C₃₁	—	2	4	no	—
74	C₂×Q₈×C₃₁	—	2	16	no	—
75	C₂×Q₈×C₃₁	—	2	4	no	—
76	C₂×Q₈×C₃₁	—	2	4	no	—
77	C₂×Q₈×C₃₁	—	4	8	no	—
78	C₂×Q₈×C₃₁	—	4	8	no	—
79	C₂×Q₈×C₃₁	—	4	16	no	—
80	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
81	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
82	C₂×Q₈×C₃₁	C₂×D₄	1	8	yes	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	—	2	8	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	2	4	no	—
89	C₂²×C₄×C₃₁	—	4	8	no	—
90	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
92	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	#41	1	2	yes	—
94	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
95	Q₁₆×C₃₁	—	2	16	no	—
96	Q₁₆×C₃₁	—	2	8	no	—
97	Q₁₆×C₃₁	C₂²⋊C₄	1	16	yes	complemented
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	2	8	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
104	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
105	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
106	C₄⋊C₄×C₃₁	—	2	4	no	—
107	C₄⋊C₄×C₃₁	—	2	4	no	—
108	C₄⋊C₄×C₃₁	—	4	8	no	—
109	C₂²⋊C₄×C₃₁	—	2	4	no	—
110	C₂²⋊C₄×C₃₁	—	4	4	no	—
111	C₄²×C₃₁	—	2	2	no	—
112	C₄²×C₃₁	—	2	2	no	—
113	C₄²×C₃₁	—	4	8	no	—
114	#22440	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	—	2	2	no	—
116	C₂³×C₃₁	—	2	2	no	—
117	C₂³×C₃₁	C₄×D₄	1	2	yes	—
118	C₂³×C₃₁	#137	1	4	yes	—
119	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
120	Q₈×C₃₁	—	2	4	no	—
121	Q₈×C₃₁	—	2	8	no	—
122	Q₈×C₃₁	—	2	4	no	—
123	Q₈×C₃₁	—	2	4	no	—
124	Q₈×C₃₁	—	4	16	no	—
125	Q₈×C₃₁	—	4	4	no	—
126	Q₈×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
127	C₂×C₄×C₃₁	—	2	4	no	—
128	C₂×C₄×C₃₁	—	2	8	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	2	4	no	—
133	C₂×C₄×C₃₁	—	2	4	no	—
134	C₂×C₄×C₃₁	—	2	4	no	—
135	C₂×C₄×C₃₁	—	2	4	no	—
136	C₂×C₄×C₃₁	—	2	4	no	—
137	C₂×C₄×C₃₁	—	4	16	no	—
138	C₂×C₄×C₃₁	—	4	4	no	—
139	C₂×C₄×C₃₁	—	4	4	no	—
140	C₂×C₄×C₃₁	—	4	4	no	—
141	C₂×C₄×C₃₁	—	4	4	no	—
142	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
145	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
147	C₈×C₃₁	—	2	2	no	—
148	C₈×C₃₁	—	4	2	no	—
149	C₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
150	C₂³×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
151	C₂×#441	C₂×C₃₁	1	2	yes	complemented
152	Q₈×C₂²⋊C₄	C₂×C₃₁	1	2	yes	—
153	#1971	C₂×C₃₁	1	2	yes	complemented
154	#1878	C₂×C₃₁	1	4	yes	complemented
155	#1807	C₂×C₃₁	1	1	yes	characteristic
156	#1787	C₂×C₃₁	1	2	yes	—
157	#1752	C₂×C₃₁	1	1	yes	characteristic
158	C₂²×C₃₁	—	2	2	no	—
159	C₂²×C₃₁	—	2	2	no	—
160	C₂²×C₃₁	—	2	8	no	—
161	C₂²×C₃₁	#441	1	4	yes	complemented
162	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂²×C₃₁	D₄²	1	2	yes	—
164	C₂²×C₃₁	C₂×#137	1	2	yes	—
165	C₂²×C₃₁	#578	1	2	yes	—
166	C₄×C₃₁	—	2	2	no	—
167	C₄×C₃₁	—	2	8	no	—
168	C₄×C₃₁	—	4	4	no	—
169	C₄×C₃₁	—	4	4	no	—
170	C₄×C₃₁	—	8	4	no	—
171	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
172	C₄×C₃₁	#518	1	1	yes	characteristic
173	C₄×C₃₁	#546	1	2	yes	—
174	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
175	C₂²×Q₁₆	—	2	8	no	—
176	C₂²×Q₁₆	C₄×C₃₁	1	8	yes	complemented
177	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
178	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
179	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
180	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
181	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	—
182	#441	—	2	8	no	—
183	C₂×#106	C₂²×C₃₁	1	2	yes	—
184	C₂×#103	C₂²×C₃₁	1	4	yes	—
185	C₂×#103	C₂²×C₃₁	1	4	yes	—
186	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
187	C₂×#97	C₂²×C₃₁	1	2	yes	—
188	#391	—	2	4	no	—
189	#391	C₂²×C₃₁	1	4	yes	—
190	#380	—	2	2	no	—
191	#380	C₂²×C₃₁	1	2	yes	—
192	C₄×C₂²⋊C₄	—	2	2	no	—
193	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
194	#340	C₂²×C₃₁	1	2	yes	—
195	#336	C₂²×C₃₁	1	2	yes	—
196	C₂×C₃₁	—	2	4	no	—
197	C₂×C₃₁	D₄×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₃₁	C₂×#441	1	1	yes	characteristic
199	C₂×C₃₁	#2950	1	1	yes	characteristic
200	C₂×C₃₁	#3270	1	4	yes	—
201	C₂²×Q₈	—	2	2	no	—
202	C₂²×Q₈	—	2	4	no	—
203	C₂²×Q₈	—	2	8	no	—
204	C₂²×Q₈	—	4	4	no	—
205	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	—
206	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
207	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
208	C₂³×C₄	—	2	2	no	—
209	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
210	C₂×Q₁₆	—	2	8	no	—
211	C₂×Q₁₆	—	2	32	no	—
212	C₂×Q₁₆	—	2	4	no	—
213	C₂×Q₁₆	C₂×C₄×C₃₁	1	8	yes	—
214	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
215	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
216	C₂×Q₁₆	D₄×C₃₁	1	8	yes	—
217	C₂²×C₈	—	2	4	no	—
218	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
219	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
220	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
221	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
222	C₄×Q₈	—	2	8	no	—
223	C₄×Q₈	—	4	4	no	—
224	C₂×C₄⋊C₄	—	2	4	no	—
225	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
228	C₂×C₄²	—	2	4	no	—
229	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
230	#106	—	2	4	no	—
231	#103	—	2	8	no	—
232	#103	—	4	4	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	#97	—	2	4	no	—
235	#95	—	2	4	no	—
236	#95	—	2	4	no	—
237	#95	—	2	2	no	—
238	#95	D₄×C₃₁	1	2	yes	—
239	C₃₁	#22440	1	1	yes	characteristic, direct product factor, Sylow
240	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
241	C₂×Q₈	—	2	8	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	4	no	—
245	C₂×Q₈	—	2	16	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	—	4	8	no	—
249	C₂×Q₈	—	4	8	no	—
250	C₂×Q₈	—	4	16	no	—
251	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
252	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
253	C₂×Q₈	C₂×D₄×C₃₁	1	8	yes	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	8	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	4	no	—
260	C₂²×C₄	—	4	8	no	—
261	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
263	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
264	C₂²×C₄	#41×C₃₁	1	2	yes	—
265	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
266	Q₁₆	—	2	16	no	—
267	Q₁₆	—	2	8	no	—
268	Q₁₆	C₂²⋊C₄×C₃₁	1	16	yes	complemented
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	8	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
275	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
276	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	4	8	no	—
280	C₂²⋊C₄	—	2	4	no	—
281	C₂²⋊C₄	—	4	4	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	4	8	no	—
285	C₂³	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	C₄×D₄×C₃₁	1	2	yes	—
288	C₂³	C₃₁×#137	1	4	yes	—
289	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
290	Q₈	—	2	4	no	—
291	Q₈	—	2	8	no	—
292	Q₈	—	2	4	no	—
293	Q₈	—	2	4	no	—
294	Q₈	—	4	16	no	—
295	Q₈	—	4	4	no	—
296	Q₈	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	4	16	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
313	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
315	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
316	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
317	C₈	—	2	2	no	—
318	C₈	—	4	2	no	—
319	C₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	8	no	—
323	C₂²	C₃₁×#441	1	4	yes	complemented
324	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
325	C₂²	D₄²×C₃₁	1	2	yes	—
326	C₂²	C₂×C₃₁×#137	1	2	yes	—
327	C₂²	C₃₁×#578	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	2	8	no	—
330	C₄	—	4	4	no	—
331	C₄	—	4	4	no	—
332	C₄	—	8	4	no	—
333	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
334	C₄	C₃₁×#518	1	1	yes	characteristic
335	C₄	C₃₁×#546	1	2	yes	—
336	C₂	—	2	4	no	—
337	C₂	D₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
338	C₂	C₂×C₃₁×#441	1	1	yes	characteristic
339	C₂	C₃₁×#2950	1	1	yes	characteristic
340	C₂	C₃₁×#3270	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#441	C₂	semidirect
4. C₃₁×#1971	C₂	semidirect
5. C₃₁×#1878	C₂	semidirect
11. C₂²×Q₁₆×C₃₁	C₄	semidirect
97. Q₁₆×C₃₁	C₂²⋊C₄	semidirect
114. #22440	C₃₁	direct (N × H)
151. C₂×#441	C₂×C₃₁	semidirect
153. #1971	C₂×C₃₁	semidirect
154. #1878	C₂×C₃₁	semidirect
161. C₂²×C₃₁	#441	semidirect
176. C₂²×Q₁₆	C₄×C₃₁	semidirect
239. C₃₁	#22440	direct (N × H)
268. Q₁₆	C₂²⋊C₄×C₃₁	semidirect
323. C₂²	C₃₁×#441	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₃₁	2
C₂×C₃₁×#441	2
Q₈×C₂²⋊C₄×C₃₁	2
C₃₁×#1971	2
C₃₁×#1878	2
C₃₁×#1807	2
C₃₁×#1787	2
C₃₁×#1752	2
#22440	31

Subgroups — C31×#22440 #21479199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22440 #21479199