Subgroups — C₃₁×#22744 #21479503

Order 7936.

714 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2355	C₂	1	2	yes	maximal
3	C₃₁×#1949	C₂	1	2	yes	maximal
4	C₃₁×#1949	C₂	1	2	yes	maximal
5	C₃₁×#1947	C₂	1	2	yes	maximal
6	C₃₁×#1836	C₂	1	2	yes	maximal
7	C₃₁×#1835	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1776	C₂	1	2	yes	maximal
9	C₄×C₃₁×#107	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#446	C₄	1	8	yes	complemented
12	C₂×C₃₁×#107	C₂²	1	2	yes	—
13	C₂×C₃₁×#107	C₂²	1	2	yes	—
14	C₂×C₃₁×#106	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#391	—	2	2	no	—
20	C₃₁×#387	—	2	2	no	—
21	C₃₁×#387	—	2	2	no	—
22	C₃₁×#384	C₂²	1	2	yes	—
23	C₃₁×#382	C₂²	1	2	yes	—
24	C₃₁×#382	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#340	C₂²	1	2	yes	—
30	C₃₁×#340	C₂²	1	1	yes	characteristic
31	C₃₁×#340	C₂²	1	1	yes	characteristic
32	C₃₁×#340	C₂²	1	1	yes	characteristic
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	1	yes	characteristic
36	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₃₁	Q₈	1	4	yes	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	—	2	2	no	—
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
55	C₃₁×#107	—	2	4	no	—
56	C₃₁×#107	—	2	4	no	—
57	C₃₁×#106	—	2	4	no	—
58	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
59	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
60	C₃₁×#97	C₂×C₄	1	8	yes	—
61	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
62	C₃₁×#95	—	2	2	no	—
63	C₃₁×#95	—	2	2	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	C₃₁×#95	—	2	2	no	—
69	C₂×Q₈×C₃₁	—	4	2	no	—
70	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	2	2	no	—
73	C₂²×C₄×C₃₁	—	2	2	no	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₄⋊C₄×C₃₁	—	2	2	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	4	4	no	—
95	C₄⋊C₄×C₃₁	—	8	2	no	—
96	C₄⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
97	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
98	C₄⋊C₄×C₃₁	#41	1	4	yes	—
99	C₄²×C₃₁	—	4	4	no	—
100	C₄²×C₃₁	—	8	2	no	—
101	C₄²×C₃₁	—	8	2	no	—
102	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
103	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
104	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
105	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
106	#22744	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₃₁	—	4	4	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
130	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
132	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
135	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
136	C₈×C₃₁	—	2	4	no	—
137	C₈×C₃₁	—	2	2	no	—
138	C₈×C₃₁	—	2	2	no	—
139	C₂×#446	C₂×C₃₁	1	1	yes	characteristic
140	#2355	C₂×C₃₁	1	2	yes	—
141	#1949	C₂×C₃₁	1	2	yes	—
142	#1949	C₂×C₃₁	1	2	yes	—
143	#1947	C₂×C₃₁	1	2	yes	—
144	#1836	C₂×C₃₁	1	2	yes	—
145	#1835	C₂×C₃₁	1	1	yes	characteristic
146	#1776	C₂×C₃₁	1	2	yes	—
147	C₄×#107	C₂×C₃₁	1	1	yes	characteristic
148	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₃₁	#548	1	2	yes	—
150	C₂²×C₃₁	#575	1	2	yes	—
151	C₂²×C₃₁	#576	1	1	yes	characteristic
152	C₂²×C₃₁	#578	1	1	yes	characteristic
153	C₄×C₃₁	—	2	4	no	—
154	C₄×C₃₁	—	2	2	no	—
155	C₄×C₃₁	—	2	2	no	—
156	C₄×C₃₁	—	2	4	no	—
157	C₄×C₃₁	—	4	4	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	—	8	2	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
162	C₄×C₃₁	#519	1	1	yes	characteristic
163	C₄×C₃₁	#557	1	2	yes	—
164	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
165	#446	C₄×C₃₁	1	8	yes	complemented
166	C₂×#107	C₂²×C₃₁	1	2	yes	—
167	C₂×#107	C₂²×C₃₁	1	2	yes	—
168	C₂×#106	C₂²×C₃₁	1	2	yes	—
169	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
170	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
171	C₂×#97	C₂²×C₃₁	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
173	#391	—	2	2	no	—
174	#387	—	2	2	no	—
175	#387	—	2	2	no	—
176	#384	C₂²×C₃₁	1	2	yes	—
177	#382	C₂²×C₃₁	1	2	yes	—
178	#382	C₂²×C₃₁	1	2	yes	—
179	C₄×C₄⋊C₄	—	2	2	no	—
180	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
181	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
182	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
183	#340	C₂²×C₃₁	1	2	yes	—
184	#340	C₂²×C₃₁	1	1	yes	characteristic
185	#340	C₂²×C₃₁	1	1	yes	characteristic
186	#340	C₂²×C₃₁	1	1	yes	characteristic
187	#340	C₂²×C₃₁	1	2	yes	—
188	#340	C₂²×C₃₁	1	2	yes	—
189	#340	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×C₃₁	#2351	1	1	yes	characteristic
191	C₂×C₃₁	#2911	1	1	yes	characteristic
192	C₂×C₃₁	#2913	1	1	yes	characteristic
193	C₂×C₃₁	#3370	1	2	yes	—
194	C₂×C₃₁	#3377	1	2	yes	—
195	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
199	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
200	C₄×Q₈	Q₈×C₃₁	1	4	yes	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
204	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
206	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
213	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
214	#107	—	2	4	no	—
215	#107	—	2	4	no	—
216	#106	—	2	4	no	—
217	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
218	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
219	#97	C₂×C₄×C₃₁	1	8	yes	—
220	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	C₃₁	#22744	1	1	yes	characteristic, direct product factor, Sylow
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
245	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
246	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	8	2	no	—
256	C₄⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
257	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
258	C₄⋊C₄	#41×C₃₁	1	4	yes	—
259	C₄²	—	4	4	no	—
260	C₄²	—	8	2	no	—
261	C₄²	—	8	2	no	—
262	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
263	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
264	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
265	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
266	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
267	Q₈	—	4	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
288	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
289	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
291	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
292	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
293	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
294	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
295	C₈	—	2	4	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
299	C₂²	C₃₁×#548	1	2	yes	—
300	C₂²	C₃₁×#575	1	2	yes	—
301	C₂²	C₃₁×#576	1	1	yes	characteristic
302	C₂²	C₃₁×#578	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	4	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
312	C₄	C₃₁×#519	1	1	yes	characteristic
313	C₄	C₃₁×#557	1	2	yes	—
314	C₂	C₃₁×#2351	1	1	yes	characteristic
315	C₂	C₃₁×#2911	1	1	yes	characteristic
316	C₂	C₃₁×#2913	1	1	yes	characteristic
317	C₂	C₃₁×#3370	1	2	yes	—
318	C₂	C₃₁×#3377	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₃₁×#446	C₄	semidirect
106. #22744	C₃₁	direct (N × H)
165. #446	C₄×C₃₁	semidirect
228. C₃₁	#22744	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×#446	2
C₃₁×#2355	2
C₃₁×#1949	2
C₃₁×#1947	2
C₃₁×#1836	2
C₃₁×#1835	2
C₃₁×#1776	2
C₄×C₃₁×#107	2
#22744	31

Subgroups — C31×#22744 #21479503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22744 #21479503