Subgroups — C₃₁×#22310 #21479069

Order 7936.

630 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2399	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1985	C₂	1	2	yes	maximal
3	C₃₁×#1984	C₂	1	2	yes	maximal
4	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1917	C₂	1	2	yes	maximal
7	C₃₁×#1916	C₂	1	2	yes	maximal
8	C₃₁×#1832	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic
10	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#398	C₄	1	2	yes	—
22	C₃₁×#398	C₄	1	2	yes	—
23	C₃₁×#395	C₄	1	2	yes	—
24	C₃₁×#395	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#376	C₂²	1	1	yes	characteristic
29	C₃₁×#336	C₂²	1	2	yes	—
30	C₃₁×#336	C₂²	1	2	yes	—
31	C₃₁×#336	C₂²	1	1	yes	characteristic
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	1	yes	characteristic
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	1	yes	characteristic
38	C₃₁×#336	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₃₁	—	4	2	no	—
47	C₄×Q₈×C₃₁	—	4	2	no	—
48	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
56	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
57	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊C₈	—	2	2	no	—
61	C₃₁×C₄⋊C₈	—	2	2	no	—
62	C₃₁×C₄⋊C₈	—	4	2	no	—
63	C₃₁×C₄⋊C₈	—	4	2	no	—
64	C₃₁×#97	—	2	2	no	—
65	C₃₁×#97	—	4	2	no	—
66	C₄×C₈×C₃₁	—	4	2	no	—
67	C₄×C₈×C₃₁	Q₈	1	4	yes	complemented
68	C₃₁×#95	C₂×C₄	1	2	yes	—
69	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
70	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
71	C₃₁×#95	C₂×C₄	1	2	yes	—
72	C₃₁×#95	C₂×C₄	1	2	yes	—
73	C₃₁×#95	C₂×C₄	1	2	yes	—
74	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
75	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
76	C₂×Q₈×C₃₁	—	2	4	no	—
77	C₂×Q₈×C₃₁	—	4	2	no	—
78	C₂×Q₈×C₃₁	—	4	2	no	—
79	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₃₁	—	2	1	no	—
89	C₂×C₈×C₃₁	—	2	1	no	—
90	C₂×C₈×C₃₁	—	2	1	no	—
91	C₂×C₈×C₃₁	—	2	1	no	—
92	C₂×C₈×C₃₁	—	2	1	no	—
93	C₂×C₈×C₃₁	—	2	1	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	1	no	—
99	C₂×C₈×C₃₁	—	2	1	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	1	no	—
102	C₂×C₈×C₃₁	—	2	1	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
105	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
106	C₄⋊C₄×C₃₁	—	2	4	no	—
107	C₄⋊C₄×C₃₁	—	2	4	no	—
108	C₄⋊C₄×C₃₁	—	4	2	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₄⋊C₄×C₃₁	—	4	2	no	—
111	C₄⋊C₄×C₃₁	—	4	2	no	—
112	C₄⋊C₄×C₃₁	—	4	2	no	—
113	C₄²×C₃₁	—	2	2	no	—
114	C₄²×C₃₁	—	2	2	no	—
115	C₄²×C₃₁	—	2	2	no	—
116	C₄²×C₃₁	—	2	2	no	—
117	C₄²×C₃₁	—	2	1	no	—
118	C₄²×C₃₁	—	2	1	no	—
119	C₄²×C₃₁	—	2	1	no	—
120	C₄²×C₃₁	—	2	1	no	—
121	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
122	C₄²×C₃₁	#41	1	2	yes	—
123	#22310	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
125	Q₈×C₃₁	—	4	8	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	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	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	2	2	no	—
145	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
146	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
147	C₂×C₄×C₃₁	#131	1	4	yes	—
148	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
153	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
154	C₈×C₃₁	—	2	2	no	—
155	C₈×C₃₁	—	2	2	no	—
156	C₈×C₃₁	—	4	1	no	—
157	C₈×C₃₁	—	4	1	no	—
158	C₈×C₃₁	—	4	2	no	—
159	C₈×C₃₁	—	4	2	no	—
160	#2399	C₂×C₃₁	1	1	yes	characteristic
161	#1985	C₂×C₃₁	1	2	yes	—
162	#1984	C₂×C₃₁	1	2	yes	—
163	#1940	C₂×C₃₁	1	1	yes	characteristic
164	#1940	C₂×C₃₁	1	1	yes	characteristic
165	#1917	C₂×C₃₁	1	2	yes	—
166	#1916	C₂×C₃₁	1	2	yes	—
167	#1832	C₂×C₃₁	1	1	yes	characteristic
168	#1770	C₂×C₃₁	1	1	yes	characteristic
169	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
170	#1746	C₂×C₃₁	1	1	yes	characteristic
171	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
172	C₂²×C₃₁	#518	1	1	yes	characteristic
173	C₂²×C₃₁	#519	1	1	yes	characteristic
174	C₂²×C₃₁	C₂×#131	1	2	yes	—
175	C₂²×C₃₁	#568	1	2	yes	—
176	C₄×C₃₁	—	2	2	no	—
177	C₄×C₃₁	—	2	2	no	—
178	C₄×C₃₁	—	2	4	no	—
179	C₄×C₃₁	—	4	2	no	—
180	C₄×C₃₁	—	4	2	no	—
181	C₄×C₃₁	—	4	2	no	—
182	C₄×C₃₁	—	4	2	no	—
183	C₄×C₃₁	—	4	2	no	—
184	C₄×C₃₁	#542	1	1	yes	characteristic
185	C₄×C₃₁	#542	1	1	yes	characteristic
186	C₄×C₃₁	#544	1	1	yes	characteristic
187	C₄×C₃₁	#544	1	1	yes	characteristic
188	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
189	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
191	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
192	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
193	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
194	C₂×#97	C₂²×C₃₁	1	2	yes	—
195	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
196	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
197	#398	C₄×C₃₁	1	2	yes	—
198	#398	C₄×C₃₁	1	2	yes	—
199	#395	C₄×C₃₁	1	2	yes	—
200	#395	C₄×C₃₁	1	2	yes	—
201	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
202	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
203	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
204	#376	C₂²×C₃₁	1	1	yes	characteristic
205	#336	C₂²×C₃₁	1	2	yes	—
206	#336	C₂²×C₃₁	1	2	yes	—
207	#336	C₂²×C₃₁	1	1	yes	characteristic
208	#336	C₂²×C₃₁	1	2	yes	—
209	#336	C₂²×C₃₁	1	2	yes	—
210	#336	C₂²×C₃₁	1	1	yes	characteristic
211	#336	C₂²×C₃₁	1	2	yes	—
212	#336	C₂²×C₃₁	1	2	yes	—
213	#336	C₂²×C₃₁	1	1	yes	characteristic
214	#336	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×C₃₁	#2331	1	1	yes	characteristic
216	C₂×C₃₁	#2960	1	1	yes	characteristic
217	C₂×C₃₁	#2961	1	1	yes	characteristic
218	C₂×C₃₁	#2976	1	2	yes	—
219	C₂×C₃₁	#2985	1	2	yes	—
220	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
221	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
222	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
223	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
225	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
227	C₄×Q₈	—	4	2	no	—
228	C₄×Q₈	—	4	2	no	—
229	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
230	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
231	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
232	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
233	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
234	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
235	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
236	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
237	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
238	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
239	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
241	C₄⋊C₈	—	2	2	no	—
242	C₄⋊C₈	—	2	2	no	—
243	C₄⋊C₈	—	4	2	no	—
244	C₄⋊C₈	—	4	2	no	—
245	#97	—	2	2	no	—
246	#97	—	4	2	no	—
247	C₄×C₈	—	4	2	no	—
248	C₄×C₈	Q₈×C₃₁	1	4	yes	complemented
249	#95	C₂×C₄×C₃₁	1	2	yes	—
250	#95	C₂×C₄×C₃₁	1	2	yes	—
251	#95	C₂×C₄×C₃₁	1	2	yes	—
252	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
253	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
254	#95	C₂×C₄×C₃₁	1	2	yes	—
255	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
256	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
257	C₃₁	#22310	1	1	yes	characteristic, direct product factor, Sylow
258	C₂×Q₈	—	2	4	no	—
259	C₂×Q₈	—	4	2	no	—
260	C₂×Q₈	—	4	2	no	—
261	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
262	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
263	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
264	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
265	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
266	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
267	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
269	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
287	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	4	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	1	no	—
303	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
304	C₄²	#41×C₃₁	1	2	yes	—
305	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
306	Q₈	—	4	8	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
327	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
328	C₂×C₄	C₃₁×#131	1	4	yes	—
329	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
330	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
331	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
332	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
333	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
334	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	2	2	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	4	2	no	—
341	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
342	C₂²	C₃₁×#518	1	1	yes	characteristic
343	C₂²	C₃₁×#519	1	1	yes	characteristic
344	C₂²	C₂×C₃₁×#131	1	2	yes	—
345	C₂²	C₃₁×#568	1	2	yes	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	4	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	C₃₁×#542	1	1	yes	characteristic
355	C₄	C₃₁×#542	1	1	yes	characteristic
356	C₄	C₃₁×#544	1	1	yes	characteristic
357	C₄	C₃₁×#544	1	1	yes	characteristic
358	C₂	C₃₁×#2331	1	1	yes	characteristic
359	C₂	C₃₁×#2960	1	1	yes	characteristic
360	C₂	C₃₁×#2961	1	1	yes	characteristic
361	C₂	C₃₁×#2976	1	2	yes	—
362	C₂	C₃₁×#2985	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
67. C₄×C₈×C₃₁	Q₈	semidirect
123. #22310	C₃₁	direct (N × H)
248. C₄×C₈	Q₈×C₃₁	semidirect
257. C₃₁	#22310	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×#2399	2
C₃₁×#1985	2
C₃₁×#1984	2
C₃₁×#1940	2
C₃₁×#1917	2
C₃₁×#1916	2
C₃₁×#1832	2
C₃₁×#1770	2
C₈×C₄⋊C₄×C₃₁	2
C₃₁×#1746	2
#22310	31

Subgroups — C31×#22310 #21479069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22310 #21479069