Subgroups — C₃₁×#33454 #21490213

Order 7936.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#509	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2951	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2177	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2175	C₂	1	2	yes	maximal, complemented
6	C₄×C₃₁×SD₃₂	C₂	1	2	yes	maximal, complemented
7	C₄×C₃₁×D₁₆	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2159	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#357	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#520	—	2	1	no	—
15	C₃₁×#518	—	2	1	no	—
16	C₃₁×#509	C₄	1	8	yes	complemented
17	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
18	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
19	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#442	—	2	2	no	—
22	C₃₁×#440	—	2	2	no	—
23	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
24	C₄×D₈×C₃₁	C₂²	1	2	yes	—
25	C₄×D₈×C₃₁	C₂²	1	2	yes	—
26	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
27	C₃₁×#427	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
29	C₃₁×#418	—	2	1	no	—
30	C₃₁×#417	—	2	1	no	—
31	C₃₁×#405	C₂²	1	1	yes	characteristic
32	C₃₁×#367	C₂²	1	2	yes	complemented
33	C₃₁×#366	C₂²	1	2	yes	complemented
34	C₃₁×#358	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#358	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#357	C₂²	1	2	yes	—
37	C₃₁×#357	C₂²	1	2	yes	—
38	C₃₁×#357	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#357	C₂²	1	1	yes	characteristic, complemented
40	C₃₁×#346	C₂²	1	2	yes	complemented
41	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
42	C₂×#41×C₃₁	—	2	1	no	—
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₃₁×#135	—	2	2	no	—
45	C₃₁×#135	C₂×C₄	1	4	yes	complemented
46	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×SD₁₆×C₃₁	—	2	1	no	—
48	C₂×D₈×C₃₁	—	2	1	no	—
49	C₂×D₈×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	complemented
51	C₂×D₈×C₃₁	C₂³	1	2	yes	—
52	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×D₄×C₃₁	—	2	2	no	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₄×D₄×C₃₁	—	4	1	no	—
60	C₄×D₄×C₃₁	—	4	1	no	—
61	C₄×D₄×C₃₁	—	4	2	no	—
62	C₃₁×#117	—	2	1	no	—
63	C₃₁×#117	—	4	1	no	—
64	C₃₁×#117	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
68	C₃₁×SD₃₂	C₂×C₄	1	8	yes	complemented
69	C₃₁×D₁₆	C₂×C₄	1	8	yes	complemented
70	C₃₁×M₃₂	C₂×C₄	1	4	yes	complemented
71	C₂×C₁₆×C₃₁	—	2	2	no	—
72	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
75	C₃₁×#107	—	2	1	no	—
76	C₃₁×#107	C₂³	1	2	yes	—
77	C₃₁×#107	C₂³	1	1	yes	characteristic
78	C₃₁×#107	C₂³	1	1	yes	characteristic
79	C₃₁×#106	—	2	1	no	—
80	C₃₁×#103	—	2	2	no	—
81	C₃₁×#102	—	2	2	no	—
82	C₃₁×#102	—	2	2	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#97	D₄	1	2	yes	—
85	C₄×C₈×C₃₁	C₂³	1	2	yes	—
86	C₂⁴×C₃₁	—	4	1	no	—
87	#41×C₃₁	—	2	4	no	—
88	#41×C₃₁	—	4	2	no	—
89	C₂×Q₈×C₃₁	—	2	1	no	—
90	C₂×D₄×C₃₁	—	2	4	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	1	no	—
93	C₂×D₄×C₃₁	—	4	1	no	—
94	C₂×D₄×C₃₁	—	4	1	no	—
95	C₂×D₄×C₃₁	—	4	2	no	—
96	C₂²×C₄×C₃₁	—	4	1	no	—
97	C₂²×C₄×C₃₁	—	4	1	no	—
98	C₂²×C₄×C₃₁	—	4	2	no	—
99	C₂²×C₄×C₃₁	—	4	2	no	—
100	C₂²×C₄×C₃₁	—	4	2	no	—
101	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	Q₁₆×C₃₁	—	2	1	no	—
103	Q₁₆×C₃₁	C₂²×C₄	1	2	yes	—
104	SD₁₆×C₃₁	—	2	4	no	—
105	D₈×C₃₁	—	2	4	no	—
106	D₈×C₃₁	—	2	2	no	—
107	D₈×C₃₁	—	2	1	no	—
108	D₈×C₃₁	C₂²×C₄	1	2	yes	—
109	D₈×C₃₁	C₂²×C₄	1	4	yes	—
110	M₁₆×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₃₁	#41	1	2	yes	—
115	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₃₁	—	2	1	no	—
117	C₄⋊C₄×C₃₁	—	2	1	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	4	1	no	—
120	C₄⋊C₄×C₃₁	—	4	2	no	—
121	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
122	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
123	C₂²⋊C₄×C₃₁	—	4	1	no	—
124	C₂²⋊C₄×C₃₁	—	4	2	no	—
125	C₂²⋊C₄×C₃₁	—	4	2	no	—
126	C₂²⋊C₄×C₃₁	—	4	2	no	—
127	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
128	C₄²×C₃₁	—	4	1	no	—
129	C₄²×C₃₁	—	4	2	no	—
130	C₄²×C₃₁	C₂×D₄	1	2	yes	—
131	C₁₆×C₃₁	—	2	1	no	—
132	C₁₆×C₃₁	—	2	1	no	—
133	C₁₆×C₃₁	C₂²×C₄	1	4	yes	—
134	#33454	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂³×C₃₁	—	4	1	no	—
136	C₂³×C₃₁	—	4	2	no	—
137	C₂³×C₃₁	—	4	4	no	—
138	C₂³×C₃₁	—	16	1	no	—
139	C₂³×C₃₁	—	16	1	no	—
140	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
141	Q₈×C₃₁	—	2	2	no	—
142	Q₈×C₃₁	—	4	1	no	—
143	D₄×C₃₁	—	2	4	no	—
144	D₄×C₃₁	—	2	2	no	—
145	D₄×C₃₁	—	4	4	no	—
146	D₄×C₃₁	—	4	2	no	—
147	D₄×C₃₁	—	4	1	no	—
148	D₄×C₃₁	—	4	4	no	—
149	C₂×C₄×C₃₁	—	4	1	no	—
150	C₂×C₄×C₃₁	—	4	4	no	—
151	C₂×C₄×C₃₁	—	4	1	no	—
152	C₂×C₄×C₃₁	—	4	1	no	—
153	C₂×C₄×C₃₁	—	4	2	no	—
154	C₂×C₄×C₃₁	—	16	2	no	—
155	C₂×C₄×C₃₁	—	16	2	no	—
156	C₂×C₄×C₃₁	—	16	2	no	—
157	C₂×C₄×C₃₁	—	16	1	no	—
158	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
159	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
160	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
161	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₃₁	#135	1	2	yes	—
163	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
164	C₈×C₃₁	—	2	2	no	—
165	C₈×C₃₁	C₄×D₄	1	2	yes	—
166	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
167	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
168	C₂×#509	C₂×C₃₁	1	1	yes	characteristic
169	#2951	C₂×C₃₁	1	1	yes	characteristic, complemented
170	#2948	C₂×C₃₁	1	1	yes	characteristic, complemented
171	#2177	C₂×C₃₁	1	2	yes	complemented
172	#2175	C₂×C₃₁	1	2	yes	complemented
173	C₄×SD₃₂	C₂×C₃₁	1	2	yes	complemented
174	C₄×D₁₆	C₂×C₃₁	1	2	yes	complemented
175	#2159	C₂×C₃₁	1	1	yes	characteristic, complemented
176	#2138	C₂×C₃₁	1	1	yes	characteristic, complemented
177	C₂×#357	C₂×C₃₁	1	1	yes	characteristic, complemented
178	#2108	C₂×C₃₁	1	1	yes	characteristic, complemented
179	C₂²×C₃₁	—	2	1	no	—
180	C₂²×C₃₁	—	2	1	no	—
181	C₂²×C₃₁	—	4	2	no	—
182	C₂²×C₃₁	—	4	4	no	—
183	C₂²×C₃₁	—	8	1	no	—
184	C₂²×C₃₁	—	8	2	no	—
185	C₂²×C₃₁	—	8	2	no	—
186	C₂²×C₃₁	—	8	1	no	—
187	C₂²×C₃₁	—	16	4	no	—
188	C₂²×C₃₁	C₄×D₈	1	2	yes	—
189	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
190	C₄×C₃₁	—	2	2	no	—
191	C₄×C₃₁	—	2	2	no	—
192	C₄×C₃₁	—	4	2	no	—
193	C₄×C₃₁	—	8	1	no	—
194	C₄×C₃₁	—	8	2	no	—
195	C₄×C₃₁	—	8	1	no	—
196	C₄×C₃₁	C₄×D₈	1	2	yes	—
197	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
198	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
199	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
200	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
201	#520	—	2	1	no	—
202	#518	—	2	1	no	—
203	#509	C₄×C₃₁	1	8	yes	complemented
204	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
205	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
206	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
207	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
208	#442	—	2	2	no	—
209	#440	—	2	2	no	—
210	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
211	C₄×D₈	C₂²×C₃₁	1	2	yes	—
212	C₄×D₈	C₂²×C₃₁	1	2	yes	—
213	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
214	#427	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
216	#418	—	2	1	no	—
217	#417	—	2	1	no	—
218	#405	C₂²×C₃₁	1	1	yes	characteristic
219	#367	C₂²×C₃₁	1	2	yes	complemented
220	#366	C₂²×C₃₁	1	2	yes	complemented
221	#358	C₂²×C₃₁	1	1	yes	characteristic, complemented
222	#358	C₂²×C₃₁	1	1	yes	characteristic, complemented
223	#357	C₂²×C₃₁	1	2	yes	—
224	#357	C₂²×C₃₁	1	2	yes	—
225	#357	C₂²×C₃₁	1	1	yes	characteristic, complemented
226	#357	C₂²×C₃₁	1	1	yes	characteristic, complemented
227	#346	C₂²×C₃₁	1	2	yes	complemented
228	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
229	C₂×C₃₁	—	2	2	no	—
230	C₂×C₃₁	—	8	2	no	—
231	C₂×C₃₁	—	8	4	no	—
232	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
233	C₂×C₃₁	#3415	1	1	yes	characteristic
234	C₂×C₃₁	#3416	1	1	yes	characteristic
235	C₂×#41	—	2	1	no	—
236	C₂²×D₄	—	2	1	no	—
237	#135	—	2	2	no	—
238	#135	C₂×C₄×C₃₁	1	4	yes	complemented
239	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
240	C₂×SD₁₆	—	2	1	no	—
241	C₂×D₈	—	2	1	no	—
242	C₂×D₈	—	2	2	no	—
243	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	complemented
244	C₂×D₈	C₂³×C₃₁	1	2	yes	—
245	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
246	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
247	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
248	C₄×Q₈	—	2	2	no	—
249	C₄×D₄	—	2	2	no	—
250	C₄×D₄	—	2	2	no	—
251	C₄×D₄	—	2	2	no	—
252	C₄×D₄	—	4	1	no	—
253	C₄×D₄	—	4	1	no	—
254	C₄×D₄	—	4	2	no	—
255	#117	—	2	1	no	—
256	#117	—	4	1	no	—
257	#117	D₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄⋊C₄	—	2	1	no	—
259	C₂×C₄⋊C₄	—	4	1	no	—
260	C₂×C₂²⋊C₄	—	4	2	no	—
261	SD₃₂	C₂×C₄×C₃₁	1	8	yes	complemented
262	D₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
263	M₃₂	C₂×C₄×C₃₁	1	4	yes	complemented
264	C₂×C₁₆	—	2	2	no	—
265	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
266	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
267	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
268	#107	—	2	1	no	—
269	#107	C₂³×C₃₁	1	2	yes	—
270	#107	C₂³×C₃₁	1	1	yes	characteristic
271	#107	C₂³×C₃₁	1	1	yes	characteristic
272	#106	—	2	1	no	—
273	#103	—	2	2	no	—
274	#102	—	2	2	no	—
275	#102	—	2	2	no	—
276	#102	—	2	2	no	—
277	#97	D₄×C₃₁	1	2	yes	—
278	C₄×C₈	C₂³×C₃₁	1	2	yes	—
279	C₃₁	#33454	1	1	yes	characteristic, direct product factor, Sylow
280	C₂⁴	—	4	1	no	—
281	#41	—	2	4	no	—
282	#41	—	4	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×D₄	—	2	4	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	4	2	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	—	4	2	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
296	Q₁₆	—	2	1	no	—
297	Q₁₆	C₂²×C₄×C₃₁	1	2	yes	—
298	SD₁₆	—	2	4	no	—
299	D₈	—	2	4	no	—
300	D₈	—	2	2	no	—
301	D₈	—	2	1	no	—
302	D₈	C₂²×C₄×C₃₁	1	2	yes	—
303	D₈	C₂²×C₄×C₃₁	1	4	yes	—
304	M₁₆	—	2	2	no	—
305	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
306	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
307	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
308	C₂×C₈	#41×C₃₁	1	2	yes	—
309	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
316	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
317	C₂²⋊C₄	—	4	1	no	—
318	C₂²⋊C₄	—	4	2	no	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
322	C₄²	—	4	1	no	—
323	C₄²	—	4	2	no	—
324	C₄²	C₂×D₄×C₃₁	1	2	yes	—
325	C₁₆	—	2	1	no	—
326	C₁₆	—	2	1	no	—
327	C₁₆	C₂²×C₄×C₃₁	1	4	yes	—
328	C₂³	—	4	1	no	—
329	C₂³	—	4	2	no	—
330	C₂³	—	4	4	no	—
331	C₂³	—	16	1	no	—
332	C₂³	—	16	1	no	—
333	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
334	Q₈	—	2	2	no	—
335	Q₈	—	4	1	no	—
336	D₄	—	2	4	no	—
337	D₄	—	2	2	no	—
338	D₄	—	4	4	no	—
339	D₄	—	4	4	no	—
340	D₄	—	4	2	no	—
341	D₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	4	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	16	2	no	—
348	C₂×C₄	—	16	2	no	—
349	C₂×C₄	—	16	2	no	—
350	C₂×C₄	—	16	1	no	—
351	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
352	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
353	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
354	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
355	C₂×C₄	C₃₁×#135	1	2	yes	—
356	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
357	C₈	—	2	2	no	—
358	C₈	C₄×D₄×C₃₁	1	2	yes	—
359	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
360	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	4	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	8	1	no	—
369	C₂²	—	16	4	no	—
370	C₂²	C₄×D₈×C₃₁	1	2	yes	—
371	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	4	2	no	—
375	C₄	—	8	1	no	—
376	C₄	—	8	2	no	—
377	C₄	—	8	1	no	—
378	C₄	C₄×D₈×C₃₁	1	2	yes	—
379	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
380	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	8	4	no	—
383	C₂	—	8	2	no	—
384	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
385	C₂	C₃₁×#3415	1	1	yes	characteristic
386	C₂	C₃₁×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#509	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2951	C₂	semidirect
3. C₃₁×#2948	C₂	semidirect
4. C₃₁×#2177	C₂	semidirect
5. C₃₁×#2175	C₂	semidirect
6. C₄×C₃₁×SD₃₂	C₂	semidirect
7. C₄×C₃₁×D₁₆	C₂	semidirect
8. C₃₁×#2159	C₂	semidirect
9. C₃₁×#2138	C₂	semidirect
10. C₂×C₃₁×#357	C₂	semidirect
11. C₃₁×#2108	C₂	semidirect
16. C₃₁×#509	C₄	semidirect
23. C₄×Q₁₆×C₃₁	C₂²	semidirect
26. C₄×D₈×C₃₁	C₂²	semidirect
32. C₃₁×#367	C₂²	semidirect
33. C₃₁×#366	C₂²	semidirect
34. C₃₁×#358	C₂²	semidirect
35. C₃₁×#358	C₂²	semidirect
38. C₃₁×#357	C₂²	semidirect
39. C₃₁×#357	C₂²	semidirect
40. C₃₁×#346	C₂²	semidirect
41. C₄×C₁₆×C₃₁	C₂²	semidirect
45. C₃₁×#135	C₂×C₄	semidirect
50. C₂×D₈×C₃₁	C₂×C₄	semidirect
68. C₃₁×SD₃₂	C₂×C₄	semidirect
69. C₃₁×D₁₆	C₂×C₄	semidirect
70. C₃₁×M₃₂	C₂×C₄	semidirect
134. #33454	C₃₁	direct (N × H)
169. #2951	C₂×C₃₁	semidirect
170. #2948	C₂×C₃₁	semidirect
171. #2177	C₂×C₃₁	semidirect
172. #2175	C₂×C₃₁	semidirect
173. C₄×SD₃₂	C₂×C₃₁	semidirect
174. C₄×D₁₆	C₂×C₃₁	semidirect
175. #2159	C₂×C₃₁	semidirect
176. #2138	C₂×C₃₁	semidirect
177. C₂×#357	C₂×C₃₁	semidirect
178. #2108	C₂×C₃₁	semidirect
203. #509	C₄×C₃₁	semidirect
210. C₄×Q₁₆	C₂²×C₃₁	semidirect
213. C₄×D₈	C₂²×C₃₁	semidirect
219. #367	C₂²×C₃₁	semidirect
220. #366	C₂²×C₃₁	semidirect
221. #358	C₂²×C₃₁	semidirect
222. #358	C₂²×C₃₁	semidirect
225. #357	C₂²×C₃₁	semidirect
226. #357	C₂²×C₃₁	semidirect
227. #346	C₂²×C₃₁	semidirect
228. C₄×C₁₆	C₂²×C₃₁	semidirect
238. #135	C₂×C₄×C₃₁	semidirect
243. C₂×D₈	C₂×C₄×C₃₁	semidirect
261. SD₃₂	C₂×C₄×C₃₁	semidirect
262. D₁₆	C₂×C₄×C₃₁	semidirect
263. M₃₂	C₂×C₄×C₃₁	semidirect
279. C₃₁	#33454	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×#509	2
C₃₁×#2951	2
C₃₁×#2948	2
C₃₁×#2177	2
C₃₁×#2175	2
C₄×C₃₁×SD₃₂	2
C₄×C₃₁×D₁₆	2
C₃₁×#2159	2
C₃₁×#2138	2
C₂×C₃₁×#357	2
C₃₁×#2108	2
#33454	31

Subgroups — C31×#33454 #21490213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33454 #21490213