Subgroups — C₃₁×#20512 #21477271

Order 7936.

1014 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#444	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#443	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2943	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2875	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1778	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1773	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1765	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1764	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×#104	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	C₂×C₃₁×#131	C₂²	1	2	yes	—
15	C₄×#41×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#444	C₄	1	8	yes	complemented
17	C₃₁×#443	C₄	1	8	yes	complemented
18	C₃₁×#442	C₄	1	2	yes	complemented
19	C₃₁×#441	C₄	1	2	yes	complemented
20	C₃₁×#440	C₄	1	4	yes	complemented
21	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
23	C₃₁×#427	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#104	C₂²	1	2	yes	—
25	C₂×C₃₁×#104	C₂²	1	2	yes	—
26	C₃₁×#416	C₂²	1	2	yes	—
27	C₃₁×#405	—	2	2	no	—
28	C₃₁×#405	—	2	2	no	—
29	C₃₁×#405	C₂²	1	2	yes	—
30	C₃₁×#405	C₂²	1	2	yes	—
31	C₄×M₁₆×C₃₁	—	2	2	no	—
32	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
33	C₂×C₃₁×#97	—	2	2	no	—
34	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₄³×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#341	C₂²	1	2	yes	—
42	C₃₁×#339	C₂²	1	2	yes	—
43	C₃₁×#339	C₂²	1	2	yes	—
44	C₂×#41×C₃₁	C₂³	1	2	yes	—
45	C₃₁×#135	C₂×C₄	1	4	yes	—
46	C₃₁×#135	C₂×C₄	1	4	yes	—
47	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
49	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₃₁×#131	—	2	4	no	—
51	C₃₁×#131	—	2	4	no	—
52	C₃₁×#131	C₂×C₄	1	8	yes	complemented
53	C₃₁×#131	C₂×C₄	1	8	yes	complemented
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
57	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
65	C₄×D₄×C₃₁	—	2	4	no	—
66	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
67	C₃₁×#117	—	2	2	no	—
68	C₃₁×#117	C₂³	1	2	yes	—
69	C₂×C₄²×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	—	2	2	no	—
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#108	C₂×C₄	1	4	yes	complemented
75	C₃₁×#108	C₂×C₄	1	4	yes	complemented
76	C₃₁×#107	C₂×C₄	1	2	yes	—
77	C₃₁×#106	C₂×C₄	1	2	yes	—
78	C₃₁×#104	C₂×C₄	1	8	yes	complemented
79	C₃₁×#104	C₂×C₄	1	8	yes	complemented
80	C₃₁×#103	C₂×C₄	1	4	yes	—
81	C₃₁×#102	C₂×C₄	1	4	yes	—
82	C₃₁×#97	—	2	2	no	—
83	C₃₁×#97	—	2	2	no	—
84	C₃₁×#97	—	2	2	no	—
85	C₃₁×#97	—	2	2	no	—
86	C₃₁×#97	C₂×C₄	1	2	yes	—
87	C₃₁×#97	C₂×C₄	1	2	yes	—
88	C₃₁×#97	C₂×C₄	1	4	yes	complemented
89	C₃₁×#97	D₄	1	2	yes	—
90	C₃₁×#97	D₄	1	2	yes	—
91	C₄×C₈×C₃₁	—	2	2	no	—
92	C₄×C₈×C₃₁	—	2	2	no	—
93	C₄×C₈×C₃₁	—	2	2	no	—
94	C₄×C₈×C₃₁	—	2	2	no	—
95	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
96	#41×C₃₁	—	2	2	no	—
97	#41×C₃₁	—	2	2	no	—
98	#41×C₃₁	C₂²×C₄	1	4	yes	—
99	#41×C₃₁	C₂²×C₄	1	4	yes	—
100	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	1	no	—
106	C₂²×C₄×C₃₁	—	2	4	no	—
107	C₂²×C₄×C₃₁	#41	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
109	Q₁₆×C₃₁	C₄²	1	4	yes	complemented
110	SD₁₆×C₃₁	C₄²	1	8	yes	complemented
111	D₈×C₃₁	C₄²	1	4	yes	complemented
112	M₁₆×C₃₁	—	2	2	no	—
113	M₁₆×C₃₁	—	2	2	no	—
114	M₁₆×C₃₁	—	2	8	no	—
115	M₁₆×C₃₁	—	2	8	no	—
116	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
117	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	8	no	—
123	C₂×C₈×C₃₁	—	2	8	no	—
124	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₃₁	#41	1	2	yes	—
135	C₂×C₈×C₃₁	#41	1	2	yes	—
136	C₄⋊C₄×C₃₁	—	2	4	no	—
137	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
138	C₂²⋊C₄×C₃₁	—	2	2	no	—
139	C₂²⋊C₄×C₃₁	—	2	4	no	—
140	C₄²×C₃₁	—	2	4	no	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	4	4	no	—
145	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
146	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
147	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
148	C₄²×C₃₁	C₂×D₄	1	2	yes	—
149	#20512	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	2	no	—
151	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
152	Q₈×C₃₁	—	2	2	no	—
153	Q₈×C₃₁	C₂×C₄²	1	4	yes	—
154	D₄×C₃₁	—	2	4	no	—
155	D₄×C₃₁	—	2	4	no	—
156	D₄×C₃₁	—	2	2	no	—
157	D₄×C₃₁	C₂×C₄²	1	4	yes	—
158	C₂×C₄×C₃₁	—	2	4	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	4	no	—
162	C₂×C₄×C₃₁	—	2	4	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	8	4	no	—
168	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
169	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
172	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
175	C₈×C₃₁	—	2	4	no	—
176	C₈×C₃₁	—	2	4	no	—
177	C₈×C₃₁	—	2	2	no	—
178	C₈×C₃₁	C₂×C₄²	1	2	yes	—
179	C₈×C₃₁	C₂×C₄²	1	2	yes	—
180	C₈×C₃₁	C₄×D₄	1	2	yes	—
181	C₈×C₃₁	C₄×D₄	1	4	yes	—
182	C₈×C₃₁	C₄×D₄	1	2	yes	—
183	C₂×#444	C₂×C₃₁	1	1	yes	characteristic
184	C₂×#443	C₂×C₃₁	1	1	yes	characteristic
185	#2943	C₂×C₃₁	1	1	yes	characteristic
186	#2875	C₂×C₃₁	1	2	yes	complemented
187	#1778	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#1773	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#1772	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1765	C₂×C₃₁	1	2	yes	complemented
191	#1764	C₂×C₃₁	1	2	yes	complemented
192	C₄×#104	C₂×C₃₁	1	2	yes	complemented
193	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
194	C₂²×C₃₁	—	2	2	no	—
195	C₂²×C₃₁	—	2	4	no	—
196	C₂²×C₃₁	—	4	4	no	—
197	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
198	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
199	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
200	C₄×C₃₁	—	2	4	no	—
201	C₄×C₃₁	—	2	4	no	—
202	C₄×C₃₁	—	2	4	no	—
203	C₄×C₃₁	—	4	2	no	—
204	C₄×C₃₁	—	4	4	no	—
205	C₄×C₃₁	C₂²×C₄²	1	1	yes	characteristic
206	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
207	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
208	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
209	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
210	C₂×#131	C₂²×C₃₁	1	2	yes	—
211	C₂×#131	C₂²×C₃₁	1	2	yes	—
212	C₄×#41	C₂²×C₃₁	1	2	yes	—
213	#444	C₄×C₃₁	1	8	yes	complemented
214	#443	C₄×C₃₁	1	8	yes	complemented
215	#442	C₄×C₃₁	1	2	yes	complemented
216	#441	C₄×C₃₁	1	2	yes	complemented
217	#440	C₄×C₃₁	1	4	yes	complemented
218	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
220	#427	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×#104	C₂²×C₃₁	1	2	yes	—
222	C₂×#104	C₂²×C₃₁	1	2	yes	—
223	#416	C₂²×C₃₁	1	2	yes	—
224	#405	—	2	2	no	—
225	#405	—	2	2	no	—
226	#405	C₂²×C₃₁	1	2	yes	—
227	#405	C₂²×C₃₁	1	2	yes	—
228	C₄×M₁₆	—	2	2	no	—
229	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
230	C₂×#97	—	2	2	no	—
231	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₄³	C₂²×C₃₁	1	1	yes	characteristic
238	#341	C₂²×C₃₁	1	2	yes	—
239	#339	C₂²×C₃₁	1	2	yes	—
240	#339	C₂²×C₃₁	1	2	yes	—
241	C₂×C₃₁	—	2	1	no	—
242	C₂×C₃₁	—	2	1	no	—
243	C₂×C₃₁	—	4	4	no	—
244	C₂×C₃₁	C₄²×D₄	1	1	yes	characteristic
245	C₂×C₃₁	#2956	1	2	yes	—
246	C₂×#41	C₂³×C₃₁	1	2	yes	—
247	#135	C₂×C₄×C₃₁	1	4	yes	—
248	#135	C₂×C₄×C₃₁	1	4	yes	—
249	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
250	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
251	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
252	#131	—	2	4	no	—
253	#131	—	2	4	no	—
254	#131	C₂×C₄×C₃₁	1	8	yes	complemented
255	#131	C₂×C₄×C₃₁	1	8	yes	complemented
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	—	2	2	no	—
258	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
259	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
260	C₂²×C₈	—	2	2	no	—
261	C₂²×C₈	—	2	2	no	—
262	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
265	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
266	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
267	C₄×D₄	—	2	4	no	—
268	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
269	#117	—	2	2	no	—
270	#117	C₂³×C₃₁	1	2	yes	—
271	C₂×C₄²	—	2	2	no	—
272	C₂×C₄²	—	2	2	no	—
273	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
276	#108	C₂×C₄×C₃₁	1	4	yes	complemented
277	#108	C₂×C₄×C₃₁	1	4	yes	complemented
278	#107	C₂×C₄×C₃₁	1	2	yes	—
279	#106	C₂×C₄×C₃₁	1	2	yes	—
280	#104	C₂×C₄×C₃₁	1	8	yes	complemented
281	#104	C₂×C₄×C₃₁	1	8	yes	complemented
282	#103	C₂×C₄×C₃₁	1	4	yes	—
283	#102	C₂×C₄×C₃₁	1	4	yes	—
284	#97	—	2	2	no	—
285	#97	—	2	2	no	—
286	#97	—	2	2	no	—
287	#97	—	2	2	no	—
288	#97	C₂×C₄×C₃₁	1	2	yes	—
289	#97	C₂×C₄×C₃₁	1	2	yes	—
290	#97	C₂×C₄×C₃₁	1	4	yes	complemented
291	#97	D₄×C₃₁	1	2	yes	—
292	#97	D₄×C₃₁	1	2	yes	—
293	C₄×C₈	—	2	2	no	—
294	C₄×C₈	—	2	2	no	—
295	C₄×C₈	—	2	2	no	—
296	C₄×C₈	—	2	2	no	—
297	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
298	C₃₁	#20512	1	1	yes	characteristic, direct product factor, Sylow
299	#41	—	2	2	no	—
300	#41	—	2	2	no	—
301	#41	C₂²×C₄×C₃₁	1	4	yes	—
302	#41	C₂²×C₄×C₃₁	1	4	yes	—
303	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	4	no	—
310	C₂²×C₄	#41×C₃₁	1	2	yes	—
311	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
312	Q₁₆	C₄²×C₃₁	1	4	yes	complemented
313	SD₁₆	C₄²×C₃₁	1	8	yes	complemented
314	D₈	C₄²×C₃₁	1	4	yes	complemented
315	M₁₆	—	2	2	no	—
316	M₁₆	—	2	2	no	—
317	M₁₆	—	2	8	no	—
318	M₁₆	—	2	8	no	—
319	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
320	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	8	no	—
326	C₂×C₈	—	2	8	no	—
327	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
328	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
329	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
330	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
331	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
332	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
333	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
334	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
336	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₂×C₈	#41×C₃₁	1	2	yes	—
338	C₂×C₈	#41×C₃₁	1	2	yes	—
339	C₄⋊C₄	—	2	4	no	—
340	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
341	C₂²⋊C₄	—	2	2	no	—
342	C₂²⋊C₄	—	2	4	no	—
343	C₄²	—	2	4	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	2	1	no	—
346	C₄²	—	2	1	no	—
347	C₄²	—	4	4	no	—
348	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
349	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
350	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
351	C₄²	C₂×D₄×C₃₁	1	2	yes	—
352	C₂³	—	2	2	no	—
353	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
354	Q₈	—	2	2	no	—
355	Q₈	C₂×C₄²×C₃₁	1	4	yes	—
356	D₄	—	2	4	no	—
357	D₄	—	2	4	no	—
358	D₄	—	2	2	no	—
359	D₄	C₂×C₄²×C₃₁	1	4	yes	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	8	4	no	—
370	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
371	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
372	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
373	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
375	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
376	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
377	C₈	—	2	4	no	—
378	C₈	—	2	4	no	—
379	C₈	—	2	2	no	—
380	C₈	C₂×C₄²×C₃₁	1	2	yes	—
381	C₈	C₂×C₄²×C₃₁	1	2	yes	—
382	C₈	C₄×D₄×C₃₁	1	2	yes	—
383	C₈	C₄×D₄×C₃₁	1	4	yes	—
384	C₈	C₄×D₄×C₃₁	1	2	yes	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	4	no	—
387	C₂²	—	4	4	no	—
388	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
389	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
390	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
391	C₄	—	2	4	no	—
392	C₄	—	2	4	no	—
393	C₄	—	2	4	no	—
394	C₄	—	4	2	no	—
395	C₄	—	4	4	no	—
396	C₄	C₂²×C₄²×C₃₁	1	1	yes	characteristic
397	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
398	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
399	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	4	4	no	—
403	C₂	C₄²×D₄×C₃₁	1	1	yes	characteristic
404	C₂	C₃₁×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#2875	C₂	semidirect
5. C₃₁×#1778	C₂	semidirect
6. C₃₁×#1773	C₂	semidirect
7. C₃₁×#1772	C₂	semidirect
8. C₃₁×#1765	C₂	semidirect
9. C₃₁×#1764	C₂	semidirect
10. C₄×C₃₁×#104	C₂	semidirect
11. C₄×C₃₁×#97	C₂	semidirect
16. C₃₁×#444	C₄	semidirect
17. C₃₁×#443	C₄	semidirect
18. C₃₁×#442	C₄	semidirect
19. C₃₁×#441	C₄	semidirect
20. C₃₁×#440	C₄	semidirect
52. C₃₁×#131	C₂×C₄	semidirect
53. C₃₁×#131	C₂×C₄	semidirect
74. C₃₁×#108	C₂×C₄	semidirect
75. C₃₁×#108	C₂×C₄	semidirect
78. C₃₁×#104	C₂×C₄	semidirect
79. C₃₁×#104	C₂×C₄	semidirect
88. C₃₁×#97	C₂×C₄	semidirect
95. C₄×C₈×C₃₁	C₂×C₄	semidirect
109. Q₁₆×C₃₁	C₄²	semidirect
110. SD₁₆×C₃₁	C₄²	semidirect
111. D₈×C₃₁	C₄²	semidirect
149. #20512	C₃₁	direct (N × H)
186. #2875	C₂×C₃₁	semidirect
187. #1778	C₂×C₃₁	semidirect
188. #1773	C₂×C₃₁	semidirect
189. #1772	C₂×C₃₁	semidirect
190. #1765	C₂×C₃₁	semidirect
191. #1764	C₂×C₃₁	semidirect
192. C₄×#104	C₂×C₃₁	semidirect
193. C₄×#97	C₂×C₃₁	semidirect
213. #444	C₄×C₃₁	semidirect
214. #443	C₄×C₃₁	semidirect
215. #442	C₄×C₃₁	semidirect
216. #441	C₄×C₃₁	semidirect
217. #440	C₄×C₃₁	semidirect
254. #131	C₂×C₄×C₃₁	semidirect
255. #131	C₂×C₄×C₃₁	semidirect
276. #108	C₂×C₄×C₃₁	semidirect
277. #108	C₂×C₄×C₃₁	semidirect
280. #104	C₂×C₄×C₃₁	semidirect
281. #104	C₂×C₄×C₃₁	semidirect
290. #97	C₂×C₄×C₃₁	semidirect
297. C₄×C₈	C₂×C₄×C₃₁	semidirect
298. C₃₁	#20512	direct (N × H)
312. Q₁₆	C₄²×C₃₁	semidirect
313. SD₁₆	C₄²×C₃₁	semidirect
314. D₈	C₄²×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#444	2
C₂×C₃₁×#443	2
C₃₁×#2943	2
C₃₁×#2875	2
C₃₁×#1778	2
C₃₁×#1773	2
C₃₁×#1772	2
C₃₁×#1765	2
C₃₁×#1764	2
C₄×C₃₁×#104	2
C₄×C₃₁×#97	2
#20512	31

Subgroups — C31×#20512 #21477271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20512 #21477271