Subgroups — C₃₁×#33244 #21490003

Order 7936.

880 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#501	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#497	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#488	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2642	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2640	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1973	C₂	1	2	yes	maximal
8	C₃₁×#1971	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1969	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1968	C₂	1	2	yes	maximal
11	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
18	C₃₁×#501	—	2	4	no	—
19	C₃₁×#497	—	2	4	no	—
20	C₃₁×#488	—	2	4	no	—
21	C₃₁×#487	—	2	4	no	—
22	C₂×C₃₁×#107	C₂²	1	2	yes	—
23	C₂×C₃₁×#106	C₂²	1	2	yes	—
24	C₂×C₃₁×#103	C₂²	1	2	yes	—
25	C₂×C₃₁×#103	C₂²	1	2	yes	—
26	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#97	C₂²	1	2	yes	—
31	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#400	—	2	2	no	—
35	C₃₁×#400	—	2	2	no	—
36	C₃₁×#397	—	2	2	no	—
37	C₃₁×#393	—	2	2	no	—
38	C₃₁×#391	C₂²	1	2	yes	—
39	C₃₁×#391	C₂²	1	2	yes	—
40	C₃₁×#391	C₂²	1	2	yes	—
41	C₃₁×#390	C₂²	1	2	yes	—
42	C₃₁×#376	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×Q₁₆×C₃₁	—	2	4	no	—
48	C₂×Q₁₆×C₃₁	—	2	4	no	—
49	C₂×Q₁₆×C₃₁	—	2	4	no	—
50	C₂×SD₁₆×C₃₁	—	2	4	no	—
51	C₂×SD₁₆×C₃₁	—	2	4	no	—
52	C₂×SD₁₆×C₃₁	—	2	4	no	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×#125	—	2	2	no	—
61	C₃₁×#125	—	2	2	no	—
62	C₃₁×#124	—	2	2	no	—
63	C₃₁×#124	—	2	2	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
75	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#107	—	2	4	no	—
78	C₃₁×#106	—	2	4	no	—
79	C₃₁×#103	—	2	4	no	—
80	C₃₁×#103	—	2	4	no	—
81	C₃₁×#103	—	2	2	no	—
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#97	—	2	4	no	—
86	C₃₁×#97	D₄	1	4	yes	—
87	C₄×C₈×C₃₁	—	2	4	no	—
88	C₄×C₈×C₃₁	D₄	1	4	yes	—
89	C₃₁×#95	—	2	4	no	—
90	C₃₁×#95	—	2	4	no	—
91	C₃₁×#95	—	2	4	no	—
92	C₃₁×#95	D₄	1	2	yes	—
93	C₃₁×#95	D₄	1	2	yes	—
94	C₂⁴×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	2	1	no	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	2	2	no	—
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	4	2	no	—
106	C₂×Q₈×C₃₁	—	8	2	no	—
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	Q₁₆×C₃₁	—	4	8	no	—
121	SD₁₆×C₃₁	—	4	8	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
127	C₂×C₈×C₃₁	#41	1	4	yes	—
128	C₂×C₈×C₃₁	#41	1	4	yes	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	—	4	2	no	—
138	C₄⋊C₄×C₃₁	—	4	2	no	—
139	C₄⋊C₄×C₃₁	—	4	2	no	—
140	C₂²⋊C₄×C₃₁	—	4	2	no	—
141	C₂²⋊C₄×C₃₁	—	4	2	no	—
142	C₂²⋊C₄×C₃₁	—	8	4	no	—
143	C₄²×C₃₁	—	2	2	no	—
144	C₄²×C₃₁	—	2	2	no	—
145	C₄²×C₃₁	C₂×D₄	1	2	yes	—
146	C₄²×C₃₁	C₂×D₄	1	2	yes	—
147	#33244	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂³×C₃₁	—	4	1	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	—	4	2	no	—
152	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
153	Q₈×C₃₁	—	4	2	no	—
154	Q₈×C₃₁	—	4	2	no	—
155	Q₈×C₃₁	—	4	2	no	—
156	Q₈×C₃₁	—	4	1	no	—
157	Q₈×C₃₁	—	4	1	no	—
158	Q₈×C₃₁	—	4	2	no	—
159	Q₈×C₃₁	—	4	2	no	—
160	Q₈×C₃₁	—	8	4	no	—
161	D₄×C₃₁	—	4	2	no	—
162	D₄×C₃₁	—	4	1	no	—
163	D₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	4	no	—
167	C₂×C₄×C₃₁	—	2	4	no	—
168	C₂×C₄×C₃₁	—	2	4	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	4	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	#127	1	2	yes	—
181	C₂×C₄×C₃₁	#127	1	2	yes	—
182	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
185	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
186	C₈×C₃₁	—	2	2	no	—
187	C₈×C₃₁	—	2	2	no	—
188	C₈×C₃₁	#124	1	8	yes	—
189	C₂×#501	C₂×C₃₁	1	1	yes	characteristic, complemented
190	C₂×#497	C₂×C₃₁	1	1	yes	characteristic
191	C₂×#488	C₂×C₃₁	1	1	yes	characteristic
192	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#2642	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#2640	C₂×C₃₁	1	1	yes	characteristic
195	#1973	C₂×C₃₁	1	2	yes	—
196	#1971	C₂×C₃₁	1	2	yes	complemented
197	#1969	C₂×C₃₁	1	2	yes	complemented
198	#1968	C₂×C₃₁	1	2	yes	—
199	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
200	C₂²×C₃₁	—	8	1	no	—
201	C₂²×C₃₁	—	8	1	no	—
202	C₂²×C₃₁	—	8	1	no	—
203	C₂²×C₃₁	—	8	4	no	—
204	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
205	C₂²×C₃₁	#532	1	2	yes	—
206	C₂²×C₃₁	#577	1	2	yes	—
207	C₂²×C₃₁	#578	1	2	yes	—
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	—	4	4	no	—
211	C₄×C₃₁	—	8	1	no	—
212	C₄×C₃₁	—	8	2	no	—
213	C₄×C₃₁	—	8	1	no	—
214	C₄×C₃₁	—	8	2	no	—
215	C₄×C₃₁	—	8	1	no	—
216	C₄×C₃₁	C₂×#124	1	2	yes	—
217	C₄×C₃₁	#532	1	1	yes	characteristic
218	C₄×C₃₁	#532	1	1	yes	characteristic
219	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
220	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
221	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
225	#501	—	2	4	no	—
226	#497	—	2	4	no	—
227	#488	—	2	4	no	—
228	#487	—	2	4	no	—
229	C₂×#107	C₂²×C₃₁	1	2	yes	—
230	C₂×#106	C₂²×C₃₁	1	2	yes	—
231	C₂×#103	C₂²×C₃₁	1	2	yes	—
232	C₂×#103	C₂²×C₃₁	1	2	yes	—
233	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×#97	C₂²×C₃₁	1	2	yes	—
238	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
240	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
241	#400	—	2	2	no	—
242	#400	—	2	2	no	—
243	#397	—	2	2	no	—
244	#393	—	2	2	no	—
245	#391	C₂²×C₃₁	1	2	yes	—
246	#391	C₂²×C₃₁	1	2	yes	—
247	#391	C₂²×C₃₁	1	2	yes	—
248	#390	C₂²×C₃₁	1	2	yes	—
249	#376	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×C₃₁	—	16	1	no	—
251	C₂×C₃₁	#2858	1	1	yes	characteristic
252	C₂×C₃₁	#3154	1	1	yes	characteristic
253	C₂×C₃₁	#3156	1	1	yes	characteristic
254	C₂×C₃₁	#3178	1	4	yes	—
255	C₂²×Q₈	—	2	1	no	—
256	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
257	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
258	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
259	C₂×Q₁₆	—	2	4	no	—
260	C₂×Q₁₆	—	2	4	no	—
261	C₂×Q₁₆	—	2	4	no	—
262	C₂×SD₁₆	—	2	4	no	—
263	C₂×SD₁₆	—	2	4	no	—
264	C₂×SD₁₆	—	2	4	no	—
265	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
266	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
268	C₄⋊Q₈	—	2	2	no	—
269	C₄⋊Q₈	—	2	2	no	—
270	C₄⋊Q₈	—	2	2	no	—
271	C₄⋊Q₈	—	2	2	no	—
272	#125	—	2	2	no	—
273	#125	—	2	2	no	—
274	#124	—	2	2	no	—
275	#124	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
282	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
283	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
284	C₂×C₂²⋊C₄	—	2	1	no	—
285	C₂×C₂²⋊C₄	—	2	1	no	—
286	C₂×C₂²⋊C₄	—	2	4	no	—
287	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
288	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
289	#107	—	2	4	no	—
290	#106	—	2	4	no	—
291	#103	—	2	4	no	—
292	#103	—	2	4	no	—
293	#103	—	2	2	no	—
294	#103	—	2	2	no	—
295	#102	—	2	2	no	—
296	#102	—	2	2	no	—
297	#97	—	2	4	no	—
298	#97	D₄×C₃₁	1	4	yes	—
299	C₄×C₈	—	2	4	no	—
300	C₄×C₈	D₄×C₃₁	1	4	yes	—
301	#95	—	2	4	no	—
302	#95	—	2	4	no	—
303	#95	—	2	4	no	—
304	#95	D₄×C₃₁	1	2	yes	—
305	#95	D₄×C₃₁	1	2	yes	—
306	C₃₁	#33244	1	1	yes	characteristic, direct product factor, Sylow
307	C₂⁴	—	2	1	no	—
308	C₂×Q₈	—	2	1	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	1	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	2	2	no	—
318	C₂×Q₈	—	4	2	no	—
319	C₂×Q₈	—	8	2	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	2	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
330	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
331	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
333	Q₁₆	—	4	8	no	—
334	SD₁₆	—	4	8	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
338	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
339	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
340	C₂×C₈	#41×C₃₁	1	4	yes	—
341	C₂×C₈	#41×C₃₁	1	4	yes	—
342	C₄⋊C₄	—	2	1	no	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₄⋊C₄	—	4	2	no	—
350	C₄⋊C₄	—	4	2	no	—
351	C₄⋊C₄	—	4	2	no	—
352	C₄⋊C₄	—	4	2	no	—
353	C₂²⋊C₄	—	4	2	no	—
354	C₂²⋊C₄	—	4	2	no	—
355	C₂²⋊C₄	—	8	4	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	2	2	no	—
358	C₄²	C₂×D₄×C₃₁	1	2	yes	—
359	C₄²	C₂×D₄×C₃₁	1	2	yes	—
360	C₂³	—	4	1	no	—
361	C₂³	—	4	2	no	—
362	C₂³	—	4	2	no	—
363	C₂³	—	4	2	no	—
364	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
365	Q₈	—	4	2	no	—
366	Q₈	—	4	2	no	—
367	Q₈	—	4	2	no	—
368	Q₈	—	4	1	no	—
369	Q₈	—	4	1	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	8	4	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	1	no	—
375	D₄	—	4	1	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	4	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	4	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	C₃₁×#127	1	2	yes	—
393	C₂×C₄	C₃₁×#127	1	2	yes	—
394	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
397	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
398	C₈	—	2	2	no	—
399	C₈	—	2	2	no	—
400	C₈	C₃₁×#124	1	8	yes	—
401	C₂²	—	8	1	no	—
402	C₂²	—	8	1	no	—
403	C₂²	—	8	1	no	—
404	C₂²	—	8	4	no	—
405	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
406	C₂²	C₃₁×#532	1	2	yes	—
407	C₂²	C₃₁×#577	1	2	yes	—
408	C₂²	C₃₁×#578	1	2	yes	—
409	C₄	—	2	2	no	—
410	C₄	—	2	2	no	—
411	C₄	—	4	4	no	—
412	C₄	—	8	1	no	—
413	C₄	—	8	2	no	—
414	C₄	—	8	2	no	—
415	C₄	—	8	1	no	—
416	C₄	—	8	1	no	—
417	C₄	C₂×C₃₁×#124	1	2	yes	—
418	C₄	C₃₁×#532	1	1	yes	characteristic
419	C₄	C₃₁×#532	1	1	yes	characteristic
420	C₂	—	16	1	no	—
421	C₂	C₃₁×#2858	1	1	yes	characteristic
422	C₂	C₃₁×#3154	1	1	yes	characteristic
423	C₂	C₃₁×#3156	1	1	yes	characteristic
424	C₂	C₃₁×#3178	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#501	C₂	semidirect
4. C₂×C₃₁×#487	C₂	semidirect
5. C₃₁×#2642	C₂	semidirect
8. C₃₁×#1971	C₂	semidirect
9. C₃₁×#1969	C₂	semidirect
11. C₃₁×#1726	C₂	semidirect
147. #33244	C₃₁	direct (N × H)
189. C₂×#501	C₂×C₃₁	semidirect
192. C₂×#487	C₂×C₃₁	semidirect
193. #2642	C₂×C₃₁	semidirect
196. #1971	C₂×C₃₁	semidirect
197. #1969	C₂×C₃₁	semidirect
199. #1726	C₂×C₃₁	semidirect
306. C₃₁	#33244	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×#501	2
C₂×C₃₁×#497	2
C₂×C₃₁×#488	2
C₂×C₃₁×#487	2
C₃₁×#2642	2
C₃₁×#2640	2
C₃₁×#1973	2
C₃₁×#1971	2
C₃₁×#1969	2
C₃₁×#1968	2
C₃₁×#1726	2
#33244	31

Subgroups — C31×#33244 #21490003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33244 #21490003