Subgroups — C₃₁×#22182 #21478941

Order 7936.

894 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal
3	C₃₁×#1928	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1921	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1892	C₂	1	2	yes	maximal
6	C₃₁×#1878	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic
9	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#467	C₄	1	8	yes	complemented
17	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	C₂²	1	2	yes	—
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₃₁×#391	C₂²	1	2	yes	—
24	C₃₁×#389	C₂²	1	2	yes	—
25	C₃₁×#387	—	2	2	no	—
26	C₃₁×#385	—	2	2	no	—
27	C₃₁×#380	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
30	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
37	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×Q₁₆×C₃₁	—	4	1	no	—
41	C₂×Q₁₆×C₃₁	—	4	1	no	—
42	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
43	C₂²×C₈×C₃₁	—	2	2	no	—
44	C₂²×C₈×C₃₁	—	4	1	no	—
45	C₂²×C₈×C₃₁	—	4	1	no	—
46	C₂²×C₈×C₃₁	—	4	1	no	—
47	C₂²×C₈×C₃₁	—	4	1	no	—
48	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×C₂²⋊Q₈	—	2	4	no	—
53	C₃₁×C₂²⋊Q₈	C₂×C₄	1	8	yes	—
54	C₄×Q₈×C₃₁	—	4	2	no	—
55	C₄×Q₈×C₃₁	—	4	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	—	2	2	no	—
65	C₂×C₄²×C₃₁	—	2	2	no	—
66	C₂×C₄²×C₃₁	—	4	1	no	—
67	C₂×C₄²×C₃₁	—	4	1	no	—
68	C₂×C₄²×C₃₁	D₄	1	2	yes	—
69	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
70	C₃₁×#107	—	4	1	no	—
71	C₃₁×#107	—	4	1	no	—
72	C₃₁×#107	C₂×C₄	1	4	yes	complemented
73	C₃₁×#103	—	4	2	no	—
74	C₃₁×#103	—	4	2	no	—
75	C₃₁×#103	C₂×C₄	1	8	yes	complemented
76	C₃₁×#95	—	2	2	no	—
77	C₃₁×#95	—	2	2	no	—
78	C₃₁×#95	—	2	2	no	—
79	C₃₁×#95	—	2	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×Q₈×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×Q₈×C₃₁	—	2	2	no	—
87	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
88	C₂²×C₄×C₃₁	—	2	1	no	—
89	C₂²×C₄×C₃₁	—	2	1	no	—
90	C₂²×C₄×C₃₁	—	2	1	no	—
91	C₂²×C₄×C₃₁	—	2	1	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	4	1	no	—
98	C₂²×C₄×C₃₁	—	4	1	no	—
99	C₂²×C₄×C₃₁	—	4	1	no	—
100	C₂²×C₄×C₃₁	—	4	1	no	—
101	C₂²×C₄×C₃₁	Q₁₆	1	4	yes	complemented
102	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₃₁	#41	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
108	Q₁₆×C₃₁	—	4	4	no	—
109	C₂×C₈×C₃₁	—	2	1	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	4	4	no	—
117	C₂×C₈×C₃₁	—	4	4	no	—
118	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
120	C₄⋊C₄×C₃₁	—	2	8	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	4	4	no	—
127	C₄⋊C₄×C₃₁	—	4	4	no	—
128	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
129	C₂²⋊C₄×C₃₁	—	2	8	no	—
130	C₂²⋊C₄×C₃₁	—	4	4	no	—
131	C₄²×C₃₁	—	2	2	no	—
132	C₄²×C₃₁	—	2	2	no	—
133	C₄²×C₃₁	—	4	2	no	—
134	C₄²×C₃₁	—	4	4	no	—
135	C₄²×C₃₁	—	4	4	no	—
136	#22182	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₃₁	—	2	1	no	—
138	C₂³×C₃₁	—	2	1	no	—
139	C₂³×C₃₁	—	2	1	no	—
140	C₂³×C₃₁	—	2	1	no	—
141	C₂³×C₃₁	C₄×D₄	1	2	yes	—
142	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
143	C₂³×C₃₁	#135	1	2	yes	—
144	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
145	Q₈×C₃₁	—	2	4	no	—
146	Q₈×C₃₁	—	2	4	no	—
147	Q₈×C₃₁	—	4	2	no	—
148	Q₈×C₃₁	—	4	2	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	4	no	—
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	2	no	—
162	C₂×C₄×C₃₁	—	4	4	no	—
163	C₂×C₄×C₃₁	—	4	4	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
175	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
176	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
179	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
180	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
181	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
182	C₈×C₃₁	—	2	2	no	—
183	C₈×C₃₁	—	2	2	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₂×#467	C₂×C₃₁	1	1	yes	characteristic
186	C₄×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	—
187	#1928	C₂×C₃₁	1	2	yes	complemented
188	#1921	C₂×C₃₁	1	2	yes	complemented
189	#1892	C₂×C₃₁	1	2	yes	—
190	#1878	C₂×C₃₁	1	2	yes	complemented
191	#1863	C₂×C₃₁	1	2	yes	complemented
192	#1808	C₂×C₃₁	1	1	yes	characteristic
193	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
194	C₂²×C₃₁	—	2	2	no	—
195	C₂²×C₃₁	—	2	2	no	—
196	C₂²×C₃₁	—	2	2	no	—
197	C₂²×C₃₁	—	2	2	no	—
198	C₂²×C₃₁	—	2	2	no	—
199	C₂²×C₃₁	—	2	2	no	—
200	C₂²×C₃₁	C₄×Q₁₆	1	4	yes	—
201	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂²×C₃₁	#534	1	1	yes	characteristic
203	C₂²×C₃₁	#537	1	1	yes	characteristic
204	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
205	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
206	C₂²×C₃₁	#575	1	1	yes	characteristic
207	C₂²×C₃₁	#575	1	1	yes	characteristic
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	4	no	—
210	C₄×C₃₁	—	4	4	no	—
211	C₄×C₃₁	—	4	2	no	—
212	C₄×C₃₁	—	4	4	no	—
213	C₄×C₃₁	—	8	2	no	—
214	C₄×C₃₁	—	8	2	no	—
215	C₄×C₃₁	#518	1	1	yes	characteristic
216	C₄×C₃₁	#520	1	1	yes	characteristic
217	C₄×C₃₁	#538	1	1	yes	characteristic
218	C₄×C₃₁	#539	1	1	yes	characteristic
219	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
220	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
221	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
223	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
224	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
225	#467	C₄×C₃₁	1	8	yes	complemented
226	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×#103	C₂²×C₃₁	1	2	yes	—
230	C₂×#103	C₂²×C₃₁	1	2	yes	—
231	C₂×#103	C₂²×C₃₁	1	2	yes	—
232	#391	C₂²×C₃₁	1	2	yes	—
233	#389	C₂²×C₃₁	1	2	yes	—
234	#387	—	2	2	no	—
235	#385	—	2	2	no	—
236	#380	C₂²×C₃₁	1	2	yes	—
237	C₄×C₄⋊C₄	—	2	2	no	—
238	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
239	C₄×C₂²⋊C₄	—	2	2	no	—
240	#340	C₂²×C₃₁	1	2	yes	—
241	#340	C₂²×C₃₁	1	2	yes	—
242	#340	C₂²×C₃₁	1	2	yes	—
243	#336	C₂²×C₃₁	1	2	yes	—
244	#336	C₂²×C₃₁	1	2	yes	—
245	C₂×C₃₁	—	2	4	no	—
246	C₂×C₃₁	#2325	1	1	yes	characteristic
247	C₂×C₃₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
248	C₂×C₃₁	#2943	1	1	yes	characteristic
249	C₂×C₃₁	#3189	1	1	yes	characteristic
250	C₂×C₃₁	#3191	1	1	yes	characteristic
251	C₂×C₃₁	#3249	1	1	yes	characteristic
252	C₂×C₃₁	#3250	1	1	yes	characteristic
253	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
254	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
255	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
256	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
257	C₂×Q₁₆	—	4	1	no	—
258	C₂×Q₁₆	—	4	1	no	—
259	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
260	C₂²×C₈	—	2	2	no	—
261	C₂²×C₈	—	4	1	no	—
262	C₂²×C₈	—	4	1	no	—
263	C₂²×C₈	—	4	1	no	—
264	C₂²×C₈	—	4	1	no	—
265	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
266	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
267	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
269	C₂²⋊Q₈	—	2	4	no	—
270	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
271	C₄×Q₈	—	4	2	no	—
272	C₄×Q₈	—	4	2	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
277	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
278	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
279	C₂×C₂²⋊C₄	—	2	2	no	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₄²	—	2	2	no	—
282	C₂×C₄²	—	2	2	no	—
283	C₂×C₄²	—	4	1	no	—
284	C₂×C₄²	—	4	1	no	—
285	C₂×C₄²	D₄×C₃₁	1	2	yes	—
286	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
287	#107	—	4	1	no	—
288	#107	—	4	1	no	—
289	#107	C₂×C₄×C₃₁	1	4	yes	complemented
290	#103	—	4	2	no	—
291	#103	—	4	2	no	—
292	#103	C₂×C₄×C₃₁	1	8	yes	complemented
293	#95	—	2	2	no	—
294	#95	—	2	2	no	—
295	#95	—	2	2	no	—
296	#95	—	2	2	no	—
297	#95	—	2	2	no	—
298	C₃₁	#22182	1	1	yes	characteristic, direct product factor, Sylow
299	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	4	1	no	—
316	C₂²×C₄	—	4	1	no	—
317	C₂²×C₄	—	4	1	no	—
318	C₂²×C₄	—	4	1	no	—
319	C₂²×C₄	Q₁₆×C₃₁	1	4	yes	complemented
320	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
321	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
324	C₂²×C₄	#41×C₃₁	1	2	yes	—
325	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
326	Q₁₆	—	4	4	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	4	4	no	—
335	C₂×C₈	—	4	4	no	—
336	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
337	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
338	C₄⋊C₄	—	2	8	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	4	4	no	—
345	C₄⋊C₄	—	4	4	no	—
346	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
347	C₂²⋊C₄	—	2	8	no	—
348	C₂²⋊C₄	—	4	4	no	—
349	C₄²	—	2	2	no	—
350	C₄²	—	2	2	no	—
351	C₄²	—	4	2	no	—
352	C₄²	—	4	4	no	—
353	C₄²	—	4	4	no	—
354	C₂³	—	2	1	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	2	1	no	—
357	C₂³	—	2	1	no	—
358	C₂³	C₄×D₄×C₃₁	1	2	yes	—
359	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
360	C₂³	C₃₁×#135	1	2	yes	—
361	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
362	Q₈	—	2	4	no	—
363	Q₈	—	2	4	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	4	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	4	no	—
380	C₂×C₄	—	4	4	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	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	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
392	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
393	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
394	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
396	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
397	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
398	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
399	C₈	—	2	2	no	—
400	C₈	—	2	2	no	—
401	C₈	—	4	2	no	—
402	C₂²	—	2	2	no	—
403	C₂²	—	2	2	no	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	2	2	no	—
408	C₂²	C₄×Q₁₆×C₃₁	1	4	yes	—
409	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
410	C₂²	C₃₁×#534	1	1	yes	characteristic
411	C₂²	C₃₁×#537	1	1	yes	characteristic
412	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
413	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
414	C₂²	C₃₁×#575	1	1	yes	characteristic
415	C₂²	C₃₁×#575	1	1	yes	characteristic
416	C₄	—	2	2	no	—
417	C₄	—	2	4	no	—
418	C₄	—	4	4	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	4	no	—
421	C₄	—	8	2	no	—
422	C₄	—	8	2	no	—
423	C₄	C₃₁×#518	1	1	yes	characteristic
424	C₄	C₃₁×#520	1	1	yes	characteristic
425	C₄	C₃₁×#538	1	1	yes	characteristic
426	C₄	C₃₁×#539	1	1	yes	characteristic
427	C₂	—	2	4	no	—
428	C₂	C₃₁×#2325	1	1	yes	characteristic
429	C₂	C₂×C₄×Q₁₆×C₃₁	1	1	yes	characteristic
430	C₂	C₃₁×#2943	1	1	yes	characteristic
431	C₂	C₃₁×#3189	1	1	yes	characteristic
432	C₂	C₃₁×#3191	1	1	yes	characteristic
433	C₂	C₃₁×#3249	1	1	yes	characteristic
434	C₂	C₃₁×#3250	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1928	C₂	semidirect
4. C₃₁×#1921	C₂	semidirect
6. C₃₁×#1878	C₂	semidirect
7. C₃₁×#1863	C₂	semidirect
16. C₃₁×#467	C₄	semidirect
42. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
72. C₃₁×#107	C₂×C₄	semidirect
75. C₃₁×#103	C₂×C₄	semidirect
101. C₂²×C₄×C₃₁	Q₁₆	semidirect
136. #22182	C₃₁	direct (N × H)
187. #1928	C₂×C₃₁	semidirect
188. #1921	C₂×C₃₁	semidirect
190. #1878	C₂×C₃₁	semidirect
191. #1863	C₂×C₃₁	semidirect
225. #467	C₄×C₃₁	semidirect
259. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
289. #107	C₂×C₄×C₃₁	semidirect
292. #103	C₂×C₄×C₃₁	semidirect
298. C₃₁	#22182	direct (N × H)
319. C₂²×C₄	Q₁₆×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×#467	2
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#1928	2
C₃₁×#1921	2
C₃₁×#1892	2
C₃₁×#1878	2
C₃₁×#1863	2
C₃₁×#1808	2
C₂×C₃₁×#336	2
#22182	31

Subgroups — C31×#22182 #21478941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22182 #21478941