Subgroups — C₃₁×#22188 #21478947

Order 7936.

870 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#470	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal
3	C₃₁×#1927	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1920	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1894	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×#1809	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×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×#470	C₄	1	8	yes	complemented
17	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#106	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	C₂×M₁₆×C₃₁	—	2	2	no	—
44	C₂×M₁₆×C₃₁	—	4	1	no	—
45	C₂×M₁₆×C₃₁	—	4	1	no	—
46	C₂×M₁₆×C₃₁	—	4	1	no	—
47	C₂×M₁₆×C₃₁	—	4	1	no	—
48	C₂×M₁₆×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×#106	—	4	1	no	—
72	C₃₁×#106	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	4	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	2	no	—
100	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	#41	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
106	Q₁₆×C₃₁	—	4	4	no	—
107	M₁₆×C₃₁	—	4	4	no	—
108	M₁₆×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	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
118	C₄⋊C₄×C₃₁	—	2	8	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	2	2	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	—	4	4	no	—
125	C₄⋊C₄×C₃₁	—	4	4	no	—
126	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
127	C₂²⋊C₄×C₃₁	—	2	8	no	—
128	C₂²⋊C₄×C₃₁	—	4	4	no	—
129	C₄²×C₃₁	—	2	2	no	—
130	C₄²×C₃₁	—	2	2	no	—
131	C₄²×C₃₁	—	4	2	no	—
132	C₄²×C₃₁	—	4	4	no	—
133	C₄²×C₃₁	—	4	4	no	—
134	#22188	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂³×C₃₁	—	2	1	no	—
136	C₂³×C₃₁	—	2	1	no	—
137	C₂³×C₃₁	—	2	2	no	—
138	C₂³×C₃₁	—	2	2	no	—
139	C₂³×C₃₁	C₄×D₄	1	2	yes	—
140	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
141	Q₈×C₃₁	—	2	4	no	—
142	Q₈×C₃₁	—	2	4	no	—
143	Q₈×C₃₁	—	4	2	no	—
144	Q₈×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	4	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	4	4	no	—
159	C₂×C₄×C₃₁	—	4	4	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	2	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	C₄×D₄	1	2	yes	—
171	C₂×C₄×C₃₁	#137	1	4	yes	—
172	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
175	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
176	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
177	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
178	C₈×C₃₁	—	2	2	no	—
179	C₈×C₃₁	—	2	2	no	—
180	C₈×C₃₁	—	4	2	no	—
181	C₂×#470	C₂×C₃₁	1	1	yes	characteristic
182	C₄×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	—
183	#1927	C₂×C₃₁	1	2	yes	complemented
184	#1920	C₂×C₃₁	1	2	yes	complemented
185	#1894	C₂×C₃₁	1	2	yes	—
186	#1878	C₂×C₃₁	1	2	yes	complemented
187	#1863	C₂×C₃₁	1	2	yes	complemented
188	#1809	C₂×C₃₁	1	1	yes	characteristic
189	#1728	C₂×C₃₁	1	1	yes	characteristic
190	C₂²×C₃₁	—	2	2	no	—
191	C₂²×C₃₁	—	2	4	no	—
192	C₂²×C₃₁	—	4	1	no	—
193	C₂²×C₃₁	—	4	1	no	—
194	C₂²×C₃₁	—	4	2	no	—
195	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂²×C₃₁	#534	1	1	yes	characteristic
197	C₂²×C₃₁	#537	1	1	yes	characteristic
198	C₂²×C₃₁	C₂×#137	1	2	yes	—
199	C₂²×C₃₁	#575	1	2	yes	—
200	C₄×C₃₁	—	2	2	no	—
201	C₄×C₃₁	—	2	4	no	—
202	C₄×C₃₁	—	4	4	no	—
203	C₄×C₃₁	—	4	2	no	—
204	C₄×C₃₁	—	4	4	no	—
205	C₄×C₃₁	—	8	2	no	—
206	C₄×C₃₁	—	8	2	no	—
207	C₄×C₃₁	#518	1	1	yes	characteristic
208	C₄×C₃₁	#520	1	1	yes	characteristic
209	C₄×C₃₁	#538	1	1	yes	characteristic
210	C₄×C₃₁	#539	1	1	yes	characteristic
211	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
212	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
213	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
214	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
215	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
216	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
217	#470	C₄×C₃₁	1	8	yes	complemented
218	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
219	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×#103	C₂²×C₃₁	1	2	yes	—
222	C₂×#103	C₂²×C₃₁	1	2	yes	—
223	C₂×#103	C₂²×C₃₁	1	2	yes	—
224	#391	C₂²×C₃₁	1	2	yes	—
225	#389	C₂²×C₃₁	1	2	yes	—
226	#387	—	2	2	no	—
227	#385	—	2	2	no	—
228	#380	C₂²×C₃₁	1	2	yes	—
229	C₄×C₄⋊C₄	—	2	2	no	—
230	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
231	C₄×C₂²⋊C₄	—	2	2	no	—
232	#340	C₂²×C₃₁	1	2	yes	—
233	#340	C₂²×C₃₁	1	2	yes	—
234	#340	C₂²×C₃₁	1	2	yes	—
235	#336	C₂²×C₃₁	1	2	yes	—
236	#336	C₂²×C₃₁	1	2	yes	—
237	C₂×C₃₁	—	4	2	no	—
238	C₂×C₃₁	#2325	1	1	yes	characteristic
239	C₂×C₃₁	C₄×#137	1	2	yes	—
240	C₂×C₃₁	#3195	1	1	yes	characteristic
241	C₂×C₃₁	#3197	1	1	yes	characteristic
242	C₂×C₃₁	#3252	1	2	yes	—
243	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
244	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
245	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
246	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
247	C₂×Q₁₆	—	4	1	no	—
248	C₂×Q₁₆	—	4	1	no	—
249	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
250	C₂×M₁₆	—	2	2	no	—
251	C₂×M₁₆	—	4	1	no	—
252	C₂×M₁₆	—	4	1	no	—
253	C₂×M₁₆	—	4	1	no	—
254	C₂×M₁₆	—	4	1	no	—
255	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
256	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
257	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
259	C₂²⋊Q₈	—	2	4	no	—
260	C₂²⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
261	C₄×Q₈	—	4	2	no	—
262	C₄×Q₈	—	4	2	no	—
263	C₂×C₄⋊C₄	—	2	2	no	—
264	C₂×C₄⋊C₄	—	2	2	no	—
265	C₂×C₄⋊C₄	—	2	2	no	—
266	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
267	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
268	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
269	C₂×C₂²⋊C₄	—	2	2	no	—
270	C₂×C₂²⋊C₄	—	2	2	no	—
271	C₂×C₄²	—	2	2	no	—
272	C₂×C₄²	—	2	2	no	—
273	C₂×C₄²	—	4	1	no	—
274	C₂×C₄²	—	4	1	no	—
275	C₂×C₄²	D₄×C₃₁	1	2	yes	—
276	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
277	#107	—	4	1	no	—
278	#106	—	4	1	no	—
279	#106	C₂×C₄×C₃₁	1	4	yes	complemented
280	#103	—	4	2	no	—
281	#103	—	4	2	no	—
282	#103	C₂×C₄×C₃₁	1	8	yes	complemented
283	#95	—	2	2	no	—
284	#95	—	2	2	no	—
285	#95	—	2	2	no	—
286	#95	—	2	2	no	—
287	#95	—	2	2	no	—
288	C₃₁	#22188	1	1	yes	characteristic, direct product factor, Sylow
289	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	4	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	4	1	no	—
306	C₂²×C₄	—	4	1	no	—
307	C₂²×C₄	—	4	2	no	—
308	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
309	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
310	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
312	C₂²×C₄	#41×C₃₁	1	2	yes	—
313	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
314	Q₁₆	—	4	4	no	—
315	M₁₆	—	4	4	no	—
316	M₁₆	—	4	4	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
325	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
326	C₄⋊C₄	—	2	8	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	4	no	—
333	C₄⋊C₄	—	4	4	no	—
334	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
335	C₂²⋊C₄	—	2	8	no	—
336	C₂²⋊C₄	—	4	4	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	2	no	—
339	C₄²	—	4	2	no	—
340	C₄²	—	4	4	no	—
341	C₄²	—	4	4	no	—
342	C₂³	—	2	1	no	—
343	C₂³	—	2	1	no	—
344	C₂³	—	2	2	no	—
345	C₂³	—	2	2	no	—
346	C₂³	C₄×D₄×C₃₁	1	2	yes	—
347	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
348	Q₈	—	2	4	no	—
349	Q₈	—	2	4	no	—
350	Q₈	—	4	2	no	—
351	Q₈	—	4	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	4	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	4	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
378	C₂×C₄	C₃₁×#137	1	4	yes	—
379	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
380	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
381	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
382	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
383	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
384	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
385	C₈	—	2	2	no	—
386	C₈	—	2	2	no	—
387	C₈	—	4	2	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	4	no	—
390	C₂²	—	4	1	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	2	no	—
393	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
394	C₂²	C₃₁×#534	1	1	yes	characteristic
395	C₂²	C₃₁×#537	1	1	yes	characteristic
396	C₂²	C₂×C₃₁×#137	1	2	yes	—
397	C₂²	C₃₁×#575	1	2	yes	—
398	C₄	—	2	2	no	—
399	C₄	—	2	4	no	—
400	C₄	—	4	4	no	—
401	C₄	—	4	2	no	—
402	C₄	—	4	4	no	—
403	C₄	—	8	2	no	—
404	C₄	—	8	2	no	—
405	C₄	C₃₁×#518	1	1	yes	characteristic
406	C₄	C₃₁×#520	1	1	yes	characteristic
407	C₄	C₃₁×#538	1	1	yes	characteristic
408	C₄	C₃₁×#539	1	1	yes	characteristic
409	C₂	—	4	2	no	—
410	C₂	C₃₁×#2325	1	1	yes	characteristic
411	C₂	C₄×C₃₁×#137	1	2	yes	—
412	C₂	C₃₁×#3195	1	1	yes	characteristic
413	C₂	C₃₁×#3197	1	1	yes	characteristic
414	C₂	C₃₁×#3252	1	2	yes	—

	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×#1927	C₂	semidirect
4. C₃₁×#1920	C₂	semidirect
6. C₃₁×#1878	C₂	semidirect
7. C₃₁×#1863	C₂	semidirect
16. C₃₁×#470	C₄	semidirect
42. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
72. C₃₁×#106	C₂×C₄	semidirect
75. C₃₁×#103	C₂×C₄	semidirect
134. #22188	C₃₁	direct (N × H)
183. #1927	C₂×C₃₁	semidirect
184. #1920	C₂×C₃₁	semidirect
186. #1878	C₂×C₃₁	semidirect
187. #1863	C₂×C₃₁	semidirect
217. #470	C₄×C₃₁	semidirect
249. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
279. #106	C₂×C₄×C₃₁	semidirect
282. #103	C₂×C₄×C₃₁	semidirect
288. C₃₁	#22188	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×#470	2
C₄×C₃₁×C₂²⋊Q₈	2
C₃₁×#1927	2
C₃₁×#1920	2
C₃₁×#1894	2
C₃₁×#1878	2
C₃₁×#1863	2
C₃₁×#1809	2
C₃₁×#1728	2
#22188	31

Subgroups — C31×#22188 #21478947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22188 #21478947