Subgroups — C₃₁×#21646 #21478405

Order 7936.

1022 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1936	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1762	C₂	1	4	yes	maximal, complemented
9	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
13	C₂×C₄×D₄×C₃₁	—	2	1	no	—
14	C₂×C₄×D₄×C₃₁	—	2	1	no	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#486	C₄	1	8	yes	complemented
17	C₂×C₃₁×#102	C₂²	1	4	yes	—
18	C₂×C₃₁×#102	C₂²	1	4	yes	—
19	C₂×C₃₁×#97	C₂²	1	2	yes	—
20	C₂×C₃₁×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#391	—	2	1	no	—
24	C₃₁×#390	—	2	1	no	—
25	C₃₁×#390	—	2	1	no	—
26	C₃₁×#390	C₂²	1	2	yes	—
27	C₃₁×#389	—	2	1	no	—
28	C₃₁×#389	C₂²	1	2	yes	—
29	C₃₁×#384	C₂²	1	1	yes	characteristic
30	C₃₁×#384	C₂²	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₄³×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#340	C₂²	1	4	yes	—
35	C₃₁×#340	C₂²	1	4	yes	—
36	C₂²×Q₈×C₃₁	—	2	1	no	—
37	C₂²×D₄×C₃₁	—	2	1	no	—
38	C₂²×D₄×C₃₁	—	2	1	no	—
39	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
40	C₂³×C₄×C₃₁	—	2	2	no	—
41	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
43	C₃₁×C₄⋊Q₈	—	2	2	no	—
44	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
45	C₃₁×#127	—	2	2	no	—
46	C₃₁×#127	C₂×C₄	1	4	yes	—
47	C₄×Q₈×C₃₁	—	4	4	no	—
48	C₄×D₄×C₃₁	—	2	8	no	—
49	C₄×D₄×C₃₁	—	4	4	no	—
50	C₄×D₄×C₃₁	—	4	4	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄²×C₃₁	—	2	2	no	—
64	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#102	—	2	8	no	—
72	C₃₁×#102	C₂×C₄	1	16	yes	complemented
73	C₃₁×#97	—	2	4	no	—
74	C₃₁×#97	—	2	4	no	—
75	C₄×C₈×C₃₁	—	2	2	no	—
76	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
77	C₃₁×#95	—	2	2	no	—
78	C₃₁×#95	—	2	2	no	—
79	C₂⁴×C₃₁	—	2	2	no	—
80	C₂×Q₈×C₃₁	—	2	4	no	—
81	C₂×Q₈×C₃₁	—	4	2	no	—
82	C₂×Q₈×C₃₁	—	4	2	no	—
83	C₂×D₄×C₃₁	—	2	4	no	—
84	C₂×D₄×C₃₁	—	2	4	no	—
85	C₂×D₄×C₃₁	—	2	4	no	—
86	C₂×D₄×C₃₁	—	2	4	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂×D₄×C₃₁	—	4	2	no	—
90	C₂×D₄×C₃₁	—	4	2	no	—
91	C₂×D₄×C₃₁	—	4	2	no	—
92	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
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	—	4	4	no	—
97	C₂²×C₄×C₃₁	—	4	4	no	—
98	C₂²×C₄×C₃₁	—	4	4	no	—
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₃₁	#41	1	4	yes	—
112	C₄⋊C₄×C₃₁	—	2	4	no	—
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	2	4	no	—
115	C₄⋊C₄×C₃₁	—	2	4	no	—
116	C₄⋊C₄×C₃₁	—	4	4	no	—
117	C₄⋊C₄×C₃₁	—	4	2	no	—
118	C₄⋊C₄×C₃₁	—	4	2	no	—
119	C₄⋊C₄×C₃₁	—	4	2	no	—
120	C₄⋊C₄×C₃₁	—	4	2	no	—
121	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
122	C₂²⋊C₄×C₃₁	—	4	4	no	—
123	C₂²⋊C₄×C₃₁	—	4	4	no	—
124	C₂²⋊C₄×C₃₁	—	4	4	no	—
125	C₄²×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	—	2	2	no	—
127	C₄²×C₃₁	—	2	2	no	—
128	C₄²×C₃₁	—	2	2	no	—
129	C₄²×C₃₁	—	2	2	no	—
130	C₄²×C₃₁	—	4	4	no	—
131	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
132	C₄²×C₃₁	#41	1	2	yes	—
133	C₄²×C₃₁	#41	1	4	yes	—
134	#21646	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂³×C₃₁	—	2	4	no	—
136	C₂³×C₃₁	—	4	2	no	—
137	C₂³×C₃₁	—	4	2	no	—
138	C₂³×C₃₁	—	4	2	no	—
139	C₂³×C₃₁	—	4	2	no	—
140	C₂³×C₃₁	—	4	2	no	—
141	C₂³×C₃₁	—	4	2	no	—
142	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
143	Q₈×C₃₁	—	4	8	no	—
144	D₄×C₃₁	—	2	8	no	—
145	D₄×C₃₁	—	4	4	no	—
146	D₄×C₃₁	—	4	8	no	—
147	D₄×C₃₁	—	4	8	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	4	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	4	2	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	—	8	8	no	—
168	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
169	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
170	C₂×C₄×C₃₁	#124	1	2	yes	—
171	C₂×C₄×C₃₁	#124	1	2	yes	—
172	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
173	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
175	C₂×C₄×C₃₁	#137	1	2	yes	—
176	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
178	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
179	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
180	C₈×C₃₁	—	2	4	no	—
181	C₈×C₃₁	—	2	4	no	—
182	C₂×#486	C₂×C₃₁	1	1	yes	characteristic
183	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
184	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
185	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#1957	C₂×C₃₁	1	1	yes	characteristic
187	#1936	C₂×C₃₁	1	4	yes	complemented
188	#1834	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#1762	C₂×C₃₁	1	4	yes	complemented
190	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
191	C₂²×C₃₁	—	4	4	no	—
192	C₂²×C₃₁	—	4	4	no	—
193	C₂²×C₃₁	—	4	4	no	—
194	C₂²×C₃₁	—	8	2	no	—
195	C₂²×C₃₁	—	8	2	no	—
196	C₂²×C₃₁	—	8	2	no	—
197	C₂²×C₃₁	—	8	2	no	—
198	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
199	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
200	C₂²×C₃₁	#532	1	1	yes	characteristic
201	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
203	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
205	C₄×C₃₁	—	2	2	no	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	4	1	no	—
208	C₄×C₃₁	—	4	1	no	—
209	C₄×C₃₁	—	4	4	no	—
210	C₄×C₃₁	—	8	2	no	—
211	C₄×C₃₁	—	8	2	no	—
212	C₄×C₃₁	#440	1	4	yes	—
213	C₄×C₃₁	#442	1	4	yes	—
214	C₄×C₃₁	C₄×#41	1	2	yes	—
215	C₄×C₃₁	#529	1	2	yes	—
216	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
217	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×C₄×Q₈	—	2	1	no	—
219	C₂×C₄×D₄	—	2	1	no	—
220	C₂×C₄×D₄	—	2	1	no	—
221	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
222	#486	C₄×C₃₁	1	8	yes	complemented
223	C₂×#102	C₂²×C₃₁	1	4	yes	—
224	C₂×#102	C₂²×C₃₁	1	4	yes	—
225	C₂×#97	C₂²×C₃₁	1	2	yes	—
226	C₂×#97	C₂²×C₃₁	1	2	yes	—
227	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
229	#391	—	2	1	no	—
230	#390	—	2	1	no	—
231	#390	—	2	1	no	—
232	#390	C₂²×C₃₁	1	2	yes	—
233	#389	—	2	1	no	—
234	#389	C₂²×C₃₁	1	2	yes	—
235	#384	C₂²×C₃₁	1	1	yes	characteristic
236	#384	C₂²×C₃₁	1	1	yes	characteristic
237	C₄×C₄⋊C₄	—	2	1	no	—
238	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
239	C₄³	C₂²×C₃₁	1	1	yes	characteristic
240	#340	C₂²×C₃₁	1	4	yes	—
241	#340	C₂²×C₃₁	1	4	yes	—
242	C₂×C₃₁	—	8	4	no	—
243	C₂×C₃₁	C₄×#124	1	1	yes	characteristic
244	C₂×C₃₁	C₂×#440	1	1	yes	characteristic
245	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
246	C₂×C₃₁	#3138	1	1	yes	characteristic
247	C₂×C₃₁	#3141	1	1	yes	characteristic
248	C₂×C₃₁	#3182	1	1	yes	characteristic
249	C₂×C₃₁	#3184	1	1	yes	characteristic
250	C₂²×Q₈	—	2	1	no	—
251	C₂²×D₄	—	2	1	no	—
252	C₂²×D₄	—	2	1	no	—
253	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
254	C₂³×C₄	—	2	2	no	—
255	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
256	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
257	C₄⋊Q₈	—	2	2	no	—
258	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
259	#127	—	2	2	no	—
260	#127	C₂×C₄×C₃₁	1	4	yes	—
261	C₄×Q₈	—	4	4	no	—
262	C₄×D₄	—	2	8	no	—
263	C₄×D₄	—	4	4	no	—
264	C₄×D₄	—	4	4	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
272	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
273	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
274	C₂×C₂²⋊C₄	—	2	2	no	—
275	C₂×C₂²⋊C₄	—	2	2	no	—
276	C₂×C₂²⋊C₄	—	2	2	no	—
277	C₂×C₄²	—	2	2	no	—
278	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
281	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
282	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
283	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
285	#102	—	2	8	no	—
286	#102	C₂×C₄×C₃₁	1	16	yes	complemented
287	#97	—	2	4	no	—
288	#97	—	2	4	no	—
289	C₄×C₈	—	2	2	no	—
290	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
291	#95	—	2	2	no	—
292	#95	—	2	2	no	—
293	C₃₁	#21646	1	1	yes	characteristic, direct product factor, Sylow
294	C₂⁴	—	2	2	no	—
295	C₂×Q₈	—	2	4	no	—
296	C₂×Q₈	—	4	2	no	—
297	C₂×Q₈	—	4	2	no	—
298	C₂×D₄	—	2	4	no	—
299	C₂×D₄	—	2	4	no	—
300	C₂×D₄	—	2	4	no	—
301	C₂×D₄	—	2	4	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	4	2	no	—
304	C₂×D₄	—	4	2	no	—
305	C₂×D₄	—	4	2	no	—
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	4	4	no	—
312	C₂²×C₄	—	4	4	no	—
313	C₂²×C₄	—	4	4	no	—
314	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
316	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
326	C₂×C₈	#41×C₃₁	1	4	yes	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	4	4	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	—	4	2	no	—
334	C₄⋊C₄	—	4	2	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
337	C₂²⋊C₄	—	4	4	no	—
338	C₂²⋊C₄	—	4	4	no	—
339	C₂²⋊C₄	—	4	4	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	4	4	no	—
346	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
347	C₄²	#41×C₃₁	1	2	yes	—
348	C₄²	#41×C₃₁	1	4	yes	—
349	C₂³	—	2	4	no	—
350	C₂³	—	4	2	no	—
351	C₂³	—	4	2	no	—
352	C₂³	—	4	2	no	—
353	C₂³	—	4	2	no	—
354	C₂³	—	4	2	no	—
355	C₂³	—	4	2	no	—
356	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
357	Q₈	—	4	8	no	—
358	D₄	—	2	8	no	—
359	D₄	—	4	8	no	—
360	D₄	—	4	8	no	—
361	D₄	—	4	4	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	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	—	4	2	no	—
378	C₂×C₄	—	4	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	8	8	no	—
382	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
383	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
384	C₂×C₄	C₃₁×#124	1	2	yes	—
385	C₂×C₄	C₃₁×#124	1	2	yes	—
386	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
387	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
388	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
389	C₂×C₄	C₃₁×#137	1	2	yes	—
390	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
392	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
393	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
394	C₈	—	2	4	no	—
395	C₈	—	2	4	no	—
396	C₂²	—	4	4	no	—
397	C₂²	—	4	4	no	—
398	C₂²	—	4	4	no	—
399	C₂²	—	8	2	no	—
400	C₂²	—	8	2	no	—
401	C₂²	—	8	2	no	—
402	C₂²	—	8	2	no	—
403	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
404	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
405	C₂²	C₃₁×#532	1	1	yes	characteristic
406	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
407	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
408	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
409	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
410	C₄	—	2	2	no	—
411	C₄	—	2	2	no	—
412	C₄	—	4	1	no	—
413	C₄	—	4	1	no	—
414	C₄	—	4	4	no	—
415	C₄	—	8	2	no	—
416	C₄	—	8	2	no	—
417	C₄	C₃₁×#440	1	4	yes	—
418	C₄	C₃₁×#442	1	4	yes	—
419	C₄	C₄×#41×C₃₁	1	2	yes	—
420	C₄	C₃₁×#529	1	2	yes	—
421	C₂	—	8	4	no	—
422	C₂	C₄×C₃₁×#124	1	1	yes	characteristic
423	C₂	C₂×C₃₁×#440	1	1	yes	characteristic
424	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
425	C₂	C₃₁×#3138	1	1	yes	characteristic
426	C₂	C₃₁×#3141	1	1	yes	characteristic
427	C₂	C₃₁×#3182	1	1	yes	characteristic
428	C₂	C₃₁×#3184	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	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
4. C₃₁×#1990	C₂	semidirect
6. C₃₁×#1936	C₂	semidirect
7. C₃₁×#1834	C₂	semidirect
8. C₃₁×#1762	C₂	semidirect
9. C₄×C₃₁×#97	C₂	semidirect
16. C₃₁×#486	C₄	semidirect
44. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
72. C₃₁×#102	C₂×C₄	semidirect
76. C₄×C₈×C₃₁	C₂×C₄	semidirect
134. #21646	C₃₁	direct (N × H)
183. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
185. #1990	C₂×C₃₁	semidirect
187. #1936	C₂×C₃₁	semidirect
188. #1834	C₂×C₃₁	semidirect
189. #1762	C₂×C₃₁	semidirect
190. C₄×#97	C₂×C₃₁	semidirect
222. #486	C₄×C₃₁	semidirect
258. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
286. #102	C₂×C₄×C₃₁	semidirect
290. C₄×C₈	C₂×C₄×C₃₁	semidirect
293. C₃₁	#21646	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×#486	2
C₄×C₃₁×C₄⋊Q₈	2
C₄×C₃₁×#127	2
C₃₁×#1990	2
C₃₁×#1957	2
C₃₁×#1936	2
C₃₁×#1834	2
C₃₁×#1762	2
C₄×C₃₁×#97	2
#21646	31

Subgroups — C31×#21646 #21478405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21646 #21478405