Subgroups — C₃₁×#17822 #21474581

Order 7936.

664 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1573	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1571	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1570	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1569	C₂	1	4	yes	maximal, complemented
8	C₂²×M₁₆×C₃₁	—	2	1	no	—
9	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#432	—	2	1	no	—
11	C₃₁×#432	—	2	1	no	—
12	C₃₁×#432	C₄	1	2	yes	—
13	C₃₁×#432	C₄	1	2	yes	—
14	C₃₁×#432	C₂²	1	2	yes	—
15	C₃₁×#432	C₂²	1	2	yes	—
16	C₃₁×#431	C₂²	1	2	yes	—
17	C₃₁×#431	C₂²	1	2	yes	—
18	C₃₁×#423	—	2	1	no	—
19	C₃₁×#423	—	2	1	no	—
20	C₃₁×#423	C₄	1	2	yes	—
21	C₃₁×#423	C₂²	1	2	yes	—
22	C₃₁×#423	C₂²	1	2	yes	—
23	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
27	C₃₁×#407	C₄	1	2	yes	—
28	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
29	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
30	C₂×C₃₁×#97	—	2	1	no	—
31	C₃₁×#335	C₂²	1	8	yes	complemented
32	C₃₁×#334	C₂²	1	8	yes	complemented
33	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
36	C₂×M₁₆×C₃₁	—	2	2	no	—
37	C₂×M₁₆×C₃₁	—	2	2	no	—
38	C₂×M₁₆×C₃₁	—	4	4	no	—
39	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₃₁	—	2	1	no	—
41	C₂²×C₈×C₃₁	—	2	1	no	—
42	C₂²×C₈×C₃₁	—	2	2	no	—
43	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
50	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
51	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×C₄⋊C₈	—	2	2	no	—
56	C₃₁×C₄⋊C₈	—	2	2	no	—
57	C₃₁×C₄⋊C₈	—	2	2	no	—
58	C₃₁×C₄⋊C₈	—	2	1	no	—
59	C₃₁×C₄⋊C₈	—	2	1	no	—
60	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
61	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
62	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
63	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
64	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
65	C₃₁×C₂²⋊C₈	—	2	2	no	—
66	C₃₁×C₂²⋊C₈	—	2	2	no	—
67	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
68	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
69	C₃₁×#97	—	2	2	no	—
70	C₃₁×#97	—	2	2	no	—
71	C₃₁×#97	C₂×C₄	1	4	yes	—
72	C₄×C₈×C₃₁	C₂³	1	4	yes	—
73	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	—	4	2	no	—
77	C₂²×C₄×C₃₁	—	4	2	no	—
78	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
96	M₁₆×C₃₁	—	4	4	no	—
97	M₁₆×C₃₁	—	4	4	no	—
98	M₁₆×C₃₁	—	4	4	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	4	no	—
102	C₂×C₈×C₃₁	—	4	4	no	—
103	C₂×C₈×C₃₁	—	4	2	no	—
104	C₂×C₈×C₃₁	—	4	2	no	—
105	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
106	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
112	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
119	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
120	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
121	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
122	#17822	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂³×C₃₁	—	2	2	no	—
124	C₂³×C₃₁	—	2	2	no	—
125	C₂³×C₃₁	—	4	1	no	—
126	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	1	no	—
137	C₂×C₄×C₃₁	—	2	1	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	4	1	no	—
143	C₂×C₄×C₃₁	—	4	1	no	—
144	C₂×C₄×C₃₁	—	4	1	no	—
145	C₂×C₄×C₃₁	—	4	1	no	—
146	C₂×C₄×C₃₁	—	4	1	no	—
147	C₂×C₄×C₃₁	—	4	1	no	—
148	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
158	C₂×C₄×C₃₁	#137	1	2	yes	—
159	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
162	C₈×C₃₁	—	2	8	no	—
163	C₈×C₃₁	—	4	2	no	—
164	C₈×C₃₁	—	4	4	no	—
165	C₈×C₃₁	—	4	2	no	—
166	#2920	C₂×C₃₁	1	1	yes	characteristic
167	C₂×#432	C₂×C₃₁	1	1	yes	characteristic
168	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
169	#1573	C₂×C₃₁	1	4	yes	complemented
170	#1571	C₂×C₃₁	1	2	yes	complemented
171	#1570	C₂×C₃₁	1	2	yes	complemented
172	#1569	C₂×C₃₁	1	4	yes	complemented
173	C₂²×C₃₁	—	2	1	no	—
174	C₂²×C₃₁	—	2	1	no	—
175	C₂²×C₃₁	—	2	1	no	—
176	C₂²×C₃₁	—	2	1	no	—
177	C₂²×C₃₁	—	2	1	no	—
178	C₂²×C₃₁	—	2	1	no	—
179	C₂²×C₃₁	—	2	1	no	—
180	C₂²×C₃₁	—	4	4	no	—
181	C₂²×C₃₁	#428	1	2	yes	—
182	C₂²×C₃₁	#430	1	2	yes	—
183	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
184	C₄×C₃₁	—	2	1	no	—
185	C₄×C₃₁	—	2	1	no	—
186	C₄×C₃₁	—	2	1	no	—
187	C₄×C₃₁	—	2	1	no	—
188	C₄×C₃₁	—	2	1	no	—
189	C₄×C₃₁	—	2	1	no	—
190	C₄×C₃₁	—	2	1	no	—
191	C₄×C₃₁	—	2	1	no	—
192	C₄×C₃₁	—	2	1	no	—
193	C₄×C₃₁	#428	1	2	yes	—
194	C₄×C₃₁	#568	1	2	yes	—
195	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
196	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
197	C₂²×M₁₆	—	2	1	no	—
198	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
199	#432	—	2	1	no	—
200	#432	—	2	1	no	—
201	#432	C₄×C₃₁	1	2	yes	—
202	#432	C₄×C₃₁	1	2	yes	—
203	#432	C₂²×C₃₁	1	2	yes	—
204	#432	C₂²×C₃₁	1	2	yes	—
205	#431	C₂²×C₃₁	1	2	yes	—
206	#431	C₂²×C₃₁	1	2	yes	—
207	#423	—	2	1	no	—
208	#423	—	2	1	no	—
209	#423	C₄×C₃₁	1	2	yes	—
210	#423	C₂²×C₃₁	1	2	yes	—
211	#423	C₂²×C₃₁	1	2	yes	—
212	C₂×C₄⋊C₈	—	2	1	no	—
213	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
214	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
216	#407	C₄×C₃₁	1	2	yes	—
217	C₂×C₂²⋊C₈	—	2	1	no	—
218	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
219	C₂×#97	—	2	1	no	—
220	#335	C₂²×C₃₁	1	8	yes	complemented
221	#334	C₂²×C₃₁	1	8	yes	complemented
222	C₂×C₃₁	—	2	2	no	—
223	C₂×C₃₁	—	2	2	no	—
224	C₂×C₃₁	—	4	1	no	—
225	C₂×C₃₁	#2905	1	1	yes	characteristic
226	C₂×C₃₁	C₂×#428	1	1	yes	characteristic
227	C₂×C₃₁	C₂×#430	1	1	yes	characteristic
228	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
229	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
230	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
231	C₂×M₁₆	—	2	2	no	—
232	C₂×M₁₆	—	2	2	no	—
233	C₂×M₁₆	—	4	4	no	—
234	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
235	C₂²×C₈	—	2	2	no	—
236	C₂²×C₈	—	2	1	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
243	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
244	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
245	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
246	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
247	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
248	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
249	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	2	2	no	—
253	C₄⋊C₈	—	2	1	no	—
254	C₄⋊C₈	—	2	1	no	—
255	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
256	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
257	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
258	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
259	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
260	C₂²⋊C₈	—	2	2	no	—
261	C₂²⋊C₈	—	2	2	no	—
262	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
263	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
264	#97	—	2	2	no	—
265	#97	—	2	2	no	—
266	#97	C₂×C₄×C₃₁	1	4	yes	—
267	C₄×C₈	C₂³×C₃₁	1	4	yes	—
268	C₃₁	#17822	1	1	yes	characteristic, direct product factor, Sylow
269	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
276	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
277	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
278	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
279	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
280	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
281	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
282	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
283	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
284	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
285	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
288	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
289	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
290	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
291	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
292	M₁₆	—	4	4	no	—
293	M₁₆	—	4	4	no	—
294	M₁₆	—	4	4	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	4	no	—
298	C₂×C₈	—	4	4	no	—
299	C₂×C₈	—	4	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
302	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
303	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
304	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
305	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
306	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
307	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
308	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
309	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
310	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
311	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
312	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
313	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
314	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
315	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
316	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
317	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
318	C₂³	—	2	2	no	—
319	C₂³	—	2	2	no	—
320	C₂³	—	4	1	no	—
321	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
322	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
323	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
344	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
345	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
346	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
347	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
348	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
349	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
350	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
351	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
352	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
353	C₂×C₄	C₃₁×#137	1	2	yes	—
354	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
356	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
357	C₈	—	2	8	no	—
358	C₈	—	4	2	no	—
359	C₈	—	4	4	no	—
360	C₈	—	4	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	4	4	no	—
369	C₂²	C₃₁×#428	1	2	yes	—
370	C₂²	C₃₁×#430	1	2	yes	—
371	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	C₃₁×#428	1	2	yes	—
382	C₄	C₃₁×#568	1	2	yes	—
383	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
384	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
385	C₂	—	2	2	no	—
386	C₂	—	2	2	no	—
387	C₂	—	4	1	no	—
388	C₂	C₃₁×#2905	1	1	yes	characteristic
389	C₂	C₂×C₃₁×#428	1	1	yes	characteristic
390	C₂	C₂×C₃₁×#430	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1573	C₂	semidirect
5. C₃₁×#1571	C₂	semidirect
6. C₃₁×#1570	C₂	semidirect
7. C₃₁×#1569	C₂	semidirect
31. C₃₁×#335	C₂²	semidirect
32. C₃₁×#334	C₂²	semidirect
122. #17822	C₃₁	direct (N × H)
169. #1573	C₂×C₃₁	semidirect
170. #1571	C₂×C₃₁	semidirect
171. #1570	C₂×C₃₁	semidirect
172. #1569	C₂×C₃₁	semidirect
220. #335	C₂²×C₃₁	semidirect
221. #334	C₂²×C₃₁	semidirect
268. C₃₁	#17822	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2920	2
C₂×C₃₁×#432	2
C₂×C₃₁×#423	2
C₃₁×#1573	2
C₃₁×#1571	2
C₃₁×#1570	2
C₃₁×#1569	2
#17822	31

Subgroups — C31×#17822 #21474581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17822 #21474581