Subgroups — C₃₁×#17830 #21474589

Order 7936.

634 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1574	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1573	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1571	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1569	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1565	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#446	C₄	1	1	yes	characteristic
15	C₃₁×#446	C₄	1	1	yes	characteristic
16	C₃₁×#436	C₄	1	1	yes	characteristic
17	C₃₁×#436	C₄	1	1	yes	characteristic
18	C₃₁×#435	C₄	1	2	yes	—
19	C₃₁×#435	C₄	1	2	yes	—
20	C₃₁×#433	—	2	2	no	—
21	C₃₁×#432	C₂²	1	2	yes	—
22	C₃₁×#431	C₂²	1	2	yes	—
23	C₃₁×#431	C₂²	1	1	yes	characteristic
24	C₃₁×#431	C₂²	1	2	yes	—
25	C₃₁×#424	—	2	1	no	—
26	C₃₁×#424	—	2	2	no	—
27	C₃₁×#423	C₂²	1	1	yes	characteristic
28	C₃₁×#423	C₂²	1	1	yes	characteristic
29	C₃₁×#423	C₂²	1	2	yes	—
30	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
33	C₃₁×#408	—	2	1	no	—
34	C₃₁×#405	—	2	1	no	—
35	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#335	C₂²	1	2	yes	—
38	C₃₁×#335	C₂²	1	2	yes	—
39	C₃₁×#335	C₂²	1	2	yes	—
40	C₃₁×#335	C₂²	1	2	yes	—
41	C₃₁×#334	C₂²	1	1	yes	characteristic
42	C₃₁×#334	C₂²	1	1	yes	characteristic
43	C₃₁×#334	C₂²	1	1	yes	characteristic
44	C₃₁×#334	C₂²	1	1	yes	characteristic
45	C₃₁×#334	C₂²	1	2	yes	—
46	C₃₁×#334	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
48	C₃₁×#131	—	4	4	no	—
49	C₂×M₁₆×C₃₁	—	2	1	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	2	1	no	—
52	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
54	C₂²×C₈×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₂²×C₈×C₃₁	—	2	1	no	—
57	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₄×Q₈×C₃₁	Q₈	1	1	yes	characteristic
61	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₄×D₄×C₃₁	Q₈	1	2	yes	—
64	C₄×D₄×C₃₁	Q₈	1	1	yes	characteristic
65	C₃₁×#117	D₄	1	1	yes	characteristic
66	C₃₁×#117	D₄	1	2	yes	—
67	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
69	C₃₁×C₄⋊C₈	—	2	2	no	—
70	C₃₁×C₄⋊C₈	—	2	1	no	—
71	C₃₁×C₄⋊C₈	—	2	1	no	—
72	C₃₁×C₄⋊C₈	—	2	2	no	—
73	C₃₁×C₄⋊C₈	—	2	1	no	—
74	C₃₁×C₄⋊C₈	—	2	1	no	—
75	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
76	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
77	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
79	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
82	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
84	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
85	C₃₁×C₂²⋊C₈	—	2	2	no	—
86	C₃₁×C₂²⋊C₈	—	2	2	no	—
87	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
88	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
89	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
90	C₃₁×#97	—	2	2	no	—
91	C₃₁×#97	C₂×C₄	1	2	yes	—
92	C₃₁×#97	C₂×C₄	1	2	yes	—
93	C₄×C₈×C₃₁	—	2	2	no	—
94	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
95	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
96	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
97	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
98	C₄×C₈×C₃₁	C₂³	1	2	yes	—
99	#41×C₃₁	—	2	2	no	—
100	#41×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
102	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
103	C₂×D₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
110	M₁₆×C₃₁	—	2	4	no	—
111	M₁₆×C₃₁	—	4	2	no	—
112	M₁₆×C₃₁	—	4	4	no	—
113	M₁₆×C₃₁	—	4	2	no	—
114	M₁₆×C₃₁	—	4	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	4	1	no	—
123	C₂×C₈×C₃₁	—	4	4	no	—
124	C₂×C₈×C₃₁	—	4	2	no	—
125	C₂×C₈×C₃₁	—	4	2	no	—
126	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
131	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
132	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
133	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
134	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
135	C₄⋊C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
136	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
137	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
138	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
139	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
140	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
141	C₄²×C₃₁	C₂×D₄	1	2	yes	—
142	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
144	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
145	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
146	#17830	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
147	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
148	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
149	Q₈×C₃₁	—	2	2	no	—
150	D₄×C₃₁	—	2	4	no	—
151	D₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	1	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	1	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
162	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
163	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
167	C₂×C₄×C₃₁	#137	1	2	yes	—
168	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
171	C₈×C₃₁	—	2	1	no	—
172	C₈×C₃₁	—	2	2	no	—
173	C₈×C₃₁	—	2	1	no	—
174	C₈×C₃₁	—	2	1	no	—
175	C₈×C₃₁	—	2	2	no	—
176	C₈×C₃₁	—	4	1	no	—
177	C₈×C₃₁	—	4	2	no	—
178	C₈×C₃₁	—	4	1	no	—
179	C₈×C₃₁	—	4	2	no	—
180	C₈×C₃₁	—	4	1	no	—
181	C₈×C₃₁	—	4	1	no	—
182	C₈×C₃₁	#131	1	2	yes	complemented
183	#2965	C₂×C₃₁	1	1	yes	characteristic
184	#2925	C₂×C₃₁	1	1	yes	characteristic
185	#2906	C₂×C₃₁	1	1	yes	characteristic
186	#1574	C₂×C₃₁	1	2	yes	complemented
187	#1573	C₂×C₃₁	1	2	yes	complemented
188	#1571	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#1570	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1569	C₂×C₃₁	1	2	yes	complemented
191	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1565	C₂×C₃₁	1	2	yes	complemented
193	C₂×#334	C₂×C₃₁	1	1	yes	characteristic, complemented
194	C₂²×C₃₁	—	2	1	no	—
195	C₂²×C₃₁	—	2	1	no	—
196	C₂²×C₃₁	—	2	2	no	—
197	C₂²×C₃₁	—	2	2	no	—
198	C₂²×C₃₁	—	2	2	no	—
199	C₂²×C₃₁	#428	1	2	yes	—
200	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
201	C₄×C₃₁	—	2	1	no	—
202	C₄×C₃₁	—	2	2	no	—
203	C₄×C₃₁	—	2	2	no	—
204	C₄×C₃₁	—	2	1	no	—
205	C₄×C₃₁	—	2	1	no	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	#428	1	2	yes	—
208	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
209	C₄×C₃₁	#568	1	1	yes	characteristic
210	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
211	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
212	C₂×#131	—	2	1	no	—
213	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
214	#446	C₄×C₃₁	1	1	yes	characteristic
215	#446	C₄×C₃₁	1	1	yes	characteristic
216	#436	C₄×C₃₁	1	1	yes	characteristic
217	#436	C₄×C₃₁	1	1	yes	characteristic
218	#435	C₄×C₃₁	1	2	yes	—
219	#435	C₄×C₃₁	1	2	yes	—
220	#433	—	2	2	no	—
221	#432	C₂²×C₃₁	1	2	yes	—
222	#431	C₂²×C₃₁	1	2	yes	—
223	#431	C₂²×C₃₁	1	1	yes	characteristic
224	#431	C₂²×C₃₁	1	2	yes	—
225	#424	—	2	1	no	—
226	#424	—	2	2	no	—
227	#423	C₂²×C₃₁	1	1	yes	characteristic
228	#423	C₂²×C₃₁	1	1	yes	characteristic
229	#423	C₂²×C₃₁	1	2	yes	—
230	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
231	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
233	#408	—	2	1	no	—
234	#405	—	2	1	no	—
235	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
236	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
237	#335	C₂²×C₃₁	1	2	yes	—
238	#335	C₂²×C₃₁	1	2	yes	—
239	#335	C₂²×C₃₁	1	2	yes	—
240	#335	C₂²×C₃₁	1	2	yes	—
241	#334	C₂²×C₃₁	1	1	yes	characteristic
242	#334	C₂²×C₃₁	1	1	yes	characteristic
243	#334	C₂²×C₃₁	1	1	yes	characteristic
244	#334	C₂²×C₃₁	1	1	yes	characteristic
245	#334	C₂²×C₃₁	1	2	yes	—
246	#334	C₂²×C₃₁	1	2	yes	—
247	C₂×C₃₁	—	2	2	no	—
248	C₂×C₃₁	—	4	2	no	—
249	C₂×C₃₁	#2906	1	1	yes	characteristic
250	C₂×C₃₁	C₂×#428	1	1	yes	characteristic
251	C₂×C₃₁	#2916	1	1	yes	characteristic
252	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
253	#131	—	4	4	no	—
254	C₂×M₁₆	—	2	2	no	—
255	C₂×M₁₆	—	2	1	no	—
256	C₂×M₁₆	—	2	1	no	—
257	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
258	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
259	C₂²×C₈	—	2	2	no	—
260	C₂²×C₈	—	2	1	no	—
261	C₂²×C₈	—	2	1	no	—
262	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
263	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
264	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₄×Q₈	Q₈×C₃₁	1	1	yes	characteristic
266	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
267	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
268	C₄×D₄	Q₈×C₃₁	1	2	yes	—
269	C₄×D₄	Q₈×C₃₁	1	1	yes	characteristic
270	#117	D₄×C₃₁	1	1	yes	characteristic
271	#117	D₄×C₃₁	1	2	yes	—
272	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
274	C₄⋊C₈	—	2	2	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	2	1	no	—
277	C₄⋊C₈	—	2	2	no	—
278	C₄⋊C₈	—	2	1	no	—
279	C₄⋊C₈	—	2	1	no	—
280	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
281	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
282	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
283	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
284	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
285	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
289	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
290	C₂²⋊C₈	—	2	2	no	—
291	C₂²⋊C₈	—	2	2	no	—
292	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
293	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
294	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
295	#97	—	2	2	no	—
296	#97	C₂×C₄×C₃₁	1	2	yes	—
297	#97	C₂×C₄×C₃₁	1	2	yes	—
298	C₄×C₈	—	2	2	no	—
299	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
300	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
301	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₄×C₈	C₂³×C₃₁	1	2	yes	—
304	C₃₁	#17830	1	1	yes	characteristic, direct product factor, Sylow
305	#41	—	2	2	no	—
306	#41	—	2	2	no	—
307	C₂×Q₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
308	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
309	C₂×D₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
312	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
313	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
314	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
315	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
316	M₁₆	—	2	4	no	—
317	M₁₆	—	4	2	no	—
318	M₁₆	—	4	4	no	—
319	M₁₆	—	4	2	no	—
320	M₁₆	—	4	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	4	1	no	—
329	C₂×C₈	—	4	4	no	—
330	C₂×C₈	—	4	2	no	—
331	C₂×C₈	—	4	2	no	—
332	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
333	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
334	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
335	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
336	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
337	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
338	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
339	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
340	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
341	C₄⋊C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
342	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
343	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
344	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
345	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
346	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
347	C₄²	C₂×D₄×C₃₁	1	2	yes	—
348	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
350	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
351	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
352	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
353	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
354	Q₈	—	2	2	no	—
355	D₄	—	2	4	no	—
356	D₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
366	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
367	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
368	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
369	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
370	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
371	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
372	C₂×C₄	C₃₁×#137	1	2	yes	—
373	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
375	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
376	C₈	—	2	1	no	—
377	C₈	—	2	2	no	—
378	C₈	—	2	1	no	—
379	C₈	—	2	1	no	—
380	C₈	—	2	2	no	—
381	C₈	—	4	1	no	—
382	C₈	—	4	2	no	—
383	C₈	—	4	1	no	—
384	C₈	—	4	2	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	C₃₁×#131	1	2	yes	complemented
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	2	no	—
393	C₂²	C₃₁×#428	1	2	yes	—
394	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
395	C₄	—	2	1	no	—
396	C₄	—	2	2	no	—
397	C₄	—	2	2	no	—
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	2	2	no	—
401	C₄	C₃₁×#428	1	2	yes	—
402	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
403	C₄	C₃₁×#568	1	1	yes	characteristic
404	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
405	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
406	C₂	—	2	2	no	—
407	C₂	—	4	2	no	—
408	C₂	C₃₁×#2906	1	1	yes	characteristic
409	C₂	C₂×C₃₁×#428	1	1	yes	characteristic
410	C₂	C₃₁×#2916	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	C₂×#41×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1574	C₂	semidirect
5. C₃₁×#1573	C₂	semidirect
6. C₃₁×#1571	C₂	semidirect
7. C₃₁×#1570	C₂	semidirect
8. C₃₁×#1569	C₂	semidirect
9. C₃₁×#1567	C₂	semidirect
10. C₃₁×#1565	C₂	semidirect
11. C₂×C₃₁×#334	C₂	semidirect
146. #17830	C₃₁	direct (N × H)
182. C₈×C₃₁	#131	semidirect
186. #1574	C₂×C₃₁	semidirect
187. #1573	C₂×C₃₁	semidirect
188. #1571	C₂×C₃₁	semidirect
189. #1570	C₂×C₃₁	semidirect
190. #1569	C₂×C₃₁	semidirect
191. #1567	C₂×C₃₁	semidirect
192. #1565	C₂×C₃₁	semidirect
193. C₂×#334	C₂×C₃₁	semidirect
304. C₃₁	#17830	direct (N × H)
387. C₈	C₃₁×#131	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2965	2
C₃₁×#2925	2
C₃₁×#2906	2
C₃₁×#1574	2
C₃₁×#1573	2
C₃₁×#1571	2
C₃₁×#1570	2
C₃₁×#1569	2
C₃₁×#1567	2
C₃₁×#1565	2
C₂×C₃₁×#334	2
#17830	31

Subgroups — C31×#17830 #21474589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17830 #21474589