Subgroups — C₃₁×#17759 #21474518

Order 7936.

1104 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3499	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2989	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1557	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1556	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1553	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1549	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1547	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1546	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1540	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#332	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×2₋¹⁺⁴	—	2	1	no	—
13	D₄×Q₈×C₃₁	—	2	1	no	—
14	D₄×Q₈×C₃₁	C₄	1	2	yes	—
15	C₃₁×#547	—	2	2	no	—
16	C₃₁×#547	—	2	1	no	—
17	C₃₁×#547	C₄	1	4	yes	—
18	C₃₁×#547	C₄	1	2	yes	—
19	C₃₁×#536	—	2	1	no	—
20	C₃₁×#536	—	2	2	no	—
21	C₃₁×#532	C₂²	1	1	yes	characteristic
22	C₃₁×#532	C₂²	1	2	yes	—
23	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
25	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#446	—	2	1	no	—
27	C₃₁×#436	—	2	1	no	—
28	C₃₁×#435	—	2	2	no	—
29	C₃₁×#433	—	2	2	no	—
30	C₃₁×#432	C₂²	1	2	yes	—
31	C₃₁×#432	C₂²	1	2	yes	—
32	C₃₁×#431	C₂²	1	1	yes	characteristic
33	C₃₁×#431	C₂²	1	2	yes	—
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₃₁×#423	C₂²	1	2	yes	—
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₃₁×#408	—	2	1	no	—
39	C₃₁×#405	—	2	1	no	—
40	C₃₁×#332	C₂²	1	2	yes	—
41	C₃₁×#332	C₂²	1	2	yes	—
42	C₃₁×#332	C₂²	1	1	yes	characteristic
43	C₃₁×#332	C₂²	1	1	yes	characteristic
44	C₃₁×#332	C₂²	1	2	yes	—
45	C₃₁×#332	C₂²	1	2	yes	—
46	C₃₁×#332	C₂²	1	2	yes	—
47	C₃₁×#331	C₂²	1	1	yes	characteristic
48	C₃₁×#331	C₂²	1	1	yes	characteristic
49	C₃₁×#331	C₂²	1	2	yes	—
50	C₃₁×2₋¹⁺⁴	—	4	4	no	—
51	C₃₁×2₋¹⁺⁴	—	4	4	no	—
52	C₂×#41×C₃₁	—	2	2	no	—
53	C₂×#41×C₃₁	—	2	4	no	—
54	C₂×#41×C₃₁	—	2	2	no	—
55	C₂×#41×C₃₁	—	2	1	no	—
56	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₃₁	—	2	2	no	—
58	C₂²×Q₈×C₃₁	—	2	2	no	—
59	C₂²×Q₈×C₃₁	—	2	1	no	—
60	C₂×M₁₆×C₃₁	—	2	1	no	—
61	C₂×M₁₆×C₃₁	—	2	2	no	—
62	C₂²×C₈×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	—	2	1	no	—
64	C₃₁×C₄⋊Q₈	—	2	2	no	—
65	C₃₁×C₄⋊Q₈	—	2	1	no	—
66	C₃₁×C₄⋊Q₈	—	2	1	no	—
67	C₃₁×C₄⋊Q₈	—	2	2	no	—
68	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
69	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
70	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
71	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
72	C₃₁×#127	C₂³	1	1	yes	characteristic
73	C₃₁×#124	—	2	2	no	—
74	C₃₁×#124	—	2	1	no	—
75	C₃₁×#123	—	2	4	no	—
76	C₃₁×#123	—	2	4	no	—
77	C₃₁×#123	—	2	4	no	—
78	C₃₁×C₂²⋊Q₈	—	2	4	no	—
79	C₃₁×C₂²⋊Q₈	—	2	2	no	—
80	C₃₁×C₂²⋊Q₈	—	2	4	no	—
81	C₃₁×C₂²⋊Q₈	—	2	2	no	—
82	C₃₁×#121	—	2	1	no	—
83	C₃₁×#121	—	2	2	no	—
84	C₃₁×#121	C₂×C₄	1	2	yes	—
85	C₃₁×#121	C₂×C₄	1	4	yes	—
86	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
88	C₄×D₄×C₃₁	—	2	1	no	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
91	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
92	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
93	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₃₁	D₄	1	2	yes	—
95	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
96	C₃₁×#117	D₄	1	1	yes	characteristic
97	C₃₁×#117	D₄	1	2	yes	—
98	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
102	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
104	C₃₁×C₄⋊C₈	—	2	1	no	—
105	C₃₁×C₄⋊C₈	—	2	1	no	—
106	C₃₁×C₄⋊C₈	—	2	1	no	—
107	C₃₁×C₄⋊C₈	—	2	2	no	—
108	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
109	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
111	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
113	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₃₁×C₂²⋊C₈	—	2	2	no	—
115	C₃₁×C₂²⋊C₈	—	2	2	no	—
116	C₃₁×C₂²⋊C₈	—	2	1	no	—
117	C₃₁×C₂²⋊C₈	—	2	2	no	—
118	C₃₁×#97	—	2	2	no	—
119	C₃₁×#97	—	2	2	no	—
120	C₄×C₈×C₃₁	—	2	2	no	—
121	C₄×C₈×C₃₁	—	2	1	no	—
122	#41×C₃₁	—	2	2	no	—
123	#41×C₃₁	—	2	2	no	—
124	#41×C₃₁	—	4	4	no	—
125	#41×C₃₁	—	4	4	no	—
126	#41×C₃₁	—	4	16	no	—
127	#41×C₃₁	—	4	8	no	—
128	#41×C₃₁	—	4	2	no	—
129	#41×C₃₁	—	4	2	no	—
130	C₂×Q₈×C₃₁	—	2	4	no	—
131	C₂×Q₈×C₃₁	—	2	2	no	—
132	C₂×Q₈×C₃₁	—	2	1	no	—
133	C₂×Q₈×C₃₁	—	2	2	no	—
134	C₂×Q₈×C₃₁	—	4	4	no	—
135	C₂×Q₈×C₃₁	—	4	4	no	—
136	C₂×Q₈×C₃₁	—	4	8	no	—
137	C₂×Q₈×C₃₁	—	4	2	no	—
138	C₂×Q₈×C₃₁	—	4	2	no	—
139	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
140	C₂×D₄×C₃₁	—	2	4	no	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
145	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
146	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
147	C₂²×C₄×C₃₁	—	2	4	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
152	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	M₁₆×C₃₁	—	4	2	no	—
158	M₁₆×C₃₁	—	8	2	no	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
161	C₂×C₈×C₃₁	—	2	2	no	—
162	C₂×C₈×C₃₁	—	2	1	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	—	4	2	no	—
166	C₂×C₈×C₃₁	—	8	1	no	—
167	C₄⋊C₄×C₃₁	—	2	4	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	2	4	no	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
174	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
175	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
176	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
177	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
178	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
179	C₂²⋊C₄×C₃₁	—	2	4	no	—
180	C₂²⋊C₄×C₃₁	—	2	2	no	—
181	C₂²⋊C₄×C₃₁	—	2	4	no	—
182	C₂²⋊C₄×C₃₁	—	2	2	no	—
183	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
184	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
185	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
186	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
187	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
188	C₄²×C₃₁	C₂×D₄	1	2	yes	—
189	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₃₁	C₂×D₄	1	2	yes	—
191	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
193	#17759	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
196	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	Q₈×C₃₁	—	2	2	no	—
198	Q₈×C₃₁	—	4	8	no	—
199	Q₈×C₃₁	—	4	1	no	—
200	Q₈×C₃₁	—	4	1	no	—
201	Q₈×C₃₁	—	4	2	no	—
202	Q₈×C₃₁	—	4	2	no	—
203	Q₈×C₃₁	—	4	4	no	—
204	D₄×C₃₁	—	2	4	no	—
205	D₄×C₃₁	—	2	2	no	—
206	D₄×C₃₁	—	4	2	no	—
207	D₄×C₃₁	—	4	8	no	—
208	D₄×C₃₁	—	4	4	no	—
209	D₄×C₃₁	—	4	1	no	—
210	D₄×C₃₁	—	4	1	no	—
211	D₄×C₃₁	—	4	1	no	—
212	D₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	—	4	8	no	—
224	C₂×C₄×C₃₁	—	4	8	no	—
225	C₂×C₄×C₃₁	—	4	4	no	—
226	C₂×C₄×C₃₁	—	4	4	no	—
227	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
228	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
230	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
231	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
234	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
235	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₈×C₃₁	—	4	1	no	—
239	C₈×C₃₁	—	4	1	no	—
240	C₈×C₃₁	—	4	2	no	—
241	C₈×C₃₁	—	4	2	no	—
242	C₈×C₃₁	—	4	1	no	—
243	C₈×C₃₁	—	4	1	no	—
244	#3499	C₂×C₃₁	1	1	yes	characteristic
245	#2989	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#2925	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1557	C₂×C₃₁	1	2	yes	complemented
248	#1556	C₂×C₃₁	1	2	yes	complemented
249	#1553	C₂×C₃₁	1	2	yes	complemented
250	#1549	C₂×C₃₁	1	2	yes	complemented
251	#1547	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1546	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#1540	C₂×C₃₁	1	1	yes	characteristic, complemented
254	C₂×#332	C₂×C₃₁	1	1	yes	characteristic, complemented
255	C₂²×C₃₁	—	2	1	no	—
256	C₂²×C₃₁	—	2	1	no	—
257	C₂²×C₃₁	—	2	2	no	—
258	C₂²×C₃₁	—	2	2	no	—
259	C₂²×C₃₁	—	2	2	no	—
260	C₂²×C₃₁	—	4	2	no	—
261	C₂²×C₃₁	—	4	2	no	—
262	C₂²×C₃₁	—	4	2	no	—
263	C₂²×C₃₁	#418	1	2	yes	—
264	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
265	C₄×C₃₁	—	2	1	no	—
266	C₄×C₃₁	—	2	1	no	—
267	C₄×C₃₁	—	2	2	no	—
268	C₄×C₃₁	—	2	2	no	—
269	C₄×C₃₁	—	2	1	no	—
270	C₄×C₃₁	—	2	2	no	—
271	C₄×C₃₁	—	4	4	no	—
272	C₄×C₃₁	—	4	2	no	—
273	C₄×C₃₁	#418	1	2	yes	—
274	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
275	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
276	C₄×C₃₁	#577	1	1	yes	characteristic
277	C₄×C₃₁	#578	1	1	yes	characteristic
278	C₂×2₋¹⁺⁴	—	2	1	no	—
279	D₄×Q₈	—	2	1	no	—
280	D₄×Q₈	C₄×C₃₁	1	2	yes	—
281	#547	—	2	2	no	—
282	#547	—	2	1	no	—
283	#547	C₄×C₃₁	1	4	yes	—
284	#547	C₄×C₃₁	1	2	yes	—
285	#536	—	2	1	no	—
286	#536	—	2	2	no	—
287	#532	C₂²×C₃₁	1	1	yes	characteristic
288	#532	C₂²×C₃₁	1	2	yes	—
289	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
291	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
292	#446	—	2	1	no	—
293	#436	—	2	1	no	—
294	#435	—	2	2	no	—
295	#433	—	2	2	no	—
296	#432	C₂²×C₃₁	1	2	yes	—
297	#432	C₂²×C₃₁	1	2	yes	—
298	#431	C₂²×C₃₁	1	1	yes	characteristic
299	#431	C₂²×C₃₁	1	2	yes	—
300	#423	C₂²×C₃₁	1	1	yes	characteristic
301	#423	C₂²×C₃₁	1	2	yes	—
302	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
304	#408	—	2	1	no	—
305	#405	—	2	1	no	—
306	#332	C₂²×C₃₁	1	2	yes	—
307	#332	C₂²×C₃₁	1	2	yes	—
308	#332	C₂²×C₃₁	1	1	yes	characteristic
309	#332	C₂²×C₃₁	1	1	yes	characteristic
310	#332	C₂²×C₃₁	1	2	yes	—
311	#332	C₂²×C₃₁	1	2	yes	—
312	#332	C₂²×C₃₁	1	2	yes	—
313	#331	C₂²×C₃₁	1	1	yes	characteristic
314	#331	C₂²×C₃₁	1	1	yes	characteristic
315	#331	C₂²×C₃₁	1	2	yes	—
316	C₂×C₃₁	—	2	2	no	—
317	C₂×C₃₁	—	4	2	no	—
318	C₂×C₃₁	—	8	2	no	—
319	C₂×C₃₁	#2889	1	1	yes	characteristic
320	C₂×C₃₁	C₂×#418	1	1	yes	characteristic
321	C₂×C₃₁	#2898	1	1	yes	characteristic
322	2₋¹⁺⁴	—	4	4	no	—
323	2₋¹⁺⁴	—	4	4	no	—
324	C₂×#41	—	2	2	no	—
325	C₂×#41	—	2	4	no	—
326	C₂×#41	—	2	2	no	—
327	C₂×#41	—	2	1	no	—
328	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
329	C₂²×Q₈	—	2	2	no	—
330	C₂²×Q₈	—	2	2	no	—
331	C₂²×Q₈	—	2	1	no	—
332	C₂×M₁₆	—	2	1	no	—
333	C₂×M₁₆	—	2	2	no	—
334	C₂²×C₈	—	2	1	no	—
335	C₂²×C₈	—	2	1	no	—
336	C₄⋊Q₈	—	2	2	no	—
337	C₄⋊Q₈	—	2	1	no	—
338	C₄⋊Q₈	—	2	1	no	—
339	C₄⋊Q₈	—	2	2	no	—
340	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
341	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
342	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
343	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
344	#127	C₂³×C₃₁	1	1	yes	characteristic
345	#124	—	2	2	no	—
346	#124	—	2	1	no	—
347	#123	—	2	4	no	—
348	#123	—	2	4	no	—
349	#123	—	2	4	no	—
350	C₂²⋊Q₈	—	2	4	no	—
351	C₂²⋊Q₈	—	2	2	no	—
352	C₂²⋊Q₈	—	2	4	no	—
353	C₂²⋊Q₈	—	2	2	no	—
354	#121	—	2	1	no	—
355	#121	—	2	2	no	—
356	#121	C₂×C₄×C₃₁	1	2	yes	—
357	#121	C₂×C₄×C₃₁	1	4	yes	—
358	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
359	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
360	C₄×D₄	—	2	1	no	—
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
363	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
364	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
365	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
366	C₄×D₄	D₄×C₃₁	1	2	yes	—
367	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
368	#117	D₄×C₃₁	1	1	yes	characteristic
369	#117	D₄×C₃₁	1	2	yes	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	2	no	—
372	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
373	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
374	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
375	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
376	C₄⋊C₈	—	2	1	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	—	2	1	no	—
379	C₄⋊C₈	—	2	2	no	—
380	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
381	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
382	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
383	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
384	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
385	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
386	C₂²⋊C₈	—	2	2	no	—
387	C₂²⋊C₈	—	2	2	no	—
388	C₂²⋊C₈	—	2	1	no	—
389	C₂²⋊C₈	—	2	2	no	—
390	#97	—	2	2	no	—
391	#97	—	2	2	no	—
392	C₄×C₈	—	2	2	no	—
393	C₄×C₈	—	2	1	no	—
394	C₃₁	#17759	1	1	yes	characteristic, direct product factor, Sylow
395	#41	—	2	2	no	—
396	#41	—	2	2	no	—
397	#41	—	4	4	no	—
398	#41	—	4	4	no	—
399	#41	—	4	16	no	—
400	#41	—	4	8	no	—
401	#41	—	4	2	no	—
402	#41	—	4	2	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	—	2	1	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	4	4	no	—
408	C₂×Q₈	—	4	4	no	—
409	C₂×Q₈	—	4	8	no	—
410	C₂×Q₈	—	4	2	no	—
411	C₂×Q₈	—	4	2	no	—
412	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
413	C₂×D₄	—	2	4	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
418	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
419	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
425	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
427	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
428	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
429	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
430	M₁₆	—	4	2	no	—
431	M₁₆	—	8	2	no	—
432	C₂×C₈	—	2	1	no	—
433	C₂×C₈	—	2	1	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	2	no	—
438	C₂×C₈	—	4	2	no	—
439	C₂×C₈	—	8	1	no	—
440	C₄⋊C₄	—	2	4	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
447	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
448	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
449	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
450	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
451	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	2	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
457	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
458	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
459	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
460	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
461	C₄²	C₂×D₄×C₃₁	1	2	yes	—
462	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₄²	C₂×D₄×C₃₁	1	2	yes	—
464	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
466	C₂³	—	2	2	no	—
467	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
468	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
469	Q₈	—	2	2	no	—
470	Q₈	—	4	8	no	—
471	Q₈	—	4	4	no	—
472	Q₈	—	4	1	no	—
473	Q₈	—	4	1	no	—
474	Q₈	—	4	2	no	—
475	Q₈	—	4	2	no	—
476	D₄	—	2	4	no	—
477	D₄	—	2	2	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	8	no	—
480	D₄	—	4	4	no	—
481	D₄	—	4	1	no	—
482	D₄	—	4	1	no	—
483	D₄	—	4	1	no	—
484	D₄	—	4	1	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	4	1	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	8	no	—
496	C₂×C₄	—	4	8	no	—
497	C₂×C₄	—	4	4	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
500	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
501	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
502	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
503	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
504	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
505	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
506	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
507	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
508	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
509	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
510	C₈	—	4	1	no	—
511	C₈	—	4	1	no	—
512	C₈	—	4	2	no	—
513	C₈	—	4	2	no	—
514	C₈	—	4	1	no	—
515	C₈	—	4	1	no	—
516	C₂²	—	2	1	no	—
517	C₂²	—	2	1	no	—
518	C₂²	—	2	2	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	2	no	—
521	C₂²	—	4	2	no	—
522	C₂²	—	4	2	no	—
523	C₂²	—	4	2	no	—
524	C₂²	C₃₁×#418	1	2	yes	—
525	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
526	C₄	—	2	1	no	—
527	C₄	—	2	1	no	—
528	C₄	—	2	2	no	—
529	C₄	—	2	2	no	—
530	C₄	—	2	1	no	—
531	C₄	—	2	2	no	—
532	C₄	—	4	4	no	—
533	C₄	—	4	2	no	—
534	C₄	C₃₁×#418	1	2	yes	—
535	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
536	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
537	C₄	C₃₁×#577	1	1	yes	characteristic
538	C₄	C₃₁×#578	1	1	yes	characteristic
539	C₂	—	2	2	no	—
540	C₂	—	4	2	no	—
541	C₂	—	8	2	no	—
542	C₂	C₃₁×#2889	1	1	yes	characteristic
543	C₂	C₂×C₃₁×#418	1	1	yes	characteristic
544	C₂	C₃₁×#2898	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2989	C₂	semidirect
3. C₃₁×#2925	C₂	semidirect
4. C₃₁×#1557	C₂	semidirect
5. C₃₁×#1556	C₂	semidirect
6. C₃₁×#1553	C₂	semidirect
7. C₃₁×#1549	C₂	semidirect
8. C₃₁×#1547	C₂	semidirect
9. C₃₁×#1546	C₂	semidirect
10. C₃₁×#1540	C₂	semidirect
11. C₂×C₃₁×#332	C₂	semidirect
193. #17759	C₃₁	direct (N × H)
245. #2989	C₂×C₃₁	semidirect
246. #2925	C₂×C₃₁	semidirect
247. #1557	C₂×C₃₁	semidirect
248. #1556	C₂×C₃₁	semidirect
249. #1553	C₂×C₃₁	semidirect
250. #1549	C₂×C₃₁	semidirect
251. #1547	C₂×C₃₁	semidirect
252. #1546	C₂×C₃₁	semidirect
253. #1540	C₂×C₃₁	semidirect
254. C₂×#332	C₂×C₃₁	semidirect
394. C₃₁	#17759	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×#3499	2
C₃₁×#2989	2
C₃₁×#2925	2
C₃₁×#1557	2
C₃₁×#1556	2
C₃₁×#1553	2
C₃₁×#1549	2
C₃₁×#1547	2
C₃₁×#1546	2
C₃₁×#1540	2
C₂×C₃₁×#332	2
#17759	31

Subgroups — C31×#17759 #21474518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17759 #21474518