Subgroups — C₃₁×#21070 #21477829

Order 7936.

1026 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#430	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#421	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2896	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#408	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1929	C₂	1	8	yes	maximal, complemented
7	C₃₁×#1733	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×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	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₃₁×#430	—	2	4	no	—
17	C₂×C₃₁×#108	C₂²	1	4	yes	complemented
18	C₃₁×#421	—	2	4	no	—
19	C₂×C₃₁×#104	C₂²	1	4	yes	—
20	C₃₁×#419	C₄	1	2	yes	complemented
21	C₃₁×#418	C₄	1	2	yes	complemented
22	C₃₁×#417	C₄	1	4	yes	complemented
23	C₃₁×#416	C₂²	1	4	yes	—
24	C₃₁×#408	—	2	1	no	—
25	C₃₁×#408	—	2	1	no	—
26	C₃₁×#408	C₂²	1	4	yes	—
27	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
28	C₃₁×#405	—	2	2	no	—
29	C₃₁×#405	C₂²	1	4	yes	complemented
30	C₂×C₃₁×#97	—	2	1	no	—
31	C₂×C₄×C₈×C₃₁	—	2	1	no	—
32	C₃₁×#339	C₂²	1	4	yes	complemented
33	C₃₁×#336	C₂²	1	4	yes	complemented
34	C₂×#41×C₃₁	—	2	1	no	—
35	C₂×#41×C₃₁	—	2	1	no	—
36	C₂×#41×C₃₁	—	2	2	no	—
37	C₂×#41×C₃₁	—	2	2	no	—
38	C₂×#41×C₃₁	—	2	2	no	—
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×#41×C₃₁	C₂³	1	2	yes	—
42	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	—	2	1	no	—
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₃₁	—	2	1	no	—
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	2	1	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
57	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
58	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₂²×C₈×C₃₁	—	2	2	no	—
61	C₂²×C₈×C₃₁	—	2	2	no	—
62	C₂²×C₈×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	D₄	1	2	yes	—
66	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
67	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
68	C₃₁×#117	—	2	1	no	—
69	C₃₁×#117	—	2	1	no	—
70	C₃₁×#117	—	2	2	no	—
71	C₃₁×#117	—	2	2	no	—
72	C₃₁×#117	C₂×C₄	1	2	yes	—
73	C₃₁×#117	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄²×C₃₁	—	2	1	no	—
79	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
80	C₃₁×#108	—	2	8	no	—
81	C₃₁×#104	—	2	8	no	—
82	C₃₁×#103	C₂×C₄	1	8	yes	—
83	C₃₁×#102	C₂×C₄	1	8	yes	—
84	C₃₁×C₂²⋊C₈	—	2	4	no	—
85	C₃₁×C₂²⋊C₈	—	2	4	no	—
86	C₃₁×#97	—	2	4	no	—
87	C₄×C₈×C₃₁	—	2	4	no	—
88	C₂⁴×C₃₁	—	2	1	no	—
89	C₂⁴×C₃₁	#41	1	1	yes	characteristic
90	#41×C₃₁	—	2	4	no	—
91	#41×C₃₁	—	2	4	no	—
92	#41×C₃₁	—	2	2	no	—
93	#41×C₃₁	—	2	2	no	—
94	#41×C₃₁	—	2	2	no	—
95	#41×C₃₁	—	2	2	no	—
96	#41×C₃₁	—	4	2	no	—
97	#41×C₃₁	—	4	2	no	—
98	#41×C₃₁	—	4	2	no	—
99	#41×C₃₁	—	4	2	no	—
100	C₂×Q₈×C₃₁	—	2	1	no	—
101	C₂×Q₈×C₃₁	—	2	2	no	—
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	4	2	no	—
110	C₂×D₄×C₃₁	—	4	2	no	—
111	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	4	2	no	—
118	C₂²×C₄×C₃₁	—	4	2	no	—
119	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	#41	1	2	yes	—
125	C₂²×C₄×C₃₁	#41	1	2	yes	—
126	C₂²×C₄×C₃₁	#41	1	2	yes	—
127	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
129	M₁₆×C₃₁	—	2	4	no	—
130	M₁₆×C₃₁	—	2	4	no	—
131	M₁₆×C₃₁	—	2	4	no	—
132	M₁₆×C₃₁	—	2	2	no	—
133	M₁₆×C₃₁	—	2	2	no	—
134	M₁₆×C₃₁	—	2	2	no	—
135	M₁₆×C₃₁	—	4	2	no	—
136	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
137	C₂×C₈×C₃₁	—	2	4	no	—
138	C₂×C₈×C₃₁	—	2	4	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	4	2	no	—
148	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
149	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
150	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	4	2	no	—
153	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
154	C₂²⋊C₄×C₃₁	—	2	2	no	—
155	C₂²⋊C₄×C₃₁	—	2	2	no	—
156	C₂²⋊C₄×C₃₁	—	4	2	no	—
157	C₄²×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	—	2	2	no	—
159	C₄²×C₃₁	—	2	2	no	—
160	C₄²×C₃₁	—	2	2	no	—
161	#21070	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	—	4	2	no	—
164	C₂³×C₃₁	—	4	4	no	—
165	C₂³×C₃₁	—	4	2	no	—
166	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
167	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
168	C₂³×C₃₁	#124	1	2	yes	—
169	C₂³×C₃₁	#126	1	2	yes	—
170	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
171	Q₈×C₃₁	—	2	2	no	—
172	Q₈×C₃₁	—	2	2	no	—
173	Q₈×C₃₁	—	4	2	no	—
174	D₄×C₃₁	—	2	2	no	—
175	D₄×C₃₁	—	2	2	no	—
176	D₄×C₃₁	—	4	2	no	—
177	D₄×C₃₁	—	4	2	no	—
178	D₄×C₃₁	—	4	4	no	—
179	D₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	2	4	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	4	4	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	4	no	—
187	C₂×C₄×C₃₁	—	4	4	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	#117	1	2	yes	—
190	C₂×C₄×C₃₁	#117	1	2	yes	—
191	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
192	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
193	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
195	C₂×C₄×C₃₁	#121	1	2	yes	—
196	C₂×C₄×C₃₁	#121	1	2	yes	—
197	C₂×C₄×C₃₁	#123	1	2	yes	—
198	C₂×C₄×C₃₁	#123	1	2	yes	—
199	C₂×C₄×C₃₁	#124	1	2	yes	—
200	C₂×C₄×C₃₁	#126	1	2	yes	—
201	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
207	C₈×C₃₁	—	2	4	no	—
208	C₈×C₃₁	—	2	4	no	—
209	C₈×C₃₁	—	2	2	no	—
210	C₈×C₃₁	—	2	2	no	—
211	C₈×C₃₁	—	4	2	no	—
212	C₂×#430	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂×#421	C₂×C₃₁	1	1	yes	characteristic
214	#2896	C₂×C₃₁	1	1	yes	characteristic
215	C₂×#408	C₂×C₃₁	1	1	yes	characteristic
216	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1929	C₂×C₃₁	1	8	yes	complemented
218	#1733	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#1729	C₂×C₃₁	1	1	yes	characteristic, complemented
220	C₂²×C₃₁	—	2	2	no	—
221	C₂²×C₃₁	—	2	2	no	—
222	C₂²×C₃₁	—	2	2	no	—
223	C₂²×C₃₁	—	4	2	no	—
224	C₂²×C₃₁	—	4	2	no	—
225	C₂²×C₃₁	—	4	4	no	—
226	C₂²×C₃₁	—	4	2	no	—
227	C₂²×C₃₁	—	8	4	no	—
228	C₂²×C₃₁	#388	1	2	yes	—
229	C₂²×C₃₁	#388	1	2	yes	—
230	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
231	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
232	C₂²×C₃₁	C₂×#126	1	1	yes	characteristic
233	C₄×C₃₁	—	4	4	no	—
234	C₄×C₃₁	—	4	2	no	—
235	C₄×C₃₁	—	8	2	no	—
236	C₄×C₃₁	#388	1	2	yes	—
237	C₄×C₃₁	#388	1	2	yes	—
238	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
239	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
240	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
241	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
242	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
243	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
244	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
245	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
246	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
249	#430	—	2	4	no	—
250	C₂×#108	C₂²×C₃₁	1	4	yes	complemented
251	#421	—	2	4	no	—
252	C₂×#104	C₂²×C₃₁	1	4	yes	—
253	#419	C₄×C₃₁	1	2	yes	complemented
254	#418	C₄×C₃₁	1	2	yes	complemented
255	#417	C₄×C₃₁	1	4	yes	complemented
256	#416	C₂²×C₃₁	1	4	yes	—
257	#408	—	2	1	no	—
258	#408	—	2	1	no	—
259	#408	C₂²×C₃₁	1	4	yes	—
260	C₂×C₂²⋊C₈	—	2	2	no	—
261	#405	—	2	2	no	—
262	#405	C₂²×C₃₁	1	4	yes	complemented
263	C₂×#97	—	2	1	no	—
264	C₂×C₄×C₈	—	2	1	no	—
265	#339	C₂²×C₃₁	1	4	yes	complemented
266	#336	C₂²×C₃₁	1	4	yes	complemented
267	C₂×C₃₁	—	2	1	no	—
268	C₂×C₃₁	—	2	2	no	—
269	C₂×C₃₁	—	2	1	no	—
270	C₂×C₃₁	—	2	2	no	—
271	C₂×C₃₁	—	8	2	no	—
272	C₂×C₃₁	C₂×#388	1	1	yes	characteristic
273	C₂×C₃₁	#2955	1	2	yes	—
274	C₂×#41	—	2	1	no	—
275	C₂×#41	—	2	1	no	—
276	C₂×#41	—	2	2	no	—
277	C₂×#41	—	2	2	no	—
278	C₂×#41	—	2	2	no	—
279	C₂×#41	—	2	2	no	—
280	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
281	C₂×#41	C₂³×C₃₁	1	2	yes	—
282	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
283	C₂²×D₄	—	2	1	no	—
284	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
285	C₂³×C₄	—	2	1	no	—
286	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
287	C₂×M₁₆	—	2	1	no	—
288	C₂×M₁₆	—	2	2	no	—
289	C₂×M₁₆	—	2	2	no	—
290	C₂×M₁₆	—	2	2	no	—
291	C₂×M₁₆	—	2	1	no	—
292	C₂×M₁₆	—	2	2	no	—
293	C₂×M₁₆	—	2	2	no	—
294	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
295	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
296	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
297	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
298	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
299	C₂²×C₈	—	2	1	no	—
300	C₂²×C₈	—	2	2	no	—
301	C₂²×C₈	—	2	2	no	—
302	C₂²×C₈	—	2	1	no	—
303	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
304	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
305	C₂²×C₈	D₄×C₃₁	1	2	yes	—
306	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
307	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
308	#117	—	2	1	no	—
309	#117	—	2	1	no	—
310	#117	—	2	2	no	—
311	#117	—	2	2	no	—
312	#117	C₂×C₄×C₃₁	1	2	yes	—
313	#117	C₂³×C₃₁	1	2	yes	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
316	C₂×C₂²⋊C₄	—	2	1	no	—
317	C₂×C₂²⋊C₄	—	2	1	no	—
318	C₂×C₄²	—	2	1	no	—
319	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
320	#108	—	2	8	no	—
321	#104	—	2	8	no	—
322	#103	C₂×C₄×C₃₁	1	8	yes	—
323	#102	C₂×C₄×C₃₁	1	8	yes	—
324	C₂²⋊C₈	—	2	4	no	—
325	C₂²⋊C₈	—	2	4	no	—
326	#97	—	2	4	no	—
327	C₄×C₈	—	2	4	no	—
328	C₃₁	#21070	1	1	yes	characteristic, direct product factor, Sylow
329	C₂⁴	—	2	1	no	—
330	C₂⁴	#41×C₃₁	1	1	yes	characteristic
331	#41	—	2	4	no	—
332	#41	—	2	4	no	—
333	#41	—	2	2	no	—
334	#41	—	2	2	no	—
335	#41	—	2	2	no	—
336	#41	—	2	2	no	—
337	#41	—	4	2	no	—
338	#41	—	4	2	no	—
339	#41	—	4	2	no	—
340	#41	—	4	2	no	—
341	C₂×Q₈	—	2	1	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	4	2	no	—
351	C₂×D₄	—	4	2	no	—
352	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	4	2	no	—
359	C₂²×C₄	—	4	2	no	—
360	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
361	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
363	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₂²×C₄	#41×C₃₁	1	2	yes	—
366	C₂²×C₄	#41×C₃₁	1	2	yes	—
367	C₂²×C₄	#41×C₃₁	1	2	yes	—
368	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
369	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
370	M₁₆	—	2	4	no	—
371	M₁₆	—	2	4	no	—
372	M₁₆	—	2	4	no	—
373	M₁₆	—	2	2	no	—
374	M₁₆	—	2	2	no	—
375	M₁₆	—	2	2	no	—
376	M₁₆	—	4	2	no	—
377	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
378	C₂×C₈	—	2	4	no	—
379	C₂×C₈	—	2	4	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	4	2	no	—
389	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
390	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
391	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	4	2	no	—
394	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
395	C₂²⋊C₄	—	2	2	no	—
396	C₂²⋊C₄	—	2	2	no	—
397	C₂²⋊C₄	—	4	2	no	—
398	C₄²	—	2	2	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	2	no	—
402	C₂³	—	2	2	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	4	no	—
405	C₂³	—	4	2	no	—
406	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
407	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
408	C₂³	C₃₁×#124	1	2	yes	—
409	C₂³	C₃₁×#126	1	2	yes	—
410	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
411	Q₈	—	2	2	no	—
412	Q₈	—	2	2	no	—
413	Q₈	—	4	2	no	—
414	D₄	—	2	2	no	—
415	D₄	—	2	2	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	2	no	—
418	D₄	—	4	4	no	—
419	D₄	—	4	2	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	4	no	—
427	C₂×C₄	—	4	4	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	C₃₁×#117	1	2	yes	—
430	C₂×C₄	C₃₁×#117	1	2	yes	—
431	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
432	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
433	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
435	C₂×C₄	C₃₁×#121	1	2	yes	—
436	C₂×C₄	C₃₁×#121	1	2	yes	—
437	C₂×C₄	C₃₁×#123	1	2	yes	—
438	C₂×C₄	C₃₁×#123	1	2	yes	—
439	C₂×C₄	C₃₁×#124	1	2	yes	—
440	C₂×C₄	C₃₁×#126	1	2	yes	—
441	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
442	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
443	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
444	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
445	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
446	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
447	C₈	—	2	4	no	—
448	C₈	—	2	4	no	—
449	C₈	—	2	2	no	—
450	C₈	—	2	2	no	—
451	C₈	—	4	2	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	2	2	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	4	4	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	8	4	no	—
460	C₂²	C₃₁×#388	1	2	yes	—
461	C₂²	C₃₁×#388	1	2	yes	—
462	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
463	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
464	C₂²	C₂×C₃₁×#126	1	1	yes	characteristic
465	C₄	—	4	4	no	—
466	C₄	—	4	2	no	—
467	C₄	—	8	2	no	—
468	C₄	C₃₁×#388	1	2	yes	—
469	C₄	C₃₁×#388	1	2	yes	—
470	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
471	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
472	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
473	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
474	C₂	—	2	1	no	—
475	C₂	—	2	2	no	—
476	C₂	—	2	1	no	—
477	C₂	—	2	2	no	—
478	C₂	—	8	2	no	—
479	C₂	C₂×C₃₁×#388	1	1	yes	characteristic
480	C₂	C₃₁×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#430	C₂	semidirect
5. C₂×C₃₁×#405	C₂	semidirect
6. C₃₁×#1929	C₂	semidirect
7. C₃₁×#1733	C₂	semidirect
8. C₃₁×#1729	C₂	semidirect
17. C₂×C₃₁×#108	C₂²	semidirect
20. C₃₁×#419	C₄	semidirect
21. C₃₁×#418	C₄	semidirect
22. C₃₁×#417	C₄	semidirect
29. C₃₁×#405	C₂²	semidirect
32. C₃₁×#339	C₂²	semidirect
33. C₃₁×#336	C₂²	semidirect
161. #21070	C₃₁	direct (N × H)
212. C₂×#430	C₂×C₃₁	semidirect
216. C₂×#405	C₂×C₃₁	semidirect
217. #1929	C₂×C₃₁	semidirect
218. #1733	C₂×C₃₁	semidirect
219. #1729	C₂×C₃₁	semidirect
250. C₂×#108	C₂²×C₃₁	semidirect
253. #419	C₄×C₃₁	semidirect
254. #418	C₄×C₃₁	semidirect
255. #417	C₄×C₃₁	semidirect
262. #405	C₂²×C₃₁	semidirect
265. #339	C₂²×C₃₁	semidirect
266. #336	C₂²×C₃₁	semidirect
328. C₃₁	#21070	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×#430	2
C₂×C₃₁×#421	2
C₃₁×#2896	2
C₂×C₃₁×#408	2
C₂×C₃₁×#405	2
C₃₁×#1929	2
C₃₁×#1733	2
C₃₁×#1729	2
#21070	31

Subgroups — C31×#21070 #21477829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21070 #21477829