Subgroups — C₃₁×#22061 #21478820

Order 7936.

914 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3002	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#408	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1975	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1938	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1929	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1876	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1871	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1855	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1789	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1752	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#135	C₂²	1	2	yes	—
14	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic, complemented
15	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#525	C₂²	1	1	yes	characteristic
18	C₃₁×#452	C₄	1	2	yes	complemented
19	C₃₁×#451	C₄	1	1	yes	characteristic, complemented
20	C₃₁×#450	C₄	1	1	yes	characteristic, complemented
21	C₃₁×#449	C₄	1	2	yes	complemented
22	C₃₁×#448	C₄	1	1	yes	characteristic, complemented
23	C₃₁×#447	C₄	1	1	yes	characteristic, complemented
24	C₃₁×#435	—	2	1	no	—
25	C₃₁×#435	—	2	1	no	—
26	C₈×D₄×C₃₁	—	2	1	no	—
27	C₈×D₄×C₃₁	—	2	1	no	—
28	C₂×C₃₁×#108	C₂²	1	2	yes	complemented
29	C₃₁×#424	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₃₁×#104	C₂²	1	2	yes	—
31	C₂×C₃₁×#104	C₂²	1	2	yes	—
32	C₂×C₃₁×#104	C₂²	1	2	yes	complemented
33	C₃₁×#416	C₂²	1	2	yes	—
34	C₃₁×#408	—	2	1	no	—
35	C₃₁×#408	—	2	1	no	—
36	C₃₁×#408	C₂²	1	2	yes	—
37	C₃₁×#408	C₂²	1	2	yes	—
38	C₃₁×#408	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#407	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
41	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
42	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×#405	C₂²	1	1	yes	characteristic, complemented
44	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
45	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
46	C₃₁×#341	C₂²	1	2	yes	complemented
47	C₃₁×#339	C₂²	1	2	yes	complemented
48	C₃₁×#336	C₂²	1	2	yes	complemented
49	C₂×#41×C₃₁	—	2	1	no	—
50	C₂×#41×C₃₁	—	2	1	no	—
51	C₂×#41×C₃₁	—	4	2	no	—
52	C₂×#41×C₃₁	—	4	2	no	—
53	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×#41×C₃₁	C₂³	1	2	yes	—
55	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂²×D₄×C₃₁	—	2	1	no	—
58	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂³×C₄×C₃₁	—	2	1	no	—
60	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#135	—	4	2	no	—
62	C₃₁×#135	—	4	2	no	—
63	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
64	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
65	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
66	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
67	C₃₁×#131	—	4	2	no	—
68	C₂×M₁₆×C₃₁	—	2	1	no	—
69	C₂×M₁₆×C₃₁	—	2	1	no	—
70	C₂×M₁₆×C₃₁	—	4	2	no	—
71	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
74	C₂²×C₈×C₃₁	—	2	1	no	—
75	C₂²×C₈×C₃₁	—	2	1	no	—
76	C₂²×C₈×C₃₁	—	4	2	no	—
77	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
82	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#123	—	2	1	no	—
85	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
86	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
87	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
88	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
89	C₃₁×#120	—	2	1	no	—
90	C₄×D₄×C₃₁	—	2	1	no	—
91	C₄×D₄×C₃₁	—	2	1	no	—
92	C₃₁×#117	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
94	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
95	C₃₁×#108	—	4	2	no	—
96	C₃₁×C₄⋊C₈	—	2	1	no	—
97	C₃₁×C₄⋊C₈	—	2	1	no	—
98	C₃₁×#104	—	4	2	no	—
99	C₃₁×#104	—	4	2	no	—
100	C₃₁×#104	—	4	2	no	—
101	C₃₁×#103	C₂×C₄	1	2	yes	—
102	C₃₁×#103	C₂×C₄	1	2	yes	—
103	C₃₁×#102	C₂×C₄	1	2	yes	—
104	C₃₁×#102	C₂×C₄	1	2	yes	—
105	C₃₁×C₂²⋊C₈	—	2	1	no	—
106	C₃₁×C₂²⋊C₈	—	2	1	no	—
107	C₃₁×C₂²⋊C₈	—	2	2	no	—
108	C₃₁×C₂²⋊C₈	—	2	2	no	—
109	C₃₁×C₂²⋊C₈	—	2	2	no	—
110	C₃₁×C₂²⋊C₈	—	2	2	no	—
111	C₃₁×C₂²⋊C₈	—	2	1	no	—
112	C₃₁×C₂²⋊C₈	—	2	1	no	—
113	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
114	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
115	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
116	C₃₁×#97	—	2	1	no	—
117	C₃₁×#97	—	4	2	no	—
118	C₄×C₈×C₃₁	—	2	1	no	—
119	C₃₁×#95	—	2	1	no	—
120	C₂⁴×C₃₁	—	2	1	no	—
121	C₂⁴×C₃₁	#41	1	1	yes	characteristic
122	#41×C₃₁	—	2	2	no	—
123	#41×C₃₁	—	2	2	no	—
124	#41×C₃₁	—	2	2	no	—
125	#41×C₃₁	—	2	2	no	—
126	#41×C₃₁	—	2	2	no	—
127	#41×C₃₁	—	2	2	no	—
128	#41×C₃₁	—	4	2	no	—
129	#41×C₃₁	—	4	2	no	—
130	#41×C₃₁	—	4	4	no	—
131	#41×C₃₁	—	4	2	no	—
132	#41×C₃₁	—	4	2	no	—
133	C₂×Q₈×C₃₁	—	2	1	no	—
134	C₂×Q₈×C₃₁	—	4	2	no	—
135	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
136	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
137	C₂×D₄×C₃₁	—	2	1	no	—
138	C₂×D₄×C₃₁	—	2	1	no	—
139	C₂×D₄×C₃₁	—	2	2	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	4	2	no	—
144	C₂×D₄×C₃₁	—	4	4	no	—
145	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
146	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	—	2	2	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	1	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	—	2	1	no	—
153	C₂²×C₄×C₃₁	—	2	2	no	—
154	C₂²×C₄×C₃₁	—	4	2	no	—
155	C₂²×C₄×C₃₁	—	4	2	no	—
156	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	Q₁₆×C₃₁	—	4	2	no	—
163	SD₁₆×C₃₁	—	4	2	no	—
164	SD₁₆×C₃₁	—	4	2	no	—
165	D₈×C₃₁	—	4	2	no	—
166	M₁₆×C₃₁	—	2	2	no	—
167	M₁₆×C₃₁	—	2	2	no	—
168	M₁₆×C₃₁	—	2	2	no	—
169	M₁₆×C₃₁	—	2	2	no	—
170	M₁₆×C₃₁	—	4	2	no	—
171	M₁₆×C₃₁	—	4	2	no	—
172	C₂×C₈×C₃₁	—	2	2	no	—
173	C₂×C₈×C₃₁	—	2	2	no	—
174	C₂×C₈×C₃₁	—	2	2	no	—
175	C₂×C₈×C₃₁	—	2	2	no	—
176	C₂×C₈×C₃₁	—	2	1	no	—
177	C₂×C₈×C₃₁	—	2	2	no	—
178	C₂×C₈×C₃₁	—	2	1	no	—
179	C₂×C₈×C₃₁	—	4	1	no	—
180	C₂×C₈×C₃₁	—	4	1	no	—
181	C₂×C₈×C₃₁	—	4	1	no	—
182	C₂×C₈×C₃₁	—	4	1	no	—
183	C₂×C₈×C₃₁	—	4	2	no	—
184	C₂×C₈×C₃₁	—	4	2	no	—
185	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
186	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
187	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₄⋊C₄×C₃₁	—	2	1	no	—
192	C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
194	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
195	C₂²⋊C₄×C₃₁	—	2	1	no	—
196	C₂²⋊C₄×C₃₁	—	2	1	no	—
197	C₂²⋊C₄×C₃₁	—	2	1	no	—
198	C₂²⋊C₄×C₃₁	—	2	1	no	—
199	C₂²⋊C₄×C₃₁	—	2	1	no	—
200	C₂²⋊C₄×C₃₁	—	4	2	no	—
201	C₄²×C₃₁	—	2	1	no	—
202	C₄²×C₃₁	—	2	2	no	—
203	C₄²×C₃₁	—	2	2	no	—
204	#22061	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂³×C₃₁	—	2	2	no	—
206	C₂³×C₃₁	—	2	2	no	—
207	C₂³×C₃₁	—	2	1	no	—
208	C₂³×C₃₁	—	2	2	no	—
209	C₂³×C₃₁	—	4	1	no	—
210	C₂³×C₃₁	—	4	2	no	—
211	C₂³×C₃₁	—	4	2	no	—
212	C₂³×C₃₁	—	4	2	no	—
213	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
214	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
215	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
216	Q₈×C₃₁	—	2	2	no	—
217	Q₈×C₃₁	—	2	2	no	—
218	Q₈×C₃₁	—	2	2	no	—
219	Q₈×C₃₁	—	4	2	no	—
220	D₄×C₃₁	—	2	2	no	—
221	D₄×C₃₁	—	2	2	no	—
222	D₄×C₃₁	—	2	2	no	—
223	D₄×C₃₁	—	4	2	no	—
224	D₄×C₃₁	—	4	2	no	—
225	D₄×C₃₁	—	4	4	no	—
226	D₄×C₃₁	—	4	2	no	—
227	D₄×C₃₁	—	4	2	no	—
228	D₄×C₃₁	—	4	2	no	—
229	D₄×C₃₁	—	4	2	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	1	no	—
234	C₂×C₄×C₃₁	—	2	1	no	—
235	C₂×C₄×C₃₁	—	2	1	no	—
236	C₂×C₄×C₃₁	—	2	1	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	1	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	4	4	no	—
242	C₂×C₄×C₃₁	—	4	2	no	—
243	C₂×C₄×C₃₁	—	4	1	no	—
244	C₂×C₄×C₃₁	—	4	2	no	—
245	C₂×C₄×C₃₁	—	4	2	no	—
246	C₂×C₄×C₃₁	—	4	1	no	—
247	C₂×C₄×C₃₁	—	4	2	no	—
248	C₂×C₄×C₃₁	—	4	2	no	—
249	C₂×C₄×C₃₁	—	4	2	no	—
250	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
252	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
253	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
255	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
256	C₈×C₃₁	—	2	2	no	—
257	C₈×C₃₁	—	2	2	no	—
258	C₈×C₃₁	—	2	1	no	—
259	C₈×C₃₁	—	2	1	no	—
260	C₈×C₃₁	—	2	1	no	—
261	C₈×C₃₁	—	2	1	no	—
262	C₈×C₃₁	—	4	1	no	—
263	C₈×C₃₁	—	4	2	no	—
264	C₈×C₃₁	—	4	1	no	—
265	#3002	C₂×C₃₁	1	1	yes	characteristic
266	#2929	C₂×C₃₁	1	1	yes	characteristic, complemented
267	C₂×#408	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#1975	C₂×C₃₁	1	2	yes	complemented
269	#1938	C₂×C₃₁	1	2	yes	complemented
270	#1929	C₂×C₃₁	1	2	yes	complemented
271	#1876	C₂×C₃₁	1	2	yes	complemented
272	#1871	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#1855	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#1789	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1752	C₂×C₃₁	1	1	yes	characteristic, complemented
276	C₂²×C₃₁	—	2	1	no	—
277	C₂²×C₃₁	—	2	1	no	—
278	C₂²×C₃₁	—	2	1	no	—
279	C₂²×C₃₁	—	2	1	no	—
280	C₂²×C₃₁	—	2	1	no	—
281	C₂²×C₃₁	—	2	1	no	—
282	C₂²×C₃₁	—	2	1	no	—
283	C₂²×C₃₁	—	4	4	no	—
284	C₂²×C₃₁	—	4	2	no	—
285	C₂²×C₃₁	—	4	1	no	—
286	C₂²×C₃₁	—	4	1	no	—
287	C₂²×C₃₁	—	4	2	no	—
288	C₂²×C₃₁	—	4	1	no	—
289	C₂²×C₃₁	—	4	2	no	—
290	C₂²×C₃₁	—	8	4	no	—
291	C₂²×C₃₁	#444	1	2	yes	—
292	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
293	C₂²×C₃₁	#538	1	2	yes	—
294	C₄×C₃₁	—	2	1	no	—
295	C₄×C₃₁	—	2	1	no	—
296	C₄×C₃₁	—	4	2	no	—
297	C₄×C₃₁	—	4	1	no	—
298	C₄×C₃₁	—	4	1	no	—
299	C₄×C₃₁	—	4	1	no	—
300	C₄×C₃₁	—	8	2	no	—
301	C₄×C₃₁	#518	1	2	yes	—
302	C₄×C₃₁	#534	1	1	yes	characteristic
303	C₄×C₃₁	#535	1	1	yes	characteristic
304	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×#135	C₂²×C₃₁	1	2	yes	—
306	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic, complemented
307	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
308	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
309	#525	C₂²×C₃₁	1	1	yes	characteristic
310	#452	C₄×C₃₁	1	2	yes	complemented
311	#451	C₄×C₃₁	1	1	yes	characteristic, complemented
312	#450	C₄×C₃₁	1	1	yes	characteristic, complemented
313	#449	C₄×C₃₁	1	2	yes	complemented
314	#448	C₄×C₃₁	1	1	yes	characteristic, complemented
315	#447	C₄×C₃₁	1	1	yes	characteristic, complemented
316	#435	—	2	1	no	—
317	#435	—	2	1	no	—
318	C₈×D₄	—	2	1	no	—
319	C₈×D₄	—	2	1	no	—
320	C₂×#108	C₂²×C₃₁	1	2	yes	complemented
321	#424	C₂²×C₃₁	1	1	yes	characteristic, complemented
322	C₂×#104	C₂²×C₃₁	1	2	yes	—
323	C₂×#104	C₂²×C₃₁	1	2	yes	—
324	C₂×#104	C₂²×C₃₁	1	2	yes	complemented
325	#416	C₂²×C₃₁	1	2	yes	—
326	#408	—	2	1	no	—
327	#408	—	2	1	no	—
328	#408	C₂²×C₃₁	1	2	yes	—
329	#408	C₂²×C₃₁	1	2	yes	—
330	#408	C₂²×C₃₁	1	1	yes	characteristic, complemented
331	#407	C₂²×C₃₁	1	1	yes	characteristic
332	C₂×C₂²⋊C₈	—	2	1	no	—
333	C₂×C₂²⋊C₈	—	2	1	no	—
334	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
335	#405	C₂²×C₃₁	1	1	yes	characteristic, complemented
336	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
337	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
338	#341	C₂²×C₃₁	1	2	yes	complemented
339	#339	C₂²×C₃₁	1	2	yes	complemented
340	#336	C₂²×C₃₁	1	2	yes	complemented
341	C₂×C₃₁	—	2	2	no	—
342	C₂×C₃₁	—	2	2	no	—
343	C₂×C₃₁	—	4	1	no	—
344	C₂×C₃₁	—	8	1	no	—
345	C₂×C₃₁	—	8	2	no	—
346	C₂×C₃₁	#2321	1	1	yes	characteristic
347	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
348	C₂×C₃₁	#2955	1	1	yes	characteristic
349	C₂×#41	—	2	1	no	—
350	C₂×#41	—	2	1	no	—
351	C₂×#41	—	4	2	no	—
352	C₂×#41	—	4	2	no	—
353	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
354	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×#41	C₂³×C₃₁	1	2	yes	—
356	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
357	C₂²×D₄	—	2	1	no	—
358	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
359	C₂³×C₄	—	2	1	no	—
360	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
361	#135	—	4	2	no	—
362	#135	—	4	2	no	—
363	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
364	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
365	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
366	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
367	#131	—	4	2	no	—
368	C₂×M₁₆	—	2	1	no	—
369	C₂×M₁₆	—	2	1	no	—
370	C₂×M₁₆	—	4	2	no	—
371	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
372	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
373	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
374	C₂²×C₈	—	2	1	no	—
375	C₂²×C₈	—	2	1	no	—
376	C₂²×C₈	—	4	2	no	—
377	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
379	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
380	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
382	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
383	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
384	#123	—	2	1	no	—
385	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
386	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
387	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
388	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
389	#120	—	2	1	no	—
390	C₄×D₄	—	2	1	no	—
391	C₄×D₄	—	2	1	no	—
392	#117	C₂³×C₃₁	1	1	yes	characteristic
393	C₂×C₂²⋊C₄	—	2	1	no	—
394	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
395	#108	—	4	2	no	—
396	C₄⋊C₈	—	2	1	no	—
397	C₄⋊C₈	—	2	1	no	—
398	#104	—	4	2	no	—
399	#104	—	4	2	no	—
400	#104	—	4	2	no	—
401	#103	C₂×C₄×C₃₁	1	2	yes	—
402	#103	C₂×C₄×C₃₁	1	2	yes	—
403	#102	C₂×C₄×C₃₁	1	2	yes	—
404	#102	C₂×C₄×C₃₁	1	2	yes	—
405	C₂²⋊C₈	—	2	1	no	—
406	C₂²⋊C₈	—	2	1	no	—
407	C₂²⋊C₈	—	2	2	no	—
408	C₂²⋊C₈	—	2	2	no	—
409	C₂²⋊C₈	—	2	2	no	—
410	C₂²⋊C₈	—	2	2	no	—
411	C₂²⋊C₈	—	2	1	no	—
412	C₂²⋊C₈	—	2	1	no	—
413	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
414	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
415	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
416	#97	—	2	1	no	—
417	#97	—	4	2	no	—
418	C₄×C₈	—	2	1	no	—
419	#95	—	2	1	no	—
420	C₃₁	#22061	1	1	yes	characteristic, direct product factor, Sylow
421	C₂⁴	—	2	1	no	—
422	C₂⁴	#41×C₃₁	1	1	yes	characteristic
423	#41	—	2	2	no	—
424	#41	—	2	2	no	—
425	#41	—	2	2	no	—
426	#41	—	2	2	no	—
427	#41	—	2	2	no	—
428	#41	—	2	2	no	—
429	#41	—	4	2	no	—
430	#41	—	4	2	no	—
431	#41	—	4	4	no	—
432	#41	—	4	2	no	—
433	#41	—	4	2	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	4	2	no	—
436	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
437	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	1	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	4	2	no	—
445	C₂×D₄	—	4	4	no	—
446	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
447	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	2	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	4	2	no	—
456	C₂²×C₄	—	4	2	no	—
457	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
458	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
459	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
462	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
463	Q₁₆	—	4	2	no	—
464	SD₁₆	—	4	2	no	—
465	SD₁₆	—	4	2	no	—
466	D₈	—	4	2	no	—
467	M₁₆	—	2	2	no	—
468	M₁₆	—	2	2	no	—
469	M₁₆	—	2	2	no	—
470	M₁₆	—	2	2	no	—
471	M₁₆	—	4	2	no	—
472	M₁₆	—	4	2	no	—
473	C₂×C₈	—	2	2	no	—
474	C₂×C₈	—	2	2	no	—
475	C₂×C₈	—	2	2	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	2	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	4	1	no	—
481	C₂×C₈	—	4	1	no	—
482	C₂×C₈	—	4	1	no	—
483	C₂×C₈	—	4	1	no	—
484	C₂×C₈	—	4	2	no	—
485	C₂×C₈	—	4	2	no	—
486	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
487	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
488	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
489	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
490	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
491	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
495	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
496	C₂²⋊C₄	—	2	1	no	—
497	C₂²⋊C₄	—	2	1	no	—
498	C₂²⋊C₄	—	2	1	no	—
499	C₂²⋊C₄	—	2	1	no	—
500	C₂²⋊C₄	—	2	1	no	—
501	C₂²⋊C₄	—	4	2	no	—
502	C₄²	—	2	1	no	—
503	C₄²	—	2	2	no	—
504	C₄²	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	2	no	—
507	C₂³	—	2	1	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	4	1	no	—
510	C₂³	—	4	2	no	—
511	C₂³	—	4	2	no	—
512	C₂³	—	4	2	no	—
513	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
514	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
515	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
516	Q₈	—	2	2	no	—
517	Q₈	—	2	2	no	—
518	Q₈	—	2	2	no	—
519	Q₈	—	4	2	no	—
520	D₄	—	2	2	no	—
521	D₄	—	2	2	no	—
522	D₄	—	2	2	no	—
523	D₄	—	4	2	no	—
524	D₄	—	4	2	no	—
525	D₄	—	4	4	no	—
526	D₄	—	4	2	no	—
527	D₄	—	4	2	no	—
528	D₄	—	4	2	no	—
529	D₄	—	4	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	4	4	no	—
542	C₂×C₄	—	4	2	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
551	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
552	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
553	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
554	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
555	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
556	C₈	—	2	2	no	—
557	C₈	—	2	2	no	—
558	C₈	—	2	1	no	—
559	C₈	—	2	1	no	—
560	C₈	—	2	1	no	—
561	C₈	—	2	1	no	—
562	C₈	—	4	1	no	—
563	C₈	—	4	2	no	—
564	C₈	—	4	1	no	—
565	C₂²	—	2	1	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	1	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	1	no	—
571	C₂²	—	2	1	no	—
572	C₂²	—	4	4	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	8	4	no	—
580	C₂²	C₃₁×#444	1	2	yes	—
581	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
582	C₂²	C₃₁×#538	1	2	yes	—
583	C₄	—	2	1	no	—
584	C₄	—	2	1	no	—
585	C₄	—	4	2	no	—
586	C₄	—	4	1	no	—
587	C₄	—	4	1	no	—
588	C₄	—	4	1	no	—
589	C₄	—	8	2	no	—
590	C₄	C₃₁×#518	1	2	yes	—
591	C₄	C₃₁×#534	1	1	yes	characteristic
592	C₄	C₃₁×#535	1	1	yes	characteristic
593	C₂	—	2	2	no	—
594	C₂	—	2	2	no	—
595	C₂	—	4	1	no	—
596	C₂	—	8	2	no	—
597	C₂	—	8	1	no	—
598	C₂	C₃₁×#2321	1	1	yes	characteristic
599	C₂	C₂×C₃₁×#444	1	1	yes	characteristic
600	C₂	C₃₁×#2955	1	1	yes	characteristic

	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	C₃₁×#3002	C₂
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
2. C₃₁×#2929	C₂	semidirect
3. C₂×C₃₁×#408	C₂	semidirect
4. C₃₁×#1975	C₂	semidirect
5. C₃₁×#1938	C₂	semidirect
6. C₃₁×#1929	C₂	semidirect
7. C₃₁×#1876	C₂	semidirect
8. C₃₁×#1871	C₂	semidirect
9. C₃₁×#1855	C₂	semidirect
10. C₃₁×#1789	C₂	semidirect
11. C₃₁×#1752	C₂	semidirect
14. C₂×C₃₁×#131	C₂²	semidirect
18. C₃₁×#452	C₄	semidirect
19. C₃₁×#451	C₄	semidirect
20. C₃₁×#450	C₄	semidirect
21. C₃₁×#449	C₄	semidirect
22. C₃₁×#448	C₄	semidirect
23. C₃₁×#447	C₄	semidirect
28. C₂×C₃₁×#108	C₂²	semidirect
29. C₃₁×#424	C₂²	semidirect
32. C₂×C₃₁×#104	C₂²	semidirect
38. C₃₁×#408	C₂²	semidirect
43. C₃₁×#405	C₂²	semidirect
44. C₂×C₃₁×#97	C₂²	semidirect
46. C₃₁×#341	C₂²	semidirect
47. C₃₁×#339	C₂²	semidirect
48. C₃₁×#336	C₂²	semidirect
204. #22061	C₃₁	direct (N × H)
266. #2929	C₂×C₃₁	semidirect
267. C₂×#408	C₂×C₃₁	semidirect
268. #1975	C₂×C₃₁	semidirect
269. #1938	C₂×C₃₁	semidirect
270. #1929	C₂×C₃₁	semidirect
271. #1876	C₂×C₃₁	semidirect
272. #1871	C₂×C₃₁	semidirect
273. #1855	C₂×C₃₁	semidirect
274. #1789	C₂×C₃₁	semidirect
275. #1752	C₂×C₃₁	semidirect
306. C₂×#131	C₂²×C₃₁	semidirect
310. #452	C₄×C₃₁	semidirect
311. #451	C₄×C₃₁	semidirect
312. #450	C₄×C₃₁	semidirect
313. #449	C₄×C₃₁	semidirect
314. #448	C₄×C₃₁	semidirect
315. #447	C₄×C₃₁	semidirect
320. C₂×#108	C₂²×C₃₁	semidirect
321. #424	C₂²×C₃₁	semidirect
324. C₂×#104	C₂²×C₃₁	semidirect
330. #408	C₂²×C₃₁	semidirect
335. #405	C₂²×C₃₁	semidirect
336. C₂×#97	C₂²×C₃₁	semidirect
338. #341	C₂²×C₃₁	semidirect
339. #339	C₂²×C₃₁	semidirect
340. #336	C₂²×C₃₁	semidirect
420. C₃₁	#22061	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×#3002	2
C₃₁×#2929	2
C₂×C₃₁×#408	2
C₃₁×#1975	2
C₃₁×#1938	2
C₃₁×#1929	2
C₃₁×#1876	2
C₃₁×#1871	2
C₃₁×#1855	2
C₃₁×#1789	2
C₃₁×#1752	2
#22061	31

Subgroups — C31×#22061 #21478820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22061 #21478820