Subgroups — C₃₁×#21880 #21478639

Order 7936.

1076 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#529	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1998	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1967	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1901	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1900	C₂	1	2	yes	maximal
8	C₃₁×#1801	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1800	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#529	—	2	1	no	—
12	C₃₁×#529	—	2	1	no	—
13	C₃₁×#529	—	2	1	no	—
14	C₃₁×#529	—	2	1	no	—
15	C₃₁×#529	—	2	1	no	—
16	C₃₁×#529	—	2	1	no	—
17	C₃₁×#529	—	2	1	no	—
18	C₃₁×#529	—	2	1	no	—
19	C₂×C₃₁×#126	C₄	1	2	yes	—
20	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#123	C₄	1	2	yes	—
23	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#117	C₄	1	2	yes	—
29	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#432	—	2	1	no	—
31	C₃₁×#432	—	2	1	no	—
32	C₃₁×#431	—	2	1	no	—
33	C₃₁×#431	—	2	1	no	—
34	C₃₁×#423	—	2	1	no	—
35	C₃₁×#423	—	2	1	no	—
36	C₃₁×#423	C₂²	1	2	yes	—
37	C₃₁×#423	C₂²	1	2	yes	—
38	C₃₁×#423	C₂²	1	2	yes	—
39	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
40	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
41	C₂×C₃₁×#101	C₂²	1	4	yes	—
42	C₂×C₃₁×#100	C₂²	1	4	yes	—
43	C₃₁×#407	C₂²	1	2	yes	—
44	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
45	C₃₁×#343	C₂²	1	4	yes	—
46	C₃₁×#343	C₂²	1	4	yes	—
47	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
51	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	4	4	no	—
56	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
57	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₃₁×#126	—	2	2	no	—
60	C₃₁×#126	—	2	2	no	—
61	C₃₁×#126	—	2	2	no	—
62	C₃₁×#126	—	2	2	no	—
63	C₃₁×#125	—	2	1	no	—
64	C₃₁×#125	—	2	1	no	—
65	C₃₁×#125	—	2	1	no	—
66	C₃₁×#125	—	2	1	no	—
67	C₃₁×#124	—	2	1	no	—
68	C₃₁×#124	—	2	1	no	—
69	C₃₁×#124	—	2	1	no	—
70	C₃₁×#124	—	2	1	no	—
71	C₃₁×#123	—	2	2	no	—
72	C₃₁×#123	—	2	2	no	—
73	C₃₁×#123	—	2	2	no	—
74	C₃₁×#123	—	2	2	no	—
75	C₃₁×C₂²⋊Q₈	—	2	1	no	—
76	C₃₁×C₂²⋊Q₈	—	2	1	no	—
77	C₃₁×C₂²⋊Q₈	—	2	1	no	—
78	C₃₁×C₂²⋊Q₈	—	2	1	no	—
79	C₃₁×#121	—	2	1	no	—
80	C₃₁×#121	—	2	1	no	—
81	C₃₁×#121	—	2	1	no	—
82	C₃₁×#121	—	2	1	no	—
83	C₄×Q₈×C₃₁	—	2	1	no	—
84	C₄×Q₈×C₃₁	—	2	1	no	—
85	C₄×Q₈×C₃₁	—	2	1	no	—
86	C₄×Q₈×C₃₁	—	2	1	no	—
87	C₄×D₄×C₃₁	—	2	2	no	—
88	C₄×D₄×C₃₁	—	2	2	no	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	—	2	1	no	—
92	C₄×D₄×C₃₁	—	2	1	no	—
93	C₄×D₄×C₃₁	—	2	1	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₃₁×#117	—	2	2	no	—
96	C₃₁×#117	—	2	2	no	—
97	C₃₁×#117	—	2	2	no	—
98	C₃₁×#117	—	2	2	no	—
99	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
106	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
108	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
109	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
110	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
111	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
123	C₃₁×C₄⋊C₈	—	2	2	no	—
124	C₃₁×C₄⋊C₈	—	2	2	no	—
125	C₃₁×C₄⋊C₈	—	2	2	no	—
126	C₃₁×C₄⋊C₈	—	2	2	no	—
127	C₃₁×C₄⋊C₈	—	2	2	no	—
128	C₃₁×C₄⋊C₈	—	2	2	no	—
129	C₃₁×#101	—	4	4	no	—
130	C₃₁×#100	—	4	4	no	—
131	C₃₁×C₂²⋊C₈	—	2	2	no	—
132	C₃₁×C₂²⋊C₈	—	2	2	no	—
133	C₃₁×#97	—	2	2	no	—
134	C₄×C₈×C₃₁	—	2	2	no	—
135	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
137	C₂×Q₈×C₃₁	—	2	1	no	—
138	C₂×Q₈×C₃₁	—	2	1	no	—
139	C₂×Q₈×C₃₁	—	2	2	no	—
140	C₂×Q₈×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	2	2	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	—	2	1	no	—
145	C₂×D₄×C₃₁	—	2	2	no	—
146	C₂×D₄×C₃₁	—	2	2	no	—
147	C₂×D₄×C₃₁	—	4	4	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	—	4	2	no	—
153	C₂²×C₄×C₃₁	—	4	2	no	—
154	C₂²×C₄×C₃₁	—	4	4	no	—
155	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
164	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
168	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
171	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	M₁₆×C₃₁	—	2	4	no	—
177	M₁₆×C₃₁	—	2	4	no	—
178	M₁₆×C₃₁	—	2	4	no	—
179	M₁₆×C₃₁	—	2	4	no	—
180	M₁₆×C₃₁	—	4	4	no	—
181	C₂×C₈×C₃₁	—	2	4	no	—
182	C₂×C₈×C₃₁	—	2	4	no	—
183	C₂×C₈×C₃₁	—	4	4	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	—	2	2	no	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	—	2	2	no	—
192	C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	2	1	no	—
195	C₄⋊C₄×C₃₁	—	2	1	no	—
196	C₂²⋊C₄×C₃₁	—	2	2	no	—
197	C₂²⋊C₄×C₃₁	—	2	2	no	—
198	C₂²⋊C₄×C₃₁	—	2	2	no	—
199	C₂²⋊C₄×C₃₁	—	2	2	no	—
200	C₂²⋊C₄×C₃₁	—	2	2	no	—
201	C₂²⋊C₄×C₃₁	—	2	2	no	—
202	C₂²⋊C₄×C₃₁	—	2	2	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	2	2	no	—
205	C₂²⋊C₄×C₃₁	—	2	2	no	—
206	C₄²×C₃₁	—	2	2	no	—
207	C₄²×C₃₁	—	2	2	no	—
208	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
209	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
210	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
211	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
212	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
213	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
214	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
215	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
216	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
218	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
221	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
222	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
223	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
224	#21880	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂³×C₃₁	—	2	2	no	—
226	C₂³×C₃₁	—	2	2	no	—
227	C₂³×C₃₁	—	2	2	no	—
228	C₂³×C₃₁	—	2	4	no	—
229	C₂³×C₃₁	—	4	1	no	—
230	C₂³×C₃₁	—	4	2	no	—
231	C₂³×C₃₁	—	4	2	no	—
232	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
235	Q₈×C₃₁	—	4	1	no	—
236	Q₈×C₃₁	—	4	1	no	—
237	Q₈×C₃₁	—	4	2	no	—
238	D₄×C₃₁	—	4	4	no	—
239	D₄×C₃₁	—	4	4	no	—
240	D₄×C₃₁	—	4	1	no	—
241	D₄×C₃₁	—	4	1	no	—
242	D₄×C₃₁	—	4	2	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	2	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	1	no	—
247	C₂×C₄×C₃₁	—	2	1	no	—
248	C₂×C₄×C₃₁	—	2	1	no	—
249	C₂×C₄×C₃₁	—	2	1	no	—
250	C₂×C₄×C₃₁	—	2	2	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	2	no	—
253	C₂×C₄×C₃₁	—	2	2	no	—
254	C₂×C₄×C₃₁	—	2	2	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	2	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	4	no	—
259	C₂×C₄×C₃₁	—	4	1	no	—
260	C₂×C₄×C₃₁	—	4	1	no	—
261	C₂×C₄×C₃₁	—	4	1	no	—
262	C₂×C₄×C₃₁	—	4	1	no	—
263	C₂×C₄×C₃₁	—	4	1	no	—
264	C₂×C₄×C₃₁	—	4	1	no	—
265	C₂×C₄×C₃₁	—	4	1	no	—
266	C₂×C₄×C₃₁	—	4	4	no	—
267	C₂×C₄×C₃₁	—	4	4	no	—
268	C₂×C₄×C₃₁	—	4	2	no	—
269	C₂×C₄×C₃₁	—	4	2	no	—
270	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
283	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
284	C₈×C₃₁	—	4	4	no	—
285	C₈×C₃₁	—	4	4	no	—
286	C₂×#529	C₂×C₃₁	1	1	yes	characteristic
287	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
288	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#1998	C₂×C₃₁	1	2	yes	complemented
290	#1967	C₂×C₃₁	1	2	yes	complemented
291	#1901	C₂×C₃₁	1	2	yes	complemented
292	#1900	C₂×C₃₁	1	2	yes	—
293	#1801	C₂×C₃₁	1	2	yes	complemented
294	#1800	C₂×C₃₁	1	2	yes	complemented
295	C₂²×C₃₁	—	2	1	no	—
296	C₂²×C₃₁	—	2	1	no	—
297	C₂²×C₃₁	—	2	1	no	—
298	C₂²×C₃₁	—	2	1	no	—
299	C₂²×C₃₁	—	2	1	no	—
300	C₂²×C₃₁	—	2	1	no	—
301	C₂²×C₃₁	—	2	1	no	—
302	C₂²×C₃₁	—	4	4	no	—
303	C₂²×C₃₁	—	4	2	no	—
304	C₂²×C₃₁	—	4	2	no	—
305	C₂²×C₃₁	—	4	2	no	—
306	C₂²×C₃₁	—	8	4	no	—
307	C₂²×C₃₁	#413	1	2	yes	—
308	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
309	C₂²×C₃₁	#520	1	2	yes	—
310	C₄×C₃₁	—	2	1	no	—
311	C₄×C₃₁	—	2	1	no	—
312	C₄×C₃₁	—	2	1	no	—
313	C₄×C₃₁	—	2	1	no	—
314	C₄×C₃₁	—	2	1	no	—
315	C₄×C₃₁	—	2	1	no	—
316	C₄×C₃₁	—	2	1	no	—
317	C₄×C₃₁	—	2	1	no	—
318	C₄×C₃₁	—	2	1	no	—
319	C₄×C₃₁	—	4	2	no	—
320	C₄×C₃₁	—	4	2	no	—
321	C₄×C₃₁	—	4	2	no	—
322	C₄×C₃₁	#413	1	2	yes	—
323	C₄×C₃₁	#535	1	1	yes	characteristic
324	C₄×C₃₁	#536	1	1	yes	characteristic
325	C₄×C₃₁	#537	1	2	yes	—
326	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
327	#529	—	2	1	no	—
328	#529	—	2	1	no	—
329	#529	—	2	1	no	—
330	#529	—	2	1	no	—
331	#529	—	2	1	no	—
332	#529	—	2	1	no	—
333	#529	—	2	1	no	—
334	#529	—	2	1	no	—
335	C₂×#126	C₄×C₃₁	1	2	yes	—
336	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×#123	C₄×C₃₁	1	2	yes	—
339	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
340	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
341	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
342	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
343	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
344	C₂×#117	C₄×C₃₁	1	2	yes	—
345	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
346	#432	—	2	1	no	—
347	#432	—	2	1	no	—
348	#431	—	2	1	no	—
349	#431	—	2	1	no	—
350	#423	—	2	1	no	—
351	#423	—	2	1	no	—
352	#423	C₂²×C₃₁	1	2	yes	—
353	#423	C₂²×C₃₁	1	2	yes	—
354	#423	C₂²×C₃₁	1	2	yes	—
355	C₂×C₄⋊C₈	—	2	1	no	—
356	C₂×C₄⋊C₈	—	2	1	no	—
357	C₂×#101	C₂²×C₃₁	1	4	yes	—
358	C₂×#100	C₂²×C₃₁	1	4	yes	—
359	#407	C₂²×C₃₁	1	2	yes	—
360	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
361	#343	C₂²×C₃₁	1	4	yes	—
362	#343	C₂²×C₃₁	1	4	yes	—
363	C₂×C₃₁	—	2	2	no	—
364	C₂×C₃₁	—	2	2	no	—
365	C₂×C₃₁	—	4	1	no	—
366	C₂×C₃₁	—	8	2	no	—
367	C₂×C₃₁	#2311	1	1	yes	characteristic
368	C₂×C₃₁	C₂×#413	1	1	yes	characteristic
369	C₂×C₃₁	C₂×#413	1	1	yes	characteristic
370	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
371	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
372	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
373	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
374	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
375	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
376	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
377	C₂×M₁₆	—	2	2	no	—
378	C₂×M₁₆	—	4	4	no	—
379	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
380	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
381	C₂²×C₈	—	2	2	no	—
382	#126	—	2	2	no	—
383	#126	—	2	2	no	—
384	#126	—	2	2	no	—
385	#126	—	2	2	no	—
386	#125	—	2	1	no	—
387	#125	—	2	1	no	—
388	#125	—	2	1	no	—
389	#125	—	2	1	no	—
390	#124	—	2	1	no	—
391	#124	—	2	1	no	—
392	#124	—	2	1	no	—
393	#124	—	2	1	no	—
394	#123	—	2	2	no	—
395	#123	—	2	2	no	—
396	#123	—	2	2	no	—
397	#123	—	2	2	no	—
398	C₂²⋊Q₈	—	2	1	no	—
399	C₂²⋊Q₈	—	2	1	no	—
400	C₂²⋊Q₈	—	2	1	no	—
401	C₂²⋊Q₈	—	2	1	no	—
402	#121	—	2	1	no	—
403	#121	—	2	1	no	—
404	#121	—	2	1	no	—
405	#121	—	2	1	no	—
406	C₄×Q₈	—	2	1	no	—
407	C₄×Q₈	—	2	1	no	—
408	C₄×Q₈	—	2	1	no	—
409	C₄×Q₈	—	2	1	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	1	no	—
418	#117	—	2	2	no	—
419	#117	—	2	2	no	—
420	#117	—	2	2	no	—
421	#117	—	2	2	no	—
422	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
423	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
424	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
425	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
426	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
428	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
431	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
432	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
433	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
434	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
435	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
436	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
438	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
439	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
440	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
441	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
442	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
443	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
446	C₄⋊C₈	—	2	2	no	—
447	C₄⋊C₈	—	2	2	no	—
448	C₄⋊C₈	—	2	2	no	—
449	C₄⋊C₈	—	2	2	no	—
450	C₄⋊C₈	—	2	2	no	—
451	C₄⋊C₈	—	2	2	no	—
452	#101	—	4	4	no	—
453	#100	—	4	4	no	—
454	C₂²⋊C₈	—	2	2	no	—
455	C₂²⋊C₈	—	2	2	no	—
456	#97	—	2	2	no	—
457	C₄×C₈	—	2	2	no	—
458	C₃₁	#21880	1	1	yes	characteristic, direct product factor, Sylow
459	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
460	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
461	C₂×Q₈	—	2	1	no	—
462	C₂×Q₈	—	2	1	no	—
463	C₂×Q₈	—	2	2	no	—
464	C₂×Q₈	—	2	2	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	—	2	2	no	—
471	C₂×D₄	—	4	4	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	4	2	no	—
477	C₂²×C₄	—	4	2	no	—
478	C₂²×C₄	—	4	4	no	—
479	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
482	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
483	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
484	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
486	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
487	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
488	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
489	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
490	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
491	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
492	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
493	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
494	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
497	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
498	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
499	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
500	M₁₆	—	2	4	no	—
501	M₁₆	—	2	4	no	—
502	M₁₆	—	2	4	no	—
503	M₁₆	—	2	4	no	—
504	M₁₆	—	4	4	no	—
505	C₂×C₈	—	2	4	no	—
506	C₂×C₈	—	2	4	no	—
507	C₂×C₈	—	4	4	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₂²⋊C₄	—	2	2	no	—
524	C₂²⋊C₄	—	2	2	no	—
525	C₂²⋊C₄	—	2	2	no	—
526	C₂²⋊C₄	—	2	2	no	—
527	C₂²⋊C₄	—	2	2	no	—
528	C₂²⋊C₄	—	2	2	no	—
529	C₂²⋊C₄	—	2	2	no	—
530	C₄²	—	2	2	no	—
531	C₄²	—	2	2	no	—
532	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
533	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
534	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
535	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
536	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
537	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
538	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
539	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
540	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
541	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
542	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
543	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
544	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
545	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
546	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
547	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
548	C₂³	—	2	2	no	—
549	C₂³	—	2	2	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	2	4	no	—
552	C₂³	—	4	1	no	—
553	C₂³	—	4	2	no	—
554	C₂³	—	4	2	no	—
555	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
556	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
557	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
558	Q₈	—	4	1	no	—
559	Q₈	—	4	1	no	—
560	Q₈	—	4	2	no	—
561	D₄	—	4	4	no	—
562	D₄	—	4	4	no	—
563	D₄	—	4	1	no	—
564	D₄	—	4	1	no	—
565	D₄	—	4	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	4	no	—
590	C₂×C₄	—	4	4	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
594	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
595	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
596	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
597	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
598	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
599	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
600	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
601	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
602	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
603	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
604	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
605	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
606	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
607	C₈	—	4	4	no	—
608	C₈	—	4	4	no	—
609	C₂²	—	2	1	no	—
610	C₂²	—	2	1	no	—
611	C₂²	—	2	1	no	—
612	C₂²	—	2	1	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	2	1	no	—
616	C₂²	—	4	4	no	—
617	C₂²	—	4	2	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	4	2	no	—
620	C₂²	—	8	4	no	—
621	C₂²	C₃₁×#413	1	2	yes	—
622	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
623	C₂²	C₃₁×#520	1	2	yes	—
624	C₄	—	2	1	no	—
625	C₄	—	2	1	no	—
626	C₄	—	2	1	no	—
627	C₄	—	2	1	no	—
628	C₄	—	2	1	no	—
629	C₄	—	2	1	no	—
630	C₄	—	2	1	no	—
631	C₄	—	2	1	no	—
632	C₄	—	2	1	no	—
633	C₄	—	4	2	no	—
634	C₄	—	4	2	no	—
635	C₄	—	4	2	no	—
636	C₄	C₃₁×#413	1	2	yes	—
637	C₄	C₃₁×#535	1	1	yes	characteristic
638	C₄	C₃₁×#536	1	1	yes	characteristic
639	C₄	C₃₁×#537	1	2	yes	—
640	C₂	—	2	2	no	—
641	C₂	—	2	2	no	—
642	C₂	—	4	1	no	—
643	C₂	—	8	2	no	—
644	C₂	C₃₁×#2311	1	1	yes	characteristic
645	C₂	C₂×C₃₁×#413	1	1	yes	characteristic
646	C₂	C₂×C₃₁×#413	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2920	C₂	semidirect
3. C₂×C₃₁×#423	C₂	semidirect
4. C₃₁×#1998	C₂	semidirect
5. C₃₁×#1967	C₂	semidirect
6. C₃₁×#1901	C₂	semidirect
8. C₃₁×#1801	C₂	semidirect
9. C₃₁×#1800	C₂	semidirect
224. #21880	C₃₁	direct (N × H)
287. #2920	C₂×C₃₁	semidirect
288. C₂×#423	C₂×C₃₁	semidirect
289. #1998	C₂×C₃₁	semidirect
290. #1967	C₂×C₃₁	semidirect
291. #1901	C₂×C₃₁	semidirect
293. #1801	C₂×C₃₁	semidirect
294. #1800	C₂×C₃₁	semidirect
458. C₃₁	#21880	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×#529	2
C₃₁×#2920	2
C₂×C₃₁×#423	2
C₃₁×#1998	2
C₃₁×#1967	2
C₃₁×#1901	2
C₃₁×#1900	2
C₃₁×#1801	2
C₃₁×#1800	2
#21880	31

Subgroups — C31×#21880 #21478639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21880 #21478639