Subgroups — C₃₁×#34141 #21490900

Order 7936.

3298 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#41×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2367	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2366	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2365	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2364	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2361	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2297	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2293	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2290	C₂	1	4	yes	maximal, complemented
11	C₄²×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
12	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#532	C₄	1	16	yes	complemented
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#124	C₂²	1	2	yes	—
19	C₂×C₃₁×#121	C₂²	1	2	yes	—
20	C₂×C₃₁×#121	C₂²	1	2	yes	—
21	C₄×#41×C₃₁	—	2	16	no	—
22	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₃₁×#117	C₂²	1	4	yes	—
29	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
33	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
34	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#391	C₂²	1	4	yes	—
36	C₃₁×#390	C₂²	1	4	yes	—
37	C₃₁×#390	C₂²	1	4	yes	—
38	C₃₁×#389	C₂²	1	4	yes	—
39	C₃₁×#388	C₂²	1	8	yes	—
40	C₃₁×#388	C₂²	1	8	yes	—
41	C₃₁×#387	C₂²	1	8	yes	—
42	C₃₁×#386	C₂²	1	8	yes	—
43	C₃₁×#386	C₂²	1	8	yes	—
44	C₃₁×#385	C₂²	1	8	yes	—
45	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
46	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
47	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
48	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
49	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
50	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
51	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
52	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
53	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
54	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
55	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
56	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
57	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
58	C₃₁×#376	C₂²	1	2	yes	—
59	C₂×C₃₁×#95	C₂²	1	2	yes	—
60	C₄³×C₃₁	C₂²	1	2	yes	—
61	C₂×#41×C₃₁	—	2	8	no	—
62	C₂×#41×C₃₁	—	2	8	no	—
63	C₂×#41×C₃₁	C₂×C₄	1	16	yes	complemented
64	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
65	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
66	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
67	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
71	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
76	C₃₁×#127	C₂×C₄	1	8	yes	complemented
77	C₃₁×#124	C₂×C₄	1	16	yes	complemented
78	C₃₁×#121	C₂×C₄	1	16	yes	complemented
79	C₃₁×#121	C₂×C₄	1	16	yes	complemented
80	C₄×Q₈×C₃₁	—	2	16	no	—
81	C₄×D₄×C₃₁	—	2	16	no	—
82	C₄×D₄×C₃₁	—	2	16	no	—
83	C₄×D₄×C₃₁	—	2	32	no	—
84	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
85	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
86	C₃₁×#117	—	2	4	no	—
87	C₃₁×#117	—	2	16	no	—
88	C₃₁×#117	D₄	1	8	yes	complemented
89	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
99	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
105	C₂×C₄²×C₃₁	—	2	8	no	—
106	C₂×C₄²×C₃₁	—	2	16	no	—
107	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
108	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
109	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
110	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
111	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
112	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
113	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
114	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
115	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
120	C₃₁×#95	C₂³	1	2	yes	—
121	C₃₁×#95	C₂³	1	4	yes	—
122	C₃₁×#95	C₂³	1	2	yes	—
123	C₃₁×#95	C₂³	1	4	yes	—
124	C₃₁×#95	C₂³	1	4	yes	—
125	C₃₁×#95	C₂³	1	8	yes	—
126	C₃₁×#95	C₂³	1	8	yes	—
127	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
128	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
129	#41×C₃₁	—	2	64	no	—
130	C₂×Q₈×C₃₁	—	2	4	no	—
131	C₂×Q₈×C₃₁	—	2	4	no	—
132	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
133	C₂×D₄×C₃₁	—	2	8	no	—
134	C₂×D₄×C₃₁	—	2	8	no	—
135	C₂×D₄×C₃₁	—	2	4	no	—
136	C₂×D₄×C₃₁	—	2	8	no	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	4	4	no	—
139	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
140	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
141	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
142	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
143	C₂²×C₄×C₃₁	—	2	8	no	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	8	no	—
146	C₂²×C₄×C₃₁	—	2	8	no	—
147	C₂²×C₄×C₃₁	—	2	16	no	—
148	C₂²×C₄×C₃₁	—	2	8	no	—
149	C₂²×C₄×C₃₁	—	2	16	no	—
150	C₂²×C₄×C₃₁	—	4	2	no	—
151	C₂²×C₄×C₃₁	—	4	8	no	—
152	C₂²×C₄×C₃₁	—	4	4	no	—
153	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
154	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
155	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
156	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₃₁	#41	1	4	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
166	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
170	C₄⋊C₄×C₃₁	—	2	16	no	—
171	C₄⋊C₄×C₃₁	—	2	8	no	—
172	C₄⋊C₄×C₃₁	—	2	4	no	—
173	C₄⋊C₄×C₃₁	—	2	4	no	—
174	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
175	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
176	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
177	C₄⋊C₄×C₃₁	#41	1	8	yes	complemented
178	C₂²⋊C₄×C₃₁	—	2	4	no	—
179	C₂²⋊C₄×C₃₁	—	2	16	no	—
180	C₂²⋊C₄×C₃₁	—	2	16	no	—
181	C₂²⋊C₄×C₃₁	—	4	8	no	—
182	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
183	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
184	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
185	C₄²×C₃₁	—	2	4	no	—
186	C₄²×C₃₁	—	2	4	no	—
187	C₄²×C₃₁	—	2	16	no	—
188	C₄²×C₃₁	—	2	8	no	—
189	C₄²×C₃₁	—	2	4	no	—
190	C₄²×C₃₁	—	2	8	no	—
191	C₄²×C₃₁	—	4	4	no	—
192	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
193	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
194	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
195	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
196	C₄²×C₃₁	C₂×D₄	1	8	yes	—
197	#34141	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	—	2	2	no	—
200	C₂³×C₃₁	—	2	4	no	—
201	C₂³×C₃₁	—	2	8	no	—
202	C₂³×C₃₁	—	2	4	no	—
203	C₂³×C₃₁	—	2	8	no	—
204	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
205	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
206	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
207	C₂³×C₃₁	C₂×#41	1	2	yes	—
208	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
209	Q₈×C₃₁	—	2	16	no	—
210	D₄×C₃₁	—	2	32	no	—
211	D₄×C₃₁	—	2	16	no	—
212	D₄×C₃₁	—	2	16	no	—
213	D₄×C₃₁	—	4	8	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	4	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	32	no	—
224	C₂×C₄×C₃₁	—	2	8	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	4	4	no	—
232	C₂×C₄×C₃₁	—	4	32	no	—
233	C₂×C₄×C₃₁	—	4	16	no	—
234	C₂×C₄×C₃₁	C₄×D₄	1	16	yes	complemented
235	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
236	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
237	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
238	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
239	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
240	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
241	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
242	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
243	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
246	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
247	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
248	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
249	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
250	C₂×C₄×#41	C₂×C₃₁	1	2	yes	complemented
251	#2367	C₂×C₃₁	1	4	yes	complemented
252	#2366	C₂×C₃₁	1	2	yes	complemented
253	#2365	C₂×C₃₁	1	2	yes	complemented
254	#2364	C₂×C₃₁	1	4	yes	complemented
255	#2361	C₂×C₃₁	1	4	yes	complemented
256	#2297	C₂×C₃₁	1	2	yes	complemented
257	#2293	C₂×C₃₁	1	2	yes	complemented
258	#2290	C₂×C₃₁	1	4	yes	complemented
259	C₄²×D₄	C₂×C₃₁	1	2	yes	complemented
260	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
261	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂²×C₃₁	—	2	2	no	—
263	C₂²×C₃₁	—	2	2	no	—
264	C₂²×C₃₁	—	2	4	no	—
265	C₂²×C₃₁	—	2	4	no	—
266	C₂²×C₃₁	—	2	8	no	—
267	C₂²×C₃₁	—	2	8	no	—
268	C₂²×C₃₁	—	2	8	no	—
269	C₂²×C₃₁	—	4	8	no	—
270	C₂²×C₃₁	—	4	8	no	—
271	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
272	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
273	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
274	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
275	C₂²×C₃₁	C₂²×#41	1	2	yes	—
276	C₂²×C₃₁	#585	1	2	yes	—
277	C₄×C₃₁	—	2	8	no	—
278	C₄×C₃₁	—	2	2	no	—
279	C₄×C₃₁	—	2	2	no	—
280	C₄×C₃₁	—	2	2	no	—
281	C₄×C₃₁	—	2	8	no	—
282	C₄×C₃₁	—	2	8	no	—
283	C₄×C₃₁	—	4	4	no	—
284	C₄×C₃₁	—	4	4	no	—
285	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
286	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
287	C₄×C₃₁	C₄×#41	1	4	yes	complemented
288	C₂²×#41	C₂²×C₃₁	1	2	yes	—
289	#532	C₄×C₃₁	1	16	yes	complemented
290	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×#124	C₂²×C₃₁	1	2	yes	—
293	C₂×#121	C₂²×C₃₁	1	2	yes	—
294	C₂×#121	C₂²×C₃₁	1	2	yes	—
295	C₄×#41	—	2	16	no	—
296	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	—
297	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
298	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
299	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
300	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
301	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
302	C₂×#117	C₂²×C₃₁	1	4	yes	—
303	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
305	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
306	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
307	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
308	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
309	#391	C₂²×C₃₁	1	4	yes	—
310	#390	C₂²×C₃₁	1	4	yes	—
311	#390	C₂²×C₃₁	1	4	yes	—
312	#389	C₂²×C₃₁	1	4	yes	—
313	#388	C₂²×C₃₁	1	8	yes	—
314	#388	C₂²×C₃₁	1	8	yes	—
315	#387	C₂²×C₃₁	1	8	yes	—
316	#386	C₂²×C₃₁	1	8	yes	—
317	#386	C₂²×C₃₁	1	8	yes	—
318	#385	C₂²×C₃₁	1	8	yes	—
319	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
320	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
321	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
322	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
323	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
324	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
325	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
326	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
327	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
328	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
329	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
330	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
331	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
332	#376	C₂²×C₃₁	1	2	yes	—
333	C₂×#95	C₂²×C₃₁	1	2	yes	—
334	C₄³	C₂²×C₃₁	1	2	yes	—
335	C₂×C₃₁	—	2	4	no	—
336	C₂×C₃₁	—	4	8	no	—
337	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
338	C₂×C₃₁	C₂×C₄×#41	1	1	yes	characteristic
339	C₂×C₃₁	#3428	1	1	yes	characteristic
340	C₂×C₃₁	D₄×#41	1	2	yes	—
341	C₂×C₃₁	#3480	1	2	yes	—
342	C₂×#41	—	2	8	no	—
343	C₂×#41	—	2	8	no	—
344	C₂×#41	C₂×C₄×C₃₁	1	16	yes	complemented
345	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
346	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
347	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
348	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
349	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
350	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
351	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
352	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
353	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
354	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
356	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
357	#127	C₂×C₄×C₃₁	1	8	yes	complemented
358	#124	C₂×C₄×C₃₁	1	16	yes	complemented
359	#121	C₂×C₄×C₃₁	1	16	yes	complemented
360	#121	C₂×C₄×C₃₁	1	16	yes	complemented
361	C₄×Q₈	—	2	16	no	—
362	C₄×D₄	—	2	16	no	—
363	C₄×D₄	—	2	16	no	—
364	C₄×D₄	—	2	32	no	—
365	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
366	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
367	#117	—	2	4	no	—
368	#117	—	2	16	no	—
369	#117	D₄×C₃₁	1	8	yes	complemented
370	C₂×C₄⋊C₄	—	2	8	no	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
374	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
375	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
378	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
386	C₂×C₄²	—	2	8	no	—
387	C₂×C₄²	—	2	16	no	—
388	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
389	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
390	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
391	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
392	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
393	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
394	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
395	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
396	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
397	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
400	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
401	#95	C₂³×C₃₁	1	2	yes	—
402	#95	C₂³×C₃₁	1	4	yes	—
403	#95	C₂³×C₃₁	1	2	yes	—
404	#95	C₂³×C₃₁	1	4	yes	—
405	#95	C₂³×C₃₁	1	4	yes	—
406	#95	C₂³×C₃₁	1	8	yes	—
407	#95	C₂³×C₃₁	1	8	yes	—
408	C₃₁	#34141	1	1	yes	characteristic, direct product factor, Sylow
409	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
410	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
411	#41	—	2	64	no	—
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	4	no	—
414	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
415	C₂×D₄	—	2	8	no	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	4	4	no	—
421	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
422	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
423	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
424	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
425	C₂²×C₄	—	2	8	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	8	no	—
428	C₂²×C₄	—	2	8	no	—
429	C₂²×C₄	—	2	16	no	—
430	C₂²×C₄	—	2	8	no	—
431	C₂²×C₄	—	2	16	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	—	4	8	no	—
434	C₂²×C₄	—	4	4	no	—
435	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
436	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
437	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
438	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
439	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
440	C₂²×C₄	#41×C₃₁	1	4	yes	—
441	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
442	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
443	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
444	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
447	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
448	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
450	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
452	C₄⋊C₄	—	2	16	no	—
453	C₄⋊C₄	—	2	8	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
457	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
458	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
459	C₄⋊C₄	#41×C₃₁	1	8	yes	complemented
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	16	no	—
462	C₂²⋊C₄	—	2	16	no	—
463	C₂²⋊C₄	—	4	8	no	—
464	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
465	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
466	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
467	C₄²	—	2	4	no	—
468	C₄²	—	2	4	no	—
469	C₄²	—	2	16	no	—
470	C₄²	—	2	8	no	—
471	C₄²	—	2	4	no	—
472	C₄²	—	2	8	no	—
473	C₄²	—	4	4	no	—
474	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
475	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
476	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
477	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
478	C₄²	C₂×D₄×C₃₁	1	8	yes	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	4	no	—
482	C₂³	—	2	8	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	8	no	—
485	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
486	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
487	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
488	C₂³	C₂×#41×C₃₁	1	2	yes	—
489	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
490	Q₈	—	2	16	no	—
491	D₄	—	2	32	no	—
492	D₄	—	2	16	no	—
493	D₄	—	2	16	no	—
494	D₄	—	4	8	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	32	no	—
505	C₂×C₄	—	2	8	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	4	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	4	4	no	—
513	C₂×C₄	—	4	32	no	—
514	C₂×C₄	—	4	16	no	—
515	C₂×C₄	C₄×D₄×C₃₁	1	16	yes	complemented
516	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
517	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
518	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
519	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
520	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
521	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
522	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
523	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
524	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
525	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
526	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
527	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
528	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
529	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
530	C₂²	—	2	2	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	2	4	no	—
534	C₂²	—	2	8	no	—
535	C₂²	—	2	8	no	—
536	C₂²	—	2	8	no	—
537	C₂²	—	4	8	no	—
538	C₂²	—	4	8	no	—
539	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
540	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
541	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
542	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
543	C₂²	C₂²×#41×C₃₁	1	2	yes	—
544	C₂²	C₃₁×#585	1	2	yes	—
545	C₄	—	2	8	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	2	no	—
548	C₄	—	2	2	no	—
549	C₄	—	2	8	no	—
550	C₄	—	2	8	no	—
551	C₄	—	4	4	no	—
552	C₄	—	4	4	no	—
553	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
554	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
555	C₄	C₄×#41×C₃₁	1	4	yes	complemented
556	C₂	—	2	4	no	—
557	C₂	—	4	8	no	—
558	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
559	C₂	C₂×C₄×#41×C₃₁	1	1	yes	characteristic
560	C₂	C₃₁×#3428	1	1	yes	characteristic
561	C₂	D₄×#41×C₃₁	1	2	yes	—
562	C₂	C₃₁×#3480	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂×C₃₁×#532	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#41×C₃₁	C₂	semidirect
3. C₃₁×#2367	C₂	semidirect
4. C₃₁×#2366	C₂	semidirect
5. C₃₁×#2365	C₂	semidirect
6. C₃₁×#2364	C₂	semidirect
7. C₃₁×#2361	C₂	semidirect
8. C₃₁×#2297	C₂	semidirect
9. C₃₁×#2293	C₂	semidirect
10. C₃₁×#2290	C₂	semidirect
11. C₄²×D₄×C₃₁	C₂	semidirect
12. C₄×C₃₁×#117	C₂	semidirect
13. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
15. C₃₁×#532	C₄	semidirect
63. C₂×#41×C₃₁	C₂×C₄	semidirect
75. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
76. C₃₁×#127	C₂×C₄	semidirect
77. C₃₁×#124	C₂×C₄	semidirect
78. C₃₁×#121	C₂×C₄	semidirect
79. C₃₁×#121	C₂×C₄	semidirect
84. C₄×D₄×C₃₁	C₂×C₄	semidirect
85. C₄×D₄×C₃₁	C₂×C₄	semidirect
88. C₃₁×#117	D₄	semidirect
107. C₂×C₄²×C₃₁	C₂×C₄	semidirect
177. C₄⋊C₄×C₃₁	#41	semidirect
197. #34141	C₃₁	direct (N × H)
234. C₂×C₄×C₃₁	C₄×D₄	semidirect
250. C₂×C₄×#41	C₂×C₃₁	semidirect
251. #2367	C₂×C₃₁	semidirect
252. #2366	C₂×C₃₁	semidirect
253. #2365	C₂×C₃₁	semidirect
254. #2364	C₂×C₃₁	semidirect
255. #2361	C₂×C₃₁	semidirect
256. #2297	C₂×C₃₁	semidirect
257. #2293	C₂×C₃₁	semidirect
258. #2290	C₂×C₃₁	semidirect
259. C₄²×D₄	C₂×C₃₁	semidirect
260. C₄×#117	C₂×C₃₁	semidirect
261. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
287. C₄×C₃₁	C₄×#41	semidirect
289. #532	C₄×C₃₁	semidirect
344. C₂×#41	C₂×C₄×C₃₁	semidirect
356. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
357. #127	C₂×C₄×C₃₁	semidirect
358. #124	C₂×C₄×C₃₁	semidirect
359. #121	C₂×C₄×C₃₁	semidirect
360. #121	C₂×C₄×C₃₁	semidirect
365. C₄×D₄	C₂×C₄×C₃₁	semidirect
366. C₄×D₄	C₂×C₄×C₃₁	semidirect
369. #117	D₄×C₃₁	semidirect
388. C₂×C₄²	C₂×C₄×C₃₁	semidirect
408. C₃₁	#34141	direct (N × H)
459. C₄⋊C₄	#41×C₃₁	semidirect
515. C₂×C₄	C₄×D₄×C₃₁	semidirect
555. C₄	C₄×#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#532	2
C₂×C₄×#41×C₃₁	2
C₃₁×#2367	2
C₃₁×#2366	2
C₃₁×#2365	2
C₃₁×#2364	2
C₃₁×#2361	2
C₃₁×#2297	2
C₃₁×#2293	2
C₃₁×#2290	2
C₄²×D₄×C₃₁	2
C₄×C₃₁×#117	2
C₂×C₄×C₄⋊C₄×C₃₁	2
#34141	31

Subgroups — C31×#34141 #21490900

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34141 #21490900