Subgroups — C₃₁×#35051 #21491810

Order 7936.

3078 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#525	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2847	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2659	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2656	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2639	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2564	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2543	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2498	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2430	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2325	C₂	1	4	yes	maximal, complemented
11	C₂²×C₃₁×#95	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#525	—	2	16	no	—
16	C₂×C₃₁×#123	C₂²	1	4	yes	—
17	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₃₁×#121	C₂²	1	4	yes	—
20	C₂×C₃₁×#121	C₂²	1	2	yes	—
21	C₂×C₃₁×#121	C₂²	1	2	yes	—
22	C₂×C₃₁×#121	C₂²	1	4	yes	—
23	C₂×C₃₁×#121	C₂²	1	4	yes	—
24	C₂×C₃₁×#120	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
27	C₂×C₃₁×#117	C₂²	1	2	yes	—
28	C₃₁×#400	C₂²	1	4	yes	complemented
29	C₃₁×#399	C₂²	1	4	yes	complemented
30	C₃₁×#398	C₂²	1	4	yes	complemented
31	C₃₁×#397	C₂²	1	4	yes	complemented
32	C₃₁×#395	C₂²	1	4	yes	complemented
33	C₃₁×#394	C₂²	1	4	yes	complemented
34	C₃₁×#393	C₂²	1	4	yes	complemented
35	C₃₁×#392	C₂²	1	4	yes	complemented
36	C₃₁×#388	C₂²	1	8	yes	complemented
37	C₃₁×#387	C₂²	1	8	yes	complemented
38	C₃₁×#386	C₂²	1	8	yes	—
39	C₃₁×#386	C₂²	1	8	yes	—
40	C₃₁×#386	C₂²	1	8	yes	—
41	C₃₁×#386	C₂²	1	8	yes	complemented
42	C₃₁×#385	C₂²	1	8	yes	complemented
43	C₃₁×#380	C₂²	1	4	yes	—
44	C₃₁×#380	C₂²	1	4	yes	—
45	C₃₁×#380	C₂²	1	4	yes	—
46	C₂×C₃₁×#95	C₂²	1	4	yes	—
47	C₂×C₃₁×#95	C₂²	1	2	yes	—
48	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
49	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×#95	C₂²	1	4	yes	—
51	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×#41×C₃₁	—	2	8	no	—
53	C₂×#41×C₃₁	—	4	8	no	—
54	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
56	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
57	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
58	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
59	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
60	C₂³×C₄×C₃₁	—	2	8	no	—
61	C₂³×C₄×C₃₁	—	4	4	no	—
62	C₂³×C₄×C₃₁	—	4	4	no	—
63	C₂³×C₄×C₃₁	—	4	4	no	—
64	C₂³×C₄×C₃₁	D₄	1	8	yes	complemented
65	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
66	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
67	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
68	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
69	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
74	C₃₁×#123	—	2	16	no	—
75	C₃₁×C₂²⋊Q₈	—	2	8	no	—
76	C₃₁×C₂²⋊Q₈	—	2	8	no	—
77	C₃₁×#121	—	2	8	no	—
78	C₃₁×#121	—	2	8	no	—
79	C₃₁×#121	—	2	16	no	—
80	C₃₁×#121	—	4	8	no	—
81	C₃₁×#121	—	4	8	no	—
82	C₃₁×#120	—	2	16	no	—
83	C₄×D₄×C₃₁	—	2	16	no	—
84	C₄×D₄×C₃₁	—	2	16	no	—
85	C₃₁×#117	—	2	8	no	—
86	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
96	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
101	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
102	C₃₁×#95	C₂³	1	4	yes	—
103	C₃₁×#95	C₂³	1	8	yes	—
104	C₃₁×#95	C₂³	1	4	yes	—
105	C₃₁×#95	C₂³	1	8	yes	—
106	C₃₁×#95	C₂³	1	8	yes	—
107	C₃₁×#95	C₂³	1	8	yes	—
108	C₂⁴×C₃₁	—	2	4	no	—
109	C₂⁴×C₃₁	—	2	2	no	—
110	C₂⁴×C₃₁	—	2	4	no	—
111	C₂⁴×C₃₁	—	2	2	no	—
112	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
113	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
114	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
115	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
116	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
117	#41×C₃₁	—	4	32	no	—
118	C₂×Q₈×C₃₁	—	2	4	no	—
119	C₂×Q₈×C₃₁	—	4	4	no	—
120	C₂×D₄×C₃₁	—	2	8	no	—
121	C₂×D₄×C₃₁	—	2	8	no	—
122	C₂×D₄×C₃₁	—	2	8	no	—
123	C₂×D₄×C₃₁	—	2	16	no	—
124	C₂×D₄×C₃₁	—	2	8	no	—
125	C₂×D₄×C₃₁	—	2	16	no	—
126	C₂×D₄×C₃₁	—	2	8	no	—
127	C₂×D₄×C₃₁	—	2	4	no	—
128	C₂×D₄×C₃₁	—	2	8	no	—
129	C₂×D₄×C₃₁	—	4	4	no	—
130	C₂×D₄×C₃₁	—	4	4	no	—
131	C₂×D₄×C₃₁	—	4	4	no	—
132	C₂²×C₄×C₃₁	—	2	8	no	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	8	no	—
135	C₂²×C₄×C₃₁	—	2	8	no	—
136	C₂²×C₄×C₃₁	—	2	8	no	—
137	C₂²×C₄×C₃₁	—	2	8	no	—
138	C₂²×C₄×C₃₁	—	2	8	no	—
139	C₂²×C₄×C₃₁	—	4	16	no	—
140	C₂²×C₄×C₃₁	—	4	2	no	—
141	C₂²×C₄×C₃₁	—	4	2	no	—
142	C₂²×C₄×C₃₁	—	4	4	no	—
143	C₂²×C₄×C₃₁	—	4	4	no	—
144	C₂²×C₄×C₃₁	—	4	2	no	—
145	C₂²×C₄×C₃₁	—	4	4	no	—
146	C₂²×C₄×C₃₁	—	4	2	no	—
147	C₂²×C₄×C₃₁	—	4	2	no	—
148	C₂²×C₄×C₃₁	—	4	2	no	—
149	C₂²×C₄×C₃₁	—	4	8	no	—
150	C₂²×C₄×C₃₁	—	4	32	no	—
151	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
155	C₂²×C₄×C₃₁	#41	1	8	yes	complemented
156	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
158	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
160	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
163	C₄⋊C₄×C₃₁	—	2	4	no	—
164	C₄⋊C₄×C₃₁	—	2	4	no	—
165	C₄⋊C₄×C₃₁	—	2	8	no	—
166	C₄⋊C₄×C₃₁	—	2	8	no	—
167	C₄⋊C₄×C₃₁	—	2	8	no	—
168	C₄⋊C₄×C₃₁	—	4	4	no	—
169	C₄⋊C₄×C₃₁	—	4	4	no	—
170	C₂²⋊C₄×C₃₁	—	2	8	no	—
171	C₂²⋊C₄×C₃₁	—	2	8	no	—
172	C₂²⋊C₄×C₃₁	—	2	8	no	—
173	C₂²⋊C₄×C₃₁	—	2	8	no	—
174	C₂²⋊C₄×C₃₁	—	2	8	no	—
175	C₂²⋊C₄×C₃₁	—	2	16	no	—
176	C₂²⋊C₄×C₃₁	—	4	8	no	—
177	C₂²⋊C₄×C₃₁	—	4	8	no	—
178	C₄²×C₃₁	—	2	8	no	—
179	#35051	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
180	C₂³×C₃₁	—	2	16	no	—
181	C₂³×C₃₁	—	2	16	no	—
182	C₂³×C₃₁	—	2	8	no	—
183	C₂³×C₃₁	—	2	8	no	—
184	C₂³×C₃₁	—	2	8	no	—
185	C₂³×C₃₁	—	2	2	no	—
186	C₂³×C₃₁	—	2	2	no	—
187	C₂³×C₃₁	—	2	2	no	—
188	C₂³×C₃₁	—	2	2	no	—
189	C₂³×C₃₁	—	2	1	no	—
190	C₂³×C₃₁	—	2	1	no	—
191	C₂³×C₃₁	—	2	1	no	—
192	C₂³×C₃₁	—	2	1	no	—
193	C₂³×C₃₁	—	2	4	no	—
194	C₂³×C₃₁	—	2	8	no	—
195	C₂³×C₃₁	—	2	4	no	—
196	C₂³×C₃₁	—	2	8	no	—
197	C₂³×C₃₁	—	2	4	no	—
198	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
199	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
200	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
201	C₂³×C₃₁	C₂×#41	1	4	yes	—
202	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
203	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
204	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
205	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
206	Q₈×C₃₁	—	4	8	no	—
207	D₄×C₃₁	—	4	8	no	—
208	D₄×C₃₁	—	4	16	no	—
209	D₄×C₃₁	—	4	16	no	—
210	D₄×C₃₁	—	4	16	no	—
211	D₄×C₃₁	—	8	8	no	—
212	C₂×C₄×C₃₁	—	2	16	no	—
213	C₂×C₄×C₃₁	—	2	4	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	4	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	8	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	4	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	8	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	4	4	no	—
228	C₂×C₄×C₃₁	—	4	8	no	—
229	C₂×C₄×C₃₁	—	4	4	no	—
230	C₂×C₄×C₃₁	—	4	8	no	—
231	C₂×C₄×C₃₁	—	4	16	no	—
232	C₂×C₄×C₃₁	—	4	16	no	—
233	C₂×C₄×C₃₁	—	4	16	no	—
234	C₂×C₄×C₃₁	—	4	8	no	—
235	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
236	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
237	C₂×C₄×C₃₁	C₂×#41	1	8	yes	—
238	C₂×#525	C₂×C₃₁	1	2	yes	complemented
239	#2847	C₂×C₃₁	1	4	yes	complemented
240	#2659	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#2656	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#2639	C₂×C₃₁	1	2	yes	complemented
243	#2564	C₂×C₃₁	1	4	yes	complemented
244	#2543	C₂×C₃₁	1	4	yes	complemented
245	#2498	C₂×C₃₁	1	4	yes	complemented
246	#2430	C₂×C₃₁	1	4	yes	complemented
247	#2325	C₂×C₃₁	1	4	yes	complemented
248	C₂²×#95	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂²×C₃₁	—	2	32	no	—
250	C₂²×C₃₁	—	2	2	no	—
251	C₂²×C₃₁	—	2	2	no	—
252	C₂²×C₃₁	—	2	2	no	—
253	C₂²×C₃₁	—	2	2	no	—
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	2	2	no	—
256	C₂²×C₃₁	—	2	4	no	—
257	C₂²×C₃₁	—	2	4	no	—
258	C₂²×C₃₁	—	2	4	no	—
259	C₂²×C₃₁	—	2	4	no	—
260	C₂²×C₃₁	—	2	4	no	—
261	C₂²×C₃₁	—	2	4	no	—
262	C₂²×C₃₁	—	4	4	no	—
263	C₂²×C₃₁	—	4	4	no	—
264	C₂²×C₃₁	—	4	4	no	—
265	C₂²×C₃₁	—	4	8	no	—
266	C₂²×C₃₁	—	4	8	no	—
267	C₂²×C₃₁	#532	1	4	yes	—
268	C₂²×C₃₁	#534	1	4	yes	—
269	C₂²×C₃₁	#537	1	4	yes	—
270	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
271	C₂²×C₃₁	C₂²×#41	1	2	yes	—
272	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
273	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
274	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
275	C₄×C₃₁	—	2	4	no	—
276	C₄×C₃₁	—	2	4	no	—
277	C₄×C₃₁	—	2	8	no	—
278	C₄×C₃₁	—	4	4	no	—
279	C₄×C₃₁	—	4	4	no	—
280	C₄×C₃₁	—	4	4	no	—
281	C₄×C₃₁	—	4	4	no	—
282	C₂²×#41	C₂²×C₃₁	1	2	yes	—
283	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
284	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
285	#525	—	2	16	no	—
286	C₂×#123	C₂²×C₃₁	1	4	yes	—
287	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
288	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
289	C₂×#121	C₂²×C₃₁	1	4	yes	—
290	C₂×#121	C₂²×C₃₁	1	2	yes	—
291	C₂×#121	C₂²×C₃₁	1	2	yes	—
292	C₂×#121	C₂²×C₃₁	1	4	yes	—
293	C₂×#121	C₂²×C₃₁	1	4	yes	—
294	C₂×#120	C₂²×C₃₁	1	4	yes	—
295	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
296	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
297	C₂×#117	C₂²×C₃₁	1	2	yes	—
298	#400	C₂²×C₃₁	1	4	yes	complemented
299	#399	C₂²×C₃₁	1	4	yes	complemented
300	#398	C₂²×C₃₁	1	4	yes	complemented
301	#397	C₂²×C₃₁	1	4	yes	complemented
302	#395	C₂²×C₃₁	1	4	yes	complemented
303	#394	C₂²×C₃₁	1	4	yes	complemented
304	#393	C₂²×C₃₁	1	4	yes	complemented
305	#392	C₂²×C₃₁	1	4	yes	complemented
306	#388	C₂²×C₃₁	1	8	yes	complemented
307	#387	C₂²×C₃₁	1	8	yes	complemented
308	#386	C₂²×C₃₁	1	8	yes	—
309	#386	C₂²×C₃₁	1	8	yes	—
310	#386	C₂²×C₃₁	1	8	yes	—
311	#386	C₂²×C₃₁	1	8	yes	complemented
312	#385	C₂²×C₃₁	1	8	yes	complemented
313	#380	C₂²×C₃₁	1	4	yes	—
314	#380	C₂²×C₃₁	1	4	yes	—
315	#380	C₂²×C₃₁	1	4	yes	—
316	C₂×#95	C₂²×C₃₁	1	4	yes	—
317	C₂×#95	C₂²×C₃₁	1	2	yes	—
318	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×#95	C₂²×C₃₁	1	4	yes	—
321	C₂×C₃₁	—	2	4	no	—
322	C₂×C₃₁	—	2	4	no	—
323	C₂×C₃₁	—	2	4	no	—
324	C₂×C₃₁	—	8	4	no	—
325	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
326	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
327	C₂×C₃₁	C₂×#537	1	1	yes	characteristic
328	C₂×C₃₁	#3490	1	2	yes	—
329	C₂×C₃₁	#3492	1	2	yes	—
330	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×#41	—	2	8	no	—
332	C₂×#41	—	4	8	no	—
333	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
334	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
335	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
336	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
337	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
338	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
339	C₂³×C₄	—	2	8	no	—
340	C₂³×C₄	—	4	4	no	—
341	C₂³×C₄	—	4	4	no	—
342	C₂³×C₄	—	4	4	no	—
343	C₂³×C₄	D₄×C₃₁	1	8	yes	complemented
344	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
345	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
346	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
347	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
348	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
349	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
350	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
351	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
353	#123	—	2	16	no	—
354	C₂²⋊Q₈	—	2	8	no	—
355	C₂²⋊Q₈	—	2	8	no	—
356	#121	—	2	8	no	—
357	#121	—	2	8	no	—
358	#121	—	2	16	no	—
359	#121	—	4	8	no	—
360	#121	—	4	8	no	—
361	#120	—	2	16	no	—
362	C₄×D₄	—	2	16	no	—
363	C₄×D₄	—	2	16	no	—
364	#117	—	2	8	no	—
365	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
372	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
373	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
374	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
380	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
381	#95	C₂³×C₃₁	1	8	yes	—
382	#95	C₂³×C₃₁	1	4	yes	—
383	#95	C₂³×C₃₁	1	4	yes	—
384	#95	C₂³×C₃₁	1	8	yes	—
385	#95	C₂³×C₃₁	1	8	yes	—
386	#95	C₂³×C₃₁	1	8	yes	—
387	C₃₁	#35051	1	1	yes	characteristic, direct product factor, Sylow
388	C₂⁴	—	2	4	no	—
389	C₂⁴	—	2	2	no	—
390	C₂⁴	—	2	4	no	—
391	C₂⁴	—	2	2	no	—
392	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
393	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
394	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
395	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
396	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
397	#41	—	4	32	no	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	—	4	4	no	—
400	C₂×D₄	—	2	8	no	—
401	C₂×D₄	—	2	8	no	—
402	C₂×D₄	—	2	8	no	—
403	C₂×D₄	—	2	16	no	—
404	C₂×D₄	—	2	8	no	—
405	C₂×D₄	—	2	16	no	—
406	C₂×D₄	—	2	8	no	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	8	no	—
409	C₂×D₄	—	4	4	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	—	4	4	no	—
412	C₂²×C₄	—	2	8	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	8	no	—
415	C₂²×C₄	—	2	8	no	—
416	C₂²×C₄	—	2	8	no	—
417	C₂²×C₄	—	2	8	no	—
418	C₂²×C₄	—	2	8	no	—
419	C₂²×C₄	—	4	16	no	—
420	C₂²×C₄	—	4	2	no	—
421	C₂²×C₄	—	4	2	no	—
422	C₂²×C₄	—	4	4	no	—
423	C₂²×C₄	—	4	4	no	—
424	C₂²×C₄	—	4	2	no	—
425	C₂²×C₄	—	4	4	no	—
426	C₂²×C₄	—	4	2	no	—
427	C₂²×C₄	—	4	2	no	—
428	C₂²×C₄	—	4	2	no	—
429	C₂²×C₄	—	4	8	no	—
430	C₂²×C₄	—	4	32	no	—
431	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
432	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
433	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
434	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
435	C₂²×C₄	#41×C₃₁	1	8	yes	complemented
436	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
437	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
438	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
439	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
441	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
442	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	4	no	—
445	C₄⋊C₄	—	2	8	no	—
446	C₄⋊C₄	—	2	8	no	—
447	C₄⋊C₄	—	2	8	no	—
448	C₄⋊C₄	—	4	4	no	—
449	C₄⋊C₄	—	4	4	no	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	—	2	8	no	—
452	C₂²⋊C₄	—	2	8	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	—	2	8	no	—
455	C₂²⋊C₄	—	2	16	no	—
456	C₂²⋊C₄	—	4	8	no	—
457	C₂²⋊C₄	—	4	8	no	—
458	C₄²	—	2	8	no	—
459	C₂³	—	2	16	no	—
460	C₂³	—	2	16	no	—
461	C₂³	—	2	8	no	—
462	C₂³	—	2	8	no	—
463	C₂³	—	2	8	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	1	no	—
469	C₂³	—	2	1	no	—
470	C₂³	—	2	1	no	—
471	C₂³	—	2	1	no	—
472	C₂³	—	2	4	no	—
473	C₂³	—	2	8	no	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	8	no	—
476	C₂³	—	2	4	no	—
477	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
478	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
479	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
480	C₂³	C₂×#41×C₃₁	1	4	yes	—
481	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
482	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
483	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
484	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
485	Q₈	—	4	8	no	—
486	D₄	—	4	8	no	—
487	D₄	—	4	16	no	—
488	D₄	—	4	16	no	—
489	D₄	—	4	16	no	—
490	D₄	—	8	8	no	—
491	C₂×C₄	—	2	16	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	8	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	8	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	4	4	no	—
507	C₂×C₄	—	4	8	no	—
508	C₂×C₄	—	4	4	no	—
509	C₂×C₄	—	4	8	no	—
510	C₂×C₄	—	4	16	no	—
511	C₂×C₄	—	4	16	no	—
512	C₂×C₄	—	4	16	no	—
513	C₂×C₄	—	4	8	no	—
514	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
515	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
516	C₂×C₄	C₂×#41×C₃₁	1	8	yes	—
517	C₂²	—	2	32	no	—
518	C₂²	—	2	2	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	2	no	—
521	C₂²	—	2	2	no	—
522	C₂²	—	2	2	no	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	2	4	no	—
526	C₂²	—	2	4	no	—
527	C₂²	—	2	4	no	—
528	C₂²	—	2	4	no	—
529	C₂²	—	2	4	no	—
530	C₂²	—	4	4	no	—
531	C₂²	—	4	4	no	—
532	C₂²	—	4	4	no	—
533	C₂²	—	4	8	no	—
534	C₂²	—	4	8	no	—
535	C₂²	C₃₁×#532	1	4	yes	—
536	C₂²	C₃₁×#534	1	4	yes	—
537	C₂²	C₃₁×#537	1	4	yes	—
538	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
539	C₂²	C₂²×#41×C₃₁	1	2	yes	—
540	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
541	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
542	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
543	C₄	—	2	4	no	—
544	C₄	—	2	4	no	—
545	C₄	—	2	8	no	—
546	C₄	—	4	4	no	—
547	C₄	—	4	4	no	—
548	C₄	—	4	4	no	—
549	C₄	—	4	4	no	—
550	C₂	—	2	4	no	—
551	C₂	—	2	4	no	—
552	C₂	—	2	4	no	—
553	C₂	—	8	4	no	—
554	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
555	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
556	C₂	C₂×C₃₁×#537	1	1	yes	characteristic
557	C₂	C₃₁×#3490	1	2	yes	—
558	C₂	C₃₁×#3492	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#525	C₂	semidirect
2. C₃₁×#2847	C₂	semidirect
3. C₃₁×#2659	C₂	semidirect
4. C₃₁×#2656	C₂	semidirect
5. C₃₁×#2639	C₂	semidirect
6. C₃₁×#2564	C₂	semidirect
7. C₃₁×#2543	C₂	semidirect
8. C₃₁×#2498	C₂	semidirect
9. C₃₁×#2430	C₂	semidirect
10. C₃₁×#2325	C₂	semidirect
11. C₂²×C₃₁×#95	C₂	semidirect
28. C₃₁×#400	C₂²	semidirect
29. C₃₁×#399	C₂²	semidirect
30. C₃₁×#398	C₂²	semidirect
31. C₃₁×#397	C₂²	semidirect
32. C₃₁×#395	C₂²	semidirect
33. C₃₁×#394	C₂²	semidirect
34. C₃₁×#393	C₂²	semidirect
35. C₃₁×#392	C₂²	semidirect
36. C₃₁×#388	C₂²	semidirect
37. C₃₁×#387	C₂²	semidirect
41. C₃₁×#386	C₂²	semidirect
42. C₃₁×#385	C₂²	semidirect
64. C₂³×C₄×C₃₁	D₄	semidirect
155. C₂²×C₄×C₃₁	#41	semidirect
179. #35051	C₃₁	direct (N × H)
238. C₂×#525	C₂×C₃₁	semidirect
239. #2847	C₂×C₃₁	semidirect
240. #2659	C₂×C₃₁	semidirect
241. #2656	C₂×C₃₁	semidirect
242. #2639	C₂×C₃₁	semidirect
243. #2564	C₂×C₃₁	semidirect
244. #2543	C₂×C₃₁	semidirect
245. #2498	C₂×C₃₁	semidirect
246. #2430	C₂×C₃₁	semidirect
247. #2325	C₂×C₃₁	semidirect
248. C₂²×#95	C₂×C₃₁	semidirect
298. #400	C₂²×C₃₁	semidirect
299. #399	C₂²×C₃₁	semidirect
300. #398	C₂²×C₃₁	semidirect
301. #397	C₂²×C₃₁	semidirect
302. #395	C₂²×C₃₁	semidirect
303. #394	C₂²×C₃₁	semidirect
304. #393	C₂²×C₃₁	semidirect
305. #392	C₂²×C₃₁	semidirect
306. #388	C₂²×C₃₁	semidirect
307. #387	C₂²×C₃₁	semidirect
311. #386	C₂²×C₃₁	semidirect
312. #385	C₂²×C₃₁	semidirect
343. C₂³×C₄	D₄×C₃₁	semidirect
387. C₃₁	#35051	direct (N × H)
435. C₂²×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×#525	2
C₃₁×#2847	2
C₃₁×#2659	2
C₃₁×#2656	2
C₃₁×#2639	2
C₃₁×#2564	2
C₃₁×#2543	2
C₃₁×#2498	2
C₃₁×#2430	2
C₃₁×#2325	2
C₂²×C₃₁×#95	2
#35051	31

Subgroups — C31×#35051 #21491810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#35051 #21491810