Subgroups — C₃₁×#60458 #21517217

Order 7936.

2824 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#576	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3488	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3062	C₂	1	6	yes	maximal, complemented
5	C₃₁×#3061	C₂	1	3	yes	maximal, complemented
6	C₃₁×#3058	C₂	1	3	yes	maximal, complemented
7	C₃₁×#3055	C₂	1	6	yes	maximal, complemented
8	C₃₁×#3050	C₂	1	3	yes	maximal, complemented
9	C₂×C₃₁×#466	C₂	1	3	yes	maximal, complemented
10	C₃₁×#2913	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2897	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	2	yes	—
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#576	—	2	8	no	—
15	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	6	yes	complemented
16	C₂×C₃₁×#135	C₂²	1	3	yes	complemented
17	C₂²×D₈×C₃₁	C₂²	1	3	yes	complemented
18	C₂×C₃₁×#131	—	2	8	no	—
19	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#131	C₂²	1	3	yes	complemented
22	C₂×C₃₁×#131	C₂²	1	3	yes	complemented
23	C₂²×M₁₆×C₃₁	C₂²	1	3	yes	complemented
24	C₂³×C₈×C₃₁	C₂²	1	3	yes	complemented
25	C₃₁×#550	—	2	2	no	—
26	C₃₁×#548	—	2	2	no	—
27	C₃₁×#546	—	2	6	no	—
28	D₄²×C₃₁	—	2	6	no	—
29	C₃₁×#537	C₂²	1	6	yes	complemented
30	C₃₁×#535	C₂²	1	6	yes	complemented
31	C₃₁×#525	C₂²	1	6	yes	complemented
32	C₂×C₃₁×#121	C₂²	1	6	yes	complemented
33	C₃₁×#520	C₂²	1	2	yes	—
34	C₃₁×#469	C₂²	1	12	yes	complemented
35	C₃₁×#468	C₂²	1	12	yes	complemented
36	C₃₁×#467	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#467	C₂²	1	3	yes	complemented
38	C₃₁×#466	C₂²	1	3	yes	complemented
39	C₃₁×#466	C₂²	1	6	yes	complemented
40	C₃₁×#466	C₂²	1	6	yes	complemented
41	C₃₁×#466	C₂²	1	6	yes	complemented
42	C₃₁×#466	C₂²	1	3	yes	complemented
43	C₃₁×#465	C₂²	1	6	yes	complemented
44	C₃₁×#465	C₂²	1	6	yes	complemented
45	C₃₁×#428	C₂²	1	6	yes	complemented
46	C₃₁×#427	C₂²	1	3	yes	complemented
47	C₂×C₃₁×#107	C₂²	1	3	yes	complemented
48	C₃₁×#418	C₂²	1	6	yes	complemented
49	C₃₁×#417	C₂²	1	6	yes	complemented
50	C₃₁×#416	C₂²	1	6	yes	complemented
51	C₂×C₃₁×#102	C₂²	1	6	yes	complemented
52	C₃₁×H(5,𝔽₂)	—	2	8	no	—
53	C₃₁×H(5,𝔽₂)	—	2	8	no	—
54	C₂×#41×C₃₁	—	2	4	no	—
55	C₂×#41×C₃₁	—	2	4	no	—
56	C₂×#41×C₃₁	—	2	2	no	—
57	C₂×#41×C₃₁	D₄	1	3	yes	—
58	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
59	C₂×#41×C₃₁	C₂³	1	6	yes	—
60	C₂×#41×C₃₁	C₂³	1	3	yes	—
61	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
63	C₂²×D₄×C₃₁	—	2	6	no	—
64	C₂²×D₄×C₃₁	—	2	6	no	—
65	C₂²×D₄×C₃₁	D₄	1	3	yes	—
66	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
67	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
68	C₃₁×AGL(1,ℤ/8)	—	2	24	no	—
69	C₃₁×#135	—	2	12	no	—
70	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
72	C₂×D₈×C₃₁	—	2	12	no	—
73	C₂×D₈×C₃₁	C₂³	1	3	yes	—
74	C₂×D₈×C₃₁	C₂³	1	6	yes	—
75	C₃₁×#131	—	2	4	no	—
76	C₃₁×#131	—	2	12	no	—
77	C₃₁×#131	—	2	12	no	—
78	C₃₁×#131	D₄	1	8	yes	complemented
79	C₂×M₁₆×C₃₁	—	2	12	no	—
80	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
81	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
82	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
83	C₂²×C₈×C₃₁	—	2	12	no	—
84	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
85	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
86	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
87	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
89	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
90	C₃₁×#127	—	2	6	no	—
91	C₃₁×#124	—	2	6	no	—
92	C₃₁×#123	—	2	6	no	—
93	C₃₁×C₂²⋊Q₈	—	2	6	no	—
94	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
95	C₃₁×C₂²⋊Q₈	C₂³	1	6	yes	—
96	C₃₁×#121	—	2	12	no	—
97	C₃₁×#121	—	2	6	no	—
98	C₃₁×#121	—	2	6	no	—
99	C₃₁×#121	—	2	6	no	—
100	C₃₁×#121	C₂³	1	6	yes	—
101	C₃₁×#121	C₂³	1	6	yes	—
102	C₃₁×#121	C₂³	1	12	yes	—
103	C₃₁×#120	—	2	6	no	—
104	C₃₁×#120	—	2	12	no	—
105	C₄×Q₈×C₃₁	—	2	2	no	—
106	C₄×D₄×C₃₁	—	2	6	no	—
107	C₄×D₄×C₃₁	—	2	6	no	—
108	C₄×D₄×C₃₁	—	2	6	no	—
109	C₃₁×#117	C₂³	1	6	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
111	C₂×C₂²⋊C₄×C₃₁	—	2	6	no	—
112	C₃₁×#107	C₂³	1	3	yes	—
113	C₃₁×#107	C₂³	1	6	yes	—
114	C₃₁×#107	C₂³	1	1	yes	characteristic
115	C₃₁×#106	C₂³	1	6	yes	—
116	C₃₁×#103	C₂³	1	2	yes	—
117	C₃₁×#103	C₂³	1	6	yes	—
118	C₃₁×#102	C₂³	1	6	yes	—
119	C₃₁×#102	C₂³	1	12	yes	—
120	C₃₁×#102	C₂³	1	6	yes	—
121	C₂⁴×C₃₁	—	2	6	no	—
122	C₂⁴×C₃₁	C₂×D₄	1	3	yes	—
123	#41×C₃₁	—	2	4	no	—
124	#41×C₃₁	—	2	12	no	—
125	#41×C₃₁	—	2	6	no	—
126	#41×C₃₁	—	2	6	no	—
127	#41×C₃₁	—	2	12	no	—
128	#41×C₃₁	—	2	12	no	—
129	#41×C₃₁	—	4	8	no	—
130	#41×C₃₁	C₂×D₄	1	4	yes	—
131	#41×C₃₁	#41	1	4	yes	—
132	C₂×Q₈×C₃₁	—	2	4	no	—
133	C₂×Q₈×C₃₁	C₂×D₄	1	3	yes	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
136	C₂×D₄×C₃₁	—	2	12	no	—
137	C₂×D₄×C₃₁	—	2	6	no	—
138	C₂×D₄×C₃₁	—	2	6	no	—
139	C₂×D₄×C₃₁	—	2	6	no	—
140	C₂×D₄×C₃₁	—	2	12	no	—
141	C₂×D₄×C₃₁	—	2	12	no	—
142	C₂×D₄×C₃₁	—	2	6	no	—
143	C₂×D₄×C₃₁	—	2	12	no	—
144	C₂×D₄×C₃₁	—	4	24	no	—
145	C₂×D₄×C₃₁	—	8	12	no	—
146	C₂×D₄×C₃₁	C₂×D₄	1	6	yes	—
147	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
148	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
149	C₂×D₄×C₃₁	C₂⁴	1	6	yes	—
150	C₂²×C₄×C₃₁	—	2	6	no	—
151	C₂²×C₄×C₃₁	—	2	6	no	—
152	C₂²×C₄×C₃₁	—	2	6	no	—
153	C₂²×C₄×C₃₁	—	2	6	no	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
156	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
160	Q₁₆×C₃₁	—	2	2	no	—
161	SD₁₆×C₃₁	—	2	12	no	—
162	D₈×C₃₁	—	2	6	no	—
163	D₈×C₃₁	—	2	12	no	—
164	M₁₆×C₃₁	—	2	6	no	—
165	M₁₆×C₃₁	—	2	12	no	—
166	M₁₆×C₃₁	C₂×D₄	1	12	yes	complemented
167	C₂×C₈×C₃₁	—	2	6	no	—
168	C₂×C₈×C₃₁	—	2	2	no	—
169	C₂×C₈×C₃₁	—	2	6	no	—
170	C₂×C₈×C₃₁	—	2	6	no	—
171	C₂×C₈×C₃₁	—	2	6	no	—
172	C₂×C₈×C₃₁	C₂×D₄	1	12	yes	complemented
173	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
175	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
176	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₄⋊C₄×C₃₁	—	2	6	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	2	6	no	—
180	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
181	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
182	C₂²⋊C₄×C₃₁	—	2	6	no	—
183	C₂²⋊C₄×C₃₁	—	2	2	no	—
184	C₂²⋊C₄×C₃₁	—	2	6	no	—
185	C₂²⋊C₄×C₃₁	—	2	6	no	—
186	C₂²⋊C₄×C₃₁	—	2	12	no	—
187	C₂²⋊C₄×C₃₁	—	2	6	no	—
188	C₄²×C₃₁	—	2	6	no	—
189	#60458	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂³×C₃₁	—	2	6	no	—
191	C₂³×C₃₁	—	2	6	no	—
192	C₂³×C₃₁	—	2	6	no	—
193	C₂³×C₃₁	—	4	3	no	—
194	C₂³×C₃₁	—	4	12	no	—
195	C₂³×C₃₁	—	8	6	no	—
196	C₂³×C₃₁	—	8	6	no	—
197	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
198	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
199	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
200	Q₈×C₃₁	—	2	6	no	—
201	Q₈×C₃₁	—	2	2	no	—
202	Q₈×C₃₁	—	2	2	no	—
203	Q₈×C₃₁	#121	1	4	yes	—
204	D₄×C₃₁	—	2	12	no	—
205	D₄×C₃₁	—	2	6	no	—
206	D₄×C₃₁	—	2	6	no	—
207	D₄×C₃₁	—	2	6	no	—
208	D₄×C₃₁	—	4	12	no	—
209	D₄×C₃₁	—	4	12	no	—
210	D₄×C₃₁	—	4	12	no	—
211	D₄×C₃₁	—	8	6	no	—
212	D₄×C₃₁	—	8	12	no	—
213	D₄×C₃₁	#121	1	12	yes	—
214	C₂×C₄×C₃₁	—	2	6	no	—
215	C₂×C₄×C₃₁	—	2	6	no	—
216	C₂×C₄×C₃₁	—	2	6	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	3	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	3	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	6	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	4	12	no	—
225	C₂×C₄×C₃₁	—	8	6	no	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
228	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
229	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
230	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
231	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
232	C₈×C₃₁	—	2	1	no	—
233	C₈×C₃₁	—	2	3	no	—
234	C₈×C₃₁	C₂²×D₄	1	2	yes	—
235	C₈×C₃₁	C₂²×D₄	1	6	yes	—
236	C₂×#576	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#3488	C₂×C₃₁	1	2	yes	complemented
239	#3062	C₂×C₃₁	1	6	yes	complemented
240	#3061	C₂×C₃₁	1	3	yes	complemented
241	#3058	C₂×C₃₁	1	3	yes	complemented
242	#3055	C₂×C₃₁	1	6	yes	complemented
243	#3050	C₂×C₃₁	1	3	yes	complemented
244	C₂×#466	C₂×C₃₁	1	3	yes	complemented
245	#2913	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#2897	C₂×C₃₁	1	2	yes	complemented
247	C₂²×C₃₁	—	2	1	no	—
248	C₂²×C₃₁	—	2	3	no	—
249	C₂²×C₃₁	—	2	1	no	—
250	C₂²×C₃₁	—	2	3	no	—
251	C₂²×C₃₁	—	2	3	no	—
252	C₂²×C₃₁	—	2	3	no	—
253	C₂²×C₃₁	—	2	3	no	—
254	C₂²×C₃₁	—	4	12	no	—
255	C₂²×C₃₁	—	4	6	no	—
256	C₂²×C₃₁	—	4	6	no	—
257	C₂²×C₃₁	—	4	6	no	—
258	C₂²×C₃₁	—	16	12	no	—
259	C₂²×C₃₁	C₂×#121	1	6	yes	—
260	C₂²×C₃₁	#576	1	2	yes	complemented
261	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
262	C₄×C₃₁	—	2	1	no	—
263	C₄×C₃₁	—	2	3	no	—
264	C₄×C₃₁	—	4	2	no	—
265	C₄×C₃₁	—	4	2	no	—
266	C₄×C₃₁	—	4	6	no	—
267	C₄×C₃₁	C₂×#121	1	6	yes	—
268	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
269	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
270	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	2	yes	—
271	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
272	#576	—	2	8	no	—
273	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	6	yes	complemented
274	C₂×#135	C₂²×C₃₁	1	3	yes	complemented
275	C₂²×D₈	C₂²×C₃₁	1	3	yes	complemented
276	C₂×#131	—	2	8	no	—
277	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×#131	C₂²×C₃₁	1	3	yes	complemented
280	C₂×#131	C₂²×C₃₁	1	3	yes	complemented
281	C₂²×M₁₆	C₂²×C₃₁	1	3	yes	complemented
282	C₂³×C₈	C₂²×C₃₁	1	3	yes	complemented
283	#550	—	2	2	no	—
284	#548	—	2	2	no	—
285	#546	—	2	6	no	—
286	D₄²	—	2	6	no	—
287	#537	C₂²×C₃₁	1	6	yes	complemented
288	#535	C₂²×C₃₁	1	6	yes	complemented
289	#525	C₂²×C₃₁	1	6	yes	complemented
290	C₂×#121	C₂²×C₃₁	1	6	yes	complemented
291	#520	C₂²×C₃₁	1	2	yes	—
292	#469	C₂²×C₃₁	1	12	yes	complemented
293	#468	C₂²×C₃₁	1	12	yes	complemented
294	#467	C₂²×C₃₁	1	1	yes	characteristic, complemented
295	#467	C₂²×C₃₁	1	3	yes	complemented
296	#466	C₂²×C₃₁	1	3	yes	complemented
297	#466	C₂²×C₃₁	1	6	yes	complemented
298	#466	C₂²×C₃₁	1	6	yes	complemented
299	#466	C₂²×C₃₁	1	6	yes	complemented
300	#466	C₂²×C₃₁	1	3	yes	complemented
301	#465	C₂²×C₃₁	1	6	yes	complemented
302	#465	C₂²×C₃₁	1	6	yes	complemented
303	#428	C₂²×C₃₁	1	6	yes	complemented
304	#427	C₂²×C₃₁	1	3	yes	complemented
305	C₂×#107	C₂²×C₃₁	1	3	yes	complemented
306	#418	C₂²×C₃₁	1	6	yes	complemented
307	#417	C₂²×C₃₁	1	6	yes	complemented
308	#416	C₂²×C₃₁	1	6	yes	complemented
309	C₂×#102	C₂²×C₃₁	1	6	yes	complemented
310	C₂×C₃₁	—	2	2	no	—
311	C₂×C₃₁	—	2	6	no	—
312	C₂×C₃₁	—	4	3	no	—
313	C₂×C₃₁	—	8	6	no	—
314	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
315	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
316	C₂×C₃₁	#3584	1	1	yes	characteristic
317	H(5,𝔽₂)	—	2	8	no	—
318	H(5,𝔽₂)	—	2	8	no	—
319	C₂×#41	—	2	4	no	—
320	C₂×#41	—	2	4	no	—
321	C₂×#41	—	2	2	no	—
322	C₂×#41	D₄×C₃₁	1	3	yes	—
323	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
324	C₂×#41	C₂³×C₃₁	1	6	yes	—
325	C₂×#41	C₂³×C₃₁	1	3	yes	—
326	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
327	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
328	C₂²×D₄	—	2	6	no	—
329	C₂²×D₄	—	2	6	no	—
330	C₂²×D₄	D₄×C₃₁	1	3	yes	—
331	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
332	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
333	AGL(1,ℤ/8)	—	2	24	no	—
334	#135	—	2	12	no	—
335	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
336	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
337	C₂×D₈	—	2	12	no	—
338	C₂×D₈	C₂³×C₃₁	1	3	yes	—
339	C₂×D₈	C₂³×C₃₁	1	6	yes	—
340	#131	—	2	4	no	—
341	#131	—	2	12	no	—
342	#131	—	2	12	no	—
343	#131	D₄×C₃₁	1	8	yes	complemented
344	C₂×M₁₆	—	2	12	no	—
345	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
346	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
347	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
348	C₂²×C₈	—	2	12	no	—
349	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
350	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
351	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
352	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
353	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
354	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
355	#127	—	2	6	no	—
356	#124	—	2	6	no	—
357	#123	—	2	6	no	—
358	C₂²⋊Q₈	—	2	6	no	—
359	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
360	C₂²⋊Q₈	C₂³×C₃₁	1	6	yes	—
361	#121	—	2	12	no	—
362	#121	—	2	6	no	—
363	#121	—	2	6	no	—
364	#121	—	2	6	no	—
365	#121	C₂³×C₃₁	1	6	yes	—
366	#121	C₂³×C₃₁	1	12	yes	—
367	#121	C₂³×C₃₁	1	6	yes	—
368	#120	—	2	6	no	—
369	#120	—	2	12	no	—
370	C₄×Q₈	—	2	2	no	—
371	C₄×D₄	—	2	6	no	—
372	C₄×D₄	—	2	6	no	—
373	C₄×D₄	—	2	6	no	—
374	#117	C₂³×C₃₁	1	6	yes	—
375	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
376	C₂×C₂²⋊C₄	—	2	6	no	—
377	#107	C₂³×C₃₁	1	3	yes	—
378	#107	C₂³×C₃₁	1	6	yes	—
379	#107	C₂³×C₃₁	1	1	yes	characteristic
380	#106	C₂³×C₃₁	1	6	yes	—
381	#103	C₂³×C₃₁	1	2	yes	—
382	#103	C₂³×C₃₁	1	6	yes	—
383	#102	C₂³×C₃₁	1	6	yes	—
384	#102	C₂³×C₃₁	1	12	yes	—
385	#102	C₂³×C₃₁	1	6	yes	—
386	C₃₁	#60458	1	1	yes	characteristic, direct product factor, Sylow
387	C₂⁴	—	2	6	no	—
388	C₂⁴	C₂×D₄×C₃₁	1	3	yes	—
389	#41	—	2	4	no	—
390	#41	—	2	12	no	—
391	#41	—	2	6	no	—
392	#41	—	2	6	no	—
393	#41	—	2	12	no	—
394	#41	—	2	12	no	—
395	#41	—	4	8	no	—
396	#41	C₂×D₄×C₃₁	1	4	yes	—
397	#41	#41×C₃₁	1	4	yes	—
398	C₂×Q₈	—	2	4	no	—
399	C₂×Q₈	C₂×D₄×C₃₁	1	3	yes	—
400	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
402	C₂×D₄	—	2	12	no	—
403	C₂×D₄	—	2	6	no	—
404	C₂×D₄	—	2	6	no	—
405	C₂×D₄	—	2	6	no	—
406	C₂×D₄	—	2	12	no	—
407	C₂×D₄	—	2	12	no	—
408	C₂×D₄	—	2	6	no	—
409	C₂×D₄	—	2	12	no	—
410	C₂×D₄	—	4	24	no	—
411	C₂×D₄	—	8	12	no	—
412	C₂×D₄	C₂×D₄×C₃₁	1	6	yes	—
413	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
414	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
415	C₂×D₄	C₂⁴×C₃₁	1	6	yes	—
416	C₂²×C₄	—	2	6	no	—
417	C₂²×C₄	—	2	6	no	—
418	C₂²×C₄	—	2	6	no	—
419	C₂²×C₄	—	2	6	no	—
420	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
421	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
422	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
423	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
425	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
426	Q₁₆	—	2	2	no	—
427	SD₁₆	—	2	12	no	—
428	D₈	—	2	6	no	—
429	D₈	—	2	12	no	—
430	M₁₆	—	2	6	no	—
431	M₁₆	—	2	12	no	—
432	M₁₆	C₂×D₄×C₃₁	1	12	yes	complemented
433	C₂×C₈	—	2	6	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	6	no	—
436	C₂×C₈	—	2	6	no	—
437	C₂×C₈	—	2	6	no	—
438	C₂×C₈	C₂×D₄×C₃₁	1	12	yes	complemented
439	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
440	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
441	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
442	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
443	C₄⋊C₄	—	2	6	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	6	no	—
446	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
447	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
448	C₂²⋊C₄	—	2	6	no	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	—	2	6	no	—
451	C₂²⋊C₄	—	2	6	no	—
452	C₂²⋊C₄	—	2	12	no	—
453	C₂²⋊C₄	—	2	6	no	—
454	C₄²	—	2	6	no	—
455	C₂³	—	2	6	no	—
456	C₂³	—	2	6	no	—
457	C₂³	—	2	6	no	—
458	C₂³	—	4	3	no	—
459	C₂³	—	4	12	no	—
460	C₂³	—	8	6	no	—
461	C₂³	—	8	6	no	—
462	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
463	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
464	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
465	Q₈	—	2	6	no	—
466	Q₈	—	2	2	no	—
467	Q₈	—	2	2	no	—
468	Q₈	C₃₁×#121	1	4	yes	—
469	D₄	—	2	12	no	—
470	D₄	—	2	6	no	—
471	D₄	—	2	6	no	—
472	D₄	—	2	6	no	—
473	D₄	—	4	12	no	—
474	D₄	—	4	12	no	—
475	D₄	—	4	12	no	—
476	D₄	—	8	6	no	—
477	D₄	—	8	12	no	—
478	D₄	C₃₁×#121	1	12	yes	—
479	C₂×C₄	—	2	6	no	—
480	C₂×C₄	—	2	6	no	—
481	C₂×C₄	—	2	6	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	3	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	3	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	6	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	4	12	no	—
490	C₂×C₄	—	8	6	no	—
491	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
493	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
494	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
495	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
496	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
497	C₈	—	2	1	no	—
498	C₈	—	2	3	no	—
499	C₈	C₂²×D₄×C₃₁	1	2	yes	—
500	C₈	C₂²×D₄×C₃₁	1	6	yes	—
501	C₂²	—	2	1	no	—
502	C₂²	—	2	3	no	—
503	C₂²	—	2	1	no	—
504	C₂²	—	2	3	no	—
505	C₂²	—	2	3	no	—
506	C₂²	—	2	3	no	—
507	C₂²	—	2	3	no	—
508	C₂²	—	4	12	no	—
509	C₂²	—	4	6	no	—
510	C₂²	—	4	6	no	—
511	C₂²	—	4	6	no	—
512	C₂²	—	16	12	no	—
513	C₂²	C₂×C₃₁×#121	1	6	yes	—
514	C₂²	C₃₁×#576	1	2	yes	complemented
515	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
516	C₄	—	2	1	no	—
517	C₄	—	2	3	no	—
518	C₄	—	4	2	no	—
519	C₄	—	4	6	no	—
520	C₄	—	4	2	no	—
521	C₄	C₂×C₃₁×#121	1	6	yes	—
522	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
523	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
524	C₂	—	2	2	no	—
525	C₂	—	2	6	no	—
526	C₂	—	4	3	no	—
527	C₂	—	8	6	no	—
528	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
529	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
530	C₂	C₃₁×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#576	C₂	semidirect
2. C₂²×C₃₁×#131	C₂	semidirect
3. C₃₁×#3488	C₂	semidirect
4. C₃₁×#3062	C₂	semidirect
5. C₃₁×#3061	C₂	semidirect
6. C₃₁×#3058	C₂	semidirect
7. C₃₁×#3055	C₂	semidirect
8. C₃₁×#3050	C₂	semidirect
9. C₂×C₃₁×#466	C₂	semidirect
10. C₃₁×#2913	C₂	semidirect
11. C₃₁×#2897	C₂	semidirect
15. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
16. C₂×C₃₁×#135	C₂²	semidirect
17. C₂²×D₈×C₃₁	C₂²	semidirect
21. C₂×C₃₁×#131	C₂²	semidirect
22. C₂×C₃₁×#131	C₂²	semidirect
23. C₂²×M₁₆×C₃₁	C₂²	semidirect
24. C₂³×C₈×C₃₁	C₂²	semidirect
29. C₃₁×#537	C₂²	semidirect
30. C₃₁×#535	C₂²	semidirect
31. C₃₁×#525	C₂²	semidirect
32. C₂×C₃₁×#121	C₂²	semidirect
34. C₃₁×#469	C₂²	semidirect
35. C₃₁×#468	C₂²	semidirect
36. C₃₁×#467	C₂²	semidirect
37. C₃₁×#467	C₂²	semidirect
38. C₃₁×#466	C₂²	semidirect
39. C₃₁×#466	C₂²	semidirect
40. C₃₁×#466	C₂²	semidirect
41. C₃₁×#466	C₂²	semidirect
42. C₃₁×#466	C₂²	semidirect
43. C₃₁×#465	C₂²	semidirect
44. C₃₁×#465	C₂²	semidirect
45. C₃₁×#428	C₂²	semidirect
46. C₃₁×#427	C₂²	semidirect
47. C₂×C₃₁×#107	C₂²	semidirect
48. C₃₁×#418	C₂²	semidirect
49. C₃₁×#417	C₂²	semidirect
50. C₃₁×#416	C₂²	semidirect
51. C₂×C₃₁×#102	C₂²	semidirect
78. C₃₁×#131	D₄	semidirect
166. M₁₆×C₃₁	C₂×D₄	semidirect
172. C₂×C₈×C₃₁	C₂×D₄	semidirect
189. #60458	C₃₁	direct (N × H)
236. C₂×#576	C₂×C₃₁	semidirect
237. C₂²×#131	C₂×C₃₁	semidirect
238. #3488	C₂×C₃₁	semidirect
239. #3062	C₂×C₃₁	semidirect
240. #3061	C₂×C₃₁	semidirect
241. #3058	C₂×C₃₁	semidirect
242. #3055	C₂×C₃₁	semidirect
243. #3050	C₂×C₃₁	semidirect
244. C₂×#466	C₂×C₃₁	semidirect
245. #2913	C₂×C₃₁	semidirect
246. #2897	C₂×C₃₁	semidirect
260. C₂²×C₃₁	#576	semidirect
273. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
274. C₂×#135	C₂²×C₃₁	semidirect
275. C₂²×D₈	C₂²×C₃₁	semidirect
279. C₂×#131	C₂²×C₃₁	semidirect
280. C₂×#131	C₂²×C₃₁	semidirect
281. C₂²×M₁₆	C₂²×C₃₁	semidirect
282. C₂³×C₈	C₂²×C₃₁	semidirect
287. #537	C₂²×C₃₁	semidirect
288. #535	C₂²×C₃₁	semidirect
289. #525	C₂²×C₃₁	semidirect
290. C₂×#121	C₂²×C₃₁	semidirect
292. #469	C₂²×C₃₁	semidirect
293. #468	C₂²×C₃₁	semidirect
294. #467	C₂²×C₃₁	semidirect
295. #467	C₂²×C₃₁	semidirect
296. #466	C₂²×C₃₁	semidirect
297. #466	C₂²×C₃₁	semidirect
298. #466	C₂²×C₃₁	semidirect
299. #466	C₂²×C₃₁	semidirect
300. #466	C₂²×C₃₁	semidirect
301. #465	C₂²×C₃₁	semidirect
302. #465	C₂²×C₃₁	semidirect
303. #428	C₂²×C₃₁	semidirect
304. #427	C₂²×C₃₁	semidirect
305. C₂×#107	C₂²×C₃₁	semidirect
306. #418	C₂²×C₃₁	semidirect
307. #417	C₂²×C₃₁	semidirect
308. #416	C₂²×C₃₁	semidirect
309. C₂×#102	C₂²×C₃₁	semidirect
343. #131	D₄×C₃₁	semidirect
386. C₃₁	#60458	direct (N × H)
432. M₁₆	C₂×D₄×C₃₁	semidirect
438. C₂×C₈	C₂×D₄×C₃₁	semidirect
514. C₂²	C₃₁×#576	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#576	2
C₂²×C₃₁×#131	2
C₃₁×#3488	2
C₃₁×#3062	2
C₃₁×#3061	2
C₃₁×#3058	2
C₃₁×#3055	2
C₃₁×#3050	2
C₂×C₃₁×#466	2
C₃₁×#2913	2
C₃₁×#2897	2
#60458	31

Subgroups — C31×#60458 #21517217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60458 #21517217