Subgroups — C₃₁×#33837 #21490596

Order 7936.

680 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3287	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3285	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2250	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2218	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2217	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2211	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2207	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2189	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2186	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2186	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2185	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2176	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#2168	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
19	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#550	—	2	1	no	—
21	C₃₁×#548	—	2	1	no	—
22	C₃₁×#510	—	2	4	no	—
23	C₃₁×#508	—	2	4	no	—
24	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
34	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#495	C₂²	1	1	yes	characteristic
36	C₃₁×#493	C₂²	1	1	yes	characteristic
37	C₃₁×#467	C₂²	1	1	yes	characteristic
38	C₃₁×#466	C₂²	1	1	yes	characteristic
39	C₃₁×#466	C₂²	1	1	yes	characteristic
40	C₃₁×#466	C₂²	1	1	yes	characteristic
41	C₃₁×#464	—	2	1	no	—
42	C₃₁×#460	—	2	1	no	—
43	C₃₁×#452	—	2	1	no	—
44	C₃₁×#449	—	2	1	no	—
45	C₃₁×#449	—	2	1	no	—
46	C₃₁×#448	—	2	1	no	—
47	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
48	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
49	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
50	C₃₁×#367	C₂²	1	1	yes	characteristic
51	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
52	C₃₁×#358	C₂²	1	1	yes	characteristic
53	C₃₁×#358	C₂²	1	1	yes	characteristic
54	C₃₁×#358	C₂²	1	1	yes	characteristic
55	C₃₁×#357	C₂²	1	1	yes	characteristic
56	C₃₁×#357	C₂²	1	1	yes	characteristic
57	C₃₁×#357	C₂²	1	1	yes	characteristic
58	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
59	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
60	C₃₁×#346	C₂²	1	1	yes	characteristic
61	C₂×#41×C₃₁	—	2	1	no	—
62	C₂×#41×C₃₁	—	2	1	no	—
63	C₂×#41×C₃₁	—	2	1	no	—
64	C₂²×Q₈×C₃₁	—	2	1	no	—
65	C₃₁×#135	—	2	2	no	—
66	C₃₁×#135	—	2	2	no	—
67	C₃₁×#135	—	2	2	no	—
68	C₃₁×#135	—	4	1	no	—
69	C₃₁×#135	—	4	1	no	—
70	C₃₁×#135	—	4	1	no	—
71	C₂×Q₁₆×C₃₁	—	2	2	no	—
72	C₂×Q₁₆×C₃₁	—	2	1	no	—
73	C₂×Q₁₆×C₃₁	—	4	1	no	—
74	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×SD₁₆×C₃₁	—	2	1	no	—
78	C₂×SD₁₆×C₃₁	—	2	1	no	—
79	C₂×SD₁₆×C₃₁	—	2	1	no	—
80	C₂×SD₁₆×C₃₁	—	2	1	no	—
81	C₂×D₈×C₃₁	—	2	1	no	—
82	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×#127	—	2	1	no	—
88	C₃₁×#127	—	4	1	no	—
89	C₃₁×#124	—	2	1	no	—
90	C₃₁×#124	—	4	1	no	—
91	C₃₁×C₂²⋊Q₈	—	2	1	no	—
92	C₃₁×C₂²⋊Q₈	—	4	1	no	—
93	C₃₁×#121	—	2	1	no	—
94	C₃₁×#121	—	2	1	no	—
95	C₃₁×#121	—	2	1	no	—
96	C₃₁×#121	—	4	1	no	—
97	C₃₁×#121	—	4	1	no	—
98	C₃₁×#121	—	4	1	no	—
99	C₄×Q₈×C₃₁	—	2	1	no	—
100	C₄×Q₈×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	—	4	1	no	—
102	C₄×D₄×C₃₁	—	4	1	no	—
103	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
104	C₃₁×Q₃₂	—	2	2	no	—
105	C₃₁×Q₃₂	D₄	1	4	yes	complemented
106	C₃₁×SD₃₂	—	2	2	no	—
107	C₃₁×SD₃₂	—	2	2	no	—
108	C₃₁×SD₃₂	—	2	2	no	—
109	C₃₁×SD₃₂	—	2	2	no	—
110	C₃₁×SD₃₂	—	4	1	no	—
111	C₃₁×D₁₆	—	2	2	no	—
112	C₃₁×D₁₆	—	4	1	no	—
113	C₃₁×M₃₂	—	2	2	no	—
114	C₂×C₁₆×C₃₁	—	2	2	no	—
115	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
116	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
117	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
119	C₃₁×#107	C₂³	1	1	yes	characteristic
120	C₃₁×#107	C₂³	1	1	yes	characteristic
121	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
122	C₃₁×#103	—	2	1	no	—
123	C₃₁×#103	—	2	1	no	—
124	C₃₁×#103	—	2	1	no	—
125	C₃₁×#102	—	2	1	no	—
126	C₃₁×#102	—	2	1	no	—
127	C₃₁×#102	—	2	1	no	—
128	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
129	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
130	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
131	#41×C₃₁	—	4	2	no	—
132	#41×C₃₁	—	4	2	no	—
133	#41×C₃₁	—	4	2	no	—
134	#41×C₃₁	—	8	1	no	—
135	#41×C₃₁	—	8	1	no	—
136	#41×C₃₁	—	8	1	no	—
137	C₂×Q₈×C₃₁	—	2	1	no	—
138	C₂×Q₈×C₃₁	—	2	1	no	—
139	C₂×Q₈×C₃₁	—	2	1	no	—
140	C₂×Q₈×C₃₁	—	4	2	no	—
141	C₂×Q₈×C₃₁	—	4	1	no	—
142	C₂×Q₈×C₃₁	—	8	1	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	—	2	1	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	—	4	1	no	—
147	C₂×D₄×C₃₁	—	4	1	no	—
148	C₂×D₄×C₃₁	—	4	1	no	—
149	C₂×D₄×C₃₁	—	4	1	no	—
150	C₂×D₄×C₃₁	—	4	1	no	—
151	C₂×D₄×C₃₁	—	4	1	no	—
152	C₂×D₄×C₃₁	—	4	1	no	—
153	C₂×D₄×C₃₁	—	4	1	no	—
154	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	—	4	1	no	—
156	C₂²×C₄×C₃₁	—	4	1	no	—
157	C₂²×C₄×C₃₁	—	4	1	no	—
158	C₂²×C₄×C₃₁	—	4	1	no	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	Q₁₆×C₃₁	—	2	1	no	—
162	Q₁₆×C₃₁	—	2	1	no	—
163	Q₁₆×C₃₁	—	2	1	no	—
164	Q₁₆×C₃₁	—	2	1	no	—
165	Q₁₆×C₃₁	—	4	2	no	—
166	Q₁₆×C₃₁	C₂×D₄	1	2	yes	—
167	Q₁₆×C₃₁	C₂×D₄	1	2	yes	—
168	SD₁₆×C₃₁	—	4	2	no	—
169	SD₁₆×C₃₁	—	4	2	no	—
170	SD₁₆×C₃₁	—	4	2	no	—
171	SD₁₆×C₃₁	—	8	1	no	—
172	D₈×C₃₁	—	2	1	no	—
173	D₈×C₃₁	—	2	1	no	—
174	D₈×C₃₁	—	2	1	no	—
175	D₈×C₃₁	—	2	1	no	—
176	D₈×C₃₁	—	2	1	no	—
177	D₈×C₃₁	—	2	1	no	—
178	D₈×C₃₁	—	8	1	no	—
179	C₂×C₈×C₃₁	—	2	1	no	—
180	C₂×C₈×C₃₁	—	2	1	no	—
181	C₂×C₈×C₃₁	—	2	1	no	—
182	C₂×C₈×C₃₁	—	2	1	no	—
183	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	—	4	1	no	—
190	C₄⋊C₄×C₃₁	—	4	1	no	—
191	C₄⋊C₄×C₃₁	—	4	1	no	—
192	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
193	C₂²⋊C₄×C₃₁	—	4	1	no	—
194	C₂²⋊C₄×C₃₁	—	4	1	no	—
195	C₂²⋊C₄×C₃₁	—	4	1	no	—
196	C₂²⋊C₄×C₃₁	—	4	1	no	—
197	C₂²⋊C₄×C₃₁	—	4	1	no	—
198	C₂²⋊C₄×C₃₁	—	4	1	no	—
199	C₂²⋊C₄×C₃₁	D₈	1	1	yes	characteristic
200	C₂²⋊C₄×C₃₁	D₈	1	1	yes	characteristic
201	C₄²×C₃₁	—	4	1	no	—
202	C₄²×C₃₁	—	4	1	no	—
203	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
204	C₁₆×C₃₁	—	2	1	no	—
205	C₁₆×C₃₁	—	2	1	no	—
206	C₁₆×C₃₁	—	2	1	no	—
207	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
208	#33837	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂³×C₃₁	—	4	1	no	—
210	C₂³×C₃₁	—	4	1	no	—
211	C₂³×C₃₁	—	4	1	no	—
212	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
213	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
214	Q₈×C₃₁	—	2	2	no	—
215	Q₈×C₃₁	—	2	2	no	—
216	Q₈×C₃₁	—	4	1	no	—
217	Q₈×C₃₁	—	4	1	no	—
218	Q₈×C₃₁	—	4	1	no	—
219	Q₈×C₃₁	—	4	1	no	—
220	Q₈×C₃₁	—	8	2	no	—
221	D₄×C₃₁	—	4	1	no	—
222	D₄×C₃₁	—	4	1	no	—
223	D₄×C₃₁	—	4	1	no	—
224	D₄×C₃₁	—	4	1	no	—
225	D₄×C₃₁	—	4	1	no	—
226	D₄×C₃₁	—	4	1	no	—
227	D₄×C₃₁	—	4	1	no	—
228	D₄×C₃₁	—	8	2	no	—
229	D₄×C₃₁	—	8	2	no	—
230	D₄×C₃₁	—	8	2	no	—
231	D₄×C₃₁	—	8	2	no	—
232	D₄×C₃₁	—	8	2	no	—
233	D₄×C₃₁	—	16	1	no	—
234	D₄×C₃₁	—	16	1	no	—
235	D₄×C₃₁	—	16	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	—	4	1	no	—
241	C₂×C₄×C₃₁	—	4	1	no	—
242	C₂×C₄×C₃₁	—	4	1	no	—
243	C₂×C₄×C₃₁	—	4	1	no	—
244	C₂×C₄×C₃₁	—	4	1	no	—
245	C₂×C₄×C₃₁	—	8	2	no	—
246	C₂×C₄×C₃₁	—	8	2	no	—
247	C₂×C₄×C₃₁	—	8	2	no	—
248	C₂×C₄×C₃₁	—	8	2	no	—
249	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
250	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
252	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
253	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
254	C₈×C₃₁	—	2	1	no	—
255	C₈×C₃₁	—	2	1	no	—
256	C₈×C₃₁	—	2	1	no	—
257	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
258	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
259	C₂×#510	C₂×C₃₁	1	1	yes	characteristic, complemented
260	C₂×#508	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#3287	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#3285	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#2250	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#2218	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#2217	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#2211	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#2207	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#2189	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#2186	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#2186	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#2185	C₂×C₃₁	1	1	yes	characteristic, complemented
272	#2176	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#2168	C₂×C₃₁	1	1	yes	characteristic, complemented
274	C₂²×C₃₁	—	2	1	no	—
275	C₂²×C₃₁	—	2	1	no	—
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	—	8	1	no	—
281	C₂²×C₃₁	—	8	1	no	—
282	C₂²×C₃₁	—	8	1	no	—
283	C₂²×C₃₁	—	8	1	no	—
284	C₂²×C₃₁	—	8	1	no	—
285	C₂²×C₃₁	—	8	1	no	—
286	C₂²×C₃₁	—	8	1	no	—
287	C₂²×C₃₁	—	8	1	no	—
288	C₂²×C₃₁	—	8	1	no	—
289	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
290	C₄×C₃₁	—	2	1	no	—
291	C₄×C₃₁	—	2	1	no	—
292	C₄×C₃₁	—	2	1	no	—
293	C₄×C₃₁	—	2	1	no	—
294	C₄×C₃₁	—	4	2	no	—
295	C₄×C₃₁	—	4	2	no	—
296	C₄×C₃₁	—	8	1	no	—
297	C₄×C₃₁	—	8	1	no	—
298	C₄×C₃₁	—	8	1	no	—
299	C₄×C₃₁	D₄²	1	1	yes	characteristic
300	C₄×C₃₁	#576	1	1	yes	characteristic
301	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
302	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
304	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
305	#550	—	2	1	no	—
306	#548	—	2	1	no	—
307	#510	—	2	4	no	—
308	#508	—	2	4	no	—
309	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
310	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
313	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
315	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
316	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
319	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
320	#495	C₂²×C₃₁	1	1	yes	characteristic
321	#493	C₂²×C₃₁	1	1	yes	characteristic
322	#467	C₂²×C₃₁	1	1	yes	characteristic
323	#466	C₂²×C₃₁	1	1	yes	characteristic
324	#466	C₂²×C₃₁	1	1	yes	characteristic
325	#466	C₂²×C₃₁	1	1	yes	characteristic
326	#464	—	2	1	no	—
327	#460	—	2	1	no	—
328	#452	—	2	1	no	—
329	#449	—	2	1	no	—
330	#449	—	2	1	no	—
331	#448	—	2	1	no	—
332	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
333	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
334	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
335	#367	C₂²×C₃₁	1	1	yes	characteristic
336	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
337	#358	C₂²×C₃₁	1	1	yes	characteristic
338	#358	C₂²×C₃₁	1	1	yes	characteristic
339	#358	C₂²×C₃₁	1	1	yes	characteristic
340	#357	C₂²×C₃₁	1	1	yes	characteristic
341	#357	C₂²×C₃₁	1	1	yes	characteristic
342	#357	C₂²×C₃₁	1	1	yes	characteristic
343	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
344	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
345	#346	C₂²×C₃₁	1	1	yes	characteristic
346	C₂×C₃₁	—	4	1	no	—
347	C₂×C₃₁	—	4	1	no	—
348	C₂×C₃₁	—	16	1	no	—
349	C₂×C₃₁	—	16	1	no	—
350	C₂×C₃₁	—	16	1	no	—
351	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
352	C₂×C₃₁	#3414	1	1	yes	characteristic
353	C₂×C₃₁	#3416	1	1	yes	characteristic
354	C₂×#41	—	2	1	no	—
355	C₂×#41	—	2	1	no	—
356	C₂×#41	—	2	1	no	—
357	C₂²×Q₈	—	2	1	no	—
358	#135	—	2	2	no	—
359	#135	—	2	2	no	—
360	#135	—	2	2	no	—
361	#135	—	4	1	no	—
362	#135	—	4	1	no	—
363	#135	—	4	1	no	—
364	C₂×Q₁₆	—	2	2	no	—
365	C₂×Q₁₆	—	2	1	no	—
366	C₂×Q₁₆	—	4	1	no	—
367	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
368	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
369	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
370	C₂×SD₁₆	—	2	1	no	—
371	C₂×SD₁₆	—	2	1	no	—
372	C₂×SD₁₆	—	2	1	no	—
373	C₂×SD₁₆	—	2	1	no	—
374	C₂×D₈	—	2	1	no	—
375	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
376	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
377	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
379	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
380	#127	—	2	1	no	—
381	#127	—	4	1	no	—
382	#124	—	2	1	no	—
383	#124	—	4	1	no	—
384	C₂²⋊Q₈	—	2	1	no	—
385	C₂²⋊Q₈	—	4	1	no	—
386	#121	—	2	1	no	—
387	#121	—	2	1	no	—
388	#121	—	2	1	no	—
389	#121	—	4	1	no	—
390	#121	—	4	1	no	—
391	#121	—	4	1	no	—
392	C₄×Q₈	—	2	1	no	—
393	C₄×Q₈	—	2	1	no	—
394	C₄×D₄	—	4	1	no	—
395	C₄×D₄	—	4	1	no	—
396	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
397	Q₃₂	—	2	2	no	—
398	Q₃₂	D₄×C₃₁	1	4	yes	complemented
399	SD₃₂	—	2	2	no	—
400	SD₃₂	—	2	2	no	—
401	SD₃₂	—	2	2	no	—
402	SD₃₂	—	2	2	no	—
403	SD₃₂	—	4	1	no	—
404	D₁₆	—	2	2	no	—
405	D₁₆	—	4	1	no	—
406	M₃₂	—	2	2	no	—
407	C₂×C₁₆	—	2	2	no	—
408	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
409	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
410	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
411	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
412	#107	C₂³×C₃₁	1	1	yes	characteristic
413	#107	C₂³×C₃₁	1	1	yes	characteristic
414	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
415	#103	—	2	1	no	—
416	#103	—	2	1	no	—
417	#103	—	2	1	no	—
418	#102	—	2	1	no	—
419	#102	—	2	1	no	—
420	#102	—	2	1	no	—
421	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
422	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
423	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
424	C₃₁	#33837	1	1	yes	characteristic, direct product factor, Sylow
425	#41	—	4	2	no	—
426	#41	—	4	2	no	—
427	#41	—	4	2	no	—
428	#41	—	8	1	no	—
429	#41	—	8	1	no	—
430	#41	—	8	1	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	4	2	no	—
435	C₂×Q₈	—	4	1	no	—
436	C₂×Q₈	—	8	1	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	4	1	no	—
441	C₂×D₄	—	4	1	no	—
442	C₂×D₄	—	4	1	no	—
443	C₂×D₄	—	4	1	no	—
444	C₂×D₄	—	4	1	no	—
445	C₂×D₄	—	4	1	no	—
446	C₂×D₄	—	4	1	no	—
447	C₂×D₄	—	4	1	no	—
448	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	—	4	1	no	—
450	C₂²×C₄	—	4	1	no	—
451	C₂²×C₄	—	4	1	no	—
452	C₂²×C₄	—	4	1	no	—
453	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
455	Q₁₆	—	2	1	no	—
456	Q₁₆	—	2	1	no	—
457	Q₁₆	—	2	1	no	—
458	Q₁₆	—	2	1	no	—
459	Q₁₆	—	4	2	no	—
460	Q₁₆	C₂×D₄×C₃₁	1	2	yes	—
461	Q₁₆	C₂×D₄×C₃₁	1	2	yes	—
462	SD₁₆	—	4	2	no	—
463	SD₁₆	—	4	2	no	—
464	SD₁₆	—	4	2	no	—
465	SD₁₆	—	8	1	no	—
466	D₈	—	2	1	no	—
467	D₈	—	2	1	no	—
468	D₈	—	2	1	no	—
469	D₈	—	2	1	no	—
470	D₈	—	2	1	no	—
471	D₈	—	2	1	no	—
472	D₈	—	8	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
480	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	4	1	no	—
484	C₄⋊C₄	—	4	1	no	—
485	C₄⋊C₄	—	4	1	no	—
486	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
487	C₂²⋊C₄	—	4	1	no	—
488	C₂²⋊C₄	—	4	1	no	—
489	C₂²⋊C₄	—	4	1	no	—
490	C₂²⋊C₄	—	4	1	no	—
491	C₂²⋊C₄	—	4	1	no	—
492	C₂²⋊C₄	—	4	1	no	—
493	C₂²⋊C₄	D₈×C₃₁	1	1	yes	characteristic
494	C₂²⋊C₄	D₈×C₃₁	1	1	yes	characteristic
495	C₄²	—	4	1	no	—
496	C₄²	—	4	1	no	—
497	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
498	C₁₆	—	2	1	no	—
499	C₁₆	—	2	1	no	—
500	C₁₆	—	2	1	no	—
501	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
502	C₂³	—	4	1	no	—
503	C₂³	—	4	1	no	—
504	C₂³	—	4	1	no	—
505	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
506	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
507	Q₈	—	2	2	no	—
508	Q₈	—	2	2	no	—
509	Q₈	—	4	1	no	—
510	Q₈	—	4	1	no	—
511	Q₈	—	4	1	no	—
512	Q₈	—	4	1	no	—
513	Q₈	—	8	2	no	—
514	D₄	—	4	1	no	—
515	D₄	—	4	1	no	—
516	D₄	—	4	1	no	—
517	D₄	—	4	1	no	—
518	D₄	—	4	1	no	—
519	D₄	—	4	1	no	—
520	D₄	—	4	1	no	—
521	D₄	—	8	2	no	—
522	D₄	—	8	2	no	—
523	D₄	—	8	2	no	—
524	D₄	—	8	2	no	—
525	D₄	—	8	2	no	—
526	D₄	—	16	1	no	—
527	D₄	—	16	1	no	—
528	D₄	—	16	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	8	2	no	—
539	C₂×C₄	—	8	2	no	—
540	C₂×C₄	—	8	2	no	—
541	C₂×C₄	—	8	2	no	—
542	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
543	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
544	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
545	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
546	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
547	C₈	—	2	1	no	—
548	C₈	—	2	1	no	—
549	C₈	—	2	1	no	—
550	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
551	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
552	C₂²	—	2	1	no	—
553	C₂²	—	2	1	no	—
554	C₂²	—	2	1	no	—
555	C₂²	—	2	1	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	2	1	no	—
558	C₂²	—	8	1	no	—
559	C₂²	—	8	1	no	—
560	C₂²	—	8	1	no	—
561	C₂²	—	8	1	no	—
562	C₂²	—	8	1	no	—
563	C₂²	—	8	1	no	—
564	C₂²	—	8	1	no	—
565	C₂²	—	8	1	no	—
566	C₂²	—	8	1	no	—
567	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
568	C₄	—	2	1	no	—
569	C₄	—	2	1	no	—
570	C₄	—	2	1	no	—
571	C₄	—	2	1	no	—
572	C₄	—	4	2	no	—
573	C₄	—	4	2	no	—
574	C₄	—	8	1	no	—
575	C₄	—	8	1	no	—
576	C₄	—	8	1	no	—
577	C₄	D₄²×C₃₁	1	1	yes	characteristic
578	C₄	C₃₁×#576	1	1	yes	characteristic
579	C₂	—	4	1	no	—
580	C₂	—	4	1	no	—
581	C₂	—	16	1	no	—
582	C₂	—	16	1	no	—
583	C₂	—	16	1	no	—
584	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
585	C₂	C₃₁×#3414	1	1	yes	characteristic
586	C₂	C₃₁×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#510	C₂	semidirect
2. C₂×C₃₁×#508	C₂	semidirect
3. C₃₁×#3287	C₂	semidirect
4. C₃₁×#3285	C₂	semidirect
5. C₃₁×#2250	C₂	semidirect
6. C₃₁×#2218	C₂	semidirect
7. C₃₁×#2217	C₂	semidirect
8. C₃₁×#2211	C₂	semidirect
9. C₃₁×#2207	C₂	semidirect
10. C₃₁×#2189	C₂	semidirect
11. C₃₁×#2186	C₂	semidirect
12. C₃₁×#2186	C₂	semidirect
13. C₃₁×#2185	C₂	semidirect
14. C₃₁×#2176	C₂	semidirect
15. C₃₁×#2168	C₂	semidirect
105. C₃₁×Q₃₂	D₄	semidirect
208. #33837	C₃₁	direct (N × H)
259. C₂×#510	C₂×C₃₁	semidirect
260. C₂×#508	C₂×C₃₁	semidirect
261. #3287	C₂×C₃₁	semidirect
262. #3285	C₂×C₃₁	semidirect
263. #2250	C₂×C₃₁	semidirect
264. #2218	C₂×C₃₁	semidirect
265. #2217	C₂×C₃₁	semidirect
266. #2211	C₂×C₃₁	semidirect
267. #2207	C₂×C₃₁	semidirect
268. #2189	C₂×C₃₁	semidirect
269. #2186	C₂×C₃₁	semidirect
270. #2186	C₂×C₃₁	semidirect
271. #2185	C₂×C₃₁	semidirect
272. #2176	C₂×C₃₁	semidirect
273. #2168	C₂×C₃₁	semidirect
398. Q₃₂	D₄×C₃₁	semidirect
424. C₃₁	#33837	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×#510	2
C₂×C₃₁×#508	2
C₃₁×#3287	2
C₃₁×#3285	2
C₃₁×#2250	2
C₃₁×#2218	2
C₃₁×#2217	2
C₃₁×#2211	2
C₃₁×#2207	2
C₃₁×#2189	2
C₃₁×#2186	2
C₃₁×#2185	2
C₃₁×#2176	2
C₃₁×#2168	2
#33837	31

Subgroups — C31×#33837 #21490596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33837 #21490596