Subgroups — C₃₁×#33836 #21490595

Order 7936.

664 conjugacy classes of subgroups.

The subgroup lattice has 568 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	C₂²×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3285	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2251	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2219	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×#2209	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2189	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2188	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2188	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	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	D₄×Q₈×C₃₁	—	2	1	no	—
21	C₃₁×#548	—	2	1	no	—
22	C₃₁×#510	—	2	4	no	—
23	C₂×C₃₁×Q₃₂	—	2	4	no	—
24	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×SD₃₂	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×#494	C₂²	1	1	yes	characteristic
37	C₃₁×#467	C₂²	1	1	yes	characteristic
38	C₃₁×#467	C₂²	1	1	yes	characteristic
39	C₃₁×#467	C₂²	1	1	yes	characteristic
40	C₃₁×#466	C₂²	1	1	yes	characteristic
41	C₃₁×#464	—	2	1	no	—
42	C₃₁×#462	—	2	1	no	—
43	C₃₁×#452	—	2	1	no	—
44	C₃₁×#451	—	2	1	no	—
45	C₃₁×#451	—	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×#358	C₂²	1	1	yes	characteristic
56	C₃₁×#358	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	C₂²×Q₈×C₃₁	—	2	1	no	—
63	C₂²×Q₈×C₃₁	—	2	1	no	—
64	C₂²×Q₈×C₃₁	—	2	1	no	—
65	C₃₁×#135	—	2	2	no	—
66	C₃₁×#135	—	4	1	no	—
67	C₂×Q₁₆×C₃₁	—	2	2	no	—
68	C₂×Q₁₆×C₃₁	—	2	2	no	—
69	C₂×Q₁₆×C₃₁	—	2	2	no	—
70	C₂×Q₁₆×C₃₁	—	2	1	no	—
71	C₂×Q₁₆×C₃₁	—	2	1	no	—
72	C₂×Q₁₆×C₃₁	—	2	1	no	—
73	C₂×Q₁₆×C₃₁	—	2	1	no	—
74	C₂×Q₁₆×C₃₁	—	4	1	no	—
75	C₂×Q₁₆×C₃₁	—	4	1	no	—
76	C₂×Q₁₆×C₃₁	—	4	1	no	—
77	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×SD₁₆×C₃₁	—	2	1	no	—
83	C₂×SD₁₆×C₃₁	—	2	1	no	—
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×C₄⋊Q₈	—	2	1	no	—
88	C₃₁×C₄⋊Q₈	—	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	—	2	1	no	—
93	C₃₁×C₂²⋊Q₈	—	2	1	no	—
94	C₃₁×C₂²⋊Q₈	—	4	1	no	—
95	C₃₁×C₂²⋊Q₈	—	4	1	no	—
96	C₃₁×C₂²⋊Q₈	—	4	1	no	—
97	C₃₁×#121	—	2	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	—	2	2	no	—
106	C₃₁×Q₃₂	—	2	2	no	—
107	C₃₁×Q₃₂	—	2	2	no	—
108	C₃₁×Q₃₂	—	4	1	no	—
109	C₃₁×Q₃₂	D₄	1	4	yes	complemented
110	C₃₁×SD₃₂	—	2	2	no	—
111	C₃₁×SD₃₂	—	2	2	no	—
112	C₃₁×SD₃₂	—	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×#103	—	2	1	no	—
126	C₃₁×#103	—	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	—	8	1	no	—
133	C₂×Q₈×C₃₁	—	2	1	no	—
134	C₂×Q₈×C₃₁	—	2	1	no	—
135	C₂×Q₈×C₃₁	—	2	1	no	—
136	C₂×Q₈×C₃₁	—	2	1	no	—
137	C₂×Q₈×C₃₁	—	2	1	no	—
138	C₂×Q₈×C₃₁	—	4	2	no	—
139	C₂×Q₈×C₃₁	—	4	2	no	—
140	C₂×Q₈×C₃₁	—	4	2	no	—
141	C₂×Q₈×C₃₁	—	4	1	no	—
142	C₂×Q₈×C₃₁	—	4	1	no	—
143	C₂×Q₈×C₃₁	—	4	1	no	—
144	C₂×Q₈×C₃₁	—	8	1	no	—
145	C₂×Q₈×C₃₁	—	8	1	no	—
146	C₂×Q₈×C₃₁	—	8	1	no	—
147	C₂×D₄×C₃₁	—	2	1	no	—
148	C₂×D₄×C₃₁	—	4	1	no	—
149	C₂×D₄×C₃₁	—	4	1	no	—
150	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	—	4	1	no	—
152	C₂²×C₄×C₃₁	—	4	1	no	—
153	C₂²×C₄×C₃₁	—	4	1	no	—
154	C₂²×C₄×C₃₁	—	4	1	no	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	Q₁₆×C₃₁	—	2	1	no	—
158	Q₁₆×C₃₁	—	2	1	no	—
159	Q₁₆×C₃₁	—	2	1	no	—
160	Q₁₆×C₃₁	—	2	1	no	—
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	—	4	2	no	—
167	Q₁₆×C₃₁	—	4	2	no	—
168	Q₁₆×C₃₁	—	8	1	no	—
169	Q₁₆×C₃₁	C₂×D₄	1	2	yes	—
170	Q₁₆×C₃₁	C₂×D₄	1	2	yes	—
171	SD₁₆×C₃₁	—	4	2	no	—
172	SD₁₆×C₃₁	—	8	1	no	—
173	D₈×C₃₁	—	2	1	no	—
174	D₈×C₃₁	—	2	1	no	—
175	C₂×C₈×C₃₁	—	2	1	no	—
176	C₂×C₈×C₃₁	—	2	1	no	—
177	C₂×C₈×C₃₁	—	2	1	no	—
178	C₂×C₈×C₃₁	—	2	1	no	—
179	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	1	no	—
185	C₄⋊C₄×C₃₁	—	4	1	no	—
186	C₄⋊C₄×C₃₁	—	4	1	no	—
187	C₄⋊C₄×C₃₁	—	4	1	no	—
188	C₄⋊C₄×C₃₁	—	4	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	#33836	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂³×C₃₁	—	4	1	no	—
210	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
211	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
212	Q₈×C₃₁	—	2	2	no	—
213	Q₈×C₃₁	—	2	2	no	—
214	Q₈×C₃₁	—	4	1	no	—
215	Q₈×C₃₁	—	4	1	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	—	4	1	no	—
221	Q₈×C₃₁	—	4	1	no	—
222	Q₈×C₃₁	—	8	2	no	—
223	Q₈×C₃₁	—	8	2	no	—
224	Q₈×C₃₁	—	8	2	no	—
225	D₄×C₃₁	—	2	2	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	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	2	1	no	—
232	C₂×C₄×C₃₁	—	2	1	no	—
233	C₂×C₄×C₃₁	—	2	1	no	—
234	C₂×C₄×C₃₁	—	4	1	no	—
235	C₂×C₄×C₃₁	—	4	1	no	—
236	C₂×C₄×C₃₁	—	4	1	no	—
237	C₂×C₄×C₃₁	—	4	1	no	—
238	C₂×C₄×C₃₁	—	4	1	no	—
239	C₂×C₄×C₃₁	—	4	1	no	—
240	C₂×C₄×C₃₁	—	4	1	no	—
241	C₂×C₄×C₃₁	—	8	2	no	—
242	C₂×C₄×C₃₁	—	8	2	no	—
243	C₂×C₄×C₃₁	—	8	2	no	—
244	C₂×C₄×C₃₁	—	8	2	no	—
245	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
247	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
248	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
249	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
250	C₈×C₃₁	—	2	1	no	—
251	C₈×C₃₁	—	2	1	no	—
252	C₈×C₃₁	—	2	1	no	—
253	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
254	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
255	C₂×#510	C₂×C₃₁	1	1	yes	characteristic, complemented
256	C₂²×Q₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
257	D₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#3285	C₂×C₃₁	1	1	yes	characteristic
259	#2251	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#2219	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#2217	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#2211	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#2209	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#2189	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#2188	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#2188	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#2185	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#2176	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#2168	C₂×C₃₁	1	1	yes	characteristic, complemented
270	C₂²×C₃₁	—	2	1	no	—
271	C₂²×C₃₁	—	2	1	no	—
272	C₂²×C₃₁	—	2	1	no	—
273	C₂²×C₃₁	—	2	1	no	—
274	C₂²×C₃₁	—	2	1	no	—
275	C₂²×C₃₁	—	8	1	no	—
276	C₂²×C₃₁	—	8	1	no	—
277	C₂²×C₃₁	—	8	1	no	—
278	C₂²×C₃₁	#510	1	2	yes	complemented
279	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
280	C₄×C₃₁	—	2	1	no	—
281	C₄×C₃₁	—	2	1	no	—
282	C₄×C₃₁	—	2	1	no	—
283	C₄×C₃₁	—	2	1	no	—
284	C₄×C₃₁	—	4	2	no	—
285	C₄×C₃₁	—	4	2	no	—
286	C₄×C₃₁	—	8	1	no	—
287	C₄×C₃₁	—	8	1	no	—
288	C₄×C₃₁	—	8	1	no	—
289	C₄×C₃₁	—	8	1	no	—
290	C₄×C₃₁	—	8	1	no	—
291	C₄×C₃₁	D₄²	1	1	yes	characteristic
292	C₄×C₃₁	#576	1	1	yes	characteristic
293	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
294	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
295	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
296	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
297	D₄×Q₈	—	2	1	no	—
298	#548	—	2	1	no	—
299	#510	—	2	4	no	—
300	C₂×Q₃₂	—	2	4	no	—
301	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
302	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
308	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
310	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
311	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
312	#495	C₂²×C₃₁	1	1	yes	characteristic
313	#494	C₂²×C₃₁	1	1	yes	characteristic
314	#467	C₂²×C₃₁	1	1	yes	characteristic
315	#467	C₂²×C₃₁	1	1	yes	characteristic
316	#467	C₂²×C₃₁	1	1	yes	characteristic
317	#466	C₂²×C₃₁	1	1	yes	characteristic
318	#464	—	2	1	no	—
319	#462	—	2	1	no	—
320	#452	—	2	1	no	—
321	#451	—	2	1	no	—
322	#451	—	2	1	no	—
323	#448	—	2	1	no	—
324	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
325	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
326	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
327	#367	C₂²×C₃₁	1	1	yes	characteristic
328	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
329	#358	C₂²×C₃₁	1	1	yes	characteristic
330	#358	C₂²×C₃₁	1	1	yes	characteristic
331	#358	C₂²×C₃₁	1	1	yes	characteristic
332	#358	C₂²×C₃₁	1	1	yes	characteristic
333	#358	C₂²×C₃₁	1	1	yes	characteristic
334	#357	C₂²×C₃₁	1	1	yes	characteristic
335	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
336	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
337	#346	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×C₃₁	—	2	2	no	—
339	C₂×C₃₁	—	4	1	no	—
340	C₂×C₃₁	—	16	1	no	—
341	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
342	C₂×C₃₁	C₂×#510	1	1	yes	characteristic
343	C₂×C₃₁	#3417	1	1	yes	characteristic
344	C₂×#41	—	2	1	no	—
345	C₂²×Q₈	—	2	1	no	—
346	C₂²×Q₈	—	2	1	no	—
347	C₂²×Q₈	—	2	1	no	—
348	#135	—	2	2	no	—
349	#135	—	4	1	no	—
350	C₂×Q₁₆	—	2	2	no	—
351	C₂×Q₁₆	—	2	2	no	—
352	C₂×Q₁₆	—	2	2	no	—
353	C₂×Q₁₆	—	2	1	no	—
354	C₂×Q₁₆	—	2	1	no	—
355	C₂×Q₁₆	—	2	1	no	—
356	C₂×Q₁₆	—	2	1	no	—
357	C₂×Q₁₆	—	4	1	no	—
358	C₂×Q₁₆	—	4	1	no	—
359	C₂×Q₁₆	—	4	1	no	—
360	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
361	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
362	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
363	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
364	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
365	C₂×SD₁₆	—	2	1	no	—
366	C₂×SD₁₆	—	2	1	no	—
367	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
368	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
369	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
370	C₄⋊Q₈	—	2	1	no	—
371	C₄⋊Q₈	—	4	1	no	—
372	#124	—	2	1	no	—
373	#124	—	4	1	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	—	2	1	no	—
377	C₂²⋊Q₈	—	4	1	no	—
378	C₂²⋊Q₈	—	4	1	no	—
379	C₂²⋊Q₈	—	4	1	no	—
380	#121	—	2	1	no	—
381	#121	—	4	1	no	—
382	C₄×Q₈	—	2	1	no	—
383	C₄×Q₈	—	2	1	no	—
384	C₄×D₄	—	4	1	no	—
385	C₄×D₄	—	4	1	no	—
386	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
387	Q₃₂	—	2	2	no	—
388	Q₃₂	—	2	2	no	—
389	Q₃₂	—	2	2	no	—
390	Q₃₂	—	2	2	no	—
391	Q₃₂	—	4	1	no	—
392	Q₃₂	D₄×C₃₁	1	4	yes	complemented
393	SD₃₂	—	2	2	no	—
394	SD₃₂	—	2	2	no	—
395	SD₃₂	—	4	1	no	—
396	M₃₂	—	2	2	no	—
397	C₂×C₁₆	—	2	2	no	—
398	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
399	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
400	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
401	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
402	#107	C₂³×C₃₁	1	1	yes	characteristic
403	#107	C₂³×C₃₁	1	1	yes	characteristic
404	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
405	#103	—	2	1	no	—
406	#103	—	2	1	no	—
407	#103	—	2	1	no	—
408	#103	—	2	1	no	—
409	#103	—	2	1	no	—
410	#102	—	2	1	no	—
411	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
412	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
413	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
414	C₃₁	#33836	1	1	yes	characteristic, direct product factor, Sylow
415	#41	—	4	2	no	—
416	#41	—	8	1	no	—
417	C₂×Q₈	—	2	1	no	—
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	2	1	no	—
421	C₂×Q₈	—	2	1	no	—
422	C₂×Q₈	—	4	2	no	—
423	C₂×Q₈	—	4	2	no	—
424	C₂×Q₈	—	4	2	no	—
425	C₂×Q₈	—	4	1	no	—
426	C₂×Q₈	—	4	1	no	—
427	C₂×Q₈	—	4	1	no	—
428	C₂×Q₈	—	8	1	no	—
429	C₂×Q₈	—	8	1	no	—
430	C₂×Q₈	—	8	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	4	1	no	—
433	C₂×D₄	—	4	1	no	—
434	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	—	4	1	no	—
436	C₂²×C₄	—	4	1	no	—
437	C₂²×C₄	—	4	1	no	—
438	C₂²×C₄	—	4	1	no	—
439	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
441	Q₁₆	—	2	1	no	—
442	Q₁₆	—	2	1	no	—
443	Q₁₆	—	2	1	no	—
444	Q₁₆	—	2	1	no	—
445	Q₁₆	—	2	1	no	—
446	Q₁₆	—	2	1	no	—
447	Q₁₆	—	2	1	no	—
448	Q₁₆	—	2	1	no	—
449	Q₁₆	—	4	2	no	—
450	Q₁₆	—	4	2	no	—
451	Q₁₆	—	4	2	no	—
452	Q₁₆	—	8	1	no	—
453	Q₁₆	C₂×D₄×C₃₁	1	2	yes	—
454	Q₁₆	C₂×D₄×C₃₁	1	2	yes	—
455	SD₁₆	—	4	2	no	—
456	SD₁₆	—	8	1	no	—
457	D₈	—	2	1	no	—
458	D₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
466	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	4	1	no	—
470	C₄⋊C₄	—	4	1	no	—
471	C₄⋊C₄	—	4	1	no	—
472	C₄⋊C₄	—	4	1	no	—
473	C₄⋊C₄	—	4	1	no	—
474	C₄⋊C₄	—	4	1	no	—
475	C₄⋊C₄	—	4	1	no	—
476	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
477	C₂²⋊C₄	—	4	1	no	—
478	C₂²⋊C₄	—	4	1	no	—
479	C₂²⋊C₄	—	4	1	no	—
480	C₂²⋊C₄	—	4	1	no	—
481	C₂²⋊C₄	—	4	1	no	—
482	C₂²⋊C₄	—	4	1	no	—
483	C₂²⋊C₄	D₈×C₃₁	1	1	yes	characteristic
484	C₂²⋊C₄	D₈×C₃₁	1	1	yes	characteristic
485	C₄²	—	4	1	no	—
486	C₄²	—	4	1	no	—
487	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₁₆	—	2	1	no	—
489	C₁₆	—	2	1	no	—
490	C₁₆	—	2	1	no	—
491	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
492	C₂³	—	4	1	no	—
493	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
494	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
495	Q₈	—	2	2	no	—
496	Q₈	—	2	2	no	—
497	Q₈	—	4	1	no	—
498	Q₈	—	4	1	no	—
499	Q₈	—	4	1	no	—
500	Q₈	—	4	1	no	—
501	Q₈	—	4	1	no	—
502	Q₈	—	4	1	no	—
503	Q₈	—	4	1	no	—
504	Q₈	—	4	1	no	—
505	Q₈	—	8	2	no	—
506	Q₈	—	8	2	no	—
507	Q₈	—	8	2	no	—
508	D₄	—	2	2	no	—
509	D₄	—	4	1	no	—
510	D₄	—	4	1	no	—
511	D₄	—	8	2	no	—
512	D₄	—	8	2	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	8	2	no	—
525	C₂×C₄	—	8	2	no	—
526	C₂×C₄	—	8	2	no	—
527	C₂×C₄	—	8	2	no	—
528	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
529	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
530	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
531	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
532	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
533	C₈	—	2	1	no	—
534	C₈	—	2	1	no	—
535	C₈	—	2	1	no	—
536	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
537	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
538	C₂²	—	2	1	no	—
539	C₂²	—	2	1	no	—
540	C₂²	—	2	1	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	8	1	no	—
544	C₂²	—	8	1	no	—
545	C₂²	—	8	1	no	—
546	C₂²	C₃₁×#510	1	2	yes	complemented
547	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
548	C₄	—	2	1	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	4	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	8	1	no	—
555	C₄	—	8	1	no	—
556	C₄	—	8	1	no	—
557	C₄	—	8	1	no	—
558	C₄	—	8	1	no	—
559	C₄	D₄²×C₃₁	1	1	yes	characteristic
560	C₄	C₃₁×#576	1	1	yes	characteristic
561	C₂	—	2	2	no	—
562	C₂	—	4	1	no	—
563	C₂	—	16	1	no	—
564	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
565	C₂	C₂×C₃₁×#510	1	1	yes	characteristic
566	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3285	C₂
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. C₂²×C₃₁×Q₃₂	C₂	semidirect
3. D₄×Q₁₆×C₃₁	C₂	semidirect
5. C₃₁×#2251	C₂	semidirect
6. C₃₁×#2219	C₂	semidirect
7. C₃₁×#2217	C₂	semidirect
8. C₃₁×#2211	C₂	semidirect
9. C₃₁×#2209	C₂	semidirect
10. C₃₁×#2189	C₂	semidirect
11. C₃₁×#2188	C₂	semidirect
12. C₃₁×#2188	C₂	semidirect
13. C₃₁×#2185	C₂	semidirect
14. C₃₁×#2176	C₂	semidirect
15. C₃₁×#2168	C₂	semidirect
109. C₃₁×Q₃₂	D₄	semidirect
208. #33836	C₃₁	direct (N × H)
255. C₂×#510	C₂×C₃₁	semidirect
256. C₂²×Q₃₂	C₂×C₃₁	semidirect
257. D₄×Q₁₆	C₂×C₃₁	semidirect
259. #2251	C₂×C₃₁	semidirect
260. #2219	C₂×C₃₁	semidirect
261. #2217	C₂×C₃₁	semidirect
262. #2211	C₂×C₃₁	semidirect
263. #2209	C₂×C₃₁	semidirect
264. #2189	C₂×C₃₁	semidirect
265. #2188	C₂×C₃₁	semidirect
266. #2188	C₂×C₃₁	semidirect
267. #2185	C₂×C₃₁	semidirect
268. #2176	C₂×C₃₁	semidirect
269. #2168	C₂×C₃₁	semidirect
278. C₂²×C₃₁	#510	semidirect
392. Q₃₂	D₄×C₃₁	semidirect
414. C₃₁	#33836	direct (N × H)
546. C₂²	C₃₁×#510	semidirect

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
C₂²×C₃₁×Q₃₂	2
D₄×Q₁₆×C₃₁	2
C₃₁×#3285	2
C₃₁×#2251	2
C₃₁×#2219	2
C₃₁×#2217	2
C₃₁×#2211	2
C₃₁×#2209	2
C₃₁×#2189	2
C₃₁×#2188	2
C₃₁×#2185	2
C₃₁×#2176	2
C₃₁×#2168	2
#33836	31

Subgroups — C31×#33836 #21490595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33836 #21490595