Subgroups — C₃₁×#17514 #21474273

Order 7936.

926 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1497	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1491	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1485	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1481	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1479	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈²×C₃₁	C₄	1	1	yes	characteristic
13	C₃₁×#557	—	2	1	no	—
14	C₃₁×#557	C₄	1	1	yes	characteristic
15	C₃₁×#554	—	2	2	no	—
16	C₃₁×#554	—	2	1	no	—
17	C₃₁×#554	C₄	1	2	yes	—
18	C₃₁×#554	C₄	1	1	yes	characteristic
19	D₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
20	D₄×Q₈×C₃₁	C₄	1	2	yes	—
21	C₃₁×#533	—	2	1	no	—
22	C₃₁×#533	—	2	2	no	—
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₃₁×#533	C₂²	1	2	yes	—
25	C₄×#41×C₃₁	—	2	1	no	—
26	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#446	—	2	1	no	—
30	C₃₁×#436	—	2	1	no	—
31	C₃₁×#435	—	2	2	no	—
32	C₃₁×#433	—	2	2	no	—
33	C₃₁×#432	C₂²	1	2	yes	—
34	C₃₁×#431	C₂²	1	2	yes	—
35	C₃₁×#431	C₂²	1	2	yes	—
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#423	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₃₁×#408	—	2	1	no	—
40	C₃₁×#405	—	2	1	no	—
41	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
44	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
45	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
46	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
51	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
52	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
54	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	—	2	2	no	—
63	C₃₁×C₄⋊Q₈	—	2	1	no	—
64	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
66	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
67	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
68	C₃₁×#125	—	2	2	no	—
69	C₃₁×#125	—	2	1	no	—
70	C₃₁×#125	—	2	2	no	—
71	C₃₁×#125	—	2	1	no	—
72	C₃₁×#125	—	2	1	no	—
73	C₃₁×#125	—	2	2	no	—
74	C₃₁×C₂²⋊Q₈	—	2	1	no	—
75	C₃₁×C₂²⋊Q₈	—	2	2	no	—
76	C₃₁×C₂²⋊Q₈	—	2	2	no	—
77	C₃₁×C₂²⋊Q₈	—	2	2	no	—
78	C₃₁×C₂²⋊Q₈	—	2	2	no	—
79	C₃₁×C₂²⋊Q₈	—	2	2	no	—
80	C₃₁×C₂²⋊Q₈	—	2	2	no	—
81	C₃₁×C₂²⋊Q₈	—	2	2	no	—
82	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
83	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
84	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
85	C₄×Q₈×C₃₁	—	2	1	no	—
86	C₄×Q₈×C₃₁	—	2	1	no	—
87	C₄×Q₈×C₃₁	—	2	2	no	—
88	C₄×Q₈×C₃₁	—	2	1	no	—
89	C₄×Q₈×C₃₁	—	2	1	no	—
90	C₄×Q₈×C₃₁	—	2	2	no	—
91	C₄×Q₈×C₃₁	—	2	2	no	—
92	C₄×Q₈×C₃₁	—	2	1	no	—
93	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
95	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
96	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
97	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
98	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
99	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
102	C₄×D₄×C₃₁	—	2	2	no	—
103	C₄×D₄×C₃₁	—	2	1	no	—
104	C₄×D₄×C₃₁	—	2	2	no	—
105	C₄×D₄×C₃₁	—	2	1	no	—
106	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
107	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
108	C₄×D₄×C₃₁	D₄	1	2	yes	—
109	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
110	C₃₁×#117	—	2	2	no	—
111	C₃₁×#117	—	2	1	no	—
112	C₃₁×#117	D₄	1	1	yes	characteristic
113	C₃₁×#117	D₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
115	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
118	C₂×C₄²×C₃₁	—	2	2	no	—
119	C₂×C₄²×C₃₁	—	2	1	no	—
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
122	C₃₁×C₄⋊C₈	—	2	1	no	—
123	C₃₁×C₄⋊C₈	—	2	1	no	—
124	C₃₁×C₄⋊C₈	—	2	2	no	—
125	C₃₁×C₄⋊C₈	—	2	1	no	—
126	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
127	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
128	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
129	C₃₁×C₂²⋊C₈	—	2	2	no	—
130	C₃₁×C₂²⋊C₈	—	2	2	no	—
131	C₃₁×C₂²⋊C₈	—	2	1	no	—
132	C₃₁×C₂²⋊C₈	—	2	2	no	—
133	C₃₁×#97	—	2	2	no	—
134	C₃₁×#97	—	2	2	no	—
135	C₄×C₈×C₃₁	—	2	1	no	—
136	C₄×C₈×C₃₁	—	2	2	no	—
137	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₃₁	C₂³	1	2	yes	—
140	#41×C₃₁	—	2	2	no	—
141	#41×C₃₁	—	2	2	no	—
142	C₂×Q₈×C₃₁	—	2	1	no	—
143	C₂×Q₈×C₃₁	—	2	2	no	—
144	C₂×Q₈×C₃₁	—	2	2	no	—
145	C₂×Q₈×C₃₁	—	2	1	no	—
146	C₂×Q₈×C₃₁	—	2	2	no	—
147	C₂×Q₈×C₃₁	—	2	4	no	—
148	C₂×Q₈×C₃₁	—	4	1	no	—
149	C₂×Q₈×C₃₁	—	4	2	no	—
150	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
151	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
153	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
154	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
155	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	—	2	2	no	—
157	C₂²×C₄×C₃₁	—	2	1	no	—
158	C₂²×C₄×C₃₁	—	2	2	no	—
159	C₂²×C₄×C₃₁	—	2	1	no	—
160	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	M₁₆×C₃₁	—	4	4	no	—
167	M₁₆×C₃₁	—	4	2	no	—
168	C₂×C₈×C₃₁	—	2	1	no	—
169	C₂×C₈×C₃₁	—	2	1	no	—
170	C₂×C₈×C₃₁	—	2	2	no	—
171	C₂×C₈×C₃₁	—	2	1	no	—
172	C₂×C₈×C₃₁	—	2	2	no	—
173	C₂×C₈×C₃₁	—	2	1	no	—
174	C₂×C₈×C₃₁	—	4	2	no	—
175	C₂×C₈×C₃₁	—	4	2	no	—
176	C₄⋊C₄×C₃₁	—	2	2	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	2	no	—
182	C₄⋊C₄×C₃₁	—	2	2	no	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	—	2	1	no	—
186	C₄⋊C₄×C₃₁	—	2	2	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	—	2	1	no	—
192	C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₄⋊C₄×C₃₁	—	2	2	no	—
194	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
195	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
197	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
200	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
201	C₂²⋊C₄×C₃₁	—	2	2	no	—
202	C₂²⋊C₄×C₃₁	—	2	2	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
205	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
206	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
207	C₄²×C₃₁	—	2	2	no	—
208	C₄²×C₃₁	—	2	1	no	—
209	C₄²×C₃₁	—	2	2	no	—
210	C₄²×C₃₁	—	2	1	no	—
211	C₄²×C₃₁	—	2	1	no	—
212	C₄²×C₃₁	—	2	2	no	—
213	C₄²×C₃₁	—	2	1	no	—
214	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
215	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
216	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₃₁	C₂×D₄	1	2	yes	—
218	C₄²×C₃₁	C₂×D₄	1	2	yes	—
219	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
221	#17514	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
223	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	Q₈×C₃₁	—	2	1	no	—
225	Q₈×C₃₁	—	2	2	no	—
226	Q₈×C₃₁	—	2	1	no	—
227	Q₈×C₃₁	—	2	2	no	—
228	Q₈×C₃₁	—	2	1	no	—
229	Q₈×C₃₁	—	2	2	no	—
230	Q₈×C₃₁	—	4	2	no	—
231	Q₈×C₃₁	—	4	4	no	—
232	Q₈×C₃₁	—	8	1	no	—
233	Q₈×C₃₁	—	8	2	no	—
234	Q₈×C₃₁	#131	1	2	yes	—
235	D₄×C₃₁	—	2	4	no	—
236	D₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	2	1	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	1	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	2	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	1	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	1	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	1	no	—
249	C₂×C₄×C₃₁	—	4	1	no	—
250	C₂×C₄×C₃₁	—	4	2	no	—
251	C₂×C₄×C₃₁	—	4	2	no	—
252	C₂×C₄×C₃₁	—	4	4	no	—
253	C₂×C₄×C₃₁	—	8	1	no	—
254	C₂×C₄×C₃₁	—	8	2	no	—
255	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
256	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
257	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
260	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₃₁	#137	1	2	yes	—
262	C₂×C₄×C₃₁	#137	1	2	yes	—
263	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
264	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
266	C₈×C₃₁	—	2	2	no	—
267	C₈×C₃₁	—	4	1	no	—
268	C₈×C₃₁	—	4	2	no	—
269	C₈×C₃₁	—	4	1	no	—
270	C₈×C₃₁	—	4	1	no	—
271	C₈×C₃₁	—	4	2	no	—
272	Q₈×#41	C₂×C₃₁	1	1	yes	characteristic
273	#2965	C₂×C₃₁	1	1	yes	characteristic
274	#2925	C₂×C₃₁	1	1	yes	characteristic
275	#1497	C₂×C₃₁	1	2	yes	complemented
276	#1493	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#1491	C₂×C₃₁	1	2	yes	complemented
278	#1487	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#1485	C₂×C₃₁	1	2	yes	complemented
280	#1481	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#1479	C₂×C₃₁	1	2	yes	complemented
282	C₂×Q₈⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
283	C₂²×C₃₁	—	2	1	no	—
284	C₂²×C₃₁	—	2	1	no	—
285	C₂²×C₃₁	—	2	2	no	—
286	C₂²×C₃₁	—	2	2	no	—
287	C₂²×C₃₁	—	2	2	no	—
288	C₂²×C₃₁	#419	1	2	yes	—
289	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
290	C₄×C₃₁	—	2	1	no	—
291	C₄×C₃₁	—	2	2	no	—
292	C₄×C₃₁	—	2	2	no	—
293	C₄×C₃₁	—	2	2	no	—
294	C₄×C₃₁	—	2	1	no	—
295	C₄×C₃₁	—	2	1	no	—
296	C₄×C₃₁	—	2	2	no	—
297	C₄×C₃₁	—	4	1	no	—
298	C₄×C₃₁	—	4	2	no	—
299	C₄×C₃₁	—	4	1	no	—
300	C₄×C₃₁	—	4	1	no	—
301	C₄×C₃₁	—	4	2	no	—
302	C₄×C₃₁	#419	1	2	yes	—
303	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
304	C₄×C₃₁	#568	1	1	yes	characteristic
305	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
306	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
307	Q₈²	C₄×C₃₁	1	1	yes	characteristic
308	#557	—	2	1	no	—
309	#557	C₄×C₃₁	1	1	yes	characteristic
310	#554	—	2	2	no	—
311	#554	—	2	1	no	—
312	#554	C₄×C₃₁	1	2	yes	—
313	#554	C₄×C₃₁	1	1	yes	characteristic
314	D₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
315	D₄×Q₈	C₄×C₃₁	1	2	yes	—
316	#533	—	2	1	no	—
317	#533	—	2	2	no	—
318	#533	C₂²×C₃₁	1	1	yes	characteristic
319	#533	C₂²×C₃₁	1	2	yes	—
320	C₄×#41	—	2	1	no	—
321	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
322	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
324	#446	—	2	1	no	—
325	#436	—	2	1	no	—
326	#435	—	2	2	no	—
327	#433	—	2	2	no	—
328	#432	C₂²×C₃₁	1	2	yes	—
329	#431	C₂²×C₃₁	1	2	yes	—
330	#431	C₂²×C₃₁	1	2	yes	—
331	#431	C₂²×C₃₁	1	1	yes	characteristic
332	#423	C₂²×C₃₁	1	1	yes	characteristic
333	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
334	#408	—	2	1	no	—
335	#405	—	2	1	no	—
336	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
337	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
338	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
339	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
340	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
341	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
342	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
343	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
344	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
345	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
346	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
347	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
348	C₂×C₃₁	—	2	2	no	—
349	C₂×C₃₁	—	4	2	no	—
350	C₂×C₃₁	#2880	1	1	yes	characteristic
351	C₂×C₃₁	C₂×#419	1	1	yes	characteristic
352	C₂×C₃₁	#2901	1	1	yes	characteristic
353	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
354	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
355	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
356	C₂×M₁₆	—	2	2	no	—
357	C₂×M₁₆	—	2	1	no	—
358	C₂²×C₈	—	2	1	no	—
359	C₂²×C₈	—	2	1	no	—
360	C₄⋊Q₈	—	2	2	no	—
361	C₄⋊Q₈	—	2	1	no	—
362	C₄⋊Q₈	—	2	2	no	—
363	C₄⋊Q₈	—	2	1	no	—
364	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
365	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
366	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
367	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
368	#125	—	2	2	no	—
369	#125	—	2	1	no	—
370	#125	—	2	2	no	—
371	#125	—	2	1	no	—
372	#125	—	2	1	no	—
373	#125	—	2	2	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	C₂²⋊Q₈	—	2	2	no	—
376	C₂²⋊Q₈	—	2	2	no	—
377	C₂²⋊Q₈	—	2	2	no	—
378	C₂²⋊Q₈	—	2	2	no	—
379	C₂²⋊Q₈	—	2	2	no	—
380	C₂²⋊Q₈	—	2	2	no	—
381	C₂²⋊Q₈	—	2	2	no	—
382	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
383	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
384	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
385	C₄×Q₈	—	2	1	no	—
386	C₄×Q₈	—	2	1	no	—
387	C₄×Q₈	—	2	2	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	—	2	2	no	—
392	C₄×Q₈	—	2	1	no	—
393	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
394	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
395	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
396	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
397	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
398	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
399	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
400	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
401	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	—	2	2	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
407	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
408	C₄×D₄	D₄×C₃₁	1	2	yes	—
409	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
410	#117	—	2	2	no	—
411	#117	—	2	1	no	—
412	#117	D₄×C₃₁	1	1	yes	characteristic
413	#117	D₄×C₃₁	1	2	yes	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	2	no	—
416	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
418	C₂×C₄²	—	2	2	no	—
419	C₂×C₄²	—	2	1	no	—
420	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
421	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
422	C₄⋊C₈	—	2	1	no	—
423	C₄⋊C₈	—	2	1	no	—
424	C₄⋊C₈	—	2	2	no	—
425	C₄⋊C₈	—	2	1	no	—
426	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
427	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
428	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
429	C₂²⋊C₈	—	2	2	no	—
430	C₂²⋊C₈	—	2	2	no	—
431	C₂²⋊C₈	—	2	1	no	—
432	C₂²⋊C₈	—	2	2	no	—
433	#97	—	2	2	no	—
434	#97	—	2	2	no	—
435	C₄×C₈	—	2	1	no	—
436	C₄×C₈	—	2	2	no	—
437	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
438	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
439	C₄×C₈	C₂³×C₃₁	1	2	yes	—
440	C₃₁	#17514	1	1	yes	characteristic, direct product factor, Sylow
441	#41	—	2	2	no	—
442	#41	—	2	2	no	—
443	C₂×Q₈	—	2	2	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	—	4	1	no	—
450	C₂×Q₈	—	4	2	no	—
451	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
452	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
453	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
454	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
455	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
456	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
462	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
464	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
466	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
467	M₁₆	—	4	4	no	—
468	M₁₆	—	4	2	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	2	2	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	—	4	2	no	—
476	C₂×C₈	—	4	2	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	2	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
496	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
497	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
498	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
499	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
500	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
501	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	2	no	—
505	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
506	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
507	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
508	C₄²	—	2	2	no	—
509	C₄²	—	2	1	no	—
510	C₄²	—	2	2	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	1	no	—
515	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
516	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
517	C₄²	C₂×D₄×C₃₁	1	2	yes	—
518	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
519	C₄²	C₂×D₄×C₃₁	1	2	yes	—
520	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
521	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
522	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
523	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
524	Q₈	—	2	2	no	—
525	Q₈	—	2	1	no	—
526	Q₈	—	2	1	no	—
527	Q₈	—	2	2	no	—
528	Q₈	—	2	1	no	—
529	Q₈	—	2	2	no	—
530	Q₈	—	4	2	no	—
531	Q₈	—	4	4	no	—
532	Q₈	—	8	1	no	—
533	Q₈	—	8	2	no	—
534	Q₈	C₃₁×#131	1	2	yes	—
535	D₄	—	2	4	no	—
536	D₄	—	2	2	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	2	no	—
552	C₂×C₄	—	4	4	no	—
553	C₂×C₄	—	8	1	no	—
554	C₂×C₄	—	8	2	no	—
555	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
556	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
557	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
558	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
559	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
560	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
561	C₂×C₄	C₃₁×#137	1	2	yes	—
562	C₂×C₄	C₃₁×#137	1	2	yes	—
563	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
564	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
565	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
566	C₈	—	2	2	no	—
567	C₈	—	4	1	no	—
568	C₈	—	4	2	no	—
569	C₈	—	4	1	no	—
570	C₈	—	4	1	no	—
571	C₈	—	4	2	no	—
572	C₂²	—	2	1	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	C₃₁×#419	1	2	yes	—
578	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
579	C₄	—	2	1	no	—
580	C₄	—	2	2	no	—
581	C₄	—	2	2	no	—
582	C₄	—	2	2	no	—
583	C₄	—	2	1	no	—
584	C₄	—	2	1	no	—
585	C₄	—	2	2	no	—
586	C₄	—	4	1	no	—
587	C₄	—	4	2	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	1	no	—
590	C₄	—	4	2	no	—
591	C₄	C₃₁×#419	1	2	yes	—
592	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
593	C₄	C₃₁×#568	1	1	yes	characteristic
594	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
595	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
596	C₂	—	2	2	no	—
597	C₂	—	4	2	no	—
598	C₂	C₃₁×#2880	1	1	yes	characteristic
599	C₂	C₂×C₃₁×#419	1	1	yes	characteristic
600	C₂	C₃₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₂×#41×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1497	C₂	semidirect
5. C₃₁×#1493	C₂	semidirect
6. C₃₁×#1491	C₂	semidirect
7. C₃₁×#1487	C₂	semidirect
8. C₃₁×#1485	C₂	semidirect
9. C₃₁×#1481	C₂	semidirect
10. C₃₁×#1479	C₂	semidirect
11. C₂×C₃₁×Q₈⋊C₈	C₂	semidirect
221. #17514	C₃₁	direct (N × H)
275. #1497	C₂×C₃₁	semidirect
276. #1493	C₂×C₃₁	semidirect
277. #1491	C₂×C₃₁	semidirect
278. #1487	C₂×C₃₁	semidirect
279. #1485	C₂×C₃₁	semidirect
280. #1481	C₂×C₃₁	semidirect
281. #1479	C₂×C₃₁	semidirect
282. C₂×Q₈⋊C₈	C₂×C₃₁	semidirect
440. C₃₁	#17514	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×#41×C₃₁	2
C₃₁×#2965	2
C₃₁×#2925	2
C₃₁×#1497	2
C₃₁×#1493	2
C₃₁×#1491	2
C₃₁×#1487	2
C₃₁×#1485	2
C₃₁×#1481	2
C₃₁×#1479	2
C₂×C₃₁×Q₈⋊C₈	2
#17514	31

Subgroups — C31×#17514 #21474273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17514 #21474273