Subgroups — C₃₁×#17541 #21474300

Order 7936.

1150 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3444	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#376	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1506	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1505	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1504	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1500	C₂	1	4	yes	maximal, complemented
8	C₃₁×#533	C₄	1	1	yes	characteristic, complemented
9	C₃₁×#533	C₄	1	1	yes	characteristic, complemented
10	C₃₁×#533	C₂²	1	1	yes	characteristic
11	C₃₁×#532	C₄	1	1	yes	characteristic, complemented
12	C₃₁×#532	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₃₁×#529	—	2	1	no	—
15	C₃₁×#529	—	2	1	no	—
16	C₃₁×#529	—	2	1	no	—
17	C₃₁×#529	—	2	1	no	—
18	C₃₁×#529	C₄	1	2	yes	complemented
19	C₃₁×#529	C₂²	1	2	yes	—
20	C₃₁×#525	—	2	1	no	—
21	C₃₁×#525	—	2	1	no	—
22	C₃₁×#525	C₂²	1	2	yes	—
23	C₄×#41×C₃₁	—	2	1	no	—
24	C₄×#41×C₃₁	—	2	1	no	—
25	C₄×#41×C₃₁	C₄	1	2	yes	complemented
26	C₂²×C₄²×C₃₁	—	2	1	no	—
27	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#432	—	2	1	no	—
29	C₃₁×#432	C₂²	1	2	yes	—
30	C₃₁×#431	—	2	1	no	—
31	C₃₁×#431	C₂²	1	2	yes	—
32	C₃₁×#423	—	2	1	no	—
33	C₃₁×#407	C₂²	1	2	yes	—
34	C₄×M₁₆×C₃₁	—	2	1	no	—
35	C₃₁×#376	—	2	1	no	—
36	C₃₁×#376	—	2	1	no	—
37	C₃₁×#376	C₂²	1	2	yes	—
38	C₃₁×#376	C₂²	1	2	yes	—
39	C₂×C₃₁×#95	—	2	1	no	—
40	C₂×C₃₁×#95	C₂²	1	2	yes	—
41	C₃₁×#328	C₂²	1	8	yes	—
42	C₃₁×#327	C₂²	1	8	yes	—
43	C₂×#41×C₃₁	—	2	1	no	—
44	C₂×#41×C₃₁	—	2	1	no	—
45	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
46	C₂³×C₄×C₃₁	—	2	2	no	—
47	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
52	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
53	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
54	C₃₁×#127	C₂×C₄	1	1	yes	characteristic, complemented
55	C₃₁×#126	—	2	2	no	—
56	C₃₁×#126	—	2	2	no	—
57	C₃₁×#125	—	2	1	no	—
58	C₃₁×#125	—	2	1	no	—
59	C₃₁×#125	C₂×C₄	1	2	yes	complemented
60	C₃₁×#124	—	2	1	no	—
61	C₃₁×#124	—	2	1	no	—
62	C₃₁×#124	C₂×C₄	1	2	yes	complemented
63	C₃₁×#123	—	2	2	no	—
64	C₃₁×#123	—	2	2	no	—
65	C₃₁×#123	—	2	2	no	—
66	C₃₁×C₂²⋊Q₈	—	2	1	no	—
67	C₃₁×C₂²⋊Q₈	—	2	1	no	—
68	C₃₁×C₂²⋊Q₈	—	2	1	no	—
69	C₃₁×C₂²⋊Q₈	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
71	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
72	C₃₁×#121	—	2	1	no	—
73	C₃₁×#121	—	2	1	no	—
74	C₃₁×#121	—	2	1	no	—
75	C₃₁×#121	—	2	1	no	—
76	C₃₁×#121	C₂×C₄	1	2	yes	—
77	C₃₁×#121	C₂³	1	2	yes	—
78	C₃₁×#120	—	2	2	no	—
79	C₄×Q₈×C₃₁	—	2	1	no	—
80	C₄×Q₈×C₃₁	—	2	1	no	—
81	C₄×Q₈×C₃₁	—	2	1	no	—
82	C₄×Q₈×C₃₁	—	2	1	no	—
83	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	complemented
84	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	complemented
85	C₄×D₄×C₃₁	—	2	2	no	—
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	—	2	2	no	—
88	C₄×D₄×C₃₁	—	2	2	no	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	—	2	1	no	—
92	C₄×D₄×C₃₁	—	2	1	no	—
93	C₄×D₄×C₃₁	—	2	1	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	complemented
96	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	complemented
97	C₃₁×#117	—	2	1	no	—
98	C₃₁×#117	—	2	1	no	—
99	C₃₁×#117	—	2	2	no	—
100	C₃₁×#117	—	2	2	no	—
101	C₃₁×#117	—	2	2	no	—
102	C₃₁×#117	C₂×C₄	1	2	yes	—
103	C₂×C₄²×C₃₁	—	2	8	no	—
104	C₂×C₄²×C₃₁	—	2	2	no	—
105	C₂×C₄²×C₃₁	—	2	2	no	—
106	C₂×C₄²×C₃₁	—	2	2	no	—
107	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
108	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
109	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
112	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
119	C₃₁×C₄⋊C₈	—	2	2	no	—
120	C₃₁×C₄⋊C₈	—	2	2	no	—
121	C₃₁×C₂²⋊C₈	—	2	2	no	—
122	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
123	C₃₁×#97	—	2	2	no	—
124	C₄×C₈×C₃₁	—	2	2	no	—
125	C₃₁×#95	—	2	2	no	—
126	C₃₁×#95	—	2	2	no	—
127	C₃₁×#95	—	2	2	no	—
128	C₃₁×#95	C₂³	1	4	yes	—
129	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
130	#41×C₃₁	—	4	4	no	—
131	#41×C₃₁	—	4	4	no	—
132	#41×C₃₁	—	4	4	no	—
133	C₂×Q₈×C₃₁	—	2	1	no	—
134	C₂×Q₈×C₃₁	—	2	1	no	—
135	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
136	C₂×D₄×C₃₁	—	2	2	no	—
137	C₂×D₄×C₃₁	—	2	2	no	—
138	C₂×D₄×C₃₁	—	2	2	no	—
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₃₁	—	2	4	no	—
143	C₂²×C₄×C₃₁	—	2	8	no	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	2	no	—
147	C₂²×C₄×C₃₁	—	2	2	no	—
148	C₂²×C₄×C₃₁	—	4	2	no	—
149	C₂²×C₄×C₃₁	—	4	2	no	—
150	C₂²×C₄×C₃₁	—	4	4	no	—
151	C₂²×C₄×C₃₁	—	4	4	no	—
152	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
170	M₁₆×C₃₁	—	4	4	no	—
171	C₂×C₈×C₃₁	—	2	4	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	2	no	—
177	C₄⋊C₄×C₃₁	—	2	2	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	2	2	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
181	C₄⋊C₄×C₃₁	—	2	2	no	—
182	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
183	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	2	2	no	—
186	C₂²⋊C₄×C₃₁	—	2	2	no	—
187	C₂²⋊C₄×C₃₁	—	2	2	no	—
188	C₂²⋊C₄×C₃₁	—	2	2	no	—
189	C₂²⋊C₄×C₃₁	—	2	2	no	—
190	C₂²⋊C₄×C₃₁	—	2	2	no	—
191	C₂²⋊C₄×C₃₁	—	2	2	no	—
192	C₂²⋊C₄×C₃₁	—	2	2	no	—
193	C₄²×C₃₁	—	2	8	no	—
194	C₄²×C₃₁	—	2	8	no	—
195	C₄²×C₃₁	—	2	2	no	—
196	C₄²×C₃₁	—	2	2	no	—
197	C₄²×C₃₁	—	2	2	no	—
198	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
199	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
200	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
201	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
202	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
204	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
205	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
206	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
207	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
208	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
209	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
210	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
214	#17541	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂³×C₃₁	—	2	2	no	—
216	C₂³×C₃₁	—	2	2	no	—
217	C₂³×C₃₁	—	2	2	no	—
218	C₂³×C₃₁	—	2	2	no	—
219	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
221	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	Q₈×C₃₁	—	4	2	no	—
223	Q₈×C₃₁	—	4	2	no	—
224	Q₈×C₃₁	—	4	2	no	—
225	D₄×C₃₁	—	4	2	no	—
226	D₄×C₃₁	—	4	2	no	—
227	D₄×C₃₁	—	4	4	no	—
228	D₄×C₃₁	—	4	4	no	—
229	D₄×C₃₁	—	4	4	no	—
230	D₄×C₃₁	—	4	2	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	8	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	—	4	1	no	—
245	C₂×C₄×C₃₁	—	4	1	no	—
246	C₂×C₄×C₃₁	—	4	1	no	—
247	C₂×C₄×C₃₁	—	4	1	no	—
248	C₂×C₄×C₃₁	—	4	1	no	—
249	C₂×C₄×C₃₁	—	4	1	no	—
250	C₂×C₄×C₃₁	—	4	1	no	—
251	C₂×C₄×C₃₁	—	4	1	no	—
252	C₂×C₄×C₃₁	—	4	4	no	—
253	C₂×C₄×C₃₁	—	4	4	no	—
254	C₂×C₄×C₃₁	—	4	16	no	—
255	C₂×C₄×C₃₁	—	4	4	no	—
256	C₂×C₄×C₃₁	—	4	4	no	—
257	C₂×C₄×C₃₁	—	4	4	no	—
258	C₂×C₄×C₃₁	#104	1	4	yes	—
259	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
261	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
263	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
264	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
271	C₈×C₃₁	—	4	4	no	—
272	#3444	C₂×C₃₁	1	1	yes	characteristic
273	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
274	C₂×#376	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1506	C₂×C₃₁	1	2	yes	complemented
276	#1505	C₂×C₃₁	1	2	yes	complemented
277	#1504	C₂×C₃₁	1	4	yes	complemented
278	#1500	C₂×C₃₁	1	4	yes	complemented
279	C₂²×C₃₁	—	2	8	no	—
280	C₂²×C₃₁	—	2	1	no	—
281	C₂²×C₃₁	—	2	1	no	—
282	C₂²×C₃₁	—	2	1	no	—
283	C₂²×C₃₁	—	2	1	no	—
284	C₂²×C₃₁	—	2	1	no	—
285	C₂²×C₃₁	—	2	1	no	—
286	C₂²×C₃₁	—	4	2	no	—
287	C₂²×C₃₁	—	4	2	no	—
288	C₂²×C₃₁	—	4	2	no	—
289	C₂²×C₃₁	#410	1	2	yes	—
290	C₂²×C₃₁	C₂×#104	1	2	yes	—
291	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
292	C₂²×C₃₁	#575	1	2	yes	—
293	C₄×C₃₁	—	2	1	no	—
294	C₄×C₃₁	—	2	1	no	—
295	C₄×C₃₁	—	2	1	no	—
296	C₄×C₃₁	—	2	1	no	—
297	C₄×C₃₁	—	2	1	no	—
298	C₄×C₃₁	—	2	1	no	—
299	C₄×C₃₁	—	2	1	no	—
300	C₄×C₃₁	—	2	1	no	—
301	C₄×C₃₁	—	2	1	no	—
302	C₄×C₃₁	—	2	1	no	—
303	C₄×C₃₁	—	4	8	no	—
304	C₄×C₃₁	—	4	2	no	—
305	C₄×C₃₁	—	4	2	no	—
306	C₄×C₃₁	—	4	2	no	—
307	C₄×C₃₁	C₂×#104	1	2	yes	—
308	C₄×C₃₁	C₂×#104	1	2	yes	—
309	#533	C₄×C₃₁	1	1	yes	characteristic, complemented
310	#533	C₄×C₃₁	1	1	yes	characteristic, complemented
311	#533	C₂²×C₃₁	1	1	yes	characteristic
312	#532	C₄×C₃₁	1	1	yes	characteristic, complemented
313	#532	C₄×C₃₁	1	1	yes	characteristic, complemented
314	#532	C₂²×C₃₁	1	1	yes	characteristic
315	#529	—	2	1	no	—
316	#529	—	2	1	no	—
317	#529	—	2	1	no	—
318	#529	—	2	1	no	—
319	#529	C₄×C₃₁	1	2	yes	complemented
320	#529	C₂²×C₃₁	1	2	yes	—
321	#525	—	2	1	no	—
322	#525	—	2	1	no	—
323	#525	C₂²×C₃₁	1	2	yes	—
324	C₄×#41	—	2	1	no	—
325	C₄×#41	—	2	1	no	—
326	C₄×#41	C₄×C₃₁	1	2	yes	complemented
327	C₂²×C₄²	—	2	1	no	—
328	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
329	#432	—	2	1	no	—
330	#432	C₂²×C₃₁	1	2	yes	—
331	#431	—	2	1	no	—
332	#431	C₂²×C₃₁	1	2	yes	—
333	#423	—	2	1	no	—
334	#407	C₂²×C₃₁	1	2	yes	—
335	C₄×M₁₆	—	2	1	no	—
336	#376	—	2	1	no	—
337	#376	—	2	1	no	—
338	#376	C₂²×C₃₁	1	2	yes	—
339	#376	C₂²×C₃₁	1	2	yes	—
340	C₂×#95	—	2	1	no	—
341	C₂×#95	C₂²×C₃₁	1	2	yes	—
342	#328	C₂²×C₃₁	1	8	yes	—
343	#327	C₂²×C₃₁	1	8	yes	—
344	C₂×C₃₁	—	2	2	no	—
345	C₂×C₃₁	—	2	2	no	—
346	C₂×C₃₁	—	2	2	no	—
347	C₂×C₃₁	—	8	2	no	—
348	C₂×C₃₁	C₂×#410	1	1	yes	characteristic
349	C₂×C₃₁	#2896	1	1	yes	characteristic
350	C₂×C₃₁	C₂²×#104	1	1	yes	characteristic
351	C₂×#41	—	2	1	no	—
352	C₂×#41	—	2	1	no	—
353	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
354	C₂³×C₄	—	2	2	no	—
355	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
356	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
358	C₂×M₁₆	—	2	2	no	—
359	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
360	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
361	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
362	#127	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
363	#126	—	2	2	no	—
364	#126	—	2	2	no	—
365	#125	—	2	1	no	—
366	#125	—	2	1	no	—
367	#125	C₂×C₄×C₃₁	1	2	yes	complemented
368	#124	—	2	1	no	—
369	#124	—	2	1	no	—
370	#124	C₂×C₄×C₃₁	1	2	yes	complemented
371	#123	—	2	2	no	—
372	#123	—	2	2	no	—
373	#123	—	2	2	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	C₂²⋊Q₈	—	2	1	no	—
376	C₂²⋊Q₈	—	2	1	no	—
377	C₂²⋊Q₈	—	2	1	no	—
378	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
379	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
380	#121	—	2	1	no	—
381	#121	—	2	1	no	—
382	#121	—	2	1	no	—
383	#121	—	2	1	no	—
384	#121	C₂×C₄×C₃₁	1	2	yes	—
385	#121	C₂³×C₃₁	1	2	yes	—
386	#120	—	2	2	no	—
387	C₄×Q₈	—	2	1	no	—
388	C₄×Q₈	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	—	2	1	no	—
391	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
392	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
393	C₄×D₄	—	2	2	no	—
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	1	no	—
400	C₄×D₄	—	2	1	no	—
401	C₄×D₄	—	2	1	no	—
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	complemented
404	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	complemented
405	#117	—	2	1	no	—
406	#117	—	2	1	no	—
407	#117	—	2	2	no	—
408	#117	—	2	2	no	—
409	#117	—	2	2	no	—
410	#117	C₂×C₄×C₃₁	1	2	yes	—
411	C₂×C₄²	—	2	8	no	—
412	C₂×C₄²	—	2	2	no	—
413	C₂×C₄²	—	2	2	no	—
414	C₂×C₄²	—	2	2	no	—
415	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
416	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
417	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
418	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
419	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
420	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
421	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
422	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
423	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
424	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
425	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
427	C₄⋊C₈	—	2	2	no	—
428	C₄⋊C₈	—	2	2	no	—
429	C₂²⋊C₈	—	2	2	no	—
430	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
431	#97	—	2	2	no	—
432	C₄×C₈	—	2	2	no	—
433	#95	—	2	2	no	—
434	#95	—	2	2	no	—
435	#95	—	2	2	no	—
436	#95	C₂³×C₃₁	1	4	yes	—
437	C₃₁	#17541	1	1	yes	characteristic, direct product factor, Sylow
438	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
439	#41	—	4	4	no	—
440	#41	—	4	4	no	—
441	#41	—	4	4	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	8	no	—
453	C₂²×C₄	—	2	4	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	4	2	no	—
458	C₂²×C₄	—	4	2	no	—
459	C₂²×C₄	—	4	4	no	—
460	C₂²×C₄	—	4	4	no	—
461	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
462	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
463	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
464	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
465	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
466	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
467	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
468	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
469	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
470	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
472	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
479	M₁₆	—	4	4	no	—
480	C₂×C₈	—	2	4	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
492	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
493	C₂²⋊C₄	—	2	2	no	—
494	C₂²⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	2	no	—
500	C₂²⋊C₄	—	2	2	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₄²	—	2	8	no	—
503	C₄²	—	2	8	no	—
504	C₄²	—	2	2	no	—
505	C₄²	—	2	2	no	—
506	C₄²	—	2	2	no	—
507	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
508	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
509	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
510	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
511	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
512	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
513	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
514	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
515	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
516	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
517	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
518	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
519	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
520	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
521	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
522	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
523	C₂³	—	2	2	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
528	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
529	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
530	Q₈	—	4	2	no	—
531	Q₈	—	4	2	no	—
532	Q₈	—	4	2	no	—
533	D₄	—	4	2	no	—
534	D₄	—	4	2	no	—
535	D₄	—	4	4	no	—
536	D₄	—	4	4	no	—
537	D₄	—	4	4	no	—
538	D₄	—	4	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	8	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	4	no	—
561	C₂×C₄	—	4	4	no	—
562	C₂×C₄	—	4	16	no	—
563	C₂×C₄	—	4	4	no	—
564	C₂×C₄	—	4	4	no	—
565	C₂×C₄	—	4	4	no	—
566	C₂×C₄	C₃₁×#104	1	4	yes	—
567	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
568	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
569	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
570	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
571	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
572	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
573	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
574	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
575	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
576	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
577	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
578	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
579	C₈	—	4	4	no	—
580	C₂²	—	2	8	no	—
581	C₂²	—	2	1	no	—
582	C₂²	—	2	1	no	—
583	C₂²	—	2	1	no	—
584	C₂²	—	2	1	no	—
585	C₂²	—	2	1	no	—
586	C₂²	—	2	1	no	—
587	C₂²	—	4	2	no	—
588	C₂²	—	4	2	no	—
589	C₂²	—	4	2	no	—
590	C₂²	C₃₁×#410	1	2	yes	—
591	C₂²	C₂×C₃₁×#104	1	2	yes	—
592	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
593	C₂²	C₃₁×#575	1	2	yes	—
594	C₄	—	2	1	no	—
595	C₄	—	2	1	no	—
596	C₄	—	2	1	no	—
597	C₄	—	2	1	no	—
598	C₄	—	2	1	no	—
599	C₄	—	2	1	no	—
600	C₄	—	2	1	no	—
601	C₄	—	2	1	no	—
602	C₄	—	2	1	no	—
603	C₄	—	2	1	no	—
604	C₄	—	4	8	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	2	no	—
607	C₄	—	4	2	no	—
608	C₄	C₂×C₃₁×#104	1	2	yes	—
609	C₄	C₂×C₃₁×#104	1	2	yes	—
610	C₂	—	2	2	no	—
611	C₂	—	2	2	no	—
612	C₂	—	2	2	no	—
613	C₂	—	8	2	no	—
614	C₂	C₂×C₃₁×#410	1	1	yes	characteristic
615	C₂	C₃₁×#2896	1	1	yes	characteristic
616	C₂	C₂²×C₃₁×#104	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₃₁×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2920	C₂	semidirect
3. C₂×C₃₁×#376	C₂	semidirect
4. C₃₁×#1506	C₂	semidirect
5. C₃₁×#1505	C₂	semidirect
6. C₃₁×#1504	C₂	semidirect
7. C₃₁×#1500	C₂	semidirect
8. C₃₁×#533	C₄	semidirect
9. C₃₁×#533	C₄	semidirect
11. C₃₁×#532	C₄	semidirect
12. C₃₁×#532	C₄	semidirect
18. C₃₁×#529	C₄	semidirect
25. C₄×#41×C₃₁	C₄	semidirect
51. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
52. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
53. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
54. C₃₁×#127	C₂×C₄	semidirect
59. C₃₁×#125	C₂×C₄	semidirect
62. C₃₁×#124	C₂×C₄	semidirect
83. C₄×Q₈×C₃₁	C₂×C₄	semidirect
84. C₄×Q₈×C₃₁	C₂×C₄	semidirect
95. C₄×D₄×C₃₁	C₂×C₄	semidirect
96. C₄×D₄×C₃₁	C₂×C₄	semidirect
214. #17541	C₃₁	direct (N × H)
273. #2920	C₂×C₃₁	semidirect
274. C₂×#376	C₂×C₃₁	semidirect
275. #1506	C₂×C₃₁	semidirect
276. #1505	C₂×C₃₁	semidirect
277. #1504	C₂×C₃₁	semidirect
278. #1500	C₂×C₃₁	semidirect
309. #533	C₄×C₃₁	semidirect
310. #533	C₄×C₃₁	semidirect
312. #532	C₄×C₃₁	semidirect
313. #532	C₄×C₃₁	semidirect
319. #529	C₄×C₃₁	semidirect
326. C₄×#41	C₄×C₃₁	semidirect
359. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
360. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
361. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
362. #127	C₂×C₄×C₃₁	semidirect
367. #125	C₂×C₄×C₃₁	semidirect
370. #124	C₂×C₄×C₃₁	semidirect
391. C₄×Q₈	C₂×C₄×C₃₁	semidirect
392. C₄×Q₈	C₂×C₄×C₃₁	semidirect
403. C₄×D₄	C₂×C₄×C₃₁	semidirect
404. C₄×D₄	C₂×C₄×C₃₁	semidirect
437. C₃₁	#17541	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3444	2
C₃₁×#2920	2
C₂×C₃₁×#376	2
C₃₁×#1506	2
C₃₁×#1505	2
C₃₁×#1504	2
C₃₁×#1500	2
#17541	31

Subgroups — C31×#17541 #21474300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17541 #21474300