Subgroups — C₃₁×#17652 #21474411

Order 7936.

1028 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3514	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2970	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2875	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1532	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1530	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1529	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1527	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1525	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1524	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1524	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#329	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#559	C₄	1	1	yes	characteristic
14	C₃₁×#556	C₄	1	1	yes	characteristic
15	C₃₁×#555	—	2	1	no	—
16	C₃₁×#551	—	2	2	no	—
17	D₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
18	C₃₁×#548	C₄	1	2	yes	—
19	C₃₁×#546	—	2	1	no	—
20	C₃₁×#546	C₄	1	2	yes	—
21	D₄²×C₃₁	C₄	1	1	yes	characteristic
22	C₃₁×#542	—	2	1	no	—
23	C₃₁×#538	—	2	2	no	—
24	C₃₁×#532	C₂²	1	1	yes	characteristic
25	C₃₁×#529	C₂²	1	1	yes	characteristic
26	C₃₁×#529	C₂²	1	2	yes	—
27	C₂×C₃₁×#124	C₂²	1	2	yes	—
28	C₃₁×#518	—	2	1	no	—
29	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#446	—	2	1	no	—
31	C₃₁×#435	—	2	2	no	—
32	C₈×D₄×C₃₁	—	2	1	no	—
33	C₃₁×#432	C₂²	1	2	yes	complemented
34	C₃₁×#432	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#405	—	2	2	no	—
36	C₃₁×#405	—	2	1	no	—
37	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
38	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
40	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
41	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
42	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
43	C₃₁×#330	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#330	C₂²	1	2	yes	complemented
45	C₃₁×#330	C₂²	1	1	yes	characteristic, complemented
46	C₃₁×#329	C₂²	1	1	yes	characteristic
47	C₃₁×#329	C₂²	1	2	yes	—
48	C₃₁×#329	C₂²	1	1	yes	characteristic
49	C₃₁×#329	C₂²	1	2	yes	complemented
50	C₃₁×#329	C₂²	1	2	yes	complemented
51	C₃₁×#329	C₂²	1	2	yes	complemented
52	C₃₁×#329	C₂²	1	2	yes	complemented
53	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
55	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
56	C₂²×D₄×C₃₁	—	2	1	no	—
57	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₃₁×#131	—	2	8	no	—
59	C₂×M₁₆×C₃₁	—	2	2	no	—
60	C₂×M₁₆×C₃₁	—	2	1	no	—
61	C₂×M₁₆×C₃₁	—	2	1	no	—
62	C₂²×C₈×C₃₁	—	2	2	no	—
63	C₂²×C₈×C₃₁	—	2	1	no	—
64	C₂²×C₈×C₃₁	—	2	2	no	—
65	C₃₁×C₄⋊Q₈	—	2	1	no	—
66	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
67	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
68	C₃₁×#126	—	2	2	no	—
69	C₃₁×#126	—	2	2	no	—
70	C₃₁×#126	—	2	1	no	—
71	C₃₁×#125	C₂³	1	1	yes	characteristic
72	C₃₁×#124	—	2	1	no	—
73	C₃₁×#124	—	2	2	no	—
74	C₃₁×#124	—	2	2	no	—
75	C₃₁×#124	—	2	2	no	—
76	C₃₁×#124	C₂×C₄	1	2	yes	—
77	C₃₁×#124	C₂×C₄	1	2	yes	—
78	C₃₁×#124	C₂³	1	1	yes	characteristic
79	C₃₁×#124	C₂³	1	2	yes	—
80	C₃₁×#123	—	2	1	no	—
81	C₃₁×#123	—	2	2	no	—
82	C₃₁×#123	—	2	2	no	—
83	C₃₁×#123	—	2	2	no	—
84	C₃₁×C₂²⋊Q₈	—	2	1	no	—
85	C₃₁×C₂²⋊Q₈	—	2	2	no	—
86	C₃₁×C₂²⋊Q₈	—	2	2	no	—
87	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
88	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
89	C₃₁×#121	—	2	2	no	—
90	C₃₁×#121	—	2	1	no	—
91	C₃₁×#121	—	2	1	no	—
92	C₃₁×#121	—	2	1	no	—
93	C₃₁×#121	—	2	2	no	—
94	C₃₁×#121	C₂×C₄	1	2	yes	—
95	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
96	C₃₁×#120	—	2	2	no	—
97	C₃₁×#120	—	2	2	no	—
98	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
99	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
100	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
101	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	—	2	2	no	—
104	C₄×D₄×C₃₁	—	2	1	no	—
105	C₄×D₄×C₃₁	—	2	1	no	—
106	C₄×D₄×C₃₁	—	2	2	no	—
107	C₄×D₄×C₃₁	—	2	1	no	—
108	C₄×D₄×C₃₁	—	2	1	no	—
109	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
110	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
111	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
112	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
113	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
114	C₄×D₄×C₃₁	D₄	1	2	yes	—
115	C₃₁×#117	—	2	1	no	—
116	C₃₁×#117	—	2	1	no	—
117	C₃₁×#117	—	2	2	no	—
118	C₃₁×#117	D₄	1	2	yes	—
119	C₃₁×#117	D₄	1	1	yes	characteristic
120	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
121	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
122	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
123	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
125	C₃₁×C₄⋊C₈	—	2	1	no	—
126	C₃₁×C₄⋊C₈	—	2	2	no	—
127	C₃₁×C₂²⋊C₈	—	2	2	no	—
128	C₃₁×C₂²⋊C₈	—	2	1	no	—
129	C₃₁×#97	—	2	2	no	—
130	C₃₁×#97	—	2	1	no	—
131	C₃₁×#97	—	2	2	no	—
132	C₃₁×#97	C₂³	1	1	yes	characteristic
133	C₃₁×#97	C₂³	1	1	yes	characteristic
134	C₃₁×#97	C₂³	1	2	yes	—
135	C₃₁×#97	C₂³	1	1	yes	characteristic
136	C₃₁×#97	C₂³	1	2	yes	—
137	C₃₁×#97	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₃₁	—	2	2	no	—
139	C₄×C₈×C₃₁	—	2	1	no	—
140	C₄×C₈×C₃₁	—	2	1	no	—
141	C₂⁴×C₃₁	—	2	2	no	—
142	#41×C₃₁	—	2	4	no	—
143	#41×C₃₁	—	8	2	no	—
144	C₂×Q₈×C₃₁	—	2	2	no	—
145	C₂×Q₈×C₃₁	—	8	2	no	—
146	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
147	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
149	C₂×D₄×C₃₁	—	2	2	no	—
150	C₂×D₄×C₃₁	—	2	1	no	—
151	C₂×D₄×C₃₁	—	2	2	no	—
152	C₂×D₄×C₃₁	—	2	1	no	—
153	C₂×D₄×C₃₁	—	2	1	no	—
154	C₂×D₄×C₃₁	—	2	2	no	—
155	C₂×D₄×C₃₁	—	4	4	no	—
156	C₂×D₄×C₃₁	—	8	2	no	—
157	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
158	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
159	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
160	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
161	C₂²×C₄×C₃₁	—	2	2	no	—
162	C₂²×C₄×C₃₁	—	2	1	no	—
163	C₂²×C₄×C₃₁	—	2	2	no	—
164	C₂²×C₄×C₃₁	—	2	2	no	—
165	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
166	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
171	M₁₆×C₃₁	—	2	8	no	—
172	M₁₆×C₃₁	—	2	4	no	—
173	M₁₆×C₃₁	—	4	2	no	—
174	C₂×C₈×C₃₁	—	2	8	no	—
175	C₂×C₈×C₃₁	—	2	4	no	—
176	C₂×C₈×C₃₁	—	2	1	no	—
177	C₂×C₈×C₃₁	—	2	2	no	—
178	C₂×C₈×C₃₁	—	2	1	no	—
179	C₂×C₈×C₃₁	—	2	1	no	—
180	C₂×C₈×C₃₁	—	2	1	no	—
181	C₂×C₈×C₃₁	—	2	2	no	—
182	C₂×C₈×C₃₁	—	4	4	no	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	1	no	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	2	2	no	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	—	2	1	no	—
190	C₄⋊C₄×C₃₁	—	2	1	no	—
191	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
193	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
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	1	yes	characteristic
197	C₂²⋊C₄×C₃₁	—	2	2	no	—
198	C₂²⋊C₄×C₃₁	—	2	1	no	—
199	C₂²⋊C₄×C₃₁	—	2	2	no	—
200	C₂²⋊C₄×C₃₁	—	2	1	no	—
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	—	2	2	no	—
205	C₂²⋊C₄×C₃₁	—	2	1	no	—
206	C₂²⋊C₄×C₃₁	—	2	2	no	—
207	C₂²⋊C₄×C₃₁	—	2	1	no	—
208	C₂²⋊C₄×C₃₁	—	2	1	no	—
209	C₂²⋊C₄×C₃₁	—	2	2	no	—
210	C₂²⋊C₄×C₃₁	—	2	2	no	—
211	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
212	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
213	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
214	C₄²×C₃₁	—	2	2	no	—
215	C₄²×C₃₁	—	2	1	no	—
216	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
217	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
218	C₄²×C₃₁	C₂×D₄	1	2	yes	—
219	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
221	C₄²×C₃₁	C₂×D₄	1	2	yes	—
222	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
223	#17652	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂³×C₃₁	—	2	1	no	—
225	C₂³×C₃₁	—	2	2	no	—
226	C₂³×C₃₁	—	4	4	no	—
227	C₂³×C₃₁	—	8	2	no	—
228	C₂³×C₃₁	—	8	2	no	—
229	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
230	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
231	Q₈×C₃₁	—	2	2	no	—
232	Q₈×C₃₁	—	4	1	no	—
233	Q₈×C₃₁	—	4	2	no	—
234	Q₈×C₃₁	—	8	2	no	—
235	D₄×C₃₁	—	2	4	no	—
236	D₄×C₃₁	—	4	1	no	—
237	D₄×C₃₁	—	4	2	no	—
238	D₄×C₃₁	—	4	4	no	—
239	D₄×C₃₁	—	4	2	no	—
240	D₄×C₃₁	—	8	1	no	—
241	D₄×C₃₁	—	8	2	no	—
242	D₄×C₃₁	—	8	1	no	—
243	D₄×C₃₁	—	8	2	no	—
244	D₄×C₃₁	#104	1	4	yes	—
245	C₂×C₄×C₃₁	—	2	4	no	—
246	C₂×C₄×C₃₁	—	2	2	no	—
247	C₂×C₄×C₃₁	—	2	1	no	—
248	C₂×C₄×C₃₁	—	2	1	no	—
249	C₂×C₄×C₃₁	—	2	1	no	—
250	C₂×C₄×C₃₁	—	2	2	no	—
251	C₂×C₄×C₃₁	—	4	2	no	—
252	C₂×C₄×C₃₁	—	4	1	no	—
253	C₂×C₄×C₃₁	—	4	2	no	—
254	C₂×C₄×C₃₁	—	4	1	no	—
255	C₂×C₄×C₃₁	—	4	2	no	—
256	C₂×C₄×C₃₁	—	8	2	no	—
257	C₂×C₄×C₃₁	—	8	2	no	—
258	C₂×C₄×C₃₁	—	8	2	no	—
259	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
260	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
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	2	yes	—
264	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
265	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
266	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
268	C₈×C₃₁	—	2	2	no	—
269	C₈×C₃₁	—	2	4	no	—
270	C₈×C₃₁	—	2	2	no	—
271	C₈×C₃₁	—	2	2	no	—
272	C₈×C₃₁	—	4	2	no	—
273	C₈×C₃₁	—	4	1	no	—
274	#3514	C₂×C₃₁	1	1	yes	characteristic
275	#2970	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#2875	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#1532	C₂×C₃₁	1	2	yes	complemented
278	#1530	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#1529	C₂×C₃₁	1	1	yes	characteristic, complemented
280	#1527	C₂×C₃₁	1	2	yes	complemented
281	#1525	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#1524	C₂×C₃₁	1	1	yes	characteristic, complemented
283	#1524	C₂×C₃₁	1	2	yes	complemented
284	C₂×#329	C₂×C₃₁	1	2	yes	complemented
285	C₂²×C₃₁	—	2	2	no	—
286	C₂²×C₃₁	—	2	2	no	—
287	C₂²×C₃₁	—	2	1	no	—
288	C₂²×C₃₁	—	2	1	no	—
289	C₂²×C₃₁	—	2	1	no	—
290	C₂²×C₃₁	—	4	2	no	—
291	C₂²×C₃₁	—	4	1	no	—
292	C₂²×C₃₁	—	4	1	no	—
293	C₂²×C₃₁	—	4	2	no	—
294	C₂²×C₃₁	—	4	2	no	—
295	C₂²×C₃₁	—	4	1	no	—
296	C₂²×C₃₁	—	16	4	no	—
297	C₂²×C₃₁	C₂×#104	1	4	yes	—
298	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
299	C₄×C₃₁	—	2	1	no	—
300	C₄×C₃₁	—	2	1	no	—
301	C₄×C₃₁	—	2	2	no	—
302	C₄×C₃₁	—	2	1	no	—
303	C₄×C₃₁	—	2	2	no	—
304	C₄×C₃₁	—	2	1	no	—
305	C₄×C₃₁	—	2	2	no	—
306	C₄×C₃₁	—	2	1	no	—
307	C₄×C₃₁	—	4	2	no	—
308	C₄×C₃₁	—	4	1	no	—
309	C₄×C₃₁	—	4	1	no	—
310	C₄×C₃₁	—	4	1	no	—
311	C₄×C₃₁	C₂×#104	1	2	yes	—
312	C₂×#131	—	2	1	no	—
313	#559	C₄×C₃₁	1	1	yes	characteristic
314	#556	C₄×C₃₁	1	1	yes	characteristic
315	#555	—	2	1	no	—
316	#551	—	2	2	no	—
317	D₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
318	#548	C₄×C₃₁	1	2	yes	—
319	#546	—	2	1	no	—
320	#546	C₄×C₃₁	1	2	yes	—
321	D₄²	C₄×C₃₁	1	1	yes	characteristic
322	#542	—	2	1	no	—
323	#538	—	2	2	no	—
324	#532	C₂²×C₃₁	1	1	yes	characteristic
325	#529	C₂²×C₃₁	1	1	yes	characteristic
326	#529	C₂²×C₃₁	1	2	yes	—
327	C₂×#124	C₂²×C₃₁	1	2	yes	—
328	#518	—	2	1	no	—
329	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
330	#446	—	2	1	no	—
331	#435	—	2	2	no	—
332	C₈×D₄	—	2	1	no	—
333	#432	C₂²×C₃₁	1	2	yes	complemented
334	#432	C₂²×C₃₁	1	1	yes	characteristic, complemented
335	#405	—	2	2	no	—
336	#405	—	2	1	no	—
337	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
338	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
339	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
340	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
341	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
342	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
343	#330	C₂²×C₃₁	1	1	yes	characteristic, complemented
344	#330	C₂²×C₃₁	1	2	yes	complemented
345	#330	C₂²×C₃₁	1	1	yes	characteristic, complemented
346	#329	C₂²×C₃₁	1	1	yes	characteristic
347	#329	C₂²×C₃₁	1	2	yes	—
348	#329	C₂²×C₃₁	1	1	yes	characteristic
349	#329	C₂²×C₃₁	1	2	yes	complemented
350	#329	C₂²×C₃₁	1	2	yes	complemented
351	#329	C₂²×C₃₁	1	2	yes	complemented
352	#329	C₂²×C₃₁	1	2	yes	complemented
353	C₂×C₃₁	—	2	4	no	—
354	C₂×C₃₁	—	4	1	no	—
355	C₂×C₃₁	—	8	2	no	—
356	C₂×C₃₁	—	8	1	no	—
357	C₂×C₃₁	#2888	1	1	yes	characteristic
358	C₂×C₃₁	C₂²×#104	1	1	yes	characteristic
359	C₂×C₃₁	#2901	1	1	yes	characteristic
360	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
361	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
362	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
363	C₂²×D₄	—	2	1	no	—
364	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
365	#131	—	2	8	no	—
366	C₂×M₁₆	—	2	2	no	—
367	C₂×M₁₆	—	2	1	no	—
368	C₂×M₁₆	—	2	1	no	—
369	C₂²×C₈	—	2	2	no	—
370	C₂²×C₈	—	2	1	no	—
371	C₂²×C₈	—	2	2	no	—
372	C₄⋊Q₈	—	2	1	no	—
373	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
374	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
375	#126	—	2	2	no	—
376	#126	—	2	2	no	—
377	#126	—	2	1	no	—
378	#125	C₂³×C₃₁	1	1	yes	characteristic
379	#124	—	2	1	no	—
380	#124	—	2	2	no	—
381	#124	—	2	2	no	—
382	#124	—	2	2	no	—
383	#124	C₂×C₄×C₃₁	1	2	yes	—
384	#124	C₂×C₄×C₃₁	1	2	yes	—
385	#124	C₂³×C₃₁	1	1	yes	characteristic
386	#124	C₂³×C₃₁	1	2	yes	—
387	#123	—	2	1	no	—
388	#123	—	2	2	no	—
389	#123	—	2	2	no	—
390	#123	—	2	2	no	—
391	C₂²⋊Q₈	—	2	1	no	—
392	C₂²⋊Q₈	—	2	2	no	—
393	C₂²⋊Q₈	—	2	2	no	—
394	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
395	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
396	#121	—	2	2	no	—
397	#121	—	2	1	no	—
398	#121	—	2	1	no	—
399	#121	—	2	1	no	—
400	#121	—	2	2	no	—
401	#121	C₂×C₄×C₃₁	1	2	yes	—
402	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
403	#120	—	2	2	no	—
404	#120	—	2	2	no	—
405	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
406	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
407	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
408	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
409	C₄×D₄	—	2	1	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	1	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
417	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
418	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
419	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
420	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
421	C₄×D₄	D₄×C₃₁	1	2	yes	—
422	#117	—	2	1	no	—
423	#117	—	2	1	no	—
424	#117	—	2	2	no	—
425	#117	D₄×C₃₁	1	2	yes	—
426	#117	D₄×C₃₁	1	1	yes	characteristic
427	C₂×C₂²⋊C₄	—	2	2	no	—
428	C₂×C₂²⋊C₄	—	2	2	no	—
429	C₂×C₂²⋊C₄	—	2	2	no	—
430	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
431	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
432	C₄⋊C₈	—	2	1	no	—
433	C₄⋊C₈	—	2	2	no	—
434	C₂²⋊C₈	—	2	2	no	—
435	C₂²⋊C₈	—	2	1	no	—
436	#97	—	2	2	no	—
437	#97	—	2	1	no	—
438	#97	—	2	2	no	—
439	#97	C₂³×C₃₁	1	1	yes	characteristic
440	#97	C₂³×C₃₁	1	1	yes	characteristic
441	#97	C₂³×C₃₁	1	2	yes	—
442	#97	C₂³×C₃₁	1	1	yes	characteristic
443	#97	C₂³×C₃₁	1	2	yes	—
444	#97	C₂³×C₃₁	1	1	yes	characteristic
445	C₄×C₈	—	2	2	no	—
446	C₄×C₈	—	2	1	no	—
447	C₄×C₈	—	2	1	no	—
448	C₃₁	#17652	1	1	yes	characteristic, direct product factor, Sylow
449	C₂⁴	—	2	2	no	—
450	#41	—	2	4	no	—
451	#41	—	8	2	no	—
452	C₂×Q₈	—	2	2	no	—
453	C₂×Q₈	—	8	2	no	—
454	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
455	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
456	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	—	4	4	no	—
464	C₂×D₄	—	8	2	no	—
465	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
466	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
467	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
468	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
474	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
476	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
478	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
479	M₁₆	—	2	8	no	—
480	M₁₆	—	2	4	no	—
481	M₁₆	—	4	2	no	—
482	C₂×C₈	—	2	8	no	—
483	C₂×C₈	—	2	4	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	2	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	2	no	—
490	C₂×C₈	—	4	4	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
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	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
503	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
504	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	1	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	1	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	1	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	1	no	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
520	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
521	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
522	C₄²	—	2	2	no	—
523	C₄²	—	2	1	no	—
524	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
525	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
526	C₄²	C₂×D₄×C₃₁	1	2	yes	—
527	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
528	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
529	C₄²	C₂×D₄×C₃₁	1	2	yes	—
530	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
531	C₂³	—	2	1	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	4	4	no	—
534	C₂³	—	8	2	no	—
535	C₂³	—	8	2	no	—
536	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
537	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
538	Q₈	—	2	2	no	—
539	Q₈	—	4	1	no	—
540	Q₈	—	4	2	no	—
541	Q₈	—	8	2	no	—
542	D₄	—	2	4	no	—
543	D₄	—	4	1	no	—
544	D₄	—	4	2	no	—
545	D₄	—	4	4	no	—
546	D₄	—	4	2	no	—
547	D₄	—	8	1	no	—
548	D₄	—	8	2	no	—
549	D₄	—	8	1	no	—
550	D₄	—	8	2	no	—
551	D₄	C₃₁×#104	1	4	yes	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	4	2	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	2	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	—	8	2	no	—
564	C₂×C₄	—	8	2	no	—
565	C₂×C₄	—	8	2	no	—
566	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
567	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
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	2	yes	—
571	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
572	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
573	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
574	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
575	C₈	—	2	2	no	—
576	C₈	—	2	4	no	—
577	C₈	—	2	2	no	—
578	C₈	—	2	2	no	—
579	C₈	—	4	2	no	—
580	C₈	—	4	1	no	—
581	C₂²	—	2	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	2	1	no	—
584	C₂²	—	2	1	no	—
585	C₂²	—	2	1	no	—
586	C₂²	—	4	2	no	—
587	C₂²	—	4	1	no	—
588	C₂²	—	4	1	no	—
589	C₂²	—	4	2	no	—
590	C₂²	—	4	2	no	—
591	C₂²	—	4	1	no	—
592	C₂²	—	16	4	no	—
593	C₂²	C₂×C₃₁×#104	1	4	yes	—
594	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
595	C₄	—	2	1	no	—
596	C₄	—	2	1	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	1	no	—
599	C₄	—	2	2	no	—
600	C₄	—	2	1	no	—
601	C₄	—	2	2	no	—
602	C₄	—	2	1	no	—
603	C₄	—	4	2	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	1	no	—
606	C₄	—	4	1	no	—
607	C₄	C₂×C₃₁×#104	1	2	yes	—
608	C₂	—	2	4	no	—
609	C₂	—	4	1	no	—
610	C₂	—	8	1	no	—
611	C₂	—	8	2	no	—
612	C₂	C₃₁×#2888	1	1	yes	characteristic
613	C₂	C₂²×C₃₁×#104	1	1	yes	characteristic
614	C₂	C₃₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3514	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2970	C₂	semidirect
3. C₃₁×#2875	C₂	semidirect
4. C₃₁×#1532	C₂	semidirect
5. C₃₁×#1530	C₂	semidirect
6. C₃₁×#1529	C₂	semidirect
7. C₃₁×#1527	C₂	semidirect
8. C₃₁×#1525	C₂	semidirect
9. C₃₁×#1524	C₂	semidirect
10. C₃₁×#1524	C₂	semidirect
11. C₂×C₃₁×#329	C₂	semidirect
33. C₃₁×#432	C₂²	semidirect
34. C₃₁×#432	C₂²	semidirect
37. C₄×M₁₆×C₃₁	C₂²	semidirect
38. C₄×M₁₆×C₃₁	C₂²	semidirect
39. C₄×M₁₆×C₃₁	C₂²	semidirect
40. C₂×C₃₁×#97	C₂²	semidirect
41. C₂×C₃₁×#97	C₂²	semidirect
42. C₂×C₃₁×#97	C₂²	semidirect
43. C₃₁×#330	C₂²	semidirect
44. C₃₁×#330	C₂²	semidirect
45. C₃₁×#330	C₂²	semidirect
49. C₃₁×#329	C₂²	semidirect
50. C₃₁×#329	C₂²	semidirect
51. C₃₁×#329	C₂²	semidirect
52. C₃₁×#329	C₂²	semidirect
223. #17652	C₃₁	direct (N × H)
275. #2970	C₂×C₃₁	semidirect
276. #2875	C₂×C₃₁	semidirect
277. #1532	C₂×C₃₁	semidirect
278. #1530	C₂×C₃₁	semidirect
279. #1529	C₂×C₃₁	semidirect
280. #1527	C₂×C₃₁	semidirect
281. #1525	C₂×C₃₁	semidirect
282. #1524	C₂×C₃₁	semidirect
283. #1524	C₂×C₃₁	semidirect
284. C₂×#329	C₂×C₃₁	semidirect
333. #432	C₂²×C₃₁	semidirect
334. #432	C₂²×C₃₁	semidirect
337. C₄×M₁₆	C₂²×C₃₁	semidirect
338. C₄×M₁₆	C₂²×C₃₁	semidirect
339. C₄×M₁₆	C₂²×C₃₁	semidirect
340. C₂×#97	C₂²×C₃₁	semidirect
341. C₂×#97	C₂²×C₃₁	semidirect
342. C₂×#97	C₂²×C₃₁	semidirect
343. #330	C₂²×C₃₁	semidirect
344. #330	C₂²×C₃₁	semidirect
345. #330	C₂²×C₃₁	semidirect
349. #329	C₂²×C₃₁	semidirect
350. #329	C₂²×C₃₁	semidirect
351. #329	C₂²×C₃₁	semidirect
352. #329	C₂²×C₃₁	semidirect
448. C₃₁	#17652	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×#3514	2
C₃₁×#2970	2
C₃₁×#2875	2
C₃₁×#1532	2
C₃₁×#1530	2
C₃₁×#1529	2
C₃₁×#1527	2
C₃₁×#1525	2
C₃₁×#1524	2
C₂×C₃₁×#329	2
#17652	31

Subgroups — C31×#17652 #21474411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17652 #21474411