Subgroups — C₃₁×#17684 #21474443

Order 7936.

932 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3517	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2970	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2923	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1536	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1535	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1532	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1530	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1530	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1529	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1525	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1524	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#557	C₄	1	1	yes	characteristic
13	C₃₁×#553	—	2	1	no	—
14	C₃₁×#553	—	2	2	no	—
15	C₃₁×#550	—	2	1	no	—
16	C₃₁×#550	C₄	1	1	yes	characteristic
17	C₃₁×#550	C₄	1	2	yes	—
18	C₃₁×#550	C₄	1	1	yes	characteristic
19	C₃₁×#547	C₄	1	1	yes	characteristic
20	C₃₁×#547	C₄	1	2	yes	—
21	C₃₁×#540	—	2	1	no	—
22	C₃₁×#540	—	2	2	no	—
23	C₃₁×#532	C₂²	1	1	yes	characteristic
24	C₃₁×#532	C₂²	1	2	yes	—
25	C₃₁×#529	C₂²	1	1	yes	characteristic
26	C₃₁×#529	C₂²	1	2	yes	—
27	C₃₁×#520	—	2	1	no	—
28	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#446	—	2	1	no	—
30	C₃₁×#435	—	2	2	no	—
31	C₈×D₄×C₃₁	—	2	1	no	—
32	C₃₁×#433	—	2	2	no	—
33	C₃₁×#432	C₂²	1	2	yes	—
34	C₃₁×#432	C₂²	1	2	yes	—
35	C₃₁×#432	C₂²	1	1	yes	characteristic
36	C₃₁×#432	C₂²	1	2	yes	—
37	C₃₁×#424	—	2	1	no	—
38	C₃₁×#408	—	2	1	no	—
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#97	C₂²	1	2	yes	—
43	C₃₁×#330	C₂²	1	1	yes	characteristic
44	C₃₁×#330	C₂²	1	1	yes	characteristic
45	C₃₁×#330	C₂²	1	2	yes	—
46	C₃₁×#329	C₂²	1	1	yes	characteristic
47	C₃₁×#329	C₂²	1	1	yes	characteristic
48	C₃₁×#329	C₂²	1	2	yes	—
49	C₃₁×#329	C₂²	1	2	yes	—
50	C₃₁×#329	C₂²	1	2	yes	—
51	C₃₁×#329	C₂²	1	2	yes	—
52	C₃₁×#329	C₂²	1	2	yes	—
53	C₂×#41×C₃₁	—	2	1	no	—
54	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
58	C₂×M₁₆×C₃₁	—	2	1	no	—
59	C₂×M₁₆×C₃₁	—	2	1	no	—
60	C₂²×C₈×C₃₁	—	2	1	no	—
61	C₂²×C₈×C₃₁	—	2	2	no	—
62	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
64	C₃₁×#127	—	2	1	no	—
65	C₃₁×#127	—	2	2	no	—
66	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
67	C₃₁×#127	C₂×C₄	1	2	yes	—
68	C₃₁×#126	—	2	1	no	—
69	C₃₁×#126	—	2	2	no	—
70	C₃₁×#125	—	2	2	no	—
71	C₃₁×#125	—	2	1	no	—
72	C₃₁×#125	C₂³	1	1	yes	characteristic
73	C₃₁×#124	C₂³	1	1	yes	characteristic
74	C₃₁×#124	C₂³	1	2	yes	—
75	C₃₁×#123	—	2	2	no	—
76	C₃₁×#123	—	2	1	no	—
77	C₃₁×#123	—	2	2	no	—
78	C₃₁×#123	—	2	2	no	—
79	C₃₁×#123	—	2	2	no	—
80	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
81	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
82	C₃₁×#121	—	2	1	no	—
83	C₃₁×#121	—	2	2	no	—
84	C₃₁×#121	—	2	2	no	—
85	C₃₁×#121	—	2	1	no	—
86	C₃₁×#121	—	2	1	no	—
87	C₃₁×#121	—	2	2	no	—
88	C₃₁×#121	—	2	2	no	—
89	C₃₁×#121	—	2	1	no	—
90	C₃₁×#121	—	2	2	no	—
91	C₃₁×#121	—	2	2	no	—
92	C₃₁×#121	—	2	2	no	—
93	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
94	C₃₁×#121	C₂×C₄	1	2	yes	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
97	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
98	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
99	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	—	2	2	no	—
102	C₄×D₄×C₃₁	—	2	2	no	—
103	C₄×D₄×C₃₁	—	2	1	no	—
104	C₄×D₄×C₃₁	—	2	1	no	—
105	C₄×D₄×C₃₁	—	2	2	no	—
106	C₄×D₄×C₃₁	—	2	1	no	—
107	C₄×D₄×C₃₁	—	2	2	no	—
108	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
109	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
110	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
111	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
112	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
113	C₄×D₄×C₃₁	D₄	1	2	yes	—
114	C₃₁×#117	—	2	1	no	—
115	C₃₁×#117	—	2	2	no	—
116	C₃₁×#117	D₄	1	1	yes	characteristic
117	C₃₁×#117	D₄	1	2	yes	—
118	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₂×C₄⋊C₄×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	2	no	—
123	C₃₁×C₄⋊C₈	—	2	2	no	—
124	C₃₁×C₄⋊C₈	—	2	1	no	—
125	C₃₁×C₄⋊C₈	—	2	2	no	—
126	C₃₁×C₂²⋊C₈	—	2	2	no	—
127	C₃₁×C₂²⋊C₈	—	2	2	no	—
128	C₃₁×C₂²⋊C₈	—	2	2	no	—
129	C₃₁×C₂²⋊C₈	—	2	1	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	1	yes	characteristic
135	C₃₁×#97	C₂³	1	2	yes	—
136	C₃₁×#97	C₂³	1	2	yes	—
137	C₃₁×#97	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₃₁	—	2	1	no	—
139	C₄×C₈×C₃₁	—	2	1	no	—
140	#41×C₃₁	—	4	2	no	—
141	#41×C₃₁	—	4	4	no	—
142	#41×C₃₁	—	8	2	no	—
143	#41×C₃₁	—	8	2	no	—
144	#41×C₃₁	—	8	2	no	—
145	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
146	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
147	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
148	C₂×D₄×C₃₁	—	2	1	no	—
149	C₂×D₄×C₃₁	—	2	2	no	—
150	C₂×D₄×C₃₁	—	2	2	no	—
151	C₂×D₄×C₃₁	—	2	2	no	—
152	C₂×D₄×C₃₁	—	2	2	no	—
153	C₂×D₄×C₃₁	—	2	2	no	—
154	C₂×D₄×C₃₁	—	2	1	no	—
155	C₂×D₄×C₃₁	—	2	1	no	—
156	C₂×D₄×C₃₁	—	2	2	no	—
157	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
158	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
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	—	2	2	no	—
166	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂²×C₄	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	2	yes	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	M₁₆×C₃₁	—	4	2	no	—
173	M₁₆×C₃₁	—	4	2	no	—
174	C₂×C₈×C₃₁	—	2	1	no	—
175	C₂×C₈×C₃₁	—	2	2	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	—	2	2	no	—
180	C₂×C₈×C₃₁	—	4	4	no	—
181	C₂×C₈×C₃₁	—	4	2	no	—
182	C₄⋊C₄×C₃₁	—	2	1	no	—
183	C₄⋊C₄×C₃₁	—	2	2	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	—	2	1	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	2	no	—
190	C₄⋊C₄×C₃₁	—	2	1	no	—
191	C₄⋊C₄×C₃₁	—	2	1	no	—
192	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
193	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	1	no	—
203	C₂²⋊C₄×C₃₁	—	2	2	no	—
204	C₂²⋊C₄×C₃₁	—	2	1	no	—
205	C₂²⋊C₄×C₃₁	—	2	2	no	—
206	C₂²⋊C₄×C₃₁	—	2	1	no	—
207	C₂²⋊C₄×C₃₁	—	2	2	no	—
208	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
209	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
210	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
211	C₄²×C₃₁	—	2	2	no	—
212	C₄²×C₃₁	—	2	1	no	—
213	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
214	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
215	C₄²×C₃₁	C₂×D₄	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	C₂⁴	1	1	yes	characteristic
220	#17684	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂³×C₃₁	—	2	2	no	—
222	C₂³×C₃₁	—	2	1	no	—
223	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
225	Q₈×C₃₁	—	2	2	no	—
226	Q₈×C₃₁	—	4	1	no	—
227	Q₈×C₃₁	—	4	2	no	—
228	D₄×C₃₁	—	2	2	no	—
229	D₄×C₃₁	—	4	2	no	—
230	D₄×C₃₁	—	4	1	no	—
231	D₄×C₃₁	—	4	2	no	—
232	D₄×C₃₁	—	4	4	no	—
233	D₄×C₃₁	—	4	2	no	—
234	D₄×C₃₁	—	8	2	no	—
235	D₄×C₃₁	—	8	1	no	—
236	D₄×C₃₁	—	8	2	no	—
237	D₄×C₃₁	—	8	1	no	—
238	D₄×C₃₁	—	8	2	no	—
239	D₄×C₃₁	—	8	2	no	—
240	D₄×C₃₁	—	8	2	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	1	no	—
243	C₂×C₄×C₃₁	—	2	2	no	—
244	C₂×C₄×C₃₁	—	2	1	no	—
245	C₂×C₄×C₃₁	—	2	1	no	—
246	C₂×C₄×C₃₁	—	4	2	no	—
247	C₂×C₄×C₃₁	—	4	1	no	—
248	C₂×C₄×C₃₁	—	4	1	no	—
249	C₂×C₄×C₃₁	—	4	2	no	—
250	C₂×C₄×C₃₁	—	4	2	no	—
251	C₂×C₄×C₃₁	—	4	4	no	—
252	C₂×C₄×C₃₁	—	4	2	no	—
253	C₂×C₄×C₃₁	—	8	2	no	—
254	C₂×C₄×C₃₁	—	8	2	no	—
255	C₂×C₄×C₃₁	—	8	2	no	—
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	2	yes	—
259	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
263	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
265	C₈×C₃₁	—	2	2	no	—
266	C₈×C₃₁	—	4	1	no	—
267	C₈×C₃₁	—	4	1	no	—
268	C₈×C₃₁	—	4	2	no	—
269	C₈×C₃₁	—	4	2	no	—
270	C₈×C₃₁	—	4	1	no	—
271	#3517	C₂×C₃₁	1	1	yes	characteristic
272	#2970	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#2923	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#1536	C₂×C₃₁	1	2	yes	complemented
275	#1535	C₂×C₃₁	1	2	yes	complemented
276	#1532	C₂×C₃₁	1	2	yes	complemented
277	#1530	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	#1525	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#1524	C₂×C₃₁	1	1	yes	characteristic, complemented
282	C₂²×C₃₁	—	2	1	no	—
283	C₂²×C₃₁	—	2	2	no	—
284	C₂²×C₃₁	—	2	1	no	—
285	C₂²×C₃₁	—	2	2	no	—
286	C₂²×C₃₁	—	2	2	no	—
287	C₂²×C₃₁	—	2	1	no	—
288	C₂²×C₃₁	—	4	2	no	—
289	C₂²×C₃₁	—	4	1	no	—
290	C₂²×C₃₁	—	4	2	no	—
291	C₂²×C₃₁	—	4	1	no	—
292	C₂²×C₃₁	—	4	2	no	—
293	C₂²×C₃₁	—	4	1	no	—
294	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
295	C₄×C₃₁	—	2	1	no	—
296	C₄×C₃₁	—	2	1	no	—
297	C₄×C₃₁	—	2	2	no	—
298	C₄×C₃₁	—	2	1	no	—
299	C₄×C₃₁	—	2	2	no	—
300	C₄×C₃₁	—	2	1	no	—
301	C₄×C₃₁	—	2	2	no	—
302	C₄×C₃₁	—	2	1	no	—
303	C₄×C₃₁	—	4	2	no	—
304	C₄×C₃₁	—	4	1	no	—
305	C₄×C₃₁	—	4	1	no	—
306	C₄×C₃₁	—	4	1	no	—
307	C₄×C₃₁	#421	1	2	yes	—
308	#557	C₄×C₃₁	1	1	yes	characteristic
309	#553	—	2	1	no	—
310	#553	—	2	2	no	—
311	#550	—	2	1	no	—
312	#550	C₄×C₃₁	1	1	yes	characteristic
313	#550	C₄×C₃₁	1	2	yes	—
314	#550	C₄×C₃₁	1	1	yes	characteristic
315	#547	C₄×C₃₁	1	1	yes	characteristic
316	#547	C₄×C₃₁	1	2	yes	—
317	#540	—	2	1	no	—
318	#540	—	2	2	no	—
319	#532	C₂²×C₃₁	1	2	yes	—
320	#532	C₂²×C₃₁	1	1	yes	characteristic
321	#529	C₂²×C₃₁	1	1	yes	characteristic
322	#529	C₂²×C₃₁	1	2	yes	—
323	#520	—	2	1	no	—
324	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
325	#446	—	2	1	no	—
326	#435	—	2	2	no	—
327	C₈×D₄	—	2	1	no	—
328	#433	—	2	2	no	—
329	#432	C₂²×C₃₁	1	2	yes	—
330	#432	C₂²×C₃₁	1	2	yes	—
331	#432	C₂²×C₃₁	1	2	yes	—
332	#432	C₂²×C₃₁	1	1	yes	characteristic
333	#424	—	2	1	no	—
334	#408	—	2	1	no	—
335	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
336	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×#97	C₂²×C₃₁	1	2	yes	—
339	#330	C₂²×C₃₁	1	1	yes	characteristic
340	#330	C₂²×C₃₁	1	1	yes	characteristic
341	#330	C₂²×C₃₁	1	2	yes	—
342	#329	C₂²×C₃₁	1	1	yes	characteristic
343	#329	C₂²×C₃₁	1	2	yes	—
344	#329	C₂²×C₃₁	1	2	yes	—
345	#329	C₂²×C₃₁	1	2	yes	—
346	#329	C₂²×C₃₁	1	1	yes	characteristic
347	#329	C₂²×C₃₁	1	2	yes	—
348	#329	C₂²×C₃₁	1	2	yes	—
349	C₂×C₃₁	—	4	1	no	—
350	C₂×C₃₁	—	4	2	no	—
351	C₂×C₃₁	—	8	2	no	—
352	C₂×C₃₁	—	8	1	no	—
353	C₂×C₃₁	#2888	1	1	yes	characteristic
354	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
355	C₂×C₃₁	#2901	1	1	yes	characteristic
356	C₂×#41	—	2	1	no	—
357	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
358	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
359	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
360	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
361	C₂×M₁₆	—	2	1	no	—
362	C₂×M₁₆	—	2	1	no	—
363	C₂²×C₈	—	2	1	no	—
364	C₂²×C₈	—	2	2	no	—
365	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
366	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
367	#127	—	2	1	no	—
368	#127	—	2	2	no	—
369	#127	C₂×C₄×C₃₁	1	2	yes	—
370	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
371	#126	—	2	1	no	—
372	#126	—	2	2	no	—
373	#125	—	2	2	no	—
374	#125	—	2	1	no	—
375	#125	C₂³×C₃₁	1	1	yes	characteristic
376	#124	C₂³×C₃₁	1	1	yes	characteristic
377	#124	C₂³×C₃₁	1	2	yes	—
378	#123	—	2	2	no	—
379	#123	—	2	1	no	—
380	#123	—	2	2	no	—
381	#123	—	2	2	no	—
382	#123	—	2	2	no	—
383	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
384	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
385	#121	—	2	1	no	—
386	#121	—	2	2	no	—
387	#121	—	2	2	no	—
388	#121	—	2	1	no	—
389	#121	—	2	1	no	—
390	#121	—	2	2	no	—
391	#121	—	2	2	no	—
392	#121	—	2	1	no	—
393	#121	—	2	2	no	—
394	#121	—	2	2	no	—
395	#121	—	2	2	no	—
396	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
397	#121	C₂×C₄×C₃₁	1	2	yes	—
398	C₄×Q₈	—	2	1	no	—
399	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
400	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
401	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
402	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	—	2	2	no	—
405	C₄×D₄	—	2	2	no	—
406	C₄×D₄	—	2	1	no	—
407	C₄×D₄	—	2	1	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	1	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
412	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
413	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
414	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
415	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
416	C₄×D₄	D₄×C₃₁	1	2	yes	—
417	#117	—	2	1	no	—
418	#117	—	2	2	no	—
419	#117	D₄×C₃₁	1	1	yes	characteristic
420	#117	D₄×C₃₁	1	2	yes	—
421	C₂×C₄⋊C₄	—	2	2	no	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
424	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
425	C₄⋊C₈	—	2	2	no	—
426	C₄⋊C₈	—	2	2	no	—
427	C₄⋊C₈	—	2	1	no	—
428	C₄⋊C₈	—	2	2	no	—
429	C₂²⋊C₈	—	2	2	no	—
430	C₂²⋊C₈	—	2	2	no	—
431	C₂²⋊C₈	—	2	2	no	—
432	C₂²⋊C₈	—	2	1	no	—
433	#97	—	2	1	no	—
434	#97	—	2	2	no	—
435	#97	C₂³×C₃₁	1	1	yes	characteristic
436	#97	C₂³×C₃₁	1	1	yes	characteristic
437	#97	C₂³×C₃₁	1	2	yes	—
438	#97	C₂³×C₃₁	1	1	yes	characteristic
439	#97	C₂³×C₃₁	1	1	yes	characteristic
440	#97	C₂³×C₃₁	1	2	yes	—
441	C₄×C₈	—	2	1	no	—
442	C₄×C₈	—	2	1	no	—
443	C₃₁	#17684	1	1	yes	characteristic, direct product factor, Sylow
444	#41	—	4	2	no	—
445	#41	—	4	4	no	—
446	#41	—	8	2	no	—
447	#41	—	8	2	no	—
448	#41	—	8	2	no	—
449	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
450	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
451	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
462	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
463	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
464	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
472	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
474	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
475	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
476	M₁₆	—	4	2	no	—
477	M₁₆	—	4	2	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	2	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	2	no	—
484	C₂×C₈	—	4	4	no	—
485	C₂×C₈	—	4	2	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
497	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
498	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
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	1	yes	characteristic
502	C₂²⋊C₄	—	2	1	no	—
503	C₂²⋊C₄	—	2	2	no	—
504	C₂²⋊C₄	—	2	1	no	—
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	1	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
513	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
514	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	1	no	—
517	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
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	2	yes	—
522	C₄²	C₂×D₄×C₃₁	1	2	yes	—
523	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
524	C₂³	—	2	2	no	—
525	C₂³	—	2	1	no	—
526	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
527	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
528	Q₈	—	2	2	no	—
529	Q₈	—	4	1	no	—
530	Q₈	—	4	2	no	—
531	D₄	—	2	2	no	—
532	D₄	—	4	2	no	—
533	D₄	—	4	1	no	—
534	D₄	—	4	2	no	—
535	D₄	—	4	4	no	—
536	D₄	—	4	2	no	—
537	D₄	—	8	2	no	—
538	D₄	—	8	1	no	—
539	D₄	—	8	2	no	—
540	D₄	—	8	1	no	—
541	D₄	—	8	2	no	—
542	D₄	—	8	2	no	—
543	D₄	—	8	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	2	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	4	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	8	2	no	—
557	C₂×C₄	—	8	2	no	—
558	C₂×C₄	—	8	2	no	—
559	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
560	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
561	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
562	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
563	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
564	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
565	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
566	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
567	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
568	C₈	—	2	2	no	—
569	C₈	—	4	1	no	—
570	C₈	—	4	1	no	—
571	C₈	—	4	2	no	—
572	C₈	—	4	2	no	—
573	C₈	—	4	1	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	1	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	4	1	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	2	no	—
585	C₂²	—	4	1	no	—
586	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
587	C₄	—	2	1	no	—
588	C₄	—	2	1	no	—
589	C₄	—	2	2	no	—
590	C₄	—	2	1	no	—
591	C₄	—	2	2	no	—
592	C₄	—	2	1	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	1	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	1	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	1	no	—
599	C₄	C₃₁×#421	1	2	yes	—
600	C₂	—	4	1	no	—
601	C₂	—	4	2	no	—
602	C₂	—	8	2	no	—
603	C₂	—	8	1	no	—
604	C₂	C₃₁×#2888	1	1	yes	characteristic
605	C₂	C₂×C₃₁×#421	1	1	yes	characteristic
606	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×#3517	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×#2923	C₂	semidirect
4. C₃₁×#1536	C₂	semidirect
5. C₃₁×#1535	C₂	semidirect
6. C₃₁×#1532	C₂	semidirect
7. C₃₁×#1530	C₂	semidirect
8. C₃₁×#1530	C₂	semidirect
9. C₃₁×#1529	C₂	semidirect
10. C₃₁×#1525	C₂	semidirect
11. C₃₁×#1524	C₂	semidirect
220. #17684	C₃₁	direct (N × H)
272. #2970	C₂×C₃₁	semidirect
273. #2923	C₂×C₃₁	semidirect
274. #1536	C₂×C₃₁	semidirect
275. #1535	C₂×C₃₁	semidirect
276. #1532	C₂×C₃₁	semidirect
277. #1530	C₂×C₃₁	semidirect
278. #1530	C₂×C₃₁	semidirect
279. #1529	C₂×C₃₁	semidirect
280. #1525	C₂×C₃₁	semidirect
281. #1524	C₂×C₃₁	semidirect
443. C₃₁	#17684	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×#3517	2
C₃₁×#2970	2
C₃₁×#2923	2
C₃₁×#1536	2
C₃₁×#1535	2
C₃₁×#1532	2
C₃₁×#1530	2
C₃₁×#1529	2
C₃₁×#1525	2
C₃₁×#1524	2
#17684	31

Subgroups — C31×#17684 #21474443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17684 #21474443