Subgroups — C₃₁×#17683 #21474442

Order 7936.

876 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3518	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×#1537	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×#1531	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1531	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1529	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1526	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1524	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈²×C₃₁	C₄	1	1	yes	characteristic
13	C₃₁×#556	—	2	1	no	—
14	C₃₁×#556	—	2	2	no	—
15	C₃₁×#556	C₄	1	2	yes	—
16	C₃₁×#556	C₄	1	1	yes	characteristic
17	C₃₁×#555	C₄	1	2	yes	—
18	C₃₁×#555	C₄	1	1	yes	characteristic
19	D₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
20	C₃₁×#548	—	2	1	no	—
21	C₃₁×#542	—	2	1	no	—
22	C₃₁×#542	—	2	2	no	—
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₃₁×#533	C₂²	1	2	yes	—
25	C₃₁×#529	C₂²	1	1	yes	characteristic
26	C₃₁×#529	C₂²	1	2	yes	—
27	C₃₁×#519	—	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×#330	C₂²	1	2	yes	—
47	C₃₁×#330	C₂²	1	2	yes	—
48	C₃₁×#330	C₂²	1	2	yes	—
49	C₃₁×#330	C₂²	1	2	yes	—
50	C₃₁×#329	C₂²	1	1	yes	characteristic
51	C₃₁×#329	C₂²	1	1	yes	characteristic
52	C₃₁×#329	C₂²	1	2	yes	—
53	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
54	C₂²×Q₈×C₃₁	—	2	1	no	—
55	C₂×M₁₆×C₃₁	—	2	1	no	—
56	C₂×M₁₆×C₃₁	—	2	1	no	—
57	C₂²×C₈×C₃₁	—	2	1	no	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
63	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
65	C₃₁×#126	—	2	2	no	—
66	C₃₁×#126	—	2	2	no	—
67	C₃₁×#126	—	2	2	no	—
68	C₃₁×#126	—	2	1	no	—
69	C₃₁×#126	—	2	2	no	—
70	C₃₁×#125	C₂³	1	1	yes	characteristic
71	C₃₁×#125	C₂³	1	2	yes	—
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	—	2	2	no	—
77	C₃₁×#124	C₂³	1	1	yes	characteristic
78	C₃₁×#123	—	2	2	no	—
79	C₃₁×#123	—	2	1	no	—
80	C₃₁×C₂²⋊Q₈	—	2	2	no	—
81	C₃₁×C₂²⋊Q₈	—	2	1	no	—
82	C₃₁×C₂²⋊Q₈	—	2	1	no	—
83	C₃₁×C₂²⋊Q₈	—	2	2	no	—
84	C₃₁×C₂²⋊Q₈	—	2	2	no	—
85	C₃₁×C₂²⋊Q₈	—	2	1	no	—
86	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
87	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
88	C₃₁×#121	—	2	1	no	—
89	C₃₁×#121	—	2	2	no	—
90	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
91	C₃₁×#121	C₂×C₄	1	2	yes	—
92	C₄×Q₈×C₃₁	—	2	1	no	—
93	C₄×Q₈×C₃₁	—	2	2	no	—
94	C₄×Q₈×C₃₁	—	2	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	—	2	2	no	—
97	C₄×Q₈×C₃₁	—	2	1	no	—
98	C₄×Q₈×C₃₁	—	2	2	no	—
99	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
100	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
101	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
102	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
103	C₄×D₄×C₃₁	—	2	1	no	—
104	C₄×D₄×C₃₁	—	2	2	no	—
105	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
106	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
107	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
108	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
109	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
110	C₄×D₄×C₃₁	D₄	1	2	yes	—
111	C₃₁×#117	—	2	2	no	—
112	C₃₁×#117	—	2	1	no	—
113	C₃₁×#117	—	2	1	no	—
114	C₃₁×#117	—	2	2	no	—
115	C₃₁×#117	C₂×C₄	1	2	yes	—
116	C₃₁×#117	C₂×C₄	1	2	yes	—
117	C₃₁×#117	C₂×C₄	1	2	yes	—
118	C₃₁×#117	D₄	1	1	yes	characteristic
119	C₃₁×#117	D₄	1	2	yes	—
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	C₂×Q₈×C₃₁	—	2	2	no	—
142	C₂×Q₈×C₃₁	—	2	1	no	—
143	C₂×Q₈×C₃₁	—	2	1	no	—
144	C₂×Q₈×C₃₁	—	2	2	no	—
145	C₂×Q₈×C₃₁	—	4	4	no	—
146	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
147	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₂×D₄×C₃₁	—	2	1	no	—
149	C₂×D₄×C₃₁	—	2	2	no	—
150	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
151	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
152	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	—	2	2	no	—
154	C₂²×C₄×C₃₁	—	2	1	no	—
155	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	M₁₆×C₃₁	—	4	2	no	—
162	M₁₆×C₃₁	—	4	2	no	—
163	C₂×C₈×C₃₁	—	2	1	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	—	2	1	no	—
166	C₂×C₈×C₃₁	—	2	1	no	—
167	C₂×C₈×C₃₁	—	2	1	no	—
168	C₂×C₈×C₃₁	—	2	2	no	—
169	C₂×C₈×C₃₁	—	4	4	no	—
170	C₂×C₈×C₃₁	—	4	2	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	—	2	1	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	2	no	—
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	—	2	2	no	—
183	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
184	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
185	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
186	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
187	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
188	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
189	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
190	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
191	C₂²⋊C₄×C₃₁	—	2	2	no	—
192	C₂²⋊C₄×C₃₁	—	2	2	no	—
193	C₂²⋊C₄×C₃₁	—	2	2	no	—
194	C₂²⋊C₄×C₃₁	—	2	1	no	—
195	C₂²⋊C₄×C₃₁	—	2	2	no	—
196	C₂²⋊C₄×C₃₁	—	2	1	no	—
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	1	no	—
202	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
203	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
204	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
205	C₄²×C₃₁	—	2	2	no	—
206	C₄²×C₃₁	—	2	2	no	—
207	C₄²×C₃₁	—	2	2	no	—
208	C₄²×C₃₁	—	2	2	no	—
209	C₄²×C₃₁	—	2	1	no	—
210	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
211	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₃₁	C₂×D₄	1	2	yes	—
215	C₄²×C₃₁	C₂×D₄	1	2	yes	—
216	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
217	#17683	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂³×C₃₁	—	2	1	no	—
219	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
221	Q₈×C₃₁	—	2	2	no	—
222	Q₈×C₃₁	—	4	1	no	—
223	Q₈×C₃₁	—	4	4	no	—
224	Q₈×C₃₁	—	4	2	no	—
225	Q₈×C₃₁	—	8	2	no	—
226	Q₈×C₃₁	—	8	1	no	—
227	D₄×C₃₁	—	2	2	no	—
228	D₄×C₃₁	—	4	2	no	—
229	D₄×C₃₁	—	4	1	no	—
230	D₄×C₃₁	—	8	2	no	—
231	D₄×C₃₁	—	8	1	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	1	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	1	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	2	1	no	—
238	C₂×C₄×C₃₁	—	4	2	no	—
239	C₂×C₄×C₃₁	—	4	1	no	—
240	C₂×C₄×C₃₁	—	4	1	no	—
241	C₂×C₄×C₃₁	—	4	2	no	—
242	C₂×C₄×C₃₁	—	4	2	no	—
243	C₂×C₄×C₃₁	—	4	4	no	—
244	C₂×C₄×C₃₁	—	4	2	no	—
245	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
247	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
248	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
249	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
250	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
252	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
254	C₈×C₃₁	—	2	2	no	—
255	C₈×C₃₁	—	4	1	no	—
256	C₈×C₃₁	—	4	1	no	—
257	C₈×C₃₁	—	4	2	no	—
258	C₈×C₃₁	—	4	2	no	—
259	C₈×C₃₁	—	4	1	no	—
260	#3518	C₂×C₃₁	1	1	yes	characteristic
261	#2970	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#2923	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#1537	C₂×C₃₁	1	2	yes	complemented
264	#1535	C₂×C₃₁	1	2	yes	complemented
265	#1532	C₂×C₃₁	1	2	yes	complemented
266	#1531	C₂×C₃₁	1	2	yes	complemented
267	#1531	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#1529	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1526	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#1524	C₂×C₃₁	1	1	yes	characteristic, complemented
271	C₂²×C₃₁	—	2	1	no	—
272	C₂²×C₃₁	—	2	2	no	—
273	C₂²×C₃₁	—	2	1	no	—
274	C₂²×C₃₁	—	2	2	no	—
275	C₂²×C₃₁	—	2	2	no	—
276	C₂²×C₃₁	—	2	1	no	—
277	C₂²×C₃₁	—	4	1	no	—
278	C₂²×C₃₁	—	4	1	no	—
279	C₂²×C₃₁	—	4	1	no	—
280	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
281	C₄×C₃₁	—	2	1	no	—
282	C₄×C₃₁	—	2	1	no	—
283	C₄×C₃₁	—	2	2	no	—
284	C₄×C₃₁	—	2	1	no	—
285	C₄×C₃₁	—	2	2	no	—
286	C₄×C₃₁	—	2	1	no	—
287	C₄×C₃₁	—	2	2	no	—
288	C₄×C₃₁	—	2	1	no	—
289	C₄×C₃₁	—	4	1	no	—
290	C₄×C₃₁	—	4	2	no	—
291	C₄×C₃₁	—	4	1	no	—
292	C₄×C₃₁	—	4	2	no	—
293	C₄×C₃₁	—	4	1	no	—
294	C₄×C₃₁	#421	1	2	yes	—
295	Q₈²	C₄×C₃₁	1	1	yes	characteristic
296	#556	—	2	1	no	—
297	#556	—	2	2	no	—
298	#556	C₄×C₃₁	1	1	yes	characteristic
299	#556	C₄×C₃₁	1	2	yes	—
300	#555	C₄×C₃₁	1	1	yes	characteristic
301	#555	C₄×C₃₁	1	2	yes	—
302	D₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
303	#548	—	2	1	no	—
304	#542	—	2	1	no	—
305	#542	—	2	2	no	—
306	#533	C₂²×C₃₁	1	2	yes	—
307	#533	C₂²×C₃₁	1	1	yes	characteristic
308	#529	C₂²×C₃₁	1	1	yes	characteristic
309	#529	C₂²×C₃₁	1	2	yes	—
310	#519	—	2	1	no	—
311	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
312	#446	—	2	1	no	—
313	#435	—	2	2	no	—
314	C₈×D₄	—	2	1	no	—
315	#433	—	2	2	no	—
316	#432	C₂²×C₃₁	1	2	yes	—
317	#432	C₂²×C₃₁	1	2	yes	—
318	#432	C₂²×C₃₁	1	2	yes	—
319	#432	C₂²×C₃₁	1	1	yes	characteristic
320	#424	—	2	1	no	—
321	#408	—	2	1	no	—
322	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
323	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
324	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×#97	C₂²×C₃₁	1	2	yes	—
326	#330	C₂²×C₃₁	1	1	yes	characteristic
327	#330	C₂²×C₃₁	1	2	yes	—
328	#330	C₂²×C₃₁	1	2	yes	—
329	#330	C₂²×C₃₁	1	2	yes	—
330	#330	C₂²×C₃₁	1	1	yes	characteristic
331	#330	C₂²×C₃₁	1	2	yes	—
332	#330	C₂²×C₃₁	1	2	yes	—
333	#329	C₂²×C₃₁	1	1	yes	characteristic
334	#329	C₂²×C₃₁	1	1	yes	characteristic
335	#329	C₂²×C₃₁	1	2	yes	—
336	C₂×C₃₁	—	4	1	no	—
337	C₂×C₃₁	—	4	2	no	—
338	C₂×C₃₁	—	8	1	no	—
339	C₂×C₃₁	#2889	1	1	yes	characteristic
340	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
341	C₂×C₃₁	#2901	1	1	yes	characteristic
342	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
343	C₂²×Q₈	—	2	1	no	—
344	C₂×M₁₆	—	2	1	no	—
345	C₂×M₁₆	—	2	1	no	—
346	C₂²×C₈	—	2	1	no	—
347	C₂²×C₈	—	2	2	no	—
348	C₄⋊Q₈	—	2	2	no	—
349	C₄⋊Q₈	—	2	1	no	—
350	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
351	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
352	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
353	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
354	#126	—	2	2	no	—
355	#126	—	2	2	no	—
356	#126	—	2	2	no	—
357	#126	—	2	1	no	—
358	#126	—	2	2	no	—
359	#125	C₂³×C₃₁	1	1	yes	characteristic
360	#125	C₂³×C₃₁	1	2	yes	—
361	#124	—	2	1	no	—
362	#124	—	2	2	no	—
363	#124	—	2	2	no	—
364	#124	—	2	2	no	—
365	#124	—	2	2	no	—
366	#124	C₂³×C₃₁	1	1	yes	characteristic
367	#123	—	2	2	no	—
368	#123	—	2	1	no	—
369	C₂²⋊Q₈	—	2	2	no	—
370	C₂²⋊Q₈	—	2	1	no	—
371	C₂²⋊Q₈	—	2	1	no	—
372	C₂²⋊Q₈	—	2	2	no	—
373	C₂²⋊Q₈	—	2	2	no	—
374	C₂²⋊Q₈	—	2	1	no	—
375	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
376	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
377	#121	—	2	1	no	—
378	#121	—	2	2	no	—
379	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
380	#121	C₂×C₄×C₃₁	1	2	yes	—
381	C₄×Q₈	—	2	1	no	—
382	C₄×Q₈	—	2	2	no	—
383	C₄×Q₈	—	2	1	no	—
384	C₄×Q₈	—	2	1	no	—
385	C₄×Q₈	—	2	2	no	—
386	C₄×Q₈	—	2	1	no	—
387	C₄×Q₈	—	2	2	no	—
388	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
389	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
390	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
391	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
392	C₄×D₄	—	2	1	no	—
393	C₄×D₄	—	2	2	no	—
394	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
395	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
396	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
397	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
398	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
399	C₄×D₄	D₄×C₃₁	1	2	yes	—
400	#117	—	2	2	no	—
401	#117	—	2	1	no	—
402	#117	—	2	1	no	—
403	#117	—	2	2	no	—
404	#117	C₂×C₄×C₃₁	1	2	yes	—
405	#117	C₂×C₄×C₃₁	1	2	yes	—
406	#117	C₂×C₄×C₃₁	1	2	yes	—
407	#117	D₄×C₃₁	1	1	yes	characteristic
408	#117	D₄×C₃₁	1	2	yes	—
409	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
410	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
411	C₄⋊C₈	—	2	2	no	—
412	C₄⋊C₈	—	2	2	no	—
413	C₄⋊C₈	—	2	1	no	—
414	C₄⋊C₈	—	2	2	no	—
415	C₂²⋊C₈	—	2	2	no	—
416	C₂²⋊C₈	—	2	2	no	—
417	C₂²⋊C₈	—	2	2	no	—
418	C₂²⋊C₈	—	2	1	no	—
419	#97	—	2	1	no	—
420	#97	—	2	2	no	—
421	#97	C₂³×C₃₁	1	1	yes	characteristic
422	#97	C₂³×C₃₁	1	1	yes	characteristic
423	#97	C₂³×C₃₁	1	1	yes	characteristic
424	#97	C₂³×C₃₁	1	2	yes	—
425	#97	C₂³×C₃₁	1	1	yes	characteristic
426	#97	C₂³×C₃₁	1	2	yes	—
427	C₄×C₈	—	2	1	no	—
428	C₄×C₈	—	2	1	no	—
429	C₃₁	#17683	1	1	yes	characteristic, direct product factor, Sylow
430	#41	—	4	2	no	—
431	C₂×Q₈	—	2	2	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	2	2	no	—
435	C₂×Q₈	—	4	4	no	—
436	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
437	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
441	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
442	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
447	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
449	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
450	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
451	M₁₆	—	4	2	no	—
452	M₁₆	—	4	2	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	—	4	4	no	—
460	C₂×C₈	—	4	2	no	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
474	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
475	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
476	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
477	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
478	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
479	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
480	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
481	C₂²⋊C₄	—	2	2	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₂²⋊C₄	—	2	2	no	—
484	C₂²⋊C₄	—	2	1	no	—
485	C₂²⋊C₄	—	2	2	no	—
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₂²⋊C₄	—	2	1	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	2	1	no	—
491	C₂²⋊C₄	—	2	1	no	—
492	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
493	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
494	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
495	C₄²	—	2	2	no	—
496	C₄²	—	2	2	no	—
497	C₄²	—	2	2	no	—
498	C₄²	—	2	2	no	—
499	C₄²	—	2	1	no	—
500	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
501	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
502	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
503	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
504	C₄²	C₂×D₄×C₃₁	1	2	yes	—
505	C₄²	C₂×D₄×C₃₁	1	2	yes	—
506	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
507	C₂³	—	2	1	no	—
508	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
509	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
510	Q₈	—	2	2	no	—
511	Q₈	—	4	1	no	—
512	Q₈	—	4	4	no	—
513	Q₈	—	4	2	no	—
514	Q₈	—	8	2	no	—
515	Q₈	—	8	1	no	—
516	D₄	—	2	2	no	—
517	D₄	—	4	2	no	—
518	D₄	—	4	1	no	—
519	D₄	—	8	2	no	—
520	D₄	—	8	1	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	4	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
535	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
536	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
537	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
538	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
539	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
540	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
543	C₈	—	2	2	no	—
544	C₈	—	4	1	no	—
545	C₈	—	4	1	no	—
546	C₈	—	4	2	no	—
547	C₈	—	4	2	no	—
548	C₈	—	4	1	no	—
549	C₂²	—	2	1	no	—
550	C₂²	—	2	2	no	—
551	C₂²	—	2	1	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	1	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	4	1	no	—
557	C₂²	—	4	1	no	—
558	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
559	C₄	—	2	1	no	—
560	C₄	—	2	1	no	—
561	C₄	—	2	2	no	—
562	C₄	—	2	1	no	—
563	C₄	—	2	2	no	—
564	C₄	—	2	1	no	—
565	C₄	—	2	2	no	—
566	C₄	—	2	1	no	—
567	C₄	—	4	1	no	—
568	C₄	—	4	2	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	2	no	—
571	C₄	—	4	1	no	—
572	C₄	C₃₁×#421	1	2	yes	—
573	C₂	—	4	1	no	—
574	C₂	—	4	2	no	—
575	C₂	—	8	1	no	—
576	C₂	C₃₁×#2889	1	1	yes	characteristic
577	C₂	C₂×C₃₁×#421	1	1	yes	characteristic
578	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×#3518	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×#1537	C₂	semidirect
5. C₃₁×#1535	C₂	semidirect
6. C₃₁×#1532	C₂	semidirect
7. C₃₁×#1531	C₂	semidirect
8. C₃₁×#1531	C₂	semidirect
9. C₃₁×#1529	C₂	semidirect
10. C₃₁×#1526	C₂	semidirect
11. C₃₁×#1524	C₂	semidirect
217. #17683	C₃₁	direct (N × H)
261. #2970	C₂×C₃₁	semidirect
262. #2923	C₂×C₃₁	semidirect
263. #1537	C₂×C₃₁	semidirect
264. #1535	C₂×C₃₁	semidirect
265. #1532	C₂×C₃₁	semidirect
266. #1531	C₂×C₃₁	semidirect
267. #1531	C₂×C₃₁	semidirect
268. #1529	C₂×C₃₁	semidirect
269. #1526	C₂×C₃₁	semidirect
270. #1524	C₂×C₃₁	semidirect
429. C₃₁	#17683	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×#3518	2
C₃₁×#2970	2
C₃₁×#2923	2
C₃₁×#1537	2
C₃₁×#1535	2
C₃₁×#1532	2
C₃₁×#1531	2
C₃₁×#1529	2
C₃₁×#1526	2
C₃₁×#1524	2
#17683	31

Subgroups — C31×#17683 #21474442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17683 #21474442