Subgroups — C₃₁×#17519 #21474278

Order 7936.

1022 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×#3482	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1494	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×#1489	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1485	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1483	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1476	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×Q₈⋊C₈	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#557	C₄	1	1	yes	characteristic
14	C₃₁×#556	—	2	1	no	—
15	C₃₁×#556	C₄	1	1	yes	characteristic
16	C₃₁×#552	—	2	2	no	—
17	C₃₁×#552	C₄	1	2	yes	—
18	D₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
19	C₃₁×#548	C₄	1	2	yes	—
20	C₃₁×#547	—	2	1	no	—
21	C₃₁×#547	C₄	1	1	yes	characteristic
22	C₃₁×#533	—	2	1	no	—
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₃₁×#529	—	2	2	no	—
25	C₃₁×#529	C₂²	1	2	yes	—
26	C₄×#41×C₃₁	—	2	1	no	—
27	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
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	—
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₃₁×#423	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
37	C₃₁×#405	—	2	2	no	—
38	C₃₁×#405	—	2	1	no	—
39	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
40	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×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	2	yes	—
45	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
46	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
47	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
52	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₃₁	—	2	1	no	—
54	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
57	C₃₁×#131	—	2	8	no	—
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂×M₁₆×C₃₁	—	2	1	no	—
60	C₂×M₁₆×C₃₁	—	2	1	no	—
61	C₂²×C₈×C₃₁	—	2	2	no	—
62	C₂²×C₈×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₃₁×C₄⋊Q₈	—	2	1	no	—
65	C₃₁×C₄⋊Q₈	—	2	1	no	—
66	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
67	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
68	C₃₁×#126	—	2	2	no	—
69	C₃₁×#126	—	2	2	no	—
70	C₃₁×#126	—	2	2	no	—
71	C₃₁×#125	—	2	2	no	—
72	C₃₁×#125	—	2	2	no	—
73	C₃₁×#125	—	2	1	no	—
74	C₃₁×#125	C₂×C₄	1	2	yes	—
75	C₃₁×#124	—	2	2	no	—
76	C₃₁×#124	C₂×C₄	1	2	yes	—
77	C₃₁×#123	—	2	2	no	—
78	C₃₁×#123	—	2	2	no	—
79	C₃₁×#123	—	2	2	no	—
80	C₃₁×C₂²⋊Q₈	—	2	1	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	C₂×C₄	1	2	yes	—
86	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
87	C₃₁×#121	—	2	2	no	—
88	C₃₁×#121	C₂×C₄	1	2	yes	—
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	2	yes	—
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	1	yes	characteristic
97	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
98	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	—	2	2	no	—
102	C₄×D₄×C₃₁	—	2	1	no	—
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	1	yes	characteristic
107	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
108	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
109	C₄×D₄×C₃₁	D₄	1	2	yes	—
110	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₃₁×#117	—	2	1	no	—
112	C₃₁×#117	—	2	1	no	—
113	C₃₁×#117	—	2	2	no	—
114	C₃₁×#117	—	2	1	no	—
115	C₃₁×#117	—	2	2	no	—
116	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
117	C₃₁×#117	D₄	1	2	yes	—
118	C₃₁×#117	D₄	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
121	C₂×C₄²×C₃₁	—	2	2	no	—
122	C₂×C₄²×C₃₁	—	2	1	no	—
123	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
124	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
125	C₃₁×C₄⋊C₈	—	2	2	no	—
126	C₃₁×C₄⋊C₈	—	2	1	no	—
127	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
128	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
129	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
130	C₃₁×C₂²⋊C₈	—	2	1	no	—
131	C₃₁×C₂²⋊C₈	—	2	2	no	—
132	C₃₁×#97	—	2	1	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	—	2	1	no	—
138	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₃₁	C₂³	1	2	yes	—
140	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
141	#41×C₃₁	—	2	4	no	—
142	#41×C₃₁	—	2	2	no	—
143	#41×C₃₁	—	4	1	no	—
144	#41×C₃₁	—	4	4	no	—
145	C₂×Q₈×C₃₁	—	2	2	no	—
146	C₂×Q₈×C₃₁	—	2	1	no	—
147	C₂×Q₈×C₃₁	—	2	4	no	—
148	C₂×Q₈×C₃₁	—	4	2	no	—
149	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
150	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
151	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
152	C₂×D₄×C₃₁	—	2	1	no	—
153	C₂×D₄×C₃₁	—	2	2	no	—
154	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
155	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	—	2	2	no	—
158	C₂²×C₄×C₃₁	—	2	2	no	—
159	C₂²×C₄×C₃₁	—	2	1	no	—
160	C₂²×C₄×C₃₁	—	2	1	no	—
161	C₂²×C₄×C₃₁	—	2	1	no	—
162	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
163	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
168	M₁₆×C₃₁	—	2	8	no	—
169	M₁₆×C₃₁	—	2	4	no	—
170	M₁₆×C₃₁	—	8	1	no	—
171	C₂×C₈×C₃₁	—	2	8	no	—
172	C₂×C₈×C₃₁	—	2	4	no	—
173	C₂×C₈×C₃₁	—	2	1	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	2	no	—
179	C₂×C₈×C₃₁	—	8	2	no	—
180	C₄⋊C₄×C₃₁	—	2	2	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	2	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	2	no	—
189	C₄⋊C₄×C₃₁	—	2	1	no	—
190	C₄⋊C₄×C₃₁	—	2	2	no	—
191	C₄⋊C₄×C₃₁	—	2	1	no	—
192	C₄⋊C₄×C₃₁	—	2	2	no	—
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	2	yes	—
197	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
199	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
200	C₂²⋊C₄×C₃₁	—	2	2	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	2	no	—
207	C₂²⋊C₄×C₃₁	—	2	1	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	1	no	—
212	C₄²×C₃₁	—	2	2	no	—
213	C₄²×C₃₁	—	2	1	no	—
214	C₄²×C₃₁	—	2	2	no	—
215	C₄²×C₃₁	—	2	1	no	—
216	C₄²×C₃₁	—	2	2	no	—
217	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
218	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
219	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₃₁	C₂×D₄	1	2	yes	—
221	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
222	C₄²×C₃₁	C₂×D₄	1	2	yes	—
223	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
224	#17519	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂³×C₃₁	—	2	1	no	—
226	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
227	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
228	Q₈×C₃₁	—	2	1	no	—
229	Q₈×C₃₁	—	2	2	no	—
230	Q₈×C₃₁	—	2	2	no	—
231	Q₈×C₃₁	—	2	2	no	—
232	Q₈×C₃₁	—	4	4	no	—
233	Q₈×C₃₁	—	4	2	no	—
234	Q₈×C₃₁	#131	1	2	yes	complemented
235	D₄×C₃₁	—	2	4	no	—
236	D₄×C₃₁	—	2	1	no	—
237	D₄×C₃₁	—	2	1	no	—
238	D₄×C₃₁	—	4	2	no	—
239	D₄×C₃₁	—	4	4	no	—
240	D₄×C₃₁	#104	1	4	yes	—
241	C₂×C₄×C₃₁	—	2	1	no	—
242	C₂×C₄×C₃₁	—	2	4	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	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	1	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	1	no	—
253	C₂×C₄×C₃₁	—	2	1	no	—
254	C₂×C₄×C₃₁	—	2	2	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	4	2	no	—
257	C₂×C₄×C₃₁	—	4	4	no	—
258	C₂×C₄×C₃₁	—	4	4	no	—
259	C₂×C₄×C₃₁	—	4	1	no	—
260	C₂×C₄×C₃₁	—	4	1	no	—
261	C₂×C₄×C₃₁	—	4	2	no	—
262	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
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	2	yes	—
266	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
267	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
271	C₈×C₃₁	—	2	2	no	—
272	C₈×C₃₁	—	2	4	no	—
273	C₈×C₃₁	—	2	2	no	—
274	C₈×C₃₁	—	4	1	no	—
275	C₈×C₃₁	—	4	2	no	—
276	C₈×C₃₁	—	4	1	no	—
277	#3482	C₂×C₃₁	1	1	yes	characteristic
278	#2990	C₂×C₃₁	1	1	yes	characteristic, complemented
279	C₄×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
280	#1494	C₂×C₃₁	1	2	yes	complemented
281	#1493	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#1491	C₂×C₃₁	1	2	yes	complemented
283	#1489	C₂×C₃₁	1	1	yes	characteristic, complemented
284	#1485	C₂×C₃₁	1	1	yes	characteristic, complemented
285	#1483	C₂×C₃₁	1	2	yes	complemented
286	#1476	C₂×C₃₁	1	1	yes	characteristic, complemented
287	C₂×Q₈⋊C₈	C₂×C₃₁	1	2	yes	complemented
288	C₂²×C₃₁	—	2	2	no	—
289	C₂²×C₃₁	—	2	2	no	—
290	C₂²×C₃₁	—	2	1	no	—
291	C₂²×C₃₁	—	2	1	no	—
292	C₂²×C₃₁	—	2	1	no	—
293	C₂²×C₃₁	—	4	1	no	—
294	C₂²×C₃₁	—	4	1	no	—
295	C₂²×C₃₁	—	4	1	no	—
296	C₂²×C₃₁	C₂×#104	1	4	yes	—
297	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
298	C₄×C₃₁	—	2	1	no	—
299	C₄×C₃₁	—	2	2	no	—
300	C₄×C₃₁	—	2	2	no	—
301	C₄×C₃₁	—	2	2	no	—
302	C₄×C₃₁	—	2	1	no	—
303	C₄×C₃₁	—	2	1	no	—
304	C₄×C₃₁	—	2	2	no	—
305	C₄×C₃₁	—	4	2	no	—
306	C₄×C₃₁	—	4	1	no	—
307	C₄×C₃₁	—	4	1	no	—
308	C₄×C₃₁	—	4	2	no	—
309	C₄×C₃₁	C₂×#104	1	2	yes	—
310	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
311	C₄×C₃₁	#568	1	1	yes	characteristic
312	C₄×C₃₁	#577	1	1	yes	characteristic
313	C₄×C₃₁	#578	1	1	yes	characteristic
314	C₂×#131	—	2	1	no	—
315	#557	C₄×C₃₁	1	1	yes	characteristic
316	#556	—	2	1	no	—
317	#556	C₄×C₃₁	1	1	yes	characteristic
318	#552	—	2	2	no	—
319	#552	C₄×C₃₁	1	2	yes	—
320	D₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
321	#548	C₄×C₃₁	1	2	yes	—
322	#547	—	2	1	no	—
323	#547	C₄×C₃₁	1	1	yes	characteristic
324	#533	—	2	1	no	—
325	#533	C₂²×C₃₁	1	1	yes	characteristic
326	#529	—	2	2	no	—
327	#529	C₂²×C₃₁	1	2	yes	—
328	C₄×#41	—	2	1	no	—
329	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
330	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
332	#446	—	2	1	no	—
333	#435	—	2	2	no	—
334	C₈×D₄	—	2	1	no	—
335	#432	C₂²×C₃₁	1	2	yes	—
336	#431	C₂²×C₃₁	1	1	yes	characteristic
337	#423	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
339	#405	—	2	2	no	—
340	#405	—	2	1	no	—
341	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
342	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
343	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
344	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
345	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
346	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
347	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
348	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
349	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
350	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
351	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
352	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
353	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
354	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
355	C₂×C₃₁	—	2	4	no	—
356	C₂×C₃₁	—	4	1	no	—
357	C₂×C₃₁	—	8	1	no	—
358	C₂×C₃₁	#2880	1	1	yes	characteristic
359	C₂×C₃₁	#2898	1	1	yes	characteristic
360	C₂×C₃₁	C₂²×#104	1	1	yes	characteristic
361	C₂×#41	—	2	1	no	—
362	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
363	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
364	C₂²×Q₈	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	—	2	1	no	—
374	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
375	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
376	#126	—	2	2	no	—
377	#126	—	2	2	no	—
378	#126	—	2	2	no	—
379	#125	—	2	2	no	—
380	#125	—	2	2	no	—
381	#125	—	2	1	no	—
382	#125	C₂×C₄×C₃₁	1	2	yes	—
383	#124	—	2	2	no	—
384	#124	C₂×C₄×C₃₁	1	2	yes	—
385	#123	—	2	2	no	—
386	#123	—	2	2	no	—
387	#123	—	2	2	no	—
388	C₂²⋊Q₈	—	2	1	no	—
389	C₂²⋊Q₈	—	2	1	no	—
390	C₂²⋊Q₈	—	2	1	no	—
391	C₂²⋊Q₈	—	2	2	no	—
392	C₂²⋊Q₈	—	2	2	no	—
393	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
394	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
395	#121	—	2	2	no	—
396	#121	C₂×C₄×C₃₁	1	2	yes	—
397	C₄×Q₈	—	2	1	no	—
398	C₄×Q₈	—	2	2	no	—
399	C₄×Q₈	—	2	2	no	—
400	C₄×Q₈	—	2	1	no	—
401	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
402	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
403	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
404	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
405	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
406	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
407	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
408	C₄×D₄	—	2	1	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	1	no	—
411	C₄×D₄	—	2	1	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
414	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
415	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
416	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
417	C₄×D₄	D₄×C₃₁	1	2	yes	—
418	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
419	#117	—	2	1	no	—
420	#117	—	2	1	no	—
421	#117	—	2	2	no	—
422	#117	—	2	1	no	—
423	#117	—	2	2	no	—
424	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
425	#117	D₄×C₃₁	1	2	yes	—
426	#117	D₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄⋊C₄	—	2	2	no	—
428	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
429	C₂×C₄²	—	2	2	no	—
430	C₂×C₄²	—	2	1	no	—
431	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
432	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
433	C₄⋊C₈	—	2	2	no	—
434	C₄⋊C₈	—	2	1	no	—
435	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
436	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
437	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
438	C₂²⋊C₈	—	2	1	no	—
439	C₂²⋊C₈	—	2	2	no	—
440	#97	—	2	1	no	—
441	#97	—	2	2	no	—
442	#97	—	2	2	no	—
443	C₄×C₈	—	2	1	no	—
444	C₄×C₈	—	2	2	no	—
445	C₄×C₈	—	2	1	no	—
446	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
447	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
448	C₄×C₈	C₂³×C₃₁	1	2	yes	—
449	C₃₁	#17519	1	1	yes	characteristic, direct product factor, Sylow
450	#41	—	2	4	no	—
451	#41	—	2	2	no	—
452	#41	—	4	1	no	—
453	#41	—	4	4	no	—
454	C₂×Q₈	—	2	2	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	—	2	4	no	—
457	C₂×Q₈	—	4	2	no	—
458	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
459	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
460	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	2	no	—
463	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
464	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
465	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
472	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
474	C₂²×C₄	C₂×D₄×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	M₁₆	—	2	8	no	—
478	M₁₆	—	2	4	no	—
479	M₁₆	—	8	1	no	—
480	C₂×C₈	—	2	8	no	—
481	C₂×C₈	—	2	4	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	2	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	2	no	—
488	C₂×C₈	—	8	2	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	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
503	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
504	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
505	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
506	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
507	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
508	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	1	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	1	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
518	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
519	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
520	C₄²	—	2	1	no	—
521	C₄²	—	2	2	no	—
522	C₄²	—	2	1	no	—
523	C₄²	—	2	2	no	—
524	C₄²	—	2	1	no	—
525	C₄²	—	2	2	no	—
526	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
527	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
528	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
529	C₄²	C₂×D₄×C₃₁	1	2	yes	—
530	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
531	C₄²	C₂×D₄×C₃₁	1	2	yes	—
532	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
533	C₂³	—	2	1	no	—
534	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
535	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
536	Q₈	—	2	2	no	—
537	Q₈	—	2	1	no	—
538	Q₈	—	2	2	no	—
539	Q₈	—	2	2	no	—
540	Q₈	—	4	4	no	—
541	Q₈	—	4	2	no	—
542	Q₈	C₃₁×#131	1	2	yes	complemented
543	D₄	—	2	4	no	—
544	D₄	—	2	1	no	—
545	D₄	—	2	1	no	—
546	D₄	—	4	2	no	—
547	D₄	—	4	4	no	—
548	D₄	C₃₁×#104	1	4	yes	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	4	2	no	—
565	C₂×C₄	—	4	4	no	—
566	C₂×C₄	—	4	4	no	—
567	C₂×C₄	—	4	2	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
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	2	yes	—
574	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
575	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
576	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
577	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
578	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
579	C₈	—	2	2	no	—
580	C₈	—	2	4	no	—
581	C₈	—	2	2	no	—
582	C₈	—	4	1	no	—
583	C₈	—	4	2	no	—
584	C₈	—	4	1	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	1	no	—
588	C₂²	—	2	1	no	—
589	C₂²	—	2	1	no	—
590	C₂²	—	4	1	no	—
591	C₂²	—	4	1	no	—
592	C₂²	—	4	1	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	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	2	1	no	—
600	C₄	—	2	1	no	—
601	C₄	—	2	2	no	—
602	C₄	—	4	2	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	1	no	—
605	C₄	—	4	2	no	—
606	C₄	C₂×C₃₁×#104	1	2	yes	—
607	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
608	C₄	C₃₁×#568	1	1	yes	characteristic
609	C₄	C₃₁×#577	1	1	yes	characteristic
610	C₄	C₃₁×#578	1	1	yes	characteristic
611	C₂	—	2	4	no	—
612	C₂	—	4	1	no	—
613	C₂	—	8	1	no	—
614	C₂	C₃₁×#2880	1	1	yes	characteristic
615	C₂	C₃₁×#2898	1	1	yes	characteristic
616	C₂	C₂²×C₃₁×#104	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	C₃₁×#3482	C₂
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
2. C₃₁×#2990	C₂	semidirect
3. C₄×C₃₁×#131	C₂	semidirect
4. C₃₁×#1494	C₂	semidirect
5. C₃₁×#1493	C₂	semidirect
6. C₃₁×#1491	C₂	semidirect
7. C₃₁×#1489	C₂	semidirect
8. C₃₁×#1485	C₂	semidirect
9. C₃₁×#1483	C₂	semidirect
10. C₃₁×#1476	C₂	semidirect
11. C₂×C₃₁×Q₈⋊C₈	C₂	semidirect
224. #17519	C₃₁	direct (N × H)
234. Q₈×C₃₁	#131	semidirect
278. #2990	C₂×C₃₁	semidirect
279. C₄×#131	C₂×C₃₁	semidirect
280. #1494	C₂×C₃₁	semidirect
281. #1493	C₂×C₃₁	semidirect
282. #1491	C₂×C₃₁	semidirect
283. #1489	C₂×C₃₁	semidirect
284. #1485	C₂×C₃₁	semidirect
285. #1483	C₂×C₃₁	semidirect
286. #1476	C₂×C₃₁	semidirect
287. C₂×Q₈⋊C₈	C₂×C₃₁	semidirect
449. C₃₁	#17519	direct (N × H)
542. Q₈	C₃₁×#131	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3482	2
C₃₁×#2990	2
C₄×C₃₁×#131	2
C₃₁×#1494	2
C₃₁×#1493	2
C₃₁×#1491	2
C₃₁×#1489	2
C₃₁×#1485	2
C₃₁×#1483	2
C₃₁×#1476	2
C₂×C₃₁×Q₈⋊C₈	2
#17519	31

Subgroups — C31×#17519 #21474278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17519 #21474278