Subgroups — C₃₁×#17704 #21474463

Order 7936.

1298 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1547	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1546	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1545	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1540	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₃₁	—	2	1	no	—
9	C₂²×M₁₆×C₃₁	—	2	1	no	—
10	C₃₁×#532	—	2	1	no	—
11	C₃₁×#532	—	2	1	no	—
12	C₃₁×#532	—	2	1	no	—
13	C₃₁×#532	—	2	1	no	—
14	C₃₁×#532	C₄	1	2	yes	—
15	C₃₁×#532	C₄	1	2	yes	—
16	C₃₁×#532	C₂²	1	2	yes	—
17	C₃₁×#532	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#124	—	2	1	no	—
21	C₂×C₃₁×#121	—	2	1	no	—
22	C₂×C₃₁×#121	C₄	1	2	yes	—
23	C₂×C₄×D₄×C₃₁	—	2	1	no	—
24	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	—
25	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#431	—	2	1	no	—
27	C₃₁×#431	—	2	1	no	—
28	C₃₁×#431	C₂²	1	2	yes	—
29	C₃₁×#431	C₂²	1	2	yes	complemented
30	C₃₁×#423	—	2	1	no	—
31	C₃₁×#423	—	2	1	no	—
32	C₃₁×#423	C₂²	1	2	yes	—
33	C₃₁×#423	C₂²	1	2	yes	complemented
34	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
35	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
37	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
38	C₂×C₄×C₈×C₃₁	—	2	1	no	—
39	C₃₁×#332	C₂²	1	8	yes	complemented
40	C₃₁×#331	C₂²	1	8	yes	complemented
41	C₂×#41×C₃₁	—	2	2	no	—
42	C₂×#41×C₃₁	—	2	2	no	—
43	C₂×#41×C₃₁	—	2	2	no	—
44	C₂×#41×C₃₁	—	2	2	no	—
45	C₂×#41×C₃₁	—	4	4	no	—
46	C₂×#41×C₃₁	—	4	4	no	—
47	C₂²×Q₈×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	—	2	2	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
52	C₂³×C₄×C₃₁	—	2	2	no	—
53	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	2	no	—
58	C₂×M₁₆×C₃₁	—	4	4	no	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₂²×C₈×C₃₁	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	—	2	1	no	—
63	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
64	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₃₁×#127	—	2	1	no	—
66	C₃₁×#127	—	2	1	no	—
67	C₃₁×#127	C₂×C₄	1	2	yes	—
68	C₃₁×#127	C₂³	1	2	yes	—
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×#124	—	2	2	no	—
71	C₃₁×#124	—	2	2	no	—
72	C₃₁×#124	—	2	2	no	—
73	C₃₁×#121	—	2	2	no	—
74	C₃₁×#121	—	2	2	no	—
75	C₃₁×#121	—	2	2	no	—
76	C₃₁×#121	—	2	2	no	—
77	C₃₁×#121	—	2	2	no	—
78	C₃₁×#121	—	2	2	no	—
79	C₃₁×#121	C₂×C₄	1	4	yes	—
80	C₃₁×#121	C₂×C₄	1	4	yes	—
81	C₄×D₄×C₃₁	—	2	2	no	—
82	C₄×D₄×C₃₁	—	2	2	no	—
83	C₄×D₄×C₃₁	—	2	2	no	—
84	C₄×D₄×C₃₁	—	2	2	no	—
85	C₄×D₄×C₃₁	—	2	2	no	—
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
88	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
89	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
90	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
91	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
93	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
105	C₃₁×C₄⋊C₈	—	2	2	no	—
106	C₃₁×C₄⋊C₈	—	2	2	no	—
107	C₃₁×C₄⋊C₈	—	2	2	no	—
108	C₃₁×C₄⋊C₈	—	2	2	no	—
109	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
110	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
111	C₃₁×C₂²⋊C₈	—	2	2	no	—
112	C₃₁×C₂²⋊C₈	—	2	2	no	—
113	C₄×C₈×C₃₁	—	2	2	no	—
114	C₄×C₈×C₃₁	—	2	2	no	—
115	C₂⁴×C₃₁	—	2	2	no	—
116	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
117	#41×C₃₁	—	4	8	no	—
118	#41×C₃₁	—	4	8	no	—
119	#41×C₃₁	—	4	4	no	—
120	#41×C₃₁	—	4	4	no	—
121	#41×C₃₁	—	4	4	no	—
122	#41×C₃₁	—	4	4	no	—
123	C₂×Q₈×C₃₁	—	2	2	no	—
124	C₂×Q₈×C₃₁	—	2	2	no	—
125	C₂×Q₈×C₃₁	—	4	2	no	—
126	C₂×Q₈×C₃₁	—	4	2	no	—
127	C₂×D₄×C₃₁	—	2	4	no	—
128	C₂×D₄×C₃₁	—	2	4	no	—
129	C₂×D₄×C₃₁	—	2	2	no	—
130	C₂×D₄×C₃₁	—	2	2	no	—
131	C₂×D₄×C₃₁	—	2	2	no	—
132	C₂×D₄×C₃₁	—	2	2	no	—
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	4	4	no	—
135	C₂×D₄×C₃₁	—	4	4	no	—
136	C₂×D₄×C₃₁	—	4	8	no	—
137	C₂×D₄×C₃₁	—	4	2	no	—
138	C₂×D₄×C₃₁	—	4	2	no	—
139	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₃₁	—	2	2	no	—
141	C₂²×C₄×C₃₁	—	2	4	no	—
142	C₂²×C₄×C₃₁	—	2	4	no	—
143	C₂²×C₄×C₃₁	—	4	2	no	—
144	C₂²×C₄×C₃₁	—	4	2	no	—
145	C₂²×C₄×C₃₁	—	4	8	no	—
146	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	D₈	1	4	yes	—
156	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
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	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂×D₄	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	M₁₆×C₃₁	—	4	4	no	—
167	M₁₆×C₃₁	—	4	4	no	—
168	C₂×C₈×C₃₁	—	2	4	no	—
169	C₂×C₈×C₃₁	—	2	4	no	—
170	C₂×C₈×C₃₁	—	4	4	no	—
171	C₂×C₈×C₃₁	—	4	4	no	—
172	C₄⋊C₄×C₃₁	—	2	2	no	—
173	C₄⋊C₄×C₃₁	—	2	2	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
176	C₂²⋊C₄×C₃₁	—	2	4	no	—
177	C₂²⋊C₄×C₃₁	—	2	4	no	—
178	C₂²⋊C₄×C₃₁	—	2	4	no	—
179	C₂²⋊C₄×C₃₁	—	2	4	no	—
180	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
181	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
182	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
183	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
184	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
185	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
186	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
187	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
188	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
189	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
196	#17704	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	4	no	—
198	C₂³×C₃₁	—	4	1	no	—
199	C₂³×C₃₁	—	4	1	no	—
200	C₂³×C₃₁	—	4	4	no	—
201	C₂³×C₃₁	—	4	4	no	—
202	C₂³×C₃₁	—	4	4	no	—
203	C₂³×C₃₁	#102	1	4	yes	—
204	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
205	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
206	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
207	Q₈×C₃₁	—	4	2	no	—
208	Q₈×C₃₁	—	4	2	no	—
209	Q₈×C₃₁	—	4	4	no	—
210	D₄×C₃₁	—	4	4	no	—
211	D₄×C₃₁	—	4	4	no	—
212	D₄×C₃₁	—	4	8	no	—
213	D₄×C₃₁	—	4	8	no	—
214	D₄×C₃₁	—	4	2	no	—
215	D₄×C₃₁	—	4	2	no	—
216	D₄×C₃₁	—	4	4	no	—
217	D₄×C₃₁	—	4	4	no	—
218	D₄×C₃₁	—	4	4	no	—
219	C₂×C₄×C₃₁	—	2	2	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	1	no	—
229	C₂×C₄×C₃₁	—	4	1	no	—
230	C₂×C₄×C₃₁	—	4	1	no	—
231	C₂×C₄×C₃₁	—	4	1	no	—
232	C₂×C₄×C₃₁	—	4	4	no	—
233	C₂×C₄×C₃₁	—	4	8	no	—
234	C₂×C₄×C₃₁	—	4	8	no	—
235	C₂×C₄×C₃₁	#102	1	4	yes	—
236	C₂×C₄×C₃₁	#102	1	4	yes	—
237	C₂×C₄×C₃₁	#102	1	4	yes	—
238	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
247	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
248	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
249	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
250	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
251	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
252	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
253	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
256	C₈×C₃₁	—	4	4	no	—
257	C₈×C₃₁	—	4	4	no	—
258	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
259	C₂×#431	C₂×C₃₁	1	1	yes	characteristic, complemented
260	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1547	C₂×C₃₁	1	4	yes	complemented
262	#1546	C₂×C₃₁	1	2	yes	complemented
263	#1545	C₂×C₃₁	1	2	yes	complemented
264	#1540	C₂×C₃₁	1	4	yes	complemented
265	C₂²×C₃₁	—	2	1	no	—
266	C₂²×C₃₁	—	2	1	no	—
267	C₂²×C₃₁	—	2	1	no	—
268	C₂²×C₃₁	—	2	1	no	—
269	C₂²×C₃₁	—	2	1	no	—
270	C₂²×C₃₁	—	2	1	no	—
271	C₂²×C₃₁	—	4	4	no	—
272	C₂²×C₃₁	—	4	4	no	—
273	C₂²×C₃₁	—	4	4	no	—
274	C₂²×C₃₁	—	4	4	no	—
275	C₂²×C₃₁	—	8	8	no	—
276	C₂²×C₃₁	C₂×#102	1	2	yes	—
277	C₂²×C₃₁	C₂×#102	1	2	yes	—
278	C₂²×C₃₁	C₂×#102	1	2	yes	—
279	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
280	C₄×C₃₁	—	2	1	no	—
281	C₄×C₃₁	—	2	1	no	—
282	C₄×C₃₁	—	2	1	no	—
283	C₄×C₃₁	—	2	1	no	—
284	C₄×C₃₁	—	2	1	no	—
285	C₄×C₃₁	—	2	1	no	—
286	C₄×C₃₁	—	4	4	no	—
287	C₄×C₃₁	#413	1	2	yes	—
288	C₄×C₃₁	C₂×#102	1	2	yes	—
289	C₄×C₃₁	C₂×#102	1	2	yes	—
290	C₄×C₃₁	C₂×#102	1	2	yes	—
291	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
292	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
293	C₄×C₃₁	#575	1	2	yes	—
294	C₂²×#41	—	2	1	no	—
295	C₂²×M₁₆	—	2	1	no	—
296	#532	—	2	1	no	—
297	#532	—	2	1	no	—
298	#532	—	2	1	no	—
299	#532	—	2	1	no	—
300	#532	C₄×C₃₁	1	2	yes	—
301	#532	C₄×C₃₁	1	2	yes	—
302	#532	C₂²×C₃₁	1	2	yes	—
303	#532	C₂²×C₃₁	1	2	yes	—
304	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
305	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
306	C₂×#124	—	2	1	no	—
307	C₂×#121	—	2	1	no	—
308	C₂×#121	C₄×C₃₁	1	2	yes	—
309	C₂×C₄×D₄	—	2	1	no	—
310	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
311	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
312	#431	—	2	1	no	—
313	#431	—	2	1	no	—
314	#431	C₂²×C₃₁	1	2	yes	—
315	#431	C₂²×C₃₁	1	2	yes	complemented
316	#423	—	2	1	no	—
317	#423	—	2	1	no	—
318	#423	C₂²×C₃₁	1	2	yes	—
319	#423	C₂²×C₃₁	1	2	yes	complemented
320	C₂×C₄⋊C₈	—	2	1	no	—
321	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
322	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
323	C₂×C₂²⋊C₈	—	2	1	no	—
324	C₂×C₄×C₈	—	2	1	no	—
325	#332	C₂²×C₃₁	1	8	yes	complemented
326	#331	C₂²×C₃₁	1	8	yes	complemented
327	C₂×C₃₁	—	2	2	no	—
328	C₂×C₃₁	—	2	2	no	—
329	C₂×C₃₁	—	2	2	no	—
330	C₂×C₃₁	—	8	4	no	—
331	C₂×C₃₁	C₂×#413	1	1	yes	characteristic
332	C₂×C₃₁	C₂²×#102	1	1	yes	characteristic
333	C₂×C₃₁	#2896	1	1	yes	characteristic
334	C₂×#41	—	2	2	no	—
335	C₂×#41	—	2	2	no	—
336	C₂×#41	—	2	2	no	—
337	C₂×#41	—	2	2	no	—
338	C₂×#41	—	4	4	no	—
339	C₂×#41	—	4	4	no	—
340	C₂²×Q₈	—	2	1	no	—
341	C₂²×D₄	—	2	2	no	—
342	C₂²×D₄	—	2	1	no	—
343	C₂²×D₄	—	2	1	no	—
344	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
345	C₂³×C₄	—	2	2	no	—
346	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
347	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
348	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
349	C₂×M₁₆	—	2	2	no	—
350	C₂×M₁₆	—	2	2	no	—
351	C₂×M₁₆	—	4	4	no	—
352	C₂²×C₈	—	2	2	no	—
353	C₂²×C₈	—	2	2	no	—
354	C₄⋊Q₈	—	2	1	no	—
355	C₄⋊Q₈	—	2	1	no	—
356	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
357	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
358	#127	—	2	1	no	—
359	#127	—	2	1	no	—
360	#127	C₂×C₄×C₃₁	1	2	yes	—
361	#127	C₂³×C₃₁	1	2	yes	—
362	#124	—	2	2	no	—
363	#124	—	2	2	no	—
364	#124	—	2	2	no	—
365	#124	—	2	2	no	—
366	#121	—	2	2	no	—
367	#121	—	2	2	no	—
368	#121	—	2	2	no	—
369	#121	—	2	2	no	—
370	#121	—	2	2	no	—
371	#121	—	2	2	no	—
372	#121	C₂×C₄×C₃₁	1	4	yes	—
373	#121	C₂×C₄×C₃₁	1	4	yes	—
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	2	no	—
377	C₄×D₄	—	2	2	no	—
378	C₄×D₄	—	2	2	no	—
379	C₄×D₄	—	2	2	no	—
380	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
381	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
384	C₂×C₂²⋊C₄	—	2	2	no	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
387	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
388	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
389	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
390	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
392	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
393	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
394	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
395	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
397	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
398	C₄⋊C₈	—	2	2	no	—
399	C₄⋊C₈	—	2	2	no	—
400	C₄⋊C₈	—	2	2	no	—
401	C₄⋊C₈	—	2	2	no	—
402	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
403	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
404	C₂²⋊C₈	—	2	2	no	—
405	C₂²⋊C₈	—	2	2	no	—
406	C₄×C₈	—	2	2	no	—
407	C₄×C₈	—	2	2	no	—
408	C₃₁	#17704	1	1	yes	characteristic, direct product factor, Sylow
409	C₂⁴	—	2	2	no	—
410	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
411	#41	—	4	8	no	—
412	#41	—	4	8	no	—
413	#41	—	4	4	no	—
414	#41	—	4	4	no	—
415	#41	—	4	4	no	—
416	#41	—	4	4	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	4	2	no	—
420	C₂×Q₈	—	4	2	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	4	4	no	—
429	C₂×D₄	—	4	4	no	—
430	C₂×D₄	—	4	8	no	—
431	C₂×D₄	—	4	2	no	—
432	C₂×D₄	—	4	2	no	—
433	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	—	4	8	no	—
440	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
442	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
443	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
444	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
447	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	D₈×C₃₁	1	4	yes	—
450	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
451	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
452	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
453	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
458	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
459	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
460	M₁₆	—	4	4	no	—
461	M₁₆	—	4	4	no	—
462	C₂×C₈	—	2	4	no	—
463	C₂×C₈	—	2	4	no	—
464	C₂×C₈	—	4	4	no	—
465	C₂×C₈	—	4	4	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	2	4	no	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
475	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
476	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
477	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
478	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
479	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
480	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
481	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
482	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
483	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
484	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
486	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
489	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
490	C₂³	—	2	4	no	—
491	C₂³	—	4	1	no	—
492	C₂³	—	4	1	no	—
493	C₂³	—	4	4	no	—
494	C₂³	—	4	4	no	—
495	C₂³	—	4	4	no	—
496	C₂³	C₃₁×#102	1	4	yes	—
497	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
498	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
499	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
500	Q₈	—	4	4	no	—
501	Q₈	—	4	2	no	—
502	Q₈	—	4	2	no	—
503	D₄	—	4	4	no	—
504	D₄	—	4	4	no	—
505	D₄	—	4	8	no	—
506	D₄	—	4	8	no	—
507	D₄	—	4	4	no	—
508	D₄	—	4	4	no	—
509	D₄	—	4	2	no	—
510	D₄	—	4	2	no	—
511	D₄	—	4	4	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	4	no	—
526	C₂×C₄	—	4	8	no	—
527	C₂×C₄	—	4	8	no	—
528	C₂×C₄	C₃₁×#102	1	4	yes	—
529	C₂×C₄	C₃₁×#102	1	4	yes	—
530	C₂×C₄	C₃₁×#102	1	4	yes	—
531	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
532	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
533	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
534	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
535	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
536	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
537	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
538	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
539	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
540	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
541	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
542	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
543	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
544	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
545	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
546	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
547	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
548	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
549	C₈	—	4	4	no	—
550	C₈	—	4	4	no	—
551	C₂²	—	2	1	no	—
552	C₂²	—	2	1	no	—
553	C₂²	—	2	1	no	—
554	C₂²	—	2	1	no	—
555	C₂²	—	2	1	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	8	8	no	—
562	C₂²	C₂×C₃₁×#102	1	2	yes	—
563	C₂²	C₂×C₃₁×#102	1	2	yes	—
564	C₂²	C₂×C₃₁×#102	1	2	yes	—
565	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
566	C₄	—	2	1	no	—
567	C₄	—	2	1	no	—
568	C₄	—	2	1	no	—
569	C₄	—	2	1	no	—
570	C₄	—	2	1	no	—
571	C₄	—	2	1	no	—
572	C₄	—	4	4	no	—
573	C₄	C₃₁×#413	1	2	yes	—
574	C₄	C₂×C₃₁×#102	1	2	yes	—
575	C₄	C₂×C₃₁×#102	1	2	yes	—
576	C₄	C₂×C₃₁×#102	1	2	yes	—
577	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
578	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
579	C₄	C₃₁×#575	1	2	yes	—
580	C₂	—	2	2	no	—
581	C₂	—	2	2	no	—
582	C₂	—	2	2	no	—
583	C₂	—	8	4	no	—
584	C₂	C₂×C₃₁×#413	1	1	yes	characteristic
585	C₂	C₂²×C₃₁×#102	1	1	yes	characteristic
586	C₂	C₃₁×#2896	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₂×C₃₁×#532	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. C₂×C₃₁×#431	C₂	semidirect
3. C₂×C₃₁×#423	C₂	semidirect
4. C₃₁×#1547	C₂	semidirect
5. C₃₁×#1546	C₂	semidirect
6. C₃₁×#1545	C₂	semidirect
7. C₃₁×#1540	C₂	semidirect
29. C₃₁×#431	C₂²	semidirect
33. C₃₁×#423	C₂²	semidirect
35. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
36. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
39. C₃₁×#332	C₂²	semidirect
40. C₃₁×#331	C₂²	semidirect
196. #17704	C₃₁	direct (N × H)
259. C₂×#431	C₂×C₃₁	semidirect
260. C₂×#423	C₂×C₃₁	semidirect
261. #1547	C₂×C₃₁	semidirect
262. #1546	C₂×C₃₁	semidirect
263. #1545	C₂×C₃₁	semidirect
264. #1540	C₂×C₃₁	semidirect
315. #431	C₂²×C₃₁	semidirect
319. #423	C₂²×C₃₁	semidirect
321. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
322. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
325. #332	C₂²×C₃₁	semidirect
326. #331	C₂²×C₃₁	semidirect
408. C₃₁	#17704	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#532	2
C₂×C₃₁×#431	2
C₂×C₃₁×#423	2
C₃₁×#1547	2
C₃₁×#1546	2
C₃₁×#1545	2
C₃₁×#1540	2
#17704	31

Subgroups — C31×#17704 #21474463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17704 #21474463