Subgroups — C₃₁×#17753 #21474512

Order 7936.

1020 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3444	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1559	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1557	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1552	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1551	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1550	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1547	C₂	1	2	yes	maximal, complemented
10	C₂²×M₁₆×C₃₁	—	2	1	no	—
11	C₃₁×#533	C₂²	1	1	yes	characteristic
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₃₁×#533	C₂²	1	1	yes	characteristic
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₃₁×#532	C₂²	1	1	yes	characteristic
16	C₃₁×#532	C₂²	1	1	yes	characteristic
17	C₃₁×#529	—	2	1	no	—
18	C₃₁×#529	—	2	1	no	—
19	C₃₁×#529	—	2	1	no	—
20	C₃₁×#529	—	2	1	no	—
21	C₃₁×#529	C₄	1	2	yes	—
22	C₃₁×#529	C₄	1	2	yes	—
23	C₃₁×#525	—	2	1	no	—
24	C₃₁×#525	—	2	1	no	—
25	C₃₁×#525	C₄	1	2	yes	—
26	C₄×#41×C₃₁	—	2	1	no	—
27	C₄×#41×C₃₁	—	2	1	no	—
28	C₄×#41×C₃₁	C₄	1	2	yes	—
29	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#432	—	2	1	no	—
31	C₃₁×#432	—	2	1	no	—
32	C₃₁×#432	C₂²	1	2	yes	—
33	C₃₁×#432	C₂²	1	2	yes	—
34	C₃₁×#423	—	2	1	no	—
35	C₃₁×#423	—	2	1	no	—
36	C₃₁×#423	C₂²	1	2	yes	—
37	C₃₁×#423	C₂²	1	2	yes	—
38	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
40	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
41	C₂×C₃₁×C₂²⋊C₈	—	2	1	no	—
42	C₂×C₃₁×#97	—	2	1	no	—
43	C₃₁×#333	C₂²	1	4	yes	complemented
44	C₃₁×#332	C₂²	1	4	yes	complemented
45	C₃₁×#332	C₂²	1	4	yes	complemented
46	C₃₁×#331	C₂²	1	4	yes	complemented
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂×#41×C₃₁	—	2	1	no	—
49	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
50	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₃₁	—	2	2	no	—
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	4	4	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
59	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₃₁×#127	C₂³	1	1	yes	characteristic
62	C₃₁×#126	—	2	2	no	—
63	C₃₁×#126	—	2	2	no	—
64	C₃₁×#125	—	2	1	no	—
65	C₃₁×#125	—	2	1	no	—
66	C₃₁×#125	C₂×C₄	1	2	yes	—
67	C₃₁×#124	—	2	1	no	—
68	C₃₁×#124	—	2	1	no	—
69	C₃₁×#124	C₂×C₄	1	2	yes	—
70	C₃₁×#123	—	2	2	no	—
71	C₃₁×#123	—	2	2	no	—
72	C₃₁×#123	—	2	2	no	—
73	C₃₁×C₂²⋊Q₈	—	2	1	no	—
74	C₃₁×C₂²⋊Q₈	—	2	1	no	—
75	C₃₁×C₂²⋊Q₈	—	2	1	no	—
76	C₃₁×C₂²⋊Q₈	—	2	1	no	—
77	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
78	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
79	C₃₁×#121	—	2	1	no	—
80	C₃₁×#121	—	2	1	no	—
81	C₃₁×#121	—	2	1	no	—
82	C₃₁×#121	—	2	1	no	—
83	C₃₁×#121	C₂×C₄	1	2	yes	—
84	C₃₁×#121	C₂×C₄	1	2	yes	—
85	C₃₁×#120	—	2	2	no	—
86	C₄×Q₈×C₃₁	—	2	1	no	—
87	C₄×Q₈×C₃₁	—	2	1	no	—
88	C₄×Q₈×C₃₁	—	2	1	no	—
89	C₄×Q₈×C₃₁	—	2	1	no	—
90	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
91	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
92	C₄×D₄×C₃₁	—	2	2	no	—
93	C₄×D₄×C₃₁	—	2	2	no	—
94	C₄×D₄×C₃₁	—	2	2	no	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	—	2	2	no	—
97	C₄×D₄×C₃₁	—	2	2	no	—
98	C₄×D₄×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	—	2	1	no	—
102	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
103	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
104	C₃₁×#117	—	2	1	no	—
105	C₃₁×#117	—	2	1	no	—
106	C₃₁×#117	—	2	2	no	—
107	C₃₁×#117	—	2	2	no	—
108	C₃₁×#117	—	2	2	no	—
109	C₃₁×#117	C₂×C₄	1	2	yes	—
110	C₂×C₄²×C₃₁	—	2	2	no	—
111	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
123	C₃₁×C₄⋊C₈	—	2	2	no	—
124	C₃₁×C₄⋊C₈	—	2	2	no	—
125	C₃₁×C₄⋊C₈	—	2	2	no	—
126	C₃₁×C₄⋊C₈	—	2	2	no	—
127	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
128	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
129	C₃₁×C₂²⋊C₈	—	2	2	no	—
130	C₃₁×C₂²⋊C₈	—	2	2	no	—
131	C₃₁×#97	—	2	2	no	—
132	C₃₁×#97	—	2	2	no	—
133	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
134	#41×C₃₁	—	4	4	no	—
135	#41×C₃₁	—	4	4	no	—
136	#41×C₃₁	—	4	2	no	—
137	#41×C₃₁	—	4	2	no	—
138	C₂×Q₈×C₃₁	—	2	1	no	—
139	C₂×Q₈×C₃₁	—	2	1	no	—
140	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	2	2	no	—
143	C₂×D₄×C₃₁	—	2	2	no	—
144	C₂×D₄×C₃₁	—	2	1	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
147	C₂²×C₄×C₃₁	—	2	2	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	—	4	2	no	—
153	C₂²×C₄×C₃₁	—	4	2	no	—
154	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	M₁₆×C₃₁	—	4	4	no	—
173	M₁₆×C₃₁	—	4	4	no	—
174	C₂×C₈×C₃₁	—	2	4	no	—
175	C₂×C₈×C₃₁	—	2	4	no	—
176	C₂×C₈×C₃₁	—	4	4	no	—
177	C₂×C₈×C₃₁	—	4	4	no	—
178	C₄⋊C₄×C₃₁	—	2	2	no	—
179	C₄⋊C₄×C₃₁	—	2	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	2	no	—
184	C₄⋊C₄×C₃₁	—	2	1	no	—
185	C₄⋊C₄×C₃₁	—	2	1	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
189	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
190	C₂²⋊C₄×C₃₁	—	2	2	no	—
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	2	no	—
195	C₂²⋊C₄×C₃₁	—	2	2	no	—
196	C₂²⋊C₄×C₃₁	—	2	2	no	—
197	C₂²⋊C₄×C₃₁	—	2	2	no	—
198	C₂²⋊C₄×C₃₁	—	2	2	no	—
199	C₄²×C₃₁	—	2	2	no	—
200	C₄²×C₃₁	—	2	2	no	—
201	C₄²×C₃₁	—	2	2	no	—
202	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
204	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
205	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
206	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
207	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
208	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
209	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
210	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
211	C₄²×C₃₁	C₂×D₄	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	1	yes	characteristic
215	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
218	#17753	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂³×C₃₁	—	2	2	no	—
220	C₂³×C₃₁	—	2	2	no	—
221	C₂³×C₃₁	—	2	2	no	—
222	C₂³×C₃₁	—	4	1	no	—
223	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
224	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	Q₈×C₃₁	—	4	2	no	—
227	Q₈×C₃₁	—	4	1	no	—
228	Q₈×C₃₁	—	4	1	no	—
229	Q₈×C₃₁	—	4	2	no	—
230	D₄×C₃₁	—	4	2	no	—
231	D₄×C₃₁	—	4	4	no	—
232	D₄×C₃₁	—	4	4	no	—
233	D₄×C₃₁	—	4	4	no	—
234	D₄×C₃₁	—	4	1	no	—
235	D₄×C₃₁	—	4	1	no	—
236	D₄×C₃₁	—	4	2	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	1	no	—
241	C₂×C₄×C₃₁	—	2	1	no	—
242	C₂×C₄×C₃₁	—	2	1	no	—
243	C₂×C₄×C₃₁	—	2	1	no	—
244	C₂×C₄×C₃₁	—	2	2	no	—
245	C₂×C₄×C₃₁	—	2	2	no	—
246	C₂×C₄×C₃₁	—	2	2	no	—
247	C₂×C₄×C₃₁	—	2	2	no	—
248	C₂×C₄×C₃₁	—	2	2	no	—
249	C₂×C₄×C₃₁	—	2	2	no	—
250	C₂×C₄×C₃₁	—	2	2	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	4	1	no	—
253	C₂×C₄×C₃₁	—	4	1	no	—
254	C₂×C₄×C₃₁	—	4	1	no	—
255	C₂×C₄×C₃₁	—	4	1	no	—
256	C₂×C₄×C₃₁	—	4	1	no	—
257	C₂×C₄×C₃₁	—	4	1	no	—
258	C₂×C₄×C₃₁	—	4	1	no	—
259	C₂×C₄×C₃₁	—	4	4	no	—
260	C₂×C₄×C₃₁	—	4	4	no	—
261	C₂×C₄×C₃₁	—	4	4	no	—
262	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
266	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
272	C₂×C₄×C₃₁	#137	1	2	yes	—
273	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
274	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
276	C₈×C₃₁	—	4	4	no	—
277	C₈×C₃₁	—	4	4	no	—
278	#3444	C₂×C₃₁	1	1	yes	characteristic
279	C₂×#432	C₂×C₃₁	1	1	yes	characteristic, complemented
280	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#1559	C₂×C₃₁	1	2	yes	complemented
282	#1557	C₂×C₃₁	1	2	yes	complemented
283	#1552	C₂×C₃₁	1	2	yes	complemented
284	#1551	C₂×C₃₁	1	2	yes	complemented
285	#1550	C₂×C₃₁	1	2	yes	complemented
286	#1547	C₂×C₃₁	1	2	yes	complemented
287	C₂²×C₃₁	—	2	1	no	—
288	C₂²×C₃₁	—	2	1	no	—
289	C₂²×C₃₁	—	2	1	no	—
290	C₂²×C₃₁	—	2	1	no	—
291	C₂²×C₃₁	—	2	1	no	—
292	C₂²×C₃₁	—	2	1	no	—
293	C₂²×C₃₁	—	2	1	no	—
294	C₂²×C₃₁	—	4	4	no	—
295	C₂²×C₃₁	—	4	2	no	—
296	C₂²×C₃₁	—	4	2	no	—
297	C₂²×C₃₁	—	4	2	no	—
298	C₂²×C₃₁	#413	1	2	yes	—
299	C₂²×C₃₁	#417	1	2	yes	—
300	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
301	C₄×C₃₁	—	2	1	no	—
302	C₄×C₃₁	—	2	1	no	—
303	C₄×C₃₁	—	2	1	no	—
304	C₄×C₃₁	—	2	1	no	—
305	C₄×C₃₁	—	2	1	no	—
306	C₄×C₃₁	—	2	1	no	—
307	C₄×C₃₁	—	2	1	no	—
308	C₄×C₃₁	—	2	1	no	—
309	C₄×C₃₁	—	2	1	no	—
310	C₄×C₃₁	—	4	2	no	—
311	C₄×C₃₁	—	4	2	no	—
312	C₄×C₃₁	—	4	2	no	—
313	C₄×C₃₁	#417	1	2	yes	—
314	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
315	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
316	C₄×C₃₁	#575	1	2	yes	—
317	C₂²×M₁₆	—	2	1	no	—
318	#533	C₂²×C₃₁	1	1	yes	characteristic
319	#533	C₂²×C₃₁	1	1	yes	characteristic
320	#533	C₂²×C₃₁	1	1	yes	characteristic
321	#532	C₂²×C₃₁	1	1	yes	characteristic
322	#532	C₂²×C₃₁	1	1	yes	characteristic
323	#532	C₂²×C₃₁	1	1	yes	characteristic
324	#529	—	2	1	no	—
325	#529	—	2	1	no	—
326	#529	—	2	1	no	—
327	#529	—	2	1	no	—
328	#529	C₄×C₃₁	1	2	yes	—
329	#529	C₄×C₃₁	1	2	yes	—
330	#525	—	2	1	no	—
331	#525	—	2	1	no	—
332	#525	C₄×C₃₁	1	2	yes	—
333	C₄×#41	—	2	1	no	—
334	C₄×#41	—	2	1	no	—
335	C₄×#41	C₄×C₃₁	1	2	yes	—
336	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
337	#432	—	2	1	no	—
338	#432	—	2	1	no	—
339	#432	C₂²×C₃₁	1	2	yes	—
340	#432	C₂²×C₃₁	1	2	yes	—
341	#423	—	2	1	no	—
342	#423	—	2	1	no	—
343	#423	C₂²×C₃₁	1	2	yes	—
344	#423	C₂²×C₃₁	1	2	yes	—
345	C₂×C₄⋊C₈	—	2	1	no	—
346	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
347	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
348	C₂×C₂²⋊C₈	—	2	1	no	—
349	C₂×#97	—	2	1	no	—
350	#333	C₂²×C₃₁	1	4	yes	complemented
351	#332	C₂²×C₃₁	1	4	yes	complemented
352	#332	C₂²×C₃₁	1	4	yes	complemented
353	#331	C₂²×C₃₁	1	4	yes	complemented
354	C₂×C₃₁	—	2	2	no	—
355	C₂×C₃₁	—	2	2	no	—
356	C₂×C₃₁	—	4	1	no	—
357	C₂×C₃₁	—	8	2	no	—
358	C₂×C₃₁	C₂×#413	1	1	yes	characteristic
359	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
360	C₂×C₃₁	#2896	1	1	yes	characteristic
361	C₂×#41	—	2	1	no	—
362	C₂×#41	—	2	1	no	—
363	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
364	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
365	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
366	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
367	C₂×M₁₆	—	2	2	no	—
368	C₂×M₁₆	—	2	2	no	—
369	C₂×M₁₆	—	4	4	no	—
370	C₂²×C₈	—	2	2	no	—
371	C₂²×C₈	—	2	2	no	—
372	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
373	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
374	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
375	#127	C₂³×C₃₁	1	1	yes	characteristic
376	#126	—	2	2	no	—
377	#126	—	2	2	no	—
378	#125	—	2	1	no	—
379	#125	—	2	1	no	—
380	#125	C₂×C₄×C₃₁	1	2	yes	—
381	#124	—	2	1	no	—
382	#124	—	2	1	no	—
383	#124	C₂×C₄×C₃₁	1	2	yes	—
384	#123	—	2	2	no	—
385	#123	—	2	2	no	—
386	#123	—	2	2	no	—
387	C₂²⋊Q₈	—	2	1	no	—
388	C₂²⋊Q₈	—	2	1	no	—
389	C₂²⋊Q₈	—	2	1	no	—
390	C₂²⋊Q₈	—	2	1	no	—
391	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
392	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
393	#121	—	2	1	no	—
394	#121	—	2	1	no	—
395	#121	—	2	1	no	—
396	#121	—	2	1	no	—
397	#121	C₂×C₄×C₃₁	1	2	yes	—
398	#121	C₂×C₄×C₃₁	1	2	yes	—
399	#120	—	2	2	no	—
400	C₄×Q₈	—	2	1	no	—
401	C₄×Q₈	—	2	1	no	—
402	C₄×Q₈	—	2	1	no	—
403	C₄×Q₈	—	2	1	no	—
404	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
405	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
406	C₄×D₄	—	2	2	no	—
407	C₄×D₄	—	2	2	no	—
408	C₄×D₄	—	2	2	no	—
409	C₄×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	2	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
417	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
418	#117	—	2	1	no	—
419	#117	—	2	1	no	—
420	#117	—	2	2	no	—
421	#117	—	2	2	no	—
422	#117	—	2	2	no	—
423	#117	C₂×C₄×C₃₁	1	2	yes	—
424	C₂×C₄²	—	2	2	no	—
425	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
426	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
428	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
431	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
432	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
435	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
437	C₄⋊C₈	—	2	2	no	—
438	C₄⋊C₈	—	2	2	no	—
439	C₄⋊C₈	—	2	2	no	—
440	C₄⋊C₈	—	2	2	no	—
441	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
442	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
443	C₂²⋊C₈	—	2	2	no	—
444	C₂²⋊C₈	—	2	2	no	—
445	#97	—	2	2	no	—
446	#97	—	2	2	no	—
447	C₃₁	#17753	1	1	yes	characteristic, direct product factor, Sylow
448	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
449	#41	—	4	4	no	—
450	#41	—	4	4	no	—
451	#41	—	4	2	no	—
452	#41	—	4	2	no	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
470	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
471	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
472	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
473	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
474	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
478	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
479	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
480	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
487	M₁₆	—	4	4	no	—
488	M₁₆	—	4	4	no	—
489	C₂×C₈	—	2	4	no	—
490	C₂×C₈	—	2	4	no	—
491	C₂×C₈	—	4	4	no	—
492	C₂×C₈	—	4	4	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
504	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	2	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	2	no	—
514	C₄²	—	2	2	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	2	no	—
517	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
518	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
519	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
520	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
521	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
522	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
523	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
524	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
525	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
526	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
527	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
528	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
529	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
530	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
531	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
532	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
533	C₂³	—	2	2	no	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	4	1	no	—
537	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
538	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
539	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
540	Q₈	—	4	2	no	—
541	Q₈	—	4	1	no	—
542	Q₈	—	4	1	no	—
543	Q₈	—	4	2	no	—
544	D₄	—	4	2	no	—
545	D₄	—	4	4	no	—
546	D₄	—	4	4	no	—
547	D₄	—	4	4	no	—
548	D₄	—	4	1	no	—
549	D₄	—	4	1	no	—
550	D₄	—	4	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	4	no	—
574	C₂×C₄	—	4	4	no	—
575	C₂×C₄	—	4	4	no	—
576	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
577	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
578	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
579	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
580	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
581	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
582	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
583	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
584	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
585	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
586	C₂×C₄	C₃₁×#137	1	2	yes	—
587	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
588	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
589	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
590	C₈	—	4	4	no	—
591	C₈	—	4	4	no	—
592	C₂²	—	2	1	no	—
593	C₂²	—	2	1	no	—
594	C₂²	—	2	1	no	—
595	C₂²	—	2	1	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	4	4	no	—
600	C₂²	—	4	2	no	—
601	C₂²	—	4	2	no	—
602	C₂²	—	4	2	no	—
603	C₂²	C₃₁×#413	1	2	yes	—
604	C₂²	C₃₁×#417	1	2	yes	—
605	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
606	C₄	—	2	1	no	—
607	C₄	—	2	1	no	—
608	C₄	—	2	1	no	—
609	C₄	—	2	1	no	—
610	C₄	—	2	1	no	—
611	C₄	—	2	1	no	—
612	C₄	—	2	1	no	—
613	C₄	—	2	1	no	—
614	C₄	—	2	1	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	2	no	—
617	C₄	—	4	2	no	—
618	C₄	C₃₁×#417	1	2	yes	—
619	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
620	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
621	C₄	C₃₁×#575	1	2	yes	—
622	C₂	—	2	2	no	—
623	C₂	—	2	2	no	—
624	C₂	—	4	1	no	—
625	C₂	—	8	2	no	—
626	C₂	C₂×C₃₁×#413	1	1	yes	characteristic
627	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
628	C₂	C₃₁×#2896	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×#3444	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#432	C₂	semidirect
3. C₂×C₃₁×#423	C₂	semidirect
4. C₃₁×#1559	C₂	semidirect
5. C₃₁×#1557	C₂	semidirect
6. C₃₁×#1552	C₂	semidirect
7. C₃₁×#1551	C₂	semidirect
8. C₃₁×#1550	C₂	semidirect
9. C₃₁×#1547	C₂	semidirect
43. C₃₁×#333	C₂²	semidirect
44. C₃₁×#332	C₂²	semidirect
45. C₃₁×#332	C₂²	semidirect
46. C₃₁×#331	C₂²	semidirect
218. #17753	C₃₁	direct (N × H)
279. C₂×#432	C₂×C₃₁	semidirect
280. C₂×#423	C₂×C₃₁	semidirect
281. #1559	C₂×C₃₁	semidirect
282. #1557	C₂×C₃₁	semidirect
283. #1552	C₂×C₃₁	semidirect
284. #1551	C₂×C₃₁	semidirect
285. #1550	C₂×C₃₁	semidirect
286. #1547	C₂×C₃₁	semidirect
350. #333	C₂²×C₃₁	semidirect
351. #332	C₂²×C₃₁	semidirect
352. #332	C₂²×C₃₁	semidirect
353. #331	C₂²×C₃₁	semidirect
447. C₃₁	#17753	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×#3444	2
C₂×C₃₁×#432	2
C₂×C₃₁×#423	2
C₃₁×#1559	2
C₃₁×#1557	2
C₃₁×#1552	2
C₃₁×#1551	2
C₃₁×#1550	2
C₃₁×#1547	2
#17753	31

Subgroups — C31×#17753 #21474512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17753 #21474512