Subgroups — C₃₁×#21424 #21478183

Order 7936.

1380 conjugacy classes of subgroups.

The subgroup lattice has 568 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	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1997	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1965	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1900	C₂	1	4	yes	maximal, complemented
7	C₄×C₃₁×#100	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₃₁	C₄	1	2	yes	—
9	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
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	—	2	1	no	—
15	C₃₁×#532	—	2	1	no	—
16	C₃₁×#532	—	2	1	no	—
17	C₃₁×#532	—	2	1	no	—
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#124	C₄	1	2	yes	—
21	C₂×C₃₁×#121	C₄	1	2	yes	—
22	C₂×C₃₁×#121	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#432	—	2	1	no	—
27	C₃₁×#432	—	2	1	no	—
28	C₃₁×#431	—	2	1	no	—
29	C₃₁×#431	—	2	1	no	—
30	C₃₁×#423	C₂²	1	2	yes	—
31	C₂×C₃₁×#100	C₂²	1	8	yes	—
32	C₃₁×#407	C₂²	1	2	yes	—
33	C₄×M₁₆×C₃₁	—	2	1	no	—
34	C₄×M₁₆×C₃₁	—	2	1	no	—
35	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
36	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
37	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
38	C₂×C₃₁×#97	—	2	1	no	—
39	C₂×C₄×C₈×C₃₁	—	2	1	no	—
40	C₃₁×#343	C₂²	1	8	yes	—
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	8	no	—
46	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
47	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂²×D₄×C₃₁	C₂×C₄	1	4	yes	—
49	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
51	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
53	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	2	8	no	—
57	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
58	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	—	2	1	no	—
63	C₃₁×C₄⋊Q₈	—	2	1	no	—
64	C₃₁×#127	—	2	1	no	—
65	C₃₁×#127	—	2	1	no	—
66	C₃₁×#127	—	2	1	no	—
67	C₃₁×#127	—	2	1	no	—
68	C₃₁×#124	—	2	2	no	—
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×#124	—	2	2	no	—
71	C₃₁×#124	—	2	2	no	—
72	C₃₁×#121	—	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	—	2	2	no	—
80	C₄×D₄×C₃₁	—	2	2	no	—
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	—	2	2	no	—
88	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
90	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
92	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	D₄	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	C₃₁×C₄⋊C₈	—	2	2	no	—
105	C₃₁×C₄⋊C₈	—	2	2	no	—
106	C₃₁×#100	—	2	16	no	—
107	C₃₁×C₂²⋊C₈	—	2	2	no	—
108	C₃₁×C₂²⋊C₈	—	2	2	no	—
109	C₃₁×#97	—	2	2	no	—
110	C₃₁×#97	—	2	2	no	—
111	C₃₁×#97	—	2	2	no	—
112	C₄×C₈×C₃₁	—	2	2	no	—
113	C₄×C₈×C₃₁	—	2	2	no	—
114	C₄×C₈×C₃₁	—	2	2	no	—
115	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
117	#41×C₃₁	—	4	8	no	—
118	#41×C₃₁	—	4	8	no	—
119	#41×C₃₁	—	4	8	no	—
120	#41×C₃₁	—	4	8	no	—
121	C₂×Q₈×C₃₁	—	2	2	no	—
122	C₂×Q₈×C₃₁	—	2	2	no	—
123	C₂×Q₈×C₃₁	—	4	4	no	—
124	C₂×D₄×C₃₁	—	2	4	no	—
125	C₂×D₄×C₃₁	—	2	4	no	—
126	C₂×D₄×C₃₁	—	2	2	no	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂×D₄×C₃₁	—	2	2	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	8	no	—
133	C₂×D₄×C₃₁	—	4	4	no	—
134	C₂×D₄×C₃₁	—	4	4	no	—
135	C₂×D₄×C₃₁	—	4	8	no	—
136	C₂²×C₄×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	—	2	4	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	4	2	no	—
141	C₂²×C₄×C₃₁	—	4	8	no	—
142	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	4	yes	—
155	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
161	M₁₆×C₃₁	—	2	8	no	—
162	M₁₆×C₃₁	—	2	4	no	—
163	M₁₆×C₃₁	—	2	8	no	—
164	M₁₆×C₃₁	#41	1	8	yes	complemented
165	C₂×C₈×C₃₁	—	2	8	no	—
166	C₂×C₈×C₃₁	—	2	4	no	—
167	C₂×C₈×C₃₁	—	2	4	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₂²⋊C₄×C₃₁	—	2	4	no	—
173	C₂²⋊C₄×C₃₁	—	2	4	no	—
174	C₂²⋊C₄×C₃₁	—	2	4	no	—
175	C₂²⋊C₄×C₃₁	—	2	4	no	—
176	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
177	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
178	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
179	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₃₁	C₂×D₄	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	#21424	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	—	2	4	no	—
196	C₂³×C₃₁	—	2	8	no	—
197	C₂³×C₃₁	—	4	1	no	—
198	C₂³×C₃₁	—	4	4	no	—
199	C₂³×C₃₁	—	4	4	no	—
200	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
203	Q₈×C₃₁	—	4	4	no	—
204	Q₈×C₃₁	—	4	4	no	—
205	D₄×C₃₁	—	4	4	no	—
206	D₄×C₃₁	—	4	4	no	—
207	D₄×C₃₁	—	4	8	no	—
208	D₄×C₃₁	—	4	8	no	—
209	D₄×C₃₁	—	4	4	no	—
210	D₄×C₃₁	—	4	4	no	—
211	D₄×C₃₁	—	4	4	no	—
212	D₄×C₃₁	—	4	4	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	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	2	no	—
226	C₂×C₄×C₃₁	—	2	4	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	8	no	—
231	C₂×C₄×C₃₁	—	4	8	no	—
232	C₂×C₄×C₃₁	—	4	4	no	—
233	C₂×C₄×C₃₁	#100	1	4	yes	complemented
234	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
235	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
247	C₈×C₃₁	—	2	8	no	—
248	C₈×C₃₁	—	4	4	no	—
249	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
250	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1997	C₂×C₃₁	1	2	yes	complemented
253	#1965	C₂×C₃₁	1	2	yes	complemented
254	#1900	C₂×C₃₁	1	4	yes	complemented
255	C₄×#100	C₂×C₃₁	1	4	yes	complemented
256	C₂²×C₃₁	—	2	1	no	—
257	C₂²×C₃₁	—	2	1	no	—
258	C₂²×C₃₁	—	2	1	no	—
259	C₂²×C₃₁	—	2	1	no	—
260	C₂²×C₃₁	—	2	1	no	—
261	C₂²×C₃₁	—	2	1	no	—
262	C₂²×C₃₁	—	2	1	no	—
263	C₂²×C₃₁	—	4	4	no	—
264	C₂²×C₃₁	—	4	4	no	—
265	C₂²×C₃₁	—	4	4	no	—
266	C₂²×C₃₁	—	4	4	no	—
267	C₂²×C₃₁	—	8	8	no	—
268	C₂²×C₃₁	C₂×#100	1	2	yes	—
269	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
270	C₂²×C₃₁	C₄×#41	1	2	yes	—
271	C₄×C₃₁	—	2	1	no	—
272	C₄×C₃₁	—	2	1	no	—
273	C₄×C₃₁	—	2	1	no	—
274	C₄×C₃₁	—	2	1	no	—
275	C₄×C₃₁	—	2	1	no	—
276	C₄×C₃₁	—	2	1	no	—
277	C₄×C₃₁	—	2	1	no	—
278	C₄×C₃₁	—	4	4	no	—
279	C₄×C₃₁	C₂×#100	1	2	yes	—
280	C₄×C₃₁	C₂×#100	1	2	yes	—
281	C₄×C₃₁	#413	1	2	yes	—
282	C₄×C₃₁	#525	1	2	yes	—
283	C₄×C₃₁	#532	1	2	yes	—
284	C₂²×#41	C₄×C₃₁	1	2	yes	—
285	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
286	#532	—	2	1	no	—
287	#532	—	2	1	no	—
288	#532	—	2	1	no	—
289	#532	—	2	1	no	—
290	#532	—	2	1	no	—
291	#532	—	2	1	no	—
292	#532	—	2	1	no	—
293	#532	—	2	1	no	—
294	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×#124	C₄×C₃₁	1	2	yes	—
297	C₂×#121	C₄×C₃₁	1	2	yes	—
298	C₂×#121	C₂²×C₃₁	1	2	yes	—
299	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
300	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
301	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
302	#432	—	2	1	no	—
303	#432	—	2	1	no	—
304	#431	—	2	1	no	—
305	#431	—	2	1	no	—
306	#423	C₂²×C₃₁	1	2	yes	—
307	C₂×#100	C₂²×C₃₁	1	8	yes	—
308	#407	C₂²×C₃₁	1	2	yes	—
309	C₄×M₁₆	—	2	1	no	—
310	C₄×M₁₆	—	2	1	no	—
311	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
312	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
313	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
314	C₂×#97	—	2	1	no	—
315	C₂×C₄×C₈	—	2	1	no	—
316	#343	C₂²×C₃₁	1	8	yes	—
317	C₂×C₃₁	—	2	2	no	—
318	C₂×C₃₁	—	2	2	no	—
319	C₂×C₃₁	—	4	1	no	—
320	C₂×C₃₁	—	8	4	no	—
321	C₂×C₃₁	#2297	1	1	yes	characteristic
322	C₂×C₃₁	C₂²×#100	1	1	yes	characteristic
323	C₂×C₃₁	C₂×#413	1	1	yes	characteristic
324	C₂×#41	—	2	2	no	—
325	C₂×#41	—	2	2	no	—
326	C₂×#41	—	2	2	no	—
327	C₂×#41	—	2	2	no	—
328	C₂×#41	—	4	8	no	—
329	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
330	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
331	C₂²×D₄	C₂×C₄×C₃₁	1	4	yes	—
332	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
333	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
334	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
335	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
336	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
337	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×M₁₆	—	2	2	no	—
339	C₂×M₁₆	—	2	8	no	—
340	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
341	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
342	C₂²×C₈	—	2	2	no	—
343	C₄⋊Q₈	—	2	1	no	—
344	C₄⋊Q₈	—	2	1	no	—
345	C₄⋊Q₈	—	2	1	no	—
346	C₄⋊Q₈	—	2	1	no	—
347	#127	—	2	1	no	—
348	#127	—	2	1	no	—
349	#127	—	2	1	no	—
350	#127	—	2	1	no	—
351	#124	—	2	2	no	—
352	#124	—	2	2	no	—
353	#124	—	2	2	no	—
354	#124	—	2	2	no	—
355	#121	—	2	2	no	—
356	#121	—	2	2	no	—
357	#121	—	2	2	no	—
358	#121	—	2	2	no	—
359	#121	—	2	2	no	—
360	#121	—	2	2	no	—
361	#121	—	2	2	no	—
362	#121	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	—	2	2	no	—
366	C₄×D₄	—	2	2	no	—
367	C₄×D₄	—	2	2	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	—	2	2	no	—
371	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
373	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
374	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
375	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
376	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
377	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
378	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
379	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
380	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
384	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
385	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
387	C₄⋊C₈	—	2	2	no	—
388	C₄⋊C₈	—	2	2	no	—
389	#100	—	2	16	no	—
390	C₂²⋊C₈	—	2	2	no	—
391	C₂²⋊C₈	—	2	2	no	—
392	#97	—	2	2	no	—
393	#97	—	2	2	no	—
394	#97	—	2	2	no	—
395	C₄×C₈	—	2	2	no	—
396	C₄×C₈	—	2	2	no	—
397	C₄×C₈	—	2	2	no	—
398	C₃₁	#21424	1	1	yes	characteristic, direct product factor, Sylow
399	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
400	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
401	#41	—	4	8	no	—
402	#41	—	4	8	no	—
403	#41	—	4	8	no	—
404	#41	—	4	8	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	4	4	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	2	2	no	—
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	4	4	no	—
418	C₂×D₄	—	4	4	no	—
419	C₂×D₄	—	4	8	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	4	2	no	—
425	C₂²×C₄	—	4	8	no	—
426	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
427	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
428	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
429	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
430	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
431	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
434	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
436	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
438	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
439	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
443	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
444	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
445	M₁₆	—	2	8	no	—
446	M₁₆	—	2	4	no	—
447	M₁₆	—	2	8	no	—
448	M₁₆	#41×C₃₁	1	8	yes	complemented
449	C₂×C₈	—	2	8	no	—
450	C₂×C₈	—	2	4	no	—
451	C₂×C₈	—	2	4	no	—
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	2	2	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	—	2	4	no	—
458	C₂²⋊C₄	—	2	4	no	—
459	C₂²⋊C₄	—	2	4	no	—
460	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
461	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
462	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
463	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
464	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
465	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
466	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
467	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
468	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
469	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
470	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
471	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
472	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
473	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₂³	—	2	2	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	4	no	—
479	C₂³	—	2	8	no	—
480	C₂³	—	4	1	no	—
481	C₂³	—	4	4	no	—
482	C₂³	—	4	4	no	—
483	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
484	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
485	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
486	Q₈	—	4	4	no	—
487	Q₈	—	4	4	no	—
488	D₄	—	4	4	no	—
489	D₄	—	4	4	no	—
490	D₄	—	4	8	no	—
491	D₄	—	4	8	no	—
492	D₄	—	4	4	no	—
493	D₄	—	4	4	no	—
494	D₄	—	4	4	no	—
495	D₄	—	4	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	8	no	—
514	C₂×C₄	—	4	8	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	C₃₁×#100	1	4	yes	complemented
517	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
518	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
519	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
520	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
521	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
522	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
523	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
524	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
525	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
526	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
529	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
530	C₈	—	2	8	no	—
531	C₈	—	4	4	no	—
532	C₂²	—	2	1	no	—
533	C₂²	—	2	1	no	—
534	C₂²	—	2	1	no	—
535	C₂²	—	2	1	no	—
536	C₂²	—	2	1	no	—
537	C₂²	—	2	1	no	—
538	C₂²	—	2	1	no	—
539	C₂²	—	4	4	no	—
540	C₂²	—	4	4	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	8	8	no	—
544	C₂²	C₂×C₃₁×#100	1	2	yes	—
545	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
546	C₂²	C₄×#41×C₃₁	1	2	yes	—
547	C₄	—	2	1	no	—
548	C₄	—	2	1	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	2	1	no	—
553	C₄	—	2	1	no	—
554	C₄	—	4	4	no	—
555	C₄	C₂×C₃₁×#100	1	2	yes	—
556	C₄	C₂×C₃₁×#100	1	2	yes	—
557	C₄	C₃₁×#413	1	2	yes	—
558	C₄	C₃₁×#525	1	2	yes	—
559	C₄	C₃₁×#532	1	2	yes	—
560	C₂	—	2	2	no	—
561	C₂	—	2	2	no	—
562	C₂	—	4	1	no	—
563	C₂	—	8	4	no	—
564	C₂	C₃₁×#2297	1	1	yes	characteristic
565	C₂	C₂²×C₃₁×#100	1	1	yes	characteristic
566	C₂	C₂×C₃₁×#413	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
prime radical	C₂×C₃₁×#532	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ 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. C₃₁×#2920	C₂	semidirect
3. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
4. C₃₁×#1997	C₂	semidirect
5. C₃₁×#1965	C₂	semidirect
6. C₃₁×#1900	C₂	semidirect
7. C₄×C₃₁×#100	C₂	semidirect
164. M₁₆×C₃₁	#41	semidirect
192. #21424	C₃₁	direct (N × H)
233. C₂×C₄×C₃₁	#100	semidirect
250. #2920	C₂×C₃₁	semidirect
251. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
252. #1997	C₂×C₃₁	semidirect
253. #1965	C₂×C₃₁	semidirect
254. #1900	C₂×C₃₁	semidirect
255. C₄×#100	C₂×C₃₁	semidirect
398. C₃₁	#21424	direct (N × H)
448. M₁₆	#41×C₃₁	semidirect
516. C₂×C₄	C₃₁×#100	semidirect

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
C₃₁×#2920	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#1997	2
C₃₁×#1965	2
C₃₁×#1900	2
C₄×C₃₁×#100	2
#21424	31

Subgroups — C31×#21424 #21478183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21424 #21478183