Subgroups — C₃₁×#17647 #21474406

Order 7936.

1316 conjugacy classes of subgroups.

The subgroup lattice has 560 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×#1530	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1527	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1525	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₃₁	C₄	1	2	yes	—
8	C₂²×M₁₆×C₃₁	—	2	1	no	—
9	C₃₁×#532	—	2	1	no	—
10	C₃₁×#532	—	2	1	no	—
11	C₃₁×#532	—	2	1	no	—
12	C₃₁×#532	—	2	1	no	—
13	C₃₁×#532	C₄	1	2	yes	—
14	C₃₁×#532	C₄	1	2	yes	—
15	C₃₁×#532	C₂²	1	2	yes	—
16	C₃₁×#532	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊Q₈	C₄	1	1	yes	characteristic
18	C₂×C₃₁×#127	C₄	1	1	yes	characteristic
19	C₂×C₃₁×#124	C₂²	1	2	yes	—
20	C₂×C₃₁×#121	—	2	1	no	—
21	C₂×C₃₁×#121	—	2	1	no	—
22	C₂×C₄×D₄×C₃₁	—	2	1	no	—
23	C₂×C₄×D₄×C₃₁	—	2	1	no	—
24	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#432	C₂²	1	2	yes	—
26	C₃₁×#432	C₂²	1	2	yes	complemented
27	C₃₁×#431	—	2	1	no	—
28	C₃₁×#431	—	2	1	no	—
29	C₃₁×#423	—	2	1	no	—
30	C₃₁×#407	—	2	1	no	—
31	C₄×M₁₆×C₃₁	—	2	1	no	—
32	C₄×M₁₆×C₃₁	—	2	1	no	—
33	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
34	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
35	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
36	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
37	C₂×C₄×C₈×C₃₁	—	2	1	no	—
38	C₃₁×#329	C₂²	1	16	yes	complemented
39	C₂×#41×C₃₁	—	2	2	no	—
40	C₂×#41×C₃₁	—	2	2	no	—
41	C₂×#41×C₃₁	—	4	8	no	—
42	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
43	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
44	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
45	C₂²×D₄×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	1	no	—
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₂³×C₄×C₃₁	—	2	2	no	—
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	8	no	—
54	C₂×M₁₆×C₃₁	—	2	2	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂²×C₈×C₃₁	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
61	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
62	C₃₁×#127	—	2	1	no	—
63	C₃₁×#127	—	2	1	no	—
64	C₃₁×#127	C₂×C₄	1	2	yes	—
65	C₃₁×#127	C₂×C₄	1	2	yes	—
66	C₃₁×#124	—	2	2	no	—
67	C₃₁×#124	—	2	2	no	—
68	C₃₁×#124	C₂×C₄	1	4	yes	—
69	C₃₁×#124	C₂³	1	4	yes	—
70	C₃₁×#121	—	2	2	no	—
71	C₃₁×#121	—	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	C₄×D₄×C₃₁	—	2	2	no	—
79	C₄×D₄×C₃₁	—	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	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
90	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
92	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₃₁	D₄	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	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
102	C₃₁×C₄⋊C₈	—	2	2	no	—
103	C₃₁×C₄⋊C₈	—	2	2	no	—
104	C₃₁×C₂²⋊C₈	—	2	2	no	—
105	C₃₁×C₂²⋊C₈	—	2	2	no	—
106	C₃₁×#97	—	2	2	no	—
107	C₃₁×#97	C₂³	1	4	yes	—
108	C₃₁×#97	C₂³	1	4	yes	—
109	C₄×C₈×C₃₁	—	2	2	no	—
110	C₄×C₈×C₃₁	—	2	2	no	—
111	C₄×C₈×C₃₁	—	2	2	no	—
112	C₂⁴×C₃₁	—	2	2	no	—
113	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
114	#41×C₃₁	—	4	8	no	—
115	#41×C₃₁	—	4	8	no	—
116	#41×C₃₁	—	4	8	no	—
117	#41×C₃₁	—	4	8	no	—
118	C₂×Q₈×C₃₁	—	2	2	no	—
119	C₂×Q₈×C₃₁	—	4	4	no	—
120	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
121	C₂×D₄×C₃₁	—	2	4	no	—
122	C₂×D₄×C₃₁	—	2	4	no	—
123	C₂×D₄×C₃₁	—	2	2	no	—
124	C₂×D₄×C₃₁	—	2	2	no	—
125	C₂×D₄×C₃₁	—	2	2	no	—
126	C₂×D₄×C₃₁	—	2	2	no	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂×D₄×C₃₁	—	4	4	no	—
129	C₂×D₄×C₃₁	—	4	4	no	—
130	C₂×D₄×C₃₁	—	4	8	no	—
131	C₂×D₄×C₃₁	—	4	4	no	—
132	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
133	C₂²×C₄×C₃₁	—	2	4	no	—
134	C₂²×C₄×C₃₁	—	2	4	no	—
135	C₂²×C₄×C₃₁	—	2	2	no	—
136	C₂²×C₄×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	—	4	2	no	—
139	C₂²×C₄×C₃₁	—	4	8	no	—
140	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂×D₄	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	M₁₆×C₃₁	—	2	8	no	—
159	M₁₆×C₃₁	—	2	8	no	—
160	M₁₆×C₃₁	—	4	4	no	—
161	C₂×C₈×C₃₁	—	2	8	no	—
162	C₂×C₈×C₃₁	—	2	4	no	—
163	C₂×C₈×C₃₁	—	2	4	no	—
164	C₄⋊C₄×C₃₁	—	2	2	no	—
165	C₄⋊C₄×C₃₁	—	2	2	no	—
166	C₄⋊C₄×C₃₁	—	2	2	no	—
167	C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₂²⋊C₄×C₃₁	—	2	4	no	—
169	C₂²⋊C₄×C₃₁	—	2	4	no	—
170	C₂²⋊C₄×C₃₁	—	2	4	no	—
171	C₂²⋊C₄×C₃₁	—	2	4	no	—
172	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
173	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
174	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
175	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₄²×C₃₁	C₂×D₄	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	C₂⁴	1	1	yes	characteristic
188	#17647	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂³×C₃₁	—	2	2	no	—
190	C₂³×C₃₁	—	2	2	no	—
191	C₂³×C₃₁	—	2	4	no	—
192	C₂³×C₃₁	—	4	1	no	—
193	C₂³×C₃₁	—	4	4	no	—
194	C₂³×C₃₁	—	4	4	no	—
195	C₂³×C₃₁	—	4	4	no	—
196	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	Q₈×C₃₁	—	4	4	no	—
200	Q₈×C₃₁	—	4	4	no	—
201	D₄×C₃₁	—	4	4	no	—
202	D₄×C₃₁	—	4	4	no	—
203	D₄×C₃₁	—	4	8	no	—
204	D₄×C₃₁	—	4	8	no	—
205	D₄×C₃₁	—	4	4	no	—
206	D₄×C₃₁	—	4	4	no	—
207	D₄×C₃₁	—	4	4	no	—
208	D₄×C₃₁	—	4	4	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	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	4	no	—
223	C₂×C₄×C₃₁	—	4	1	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	—	4	1	no	—
226	C₂×C₄×C₃₁	—	4	8	no	—
227	C₂×C₄×C₃₁	—	4	4	no	—
228	C₂×C₄×C₃₁	—	4	8	no	—
229	C₂×C₄×C₃₁	#100	1	4	yes	—
230	C₂×C₄×C₃₁	#104	1	4	yes	—
231	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂³×C₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
243	C₈×C₃₁	—	2	8	no	—
244	C₈×C₃₁	—	4	4	no	—
245	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
246	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
247	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#1530	C₂×C₃₁	1	4	yes	complemented
249	#1527	C₂×C₃₁	1	4	yes	complemented
250	#1525	C₂×C₃₁	1	4	yes	complemented
251	C₂²×C₃₁	—	2	1	no	—
252	C₂²×C₃₁	—	2	1	no	—
253	C₂²×C₃₁	—	2	1	no	—
254	C₂²×C₃₁	—	2	1	no	—
255	C₂²×C₃₁	—	2	1	no	—
256	C₂²×C₃₁	—	2	1	no	—
257	C₂²×C₃₁	—	2	1	no	—
258	C₂²×C₃₁	—	4	4	no	—
259	C₂²×C₃₁	—	4	4	no	—
260	C₂²×C₃₁	—	4	4	no	—
261	C₂²×C₃₁	—	4	4	no	—
262	C₂²×C₃₁	—	8	8	no	—
263	C₂²×C₃₁	C₂×#100	1	2	yes	—
264	C₂²×C₃₁	C₂×#104	1	2	yes	—
265	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
266	C₄×C₃₁	—	2	1	no	—
267	C₄×C₃₁	—	2	1	no	—
268	C₄×C₃₁	—	2	1	no	—
269	C₄×C₃₁	—	2	1	no	—
270	C₄×C₃₁	—	2	1	no	—
271	C₄×C₃₁	—	2	1	no	—
272	C₄×C₃₁	—	2	1	no	—
273	C₄×C₃₁	—	4	4	no	—
274	C₄×C₃₁	C₂×#100	1	2	yes	—
275	C₄×C₃₁	C₂×#100	1	2	yes	—
276	C₄×C₃₁	C₂×#104	1	2	yes	—
277	C₄×C₃₁	C₂×#104	1	2	yes	—
278	C₄×C₃₁	#421	1	2	yes	—
279	C₂²×#41	C₄×C₃₁	1	2	yes	—
280	C₂²×M₁₆	—	2	1	no	—
281	#532	—	2	1	no	—
282	#532	—	2	1	no	—
283	#532	—	2	1	no	—
284	#532	—	2	1	no	—
285	#532	C₄×C₃₁	1	2	yes	—
286	#532	C₄×C₃₁	1	2	yes	—
287	#532	C₂²×C₃₁	1	2	yes	—
288	#532	C₂²×C₃₁	1	2	yes	—
289	C₂×C₄⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
290	C₂×#127	C₄×C₃₁	1	1	yes	characteristic
291	C₂×#124	C₂²×C₃₁	1	2	yes	—
292	C₂×#121	—	2	1	no	—
293	C₂×#121	—	2	1	no	—
294	C₂×C₄×D₄	—	2	1	no	—
295	C₂×C₄×D₄	—	2	1	no	—
296	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
297	#432	C₂²×C₃₁	1	2	yes	—
298	#432	C₂²×C₃₁	1	2	yes	complemented
299	#431	—	2	1	no	—
300	#431	—	2	1	no	—
301	#423	—	2	1	no	—
302	#407	—	2	1	no	—
303	C₄×M₁₆	—	2	1	no	—
304	C₄×M₁₆	—	2	1	no	—
305	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
306	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
307	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
308	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
309	C₂×C₄×C₈	—	2	1	no	—
310	#329	C₂²×C₃₁	1	16	yes	complemented
311	C₂×C₃₁	—	2	2	no	—
312	C₂×C₃₁	—	2	2	no	—
313	C₂×C₃₁	—	4	1	no	—
314	C₂×C₃₁	—	8	4	no	—
315	C₂×C₃₁	C₂²×#100	1	1	yes	characteristic
316	C₂×C₃₁	C₂²×#104	1	1	yes	characteristic
317	C₂×C₃₁	C₂×#421	1	1	yes	characteristic
318	C₂×#41	—	2	2	no	—
319	C₂×#41	—	2	2	no	—
320	C₂×#41	—	4	8	no	—
321	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
322	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
323	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
324	C₂²×D₄	—	2	2	no	—
325	C₂²×D₄	—	2	1	no	—
326	C₂²×D₄	—	2	1	no	—
327	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
328	C₂³×C₄	—	2	2	no	—
329	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
330	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
331	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
332	C₂×M₁₆	—	2	8	no	—
333	C₂×M₁₆	—	2	2	no	—
334	C₂×M₁₆	—	2	2	no	—
335	C₂×M₁₆	—	2	2	no	—
336	C₂²×C₈	—	2	2	no	—
337	C₄⋊Q₈	—	2	1	no	—
338	C₄⋊Q₈	—	2	1	no	—
339	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
340	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
341	#127	—	2	1	no	—
342	#127	—	2	1	no	—
343	#127	C₂×C₄×C₃₁	1	2	yes	—
344	#127	C₂×C₄×C₃₁	1	2	yes	—
345	#124	—	2	2	no	—
346	#124	—	2	2	no	—
347	#124	C₂×C₄×C₃₁	1	4	yes	—
348	#124	C₂³×C₃₁	1	4	yes	—
349	#121	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	2	2	no	—
352	#121	—	2	2	no	—
353	#121	—	2	2	no	—
354	#121	—	2	2	no	—
355	#121	—	2	2	no	—
356	#121	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	—	2	2	no	—
362	C₄×D₄	—	2	2	no	—
363	C₄×D₄	—	2	2	no	—
364	C₄×D₄	—	2	2	no	—
365	C₂×C₄⋊C₄	—	2	1	no	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₂²⋊C₄	—	2	2	no	—
368	C₂×C₂²⋊C₄	—	2	2	no	—
369	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
370	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
371	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
372	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
373	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
374	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
375	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
376	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
377	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
378	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
379	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
380	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
381	C₄⋊C₈	—	2	2	no	—
382	C₄⋊C₈	—	2	2	no	—
383	C₂²⋊C₈	—	2	2	no	—
384	C₂²⋊C₈	—	2	2	no	—
385	#97	—	2	2	no	—
386	#97	C₂³×C₃₁	1	4	yes	—
387	#97	C₂³×C₃₁	1	4	yes	—
388	C₄×C₈	—	2	2	no	—
389	C₄×C₈	—	2	2	no	—
390	C₄×C₈	—	2	2	no	—
391	C₃₁	#17647	1	1	yes	characteristic, direct product factor, Sylow
392	C₂⁴	—	2	2	no	—
393	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
394	#41	—	4	8	no	—
395	#41	—	4	8	no	—
396	#41	—	4	8	no	—
397	#41	—	4	8	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	4	4	no	—
400	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	4	4	no	—
409	C₂×D₄	—	4	4	no	—
410	C₂×D₄	—	4	8	no	—
411	C₂×D₄	—	4	4	no	—
412	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	4	2	no	—
419	C₂²×C₄	—	4	8	no	—
420	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
421	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
422	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
423	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
425	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
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	C₂×D₄×C₃₁	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
434	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
436	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
438	M₁₆	—	2	8	no	—
439	M₁₆	—	2	8	no	—
440	M₁₆	—	4	4	no	—
441	C₂×C₈	—	2	8	no	—
442	C₂×C₈	—	2	4	no	—
443	C₂×C₈	—	2	4	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	2	no	—
448	C₂²⋊C₄	—	2	4	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
453	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
454	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
455	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
456	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
457	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
458	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
459	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
460	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
461	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
466	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
467	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
468	C₂³	—	2	2	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	4	no	—
471	C₂³	—	4	1	no	—
472	C₂³	—	4	4	no	—
473	C₂³	—	4	4	no	—
474	C₂³	—	4	4	no	—
475	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
476	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
477	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
478	Q₈	—	4	4	no	—
479	Q₈	—	4	4	no	—
480	D₄	—	4	4	no	—
481	D₄	—	4	4	no	—
482	D₄	—	4	8	no	—
483	D₄	—	4	8	no	—
484	D₄	—	4	4	no	—
485	D₄	—	4	4	no	—
486	D₄	—	4	4	no	—
487	D₄	—	4	4	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	2	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	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	4	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	8	no	—
506	C₂×C₄	—	4	4	no	—
507	C₂×C₄	—	4	8	no	—
508	C₂×C₄	C₃₁×#100	1	4	yes	—
509	C₂×C₄	C₃₁×#104	1	4	yes	—
510	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
511	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
512	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
513	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
514	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
515	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
516	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
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	C₂³×C₄×C₃₁	1	1	yes	characteristic
520	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
521	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
522	C₈	—	2	8	no	—
523	C₈	—	4	4	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	1	no	—
527	C₂²	—	2	1	no	—
528	C₂²	—	2	1	no	—
529	C₂²	—	2	1	no	—
530	C₂²	—	2	1	no	—
531	C₂²	—	4	4	no	—
532	C₂²	—	4	4	no	—
533	C₂²	—	4	4	no	—
534	C₂²	—	4	4	no	—
535	C₂²	—	8	8	no	—
536	C₂²	C₂×C₃₁×#100	1	2	yes	—
537	C₂²	C₂×C₃₁×#104	1	2	yes	—
538	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
539	C₄	—	2	1	no	—
540	C₄	—	2	1	no	—
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	1	no	—
544	C₄	—	2	1	no	—
545	C₄	—	2	1	no	—
546	C₄	—	4	4	no	—
547	C₄	C₂×C₃₁×#100	1	2	yes	—
548	C₄	C₂×C₃₁×#100	1	2	yes	—
549	C₄	C₂×C₃₁×#104	1	2	yes	—
550	C₄	C₂×C₃₁×#104	1	2	yes	—
551	C₄	C₃₁×#421	1	2	yes	—
552	C₂	—	2	2	no	—
553	C₂	—	2	2	no	—
554	C₂	—	4	1	no	—
555	C₂	—	8	4	no	—
556	C₂	C₂²×C₃₁×#100	1	1	yes	characteristic
557	C₂	C₂²×C₃₁×#104	1	1	yes	characteristic
558	C₂	C₂×C₃₁×#421	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. C₃₁×#2920	C₂	semidirect
3. C₂×C₄×M₁₆×C₃₁	C₂	semidirect
4. C₃₁×#1530	C₂	semidirect
5. C₃₁×#1527	C₂	semidirect
6. C₃₁×#1525	C₂	semidirect
26. C₃₁×#432	C₂²	semidirect
34. C₄×M₁₆×C₃₁	C₂²	semidirect
35. C₄×M₁₆×C₃₁	C₂²	semidirect
36. C₂×C₃₁×#97	C₂²	semidirect
38. C₃₁×#329	C₂²	semidirect
188. #17647	C₃₁	direct (N × H)
246. #2920	C₂×C₃₁	semidirect
247. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
248. #1530	C₂×C₃₁	semidirect
249. #1527	C₂×C₃₁	semidirect
250. #1525	C₂×C₃₁	semidirect
298. #432	C₂²×C₃₁	semidirect
306. C₄×M₁₆	C₂²×C₃₁	semidirect
307. C₄×M₁₆	C₂²×C₃₁	semidirect
308. C₂×#97	C₂²×C₃₁	semidirect
310. #329	C₂²×C₃₁	semidirect
391. C₃₁	#17647	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
C₃₁×#2920	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#1530	2
C₃₁×#1527	2
C₃₁×#1525	2
#17647	31

Subgroups — C31×#17647 #21474406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17647 #21474406