Subgroups — C₃₁×#17944 #21474703

Order 7936.

602 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3107	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1591	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1590	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1588	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1587	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1574	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1489	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1485	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#485	C₄	1	2	yes	—
15	C₃₁×#475	C₄	1	1	yes	characteristic
16	C₃₁×#474	C₄	1	1	yes	characteristic
17	C₃₁×#462	C₄	1	1	yes	characteristic
18	C₃₁×#461	C₄	1	1	yes	characteristic
19	C₃₁×#452	C₄	1	2	yes	—
20	C₃₁×#446	—	2	1	no	—
21	C₃₁×#446	C₂²	1	1	yes	characteristic
22	C₃₁×#446	C₂²	1	2	yes	—
23	C₃₁×#446	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₃₁	—	2	1	no	—
25	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
27	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#436	C₂²	1	1	yes	characteristic
29	C₃₁×#436	C₂²	1	1	yes	characteristic
30	C₃₁×#435	—	2	1	no	—
31	C₃₁×#435	—	2	1	no	—
32	C₃₁×#431	—	2	1	no	—
33	C₃₁×#431	C₂²	1	1	yes	characteristic
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₃₁×#424	—	2	1	no	—
36	C₃₁×#423	C₂²	1	1	yes	characteristic
37	C₃₁×#405	—	2	1	no	—
38	C₄×M₁₆×C₃₁	—	2	1	no	—
39	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#335	C₂²	1	2	yes	—
42	C₃₁×#334	C₂²	1	2	yes	—
43	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
45	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
46	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
47	C₃₁×#322	C₂²	1	2	yes	—
48	C₈²×C₃₁	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₃₁	—	2	1	no	—
53	C₂×M₁₆×C₃₁	—	2	1	no	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂×M₁₆×C₃₁	—	2	1	no	—
56	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
59	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
60	C₃₁×#125	—	2	1	no	—
61	C₃₁×C₂²⋊Q₈	—	2	1	no	—
62	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
63	C₄×Q₈×C₃₁	—	2	1	no	—
64	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
66	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₃₁	—	2	1	no	—
68	C₄×D₄×C₃₁	—	2	1	no	—
69	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#117	—	2	1	no	—
72	C₃₁×#117	—	2	1	no	—
73	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#107	C₂×C₄	1	2	yes	—
77	C₃₁×#106	C₂×C₄	1	2	yes	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	2	1	no	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
84	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
85	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
86	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₃₁×#103	C₂×C₄	1	2	yes	—
89	C₃₁×#103	C₂×C₄	1	2	yes	—
90	C₃₁×#103	C₂×C₄	1	2	yes	—
91	C₃₁×#102	C₂×C₄	1	2	yes	—
92	C₃₁×C₂²⋊C₈	—	2	1	no	—
93	C₃₁×C₂²⋊C₈	—	2	1	no	—
94	C₃₁×C₂²⋊C₈	—	2	1	no	—
95	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
96	C₃₁×#97	—	2	1	no	—
97	C₃₁×#97	—	2	1	no	—
98	C₃₁×#97	—	2	2	no	—
99	C₃₁×#97	—	2	1	no	—
100	C₄×C₈×C₃₁	—	2	2	no	—
101	C₄×C₈×C₃₁	—	2	1	no	—
102	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
103	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
104	C₄×C₈×C₃₁	C₂³	1	2	yes	—
105	C₄×C₈×C₃₁	C₂³	1	2	yes	—
106	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
107	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
108	#41×C₃₁	—	4	1	no	—
109	#41×C₃₁	—	4	1	no	—
110	C₂×Q₈×C₃₁	—	2	1	no	—
111	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
121	Q₁₆×C₃₁	—	4	2	no	—
122	SD₁₆×C₃₁	—	4	2	no	—
123	M₁₆×C₃₁	—	2	4	no	—
124	M₁₆×C₃₁	—	4	2	no	—
125	M₁₆×C₃₁	—	4	2	no	—
126	M₁₆×C₃₁	—	4	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	4	1	no	—
133	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
134	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
135	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
147	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
149	C₂²⋊C₄×C₃₁	—	2	2	no	—
150	C₂²⋊C₄×C₃₁	—	2	1	no	—
151	C₂²⋊C₄×C₃₁	—	2	1	no	—
152	C₂²⋊C₄×C₃₁	—	2	1	no	—
153	C₄²×C₃₁	—	2	1	no	—
154	C₄²×C₃₁	—	2	1	no	—
155	C₄²×C₃₁	—	2	2	no	—
156	C₄²×C₃₁	—	2	1	no	—
157	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₃₁	#41	1	1	yes	characteristic
160	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
161	#17944	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂³×C₃₁	—	2	1	no	—
163	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
164	Q₈×C₃₁	—	2	2	no	—
165	Q₈×C₃₁	—	2	1	no	—
166	Q₈×C₃₁	—	2	2	no	—
167	Q₈×C₃₁	—	2	1	no	—
168	Q₈×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	2	1	no	—
170	D₄×C₃₁	—	2	1	no	—
171	D₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
192	C₈×C₃₁	—	2	4	no	—
193	C₈×C₃₁	—	2	2	no	—
194	C₈×C₃₁	—	2	2	no	—
195	C₈×C₃₁	—	2	1	no	—
196	C₈×C₃₁	—	2	1	no	—
197	C₈×C₃₁	—	2	1	no	—
198	C₈×C₃₁	—	4	1	no	—
199	C₈×C₃₁	—	4	1	no	—
200	C₈×C₃₁	—	4	2	no	—
201	C₈×C₃₁	#137	1	2	yes	—
202	#3107	C₂×C₃₁	1	1	yes	characteristic
203	#2965	C₂×C₃₁	1	1	yes	characteristic
204	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1591	C₂×C₃₁	1	2	yes	complemented
206	#1590	C₂×C₃₁	1	2	yes	complemented
207	#1588	C₂×C₃₁	1	2	yes	complemented
208	#1587	C₂×C₃₁	1	2	yes	complemented
209	#1574	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#1489	C₂×C₃₁	1	1	yes	characteristic
211	#1485	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂²×C₃₁	—	2	1	no	—
214	C₂²×C₃₁	—	2	1	no	—
215	C₂²×C₃₁	—	2	1	no	—
216	C₂²×C₃₁	—	4	1	no	—
217	C₂²×C₃₁	—	4	1	no	—
218	C₂²×C₃₁	—	4	1	no	—
219	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
220	C₄×C₃₁	—	2	1	no	—
221	C₄×C₃₁	—	2	1	no	—
222	C₄×C₃₁	—	2	1	no	—
223	C₄×C₃₁	—	4	1	no	—
224	C₄×C₃₁	—	4	2	no	—
225	C₄×C₃₁	—	4	1	no	—
226	C₄×C₃₁	—	4	1	no	—
227	C₄×C₃₁	—	4	2	no	—
228	C₄×C₃₁	#443	1	2	yes	—
229	C₄×C₃₁	#568	1	1	yes	characteristic
230	C₄×C₃₁	#568	1	1	yes	characteristic
231	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
232	C₄×C₃₁	#575	1	1	yes	characteristic
233	#533	C₂²×C₃₁	1	1	yes	characteristic
234	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
235	#485	C₄×C₃₁	1	2	yes	—
236	#475	C₄×C₃₁	1	1	yes	characteristic
237	#474	C₄×C₃₁	1	1	yes	characteristic
238	#462	C₄×C₃₁	1	1	yes	characteristic
239	#461	C₄×C₃₁	1	1	yes	characteristic
240	#452	C₄×C₃₁	1	2	yes	—
241	#446	—	2	1	no	—
242	#446	C₂²×C₃₁	1	1	yes	characteristic
243	#446	C₂²×C₃₁	1	2	yes	—
244	#446	C₂²×C₃₁	1	1	yes	characteristic
245	C₈×Q₈	—	2	1	no	—
246	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
247	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
248	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
249	#436	C₂²×C₃₁	1	1	yes	characteristic
250	#436	C₂²×C₃₁	1	1	yes	characteristic
251	#435	—	2	1	no	—
252	#435	—	2	1	no	—
253	#431	—	2	1	no	—
254	#431	C₂²×C₃₁	1	1	yes	characteristic
255	#431	C₂²×C₃₁	1	1	yes	characteristic
256	#424	—	2	1	no	—
257	#423	C₂²×C₃₁	1	1	yes	characteristic
258	#405	—	2	1	no	—
259	C₄×M₁₆	—	2	1	no	—
260	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
262	#335	C₂²×C₃₁	1	2	yes	—
263	#334	C₂²×C₃₁	1	2	yes	—
264	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
265	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
266	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
267	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
268	#322	C₂²×C₃₁	1	2	yes	—
269	C₈²	C₂²×C₃₁	1	2	yes	—
270	C₂×C₃₁	—	4	1	no	—
271	C₂×C₃₁	—	8	1	no	—
272	C₂×C₃₁	#2930	1	1	yes	characteristic
273	C₂×C₃₁	C₄×#137	1	1	yes	characteristic
274	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
275	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
276	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
277	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
278	C₂×M₁₆	—	2	1	no	—
279	C₂×M₁₆	—	2	1	no	—
280	C₂×M₁₆	—	2	1	no	—
281	C₂×M₁₆	—	2	1	no	—
282	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
283	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
284	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
285	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
286	#125	—	2	1	no	—
287	C₂²⋊Q₈	—	2	1	no	—
288	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
291	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
293	C₄×D₄	—	2	1	no	—
294	C₄×D₄	—	2	1	no	—
295	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
296	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
297	#117	—	2	1	no	—
298	#117	—	2	1	no	—
299	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
300	C₂×C₄²	—	2	1	no	—
301	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
302	#107	C₂×C₄×C₃₁	1	2	yes	—
303	#106	C₂×C₄×C₃₁	1	2	yes	—
304	C₄⋊C₈	—	2	2	no	—
305	C₄⋊C₈	—	2	2	no	—
306	C₄⋊C₈	—	2	1	no	—
307	C₄⋊C₈	—	2	1	no	—
308	C₄⋊C₈	—	2	1	no	—
309	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
310	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
311	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
312	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
313	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
314	#103	C₂×C₄×C₃₁	1	2	yes	—
315	#103	C₂×C₄×C₃₁	1	2	yes	—
316	#103	C₂×C₄×C₃₁	1	2	yes	—
317	#102	C₂×C₄×C₃₁	1	2	yes	—
318	C₂²⋊C₈	—	2	1	no	—
319	C₂²⋊C₈	—	2	1	no	—
320	C₂²⋊C₈	—	2	1	no	—
321	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
322	#97	—	2	1	no	—
323	#97	—	2	1	no	—
324	#97	—	2	2	no	—
325	#97	—	2	1	no	—
326	C₄×C₈	—	2	2	no	—
327	C₄×C₈	—	2	1	no	—
328	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
329	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
330	C₄×C₈	C₂³×C₃₁	1	2	yes	—
331	C₄×C₈	C₂³×C₃₁	1	2	yes	—
332	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
333	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
334	C₃₁	#17944	1	1	yes	characteristic, direct product factor, Sylow
335	#41	—	4	1	no	—
336	#41	—	4	1	no	—
337	C₂×Q₈	—	2	1	no	—
338	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
339	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
347	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
348	Q₁₆	—	4	2	no	—
349	SD₁₆	—	4	2	no	—
350	M₁₆	—	2	4	no	—
351	M₁₆	—	4	2	no	—
352	M₁₆	—	4	2	no	—
353	M₁₆	—	4	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	4	1	no	—
360	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
361	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
362	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
374	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
375	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
376	C₂²⋊C₄	—	2	2	no	—
377	C₂²⋊C₄	—	2	1	no	—
378	C₂²⋊C₄	—	2	1	no	—
379	C₂²⋊C₄	—	2	1	no	—
380	C₄²	—	2	1	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	1	no	—
384	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
385	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
386	C₄²	#41×C₃₁	1	1	yes	characteristic
387	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
388	C₂³	—	2	1	no	—
389	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
390	Q₈	—	2	2	no	—
391	Q₈	—	2	1	no	—
392	Q₈	—	2	2	no	—
393	Q₈	—	2	1	no	—
394	Q₈	—	4	2	no	—
395	D₄	—	2	1	no	—
396	D₄	—	2	1	no	—
397	D₄	—	4	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
415	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
418	C₈	—	2	4	no	—
419	C₈	—	2	2	no	—
420	C₈	—	2	2	no	—
421	C₈	—	2	1	no	—
422	C₈	—	2	1	no	—
423	C₈	—	2	1	no	—
424	C₈	—	4	1	no	—
425	C₈	—	4	1	no	—
426	C₈	—	4	2	no	—
427	C₈	C₃₁×#137	1	2	yes	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	2	1	no	—
431	C₂²	—	4	1	no	—
432	C₂²	—	4	1	no	—
433	C₂²	—	4	1	no	—
434	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
435	C₄	—	2	1	no	—
436	C₄	—	2	1	no	—
437	C₄	—	2	1	no	—
438	C₄	—	4	1	no	—
439	C₄	—	4	2	no	—
440	C₄	—	4	1	no	—
441	C₄	—	4	1	no	—
442	C₄	—	4	2	no	—
443	C₄	C₃₁×#443	1	2	yes	—
444	C₄	C₃₁×#568	1	1	yes	characteristic
445	C₄	C₃₁×#568	1	1	yes	characteristic
446	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
447	C₄	C₃₁×#575	1	1	yes	characteristic
448	C₂	—	4	1	no	—
449	C₂	—	8	1	no	—
450	C₂	C₃₁×#2930	1	1	yes	characteristic
451	C₂	C₄×C₃₁×#137	1	1	yes	characteristic
452	C₂	C₂×C₃₁×#443	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	C₂×C₄×D₄
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2960	C₂	semidirect
4. C₃₁×#1591	C₂	semidirect
5. C₃₁×#1590	C₂	semidirect
6. C₃₁×#1588	C₂	semidirect
7. C₃₁×#1587	C₂	semidirect
8. C₃₁×#1574	C₂	semidirect
10. C₃₁×#1485	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
161. #17944	C₃₁	direct (N × H)
204. #2960	C₂×C₃₁	semidirect
205. #1591	C₂×C₃₁	semidirect
206. #1590	C₂×C₃₁	semidirect
207. #1588	C₂×C₃₁	semidirect
208. #1587	C₂×C₃₁	semidirect
209. #1574	C₂×C₃₁	semidirect
211. #1485	C₂×C₃₁	semidirect
212. C₈×M₁₆	C₂×C₃₁	semidirect
334. C₃₁	#17944	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×#3107	2
C₃₁×#2965	2
C₃₁×#2960	2
C₃₁×#1591	2
C₃₁×#1590	2
C₃₁×#1588	2
C₃₁×#1587	2
C₃₁×#1574	2
C₃₁×#1489	2
C₃₁×#1485	2
C₈×M₁₆×C₃₁	2
#17944	31

Subgroups — C31×#17944 #21474703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17944 #21474703