Subgroups — C₃₁×#19683 #21476442

Order 7936.

724 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3209	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3206	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3167	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3141	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1723	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1722	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1718	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1714	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃₁×#501	—	2	2	no	—
17	C₃₁×#500	—	2	2	no	—
18	C₃₁×#500	C₂²	1	2	yes	—
19	C₃₁×#500	C₂²	1	2	yes	—
20	C₃₁×#500	C₂²	1	2	yes	—
21	C₃₁×#498	C₂²	1	2	yes	—
22	C₃₁×#498	C₂²	1	1	yes	characteristic
23	C₃₁×#494	C₂²	1	2	yes	—
24	C₃₁×#492	C₂²	1	1	yes	characteristic
25	C₃₁×#492	C₂²	1	2	yes	—
26	C₃₁×#489	—	2	2	no	—
27	C₃₁×#487	C₂²	1	2	yes	—
28	C₃₁×#487	C₂²	1	1	yes	characteristic
29	C₃₁×#487	C₂²	1	2	yes	—
30	C₃₁×#487	C₂²	1	2	yes	—
31	C₃₁×#486	C₂²	1	2	yes	—
32	C₃₁×#486	C₂²	1	1	yes	characteristic
33	C₃₁×#465	—	2	2	no	—
34	C₃₁×#462	—	2	2	no	—
35	C₃₁×#461	—	2	2	no	—
36	C₃₁×#451	—	2	2	no	—
37	C₃₁×#448	—	2	2	no	—
38	C₄×SD₁₆×C₃₁	—	2	2	no	—
39	C₃₁×#431	C₂²	1	1	yes	characteristic
40	C₃₁×#428	—	2	2	no	—
41	C₂×C₃₁×#107	—	2	2	no	—
42	C₃₁×#417	—	2	2	no	—
43	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#322	C₂²	1	2	yes	—
46	C₃₁×#322	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	—	2	1	no	—
49	C₂²×Q₈×C₃₁	—	2	1	no	—
50	C₂×Q₁₆×C₃₁	—	2	4	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	2	4	no	—
53	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
54	C₂²×C₈×C₃₁	D₄	1	2	yes	—
55	C₃₁×C₄⋊Q₈	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₃₁×#127	C₂³	1	1	yes	characteristic
67	C₃₁×#125	—	2	1	no	—
68	C₃₁×#124	—	2	1	no	—
69	C₃₁×C₂²⋊Q₈	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	—	2	1	no	—
71	C₃₁×#121	—	2	1	no	—
72	C₃₁×#121	—	2	1	no	—
73	C₄×Q₈×C₃₁	—	2	1	no	—
74	C₄×Q₈×C₃₁	—	2	1	no	—
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	—	2	1	no	—
77	C₃₁×#117	—	2	1	no	—
78	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#107	—	2	4	no	—
84	C₃₁×#107	—	2	4	no	—
85	C₃₁×#107	—	2	4	no	—
86	C₃₁×#106	—	2	4	no	—
87	C₃₁×#106	—	2	2	no	—
88	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
89	C₃₁×#103	—	2	4	no	—
90	C₃₁×#103	—	2	2	no	—
91	C₃₁×#103	—	2	4	no	—
92	C₃₁×#103	—	2	4	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
96	C₃₁×#97	D₄	1	2	yes	—
97	C₄×C₈×C₃₁	D₄	1	2	yes	—
98	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	2	yes	—
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	#41×C₃₁	—	4	4	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	4	1	no	—
109	C₂×Q₈×C₃₁	—	4	1	no	—
110	C₂×Q₈×C₃₁	—	8	2	no	—
111	C₂×Q₈×C₃₁	—	8	2	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	4	1	no	—
116	C₂²×C₄×C₃₁	—	4	1	no	—
117	C₂²×C₄×C₃₁	—	4	1	no	—
118	C₂²×C₄×C₃₁	—	4	1	no	—
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	Q₁₆×C₃₁	—	8	4	no	—
123	SD₁₆×C₃₁	—	4	4	no	—
124	SD₁₆×C₃₁	—	8	4	no	—
125	M₁₆×C₃₁	—	2	4	no	—
126	C₂×C₈×C₃₁	—	2	4	no	—
127	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
129	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
130	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	1	no	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	4	1	no	—
142	C₄⋊C₄×C₃₁	—	4	1	no	—
143	C₄⋊C₄×C₃₁	—	4	1	no	—
144	C₄⋊C₄×C₃₁	—	4	1	no	—
145	C₄⋊C₄×C₃₁	—	4	1	no	—
146	C₄⋊C₄×C₃₁	—	4	1	no	—
147	C₂²⋊C₄×C₃₁	—	4	1	no	—
148	C₂²⋊C₄×C₃₁	—	4	1	no	—
149	C₂²⋊C₄×C₃₁	—	4	1	no	—
150	C₄²×C₃₁	—	4	1	no	—
151	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
155	#19683	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	4	1	no	—
157	C₂³×C₃₁	#127	1	1	yes	characteristic
158	Q₈×C₃₁	—	4	2	no	—
159	Q₈×C₃₁	—	4	2	no	—
160	Q₈×C₃₁	—	4	4	no	—
161	Q₈×C₃₁	—	4	2	no	—
162	Q₈×C₃₁	—	4	4	no	—
163	Q₈×C₃₁	—	8	2	no	—
164	Q₈×C₃₁	—	8	2	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	4	2	no	—
168	D₄×C₃₁	—	8	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	8	2	no	—
177	C₂×C₄×C₃₁	—	8	2	no	—
178	C₂×C₄×C₃₁	—	8	2	no	—
179	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
180	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
181	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
182	C₂×C₄×C₃₁	#137	1	4	yes	—
183	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₈×C₃₁	—	2	2	no	—
187	C₈×C₃₁	—	2	4	no	—
188	C₈×C₃₁	—	2	4	no	—
189	C₈×C₃₁	#137	1	4	yes	—
190	#3209	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#3206	C₂×C₃₁	1	1	yes	characteristic
192	#3167	C₂×C₃₁	1	1	yes	characteristic
193	#3165	C₂×C₃₁	1	1	yes	characteristic, complemented
194	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#3141	C₂×C₃₁	1	1	yes	characteristic
196	#1723	C₂×C₃₁	1	2	yes	complemented
197	#1722	C₂×C₃₁	1	2	yes	complemented
198	#1718	C₂×C₃₁	1	2	yes	complemented
199	#1714	C₂×C₃₁	1	2	yes	complemented
200	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
201	C₂²×C₃₁	—	2	1	no	—
202	C₂²×C₃₁	—	2	1	no	—
203	C₂²×C₃₁	—	8	1	no	—
204	C₂²×C₃₁	—	8	1	no	—
205	C₂²×C₃₁	—	8	1	no	—
206	C₂²×C₃₁	#497	1	2	yes	—
207	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
208	C₄×C₃₁	—	2	1	no	—
209	C₄×C₃₁	—	2	1	no	—
210	C₄×C₃₁	—	8	1	no	—
211	C₄×C₃₁	—	8	1	no	—
212	C₄×C₃₁	—	8	2	no	—
213	C₄×C₃₁	—	8	2	no	—
214	C₄×C₃₁	—	8	1	no	—
215	C₄×C₃₁	#497	1	2	yes	—
216	C₄×C₃₁	C₂×#137	1	2	yes	—
217	C₄×C₃₁	C₂×#137	1	2	yes	—
218	C₄×C₃₁	#576	1	2	yes	—
219	#533	C₂²×C₃₁	1	1	yes	characteristic
220	#532	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
223	#501	—	2	2	no	—
224	#500	—	2	2	no	—
225	#500	C₂²×C₃₁	1	2	yes	—
226	#500	C₂²×C₃₁	1	2	yes	—
227	#500	C₂²×C₃₁	1	2	yes	—
228	#498	C₂²×C₃₁	1	2	yes	—
229	#498	C₂²×C₃₁	1	1	yes	characteristic
230	#494	C₂²×C₃₁	1	2	yes	—
231	#492	C₂²×C₃₁	1	1	yes	characteristic
232	#492	C₂²×C₃₁	1	2	yes	—
233	#489	—	2	2	no	—
234	#487	C₂²×C₃₁	1	2	yes	—
235	#487	C₂²×C₃₁	1	1	yes	characteristic
236	#487	C₂²×C₃₁	1	2	yes	—
237	#487	C₂²×C₃₁	1	2	yes	—
238	#486	C₂²×C₃₁	1	2	yes	—
239	#486	C₂²×C₃₁	1	1	yes	characteristic
240	#465	—	2	2	no	—
241	#462	—	2	2	no	—
242	#461	—	2	2	no	—
243	#451	—	2	2	no	—
244	#448	—	2	2	no	—
245	C₄×SD₁₆	—	2	2	no	—
246	#431	C₂²×C₃₁	1	1	yes	characteristic
247	#428	—	2	2	no	—
248	C₂×#107	—	2	2	no	—
249	#417	—	2	2	no	—
250	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
252	#322	C₂²×C₃₁	1	2	yes	—
253	#322	C₂²×C₃₁	1	2	yes	—
254	C₂×C₃₁	—	2	2	no	—
255	C₂×C₃₁	—	16	1	no	—
256	C₂×C₃₁	C₂×#497	1	1	yes	characteristic
257	C₂×C₃₁	#3152	1	1	yes	characteristic
258	C₂×C₃₁	#3155	1	1	yes	characteristic
259	C₂×#41	—	2	1	no	—
260	C₂²×Q₈	—	2	1	no	—
261	C₂²×Q₈	—	2	1	no	—
262	C₂×Q₁₆	—	2	4	no	—
263	C₂×SD₁₆	—	2	2	no	—
264	C₂×SD₁₆	—	2	4	no	—
265	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
266	C₂²×C₈	D₄×C₃₁	1	2	yes	—
267	C₄⋊Q₈	—	2	1	no	—
268	C₄⋊Q₈	—	2	1	no	—
269	C₄⋊Q₈	—	2	1	no	—
270	C₄⋊Q₈	—	2	1	no	—
271	C₄⋊Q₈	—	2	1	no	—
272	C₄⋊Q₈	—	2	1	no	—
273	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
274	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
275	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
278	#127	C₂³×C₃₁	1	1	yes	characteristic
279	#125	—	2	1	no	—
280	#124	—	2	1	no	—
281	C₂²⋊Q₈	—	2	1	no	—
282	C₂²⋊Q₈	—	2	1	no	—
283	#121	—	2	1	no	—
284	#121	—	2	1	no	—
285	C₄×Q₈	—	2	1	no	—
286	C₄×Q₈	—	2	1	no	—
287	C₄×D₄	—	2	1	no	—
288	C₄×D₄	—	2	1	no	—
289	#117	—	2	1	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
295	#107	—	2	4	no	—
296	#107	—	2	4	no	—
297	#107	—	2	4	no	—
298	#106	—	2	4	no	—
299	#106	—	2	2	no	—
300	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
301	#103	—	2	4	no	—
302	#103	—	2	2	no	—
303	#103	—	2	4	no	—
304	#103	—	2	4	no	—
305	#102	—	2	4	no	—
306	#102	—	2	2	no	—
307	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
308	#97	D₄×C₃₁	1	2	yes	—
309	C₄×C₈	D₄×C₃₁	1	2	yes	—
310	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
311	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₄×C₈	C₂³×C₃₁	1	2	yes	—
313	C₄×C₈	C₂³×C₃₁	1	2	yes	—
314	C₃₁	#19683	1	1	yes	characteristic, direct product factor, Sylow
315	#41	—	4	4	no	—
316	C₂×Q₈	—	2	1	no	—
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	—	2	2	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×Q₈	—	4	1	no	—
322	C₂×Q₈	—	4	1	no	—
323	C₂×Q₈	—	8	2	no	—
324	C₂×Q₈	—	8	2	no	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	—	4	1	no	—
329	C₂²×C₄	—	4	1	no	—
330	C₂²×C₄	—	4	1	no	—
331	C₂²×C₄	—	4	1	no	—
332	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
335	Q₁₆	—	8	4	no	—
336	SD₁₆	—	4	4	no	—
337	SD₁₆	—	8	4	no	—
338	M₁₆	—	2	4	no	—
339	C₂×C₈	—	2	4	no	—
340	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
341	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
342	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
343	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	4	1	no	—
355	C₄⋊C₄	—	4	1	no	—
356	C₄⋊C₄	—	4	1	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	—	4	1	no	—
359	C₄⋊C₄	—	4	1	no	—
360	C₂²⋊C₄	—	4	1	no	—
361	C₂²⋊C₄	—	4	1	no	—
362	C₂²⋊C₄	—	4	1	no	—
363	C₄²	—	4	1	no	—
364	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
366	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
368	C₂³	—	4	1	no	—
369	C₂³	C₃₁×#127	1	1	yes	characteristic
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	4	4	no	—
373	Q₈	—	4	2	no	—
374	Q₈	—	4	4	no	—
375	Q₈	—	8	2	no	—
376	Q₈	—	8	2	no	—
377	D₄	—	4	2	no	—
378	D₄	—	4	2	no	—
379	D₄	—	4	2	no	—
380	D₄	—	8	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	8	2	no	—
389	C₂×C₄	—	8	2	no	—
390	C₂×C₄	—	8	2	no	—
391	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
392	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
393	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
394	C₂×C₄	C₃₁×#137	1	4	yes	—
395	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
397	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
398	C₈	—	2	2	no	—
399	C₈	—	2	4	no	—
400	C₈	—	2	4	no	—
401	C₈	C₃₁×#137	1	4	yes	—
402	C₂²	—	2	1	no	—
403	C₂²	—	2	1	no	—
404	C₂²	—	8	1	no	—
405	C₂²	—	8	1	no	—
406	C₂²	—	8	1	no	—
407	C₂²	C₃₁×#497	1	2	yes	—
408	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
409	C₄	—	2	1	no	—
410	C₄	—	2	1	no	—
411	C₄	—	8	1	no	—
412	C₄	—	8	1	no	—
413	C₄	—	8	2	no	—
414	C₄	—	8	2	no	—
415	C₄	—	8	1	no	—
416	C₄	C₃₁×#497	1	2	yes	—
417	C₄	C₂×C₃₁×#137	1	2	yes	—
418	C₄	C₂×C₃₁×#137	1	2	yes	—
419	C₄	C₃₁×#576	1	2	yes	—
420	C₂	—	2	2	no	—
421	C₂	—	16	1	no	—
422	C₂	C₂×C₃₁×#497	1	1	yes	characteristic
423	C₂	C₃₁×#3152	1	1	yes	characteristic
424	C₂	C₃₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3209	C₂	semidirect
4. C₃₁×#3165	C₂	semidirect
5. C₂×C₃₁×#500	C₂	semidirect
7. C₃₁×#1723	C₂	semidirect
8. C₃₁×#1722	C₂	semidirect
9. C₃₁×#1718	C₂	semidirect
10. C₃₁×#1714	C₂	semidirect
11. C₃₁×#1451	C₂	semidirect
155. #19683	C₃₁	direct (N × H)
190. #3209	C₂×C₃₁	semidirect
193. #3165	C₂×C₃₁	semidirect
194. C₂×#500	C₂×C₃₁	semidirect
196. #1723	C₂×C₃₁	semidirect
197. #1722	C₂×C₃₁	semidirect
198. #1718	C₂×C₃₁	semidirect
199. #1714	C₂×C₃₁	semidirect
200. #1451	C₂×C₃₁	semidirect
314. C₃₁	#19683	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×#3209	2
C₃₁×#3206	2
C₃₁×#3167	2
C₃₁×#3165	2
C₂×C₃₁×#500	2
C₃₁×#3141	2
C₃₁×#1723	2
C₃₁×#1722	2
C₃₁×#1718	2
C₃₁×#1714	2
C₃₁×#1451	2
#19683	31

Subgroups — C31×#19683 #21476442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19683 #21476442