Subgroups — C₃₁×#23717 #21480476

Order 7936.

752 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	C₄×C₃₁×#137	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2089	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2078	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#2064	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2052	C₂	1	2	yes	maximal
8	C₃₁×#2047	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2020	C₂	1	1	yes	maximal, characteristic
10	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#101	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#441	—	2	2	no	—
17	C₃₁×#440	—	2	2	no	—
18	C₄×Q₁₆×C₃₁	—	2	2	no	—
19	C₄×Q₁₆×C₃₁	—	2	2	no	—
20	C₄×Q₁₆×C₃₁	—	2	2	no	—
21	C₄×SD₁₆×C₃₁	—	2	2	no	—
22	C₃₁×#433	—	2	2	no	—
23	C₃₁×#431	—	2	2	no	—
24	C₂×C₃₁×#108	C₂²	1	2	yes	—
25	C₃₁×#428	C₂²	1	2	yes	—
26	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
27	C₃₁×#424	C₂²	1	1	yes	characteristic
28	C₃₁×#424	C₂²	1	1	yes	characteristic
29	C₃₁×#419	C₂²	1	2	yes	—
30	C₃₁×#417	C₂²	1	2	yes	—
31	C₂×C₃₁×#103	C₂²	1	2	yes	—
32	C₂×C₃₁×#101	C₂²	1	2	yes	—
33	C₂×C₃₁×#101	C₂²	1	2	yes	—
34	C₃₁×#408	C₂²	1	1	yes	characteristic
35	C₃₁×#408	C₂²	1	1	yes	characteristic
36	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#343	C₂²	1	2	yes	—
40	C₃₁×#343	C₂²	1	2	yes	—
41	C₃₁×#341	C₂²	1	1	yes	characteristic
42	C₃₁×#341	C₂²	1	1	yes	characteristic
43	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
45	C₃₁×#137	—	2	4	no	—
46	C₃₁×#137	—	2	4	no	—
47	C₂×Q₁₆×C₃₁	—	2	2	no	—
48	C₂×Q₁₆×C₃₁	—	2	4	no	—
49	C₂×Q₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
54	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₄×Q₈×C₃₁	—	2	2	no	—
58	C₄×Q₈×C₃₁	—	2	2	no	—
59	C₄×Q₈×C₃₁	—	2	2	no	—
60	C₄×Q₈×C₃₁	—	2	2	no	—
61	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
62	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
63	C₄×D₄×C₃₁	—	2	1	no	—
64	C₄×D₄×C₃₁	—	2	1	no	—
65	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
66	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
67	C₃₁×#117	—	2	1	no	—
68	C₃₁×#117	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#108	—	2	4	no	—
75	C₃₁×#107	—	2	2	no	—
76	C₃₁×#107	—	2	2	no	—
77	C₃₁×#106	—	2	2	no	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×#103	—	2	2	no	—
81	C₃₁×#103	—	2	4	no	—
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#101	—	2	4	no	—
86	C₃₁×#101	—	2	4	no	—
87	C₃₁×C₂²⋊C₈	—	2	2	no	—
88	C₃₁×C₂²⋊C₈	—	2	2	no	—
89	C₃₁×#97	—	2	1	no	—
90	C₃₁×#97	—	2	2	no	—
91	C₄×C₈×C₃₁	—	2	1	no	—
92	C₄×C₈×C₃₁	—	2	2	no	—
93	C₄×C₈×C₃₁	D₄	1	2	yes	—
94	C₄×C₈×C₃₁	D₄	1	2	yes	—
95	#41×C₃₁	—	2	2	no	—
96	#41×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	2	2	no	—
102	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×Q₈×C₃₁	#41	1	4	yes	—
104	C₂×D₄×C₃₁	—	2	1	no	—
105	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
114	Q₁₆×C₃₁	—	2	8	no	—
115	Q₁₆×C₃₁	—	2	8	no	—
116	SD₁₆×C₃₁	—	2	8	no	—
117	M₁₆×C₃₁	—	2	2	no	—
118	M₁₆×C₃₁	—	2	2	no	—
119	M₁₆×C₃₁	—	2	2	no	—
120	M₁₆×C₃₁	—	2	2	no	—
121	M₁₆×C₃₁	—	2	2	no	—
122	M₁₆×C₃₁	#41	1	4	yes	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
131	C₄⋊C₄×C₃₁	—	2	1	no	—
132	C₄⋊C₄×C₃₁	—	2	1	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²⋊C₄×C₃₁	—	2	1	no	—
140	C₂²⋊C₄×C₃₁	—	2	1	no	—
141	C₂²⋊C₄×C₃₁	—	2	1	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	4	2	no	—
146	C₄²×C₃₁	—	4	2	no	—
147	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
151	#23717	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂³×C₃₁	—	2	1	no	—
153	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
154	Q₈×C₃₁	—	2	4	no	—
155	Q₈×C₃₁	—	2	4	no	—
156	Q₈×C₃₁	—	2	2	no	—
157	Q₈×C₃₁	—	4	2	no	—
158	Q₈×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	2	2	no	—
160	D₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	8	2	no	—
175	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
176	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
177	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
178	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
182	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
183	C₈×C₃₁	—	2	2	no	—
184	C₈×C₃₁	—	2	4	no	—
185	C₈×C₃₁	—	2	2	no	—
186	C₈×C₃₁	—	4	1	no	—
187	C₈×C₃₁	—	4	2	no	—
188	C₈×C₃₁	—	4	1	no	—
189	C₄×#137	C₂×C₃₁	1	2	yes	—
190	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#2922	C₂×C₃₁	1	1	yes	characteristic
192	#2089	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#2078	C₂×C₃₁	1	1	yes	characteristic
194	#2064	C₂×C₃₁	1	2	yes	complemented
195	#2052	C₂×C₃₁	1	2	yes	—
196	#2047	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#2020	C₂×C₃₁	1	1	yes	characteristic
198	C₄×#108	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₄×#101	C₂×C₃₁	1	2	yes	complemented
200	C₂²×C₃₁	—	2	1	no	—
201	C₂²×C₃₁	—	2	1	no	—
202	C₂²×C₃₁	—	4	1	no	—
203	C₂²×C₃₁	—	4	1	no	—
204	C₂²×C₃₁	—	4	1	no	—
205	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
206	C₂²×C₃₁	#529	1	2	yes	—
207	C₄×C₃₁	—	2	1	no	—
208	C₄×C₃₁	—	2	1	no	—
209	C₄×C₃₁	—	2	1	no	—
210	C₄×C₃₁	—	4	1	no	—
211	C₄×C₃₁	—	4	4	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	—	4	1	no	—
214	C₄×C₃₁	—	8	2	no	—
215	C₄×C₃₁	#473	1	2	yes	—
216	C₄×C₃₁	#525	1	1	yes	characteristic
217	C₄×C₃₁	#525	1	2	yes	—
218	C₄×C₃₁	#532	1	1	yes	characteristic
219	C₂×#137	C₂²×C₃₁	1	2	yes	—
220	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
221	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
223	#441	—	2	2	no	—
224	#440	—	2	2	no	—
225	C₄×Q₁₆	—	2	2	no	—
226	C₄×Q₁₆	—	2	2	no	—
227	C₄×Q₁₆	—	2	2	no	—
228	C₄×SD₁₆	—	2	2	no	—
229	#433	—	2	2	no	—
230	#431	—	2	2	no	—
231	C₂×#108	C₂²×C₃₁	1	2	yes	—
232	#428	C₂²×C₃₁	1	2	yes	—
233	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
234	#424	C₂²×C₃₁	1	1	yes	characteristic
235	#424	C₂²×C₃₁	1	1	yes	characteristic
236	#419	C₂²×C₃₁	1	2	yes	—
237	#417	C₂²×C₃₁	1	2	yes	—
238	C₂×#103	C₂²×C₃₁	1	2	yes	—
239	C₂×#101	C₂²×C₃₁	1	2	yes	—
240	C₂×#101	C₂²×C₃₁	1	2	yes	—
241	#408	C₂²×C₃₁	1	1	yes	characteristic
242	#408	C₂²×C₃₁	1	1	yes	characteristic
243	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
244	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
245	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
246	#343	C₂²×C₃₁	1	2	yes	—
247	#343	C₂²×C₃₁	1	2	yes	—
248	#341	C₂²×C₃₁	1	1	yes	characteristic
249	#341	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×C₃₁	—	2	2	no	—
251	C₂×C₃₁	—	8	1	no	—
252	C₂×C₃₁	#2392	1	1	yes	characteristic
253	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
254	C₂×C₃₁	#3067	1	1	yes	characteristic
255	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
256	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
257	#137	—	2	4	no	—
258	#137	—	2	4	no	—
259	C₂×Q₁₆	—	2	2	no	—
260	C₂×Q₁₆	—	2	4	no	—
261	C₂×Q₁₆	—	2	2	no	—
262	C₂×SD₁₆	—	2	2	no	—
263	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
264	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
265	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
266	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
267	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
269	C₄×Q₈	—	2	2	no	—
270	C₄×Q₈	—	2	2	no	—
271	C₄×Q₈	—	2	2	no	—
272	C₄×Q₈	—	2	2	no	—
273	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
274	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
275	C₄×D₄	—	2	1	no	—
276	C₄×D₄	—	2	1	no	—
277	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
278	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
279	#117	—	2	1	no	—
280	#117	C₂³×C₃₁	1	1	yes	characteristic
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
283	C₂×C₄²	—	2	1	no	—
284	C₂×C₄²	—	2	2	no	—
285	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
286	#108	—	2	4	no	—
287	#107	—	2	2	no	—
288	#107	—	2	2	no	—
289	#106	—	2	2	no	—
290	C₄⋊C₈	—	2	2	no	—
291	C₄⋊C₈	—	2	2	no	—
292	#103	—	2	2	no	—
293	#103	—	2	4	no	—
294	#103	—	2	2	no	—
295	#103	—	2	2	no	—
296	#102	—	2	2	no	—
297	#101	—	2	4	no	—
298	#101	—	2	4	no	—
299	C₂²⋊C₈	—	2	2	no	—
300	C₂²⋊C₈	—	2	2	no	—
301	#97	—	2	1	no	—
302	#97	—	2	2	no	—
303	C₄×C₈	—	2	1	no	—
304	C₄×C₈	—	2	2	no	—
305	C₄×C₈	D₄×C₃₁	1	2	yes	—
306	C₄×C₈	D₄×C₃₁	1	2	yes	—
307	C₃₁	#23717	1	1	yes	characteristic, direct product factor, Sylow
308	#41	—	2	2	no	—
309	#41	—	2	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
316	C₂×Q₈	#41×C₃₁	1	4	yes	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	2	1	no	—
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
326	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
327	Q₁₆	—	2	8	no	—
328	Q₁₆	—	2	8	no	—
329	SD₁₆	—	2	8	no	—
330	M₁₆	—	2	2	no	—
331	M₁₆	—	2	2	no	—
332	M₁₆	—	2	2	no	—
333	M₁₆	—	2	2	no	—
334	M₁₆	—	2	2	no	—
335	M₁₆	#41×C₃₁	1	4	yes	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
343	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₄⋊C₄	—	4	2	no	—
350	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
351	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂²⋊C₄	—	2	1	no	—
353	C₂²⋊C₄	—	2	1	no	—
354	C₂²⋊C₄	—	2	1	no	—
355	C₄²	—	2	1	no	—
356	C₄²	—	2	1	no	—
357	C₄²	—	2	1	no	—
358	C₄²	—	4	2	no	—
359	C₄²	—	4	2	no	—
360	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂³	—	2	1	no	—
365	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
366	Q₈	—	2	4	no	—
367	Q₈	—	2	4	no	—
368	Q₈	—	2	2	no	—
369	Q₈	—	4	2	no	—
370	Q₈	—	4	2	no	—
371	D₄	—	2	2	no	—
372	D₄	—	4	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	8	2	no	—
387	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
388	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
389	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
390	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
392	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
393	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
394	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
395	C₈	—	2	2	no	—
396	C₈	—	2	4	no	—
397	C₈	—	2	2	no	—
398	C₈	—	4	1	no	—
399	C₈	—	4	2	no	—
400	C₈	—	4	1	no	—
401	C₂²	—	2	1	no	—
402	C₂²	—	2	1	no	—
403	C₂²	—	4	1	no	—
404	C₂²	—	4	1	no	—
405	C₂²	—	4	1	no	—
406	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
407	C₂²	C₃₁×#529	1	2	yes	—
408	C₄	—	2	1	no	—
409	C₄	—	2	1	no	—
410	C₄	—	2	1	no	—
411	C₄	—	4	1	no	—
412	C₄	—	4	4	no	—
413	C₄	—	4	1	no	—
414	C₄	—	4	1	no	—
415	C₄	—	8	2	no	—
416	C₄	C₃₁×#473	1	2	yes	—
417	C₄	C₃₁×#525	1	1	yes	characteristic
418	C₄	C₃₁×#525	1	2	yes	—
419	C₄	C₃₁×#532	1	1	yes	characteristic
420	C₂	—	2	2	no	—
421	C₂	—	8	1	no	—
422	C₂	C₃₁×#2392	1	1	yes	characteristic
423	C₂	C₂×C₃₁×#473	1	1	yes	characteristic
424	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×#121
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
4. C₃₁×#2089	C₂	semidirect
6. C₃₁×#2064	C₂	semidirect
8. C₃₁×#2047	C₂	semidirect
10. C₄×C₃₁×#108	C₂	semidirect
11. C₄×C₃₁×#101	C₂	semidirect
151. #23717	C₃₁	direct (N × H)
190. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
192. #2089	C₂×C₃₁	semidirect
194. #2064	C₂×C₃₁	semidirect
196. #2047	C₂×C₃₁	semidirect
198. C₄×#108	C₂×C₃₁	semidirect
199. C₄×#101	C₂×C₃₁	semidirect
307. C₃₁	#23717	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₃₁×#137	2
C₂×C₄×Q₁₆×C₃₁	2
C₃₁×#2922	2
C₃₁×#2089	2
C₃₁×#2078	2
C₃₁×#2064	2
C₃₁×#2052	2
C₃₁×#2047	2
C₃₁×#2020	2
C₄×C₃₁×#108	2
C₄×C₃₁×#101	2
#23717	31

Subgroups — C31×#23717 #21480476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23717 #21480476