Subgroups — C₃₁×#22234 #21478993

Order 7936.

714 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1994	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1962	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1929	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1866	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#131	C₂²	1	2	yes	—
11	C₃₁×#533	C₂²	1	1	yes	characteristic
12	C₃₁×#532	C₂²	1	1	yes	characteristic
13	C₃₁×#488	C₄	1	4	yes	complemented
14	C₃₁×#487	C₄	1	1	yes	characteristic, complemented
15	C₃₁×#487	C₄	1	1	yes	characteristic, complemented
16	C₃₁×#486	C₄	1	1	yes	characteristic, complemented
17	C₃₁×#486	C₄	1	1	yes	characteristic, complemented
18	C₃₁×#446	—	2	1	no	—
19	C₃₁×#446	—	2	1	no	—
20	C₈×Q₈×C₃₁	—	2	1	no	—
21	C₈×Q₈×C₃₁	—	2	1	no	—
22	C₃₁×#436	—	2	1	no	—
23	C₃₁×#436	—	2	1	no	—
24	C₈×D₄×C₃₁	—	2	1	no	—
25	C₈×D₄×C₃₁	—	2	1	no	—
26	C₂×C₃₁×#108	C₂²	1	4	yes	—
27	C₃₁×#424	C₂²	1	2	yes	—
28	C₃₁×#423	C₂²	1	1	yes	characteristic
29	C₃₁×#423	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#104	C₂²	1	4	yes	—
31	C₃₁×#416	C₂²	1	4	yes	—
32	C₃₁×#408	C₂²	1	2	yes	—
33	C₃₁×#405	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₄³×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#339	C₂²	1	4	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₂×#41×C₃₁	C₂³	1	2	yes	—
41	C₃₁×#131	—	4	4	no	—
42	C₂×M₁₆×C₃₁	—	2	2	no	—
43	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
44	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	—	2	2	no	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
50	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
51	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
52	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
53	C₃₁×#125	C₂×C₄	1	2	yes	—
54	C₃₁×#124	C₂×C₄	1	2	yes	—
55	C₃₁×C₂²⋊Q₈	—	2	1	no	—
56	C₃₁×C₂²⋊Q₈	—	2	1	no	—
57	C₃₁×#121	—	2	1	no	—
58	C₃₁×#121	—	2	1	no	—
59	C₄×Q₈×C₃₁	—	2	1	no	—
60	C₄×Q₈×C₃₁	—	2	1	no	—
61	C₄×D₄×C₃₁	—	2	1	no	—
62	C₄×D₄×C₃₁	—	2	1	no	—
63	C₃₁×#117	C₂³	1	2	yes	—
64	C₂×C₄²×C₃₁	—	2	1	no	—
65	C₂×C₄²×C₃₁	—	2	1	no	—
66	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#108	—	4	4	no	—
69	C₃₁×C₄⋊C₈	—	2	2	no	—
70	C₃₁×C₄⋊C₈	—	2	2	no	—
71	C₃₁×C₄⋊C₈	—	2	1	no	—
72	C₃₁×C₄⋊C₈	—	2	1	no	—
73	C₃₁×C₄⋊C₈	—	2	1	no	—
74	C₃₁×C₄⋊C₈	—	2	1	no	—
75	C₃₁×#104	—	4	4	no	—
76	C₃₁×#103	C₂×C₄	1	4	yes	—
77	C₃₁×#103	C₂×C₄	1	4	yes	—
78	C₃₁×#102	C₂×C₄	1	4	yes	—
79	C₃₁×#102	C₂×C₄	1	4	yes	—
80	C₃₁×C₂²⋊C₈	—	2	2	no	—
81	C₃₁×C₂²⋊C₈	—	2	2	no	—
82	C₃₁×#97	—	2	2	no	—
83	C₄×C₈×C₃₁	—	2	2	no	—
84	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
86	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
87	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
88	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
89	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
90	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
91	#41×C₃₁	—	2	4	no	—
92	#41×C₃₁	—	2	4	no	—
93	C₂×Q₈×C₃₁	—	2	1	no	—
94	C₂×Q₈×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
105	M₁₆×C₃₁	—	2	4	no	—
106	M₁₆×C₃₁	—	2	4	no	—
107	M₁₆×C₃₁	—	4	4	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	4	4	no	—
115	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄×C₃₁	—	2	1	no	—
122	C₄⋊C₄×C₃₁	—	2	1	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
128	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
129	C₂²⋊C₄×C₃₁	—	2	2	no	—
130	C₂²⋊C₄×C₃₁	—	2	2	no	—
131	C₂²⋊C₄×C₃₁	—	2	2	no	—
132	C₄²×C₃₁	—	2	4	no	—
133	C₄²×C₃₁	—	2	2	no	—
134	C₄²×C₃₁	—	2	2	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	—	4	2	no	—
137	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
138	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
141	#22234	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂³×C₃₁	—	2	2	no	—
143	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
144	Q₈×C₃₁	—	2	4	no	—
145	Q₈×C₃₁	—	4	2	no	—
146	Q₈×C₃₁	—	4	2	no	—
147	D₄×C₃₁	—	2	4	no	—
148	D₄×C₃₁	—	4	4	no	—
149	D₄×C₃₁	—	4	2	no	—
150	D₄×C₃₁	—	4	2	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	4	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
162	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
168	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
169	C₈×C₃₁	—	2	2	no	—
170	C₈×C₃₁	—	2	2	no	—
171	C₈×C₃₁	—	2	1	no	—
172	C₈×C₃₁	—	2	1	no	—
173	C₈×C₃₁	—	2	1	no	—
174	C₈×C₃₁	—	2	1	no	—
175	C₈×C₃₁	—	4	2	no	—
176	C₈×C₃₁	—	4	2	no	—
177	#3131	C₂×C₃₁	1	1	yes	characteristic
178	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#2931	C₂×C₃₁	1	1	yes	characteristic
180	#1994	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#1962	C₂×C₃₁	1	1	yes	characteristic
182	#1929	C₂×C₃₁	1	4	yes	complemented
183	#1866	C₂×C₃₁	1	4	yes	complemented
184	#1848	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
186	C₂²×C₃₁	—	2	1	no	—
187	C₂²×C₃₁	—	2	1	no	—
188	C₂²×C₃₁	—	4	2	no	—
189	C₂²×C₃₁	—	4	2	no	—
190	C₂²×C₃₁	—	4	2	no	—
191	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂²×C₃₁	#542	1	2	yes	—
193	C₄×C₃₁	—	2	1	no	—
194	C₄×C₃₁	—	2	1	no	—
195	C₄×C₃₁	—	4	4	no	—
196	C₄×C₃₁	—	4	2	no	—
197	C₄×C₃₁	—	4	2	no	—
198	C₄×C₃₁	—	4	2	no	—
199	C₄×C₃₁	#443	1	2	yes	—
200	C₄×C₃₁	#444	1	2	yes	—
201	C₄×C₃₁	#520	1	2	yes	—
202	C₄×C₃₁	#535	1	1	yes	characteristic
203	C₄×C₃₁	#536	1	1	yes	characteristic
204	C₂×#131	C₂²×C₃₁	1	2	yes	—
205	#533	C₂²×C₃₁	1	1	yes	characteristic
206	#532	C₂²×C₃₁	1	1	yes	characteristic
207	#488	C₄×C₃₁	1	4	yes	complemented
208	#487	C₄×C₃₁	1	1	yes	characteristic, complemented
209	#487	C₄×C₃₁	1	1	yes	characteristic, complemented
210	#486	C₄×C₃₁	1	1	yes	characteristic, complemented
211	#486	C₄×C₃₁	1	1	yes	characteristic, complemented
212	#446	—	2	1	no	—
213	#446	—	2	1	no	—
214	C₈×Q₈	—	2	1	no	—
215	C₈×Q₈	—	2	1	no	—
216	#436	—	2	1	no	—
217	#436	—	2	1	no	—
218	C₈×D₄	—	2	1	no	—
219	C₈×D₄	—	2	1	no	—
220	C₂×#108	C₂²×C₃₁	1	4	yes	—
221	#424	C₂²×C₃₁	1	2	yes	—
222	#423	C₂²×C₃₁	1	1	yes	characteristic
223	#423	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×#104	C₂²×C₃₁	1	4	yes	—
225	#416	C₂²×C₃₁	1	4	yes	—
226	#408	C₂²×C₃₁	1	2	yes	—
227	#405	C₂²×C₃₁	1	2	yes	—
228	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₄³	C₂²×C₃₁	1	1	yes	characteristic
231	#339	C₂²×C₃₁	1	4	yes	—
232	#336	C₂²×C₃₁	1	2	yes	—
233	#336	C₂²×C₃₁	1	2	yes	—
234	C₂×C₃₁	—	2	2	no	—
235	C₂×C₃₁	—	8	2	no	—
236	C₂×C₃₁	#2326	1	1	yes	characteristic
237	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
238	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
239	C₂×#41	C₂³×C₃₁	1	2	yes	—
240	#131	—	4	4	no	—
241	C₂×M₁₆	—	2	2	no	—
242	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
243	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
244	C₂²×C₈	—	2	2	no	—
245	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
246	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
247	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
248	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
249	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
250	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
251	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
252	#125	C₂×C₄×C₃₁	1	2	yes	—
253	#124	C₂×C₄×C₃₁	1	2	yes	—
254	C₂²⋊Q₈	—	2	1	no	—
255	C₂²⋊Q₈	—	2	1	no	—
256	#121	—	2	1	no	—
257	#121	—	2	1	no	—
258	C₄×Q₈	—	2	1	no	—
259	C₄×Q₈	—	2	1	no	—
260	C₄×D₄	—	2	1	no	—
261	C₄×D₄	—	2	1	no	—
262	#117	C₂³×C₃₁	1	2	yes	—
263	C₂×C₄²	—	2	1	no	—
264	C₂×C₄²	—	2	1	no	—
265	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
266	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
267	#108	—	4	4	no	—
268	C₄⋊C₈	—	2	2	no	—
269	C₄⋊C₈	—	2	2	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	2	1	no	—
274	#104	—	4	4	no	—
275	#103	C₂×C₄×C₃₁	1	4	yes	—
276	#103	C₂×C₄×C₃₁	1	4	yes	—
277	#102	C₂×C₄×C₃₁	1	4	yes	—
278	#102	C₂×C₄×C₃₁	1	4	yes	—
279	C₂²⋊C₈	—	2	2	no	—
280	C₂²⋊C₈	—	2	2	no	—
281	#97	—	2	2	no	—
282	C₄×C₈	—	2	2	no	—
283	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
284	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
285	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
286	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
287	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
288	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
289	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
290	C₃₁	#22234	1	1	yes	characteristic, direct product factor, Sylow
291	#41	—	2	4	no	—
292	#41	—	2	4	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×Q₈	—	2	1	no	—
295	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	1	no	—
298	C₂×D₄	—	2	1	no	—
299	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
303	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
304	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
305	M₁₆	—	2	4	no	—
306	M₁₆	—	2	4	no	—
307	M₁₆	—	4	4	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	4	4	no	—
315	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
316	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
317	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
328	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
329	C₂²⋊C₄	—	2	2	no	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₄²	—	2	4	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	4	2	no	—
337	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
338	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
339	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂³	—	2	2	no	—
342	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
343	Q₈	—	2	4	no	—
344	Q₈	—	4	2	no	—
345	Q₈	—	4	2	no	—
346	D₄	—	2	4	no	—
347	D₄	—	4	4	no	—
348	D₄	—	4	2	no	—
349	D₄	—	4	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
361	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
362	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
363	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
364	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
365	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
366	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
367	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
368	C₈	—	2	2	no	—
369	C₈	—	2	2	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	1	no	—
374	C₈	—	4	2	no	—
375	C₈	—	4	2	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	2	no	—
380	C₂²	—	4	2	no	—
381	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
382	C₂²	C₃₁×#542	1	2	yes	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	4	4	no	—
386	C₄	—	4	2	no	—
387	C₄	—	4	2	no	—
388	C₄	—	4	2	no	—
389	C₄	C₃₁×#443	1	2	yes	—
390	C₄	C₃₁×#444	1	2	yes	—
391	C₄	C₃₁×#520	1	2	yes	—
392	C₄	C₃₁×#535	1	1	yes	characteristic
393	C₄	C₃₁×#536	1	1	yes	characteristic
394	C₂	—	2	2	no	—
395	C₂	—	8	2	no	—
396	C₂	C₃₁×#2326	1	1	yes	characteristic
397	C₂	C₂×C₃₁×#443	1	1	yes	characteristic
398	C₂	C₂×C₃₁×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#532	C₂²
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
2. C₃₁×#2960	C₂	semidirect
4. C₃₁×#1994	C₂	semidirect
6. C₃₁×#1929	C₂	semidirect
7. C₃₁×#1866	C₂	semidirect
8. C₃₁×#1848	C₂	semidirect
9. C₃₁×#1744	C₂	semidirect
13. C₃₁×#488	C₄	semidirect
14. C₃₁×#487	C₄	semidirect
15. C₃₁×#487	C₄	semidirect
16. C₃₁×#486	C₄	semidirect
17. C₃₁×#486	C₄	semidirect
141. #22234	C₃₁	direct (N × H)
178. #2960	C₂×C₃₁	semidirect
180. #1994	C₂×C₃₁	semidirect
182. #1929	C₂×C₃₁	semidirect
183. #1866	C₂×C₃₁	semidirect
184. #1848	C₂×C₃₁	semidirect
185. #1744	C₂×C₃₁	semidirect
207. #488	C₄×C₃₁	semidirect
208. #487	C₄×C₃₁	semidirect
209. #487	C₄×C₃₁	semidirect
210. #486	C₄×C₃₁	semidirect
211. #486	C₄×C₃₁	semidirect
290. C₃₁	#22234	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×#3131	2
C₃₁×#2960	2
C₃₁×#2931	2
C₃₁×#1994	2
C₃₁×#1962	2
C₃₁×#1929	2
C₃₁×#1866	2
C₃₁×#1848	2
C₃₁×#1744	2
#22234	31

Subgroups — C31×#22234 #21478993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22234 #21478993