Subgroups — C₃₁×#24065 #21480824

Order 7936.

1072 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#485	C₂	1	2	yes	maximal
3	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2058	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2057	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2043	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2043	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1873	C₂	1	2	yes	maximal
9	C₃₁×#1865	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1817	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#533	—	2	8	no	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#125	—	2	1	no	—
16	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
17	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₃₁×#117	—	2	1	no	—
21	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#485	—	2	8	no	—
25	C₂×C₃₁×#107	C₂²	1	2	yes	—
26	C₂×C₃₁×#106	C₂²	1	2	yes	—
27	C₂×C₃₁×#103	C₂²	1	2	yes	—
28	C₂×C₃₁×#103	C₂²	1	2	yes	—
29	C₂×C₃₁×#103	C₂²	1	2	yes	—
30	C₂×C₃₁×#103	C₂²	1	2	yes	—
31	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₃₁×#389	C₂²	1	1	yes	characteristic
33	C₃₁×#389	C₂²	1	1	yes	characteristic
34	C₃₁×#389	C₂²	1	1	yes	characteristic
35	C₃₁×#389	C₂²	1	1	yes	characteristic
36	C₃₁×#385	—	2	1	no	—
37	C₃₁×#385	—	2	1	no	—
38	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
39	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₃₁×#336	C₂²	1	1	yes	characteristic
44	C₃₁×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	—	2	1	no	—
46	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
47	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₃₁×C₄⋊Q₈	—	2	4	no	—
53	C₃₁×C₄⋊Q₈	—	2	4	no	—
54	C₃₁×#125	—	2	8	no	—
55	C₃₁×C₂²⋊Q₈	—	2	8	no	—
56	C₃₁×C₂²⋊Q₈	—	2	4	no	—
57	C₃₁×C₂²⋊Q₈	D₄	1	8	yes	—
58	C₄×Q₈×C₃₁	—	2	8	no	—
59	C₄×Q₈×C₃₁	—	2	4	no	—
60	C₄×Q₈×C₃₁	—	2	4	no	—
61	C₃₁×#117	—	2	8	no	—
62	C₃₁×#117	—	2	2	no	—
63	C₃₁×#117	—	2	2	no	—
64	C₃₁×#117	—	2	4	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄²×C₃₁	—	2	4	no	—
83	C₂×C₄²×C₃₁	—	2	2	no	—
84	C₂×C₄²×C₃₁	—	2	1	no	—
85	C₂×C₄²×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
90	C₃₁×#107	—	2	4	no	—
91	C₃₁×#106	—	2	4	no	—
92	C₃₁×#103	—	2	4	no	—
93	C₃₁×#103	—	2	4	no	—
94	C₃₁×#103	—	2	4	no	—
95	C₃₁×#103	—	2	4	no	—
96	C₃₁×C₂²⋊C₈	—	2	4	no	—
97	C₃₁×#95	—	2	1	no	—
98	C₃₁×#95	—	2	1	no	—
99	C₃₁×#95	—	2	1	no	—
100	C₃₁×#95	—	2	1	no	—
101	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×Q₈×C₃₁	—	2	4	no	—
103	C₂×Q₈×C₃₁	—	2	2	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	4	2	no	—
109	C₂×Q₈×C₃₁	—	4	2	no	—
110	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	4	1	no	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	4	no	—
140	C₄⋊C₄×C₃₁	—	2	4	no	—
141	C₄⋊C₄×C₃₁	—	2	8	no	—
142	C₄⋊C₄×C₃₁	—	2	4	no	—
143	C₄⋊C₄×C₃₁	—	2	8	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	1	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	4	2	no	—
150	C₄⋊C₄×C₃₁	—	4	2	no	—
151	C₄⋊C₄×C₃₁	—	4	2	no	—
152	C₄⋊C₄×C₃₁	—	4	2	no	—
153	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
154	C₄⋊C₄×C₃₁	#41	1	2	yes	—
155	C₄⋊C₄×C₃₁	#41	1	2	yes	—
156	C₂²⋊C₄×C₃₁	—	2	4	no	—
157	C₂²⋊C₄×C₃₁	—	2	8	no	—
158	C₂²⋊C₄×C₃₁	—	4	2	no	—
159	C₄²×C₃₁	—	2	4	no	—
160	C₄²×C₃₁	—	2	8	no	—
161	C₄²×C₃₁	—	2	2	no	—
162	C₄²×C₃₁	—	2	2	no	—
163	C₄²×C₃₁	—	2	2	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	4	2	no	—
166	#24065	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₃₁	—	2	1	no	—
168	C₂³×C₃₁	—	2	1	no	—
169	C₂³×C₃₁	—	2	1	no	—
170	C₂³×C₃₁	—	2	1	no	—
171	C₂³×C₃₁	—	2	1	no	—
172	C₂³×C₃₁	—	2	1	no	—
173	C₂³×C₃₁	#120	1	2	yes	—
174	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	Q₈×C₃₁	—	2	4	no	—
176	Q₈×C₃₁	—	2	4	no	—
177	Q₈×C₃₁	—	4	8	no	—
178	Q₈×C₃₁	—	4	2	no	—
179	Q₈×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	2	8	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	4	no	—
190	C₂×C₄×C₃₁	—	2	4	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	4	no	—
193	C₂×C₄×C₃₁	—	2	4	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	4	4	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	#120	1	2	yes	—
214	C₂×C₄×C₃₁	#135	1	2	yes	—
215	C₂×C₄×C₃₁	#137	1	2	yes	—
216	C₂×C₄×C₃₁	#137	1	2	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
221	C₈×C₃₁	—	4	2	no	—
222	C₈×C₃₁	—	4	2	no	—
223	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
224	C₂×#485	C₂×C₃₁	1	2	yes	—
225	#2296	C₂×C₃₁	1	1	yes	characteristic
226	#2058	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#2057	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#2043	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#2043	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1873	C₂×C₃₁	1	2	yes	—
231	#1865	C₂×C₃₁	1	2	yes	complemented
232	#1863	C₂×C₃₁	1	2	yes	complemented
233	#1817	C₂×C₃₁	1	1	yes	characteristic
234	C₂²×C₃₁	—	2	2	no	—
235	C₂²×C₃₁	—	2	2	no	—
236	C₂²×C₃₁	—	2	2	no	—
237	C₂²×C₃₁	—	4	1	no	—
238	C₂²×C₃₁	—	4	1	no	—
239	C₂²×C₃₁	—	4	1	no	—
240	C₂²×C₃₁	—	4	1	no	—
241	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
242	C₂²×C₃₁	#525	1	1	yes	characteristic
243	C₂²×C₃₁	#536	1	1	yes	characteristic
244	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
245	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
246	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
247	C₂²×C₃₁	#575	1	1	yes	characteristic
248	C₄×C₃₁	—	2	2	no	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	2	2	no	—
251	C₄×C₃₁	—	2	2	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	4	2	no	—
254	C₄×C₃₁	—	4	4	no	—
255	C₄×C₃₁	—	4	1	no	—
256	C₄×C₃₁	—	4	4	no	—
257	C₄×C₃₁	—	8	1	no	—
258	C₄×C₃₁	—	8	1	no	—
259	C₄×C₃₁	#547	1	2	yes	—
260	C₄×C₃₁	#547	1	2	yes	—
261	#533	—	2	8	no	—
262	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
263	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
264	C₂×#125	—	2	1	no	—
265	C₂×C₂²⋊Q₈	—	2	1	no	—
266	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
267	C₂×C₄×Q₈	—	2	1	no	—
268	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
269	C₂×#117	—	2	1	no	—
270	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
272	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
273	#485	—	2	8	no	—
274	C₂×#107	C₂²×C₃₁	1	2	yes	—
275	C₂×#106	C₂²×C₃₁	1	2	yes	—
276	C₂×#103	C₂²×C₃₁	1	2	yes	—
277	C₂×#103	C₂²×C₃₁	1	2	yes	—
278	C₂×#103	C₂²×C₃₁	1	2	yes	—
279	C₂×#103	C₂²×C₃₁	1	2	yes	—
280	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
281	#389	C₂²×C₃₁	1	1	yes	characteristic
282	#389	C₂²×C₃₁	1	1	yes	characteristic
283	#389	C₂²×C₃₁	1	1	yes	characteristic
284	#389	C₂²×C₃₁	1	1	yes	characteristic
285	#385	—	2	1	no	—
286	#385	—	2	1	no	—
287	C₄×C₄⋊C₄	—	2	1	no	—
288	C₄×C₄⋊C₄	—	2	1	no	—
289	#340	C₂²×C₃₁	1	2	yes	—
290	#340	C₂²×C₃₁	1	2	yes	—
291	#336	C₂²×C₃₁	1	1	yes	characteristic
292	#336	C₂²×C₃₁	1	1	yes	characteristic
293	#336	C₂²×C₃₁	1	2	yes	—
294	C₂×C₃₁	—	4	2	no	—
295	C₂×C₃₁	#2406	1	1	yes	characteristic
296	C₂×C₃₁	#3002	1	1	yes	characteristic
297	C₂×C₃₁	#3004	1	1	yes	characteristic
298	C₂×C₃₁	#3107	1	1	yes	characteristic
299	C₂×C₃₁	#3116	1	1	yes	characteristic
300	C₂×C₃₁	#3231	1	1	yes	characteristic
301	C₂×C₃₁	#3237	1	1	yes	characteristic
302	C₂²×Q₈	—	2	1	no	—
303	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
304	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
305	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
306	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
307	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
308	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
309	C₄⋊Q₈	—	2	4	no	—
310	C₄⋊Q₈	—	2	4	no	—
311	#125	—	2	8	no	—
312	C₂²⋊Q₈	—	2	8	no	—
313	C₂²⋊Q₈	—	2	4	no	—
314	C₂²⋊Q₈	D₄×C₃₁	1	8	yes	—
315	C₄×Q₈	—	2	8	no	—
316	C₄×Q₈	—	2	4	no	—
317	C₄×Q₈	—	2	4	no	—
318	#117	—	2	8	no	—
319	#117	—	2	2	no	—
320	#117	—	2	2	no	—
321	#117	—	2	4	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	2	no	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	—	2	1	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
332	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
333	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₂²⋊C₄	—	2	1	no	—
338	C₂×C₂²⋊C₄	—	2	1	no	—
339	C₂×C₄²	—	2	2	no	—
340	C₂×C₄²	—	2	1	no	—
341	C₂×C₄²	—	2	4	no	—
342	C₂×C₄²	—	2	1	no	—
343	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
344	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
347	#107	—	2	4	no	—
348	#106	—	2	4	no	—
349	#103	—	2	4	no	—
350	#103	—	2	4	no	—
351	#103	—	2	4	no	—
352	#103	—	2	4	no	—
353	C₂²⋊C₈	—	2	4	no	—
354	#95	—	2	1	no	—
355	#95	—	2	1	no	—
356	#95	—	2	1	no	—
357	#95	—	2	1	no	—
358	C₃₁	#24065	1	1	yes	characteristic, direct product factor, Sylow
359	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₂×Q₈	—	2	4	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	—	2	2	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	2	no	—
366	C₂×Q₈	—	4	2	no	—
367	C₂×Q₈	—	4	2	no	—
368	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	4	1	no	—
384	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	8	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	8	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	1	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	4	2	no	—
408	C₄⋊C₄	—	4	2	no	—
409	C₄⋊C₄	—	4	2	no	—
410	C₄⋊C₄	—	4	2	no	—
411	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
412	C₄⋊C₄	#41×C₃₁	1	2	yes	—
413	C₄⋊C₄	#41×C₃₁	1	2	yes	—
414	C₂²⋊C₄	—	2	4	no	—
415	C₂²⋊C₄	—	2	8	no	—
416	C₂²⋊C₄	—	4	2	no	—
417	C₄²	—	2	4	no	—
418	C₄²	—	2	8	no	—
419	C₄²	—	2	2	no	—
420	C₄²	—	2	2	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	2	no	—
423	C₄²	—	4	2	no	—
424	C₂³	—	2	1	no	—
425	C₂³	—	2	1	no	—
426	C₂³	—	2	1	no	—
427	C₂³	—	2	1	no	—
428	C₂³	—	2	1	no	—
429	C₂³	—	2	1	no	—
430	C₂³	C₃₁×#120	1	2	yes	—
431	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
432	Q₈	—	2	4	no	—
433	Q₈	—	2	4	no	—
434	Q₈	—	4	8	no	—
435	Q₈	—	4	2	no	—
436	Q₈	—	4	2	no	—
437	C₂×C₄	—	2	8	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	C₃₁×#120	1	2	yes	—
471	C₂×C₄	C₃₁×#135	1	2	yes	—
472	C₂×C₄	C₃₁×#137	1	2	yes	—
473	C₂×C₄	C₃₁×#137	1	2	yes	—
474	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
475	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
476	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
477	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
478	C₈	—	4	2	no	—
479	C₈	—	4	2	no	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	4	1	no	—
484	C₂²	—	4	1	no	—
485	C₂²	—	4	1	no	—
486	C₂²	—	4	1	no	—
487	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
488	C₂²	C₃₁×#525	1	1	yes	characteristic
489	C₂²	C₃₁×#536	1	1	yes	characteristic
490	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
491	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
492	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
493	C₂²	C₃₁×#575	1	1	yes	characteristic
494	C₄	—	2	2	no	—
495	C₄	—	2	2	no	—
496	C₄	—	2	2	no	—
497	C₄	—	2	2	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	2	no	—
500	C₄	—	4	4	no	—
501	C₄	—	4	1	no	—
502	C₄	—	4	4	no	—
503	C₄	—	8	1	no	—
504	C₄	—	8	1	no	—
505	C₄	C₃₁×#547	1	2	yes	—
506	C₄	C₃₁×#547	1	2	yes	—
507	C₂	—	4	2	no	—
508	C₂	C₃₁×#2406	1	1	yes	characteristic
509	C₂	C₃₁×#3002	1	1	yes	characteristic
510	C₂	C₃₁×#3004	1	1	yes	characteristic
511	C₂	C₃₁×#3107	1	1	yes	characteristic
512	C₂	C₃₁×#3116	1	1	yes	characteristic
513	C₂	C₃₁×#3231	1	1	yes	characteristic
514	C₂	C₃₁×#3237	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#2058	C₂	semidirect
5. C₃₁×#2057	C₂	semidirect
6. C₃₁×#2043	C₂	semidirect
7. C₃₁×#2043	C₂	semidirect
9. C₃₁×#1865	C₂	semidirect
10. C₃₁×#1863	C₂	semidirect
166. #24065	C₃₁	direct (N × H)
226. #2058	C₂×C₃₁	semidirect
227. #2057	C₂×C₃₁	semidirect
228. #2043	C₂×C₃₁	semidirect
229. #2043	C₂×C₃₁	semidirect
231. #1865	C₂×C₃₁	semidirect
232. #1863	C₂×C₃₁	semidirect
358. C₃₁	#24065	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×#533	2
C₂×C₃₁×#485	2
C₃₁×#2296	2
C₃₁×#2058	2
C₃₁×#2057	2
C₃₁×#2043	2
C₃₁×#1873	2
C₃₁×#1865	2
C₃₁×#1863	2
C₃₁×#1817	2
#24065	31

Subgroups — C31×#24065 #21480824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24065 #21480824