Subgroups — C₃₁×#24135 #21480894

Order 7936.

1076 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#556	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#441	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2536	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2036	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2001	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1955	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1873	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1854	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1805	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#556	—	2	4	no	—
14	C₃₁×#556	—	2	4	no	—
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#126	—	2	2	no	—
17	C₂×C₃₁×#124	—	2	1	no	—
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
21	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#117	C₂²	1	2	yes	—
25	C₃₁×#477	—	2	4	no	—
26	C₃₁×#441	—	2	4	no	—
27	C₂×C₃₁×#107	C₂²	1	2	yes	—
28	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#103	C₂²	1	2	yes	—
32	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#103	C₂²	1	2	yes	—
34	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
36	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
37	C₃₁×#398	—	2	1	no	—
38	C₃₁×#394	—	2	1	no	—
39	C₃₁×#388	—	2	2	no	—
40	C₃₁×#382	C₂²	1	1	yes	characteristic
41	C₃₁×#380	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	2	yes	—
44	C₃₁×#340	C₂²	1	2	yes	—
45	C₂²×Q₈×C₃₁	—	2	1	no	—
46	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₃₁	—	2	2	no	—
51	C₂×Q₁₆×C₃₁	—	2	2	no	—
52	C₂×Q₁₆×C₃₁	—	2	2	no	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
58	C₃₁×#126	—	2	16	no	—
59	C₃₁×#124	—	2	8	no	—
60	C₃₁×C₂²⋊Q₈	—	2	4	no	—
61	C₃₁×C₂²⋊Q₈	—	2	4	no	—
62	C₃₁×C₂²⋊Q₈	—	2	4	no	—
63	C₃₁×C₂²⋊Q₈	—	4	2	no	—
64	C₄×Q₈×C₃₁	—	2	8	no	—
65	C₄×Q₈×C₃₁	—	2	2	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×Q₈×C₃₁	—	2	2	no	—
68	C₄×Q₈×C₃₁	D₄	1	4	yes	—
69	C₄×D₄×C₃₁	—	2	2	no	—
70	C₄×D₄×C₃₁	—	2	2	no	—
71	C₃₁×#117	—	2	4	no	—
72	C₃₁×#117	—	2	4	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
88	C₂×C₄²×C₃₁	—	2	2	no	—
89	C₂×C₄²×C₃₁	—	2	1	no	—
90	C₂×C₄²×C₃₁	—	2	1	no	—
91	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
92	C₃₁×#107	—	2	4	no	—
93	C₃₁×#106	—	2	2	no	—
94	C₃₁×C₄⋊C₈	—	2	2	no	—
95	C₃₁×#103	—	2	2	no	—
96	C₃₁×#103	—	2	2	no	—
97	C₃₁×#103	—	2	4	no	—
98	C₃₁×#103	—	4	2	no	—
99	C₃₁×C₂²⋊C₈	—	2	4	no	—
100	C₃₁×C₂²⋊C₈	—	4	2	no	—
101	C₃₁×#97	—	2	2	no	—
102	C₃₁×#95	—	2	2	no	—
103	C₃₁×#95	—	2	1	no	—
104	C₃₁×#95	—	2	1	no	—
105	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×Q₈×C₃₁	—	2	4	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	—	2	1	no	—
111	C₂×Q₈×C₃₁	—	2	1	no	—
112	C₂×Q₈×C₃₁	—	2	1	no	—
113	C₂×Q₈×C₃₁	—	2	1	no	—
114	C₂×Q₈×C₃₁	—	2	1	no	—
115	C₂×Q₈×C₃₁	—	2	1	no	—
116	C₂×Q₈×C₃₁	—	2	1	no	—
117	C₂×Q₈×C₃₁	—	4	4	no	—
118	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
119	C₂×Q₈×C₃₁	#41	1	2	yes	—
120	C₂×D₄×C₃₁	—	2	2	no	—
121	C₂×D₄×C₃₁	—	4	2	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	2	no	—
126	C₂²×C₄×C₃₁	—	2	2	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	2	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	2	1	no	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
138	Q₁₆×C₃₁	—	2	8	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	1	no	—
147	C₂×C₈×C₃₁	#41	1	2	yes	—
148	C₄⋊C₄×C₃₁	—	2	8	no	—
149	C₄⋊C₄×C₃₁	—	2	4	no	—
150	C₄⋊C₄×C₃₁	—	2	4	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	1	no	—
155	C₄⋊C₄×C₃₁	—	2	1	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	—	4	4	no	—
159	C₄⋊C₄×C₃₁	—	4	2	no	—
160	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
161	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
162	C₂²⋊C₄×C₃₁	—	2	8	no	—
163	C₂²⋊C₄×C₃₁	—	2	8	no	—
164	C₂²⋊C₄×C₃₁	—	2	4	no	—
165	C₂²⋊C₄×C₃₁	—	4	4	no	—
166	C₂²⋊C₄×C₃₁	—	4	4	no	—
167	C₄²×C₃₁	—	2	8	no	—
168	C₄²×C₃₁	—	2	4	no	—
169	C₄²×C₃₁	—	4	2	no	—
170	C₄²×C₃₁	C₂×D₄	1	2	yes	—
171	C₄²×C₃₁	C₂×D₄	1	2	yes	—
172	#24135	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₃₁	—	2	2	no	—
174	C₂³×C₃₁	—	2	2	no	—
175	C₂³×C₃₁	—	2	2	no	—
176	C₂³×C₃₁	—	2	2	no	—
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	2	2	no	—
180	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₃₁	—	2	2	no	—
182	Q₈×C₃₁	—	2	2	no	—
183	Q₈×C₃₁	—	2	2	no	—
184	Q₈×C₃₁	—	2	2	no	—
185	Q₈×C₃₁	—	4	8	no	—
186	Q₈×C₃₁	—	4	1	no	—
187	Q₈×C₃₁	—	4	1	no	—
188	Q₈×C₃₁	—	4	1	no	—
189	Q₈×C₃₁	—	4	1	no	—
190	D₄×C₃₁	—	4	4	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	4	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	4	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	#120	1	2	yes	—
216	C₂×C₄×C₃₁	#120	1	2	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
222	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
223	C₈×C₃₁	—	2	2	no	—
224	C₈×C₃₁	—	4	1	no	—
225	C₈×C₃₁	—	4	2	no	—
226	C₂×#556	C₂×C₃₁	1	1	yes	characteristic
227	C₂×#477	C₂×C₃₁	1	1	yes	characteristic, complemented
228	C₂×#441	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#2536	C₂×C₃₁	1	1	yes	characteristic
230	#2036	C₂×C₃₁	1	2	yes	complemented
231	#2001	C₂×C₃₁	1	2	yes	complemented
232	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#1955	C₂×C₃₁	1	1	yes	characteristic
234	#1873	C₂×C₃₁	1	2	yes	complemented
235	#1854	C₂×C₃₁	1	2	yes	complemented
236	#1805	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₂²×C₃₁	—	4	2	no	—
238	C₂²×C₃₁	—	4	2	no	—
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	—	4	2	no	—
241	C₂²×C₃₁	—	4	2	no	—
242	C₂²×C₃₁	—	4	2	no	—
243	C₂²×C₃₁	—	4	2	no	—
244	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
245	C₂²×C₃₁	#546	1	1	yes	characteristic
246	C₂²×C₃₁	#547	1	1	yes	characteristic
247	C₂²×C₃₁	#575	1	2	yes	—
248	C₂²×C₃₁	#578	1	2	yes	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	2	2	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	4	2	no	—
254	C₄×C₃₁	—	4	4	no	—
255	C₄×C₃₁	—	4	2	no	—
256	C₄×C₃₁	—	8	1	no	—
257	C₄×C₃₁	—	8	2	no	—
258	C₄×C₃₁	#525	1	1	yes	characteristic
259	C₄×C₃₁	#537	1	1	yes	characteristic
260	C₄×C₃₁	#548	1	1	yes	characteristic
261	C₄×C₃₁	#550	1	1	yes	characteristic
262	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
263	#556	—	2	4	no	—
264	#556	—	2	4	no	—
265	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×#126	—	2	2	no	—
267	C₂×#124	—	2	1	no	—
268	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
269	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
270	C₂×C₄×Q₈	—	2	1	no	—
271	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#117	C₂²×C₃₁	1	2	yes	—
275	#477	—	2	4	no	—
276	#441	—	2	4	no	—
277	C₂×#107	C₂²×C₃₁	1	2	yes	—
278	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
279	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×#103	C₂²×C₃₁	1	2	yes	—
281	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
283	C₂×#103	C₂²×C₃₁	1	2	yes	—
284	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
285	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
286	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
287	#398	—	2	1	no	—
288	#394	—	2	1	no	—
289	#388	—	2	2	no	—
290	#382	C₂²×C₃₁	1	1	yes	characteristic
291	#380	C₂²×C₃₁	1	2	yes	—
292	#340	C₂²×C₃₁	1	2	yes	—
293	#340	C₂²×C₃₁	1	2	yes	—
294	#340	C₂²×C₃₁	1	2	yes	—
295	C₂×C₃₁	—	8	2	no	—
296	C₂×C₃₁	#2407	1	1	yes	characteristic
297	C₂×C₃₁	#3005	1	1	yes	characteristic
298	C₂×C₃₁	#3011	1	1	yes	characteristic
299	C₂×C₃₁	#3313	1	2	yes	—
300	C₂×C₃₁	#3349	1	2	yes	—
301	C₂²×Q₈	—	2	1	no	—
302	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
304	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
305	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
306	C₂×Q₁₆	—	2	2	no	—
307	C₂×Q₁₆	—	2	2	no	—
308	C₂×Q₁₆	—	2	2	no	—
309	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
310	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
311	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
314	#126	—	2	16	no	—
315	#124	—	2	8	no	—
316	C₂²⋊Q₈	—	2	4	no	—
317	C₂²⋊Q₈	—	2	4	no	—
318	C₂²⋊Q₈	—	2	4	no	—
319	C₂²⋊Q₈	—	4	2	no	—
320	C₄×Q₈	—	2	8	no	—
321	C₄×Q₈	—	2	2	no	—
322	C₄×Q₈	—	2	2	no	—
323	C₄×Q₈	—	2	2	no	—
324	C₄×Q₈	D₄×C₃₁	1	4	yes	—
325	C₄×D₄	—	2	2	no	—
326	C₄×D₄	—	2	2	no	—
327	#117	—	2	4	no	—
328	#117	—	2	4	no	—
329	C₂×C₄⋊C₄	—	2	2	no	—
330	C₂×C₄⋊C₄	—	2	1	no	—
331	C₂×C₄⋊C₄	—	2	1	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	2	no	—
334	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
339	C₂×C₂²⋊C₄	—	2	2	no	—
340	C₂×C₂²⋊C₄	—	2	2	no	—
341	C₂×C₂²⋊C₄	—	2	2	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
344	C₂×C₄²	—	2	2	no	—
345	C₂×C₄²	—	2	1	no	—
346	C₂×C₄²	—	2	1	no	—
347	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
348	#107	—	2	4	no	—
349	#106	—	2	2	no	—
350	C₄⋊C₈	—	2	2	no	—
351	#103	—	2	2	no	—
352	#103	—	2	2	no	—
353	#103	—	2	4	no	—
354	#103	—	4	2	no	—
355	C₂²⋊C₈	—	2	4	no	—
356	C₂²⋊C₈	—	4	2	no	—
357	#97	—	2	2	no	—
358	#95	—	2	2	no	—
359	#95	—	2	1	no	—
360	#95	—	2	1	no	—
361	C₃₁	#24135	1	1	yes	characteristic, direct product factor, Sylow
362	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	2	1	no	—
365	C₂×Q₈	—	2	1	no	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	2	1	no	—
368	C₂×Q₈	—	2	1	no	—
369	C₂×Q₈	—	2	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	1	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	—	4	4	no	—
375	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
376	C₂×Q₈	#41×C₃₁	1	2	yes	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	4	2	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
392	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
394	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
395	Q₁₆	—	2	8	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	#41×C₃₁	1	2	yes	—
405	C₄⋊C₄	—	2	8	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	2	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	4	4	no	—
416	C₄⋊C₄	—	4	2	no	—
417	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
418	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
419	C₂²⋊C₄	—	2	8	no	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	4	no	—
422	C₂²⋊C₄	—	4	4	no	—
423	C₂²⋊C₄	—	4	4	no	—
424	C₄²	—	2	8	no	—
425	C₄²	—	2	4	no	—
426	C₄²	—	4	2	no	—
427	C₄²	C₂×D₄×C₃₁	1	2	yes	—
428	C₄²	C₂×D₄×C₃₁	1	2	yes	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	2	no	—
432	C₂³	—	2	2	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	2	no	—
436	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
437	Q₈	—	2	2	no	—
438	Q₈	—	2	2	no	—
439	Q₈	—	2	2	no	—
440	Q₈	—	2	2	no	—
441	Q₈	—	4	8	no	—
442	Q₈	—	4	1	no	—
443	Q₈	—	4	1	no	—
444	Q₈	—	4	1	no	—
445	Q₈	—	4	1	no	—
446	D₄	—	4	4	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	1	no	—
458	C₂×C₄	—	4	1	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	4	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	C₃₁×#120	1	2	yes	—
472	C₂×C₄	C₃₁×#120	1	2	yes	—
473	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
474	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
475	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
476	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
477	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
478	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
479	C₈	—	2	2	no	—
480	C₈	—	4	1	no	—
481	C₈	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	4	2	no	—
489	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
490	C₂²	C₃₁×#546	1	1	yes	characteristic
491	C₂²	C₃₁×#547	1	1	yes	characteristic
492	C₂²	C₃₁×#575	1	2	yes	—
493	C₂²	C₃₁×#578	1	2	yes	—
494	C₄	—	2	2	no	—
495	C₄	—	2	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	2	no	—
499	C₄	—	4	4	no	—
500	C₄	—	4	2	no	—
501	C₄	—	8	1	no	—
502	C₄	—	8	2	no	—
503	C₄	C₃₁×#525	1	1	yes	characteristic
504	C₄	C₃₁×#537	1	1	yes	characteristic
505	C₄	C₃₁×#548	1	1	yes	characteristic
506	C₄	C₃₁×#550	1	1	yes	characteristic
507	C₂	—	8	2	no	—
508	C₂	C₃₁×#2407	1	1	yes	characteristic
509	C₂	C₃₁×#3005	1	1	yes	characteristic
510	C₂	C₃₁×#3011	1	1	yes	characteristic
511	C₂	C₃₁×#3313	1	2	yes	—
512	C₂	C₃₁×#3349	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#477	C₂	semidirect
3. C₂×C₃₁×#441	C₂	semidirect
5. C₃₁×#2036	C₂	semidirect
6. C₃₁×#2001	C₂	semidirect
7. C₃₁×#1993	C₂	semidirect
9. C₃₁×#1873	C₂	semidirect
10. C₃₁×#1854	C₂	semidirect
11. C₃₁×#1805	C₂	semidirect
172. #24135	C₃₁	direct (N × H)
227. C₂×#477	C₂×C₃₁	semidirect
228. C₂×#441	C₂×C₃₁	semidirect
230. #2036	C₂×C₃₁	semidirect
231. #2001	C₂×C₃₁	semidirect
232. #1993	C₂×C₃₁	semidirect
234. #1873	C₂×C₃₁	semidirect
235. #1854	C₂×C₃₁	semidirect
236. #1805	C₂×C₃₁	semidirect
361. C₃₁	#24135	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×#556	2
C₂×C₃₁×#477	2
C₂×C₃₁×#441	2
C₃₁×#2536	2
C₃₁×#2036	2
C₃₁×#2001	2
C₃₁×#1993	2
C₃₁×#1955	2
C₃₁×#1873	2
C₃₁×#1854	2
C₃₁×#1805	2
#24135	31

Subgroups — C31×#24135 #21480894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24135 #21480894