Subgroups — C₃₁×#33199 #21489958

Order 7936.

930 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#478	C₂	1	2	yes	maximal
2	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1989	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1949	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1947	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1804	C₂	1	2	yes	maximal
10	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×#125	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₃₁	—	2	1	no	—
17	C₃₁×#478	—	2	8	no	—
18	C₂×C₃₁×#107	C₂²	1	2	yes	—
19	C₂×C₃₁×#106	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₃₁×#389	—	2	1	no	—
27	C₃₁×#389	—	2	1	no	—
28	C₃₁×#389	—	2	1	no	—
29	C₃₁×#389	—	2	1	no	—
30	C₃₁×#384	C₂²	1	1	yes	characteristic
31	C₃₁×#384	C₂²	1	1	yes	characteristic
32	C₃₁×#382	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
37	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
38	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
39	C₄³×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#340	C₂²	1	2	yes	—
44	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	—	2	1	no	—
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂³×C₄×C₃₁	—	2	1	no	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#125	—	2	4	no	—
51	C₃₁×#125	—	2	4	no	—
52	C₄×D₄×C₃₁	—	2	2	no	—
53	C₄×D₄×C₃₁	—	2	2	no	—
54	C₄×D₄×C₃₁	—	2	2	no	—
55	C₄×D₄×C₃₁	D₄	1	4	yes	—
56	C₄×D₄×C₃₁	Q₈	1	4	yes	—
57	C₄×D₄×C₃₁	Q₈	1	4	yes	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	—	4	4	no	—
76	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#107	—	2	4	no	—
84	C₃₁×#106	—	2	4	no	—
85	C₃₁×C₄⋊C₈	—	2	4	no	—
86	C₃₁×C₄⋊C₈	—	2	2	no	—
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×C₄⋊C₈	—	2	4	no	—
89	C₃₁×#102	—	2	4	no	—
90	C₃₁×#102	—	2	4	no	—
91	C₃₁×#95	—	2	1	no	—
92	C₃₁×#95	—	2	1	no	—
93	C₃₁×#95	—	2	1	no	—
94	C₃₁×#95	—	2	2	no	—
95	C₃₁×#95	—	2	2	no	—
96	C₂⁴×C₃₁	—	2	1	no	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	—	2	1	no	—
100	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
101	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
102	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	1	no	—
106	C₂²×C₄×C₃₁	—	2	4	no	—
107	C₂²×C₄×C₃₁	—	2	4	no	—
108	C₂²×C₄×C₃₁	—	2	4	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	C₂²×C₄×C₃₁	—	4	2	no	—
113	C₂²×C₄×C₃₁	—	4	2	no	—
114	C₂²×C₄×C₃₁	—	4	2	no	—
115	C₂²×C₄×C₃₁	—	4	2	no	—
116	C₂²×C₄×C₃₁	—	4	2	no	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
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	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	1	no	—
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	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	4	2	no	—
138	C₄⋊C₄×C₃₁	—	4	2	no	—
139	C₄⋊C₄×C₃₁	—	4	4	no	—
140	C₄⋊C₄×C₃₁	—	4	4	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	—	4	2	no	—
143	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
144	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
145	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
146	C₂²⋊C₄×C₃₁	—	2	4	no	—
147	C₂²⋊C₄×C₃₁	—	2	4	no	—
148	C₂²⋊C₄×C₃₁	—	2	4	no	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	2	2	no	—
151	C₄²×C₃₁	—	4	1	no	—
152	C₄²×C₃₁	—	4	1	no	—
153	C₄²×C₃₁	—	4	2	no	—
154	C₄²×C₃₁	—	4	2	no	—
155	C₄²×C₃₁	—	4	2	no	—
156	C₄²×C₃₁	C₂×D₄	1	2	yes	—
157	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
158	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
159	C₄²×C₃₁	#41	1	2	yes	—
160	C₄²×C₃₁	#41	1	2	yes	—
161	C₄²×C₃₁	#41	1	2	yes	—
162	#33199	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₃₁	—	2	2	no	—
164	C₂³×C₃₁	—	2	2	no	—
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	4	1	no	—
167	C₂³×C₃₁	—	4	1	no	—
168	C₂³×C₃₁	—	4	1	no	—
169	C₂³×C₃₁	—	4	1	no	—
170	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
171	D₄×C₃₁	—	2	2	no	—
172	D₄×C₃₁	—	2	2	no	—
173	D₄×C₃₁	—	2	2	no	—
174	D₄×C₃₁	C₄⋊Q₈	1	4	yes	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	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	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	2	2	no	—
188	C₂×C₄×C₃₁	—	4	8	no	—
189	C₂×C₄×C₃₁	—	4	8	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	4	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	8	no	—
197	C₂×C₄×C₃₁	—	4	2	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	#121	1	2	yes	—
204	C₂×C₄×C₃₁	#121	1	2	yes	—
205	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
206	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
207	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
208	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
209	C₂×C₄×C₃₁	#135	1	2	yes	—
210	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
211	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₈×C₃₁	—	4	2	no	—
218	C₈×C₃₁	—	4	2	no	—
219	C₂×#478	C₂×C₃₁	1	2	yes	—
220	#2859	C₂×C₃₁	1	1	yes	characteristic, complemented
221	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
222	#1989	C₂×C₃₁	1	2	yes	complemented
223	#1949	C₂×C₃₁	1	2	yes	complemented
224	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1947	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#1804	C₂×C₃₁	1	2	yes	—
228	#1802	C₂×C₃₁	1	1	yes	characteristic
229	#1802	C₂×C₃₁	1	1	yes	characteristic
230	C₂²×C₃₁	—	2	4	no	—
231	C₂²×C₃₁	—	4	2	no	—
232	C₂²×C₃₁	—	4	2	no	—
233	C₂²×C₃₁	—	4	2	no	—
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
238	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
239	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
240	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
241	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂²×C₃₁	#575	1	1	yes	characteristic
243	C₂²×C₃₁	#575	1	1	yes	characteristic
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	2	2	no	—
246	C₄×C₃₁	—	2	2	no	—
247	C₄×C₃₁	—	2	2	no	—
248	C₄×C₃₁	—	2	2	no	—
249	C₄×C₃₁	—	4	2	no	—
250	C₄×C₃₁	—	4	2	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	—	8	2	no	—
253	C₄×C₃₁	—	8	2	no	—
254	C₄×C₃₁	#478	1	4	yes	complemented
255	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
256	C₄×C₃₁	#532	1	1	yes	characteristic
257	C₄×C₃₁	#533	1	1	yes	characteristic
258	C₄×C₃₁	#533	1	1	yes	characteristic
259	C₂×#125	C₂²×C₃₁	1	2	yes	—
260	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
263	C₂²×C₄²	—	2	1	no	—
264	#478	—	2	8	no	—
265	C₂×#107	C₂²×C₃₁	1	2	yes	—
266	C₂×#106	C₂²×C₃₁	1	2	yes	—
267	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
268	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
271	C₂×#102	C₂²×C₃₁	1	2	yes	—
272	C₂×#102	C₂²×C₃₁	1	2	yes	—
273	#389	—	2	1	no	—
274	#389	—	2	1	no	—
275	#389	—	2	1	no	—
276	#389	—	2	1	no	—
277	#384	C₂²×C₃₁	1	1	yes	characteristic
278	#384	C₂²×C₃₁	1	1	yes	characteristic
279	#382	C₂²×C₃₁	1	2	yes	—
280	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
281	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
282	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
283	C₄×C₂²⋊C₄	—	2	1	no	—
284	C₄×C₂²⋊C₄	—	2	1	no	—
285	C₄×C₂²⋊C₄	—	2	1	no	—
286	C₄³	C₂²×C₃₁	1	1	yes	characteristic
287	#340	C₂²×C₃₁	1	2	yes	—
288	#340	C₂²×C₃₁	1	2	yes	—
289	#340	C₂²×C₃₁	1	2	yes	—
290	#340	C₂²×C₃₁	1	2	yes	—
291	C₂×C₃₁	—	4	4	no	—
292	C₂×C₃₁	#2857	1	1	yes	characteristic
293	C₂×C₃₁	#3035	1	1	yes	characteristic
294	C₂×C₃₁	#3036	1	1	yes	characteristic
295	C₂×C₃₁	C₂×#478	1	1	yes	characteristic
296	C₂×C₃₁	#3076	1	1	yes	characteristic
297	C₂×C₃₁	#3077	1	1	yes	characteristic
298	C₂×C₃₁	#3080	1	1	yes	characteristic
299	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
300	C₂³×C₄	—	2	1	no	—
301	C₂³×C₄	—	2	1	no	—
302	C₂³×C₄	—	2	1	no	—
303	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
304	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
305	#125	—	2	4	no	—
306	#125	—	2	4	no	—
307	C₄×D₄	—	2	2	no	—
308	C₄×D₄	—	2	2	no	—
309	C₄×D₄	—	2	2	no	—
310	C₄×D₄	D₄×C₃₁	1	4	yes	—
311	C₄×D₄	Q₈×C₃₁	1	4	yes	—
312	C₄×D₄	Q₈×C₃₁	1	4	yes	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
320	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
321	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
322	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₂×C₂²⋊C₄	—	2	1	no	—
325	C₂×C₂²⋊C₄	—	2	1	no	—
326	C₂×C₂²⋊C₄	—	2	1	no	—
327	C₂×C₄²	—	2	1	no	—
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	—	2	1	no	—
330	C₂×C₄²	—	4	4	no	—
331	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
332	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
336	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
337	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
338	#107	—	2	4	no	—
339	#106	—	2	4	no	—
340	C₄⋊C₈	—	2	4	no	—
341	C₄⋊C₈	—	2	2	no	—
342	C₄⋊C₈	—	2	2	no	—
343	C₄⋊C₈	—	2	4	no	—
344	#102	—	2	4	no	—
345	#102	—	2	4	no	—
346	#95	—	2	1	no	—
347	#95	—	2	1	no	—
348	#95	—	2	1	no	—
349	#95	—	2	2	no	—
350	#95	—	2	2	no	—
351	C₃₁	#33199	1	1	yes	characteristic, direct product factor, Sylow
352	C₂⁴	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
357	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
358	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	4	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	2	4	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	—	4	2	no	—
370	C₂²×C₄	—	4	2	no	—
371	C₂²×C₄	—	4	2	no	—
372	C₂²×C₄	—	4	2	no	—
373	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	4	2	no	—
394	C₄⋊C₄	—	4	2	no	—
395	C₄⋊C₄	—	4	4	no	—
396	C₄⋊C₄	—	4	4	no	—
397	C₄⋊C₄	—	4	2	no	—
398	C₄⋊C₄	—	4	2	no	—
399	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
400	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
401	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
402	C₂²⋊C₄	—	2	4	no	—
403	C₂²⋊C₄	—	2	4	no	—
404	C₂²⋊C₄	—	2	4	no	—
405	C₄²	—	2	2	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	4	1	no	—
408	C₄²	—	4	1	no	—
409	C₄²	—	4	2	no	—
410	C₄²	—	4	2	no	—
411	C₄²	—	4	2	no	—
412	C₄²	C₂×D₄×C₃₁	1	2	yes	—
413	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
414	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
415	C₄²	#41×C₃₁	1	2	yes	—
416	C₄²	#41×C₃₁	1	2	yes	—
417	C₄²	#41×C₃₁	1	2	yes	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	1	no	—
423	C₂³	—	4	1	no	—
424	C₂³	—	4	1	no	—
425	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
426	D₄	—	2	2	no	—
427	D₄	—	2	2	no	—
428	D₄	—	2	2	no	—
429	D₄	C₃₁×C₄⋊Q₈	1	4	yes	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	1	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	4	8	no	—
444	C₂×C₄	—	4	8	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	—	4	1	no	—
451	C₂×C₄	—	4	8	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	C₃₁×#121	1	2	yes	—
459	C₂×C₄	C₃₁×#121	1	2	yes	—
460	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
461	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
462	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
463	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
464	C₂×C₄	C₃₁×#135	1	2	yes	—
465	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
466	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
467	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
468	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
472	C₈	—	4	2	no	—
473	C₈	—	4	2	no	—
474	C₂²	—	2	4	no	—
475	C₂²	—	4	2	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	2	no	—
481	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
482	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
483	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
484	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
485	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
486	C₂²	C₃₁×#575	1	1	yes	characteristic
487	C₂²	C₃₁×#575	1	1	yes	characteristic
488	C₄	—	2	2	no	—
489	C₄	—	2	2	no	—
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	2	no	—
493	C₄	—	4	2	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	8	2	no	—
497	C₄	—	8	2	no	—
498	C₄	C₃₁×#478	1	4	yes	complemented
499	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
500	C₄	C₃₁×#532	1	1	yes	characteristic
501	C₄	C₃₁×#533	1	1	yes	characteristic
502	C₄	C₃₁×#533	1	1	yes	characteristic
503	C₂	—	4	4	no	—
504	C₂	C₃₁×#2857	1	1	yes	characteristic
505	C₂	C₃₁×#3035	1	1	yes	characteristic
506	C₂	C₃₁×#3036	1	1	yes	characteristic
507	C₂	C₂×C₃₁×#478	1	1	yes	characteristic
508	C₂	C₃₁×#3076	1	1	yes	characteristic
509	C₂	C₃₁×#3077	1	1	yes	characteristic
510	C₂	C₃₁×#3080	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2859	C₂	semidirect
4. C₃₁×#1989	C₂	semidirect
5. C₃₁×#1949	C₂	semidirect
6. C₃₁×#1948	C₂	semidirect
7. C₃₁×#1947	C₂	semidirect
8. C₃₁×#1842	C₂	semidirect
162. #33199	C₃₁	direct (N × H)
220. #2859	C₂×C₃₁	semidirect
222. #1989	C₂×C₃₁	semidirect
223. #1949	C₂×C₃₁	semidirect
224. #1948	C₂×C₃₁	semidirect
225. #1947	C₂×C₃₁	semidirect
226. #1842	C₂×C₃₁	semidirect
254. C₄×C₃₁	#478	semidirect
351. C₃₁	#33199	direct (N × H)
498. C₄	C₃₁×#478	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#478	2
C₃₁×#2859	2
C₄²×D₄×C₃₁	2
C₃₁×#1989	2
C₃₁×#1949	2
C₃₁×#1948	2
C₃₁×#1947	2
C₃₁×#1842	2
C₃₁×#1804	2
C₃₁×#1802	2
#33199	31

Subgroups — C31×#33199 #21489958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33199 #21489958