Subgroups — C₃₁×#29260 #21486019

Order 7936.

800 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2095	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2094	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2024	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2023	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1983	C₂	1	2	yes	maximal
8	C₃₁×#1949	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1865	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1862	C₂	1	2	yes	maximal
11	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×#103	C₂²	1	2	yes	—
21	C₂×C₃₁×#103	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₃₁×#391	—	2	1	no	—
25	C₃₁×#391	C₂²	1	1	yes	characteristic
26	C₃₁×#390	C₂²	1	1	yes	characteristic
27	C₃₁×#389	—	2	1	no	—
28	C₃₁×#389	C₂²	1	2	yes	—
29	C₃₁×#388	—	2	1	no	—
30	C₃₁×#388	—	2	1	no	—
31	C₃₁×#384	C₂²	1	1	yes	characteristic
32	C₃₁×#382	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
36	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
37	C₄³×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₃₁×C₄⋊Q₈	—	2	4	no	—
50	C₃₁×#124	—	2	2	no	—
51	C₃₁×#124	Q₈	1	4	yes	complemented
52	C₄×Q₈×C₃₁	—	4	2	no	—
53	C₄×Q₈×C₃₁	—	4	4	no	—
54	C₄×D₄×C₃₁	—	4	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄²×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#107	—	4	2	no	—
79	C₃₁×#106	—	4	2	no	—
80	C₃₁×C₄⋊C₈	—	2	4	no	—
81	C₃₁×#103	—	2	4	no	—
82	C₃₁×#103	—	4	2	no	—
83	C₃₁×#102	—	2	4	no	—
84	C₃₁×#102	—	4	2	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	—	2	1	no	—
88	C₃₁×#95	—	2	1	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₃₁×#95	—	2	2	no	—
91	C₂⁴×C₃₁	—	2	1	no	—
92	C₂×Q₈×C₃₁	—	2	4	no	—
93	C₂×Q₈×C₃₁	—	2	1	no	—
94	C₂×Q₈×C₃₁	—	2	2	no	—
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	4	2	no	—
97	C₂×Q₈×C₃₁	—	4	2	no	—
98	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	1	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	1	no	—
106	C₂²×C₄×C₃₁	—	2	1	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	4	2	no	—
109	C₂²×C₄×C₃₁	—	4	2	no	—
110	C₂²×C₄×C₃₁	—	4	2	no	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	4	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	4	2	no	—
131	C₄⋊C₄×C₃₁	—	4	2	no	—
132	C₄⋊C₄×C₃₁	—	4	2	no	—
133	C₄⋊C₄×C₃₁	—	4	4	no	—
134	C₄⋊C₄×C₃₁	#41	1	4	yes	—
135	C₂²⋊C₄×C₃₁	—	2	4	no	—
136	C₂²⋊C₄×C₃₁	—	2	4	no	—
137	C₂²⋊C₄×C₃₁	—	4	2	no	—
138	C₂²⋊C₄×C₃₁	—	4	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	2	2	no	—
143	C₄²×C₃₁	—	2	2	no	—
144	C₄²×C₃₁	—	2	2	no	—
145	C₄²×C₃₁	—	4	1	no	—
146	C₄²×C₃₁	—	4	2	no	—
147	C₄²×C₃₁	—	4	4	no	—
148	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
149	C₄²×C₃₁	#41	1	2	yes	—
150	#29260	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂³×C₃₁	—	2	2	no	—
152	C₂³×C₃₁	—	4	1	no	—
153	C₂³×C₃₁	—	4	1	no	—
154	C₂³×C₃₁	—	4	1	no	—
155	C₂³×C₃₁	—	4	1	no	—
156	C₂³×C₃₁	—	4	1	no	—
157	C₂³×C₃₁	—	4	1	no	—
158	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₃₁	—	2	4	no	—
160	Q₈×C₃₁	—	4	8	no	—
161	Q₈×C₃₁	—	4	2	no	—
162	D₄×C₃₁	—	2	4	no	—
163	D₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	4	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	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	1	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	1	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	1	no	—
192	C₂×C₄×C₃₁	—	8	4	no	—
193	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
194	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
195	C₂×C₄×C₃₁	#135	1	2	yes	—
196	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
197	C₂×C₄×C₃₁	#137	1	2	yes	—
198	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
200	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
201	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
202	C₈×C₃₁	—	4	2	no	—
203	C₈×C₃₁	—	4	2	no	—
204	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
205	C₄×#124	C₂×C₃₁	1	1	yes	characteristic
206	#2095	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#2094	C₂×C₃₁	1	1	yes	characteristic
208	#2024	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#2023	C₂×C₃₁	1	1	yes	characteristic
210	#1983	C₂×C₃₁	1	2	yes	—
211	#1949	C₂×C₃₁	1	2	yes	complemented
212	#1865	C₂×C₃₁	1	2	yes	complemented
213	#1862	C₂×C₃₁	1	2	yes	—
214	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	8	1	no	—
219	C₂²×C₃₁	—	8	1	no	—
220	C₂²×C₃₁	—	8	1	no	—
221	C₂²×C₃₁	—	8	1	no	—
222	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
223	C₂²×C₃₁	#525	1	1	yes	characteristic
224	C₂²×C₃₁	#535	1	1	yes	characteristic
225	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
226	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
227	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
228	C₂²×C₃₁	#575	1	1	yes	characteristic
229	C₄×C₃₁	—	2	2	no	—
230	C₄×C₃₁	—	2	2	no	—
231	C₄×C₃₁	—	2	2	no	—
232	C₄×C₃₁	—	2	2	no	—
233	C₄×C₃₁	—	2	2	no	—
234	C₄×C₃₁	—	4	2	no	—
235	C₄×C₃₁	—	4	4	no	—
236	C₄×C₃₁	—	4	1	no	—
237	C₄×C₃₁	—	8	1	no	—
238	C₄×C₃₁	—	8	2	no	—
239	C₄×C₃₁	—	8	1	no	—
240	C₄×C₃₁	#554	1	2	yes	—
241	C₄×C₃₁	#559	1	2	yes	—
242	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×C₄×Q₈	—	2	1	no	—
245	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×#107	C₂²×C₃₁	1	2	yes	—
248	C₂×#106	C₂²×C₃₁	1	2	yes	—
249	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
250	C₂×#103	C₂²×C₃₁	1	2	yes	—
251	C₂×#103	C₂²×C₃₁	1	2	yes	—
252	C₂×#102	C₂²×C₃₁	1	2	yes	—
253	C₂×#102	C₂²×C₃₁	1	2	yes	—
254	#391	—	2	1	no	—
255	#391	C₂²×C₃₁	1	1	yes	characteristic
256	#390	C₂²×C₃₁	1	1	yes	characteristic
257	#389	—	2	1	no	—
258	#389	C₂²×C₃₁	1	2	yes	—
259	#388	—	2	1	no	—
260	#388	—	2	1	no	—
261	#384	C₂²×C₃₁	1	1	yes	characteristic
262	#382	C₂²×C₃₁	1	1	yes	characteristic
263	C₄×C₄⋊C₄	—	2	1	no	—
264	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
265	C₄×C₂²⋊C₄	—	2	1	no	—
266	C₄×C₂²⋊C₄	—	2	1	no	—
267	C₄³	C₂²×C₃₁	1	1	yes	characteristic
268	#340	C₂²×C₃₁	1	2	yes	—
269	#340	C₂²×C₃₁	1	2	yes	—
270	#336	C₂²×C₃₁	1	1	yes	characteristic
271	#336	C₂²×C₃₁	1	1	yes	characteristic
272	#336	C₂²×C₃₁	1	2	yes	—
273	C₂×C₃₁	—	8	2	no	—
274	C₂×C₃₁	#2580	1	1	yes	characteristic
275	C₂×C₃₁	#3076	1	1	yes	characteristic
276	C₂×C₃₁	#3079	1	1	yes	characteristic
277	C₂×C₃₁	#3105	1	1	yes	characteristic
278	C₂×C₃₁	#3117	1	1	yes	characteristic
279	C₂×C₃₁	#3210	1	1	yes	characteristic
280	C₂×C₃₁	#3213	1	1	yes	characteristic
281	C₂²×Q₈	—	2	1	no	—
282	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
283	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
284	C₂³×C₄	—	2	1	no	—
285	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
286	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
287	C₄⋊Q₈	—	2	4	no	—
288	#124	—	2	2	no	—
289	#124	Q₈×C₃₁	1	4	yes	complemented
290	C₄×Q₈	—	4	2	no	—
291	C₄×Q₈	—	4	4	no	—
292	C₄×D₄	—	4	2	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	2	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
301	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₂²⋊C₄	—	2	1	no	—
304	C₂×C₂²⋊C₄	—	2	1	no	—
305	C₂×C₂²⋊C₄	—	2	1	no	—
306	C₂×C₂²⋊C₄	—	2	1	no	—
307	C₂×C₄²	—	2	2	no	—
308	C₂×C₄²	—	2	1	no	—
309	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
310	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
311	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
312	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
316	#107	—	4	2	no	—
317	#106	—	4	2	no	—
318	C₄⋊C₈	—	2	4	no	—
319	#103	—	2	4	no	—
320	#103	—	4	2	no	—
321	#102	—	2	4	no	—
322	#102	—	4	2	no	—
323	#95	—	2	1	no	—
324	#95	—	2	1	no	—
325	#95	—	2	1	no	—
326	#95	—	2	1	no	—
327	#95	—	2	2	no	—
328	#95	—	2	2	no	—
329	C₃₁	#29260	1	1	yes	characteristic, direct product factor, Sylow
330	C₂⁴	—	2	1	no	—
331	C₂×Q₈	—	2	4	no	—
332	C₂×Q₈	—	2	1	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	4	2	no	—
336	C₂×Q₈	—	4	2	no	—
337	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	2	1	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	—	4	2	no	—
349	C₂²×C₄	—	4	2	no	—
350	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₄⋊C₄	—	4	2	no	—
371	C₄⋊C₄	—	4	2	no	—
372	C₄⋊C₄	—	4	4	no	—
373	C₄⋊C₄	#41×C₃₁	1	4	yes	—
374	C₂²⋊C₄	—	2	4	no	—
375	C₂²⋊C₄	—	2	4	no	—
376	C₂²⋊C₄	—	4	2	no	—
377	C₂²⋊C₄	—	4	2	no	—
378	C₂²⋊C₄	—	4	2	no	—
379	C₄²	—	2	2	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	2	2	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	4	1	no	—
385	C₄²	—	4	2	no	—
386	C₄²	—	4	4	no	—
387	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
388	C₄²	#41×C₃₁	1	2	yes	—
389	C₂³	—	2	2	no	—
390	C₂³	—	4	1	no	—
391	C₂³	—	4	1	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	1	no	—
394	C₂³	—	4	1	no	—
395	C₂³	—	4	1	no	—
396	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
397	Q₈	—	2	4	no	—
398	Q₈	—	4	8	no	—
399	Q₈	—	4	2	no	—
400	D₄	—	2	4	no	—
401	D₄	—	4	2	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	8	4	no	—
431	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
432	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
433	C₂×C₄	C₃₁×#135	1	2	yes	—
434	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
435	C₂×C₄	C₃₁×#137	1	2	yes	—
436	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
438	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
439	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
440	C₈	—	4	2	no	—
441	C₈	—	4	2	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	8	1	no	—
446	C₂²	—	8	1	no	—
447	C₂²	—	8	1	no	—
448	C₂²	—	8	1	no	—
449	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
450	C₂²	C₃₁×#525	1	1	yes	characteristic
451	C₂²	C₃₁×#535	1	1	yes	characteristic
452	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
453	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
454	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
455	C₂²	C₃₁×#575	1	1	yes	characteristic
456	C₄	—	2	2	no	—
457	C₄	—	2	2	no	—
458	C₄	—	2	2	no	—
459	C₄	—	2	2	no	—
460	C₄	—	2	2	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	4	no	—
463	C₄	—	4	1	no	—
464	C₄	—	8	1	no	—
465	C₄	—	8	1	no	—
466	C₄	—	8	2	no	—
467	C₄	C₃₁×#554	1	2	yes	—
468	C₄	C₃₁×#559	1	2	yes	—
469	C₂	—	8	2	no	—
470	C₂	C₃₁×#2580	1	1	yes	characteristic
471	C₂	C₃₁×#3076	1	1	yes	characteristic
472	C₂	C₃₁×#3079	1	1	yes	characteristic
473	C₂	C₃₁×#3105	1	1	yes	characteristic
474	C₂	C₃₁×#3117	1	1	yes	characteristic
475	C₂	C₃₁×#3210	1	1	yes	characteristic
476	C₂	C₃₁×#3213	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
1. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
3. C₃₁×#2095	C₂	semidirect
5. C₃₁×#2024	C₂	semidirect
8. C₃₁×#1949	C₂	semidirect
9. C₃₁×#1865	C₂	semidirect
11. C₃₁×#1831	C₂	semidirect
51. C₃₁×#124	Q₈	semidirect
150. #29260	C₃₁	direct (N × H)
204. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
206. #2095	C₂×C₃₁	semidirect
208. #2024	C₂×C₃₁	semidirect
211. #1949	C₂×C₃₁	semidirect
212. #1865	C₂×C₃₁	semidirect
214. #1831	C₂×C₃₁	semidirect
289. #124	Q₈×C₃₁	semidirect
329. C₃₁	#29260	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×C₄⋊Q₈	2
C₄×C₃₁×#124	2
C₃₁×#2095	2
C₃₁×#2094	2
C₃₁×#2024	2
C₃₁×#2023	2
C₃₁×#1983	2
C₃₁×#1949	2
C₃₁×#1865	2
C₃₁×#1862	2
C₃₁×#1831	2
#29260	31

Subgroups — C31×#29260 #21486019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29260 #21486019