Subgroups — C₃₁×#22116 #21478875

Order 7936.

836 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#474	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1943	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1936	C₂	1	2	yes	maximal
8	C₃₁×#1926	C₂	1	2	yes	maximal
9	C₃₁×#1921	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	—	2	1	no	—
16	C₃₁×#474	C₄	1	8	yes	complemented
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#97	C₂²	1	2	yes	—
24	C₃₁×#391	—	2	1	no	—
25	C₃₁×#390	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×#389	C₂²	1	1	yes	characteristic
30	C₃₁×#389	C₂²	1	1	yes	characteristic
31	C₃₁×#388	—	2	1	no	—
32	C₃₁×#388	—	2	1	no	—
33	C₃₁×#384	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
38	C₄³×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₂²×Q₈×C₃₁	—	2	1	no	—
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	2	no	—
50	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
51	C₄×Q₈×C₃₁	—	4	4	no	—
52	C₄×D₄×C₃₁	—	2	2	no	—
53	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
54	C₃₁×#117	—	4	4	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
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×C₂²⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#107	C₂×C₄	1	8	yes	—
81	C₃₁×#106	—	4	2	no	—
82	C₃₁×C₄⋊C₈	—	2	2	no	—
83	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
84	C₃₁×#102	—	4	2	no	—
85	C₃₁×#102	C₂×C₄	1	8	yes	—
86	C₃₁×#97	—	4	2	no	—
87	C₃₁×#95	—	2	1	no	—
88	C₃₁×#95	—	2	1	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₃₁×#95	—	2	1	no	—
91	C₃₁×#95	—	2	2	no	—
92	C₂⁴×C₃₁	—	2	1	no	—
93	C₂×Q₈×C₃₁	—	2	4	no	—
94	C₂×Q₈×C₃₁	—	4	2	no	—
95	C₂×Q₈×C₃₁	—	4	2	no	—
96	C₂×D₄×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	—	2	1	no	—
101	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	4	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	C₂×D₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
117	C₂×C₈×C₃₁	—	2	2	no	—
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	C₂²×C₄	1	4	yes	—
123	C₄⋊C₄×C₃₁	—	2	4	no	—
124	C₄⋊C₄×C₃₁	—	2	1	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	2	1	no	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	1	no	—
132	C₄⋊C₄×C₃₁	—	4	2	no	—
133	C₄⋊C₄×C₃₁	—	4	2	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	—	4	4	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
138	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
139	C₂²⋊C₄×C₃₁	—	2	4	no	—
140	C₂²⋊C₄×C₃₁	—	4	2	no	—
141	C₂²⋊C₄×C₃₁	—	4	2	no	—
142	C₂²⋊C₄×C₃₁	—	4	2	no	—
143	C₂²⋊C₄×C₃₁	—	4	2	no	—
144	C₄²×C₃₁	—	2	2	no	—
145	C₄²×C₃₁	—	2	2	no	—
146	C₄²×C₃₁	—	2	2	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	—	2	1	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	4	1	no	—
152	C₄²×C₃₁	—	4	4	no	—
153	C₄²×C₃₁	—	4	2	no	—
154	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
155	C₄²×C₃₁	#41	1	2	yes	—
156	#22116	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂³×C₃₁	—	2	2	no	—
158	C₂³×C₃₁	—	4	1	no	—
159	C₂³×C₃₁	—	4	1	no	—
160	C₂³×C₃₁	—	4	1	no	—
161	C₂³×C₃₁	—	4	1	no	—
162	C₂³×C₃₁	—	4	1	no	—
163	C₂³×C₃₁	—	4	1	no	—
164	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
165	Q₈×C₃₁	—	4	8	no	—
166	D₄×C₃₁	—	2	2	no	—
167	D₄×C₃₁	—	2	2	no	—
168	D₄×C₃₁	—	4	1	no	—
169	D₄×C₃₁	—	4	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	—	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	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	8	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	2	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	2	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
200	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
201	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
202	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
203	C₂×C₄×C₃₁	#137	1	2	yes	—
204	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
207	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
208	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
209	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
210	C₈×C₃₁	—	2	4	no	—
211	C₈×C₃₁	—	4	2	no	—
212	C₂×#474	C₂×C₃₁	1	1	yes	characteristic
213	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#2293	C₂×C₃₁	1	1	yes	characteristic
215	#1988	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1943	C₂×C₃₁	1	2	yes	complemented
218	#1936	C₂×C₃₁	1	2	yes	—
219	#1926	C₂×C₃₁	1	2	yes	—
220	#1921	C₂×C₃₁	1	2	yes	complemented
221	#1802	C₂×C₃₁	1	1	yes	characteristic
222	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
223	C₂²×C₃₁	—	4	2	no	—
224	C₂²×C₃₁	—	4	2	no	—
225	C₂²×C₃₁	—	4	2	no	—
226	C₂²×C₃₁	—	8	1	no	—
227	C₂²×C₃₁	—	8	1	no	—
228	C₂²×C₃₁	—	8	1	no	—
229	C₂²×C₃₁	—	8	1	no	—
230	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
231	C₂²×C₃₁	#536	1	1	yes	characteristic
232	C₂²×C₃₁	#537	1	1	yes	characteristic
233	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
236	C₂²×C₃₁	#575	1	1	yes	characteristic
237	C₄×C₃₁	—	2	2	no	—
238	C₄×C₃₁	—	2	2	no	—
239	C₄×C₃₁	—	4	2	no	—
240	C₄×C₃₁	—	4	4	no	—
241	C₄×C₃₁	—	4	1	no	—
242	C₄×C₃₁	—	4	1	no	—
243	C₄×C₃₁	—	4	1	no	—
244	C₄×C₃₁	—	8	1	no	—
245	C₄×C₃₁	—	8	2	no	—
246	C₄×C₃₁	—	8	1	no	—
247	C₄×C₃₁	#442	1	4	yes	—
248	C₄×C₃₁	#519	1	1	yes	characteristic
249	C₄×C₃₁	#520	1	1	yes	characteristic
250	C₄×C₃₁	#541	1	1	yes	characteristic
251	C₄×C₃₁	#542	1	1	yes	characteristic
252	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×C₄×Q₈	—	2	1	no	—
254	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×#117	—	2	1	no	—
256	#474	C₄×C₃₁	1	8	yes	complemented
257	C₂×#107	C₂²×C₃₁	1	2	yes	—
258	C₂×#106	C₂²×C₃₁	1	2	yes	—
259	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×#102	C₂²×C₃₁	1	2	yes	—
262	C₂×#102	C₂²×C₃₁	1	2	yes	—
263	C₂×#97	C₂²×C₃₁	1	2	yes	—
264	#391	—	2	1	no	—
265	#390	C₂²×C₃₁	1	1	yes	characteristic
266	#390	C₂²×C₃₁	1	1	yes	characteristic
267	#389	—	2	1	no	—
268	#389	C₂²×C₃₁	1	2	yes	—
269	#389	C₂²×C₃₁	1	1	yes	characteristic
270	#389	C₂²×C₃₁	1	1	yes	characteristic
271	#388	—	2	1	no	—
272	#388	—	2	1	no	—
273	#384	C₂²×C₃₁	1	1	yes	characteristic
274	C₄×C₄⋊C₄	—	2	1	no	—
275	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
276	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
277	C₄×C₂²⋊C₄	—	2	1	no	—
278	C₄³	C₂²×C₃₁	1	1	yes	characteristic
279	#340	C₂²×C₃₁	1	2	yes	—
280	#340	C₂²×C₃₁	1	2	yes	—
281	#340	C₂²×C₃₁	1	2	yes	—
282	#340	C₂²×C₃₁	1	2	yes	—
283	#336	C₂²×C₃₁	1	2	yes	—
284	C₂×C₃₁	—	8	2	no	—
285	C₂×C₃₁	#2323	1	1	yes	characteristic
286	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
287	C₂×C₃₁	C₄×#137	1	1	yes	characteristic
288	C₂×C₃₁	#3235	1	1	yes	characteristic
289	C₂×C₃₁	#3239	1	1	yes	characteristic
290	C₂×C₃₁	#3251	1	1	yes	characteristic
291	C₂×C₃₁	#3254	1	1	yes	characteristic
292	C₂²×Q₈	—	2	1	no	—
293	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
294	C₂³×C₄	—	2	1	no	—
295	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
296	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
297	C₄⋊Q₈	—	2	2	no	—
298	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
299	C₄×Q₈	—	4	4	no	—
300	C₄×D₄	—	2	2	no	—
301	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
302	#117	—	4	4	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	—	2	2	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
311	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
312	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
313	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
314	C₂×C₂²⋊C₄	—	2	1	no	—
315	C₂×C₂²⋊C₄	—	2	1	no	—
316	C₂×C₂²⋊C₄	—	2	1	no	—
317	C₂×C₂²⋊C₄	—	2	1	no	—
318	C₂×C₂²⋊C₄	—	2	1	no	—
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	—	2	2	no	—
321	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
322	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
326	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
327	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
328	#107	C₂×C₄×C₃₁	1	8	yes	—
329	#106	—	4	2	no	—
330	C₄⋊C₈	—	2	2	no	—
331	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
332	#102	—	4	2	no	—
333	#102	C₂×C₄×C₃₁	1	8	yes	—
334	#97	—	4	2	no	—
335	#95	—	2	1	no	—
336	#95	—	2	1	no	—
337	#95	—	2	2	no	—
338	#95	—	2	1	no	—
339	#95	—	2	2	no	—
340	C₃₁	#22116	1	1	yes	characteristic, direct product factor, Sylow
341	C₂⁴	—	2	1	no	—
342	C₂×Q₈	—	2	4	no	—
343	C₂×Q₈	—	4	2	no	—
344	C₂×Q₈	—	4	2	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	1	no	—
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	4	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	4	2	no	—
358	C₂²×C₄	—	4	2	no	—
359	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
372	C₄⋊C₄	—	2	4	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	4	2	no	—
382	C₄⋊C₄	—	4	2	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	—	4	4	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
387	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
388	C₂²⋊C₄	—	2	4	no	—
389	C₂²⋊C₄	—	4	2	no	—
390	C₂²⋊C₄	—	4	2	no	—
391	C₂²⋊C₄	—	4	2	no	—
392	C₂²⋊C₄	—	4	2	no	—
393	C₄²	—	2	2	no	—
394	C₄²	—	2	2	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	2	1	no	—
398	C₄²	—	2	1	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	4	1	no	—
401	C₄²	—	4	4	no	—
402	C₄²	—	4	2	no	—
403	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
404	C₄²	#41×C₃₁	1	2	yes	—
405	C₂³	—	2	2	no	—
406	C₂³	—	4	1	no	—
407	C₂³	—	4	1	no	—
408	C₂³	—	4	1	no	—
409	C₂³	—	4	1	no	—
410	C₂³	—	4	1	no	—
411	C₂³	—	4	1	no	—
412	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
413	Q₈	—	4	8	no	—
414	D₄	—	2	2	no	—
415	D₄	—	2	2	no	—
416	D₄	—	4	1	no	—
417	D₄	—	4	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	8	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
448	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
449	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
450	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
451	C₂×C₄	C₃₁×#137	1	2	yes	—
452	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
453	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
454	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
455	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
456	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
457	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
458	C₈	—	2	4	no	—
459	C₈	—	4	2	no	—
460	C₂²	—	4	2	no	—
461	C₂²	—	4	2	no	—
462	C₂²	—	4	2	no	—
463	C₂²	—	8	1	no	—
464	C₂²	—	8	1	no	—
465	C₂²	—	8	1	no	—
466	C₂²	—	8	1	no	—
467	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
468	C₂²	C₃₁×#536	1	1	yes	characteristic
469	C₂²	C₃₁×#537	1	1	yes	characteristic
470	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
471	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
472	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
473	C₂²	C₃₁×#575	1	1	yes	characteristic
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	4	2	no	—
477	C₄	—	4	4	no	—
478	C₄	—	4	1	no	—
479	C₄	—	4	1	no	—
480	C₄	—	4	1	no	—
481	C₄	—	8	1	no	—
482	C₄	—	8	2	no	—
483	C₄	—	8	1	no	—
484	C₄	C₃₁×#442	1	4	yes	—
485	C₄	C₃₁×#519	1	1	yes	characteristic
486	C₄	C₃₁×#520	1	1	yes	characteristic
487	C₄	C₃₁×#541	1	1	yes	characteristic
488	C₄	C₃₁×#542	1	1	yes	characteristic
489	C₂	—	8	2	no	—
490	C₂	C₃₁×#2323	1	1	yes	characteristic
491	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
492	C₂	C₄×C₃₁×#137	1	1	yes	characteristic
493	C₂	C₃₁×#3235	1	1	yes	characteristic
494	C₂	C₃₁×#3239	1	1	yes	characteristic
495	C₂	C₃₁×#3251	1	1	yes	characteristic
496	C₂	C₃₁×#3254	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂×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. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
4. C₃₁×#1988	C₂	semidirect
5. C₃₁×#1948	C₂	semidirect
6. C₃₁×#1943	C₂	semidirect
9. C₃₁×#1921	C₂	semidirect
11. C₃₁×#1767	C₂	semidirect
16. C₃₁×#474	C₄	semidirect
156. #22116	C₃₁	direct (N × H)
213. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
215. #1988	C₂×C₃₁	semidirect
216. #1948	C₂×C₃₁	semidirect
217. #1943	C₂×C₃₁	semidirect
220. #1921	C₂×C₃₁	semidirect
222. #1767	C₂×C₃₁	semidirect
256. #474	C₄×C₃₁	semidirect
340. C₃₁	#22116	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×#474	2
C₄×C₃₁×C₄⋊Q₈	2
C₃₁×#2293	2
C₃₁×#1988	2
C₃₁×#1948	2
C₃₁×#1943	2
C₃₁×#1936	2
C₃₁×#1926	2
C₃₁×#1921	2
C₃₁×#1802	2
C₃₁×#1767	2
#22116	31

Subgroups — C31×#22116 #21478875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22116 #21478875