Subgroups — C₃₁×#22796 #21479555

Order 7936.

828 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#433	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1954	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1954	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1924	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1923	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1798	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#117	C₂²	1	2	yes	—
14	C₃₁×#433	—	2	8	no	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×#97	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#390	C₄	1	1	yes	characteristic
30	C₃₁×#390	C₄	1	1	yes	characteristic
31	C₃₁×#389	C₄	1	1	yes	characteristic
32	C₃₁×#389	C₄	1	1	yes	characteristic
33	C₃₁×#388	C₄	1	2	yes	—
34	C₃₁×#388	C₄	1	2	yes	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
37	C₄³×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₄×D₄×C₃₁	—	2	2	no	—
51	C₄×D₄×C₃₁	—	2	2	no	—
52	C₃₁×#117	—	2	4	no	—
53	C₃₁×#117	—	2	4	no	—
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
69	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×C₄⋊C₈	—	2	4	no	—
73	C₃₁×C₄⋊C₈	—	2	2	no	—
74	C₃₁×C₄⋊C₈	—	2	2	no	—
75	C₃₁×C₄⋊C₈	—	2	2	no	—
76	C₃₁×C₄⋊C₈	—	4	2	no	—
77	C₃₁×C₄⋊C₈	D₄	1	4	yes	complemented
78	C₃₁×C₂²⋊C₈	—	2	4	no	—
79	C₃₁×C₂²⋊C₈	—	2	4	no	—
80	C₃₁×C₂²⋊C₈	—	4	2	no	—
81	C₃₁×C₂²⋊C₈	—	4	2	no	—
82	C₃₁×#97	—	2	4	no	—
83	C₄×C₈×C₃₁	—	2	4	no	—
84	C₄×C₈×C₃₁	—	2	2	no	—
85	C₄×C₈×C₃₁	—	2	2	no	—
86	C₃₁×#95	C₂×C₄	1	2	yes	—
87	C₃₁×#95	C₂×C₄	1	2	yes	—
88	C₃₁×#95	C₂×C₄	1	2	yes	—
89	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₃₁	—	2	1	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	4	no	—
129	C₄⋊C₄×C₃₁	—	2	4	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₂²⋊C₄×C₃₁	—	2	4	no	—
133	C₂²⋊C₄×C₃₁	—	2	4	no	—
134	C₂²⋊C₄×C₃₁	—	2	4	no	—
135	C₂²⋊C₄×C₃₁	—	2	4	no	—
136	C₂²⋊C₄×C₃₁	—	4	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	2	2	no	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	—	2	2	no	—
141	C₄²×C₃₁	—	2	2	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	4	1	no	—
145	C₄²×C₃₁	C₂×D₄	1	2	yes	—
146	C₄²×C₃₁	#41	1	2	yes	—
147	C₄²×C₃₁	#41	1	2	yes	—
148	C₄²×C₃₁	#41	1	2	yes	—
149	#22796	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	2	no	—
151	C₂³×C₃₁	—	2	2	no	—
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	2	2	no	—
155	C₂³×C₃₁	—	2	2	no	—
156	C₂³×C₃₁	—	2	2	no	—
157	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
158	D₄×C₃₁	—	4	4	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	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	2	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	#117	1	2	yes	—
184	C₂×C₄×C₃₁	#117	1	2	yes	—
185	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
186	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
187	C₂×C₄×C₃₁	#131	1	2	yes	—
188	C₂×C₄×C₃₁	#131	1	2	yes	—
189	C₂×C₄×C₃₁	#131	1	2	yes	—
190	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
195	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
196	C₈×C₃₁	—	2	2	no	—
197	C₈×C₃₁	—	2	2	no	—
198	C₈×C₃₁	—	4	2	no	—
199	C₈×C₃₁	—	4	2	no	—
200	C₈×C₃₁	—	4	1	no	—
201	C₈×C₃₁	—	4	1	no	—
202	C₂×#433	C₂×C₃₁	1	2	yes	complemented
203	#2293	C₂×C₃₁	1	1	yes	characteristic
204	#1954	C₂×C₃₁	1	1	yes	characteristic
205	#1954	C₂×C₃₁	1	1	yes	characteristic
206	#1924	C₂×C₃₁	1	2	yes	complemented
207	#1923	C₂×C₃₁	1	2	yes	complemented
208	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#1798	C₂×C₃₁	1	2	yes	complemented
210	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
211	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂²×C₃₁	—	4	2	no	—
214	C₂²×C₃₁	—	4	2	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	4	2	no	—
220	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
221	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂²×C₃₁	#520	1	1	yes	characteristic
223	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
224	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
225	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
226	C₂²×C₃₁	#568	1	1	yes	characteristic
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	2	2	no	—
230	C₄×C₃₁	—	2	2	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	4	2	no	—
234	C₄×C₃₁	—	4	1	no	—
235	C₄×C₃₁	#433	1	4	yes	complemented
236	C₄×C₃₁	#547	1	1	yes	characteristic
237	C₄×C₃₁	#550	1	1	yes	characteristic
238	C₄×C₃₁	#555	1	1	yes	characteristic
239	C₄×C₃₁	#556	1	1	yes	characteristic
240	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#117	C₂²×C₃₁	1	2	yes	—
242	#433	—	2	8	no	—
243	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
245	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
248	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
250	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
251	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
252	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
253	C₂×#97	C₂²×C₃₁	1	2	yes	—
254	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
255	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
257	#390	C₄×C₃₁	1	1	yes	characteristic
258	#390	C₄×C₃₁	1	1	yes	characteristic
259	#389	C₄×C₃₁	1	1	yes	characteristic
260	#389	C₄×C₃₁	1	1	yes	characteristic
261	#388	C₄×C₃₁	1	2	yes	—
262	#388	C₄×C₃₁	1	2	yes	—
263	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
264	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
265	C₄³	C₂²×C₃₁	1	1	yes	characteristic
266	#336	C₂²×C₃₁	1	2	yes	—
267	#336	C₂²×C₃₁	1	2	yes	—
268	#336	C₂²×C₃₁	1	1	yes	characteristic
269	#336	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₃₁	—	8	2	no	—
271	C₂×C₃₁	#2354	1	1	yes	characteristic
272	C₂×C₃₁	C₂×#433	1	1	yes	characteristic
273	C₂×C₃₁	#2925	1	1	yes	characteristic
274	C₂×C₃₁	#2931	1	1	yes	characteristic
275	C₂×C₃₁	#2935	1	1	yes	characteristic
276	C₂×C₃₁	#2993	1	1	yes	characteristic
277	C₂×C₃₁	#2994	1	1	yes	characteristic
278	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
279	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
280	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
283	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
284	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
285	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₄×D₄	—	2	2	no	—
287	C₄×D₄	—	2	2	no	—
288	#117	—	2	4	no	—
289	#117	—	2	4	no	—
290	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
291	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
292	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
295	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
296	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
297	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
298	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
299	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
300	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
302	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
305	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
306	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
307	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
308	C₄⋊C₈	—	2	4	no	—
309	C₄⋊C₈	—	2	2	no	—
310	C₄⋊C₈	—	2	2	no	—
311	C₄⋊C₈	—	2	2	no	—
312	C₄⋊C₈	—	4	2	no	—
313	C₄⋊C₈	D₄×C₃₁	1	4	yes	complemented
314	C₂²⋊C₈	—	2	4	no	—
315	C₂²⋊C₈	—	2	4	no	—
316	C₂²⋊C₈	—	4	2	no	—
317	C₂²⋊C₈	—	4	2	no	—
318	#97	—	2	4	no	—
319	C₄×C₈	—	2	4	no	—
320	C₄×C₈	—	2	2	no	—
321	C₄×C₈	—	2	2	no	—
322	#95	C₂×C₄×C₃₁	1	2	yes	—
323	#95	C₂×C₄×C₃₁	1	2	yes	—
324	#95	C₂×C₄×C₃₁	1	2	yes	—
325	C₃₁	#22796	1	1	yes	characteristic, direct product factor, Sylow
326	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
337	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
338	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
339	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
340	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
342	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
343	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
345	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
363	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	4	no	—
366	C₄⋊C₄	—	2	4	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₂²⋊C₄	—	2	4	no	—
370	C₂²⋊C₄	—	2	4	no	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₂²⋊C₄	—	4	2	no	—
374	C₄²	—	2	2	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	2	1	no	—
380	C₄²	—	2	1	no	—
381	C₄²	—	4	1	no	—
382	C₄²	C₂×D₄×C₃₁	1	2	yes	—
383	C₄²	#41×C₃₁	1	2	yes	—
384	C₄²	#41×C₃₁	1	2	yes	—
385	C₄²	#41×C₃₁	1	2	yes	—
386	C₂³	—	2	2	no	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	2	no	—
393	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
394	D₄	—	4	4	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	C₃₁×#117	1	2	yes	—
420	C₂×C₄	C₃₁×#117	1	2	yes	—
421	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
422	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
423	C₂×C₄	C₃₁×#131	1	2	yes	—
424	C₂×C₄	C₃₁×#131	1	2	yes	—
425	C₂×C₄	C₃₁×#131	1	2	yes	—
426	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
431	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
432	C₈	—	2	2	no	—
433	C₈	—	2	2	no	—
434	C₈	—	4	2	no	—
435	C₈	—	4	2	no	—
436	C₈	—	4	1	no	—
437	C₈	—	4	1	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	2	no	—
445	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
446	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
447	C₂²	C₃₁×#520	1	1	yes	characteristic
448	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
449	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
450	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
451	C₂²	C₃₁×#568	1	1	yes	characteristic
452	C₄	—	2	2	no	—
453	C₄	—	2	2	no	—
454	C₄	—	2	2	no	—
455	C₄	—	2	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	4	2	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	1	no	—
460	C₄	C₃₁×#433	1	4	yes	complemented
461	C₄	C₃₁×#547	1	1	yes	characteristic
462	C₄	C₃₁×#550	1	1	yes	characteristic
463	C₄	C₃₁×#555	1	1	yes	characteristic
464	C₄	C₃₁×#556	1	1	yes	characteristic
465	C₂	—	8	2	no	—
466	C₂	C₃₁×#2354	1	1	yes	characteristic
467	C₂	C₂×C₃₁×#433	1	1	yes	characteristic
468	C₂	C₃₁×#2925	1	1	yes	characteristic
469	C₂	C₃₁×#2931	1	1	yes	characteristic
470	C₂	C₃₁×#2935	1	1	yes	characteristic
471	C₂	C₃₁×#2993	1	1	yes	characteristic
472	C₂	C₃₁×#2994	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#433	C₂	semidirect
5. C₃₁×#1924	C₂	semidirect
6. C₃₁×#1923	C₂	semidirect
7. C₃₁×#1842	C₂	semidirect
8. C₃₁×#1798	C₂	semidirect
9. C₃₁×#1770	C₂	semidirect
10. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
11. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
77. C₃₁×C₄⋊C₈	D₄	semidirect
149. #22796	C₃₁	direct (N × H)
202. C₂×#433	C₂×C₃₁	semidirect
206. #1924	C₂×C₃₁	semidirect
207. #1923	C₂×C₃₁	semidirect
208. #1842	C₂×C₃₁	semidirect
209. #1798	C₂×C₃₁	semidirect
210. #1770	C₂×C₃₁	semidirect
211. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
212. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
235. C₄×C₃₁	#433	semidirect
313. C₄⋊C₈	D₄×C₃₁	semidirect
325. C₃₁	#22796	direct (N × H)
460. C₄	C₃₁×#433	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#433	2
C₃₁×#2293	2
C₃₁×#1954	2
C₃₁×#1924	2
C₃₁×#1923	2
C₃₁×#1842	2
C₃₁×#1798	2
C₃₁×#1770	2
C₈×C₄⋊C₄×C₃₁	2
C₄×C₃₁×C₄⋊C₈	2
#22796	31

Subgroups — C31×#22796 #21479555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22796 #21479555