Subgroups — C₃₁×#22652 #21479411

Order 7936.

964 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#435	C₂	1	2	yes	maximal
2	C₂×C₃₁×#424	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1940	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1851	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1850	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×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#117	C₂²	1	2	yes	—
14	C₃₁×#435	—	2	8	no	—
15	C₃₁×#424	—	2	8	no	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₂×C₃₁×#97	C₂²	1	2	yes	—
27	C₃₁×#390	C₄	1	1	yes	characteristic
28	C₃₁×#390	C₄	1	1	yes	characteristic
29	C₃₁×#389	C₄	1	1	yes	characteristic
30	C₃₁×#389	C₄	1	1	yes	characteristic
31	C₃₁×#388	C₄	1	2	yes	—
32	C₃₁×#388	C₄	1	2	yes	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
35	C₄³×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂×M₁₆×C₃₁	—	2	4	no	—
42	C₂×M₁₆×C₃₁	—	2	4	no	—
43	C₂×M₁₆×C₃₁	—	4	2	no	—
44	C₂²×C₈×C₃₁	—	2	4	no	—
45	C₂²×C₈×C₃₁	—	2	4	no	—
46	C₂²×C₈×C₃₁	—	4	2	no	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₄×D₄×C₃₁	D₄	1	4	yes	—
52	C₄×D₄×C₃₁	Q₈	1	4	yes	—
53	C₃₁×#117	—	2	4	no	—
54	C₃₁×#117	—	2	4	no	—
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
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×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
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	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×C₄⋊C₈	—	2	4	no	—
74	C₃₁×C₄⋊C₈	—	2	4	no	—
75	C₃₁×C₄⋊C₈	—	2	4	no	—
76	C₃₁×C₄⋊C₈	—	2	4	no	—
77	C₃₁×C₄⋊C₈	—	2	2	no	—
78	C₃₁×C₄⋊C₈	—	4	1	no	—
79	C₃₁×C₄⋊C₈	—	4	1	no	—
80	C₃₁×C₄⋊C₈	D₄	1	4	yes	complemented
81	C₃₁×C₂²⋊C₈	—	2	4	no	—
82	C₃₁×C₂²⋊C₈	—	2	4	no	—
83	C₃₁×#97	—	2	4	no	—
84	C₃₁×#95	C₂×C₄	1	2	yes	—
85	C₃₁×#95	C₂×C₄	1	2	yes	—
86	C₃₁×#95	C₂×C₄	1	2	yes	—
87	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₃₁	—	4	1	no	—
89	C₂×D₄×C₃₁	—	4	1	no	—
90	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
105	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
106	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
107	M₁₆×C₃₁	—	4	8	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	4	8	no	—
120	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
121	C₄⋊C₄×C₃₁	—	2	4	no	—
122	C₄⋊C₄×C₃₁	—	2	4	no	—
123	C₄⋊C₄×C₃₁	—	4	1	no	—
124	C₄⋊C₄×C₃₁	—	4	1	no	—
125	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
126	C₂²⋊C₄×C₃₁	—	2	4	no	—
127	C₂²⋊C₄×C₃₁	—	2	4	no	—
128	C₂²⋊C₄×C₃₁	—	2	4	no	—
129	C₂²⋊C₄×C₃₁	—	4	2	no	—
130	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
131	C₄²×C₃₁	—	2	2	no	—
132	C₄²×C₃₁	—	2	2	no	—
133	C₄²×C₃₁	—	2	2	no	—
134	C₄²×C₃₁	—	2	1	no	—
135	C₄²×C₃₁	—	2	1	no	—
136	C₄²×C₃₁	—	4	1	no	—
137	C₄²×C₃₁	—	4	1	no	—
138	C₄²×C₃₁	—	4	1	no	—
139	C₄²×C₃₁	C₂×D₄	1	2	yes	—
140	C₄²×C₃₁	C₂×D₄	1	2	yes	—
141	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
142	C₄²×C₃₁	#41	1	2	yes	—
143	#22652	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	2	2	no	—
146	C₂³×C₃₁	—	2	2	no	—
147	C₂³×C₃₁	—	2	2	no	—
148	C₂³×C₃₁	—	2	2	no	—
149	C₂³×C₃₁	—	2	2	no	—
150	C₂³×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
151	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
152	D₄×C₃₁	—	4	4	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	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	4	no	—
166	C₂×C₄×C₃₁	—	2	4	no	—
167	C₂×C₄×C₃₁	—	2	4	no	—
168	C₂×C₄×C₃₁	—	2	4	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	1	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
179	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
180	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
181	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₃₁	#131	1	2	yes	—
184	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
190	C₈×C₃₁	—	2	4	no	—
191	C₈×C₃₁	—	4	2	no	—
192	C₈×C₃₁	—	4	2	no	—
193	C₈×C₃₁	—	4	2	no	—
194	C₂×#435	C₂×C₃₁	1	2	yes	—
195	C₂×#424	C₂×C₃₁	1	2	yes	complemented
196	#2293	C₂×C₃₁	1	1	yes	characteristic
197	#1940	C₂×C₃₁	1	2	yes	complemented
198	#1851	C₂×C₃₁	1	2	yes	complemented
199	#1850	C₂×C₃₁	1	2	yes	complemented
200	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1798	C₂×C₃₁	1	2	yes	complemented
202	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂²×C₃₁	—	2	4	no	—
204	C₂²×C₃₁	—	2	4	no	—
205	C₂²×C₃₁	—	2	4	no	—
206	C₂²×C₃₁	—	4	2	no	—
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	4	2	no	—
209	C₂²×C₃₁	—	4	2	no	—
210	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
211	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
212	C₂²×C₃₁	#520	1	1	yes	characteristic
213	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
214	C₂²×C₃₁	#568	1	1	yes	characteristic
215	C₂²×C₃₁	#568	1	1	yes	characteristic
216	C₂²×C₃₁	#568	1	1	yes	characteristic
217	C₄×C₃₁	—	2	4	no	—
218	C₄×C₃₁	—	2	2	no	—
219	C₄×C₃₁	—	2	2	no	—
220	C₄×C₃₁	—	2	2	no	—
221	C₄×C₃₁	—	2	2	no	—
222	C₄×C₃₁	—	2	2	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	—	4	2	no	—
225	C₄×C₃₁	—	4	1	no	—
226	C₄×C₃₁	D₄²	1	1	yes	characteristic
227	C₄×C₃₁	#547	1	1	yes	characteristic
228	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
229	C₄×C₃₁	#554	1	1	yes	characteristic
230	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
231	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
232	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×#117	C₂²×C₃₁	1	2	yes	—
234	#435	—	2	8	no	—
235	#424	—	2	8	no	—
236	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
238	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
239	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
240	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
241	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
245	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
246	C₂×#97	C₂²×C₃₁	1	2	yes	—
247	#390	C₄×C₃₁	1	1	yes	characteristic
248	#390	C₄×C₃₁	1	1	yes	characteristic
249	#389	C₄×C₃₁	1	1	yes	characteristic
250	#389	C₄×C₃₁	1	1	yes	characteristic
251	#388	C₄×C₃₁	1	2	yes	—
252	#388	C₄×C₃₁	1	2	yes	—
253	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
254	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
255	C₄³	C₂²×C₃₁	1	1	yes	characteristic
256	#336	C₂²×C₃₁	1	2	yes	—
257	#336	C₂²×C₃₁	1	2	yes	—
258	#336	C₂²×C₃₁	1	2	yes	—
259	C₂×C₃₁	—	4	4	no	—
260	C₂×C₃₁	D₄×C₄⋊C₄	1	1	yes	characteristic
261	C₂×C₃₁	#2905	1	1	yes	characteristic
262	C₂×C₃₁	#2906	1	1	yes	characteristic
263	C₂×C₃₁	#2932	1	1	yes	characteristic
264	C₂×C₃₁	#2935	1	1	yes	characteristic
265	C₂×C₃₁	#2992	1	1	yes	characteristic
266	C₂×C₃₁	#2995	1	1	yes	characteristic
267	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
268	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
269	C₂×M₁₆	—	2	4	no	—
270	C₂×M₁₆	—	2	4	no	—
271	C₂×M₁₆	—	4	2	no	—
272	C₂²×C₈	—	2	4	no	—
273	C₂²×C₈	—	2	4	no	—
274	C₂²×C₈	—	4	2	no	—
275	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
276	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
277	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
278	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
279	C₄×D₄	D₄×C₃₁	1	4	yes	—
280	C₄×D₄	Q₈×C₃₁	1	4	yes	—
281	#117	—	2	4	no	—
282	#117	—	2	4	no	—
283	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
284	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
285	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
286	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
288	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
289	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
290	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
291	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
292	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
293	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
298	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
300	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
301	C₄⋊C₈	—	2	4	no	—
302	C₄⋊C₈	—	2	4	no	—
303	C₄⋊C₈	—	2	4	no	—
304	C₄⋊C₈	—	2	4	no	—
305	C₄⋊C₈	—	2	2	no	—
306	C₄⋊C₈	—	4	1	no	—
307	C₄⋊C₈	—	4	1	no	—
308	C₄⋊C₈	D₄×C₃₁	1	4	yes	complemented
309	C₂²⋊C₈	—	2	4	no	—
310	C₂²⋊C₈	—	2	4	no	—
311	#97	—	2	4	no	—
312	#95	C₂×C₄×C₃₁	1	2	yes	—
313	#95	C₂×C₄×C₃₁	1	2	yes	—
314	#95	C₂×C₄×C₃₁	1	2	yes	—
315	C₃₁	#22652	1	1	yes	characteristic, direct product factor, Sylow
316	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
317	C₂×D₄	—	4	1	no	—
318	C₂×D₄	—	4	1	no	—
319	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
320	C₂²×C₄	—	2	2	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	2	no	—
323	C₂²×C₄	—	2	2	no	—
324	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
326	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
327	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
328	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
329	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
330	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
331	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
332	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
334	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
335	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
336	M₁₆	—	4	8	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	4	8	no	—
349	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
350	C₄⋊C₄	—	2	4	no	—
351	C₄⋊C₄	—	2	4	no	—
352	C₄⋊C₄	—	4	1	no	—
353	C₄⋊C₄	—	4	1	no	—
354	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
355	C₂²⋊C₄	—	2	4	no	—
356	C₂²⋊C₄	—	2	4	no	—
357	C₂²⋊C₄	—	2	4	no	—
358	C₂²⋊C₄	—	4	2	no	—
359	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
360	C₄²	—	2	2	no	—
361	C₄²	—	2	2	no	—
362	C₄²	—	2	2	no	—
363	C₄²	—	2	1	no	—
364	C₄²	—	2	1	no	—
365	C₄²	—	4	1	no	—
366	C₄²	—	4	1	no	—
367	C₄²	—	4	1	no	—
368	C₄²	C₂×D₄×C₃₁	1	2	yes	—
369	C₄²	C₂×D₄×C₃₁	1	2	yes	—
370	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
371	C₄²	#41×C₃₁	1	2	yes	—
372	C₂³	—	2	2	no	—
373	C₂³	—	2	2	no	—
374	C₂³	—	2	2	no	—
375	C₂³	—	2	2	no	—
376	C₂³	—	2	2	no	—
377	C₂³	—	2	2	no	—
378	C₂³	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
379	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
380	D₄	—	4	4	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	4	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	2	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	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
407	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
408	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
409	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
410	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
411	C₂×C₄	C₃₁×#131	1	2	yes	—
412	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
416	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
417	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
418	C₈	—	2	4	no	—
419	C₈	—	4	2	no	—
420	C₈	—	4	2	no	—
421	C₈	—	4	2	no	—
422	C₂²	—	2	4	no	—
423	C₂²	—	2	4	no	—
424	C₂²	—	2	4	no	—
425	C₂²	—	4	2	no	—
426	C₂²	—	4	2	no	—
427	C₂²	—	4	2	no	—
428	C₂²	—	4	2	no	—
429	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
430	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
431	C₂²	C₃₁×#520	1	1	yes	characteristic
432	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
433	C₂²	C₃₁×#568	1	1	yes	characteristic
434	C₂²	C₃₁×#568	1	1	yes	characteristic
435	C₂²	C₃₁×#568	1	1	yes	characteristic
436	C₄	—	2	4	no	—
437	C₄	—	2	2	no	—
438	C₄	—	2	2	no	—
439	C₄	—	2	2	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	2	no	—
442	C₄	—	4	2	no	—
443	C₄	—	4	2	no	—
444	C₄	—	4	1	no	—
445	C₄	D₄²×C₃₁	1	1	yes	characteristic
446	C₄	C₃₁×#547	1	1	yes	characteristic
447	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
448	C₄	C₃₁×#554	1	1	yes	characteristic
449	C₂	—	4	4	no	—
450	C₂	D₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
451	C₂	C₃₁×#2905	1	1	yes	characteristic
452	C₂	C₃₁×#2906	1	1	yes	characteristic
453	C₂	C₃₁×#2932	1	1	yes	characteristic
454	C₂	C₃₁×#2935	1	1	yes	characteristic
455	C₂	C₃₁×#2992	1	1	yes	characteristic
456	C₂	C₃₁×#2995	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
2. C₂×C₃₁×#424	C₂	semidirect
4. C₃₁×#1940	C₂	semidirect
5. C₃₁×#1851	C₂	semidirect
6. C₃₁×#1850	C₂	semidirect
7. C₃₁×#1842	C₂	semidirect
8. C₃₁×#1798	C₂	semidirect
9. C₃₁×#1767	C₂	semidirect
80. C₃₁×C₄⋊C₈	D₄	semidirect
143. #22652	C₃₁	direct (N × H)
195. C₂×#424	C₂×C₃₁	semidirect
197. #1940	C₂×C₃₁	semidirect
198. #1851	C₂×C₃₁	semidirect
199. #1850	C₂×C₃₁	semidirect
200. #1842	C₂×C₃₁	semidirect
201. #1798	C₂×C₃₁	semidirect
202. #1767	C₂×C₃₁	semidirect
308. C₄⋊C₈	D₄×C₃₁	semidirect
315. C₃₁	#22652	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×#435	2
C₂×C₃₁×#424	2
C₃₁×#2293	2
C₃₁×#1940	2
C₃₁×#1851	2
C₃₁×#1850	2
C₃₁×#1842	2
C₃₁×#1798	2
C₃₁×#1767	2
#22652	31

Subgroups — C31×#22652 #21479411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22652 #21479411