Subgroups — C₃₁×#20491 #21477250

Order 7936.

1246 conjugacy classes of subgroups.

The subgroup lattice has 464 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×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1771	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1753	C₂	1	2	yes	maximal, complemented
8	C₈×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1749	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1749	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#435	C₄	1	16	yes	complemented
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₄×M₁₆×C₃₁	—	2	4	no	—
23	C₂×C₃₁×#97	—	2	4	no	—
24	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#97	C₂²	1	2	yes	—
26	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#97	C₂²	1	2	yes	—
29	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
35	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₄³×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₃₁	—	2	4	no	—
48	C₂×M₁₆×C₃₁	—	2	4	no	—
49	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	complemented
50	C₂²×C₈×C₃₁	—	2	4	no	—
51	C₂²×C₈×C₃₁	—	2	4	no	—
52	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	complemented
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
57	C₄×D₄×C₃₁	—	2	8	no	—
58	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₃₁	—	2	2	no	—
66	C₂×C₄²×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	—	2	2	no	—
68	C₂×C₄²×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
71	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂³	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×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
79	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	complemented
80	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	complemented
81	C₃₁×#97	—	2	4	no	—
82	C₃₁×#97	—	2	2	no	—
83	C₃₁×#97	—	2	2	no	—
84	C₃₁×#97	—	2	2	no	—
85	C₃₁×#97	C₂×C₄	1	8	yes	complemented
86	C₃₁×#97	D₄	1	4	yes	complemented
87	C₄×C₈×C₃₁	—	2	4	no	—
88	C₄×C₈×C₃₁	—	2	4	no	—
89	C₃₁×#95	C₂×C₄	1	2	yes	—
90	C₃₁×#95	C₂×C₄	1	2	yes	—
91	C₃₁×#95	C₂×C₄	1	2	yes	—
92	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×D₄×C₃₁	—	2	4	no	—
95	C₂×D₄×C₃₁	C₄²	1	4	yes	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	4	no	—
103	C₂²×C₄×C₃₁	—	2	4	no	—
104	C₂²×C₄×C₃₁	—	2	4	no	—
105	C₂²×C₄×C₃₁	—	2	4	no	—
106	C₂²×C₄×C₃₁	—	4	2	no	—
107	C₂²×C₄×C₃₁	—	4	2	no	—
108	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
114	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
115	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
119	M₁₆×C₃₁	—	2	16	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	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	16	no	—
127	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
128	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
129	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
130	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
131	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
132	C₂×C₈×C₃₁	#41	1	4	yes	—
133	C₄⋊C₄×C₃₁	—	2	4	no	—
134	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
135	C₂²⋊C₄×C₃₁	—	2	4	no	—
136	C₂²⋊C₄×C₃₁	C₄²	1	4	yes	—
137	C₂²⋊C₄×C₃₁	C₄²	1	4	yes	—
138	C₂²⋊C₄×C₃₁	M₁₆	1	8	yes	complemented
139	C₄²×C₃₁	—	2	4	no	—
140	C₄²×C₃₁	—	2	4	no	—
141	C₄²×C₃₁	—	2	4	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	2	2	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	4	2	no	—
148	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
149	C₄²×C₃₁	C₂×D₄	1	2	yes	—
150	C₄²×C₃₁	#41	1	2	yes	—
151	#20491	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂³×C₃₁	—	2	1	no	—
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	2	2	no	—
155	C₂³×C₃₁	—	2	1	no	—
156	C₂³×C₃₁	—	2	1	no	—
157	C₂³×C₃₁	—	2	2	no	—
158	C₂³×C₃₁	—	2	2	no	—
159	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
160	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
161	C₂³×C₃₁	C₂×M₁₆	1	4	yes	—
162	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
163	D₄×C₃₁	—	2	8	no	—
164	C₂×C₄×C₃₁	—	2	2	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	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	2	4	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	4	8	no	—
184	C₂×C₄×C₃₁	—	4	8	no	—
185	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
186	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
187	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
188	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
189	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
190	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
191	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
192	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
193	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
195	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
196	C₈×C₃₁	—	2	4	no	—
197	C₈×C₃₁	—	2	4	no	—
198	C₈×C₃₁	—	2	4	no	—
199	C₈×C₃₁	C₄×D₄	1	8	yes	complemented
200	C₂×#435	C₂×C₃₁	1	2	yes	—
201	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂²×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
204	#1771	C₂×C₃₁	1	2	yes	complemented
205	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1753	C₂×C₃₁	1	2	yes	complemented
207	C₈×C₂²⋊C₄	C₂×C₃₁	1	2	yes	complemented
208	#1749	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#1749	C₂×C₃₁	1	1	yes	characteristic, complemented
210	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
211	C₂²×C₃₁	—	2	2	no	—
212	C₂²×C₃₁	—	2	4	no	—
213	C₂²×C₃₁	—	2	4	no	—
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	2	2	no	—
216	C₂²×C₃₁	—	2	2	no	—
217	C₂²×C₃₁	—	2	2	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	4	2	no	—
220	C₂²×C₃₁	C₄×M₁₆	1	4	yes	complemented
221	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
222	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
223	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
224	C₂²×C₃₁	C₂²×M₁₆	1	2	yes	—
225	C₂²×C₃₁	#568	1	2	yes	—
226	C₄×C₃₁	—	2	2	no	—
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	2	4	no	—
229	C₄×C₃₁	—	2	4	no	—
230	C₄×C₃₁	—	2	4	no	—
231	C₄×C₃₁	—	2	2	no	—
232	C₄×C₃₁	—	2	2	no	—
233	C₄×C₃₁	—	4	2	no	—
234	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
235	C₄×C₃₁	C₄×#41	1	2	yes	—
236	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
237	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
238	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
239	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
240	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
241	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
242	#435	C₄×C₃₁	1	16	yes	complemented
243	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
244	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
245	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
246	C₄×M₁₆	—	2	4	no	—
247	C₂×#97	—	2	4	no	—
248	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#97	C₂²×C₃₁	1	2	yes	—
250	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×#97	C₂²×C₃₁	1	2	yes	—
253	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
255	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
256	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
257	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
258	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
259	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
260	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
261	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
262	C₄³	C₂²×C₃₁	1	1	yes	characteristic
263	#336	C₂²×C₃₁	1	2	yes	—
264	#336	C₂²×C₃₁	1	2	yes	—
265	#336	C₂²×C₃₁	1	2	yes	—
266	C₂×C₃₁	—	2	4	no	—
267	C₂×C₃₁	—	4	2	no	—
268	C₂×C₃₁	C₄²×D₄	1	1	yes	characteristic
269	C₂×C₃₁	C₂×C₄×M₁₆	1	1	yes	characteristic
270	C₂×C₃₁	#2875	1	1	yes	characteristic
271	C₂×C₃₁	D₄×M₁₆	1	2	yes	—
272	C₂×C₃₁	#2970	1	2	yes	—
273	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
274	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
275	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
276	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
277	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
278	C₂×M₁₆	—	2	4	no	—
279	C₂×M₁₆	—	2	4	no	—
280	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
281	C₂²×C₈	—	2	4	no	—
282	C₂²×C₈	—	2	4	no	—
283	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	complemented
284	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
285	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
286	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
287	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
288	C₄×D₄	—	2	8	no	—
289	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
290	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
291	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
292	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
293	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
294	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
295	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
296	C₂×C₄²	—	2	2	no	—
297	C₂×C₄²	—	2	2	no	—
298	C₂×C₄²	—	2	2	no	—
299	C₂×C₄²	—	2	2	no	—
300	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
301	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
302	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
303	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
304	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
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	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
309	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
310	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
311	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	complemented
312	#97	—	2	4	no	—
313	#97	—	2	2	no	—
314	#97	—	2	2	no	—
315	#97	—	2	2	no	—
316	#97	C₂×C₄×C₃₁	1	8	yes	complemented
317	#97	D₄×C₃₁	1	4	yes	complemented
318	C₄×C₈	—	2	4	no	—
319	C₄×C₈	—	2	4	no	—
320	#95	C₂×C₄×C₃₁	1	2	yes	—
321	#95	C₂×C₄×C₃₁	1	2	yes	—
322	#95	C₂×C₄×C₃₁	1	2	yes	—
323	C₃₁	#20491	1	1	yes	characteristic, direct product factor, Sylow
324	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
325	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
326	C₂×D₄	—	2	4	no	—
327	C₂×D₄	C₄²×C₃₁	1	4	yes	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	4	2	no	—
339	C₂²×C₄	—	4	2	no	—
340	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
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	C₂²×C₄×C₃₁	1	4	yes	—
346	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
347	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
348	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
349	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
350	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
351	M₁₆	—	2	16	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	16	no	—
359	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
360	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
361	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
362	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
363	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
364	C₂×C₈	#41×C₃₁	1	4	yes	—
365	C₄⋊C₄	—	2	4	no	—
366	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
367	C₂²⋊C₄	—	2	4	no	—
368	C₂²⋊C₄	C₄²×C₃₁	1	4	yes	—
369	C₂²⋊C₄	C₄²×C₃₁	1	4	yes	—
370	C₂²⋊C₄	M₁₆×C₃₁	1	8	yes	complemented
371	C₄²	—	2	4	no	—
372	C₄²	—	2	4	no	—
373	C₄²	—	2	4	no	—
374	C₄²	—	2	1	no	—
375	C₄²	—	2	1	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	1	no	—
378	C₄²	—	2	1	no	—
379	C₄²	—	4	2	no	—
380	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
381	C₄²	C₂×D₄×C₃₁	1	2	yes	—
382	C₄²	#41×C₃₁	1	2	yes	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	1	no	—
387	C₂³	—	2	1	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
391	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
392	C₂³	C₂×M₁₆×C₃₁	1	4	yes	—
393	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
394	D₄	—	2	8	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	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	4	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	4	8	no	—
415	C₂×C₄	—	4	8	no	—
416	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
417	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
418	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
419	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
420	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
421	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
422	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
423	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
424	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
426	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
427	C₈	—	2	4	no	—
428	C₈	—	2	4	no	—
429	C₈	—	2	4	no	—
430	C₈	C₄×D₄×C₃₁	1	8	yes	complemented
431	C₂²	—	2	2	no	—
432	C₂²	—	2	4	no	—
433	C₂²	—	2	4	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	2	no	—
440	C₂²	C₄×M₁₆×C₃₁	1	4	yes	complemented
441	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
442	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
443	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
444	C₂²	C₂²×M₁₆×C₃₁	1	2	yes	—
445	C₂²	C₃₁×#568	1	2	yes	—
446	C₄	—	2	2	no	—
447	C₄	—	2	2	no	—
448	C₄	—	2	4	no	—
449	C₄	—	2	4	no	—
450	C₄	—	2	4	no	—
451	C₄	—	2	2	no	—
452	C₄	—	2	2	no	—
453	C₄	—	4	2	no	—
454	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
455	C₄	C₄×#41×C₃₁	1	2	yes	—
456	C₂	—	2	4	no	—
457	C₂	—	4	2	no	—
458	C₂	C₄²×D₄×C₃₁	1	1	yes	characteristic
459	C₂	C₂×C₄×M₁₆×C₃₁	1	1	yes	characteristic
460	C₂	C₃₁×#2875	1	1	yes	characteristic
461	C₂	D₄×M₁₆×C₃₁	1	2	yes	—
462	C₂	C₃₁×#2970	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂² ▷ 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×C₄×M₁₆×C₃₁	C₂	semidirect
3. C₂²×C₃₁×#97	C₂	semidirect
5. C₃₁×#1771	C₂	semidirect
6. C₃₁×#1767	C₂	semidirect
7. C₃₁×#1753	C₂	semidirect
8. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
9. C₃₁×#1749	C₂	semidirect
10. C₃₁×#1749	C₂	semidirect
11. C₄×C₃₁×#97	C₂	semidirect
18. C₃₁×#435	C₄	semidirect
49. C₂×M₁₆×C₃₁	C₂×C₄	semidirect
52. C₂²×C₈×C₃₁	C₂×C₄	semidirect
78. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
79. C₃₁×C₂²⋊C₈	C₂×C₄	semidirect
80. C₃₁×C₂²⋊C₈	C₂×C₄	semidirect
85. C₃₁×#97	C₂×C₄	semidirect
86. C₃₁×#97	D₄	semidirect
138. C₂²⋊C₄×C₃₁	M₁₆	semidirect
151. #20491	C₃₁	direct (N × H)
199. C₈×C₃₁	C₄×D₄	semidirect
201. C₂×C₄×M₁₆	C₂×C₃₁	semidirect
202. C₂²×#97	C₂×C₃₁	semidirect
204. #1771	C₂×C₃₁	semidirect
205. #1767	C₂×C₃₁	semidirect
206. #1753	C₂×C₃₁	semidirect
207. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
208. #1749	C₂×C₃₁	semidirect
209. #1749	C₂×C₃₁	semidirect
210. C₄×#97	C₂×C₃₁	semidirect
220. C₂²×C₃₁	C₄×M₁₆	semidirect
242. #435	C₄×C₃₁	semidirect
280. C₂×M₁₆	C₂×C₄×C₃₁	semidirect
283. C₂²×C₈	C₂×C₄×C₃₁	semidirect
309. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
310. C₂²⋊C₈	C₂×C₄×C₃₁	semidirect
311. C₂²⋊C₈	C₂×C₄×C₃₁	semidirect
316. #97	C₂×C₄×C₃₁	semidirect
317. #97	D₄×C₃₁	semidirect
323. C₃₁	#20491	direct (N × H)
370. C₂²⋊C₄	M₁₆×C₃₁	semidirect
430. C₈	C₄×D₄×C₃₁	semidirect
440. C₂²	C₄×M₁₆×C₃₁	semidirect

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×C₄×M₁₆×C₃₁	2
C₂²×C₃₁×#97	2
C₄²×D₄×C₃₁	2
C₃₁×#1771	2
C₃₁×#1767	2
C₃₁×#1753	2
C₈×C₂²⋊C₄×C₃₁	2
C₃₁×#1749	2
C₄×C₃₁×#97	2
#20491	31

Subgroups — C31×#20491 #21477250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20491 #21477250