Subgroups — C₃₁×#20495 #21477254

Order 7936.

1130 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#405	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1753	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1752	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1750	C₂	1	2	yes	maximal, 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	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	C₂²	1	2	yes	—
16	C₃₁×#436	C₄	1	8	yes	complemented
17	C₈×D₄×C₃₁	C₄	1	8	yes	complemented
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₃₁×#405	—	2	8	no	—
23	C₂×C₃₁×#97	C₂²	1	2	yes	—
24	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#390	C₄	1	2	yes	—
33	C₃₁×#389	C₄	1	2	yes	—
34	C₃₁×#388	C₄	1	4	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	2	yes	—
41	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
43	C₂×M₁₆×C₃₁	—	2	4	no	—
44	C₂×M₁₆×C₃₁	—	2	4	no	—
45	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
46	C₂²×C₈×C₃₁	—	2	4	no	—
47	C₂²×C₈×C₃₁	—	2	4	no	—
48	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
56	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
57	C₃₁×#117	—	2	4	no	—
58	C₃₁×#117	—	2	4	no	—
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
63	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
64	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
72	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
73	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
74	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
75	C₃₁×#97	—	2	4	no	—
76	C₃₁×#97	—	2	2	no	—
77	C₃₁×#97	—	2	2	no	—
78	C₃₁×#97	—	2	2	no	—
79	C₃₁×#97	D₄	1	4	yes	complemented
80	C₄×C₈×C₃₁	—	2	4	no	—
81	C₄×C₈×C₃₁	—	2	4	no	—
82	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
83	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
84	C₃₁×#95	C₂×C₄	1	4	yes	—
85	C₃₁×#95	C₂×C₄	1	2	yes	—
86	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
87	C₂×D₄×C₃₁	—	2	4	no	—
88	C₂×D₄×C₃₁	C₄²	1	4	yes	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
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	4	yes	—
98	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
99	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
102	M₁₆×C₃₁	—	2	16	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	1	no	—
106	C₂×C₈×C₃₁	—	2	1	no	—
107	C₂×C₈×C₃₁	—	2	1	no	—
108	C₂×C₈×C₃₁	—	2	1	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	16	no	—
112	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
114	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
115	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
116	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₃₁	#41	1	2	yes	—
121	C₂×C₈×C₃₁	#41	1	2	yes	—
122	C₄⋊C₄×C₃₁	—	2	8	no	—
123	C₄⋊C₄×C₃₁	—	2	4	no	—
124	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
125	C₂²⋊C₄×C₃₁	—	2	8	no	—
126	C₂²⋊C₄×C₃₁	—	4	4	no	—
127	C₂²⋊C₄×C₃₁	C₄²	1	8	yes	—
128	C₄²×C₃₁	—	2	4	no	—
129	C₄²×C₃₁	—	2	2	no	—
130	C₄²×C₃₁	—	2	1	no	—
131	C₄²×C₃₁	—	2	1	no	—
132	C₄²×C₃₁	—	2	2	no	—
133	C₄²×C₃₁	—	4	2	no	—
134	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
135	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
136	C₄²×C₃₁	C₂×D₄	1	2	yes	—
137	C₄²×C₃₁	#41	1	2	yes	—
138	#20495	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂³×C₃₁	—	2	2	no	—
140	C₂³×C₃₁	—	2	2	no	—
141	C₂³×C₃₁	—	2	4	no	—
142	C₂³×C₃₁	—	2	4	no	—
143	C₂³×C₃₁	C₂×C₄²	1	4	yes	—
144	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
145	D₄×C₃₁	—	2	8	no	—
146	C₂×C₄×C₃₁	—	2	2	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	4	no	—
152	C₂×C₄×C₃₁	—	2	4	no	—
153	C₂×C₄×C₃₁	—	2	4	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	2	4	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	4	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	4	no	—
161	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
162	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
163	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
164	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
165	C₂×C₄×C₃₁	#131	1	4	yes	—
166	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
172	C₈×C₃₁	—	2	4	no	—
173	C₈×C₃₁	—	2	4	no	—
174	C₈×C₃₁	—	2	2	no	—
175	C₈×C₃₁	—	2	2	no	—
176	C₈×C₃₁	C₄×D₄	1	4	yes	—
177	C₈×C₃₁	C₄×D₄	1	4	yes	—
178	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
179	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
180	C₂×#405	C₂×C₃₁	1	2	yes	complemented
181	#2293	C₂×C₃₁	1	1	yes	characteristic
182	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
183	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
184	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#1753	C₂×C₃₁	1	2	yes	complemented
186	#1752	C₂×C₃₁	1	2	yes	complemented
187	#1750	C₂×C₃₁	1	2	yes	complemented
188	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
189	C₂²×C₃₁	—	2	4	no	—
190	C₂²×C₃₁	—	2	4	no	—
191	C₂²×C₃₁	—	2	4	no	—
192	C₂²×C₃₁	—	4	4	no	—
193	C₂²×C₃₁	—	4	4	no	—
194	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
195	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
196	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
197	C₂²×C₃₁	C₂×#131	1	2	yes	—
198	C₂²×C₃₁	#568	1	2	yes	—
199	C₄×C₃₁	—	2	2	no	—
200	C₄×C₃₁	—	2	4	no	—
201	C₄×C₃₁	—	2	4	no	—
202	C₄×C₃₁	—	4	4	no	—
203	C₄×C₃₁	—	4	2	no	—
204	C₄×C₃₁	#405	1	4	yes	complemented
205	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
206	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
207	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
208	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
209	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
210	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
211	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
212	C₂×#117	C₂²×C₃₁	1	2	yes	—
213	#436	C₄×C₃₁	1	8	yes	complemented
214	C₈×D₄	C₄×C₃₁	1	8	yes	complemented
215	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
217	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
218	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
219	#405	—	2	8	no	—
220	C₂×#97	C₂²×C₃₁	1	2	yes	—
221	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
225	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
226	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
227	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
229	#390	C₄×C₃₁	1	2	yes	—
230	#389	C₄×C₃₁	1	2	yes	—
231	#388	C₄×C₃₁	1	4	yes	—
232	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
233	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
234	C₄³	C₂²×C₃₁	1	1	yes	characteristic
235	#336	C₂²×C₃₁	1	2	yes	—
236	#336	C₂²×C₃₁	1	2	yes	—
237	#336	C₂²×C₃₁	1	2	yes	—
238	C₂×C₃₁	—	4	4	no	—
239	C₂×C₃₁	C₄²×D₄	1	1	yes	characteristic
240	C₂×C₃₁	C₂×#405	1	1	yes	characteristic
241	C₂×C₃₁	#2875	1	1	yes	characteristic
242	C₂×C₃₁	#2935	1	2	yes	—
243	C₂×C₃₁	#2969	1	2	yes	—
244	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
245	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
246	C₂×M₁₆	—	2	4	no	—
247	C₂×M₁₆	—	2	4	no	—
248	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
249	C₂²×C₈	—	2	4	no	—
250	C₂²×C₈	—	2	4	no	—
251	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
252	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
253	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
254	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
255	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
256	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
258	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
259	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
260	#117	—	2	4	no	—
261	#117	—	2	4	no	—
262	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
263	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
264	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
266	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
267	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
268	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
269	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
270	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
271	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
272	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
274	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
275	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
276	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
277	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
278	#97	—	2	4	no	—
279	#97	—	2	2	no	—
280	#97	—	2	2	no	—
281	#97	—	2	2	no	—
282	#97	D₄×C₃₁	1	4	yes	complemented
283	C₄×C₈	—	2	4	no	—
284	C₄×C₈	—	2	4	no	—
285	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
286	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
287	#95	C₂×C₄×C₃₁	1	4	yes	—
288	#95	C₂×C₄×C₃₁	1	2	yes	—
289	C₃₁	#20495	1	1	yes	characteristic, direct product factor, Sylow
290	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
291	C₂×D₄	—	2	4	no	—
292	C₂×D₄	C₄²×C₃₁	1	4	yes	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
298	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
299	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
300	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
301	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
302	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
303	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
304	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
305	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
306	M₁₆	—	2	16	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	16	no	—
316	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
317	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
318	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
319	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
320	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
321	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
322	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
323	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
324	C₂×C₈	#41×C₃₁	1	2	yes	—
325	C₂×C₈	#41×C₃₁	1	2	yes	—
326	C₄⋊C₄	—	2	8	no	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
329	C₂²⋊C₄	—	2	8	no	—
330	C₂²⋊C₄	—	4	4	no	—
331	C₂²⋊C₄	C₄²×C₃₁	1	8	yes	—
332	C₄²	—	2	4	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	2	1	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	4	2	no	—
338	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
339	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
340	C₄²	C₂×D₄×C₃₁	1	2	yes	—
341	C₄²	#41×C₃₁	1	2	yes	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	2	no	—
344	C₂³	—	2	4	no	—
345	C₂³	—	2	4	no	—
346	C₂³	C₂×C₄²×C₃₁	1	4	yes	—
347	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
348	D₄	—	2	8	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	4	no	—
355	C₂×C₄	—	2	4	no	—
356	C₂×C₄	—	2	4	no	—
357	C₂×C₄	—	2	4	no	—
358	C₂×C₄	—	2	4	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
365	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
366	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
367	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
368	C₂×C₄	C₃₁×#131	1	4	yes	—
369	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
370	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
371	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
372	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
373	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
374	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
375	C₈	—	2	4	no	—
376	C₈	—	2	4	no	—
377	C₈	—	2	2	no	—
378	C₈	—	2	2	no	—
379	C₈	C₄×D₄×C₃₁	1	4	yes	—
380	C₈	C₄×D₄×C₃₁	1	4	yes	—
381	C₂²	—	2	4	no	—
382	C₂²	—	2	4	no	—
383	C₂²	—	2	4	no	—
384	C₂²	—	4	4	no	—
385	C₂²	—	4	4	no	—
386	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
387	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
388	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
389	C₂²	C₂×C₃₁×#131	1	2	yes	—
390	C₂²	C₃₁×#568	1	2	yes	—
391	C₄	—	2	2	no	—
392	C₄	—	2	4	no	—
393	C₄	—	2	4	no	—
394	C₄	—	4	4	no	—
395	C₄	—	4	2	no	—
396	C₄	C₃₁×#405	1	4	yes	complemented
397	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
398	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
399	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
400	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
401	C₂	—	4	4	no	—
402	C₂	C₄²×D₄×C₃₁	1	1	yes	characteristic
403	C₂	C₂×C₃₁×#405	1	1	yes	characteristic
404	C₂	C₃₁×#2875	1	1	yes	characteristic
405	C₂	C₃₁×#2935	1	2	yes	—
406	C₂	C₃₁×#2969	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
3. C₂×C₃₁×#405	C₂	semidirect
5. C₃₁×#1770	C₂	semidirect
6. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
7. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
8. C₃₁×#1753	C₂	semidirect
9. C₃₁×#1752	C₂	semidirect
10. C₃₁×#1750	C₂	semidirect
11. C₄×C₃₁×#97	C₂	semidirect
16. C₃₁×#436	C₄	semidirect
17. C₈×D₄×C₃₁	C₄	semidirect
79. C₃₁×#97	D₄	semidirect
138. #20495	C₃₁	direct (N × H)
180. C₂×#405	C₂×C₃₁	semidirect
182. #1770	C₂×C₃₁	semidirect
183. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
184. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
185. #1753	C₂×C₃₁	semidirect
186. #1752	C₂×C₃₁	semidirect
187. #1750	C₂×C₃₁	semidirect
188. C₄×#97	C₂×C₃₁	semidirect
204. C₄×C₃₁	#405	semidirect
213. #436	C₄×C₃₁	semidirect
214. C₈×D₄	C₄×C₃₁	semidirect
282. #97	D₄×C₃₁	semidirect
289. C₃₁	#20495	direct (N × H)
396. C₄	C₃₁×#405	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#436	2
C₂×C₈×D₄×C₃₁	2
C₂×C₃₁×#405	2
C₃₁×#2293	2
C₃₁×#1770	2
C₈×C₄⋊C₄×C₃₁	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1753	2
C₃₁×#1752	2
C₃₁×#1750	2
C₄×C₃₁×#97	2
#20495	31

Subgroups — C31×#20495 #21477254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20495 #21477254