Subgroups — C₃₁×#20562 #21477321

Order 7936.

1002 conjugacy classes of subgroups.

The subgroup lattice has 398 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	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₃₁×C₄⋊C₈	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×#1750	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1729	C₂	1	2	yes	maximal, 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	C₂×C₃₁×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#390	C₄	1	2	yes	—
28	C₃₁×#389	C₄	1	2	yes	—
29	C₃₁×#388	C₄	1	4	yes	—
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₄³×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#336	C₂²	1	1	yes	characteristic
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	1	yes	characteristic
38	C₃₁×#336	C₂²	1	1	yes	characteristic
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂×M₁₆×C₃₁	—	4	2	no	—
43	C₂×M₁₆×C₃₁	—	4	2	no	—
44	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₃₁	—	4	2	no	—
46	C₂²×C₈×C₃₁	—	4	2	no	—
47	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
55	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
56	C₃₁×#117	—	4	2	no	—
57	C₃₁×#117	—	4	2	no	—
58	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
62	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
63	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
64	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
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	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
71	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
72	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
73	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
74	C₃₁×#97	—	4	2	no	—
75	C₄×C₈×C₃₁	—	4	2	no	—
76	C₄×C₈×C₃₁	—	4	2	no	—
77	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
78	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
79	C₃₁×#95	C₂×C₄	1	4	yes	—
80	C₃₁×#95	C₂×C₄	1	2	yes	—
81	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
82	C₂×D₄×C₃₁	—	2	4	no	—
83	C₂×D₄×C₃₁	C₄²	1	4	yes	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
89	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
93	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
94	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
97	M₁₆×C₃₁	—	4	8	no	—
98	C₂×C₈×C₃₁	—	2	1	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	2	1	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	1	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	1	no	—
110	C₂×C₈×C₃₁	—	4	8	no	—
111	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
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	2	yes	—
115	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
116	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
117	C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₄⋊C₄×C₃₁	—	4	4	no	—
119	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
120	C₂²⋊C₄×C₃₁	—	2	8	no	—
121	C₂²⋊C₄×C₃₁	—	4	4	no	—
122	C₂²⋊C₄×C₃₁	C₄²	1	8	yes	—
123	C₄²×C₃₁	—	2	4	no	—
124	C₄²×C₃₁	—	2	4	no	—
125	C₄²×C₃₁	—	2	1	no	—
126	C₄²×C₃₁	—	2	2	no	—
127	C₄²×C₃₁	—	2	2	no	—
128	C₄²×C₃₁	—	2	1	no	—
129	C₄²×C₃₁	—	2	1	no	—
130	C₄²×C₃₁	—	2	1	no	—
131	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
132	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
133	#20562	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂³×C₃₁	—	2	2	no	—
135	C₂³×C₃₁	—	2	2	no	—
136	C₂³×C₃₁	—	2	4	no	—
137	C₂³×C₃₁	—	2	4	no	—
138	C₂³×C₃₁	C₂×C₄²	1	4	yes	—
139	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
140	D₄×C₃₁	—	2	8	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	2	2	no	—
143	C₂×C₄×C₃₁	—	2	2	no	—
144	C₂×C₄×C₃₁	—	2	2	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	4	no	—
147	C₂×C₄×C₃₁	—	2	4	no	—
148	C₂×C₄×C₃₁	—	2	4	no	—
149	C₂×C₄×C₃₁	—	2	4	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	4	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	4	no	—
155	C₂×C₄×C₃₁	—	2	4	no	—
156	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
157	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
158	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
159	C₂×C₄×C₃₁	#131	1	4	yes	—
160	C₂×C₄×C₃₁	#131	1	4	yes	—
161	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
167	C₈×C₃₁	—	2	2	no	—
168	C₈×C₃₁	—	2	4	no	—
169	C₈×C₃₁	—	2	4	no	—
170	C₈×C₃₁	—	2	2	no	—
171	C₈×C₃₁	—	2	2	no	—
172	C₈×C₃₁	—	2	2	no	—
173	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
174	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
175	#2293	C₂×C₃₁	1	1	yes	characteristic
176	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
177	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
178	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#1753	C₂×C₃₁	1	2	yes	complemented
180	C₈×C₂²⋊C₄	C₂×C₃₁	1	2	yes	complemented
181	#1750	C₂×C₃₁	1	2	yes	complemented
182	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#1729	C₂×C₃₁	1	2	yes	complemented
184	C₂²×C₃₁	—	2	4	no	—
185	C₂²×C₃₁	—	2	4	no	—
186	C₂²×C₃₁	—	2	4	no	—
187	C₂²×C₃₁	—	4	4	no	—
188	C₂²×C₃₁	—	4	4	no	—
189	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
190	C₂²×C₃₁	#518	1	1	yes	characteristic
191	C₂²×C₃₁	#520	1	1	yes	characteristic
192	C₂²×C₃₁	C₂×#131	1	2	yes	—
193	C₂²×C₃₁	C₂×#131	1	2	yes	—
194	C₄×C₃₁	—	2	2	no	—
195	C₄×C₃₁	—	2	4	no	—
196	C₄×C₃₁	—	2	4	no	—
197	C₄×C₃₁	—	4	2	no	—
198	C₄×C₃₁	—	4	4	no	—
199	C₄×C₃₁	#405	1	4	yes	—
200	C₄×C₃₁	#518	1	1	yes	characteristic
201	C₄×C₃₁	#518	1	1	yes	characteristic
202	C₄×C₃₁	#520	1	1	yes	characteristic
203	C₄×C₃₁	#520	1	1	yes	characteristic
204	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
205	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
206	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
207	C₂×#117	C₂²×C₃₁	1	2	yes	—
208	#436	C₄×C₃₁	1	8	yes	complemented
209	C₈×D₄	C₄×C₃₁	1	8	yes	complemented
210	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
211	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
212	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
213	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
214	C₂×#97	C₂²×C₃₁	1	2	yes	—
215	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
216	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
217	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
218	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
219	#390	C₄×C₃₁	1	2	yes	—
220	#389	C₄×C₃₁	1	2	yes	—
221	#388	C₄×C₃₁	1	4	yes	—
222	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
223	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
224	C₄³	C₂²×C₃₁	1	1	yes	characteristic
225	#336	C₂²×C₃₁	1	1	yes	characteristic
226	#336	C₂²×C₃₁	1	2	yes	—
227	#336	C₂²×C₃₁	1	2	yes	—
228	#336	C₂²×C₃₁	1	2	yes	—
229	#336	C₂²×C₃₁	1	1	yes	characteristic
230	#336	C₂²×C₃₁	1	1	yes	characteristic
231	#336	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×C₃₁	—	4	4	no	—
233	C₂×C₃₁	#2275	1	1	yes	characteristic
234	C₂×C₃₁	C₂×#405	1	1	yes	characteristic
235	C₂×C₃₁	C₄×#131	1	1	yes	characteristic
236	C₂×C₃₁	#2979	1	2	yes	—
237	C₂×C₃₁	#2994	1	2	yes	—
238	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
239	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
240	C₂×M₁₆	—	4	2	no	—
241	C₂×M₁₆	—	4	2	no	—
242	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
243	C₂²×C₈	—	4	2	no	—
244	C₂²×C₈	—	4	2	no	—
245	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
246	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
247	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
248	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
249	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
250	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
251	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
252	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
253	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
254	#117	—	4	2	no	—
255	#117	—	4	2	no	—
256	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
257	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
258	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
259	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
260	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
261	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
262	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
263	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
264	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
265	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
266	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
268	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
269	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
270	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
271	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
272	#97	—	4	2	no	—
273	C₄×C₈	—	4	2	no	—
274	C₄×C₈	—	4	2	no	—
275	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
276	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
277	#95	C₂×C₄×C₃₁	1	4	yes	—
278	#95	C₂×C₄×C₃₁	1	2	yes	—
279	C₃₁	#20562	1	1	yes	characteristic, direct product factor, Sylow
280	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
281	C₂×D₄	—	2	4	no	—
282	C₂×D₄	C₄²×C₃₁	1	4	yes	—
283	C₂²×C₄	—	2	2	no	—
284	C₂²×C₄	—	2	2	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
288	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
289	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
290	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
291	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
292	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
293	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
294	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
295	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
296	M₁₆	—	4	8	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	4	8	no	—
310	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
311	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
312	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
313	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
314	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
315	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
316	C₄⋊C₄	—	2	4	no	—
317	C₄⋊C₄	—	4	4	no	—
318	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
319	C₂²⋊C₄	—	2	8	no	—
320	C₂²⋊C₄	—	4	4	no	—
321	C₂²⋊C₄	C₄²×C₃₁	1	8	yes	—
322	C₄²	—	2	4	no	—
323	C₄²	—	2	4	no	—
324	C₄²	—	2	1	no	—
325	C₄²	—	2	2	no	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	1	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	2	1	no	—
330	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
331	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
332	C₂³	—	2	2	no	—
333	C₂³	—	2	2	no	—
334	C₂³	—	2	4	no	—
335	C₂³	—	2	4	no	—
336	C₂³	C₂×C₄²×C₃₁	1	4	yes	—
337	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
338	D₄	—	2	8	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	4	no	—
346	C₂×C₄	—	2	4	no	—
347	C₂×C₄	—	2	4	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	4	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	4	no	—
353	C₂×C₄	—	2	4	no	—
354	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
355	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
356	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
357	C₂×C₄	C₃₁×#131	1	4	yes	—
358	C₂×C₄	C₃₁×#131	1	4	yes	—
359	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
360	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
361	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
362	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
363	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
364	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
365	C₈	—	2	2	no	—
366	C₈	—	2	4	no	—
367	C₈	—	2	4	no	—
368	C₈	—	2	2	no	—
369	C₈	—	2	2	no	—
370	C₈	—	2	2	no	—
371	C₂²	—	2	4	no	—
372	C₂²	—	2	4	no	—
373	C₂²	—	2	4	no	—
374	C₂²	—	4	4	no	—
375	C₂²	—	4	4	no	—
376	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
377	C₂²	C₃₁×#518	1	1	yes	characteristic
378	C₂²	C₃₁×#520	1	1	yes	characteristic
379	C₂²	C₂×C₃₁×#131	1	2	yes	—
380	C₂²	C₂×C₃₁×#131	1	2	yes	—
381	C₄	—	2	2	no	—
382	C₄	—	2	4	no	—
383	C₄	—	2	4	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	4	no	—
386	C₄	C₃₁×#405	1	4	yes	—
387	C₄	C₃₁×#518	1	1	yes	characteristic
388	C₄	C₃₁×#518	1	1	yes	characteristic
389	C₄	C₃₁×#520	1	1	yes	characteristic
390	C₄	C₃₁×#520	1	1	yes	characteristic
391	C₂	—	4	4	no	—
392	C₂	C₃₁×#2275	1	1	yes	characteristic
393	C₂	C₂×C₃₁×#405	1	1	yes	characteristic
394	C₂	C₄×C₃₁×#131	1	1	yes	characteristic
395	C₂	C₃₁×#2979	1	2	yes	—
396	C₂	C₃₁×#2994	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
4. C₃₁×#1770	C₂	semidirect
5. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
6. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
7. C₃₁×#1753	C₂	semidirect
8. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
9. C₃₁×#1750	C₂	semidirect
10. C₃₁×#1744	C₂	semidirect
11. C₃₁×#1729	C₂	semidirect
16. C₃₁×#436	C₄	semidirect
17. C₈×D₄×C₃₁	C₄	semidirect
133. #20562	C₃₁	direct (N × H)
176. #1770	C₂×C₃₁	semidirect
177. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
178. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
179. #1753	C₂×C₃₁	semidirect
180. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
181. #1750	C₂×C₃₁	semidirect
182. #1744	C₂×C₃₁	semidirect
183. #1729	C₂×C₃₁	semidirect
208. #436	C₄×C₃₁	semidirect
209. C₈×D₄	C₄×C₃₁	semidirect
279. C₃₁	#20562	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×#436	2
C₂×C₈×D₄×C₃₁	2
C₃₁×#2293	2
C₃₁×#1770	2
C₈×C₄⋊C₄×C₃₁	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1753	2
C₈×C₂²⋊C₄×C₃₁	2
C₃₁×#1750	2
C₃₁×#1744	2
C₃₁×#1729	2
#20562	31

Subgroups — C31×#20562 #21477321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20562 #21477321