Subgroups — C₃₁×#22797 #21479556

Order 7936.

960 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1955	C₂	1	2	yes	maximal
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×#1797	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1771	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#432	—	2	4	no	—
18	C₃₁×#431	—	2	4	no	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₂×C₃₁×#97	C₂²	1	2	yes	—
30	C₂×C₃₁×#97	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
36	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₄³×C₃₁	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₃₁×#336	C₂²	1	2	yes	—
45	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
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	C₄×D₄×C₃₁	—	2	4	no	—
57	C₄×D₄×C₃₁	—	2	4	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊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	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	—	2	2	no	—
71	C₂×C₄²×C₃₁	—	2	2	no	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	—	2	2	no	—
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×C₄⋊C₈	—	2	4	no	—
89	C₃₁×C₄⋊C₈	—	2	2	no	—
90	C₃₁×C₄⋊C₈	—	4	2	no	—
91	C₃₁×C₄⋊C₈	D₄	1	4	yes	complemented
92	C₃₁×C₂²⋊C₈	—	2	4	no	—
93	C₃₁×C₂²⋊C₈	—	2	4	no	—
94	C₃₁×C₂²⋊C₈	—	4	2	no	—
95	C₃₁×C₂²⋊C₈	—	4	2	no	—
96	C₃₁×#97	—	2	4	no	—
97	C₃₁×#97	—	2	4	no	—
98	C₄×C₈×C₃₁	—	2	4	no	—
99	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
100	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
101	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
102	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
103	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
104	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
105	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	4	1	no	—
132	C₂²×C₄×C₃₁	—	4	1	no	—
133	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
134	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	2	2	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	2	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	2	2	no	—
156	C₂×C₈×C₃₁	—	2	2	no	—
157	C₂×C₈×C₃₁	—	2	2	no	—
158	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	2	2	no	—
161	C₄⋊C₄×C₃₁	—	2	2	no	—
162	C₂²⋊C₄×C₃₁	—	2	2	no	—
163	C₂²⋊C₄×C₃₁	—	2	2	no	—
164	C₂²⋊C₄×C₃₁	—	2	2	no	—
165	C₂²⋊C₄×C₃₁	—	2	2	no	—
166	C₂²⋊C₄×C₃₁	—	2	2	no	—
167	C₂²⋊C₄×C₃₁	M₁₆	1	4	yes	—
168	C₄²×C₃₁	—	2	2	no	—
169	C₄²×C₃₁	—	2	2	no	—
170	C₄²×C₃₁	—	2	2	no	—
171	C₄²×C₃₁	—	2	2	no	—
172	C₄²×C₃₁	—	2	2	no	—
173	C₄²×C₃₁	—	2	2	no	—
174	C₄²×C₃₁	—	2	2	no	—
175	C₄²×C₃₁	—	2	1	no	—
176	C₄²×C₃₁	—	2	1	no	—
177	C₄²×C₃₁	—	2	1	no	—
178	C₄²×C₃₁	—	2	1	no	—
179	C₄²×C₃₁	—	2	1	no	—
180	C₄²×C₃₁	—	2	1	no	—
181	C₄²×C₃₁	—	4	1	no	—
182	C₄²×C₃₁	C₂×D₄	1	2	yes	—
183	C₄²×C₃₁	#41	1	2	yes	—
184	C₄²×C₃₁	#41	1	2	yes	—
185	C₄²×C₃₁	#41	1	2	yes	—
186	#22797	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂³×C₃₁	—	2	1	no	—
188	C₂³×C₃₁	—	2	1	no	—
189	C₂³×C₃₁	—	2	1	no	—
190	C₂³×C₃₁	—	2	1	no	—
191	C₂³×C₃₁	—	2	1	no	—
192	C₂³×C₃₁	—	2	1	no	—
193	C₂³×C₃₁	—	2	1	no	—
194	C₂³×C₃₁	—	2	1	no	—
195	C₂³×C₃₁	—	2	1	no	—
196	C₂³×C₃₁	—	2	1	no	—
197	C₂³×C₃₁	—	2	1	no	—
198	C₂³×C₃₁	—	2	1	no	—
199	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
200	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
201	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
202	D₄×C₃₁	—	2	8	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	2	1	no	—
214	C₂×C₄×C₃₁	—	2	1	no	—
215	C₂×C₄×C₃₁	—	2	8	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	1	no	—
221	C₂×C₄×C₃₁	—	2	1	no	—
222	C₂×C₄×C₃₁	—	2	1	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	1	no	—
225	C₂×C₄×C₃₁	—	2	1	no	—
226	C₂×C₄×C₃₁	—	2	1	no	—
227	C₂×C₄×C₃₁	—	2	1	no	—
228	C₂×C₄×C₃₁	—	2	1	no	—
229	C₂×C₄×C₃₁	—	2	1	no	—
230	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	4	4	no	—
232	C₂×C₄×C₃₁	—	4	4	no	—
233	C₂×C₄×C₃₁	—	4	1	no	—
234	C₂×C₄×C₃₁	—	4	1	no	—
235	C₂×C₄×C₃₁	—	4	1	no	—
236	C₂×C₄×C₃₁	—	4	1	no	—
237	C₂×C₄×C₃₁	—	4	1	no	—
238	C₂×C₄×C₃₁	—	4	1	no	—
239	C₂×C₄×C₃₁	—	4	1	no	—
240	C₂×C₄×C₃₁	—	4	1	no	—
241	C₂×C₄×C₃₁	—	4	4	no	—
242	C₂×C₄×C₃₁	#117	1	2	yes	—
243	C₂×C₄×C₃₁	#117	1	2	yes	—
244	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
245	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
246	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
247	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
248	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
249	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
250	C₂×C₄×C₃₁	#131	1	2	yes	—
251	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
255	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
256	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
257	C₈×C₃₁	—	2	4	no	—
258	C₈×C₃₁	—	4	2	no	—
259	C₈×C₃₁	—	4	2	no	—
260	C₈×C₃₁	—	4	2	no	—
261	C₂×#432	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂×#431	C₂×C₃₁	1	1	yes	characteristic, complemented
263	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
264	#1955	C₂×C₃₁	1	2	yes	—
265	#1924	C₂×C₃₁	1	2	yes	complemented
266	#1923	C₂×C₃₁	1	2	yes	complemented
267	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#1797	C₂×C₃₁	1	1	yes	characteristic, complemented
269	#1797	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#1771	C₂×C₃₁	1	2	yes	complemented
271	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₂²×C₃₁	—	2	2	no	—
273	C₂²×C₃₁	—	2	2	no	—
274	C₂²×C₃₁	—	2	2	no	—
275	C₂²×C₃₁	—	2	2	no	—
276	C₂²×C₃₁	—	2	2	no	—
277	C₂²×C₃₁	—	2	2	no	—
278	C₂²×C₃₁	—	4	1	no	—
279	C₂²×C₃₁	—	4	1	no	—
280	C₂²×C₃₁	—	4	1	no	—
281	C₂²×C₃₁	—	4	1	no	—
282	C₂²×C₃₁	—	4	1	no	—
283	C₂²×C₃₁	—	4	1	no	—
284	C₂²×C₃₁	—	4	1	no	—
285	C₂²×C₃₁	—	4	1	no	—
286	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
287	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
288	C₂²×C₃₁	#520	1	1	yes	characteristic
289	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
290	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
291	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
292	C₂²×C₃₁	#568	1	1	yes	characteristic
293	C₄×C₃₁	—	2	2	no	—
294	C₄×C₃₁	—	2	2	no	—
295	C₄×C₃₁	—	2	2	no	—
296	C₄×C₃₁	—	2	2	no	—
297	C₄×C₃₁	—	2	2	no	—
298	C₄×C₃₁	—	2	2	no	—
299	C₄×C₃₁	—	4	1	no	—
300	C₄×C₃₁	—	4	1	no	—
301	C₄×C₃₁	—	4	1	no	—
302	C₄×C₃₁	—	4	1	no	—
303	C₄×C₃₁	—	4	1	no	—
304	C₄×C₃₁	#435	1	4	yes	—
305	C₄×C₃₁	#547	1	1	yes	characteristic
306	C₄×C₃₁	#550	1	1	yes	characteristic
307	C₄×C₃₁	#555	1	1	yes	characteristic
308	C₄×C₃₁	#556	1	1	yes	characteristic
309	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
310	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
311	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
312	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
313	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
314	#432	—	2	4	no	—
315	#431	—	2	4	no	—
316	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
320	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
321	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
322	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
323	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
324	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
325	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
326	C₂×#97	C₂²×C₃₁	1	2	yes	—
327	C₂×#97	C₂²×C₃₁	1	2	yes	—
328	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
329	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
330	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
331	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
332	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
333	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
334	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
335	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
336	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
337	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
338	C₄³	C₂²×C₃₁	1	1	yes	characteristic
339	#336	C₂²×C₃₁	1	2	yes	—
340	#336	C₂²×C₃₁	1	2	yes	—
341	#336	C₂²×C₃₁	1	2	yes	—
342	C₂×C₃₁	—	4	2	no	—
343	C₂×C₃₁	—	4	2	no	—
344	C₂×C₃₁	#2354	1	1	yes	characteristic
345	C₂×C₃₁	#2920	1	1	yes	characteristic
346	C₂×C₃₁	#2925	1	1	yes	characteristic
347	C₂×C₃₁	C₂×#435	1	1	yes	characteristic
348	C₂×C₃₁	D₄×M₁₆	1	1	yes	characteristic
349	C₂×C₃₁	#2989	1	1	yes	characteristic
350	C₂×C₃₁	#2990	1	1	yes	characteristic
351	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
352	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
353	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
354	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
355	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
358	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
359	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
360	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
361	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
362	C₄×D₄	—	2	4	no	—
363	C₄×D₄	—	2	4	no	—
364	C₄×D₄	—	2	2	no	—
365	C₄×D₄	—	2	2	no	—
366	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
369	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
370	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
371	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
372	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
373	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
374	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
375	C₂×C₄²	—	2	2	no	—
376	C₂×C₄²	—	2	2	no	—
377	C₂×C₄²	—	2	2	no	—
378	C₂×C₄²	—	2	2	no	—
379	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
380	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
384	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
385	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
391	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
392	C₄⋊C₈	—	2	2	no	—
393	C₄⋊C₈	—	2	2	no	—
394	C₄⋊C₈	—	2	4	no	—
395	C₄⋊C₈	—	2	2	no	—
396	C₄⋊C₈	—	4	2	no	—
397	C₄⋊C₈	D₄×C₃₁	1	4	yes	complemented
398	C₂²⋊C₈	—	2	4	no	—
399	C₂²⋊C₈	—	2	4	no	—
400	C₂²⋊C₈	—	4	2	no	—
401	C₂²⋊C₈	—	4	2	no	—
402	#97	—	2	4	no	—
403	#97	—	2	4	no	—
404	C₄×C₈	—	2	4	no	—
405	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
406	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
407	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
408	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
409	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
410	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
411	C₃₁	#22797	1	1	yes	characteristic, direct product factor, Sylow
412	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
413	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	4	1	no	—
439	C₂²×C₄	—	4	1	no	—
440	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
441	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
442	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
443	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
444	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
447	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
450	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
451	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
452	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
453	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	2	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	2	no	—
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	2	2	no	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	—	2	2	no	—
465	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
466	C₄⋊C₄	—	2	2	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₂²⋊C₄	—	2	2	no	—
470	C₂²⋊C₄	—	2	2	no	—
471	C₂²⋊C₄	—	2	2	no	—
472	C₂²⋊C₄	—	2	2	no	—
473	C₂²⋊C₄	—	2	2	no	—
474	C₂²⋊C₄	M₁₆×C₃₁	1	4	yes	—
475	C₄²	—	2	2	no	—
476	C₄²	—	2	2	no	—
477	C₄²	—	2	2	no	—
478	C₄²	—	2	2	no	—
479	C₄²	—	2	2	no	—
480	C₄²	—	2	2	no	—
481	C₄²	—	2	2	no	—
482	C₄²	—	2	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	4	1	no	—
489	C₄²	C₂×D₄×C₃₁	1	2	yes	—
490	C₄²	#41×C₃₁	1	2	yes	—
491	C₄²	#41×C₃₁	1	2	yes	—
492	C₄²	#41×C₃₁	1	2	yes	—
493	C₂³	—	2	1	no	—
494	C₂³	—	2	1	no	—
495	C₂³	—	2	1	no	—
496	C₂³	—	2	1	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	1	no	—
499	C₂³	—	2	1	no	—
500	C₂³	—	2	1	no	—
501	C₂³	—	2	1	no	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	2	1	no	—
505	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
506	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
507	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
508	D₄	—	2	8	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	8	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	4	4	no	—
538	C₂×C₄	—	4	4	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	C₃₁×#117	1	2	yes	—
549	C₂×C₄	C₃₁×#117	1	2	yes	—
550	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
551	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
552	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
553	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
554	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
555	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
556	C₂×C₄	C₃₁×#131	1	2	yes	—
557	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
558	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
559	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
560	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
561	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
562	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
563	C₈	—	2	4	no	—
564	C₈	—	4	2	no	—
565	C₈	—	4	2	no	—
566	C₈	—	4	2	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	1	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
582	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
583	C₂²	C₃₁×#520	1	1	yes	characteristic
584	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
585	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
586	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
587	C₂²	C₃₁×#568	1	1	yes	characteristic
588	C₄	—	2	2	no	—
589	C₄	—	2	2	no	—
590	C₄	—	2	2	no	—
591	C₄	—	2	2	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	4	1	no	—
595	C₄	—	4	1	no	—
596	C₄	—	4	1	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	1	no	—
599	C₄	C₃₁×#435	1	4	yes	—
600	C₄	C₃₁×#547	1	1	yes	characteristic
601	C₄	C₃₁×#550	1	1	yes	characteristic
602	C₄	C₃₁×#555	1	1	yes	characteristic
603	C₄	C₃₁×#556	1	1	yes	characteristic
604	C₂	—	4	2	no	—
605	C₂	—	4	2	no	—
606	C₂	C₃₁×#2354	1	1	yes	characteristic
607	C₂	C₃₁×#2920	1	1	yes	characteristic
608	C₂	C₃₁×#2925	1	1	yes	characteristic
609	C₂	C₂×C₃₁×#435	1	1	yes	characteristic
610	C₂	D₄×M₁₆×C₃₁	1	1	yes	characteristic
611	C₂	C₃₁×#2989	1	1	yes	characteristic
612	C₂	C₃₁×#2990	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×#432	C₂	semidirect
2. C₂×C₃₁×#431	C₂	semidirect
5. C₃₁×#1924	C₂	semidirect
6. C₃₁×#1923	C₂	semidirect
7. C₃₁×#1842	C₂	semidirect
8. C₃₁×#1797	C₂	semidirect
9. C₃₁×#1797	C₂	semidirect
10. C₃₁×#1771	C₂	semidirect
11. C₃₁×#1767	C₂	semidirect
91. C₃₁×C₄⋊C₈	D₄	semidirect
186. #22797	C₃₁	direct (N × H)
261. C₂×#432	C₂×C₃₁	semidirect
262. C₂×#431	C₂×C₃₁	semidirect
265. #1924	C₂×C₃₁	semidirect
266. #1923	C₂×C₃₁	semidirect
267. #1842	C₂×C₃₁	semidirect
268. #1797	C₂×C₃₁	semidirect
269. #1797	C₂×C₃₁	semidirect
270. #1771	C₂×C₃₁	semidirect
271. #1767	C₂×C₃₁	semidirect
397. C₄⋊C₈	D₄×C₃₁	semidirect
411. C₃₁	#22797	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×#432	2
C₂×C₃₁×#431	2
C₄²×D₄×C₃₁	2
C₃₁×#1955	2
C₃₁×#1924	2
C₃₁×#1923	2
C₃₁×#1842	2
C₃₁×#1797	2
C₃₁×#1771	2
C₃₁×#1767	2
#22797	31

Subgroups — C31×#22797 #21479556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22797 #21479556