Subgroups — C₃₁×#34273 #21491032

Order 7936.

3578 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#534	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#518	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2359	C₂	1	8	yes	maximal, complemented
4	C₃₁×#2358	C₂	1	8	yes	maximal, complemented
5	C₃₁×#2325	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2315	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃₁×#121	C₂	1	4	yes	maximal, complemented
8	C₂×C₃₁×#386	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#534	C₄	1	16	yes	complemented
13	C₂×C₃₁×#123	C₂²	1	4	yes	—
14	C₂×C₃₁×#121	C₂²	1	4	yes	—
15	C₂×C₃₁×#120	C₂²	1	4	yes	—
16	C₃₁×#518	—	2	16	no	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
18	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	complemented
19	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
20	C₂×C₃₁×#117	C₂²	1	4	yes	complemented
21	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
26	C₂²×C₄²×C₃₁	C₂²	1	2	yes	complemented
27	C₃₁×#390	C₂²	1	4	yes	complemented
28	C₃₁×#389	C₂²	1	4	yes	complemented
29	C₃₁×#388	C₂²	1	8	yes	complemented
30	C₃₁×#388	C₂²	1	8	yes	complemented
31	C₃₁×#387	C₂²	1	8	yes	complemented
32	C₃₁×#386	C₂²	1	8	yes	—
33	C₃₁×#386	C₂²	1	4	yes	complemented
34	C₃₁×#386	C₂²	1	8	yes	complemented
35	C₃₁×#386	C₂²	1	4	yes	complemented
36	C₃₁×#385	C₂²	1	8	yes	complemented
37	C₃₁×#385	C₂²	1	8	yes	complemented
38	C₃₁×#380	C₂²	1	4	yes	complemented
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
40	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
41	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
42	C₂×C₃₁×#95	C₂²	1	2	yes	complemented
43	C₂×C₃₁×#95	C₂²	1	4	yes	complemented
44	C₂⁵×C₃₁	C₂³	1	2	yes	—
45	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₃₁	—	2	8	no	—
47	C₂²×D₄×C₃₁	—	2	8	no	—
48	C₂²×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
49	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
50	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
51	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
52	C₂³×C₄×C₃₁	—	2	4	no	—
53	C₂³×C₄×C₃₁	—	4	4	no	—
54	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
56	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
59	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
62	C₃₁×#123	C₂×C₄	1	32	yes	complemented
63	C₃₁×#121	C₂×C₄	1	32	yes	complemented
64	C₃₁×#120	C₂×C₄	1	32	yes	complemented
65	C₄×D₄×C₃₁	—	2	32	no	—
66	C₄×D₄×C₃₁	—	2	32	no	—
67	C₄×D₄×C₃₁	—	4	8	no	—
68	C₃₁×#117	—	2	16	no	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
76	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	complemented
77	C₂×C₂²⋊C₄×C₃₁	D₄	1	8	yes	complemented
78	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
79	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
80	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
82	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
84	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
86	C₂×C₄²×C₃₁	—	4	4	no	—
87	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
88	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
89	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
90	C₃₁×#95	C₂³	1	8	yes	—
91	C₃₁×#95	C₂³	1	8	yes	—
92	C₃₁×#95	C₂³	1	8	yes	—
93	C₃₁×#95	C₂³	1	8	yes	—
94	C₂⁴×C₃₁	—	2	4	no	—
95	C₂⁴×C₃₁	—	2	4	no	—
96	C₂⁴×C₃₁	—	4	2	no	—
97	C₂⁴×C₃₁	—	4	2	no	—
98	C₂⁴×C₃₁	—	4	4	no	—
99	C₂⁴×C₃₁	—	4	4	no	—
100	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
101	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
102	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
103	C₂⁴×C₃₁	#41	1	4	yes	—
104	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
105	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
106	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
107	C₂×D₄×C₃₁	—	2	64	no	—
108	C₂×D₄×C₃₁	—	2	8	no	—
109	C₂×D₄×C₃₁	—	2	8	no	—
110	C₂×D₄×C₃₁	—	2	8	no	—
111	C₂×D₄×C₃₁	—	2	8	no	—
112	C₂×D₄×C₃₁	—	4	4	no	—
113	C₂×D₄×C₃₁	—	4	4	no	—
114	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
115	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
116	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	complemented
117	C₂²×C₄×C₃₁	—	2	4	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	4	no	—
120	C₂²×C₄×C₃₁	—	2	4	no	—
121	C₂²×C₄×C₃₁	—	2	8	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	8	no	—
124	C₂²×C₄×C₃₁	—	2	8	no	—
125	C₂²×C₄×C₃₁	—	2	16	no	—
126	C₂²×C₄×C₃₁	—	4	2	no	—
127	C₂²×C₄×C₃₁	—	4	2	no	—
128	C₂²×C₄×C₃₁	—	4	4	no	—
129	C₂²×C₄×C₃₁	—	4	2	no	—
130	C₂²×C₄×C₃₁	—	4	2	no	—
131	C₂²×C₄×C₃₁	—	4	2	no	—
132	C₂²×C₄×C₃₁	—	4	4	no	—
133	C₂²×C₄×C₃₁	—	4	8	no	—
134	C₂²×C₄×C₃₁	—	4	8	no	—
135	C₂²×C₄×C₃₁	—	4	4	no	—
136	C₂²×C₄×C₃₁	—	4	16	no	—
137	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
138	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
139	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
140	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
142	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
146	C₄⋊C₄×C₃₁	—	2	16	no	—
147	C₄⋊C₄×C₃₁	—	4	4	no	—
148	C₄⋊C₄×C₃₁	—	4	4	no	—
149	C₄⋊C₄×C₃₁	C₂²×C₄	1	16	yes	complemented
150	C₂²⋊C₄×C₃₁	—	2	8	no	—
151	C₂²⋊C₄×C₃₁	—	2	8	no	—
152	C₂²⋊C₄×C₃₁	—	2	16	no	—
153	C₂²⋊C₄×C₃₁	—	2	16	no	—
154	C₂²⋊C₄×C₃₁	—	4	8	no	—
155	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
156	C₂²⋊C₄×C₃₁	C₂²×C₄	1	32	yes	complemented
157	C₂²⋊C₄×C₃₁	C₂×D₄	1	16	yes	—
158	C₄²×C₃₁	—	2	16	no	—
159	C₄²×C₃₁	—	2	8	no	—
160	C₄²×C₃₁	—	4	4	no	—
161	#34273	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	—	2	4	no	—
164	C₂³×C₃₁	—	2	2	no	—
165	C₂³×C₃₁	—	2	4	no	—
166	C₂³×C₃₁	—	2	4	no	—
167	C₂³×C₃₁	—	2	4	no	—
168	C₂³×C₃₁	—	2	8	no	—
169	C₂³×C₃₁	—	2	8	no	—
170	C₂³×C₃₁	—	2	16	no	—
171	C₂³×C₃₁	—	4	4	no	—
172	C₂³×C₃₁	—	4	8	no	—
173	C₂³×C₃₁	—	4	8	no	—
174	C₂³×C₃₁	—	4	4	no	—
175	C₂³×C₃₁	—	4	16	no	—
176	C₂³×C₃₁	—	8	4	no	—
177	C₂³×C₃₁	—	8	4	no	—
178	C₂³×C₃₁	—	8	8	no	—
179	C₂³×C₃₁	C₄×D₄	1	16	yes	complemented
180	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
181	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
182	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
183	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
184	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
185	C₂³×C₃₁	C₂×#41	1	4	yes	—
186	C₂³×C₃₁	C₂×#41	1	2	yes	—
187	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
188	D₄×C₃₁	—	2	32	no	—
189	D₄×C₃₁	—	2	32	no	—
190	D₄×C₃₁	—	4	16	no	—
191	C₂×C₄×C₃₁	—	2	8	no	—
192	C₂×C₄×C₃₁	—	2	16	no	—
193	C₂×C₄×C₃₁	—	2	4	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	4	no	—
196	C₂×C₄×C₃₁	—	2	4	no	—
197	C₂×C₄×C₃₁	—	2	4	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	2	4	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	8	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	2	8	no	—
204	C₂×C₄×C₃₁	—	2	4	no	—
205	C₂×C₄×C₃₁	—	2	8	no	—
206	C₂×C₄×C₃₁	—	4	8	no	—
207	C₂×C₄×C₃₁	—	4	8	no	—
208	C₂×C₄×C₃₁	—	4	16	no	—
209	C₂×C₄×C₃₁	—	4	4	no	—
210	C₂×C₄×C₃₁	—	8	16	no	—
211	C₂×C₄×C₃₁	—	8	8	no	—
212	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
213	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
214	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
215	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
216	C₂×#534	C₂×C₃₁	1	1	yes	characteristic
217	C₂×#518	C₂×C₃₁	1	2	yes	complemented
218	#2359	C₂×C₃₁	1	8	yes	complemented
219	#2358	C₂×C₃₁	1	8	yes	complemented
220	#2325	C₂×C₃₁	1	4	yes	complemented
221	#2315	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₄×#121	C₂×C₃₁	1	4	yes	complemented
223	C₂×#386	C₂×C₃₁	1	2	yes	complemented
224	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
225	C₂²×C₃₁	—	2	4	no	—
226	C₂²×C₃₁	—	2	8	no	—
227	C₂²×C₃₁	—	2	2	no	—
228	C₂²×C₃₁	—	2	4	no	—
229	C₂²×C₃₁	—	2	4	no	—
230	C₂²×C₃₁	—	2	2	no	—
231	C₂²×C₃₁	—	2	4	no	—
232	C₂²×C₃₁	—	2	8	no	—
233	C₂²×C₃₁	—	2	8	no	—
234	C₂²×C₃₁	—	2	16	no	—
235	C₂²×C₃₁	—	4	16	no	—
236	C₂²×C₃₁	—	4	4	no	—
237	C₂²×C₃₁	—	4	4	no	—
238	C₂²×C₃₁	—	4	8	no	—
239	C₂²×C₃₁	—	8	16	no	—
240	C₂²×C₃₁	C₂×C₄×D₄	1	8	yes	—
241	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
242	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
243	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
244	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
245	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
246	C₂²×C₃₁	#585	1	2	yes	—
247	C₄×C₃₁	—	2	8	no	—
248	C₄×C₃₁	—	2	8	no	—
249	C₄×C₃₁	—	2	4	no	—
250	C₄×C₃₁	—	2	8	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	—	4	8	no	—
253	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
254	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
255	#534	C₄×C₃₁	1	16	yes	complemented
256	C₂×#123	C₂²×C₃₁	1	4	yes	—
257	C₂×#121	C₂²×C₃₁	1	4	yes	—
258	C₂×#120	C₂²×C₃₁	1	4	yes	—
259	#518	—	2	16	no	—
260	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
261	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	complemented
262	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
263	C₂×#117	C₂²×C₃₁	1	4	yes	complemented
264	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
265	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
266	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
267	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
268	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
269	C₂²×C₄²	C₂²×C₃₁	1	2	yes	complemented
270	#390	C₂²×C₃₁	1	4	yes	complemented
271	#389	C₂²×C₃₁	1	4	yes	complemented
272	#388	C₂²×C₃₁	1	8	yes	complemented
273	#388	C₂²×C₃₁	1	8	yes	complemented
274	#387	C₂²×C₃₁	1	8	yes	complemented
275	#386	C₂²×C₃₁	1	8	yes	—
276	#386	C₂²×C₃₁	1	4	yes	complemented
277	#386	C₂²×C₃₁	1	8	yes	complemented
278	#386	C₂²×C₃₁	1	4	yes	complemented
279	#385	C₂²×C₃₁	1	8	yes	complemented
280	#385	C₂²×C₃₁	1	8	yes	complemented
281	#380	C₂²×C₃₁	1	4	yes	complemented
282	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
283	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
284	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
285	C₂×#95	C₂²×C₃₁	1	2	yes	complemented
286	C₂×#95	C₂²×C₃₁	1	4	yes	complemented
287	C₂×C₃₁	—	2	8	no	—
288	C₂×C₃₁	—	2	4	no	—
289	C₂×C₃₁	—	4	8	no	—
290	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
291	C₂×C₃₁	C₄×H(5,𝔽₂)	1	2	yes	—
292	C₂×C₃₁	#3490	1	1	yes	characteristic
293	C₂×C₃₁	#3491	1	1	yes	characteristic
294	C₂×C₃₁	#3496	1	2	yes	—
295	C₂⁵	C₂³×C₃₁	1	2	yes	—
296	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²×D₄	—	2	8	no	—
298	C₂²×D₄	—	2	8	no	—
299	C₂²×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
300	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
301	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
302	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
303	C₂³×C₄	—	2	4	no	—
304	C₂³×C₄	—	4	4	no	—
305	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
306	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
307	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
308	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
309	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
310	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
311	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
312	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
313	#123	C₂×C₄×C₃₁	1	32	yes	complemented
314	#121	C₂×C₄×C₃₁	1	32	yes	complemented
315	#120	C₂×C₄×C₃₁	1	32	yes	complemented
316	C₄×D₄	—	2	32	no	—
317	C₄×D₄	—	2	32	no	—
318	C₄×D₄	—	4	8	no	—
319	#117	—	2	16	no	—
320	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
321	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
322	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
323	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
324	C₂×C₂²⋊C₄	—	2	8	no	—
325	C₂×C₂²⋊C₄	—	2	16	no	—
326	C₂×C₂²⋊C₄	—	4	8	no	—
327	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	8	yes	complemented
328	C₂×C₂²⋊C₄	D₄×C₃₁	1	8	yes	complemented
329	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
330	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
332	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
333	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
334	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
335	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
336	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
337	C₂×C₄²	—	4	4	no	—
338	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
339	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
340	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
341	#95	C₂³×C₃₁	1	8	yes	—
342	#95	C₂³×C₃₁	1	8	yes	—
343	#95	C₂³×C₃₁	1	8	yes	—
344	#95	C₂³×C₃₁	1	8	yes	—
345	C₃₁	#34273	1	1	yes	characteristic, direct product factor, Sylow
346	C₂⁴	—	2	4	no	—
347	C₂⁴	—	2	4	no	—
348	C₂⁴	—	4	2	no	—
349	C₂⁴	—	4	2	no	—
350	C₂⁴	—	4	4	no	—
351	C₂⁴	—	4	4	no	—
352	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
353	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
354	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
355	C₂⁴	#41×C₃₁	1	4	yes	—
356	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
357	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
358	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
359	C₂×D₄	—	2	64	no	—
360	C₂×D₄	—	2	8	no	—
361	C₂×D₄	—	2	8	no	—
362	C₂×D₄	—	2	8	no	—
363	C₂×D₄	—	2	8	no	—
364	C₂×D₄	—	4	4	no	—
365	C₂×D₄	—	4	4	no	—
366	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
367	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
368	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	complemented
369	C₂²×C₄	—	2	4	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	4	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	8	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	8	no	—
376	C₂²×C₄	—	2	8	no	—
377	C₂²×C₄	—	2	16	no	—
378	C₂²×C₄	—	4	2	no	—
379	C₂²×C₄	—	4	2	no	—
380	C₂²×C₄	—	4	4	no	—
381	C₂²×C₄	—	4	2	no	—
382	C₂²×C₄	—	4	2	no	—
383	C₂²×C₄	—	4	2	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	4	8	no	—
386	C₂²×C₄	—	4	8	no	—
387	C₂²×C₄	—	4	4	no	—
388	C₂²×C₄	—	4	16	no	—
389	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
390	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
391	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
392	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
393	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
394	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
395	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
396	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
397	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
398	C₄⋊C₄	—	2	16	no	—
399	C₄⋊C₄	—	4	4	no	—
400	C₄⋊C₄	—	4	4	no	—
401	C₄⋊C₄	C₂²×C₄×C₃₁	1	16	yes	complemented
402	C₂²⋊C₄	—	2	8	no	—
403	C₂²⋊C₄	—	2	8	no	—
404	C₂²⋊C₄	—	2	16	no	—
405	C₂²⋊C₄	—	2	16	no	—
406	C₂²⋊C₄	—	4	8	no	—
407	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
408	C₂²⋊C₄	C₂²×C₄×C₃₁	1	32	yes	complemented
409	C₂²⋊C₄	C₂×D₄×C₃₁	1	16	yes	—
410	C₄²	—	2	16	no	—
411	C₄²	—	2	8	no	—
412	C₄²	—	4	4	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	2	4	no	—
417	C₂³	—	2	4	no	—
418	C₂³	—	2	4	no	—
419	C₂³	—	2	8	no	—
420	C₂³	—	2	8	no	—
421	C₂³	—	2	16	no	—
422	C₂³	—	4	4	no	—
423	C₂³	—	4	8	no	—
424	C₂³	—	4	8	no	—
425	C₂³	—	4	4	no	—
426	C₂³	—	4	16	no	—
427	C₂³	—	8	4	no	—
428	C₂³	—	8	4	no	—
429	C₂³	—	8	8	no	—
430	C₂³	C₄×D₄×C₃₁	1	16	yes	complemented
431	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
432	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
433	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
434	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
435	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
436	C₂³	C₂×#41×C₃₁	1	4	yes	—
437	C₂³	C₂×#41×C₃₁	1	2	yes	—
438	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
439	D₄	—	2	32	no	—
440	D₄	—	2	32	no	—
441	D₄	—	4	16	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	16	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	8	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	8	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	8	no	—
457	C₂×C₄	—	4	8	no	—
458	C₂×C₄	—	4	8	no	—
459	C₂×C₄	—	4	16	no	—
460	C₂×C₄	—	4	4	no	—
461	C₂×C₄	—	8	16	no	—
462	C₂×C₄	—	8	8	no	—
463	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
464	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
465	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
466	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
467	C₂²	—	2	4	no	—
468	C₂²	—	2	8	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	4	no	—
471	C₂²	—	2	4	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	4	no	—
474	C₂²	—	2	8	no	—
475	C₂²	—	2	8	no	—
476	C₂²	—	2	16	no	—
477	C₂²	—	4	16	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	8	no	—
481	C₂²	—	8	16	no	—
482	C₂²	C₂×C₄×D₄×C₃₁	1	8	yes	—
483	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
484	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
485	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
486	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
487	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
488	C₂²	C₃₁×#585	1	2	yes	—
489	C₄	—	2	8	no	—
490	C₄	—	2	8	no	—
491	C₄	—	2	4	no	—
492	C₄	—	2	8	no	—
493	C₄	—	4	2	no	—
494	C₄	—	4	8	no	—
495	C₂	—	2	8	no	—
496	C₂	—	2	4	no	—
497	C₂	—	4	8	no	—
498	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
499	C₂	C₄×C₃₁×H(5,𝔽₂)	1	2	yes	—
500	C₂	C₃₁×#3490	1	1	yes	characteristic
501	C₂	C₃₁×#3491	1	1	yes	characteristic
502	C₂	C₃₁×#3496	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#534	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₃₁×#518	C₂	semidirect
3. C₃₁×#2359	C₂	semidirect
4. C₃₁×#2358	C₂	semidirect
5. C₃₁×#2325	C₂	semidirect
6. C₃₁×#2315	C₂	semidirect
7. C₄×C₃₁×#121	C₂	semidirect
8. C₂×C₃₁×#386	C₂	semidirect
9. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
12. C₃₁×#534	C₄	semidirect
18. C₂×C₄×D₄×C₃₁	C₂²	semidirect
19. C₂×C₄×D₄×C₃₁	C₂²	semidirect
20. C₂×C₃₁×#117	C₂²	semidirect
24. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
25. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
26. C₂²×C₄²×C₃₁	C₂²	semidirect
27. C₃₁×#390	C₂²	semidirect
28. C₃₁×#389	C₂²	semidirect
29. C₃₁×#388	C₂²	semidirect
30. C₃₁×#388	C₂²	semidirect
31. C₃₁×#387	C₂²	semidirect
33. C₃₁×#386	C₂²	semidirect
34. C₃₁×#386	C₂²	semidirect
35. C₃₁×#386	C₂²	semidirect
36. C₃₁×#385	C₂²	semidirect
37. C₃₁×#385	C₂²	semidirect
38. C₃₁×#380	C₂²	semidirect
39. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
40. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
41. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
42. C₂×C₃₁×#95	C₂²	semidirect
43. C₂×C₃₁×#95	C₂²	semidirect
48. C₂²×D₄×C₃₁	C₂×C₄	semidirect
62. C₃₁×#123	C₂×C₄	semidirect
63. C₃₁×#121	C₂×C₄	semidirect
64. C₃₁×#120	C₂×C₄	semidirect
76. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
77. C₂×C₂²⋊C₄×C₃₁	D₄	semidirect
116. C₂×D₄×C₃₁	C₂²×C₄	semidirect
149. C₄⋊C₄×C₃₁	C₂²×C₄	semidirect
156. C₂²⋊C₄×C₃₁	C₂²×C₄	semidirect
161. #34273	C₃₁	direct (N × H)
179. C₂³×C₃₁	C₄×D₄	semidirect
217. C₂×#518	C₂×C₃₁	semidirect
218. #2359	C₂×C₃₁	semidirect
219. #2358	C₂×C₃₁	semidirect
220. #2325	C₂×C₃₁	semidirect
221. #2315	C₂×C₃₁	semidirect
222. C₄×#121	C₂×C₃₁	semidirect
223. C₂×#386	C₂×C₃₁	semidirect
224. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
255. #534	C₄×C₃₁	semidirect
261. C₂×C₄×D₄	C₂²×C₃₁	semidirect
262. C₂×C₄×D₄	C₂²×C₃₁	semidirect
263. C₂×#117	C₂²×C₃₁	semidirect
267. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
268. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
269. C₂²×C₄²	C₂²×C₃₁	semidirect
270. #390	C₂²×C₃₁	semidirect
271. #389	C₂²×C₃₁	semidirect
272. #388	C₂²×C₃₁	semidirect
273. #388	C₂²×C₃₁	semidirect
274. #387	C₂²×C₃₁	semidirect
276. #386	C₂²×C₃₁	semidirect
277. #386	C₂²×C₃₁	semidirect
278. #386	C₂²×C₃₁	semidirect
279. #385	C₂²×C₃₁	semidirect
280. #385	C₂²×C₃₁	semidirect
281. #380	C₂²×C₃₁	semidirect
282. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
283. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
284. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
285. C₂×#95	C₂²×C₃₁	semidirect
286. C₂×#95	C₂²×C₃₁	semidirect
299. C₂²×D₄	C₂×C₄×C₃₁	semidirect
313. #123	C₂×C₄×C₃₁	semidirect
314. #121	C₂×C₄×C₃₁	semidirect
315. #120	C₂×C₄×C₃₁	semidirect
327. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
328. C₂×C₂²⋊C₄	D₄×C₃₁	semidirect
345. C₃₁	#34273	direct (N × H)
368. C₂×D₄	C₂²×C₄×C₃₁	semidirect
401. C₄⋊C₄	C₂²×C₄×C₃₁	semidirect
408. C₂²⋊C₄	C₂²×C₄×C₃₁	semidirect
430. C₂³	C₄×D₄×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×#534	2
C₂×C₃₁×#518	2
C₃₁×#2359	2
C₃₁×#2358	2
C₃₁×#2325	2
C₃₁×#2315	2
C₄×C₃₁×#121	2
C₂×C₃₁×#386	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
#34273	31

Subgroups — C31×#34273 #21491032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34273 #21491032