Subgroups — C₃₁×#34984 #21491743

Order 7936.

3690 conjugacy classes of subgroups.

The subgroup lattice has 640 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	2	yes	maximal, complemented
2	C₂×C₃₁×#534	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2854	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2847	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2618	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2617	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2425	C₂	1	8	yes	maximal, complemented
9	C₃₁×#2424	C₂	1	8	yes	maximal, complemented
10	C₃₁×#2315	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#386	C₂	1	2	yes	maximal, complemented
12	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
13	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
16	C₃₁×#534	—	2	16	no	—
17	C₃₁×#534	—	2	8	no	—
18	C₂×C₃₁×#123	C₂²	1	4	yes	complemented
19	C₂×C₃₁×#123	C₂²	1	4	yes	complemented
20	C₂×C₃₁×#123	C₂²	1	4	yes	complemented
21	C₂×C₃₁×#121	C₂²	1	2	yes	—
22	C₂×C₃₁×#121	C₂²	1	2	yes	—
23	C₂×C₃₁×#121	C₂²	1	8	yes	—
24	C₂×C₃₁×#121	C₂²	1	4	yes	complemented
25	C₂×C₃₁×#120	C₂²	1	4	yes	complemented
26	C₂×C₃₁×#120	C₂²	1	4	yes	complemented
27	C₂×C₃₁×#120	C₂²	1	4	yes	complemented
28	C₂×C₄×D₄×C₃₁	—	2	8	no	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	complemented
30	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
31	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
34	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
35	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#398	C₂²	1	4	yes	complemented
37	C₃₁×#397	C₂²	1	4	yes	complemented
38	C₃₁×#396	C₂²	1	8	yes	complemented
39	C₃₁×#394	C₂²	1	8	yes	complemented
40	C₃₁×#394	C₂²	1	4	yes	complemented
41	C₃₁×#392	C₂²	1	4	yes	complemented
42	C₃₁×#390	C₂²	1	4	yes	complemented
43	C₃₁×#389	C₂²	1	4	yes	complemented
44	C₃₁×#388	C₂²	1	8	yes	complemented
45	C₃₁×#386	C₂²	1	4	yes	complemented
46	C₃₁×#386	C₂²	1	8	yes	complemented
47	C₃₁×#386	C₂²	1	4	yes	complemented
48	C₃₁×#386	C₂²	1	8	yes	complemented
49	C₃₁×#385	C₂²	1	8	yes	complemented
50	C₃₁×#380	C₂²	1	2	yes	complemented
51	C₃₁×#380	C₂²	1	2	yes	complemented
52	C₃₁×#380	C₂²	1	4	yes	complemented
53	C₂×C₃₁×#95	C₂²	1	4	yes	complemented
54	C₂⁵×C₃₁	C₂³	1	2	yes	—
55	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
56	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₃₁	—	2	4	no	—
58	C₂²×D₄×C₃₁	—	2	4	no	—
59	C₂²×D₄×C₃₁	—	2	8	no	—
60	C₂²×D₄×C₃₁	—	2	8	no	—
61	C₂²×D₄×C₃₁	—	2	8	no	—
62	C₂²×D₄×C₃₁	D₄	1	8	yes	complemented
63	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
64	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
65	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
66	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
67	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂³×C₄×C₃₁	—	2	4	no	—
70	C₂³×C₄×C₃₁	—	2	4	no	—
71	C₂³×C₄×C₃₁	—	4	4	no	—
72	C₂³×C₄×C₃₁	—	4	4	no	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
76	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
77	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
79	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₃₁×#123	—	2	16	no	—
83	C₃₁×#123	—	2	16	no	—
84	C₃₁×#123	—	2	16	no	—
85	C₃₁×#121	—	2	32	no	—
86	C₃₁×#121	—	2	8	no	—
87	C₃₁×#121	—	2	8	no	—
88	C₃₁×#121	—	4	8	no	—
89	C₃₁×#120	—	2	16	no	—
90	C₃₁×#120	—	2	16	no	—
91	C₃₁×#120	—	2	16	no	—
92	C₄×D₄×C₃₁	—	2	32	no	—
93	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
101	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
102	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
103	C₂×C₂²⋊C₄×C₃₁	D₄	1	8	yes	complemented
104	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
106	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
108	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
111	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₂×C₄²×C₃₁	—	2	4	no	—
115	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
116	C₃₁×#95	C₂³	1	8	yes	—
117	C₃₁×#95	C₂³	1	4	yes	—
118	C₃₁×#95	C₂³	1	4	yes	—
119	C₃₁×#95	C₂³	1	8	yes	—
120	C₂⁴×C₃₁	—	2	2	no	—
121	C₂⁴×C₃₁	—	2	2	no	—
122	C₂⁴×C₃₁	—	2	4	no	—
123	C₂⁴×C₃₁	—	2	4	no	—
124	C₂⁴×C₃₁	—	2	4	no	—
125	C₂⁴×C₃₁	—	4	2	no	—
126	C₂⁴×C₃₁	—	4	2	no	—
127	C₂⁴×C₃₁	—	4	2	no	—
128	C₂⁴×C₃₁	—	4	2	no	—
129	C₂⁴×C₃₁	—	4	4	no	—
130	C₂⁴×C₃₁	—	4	4	no	—
131	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
132	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
133	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
134	C₂⁴×C₃₁	#41	1	4	yes	—
135	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
136	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
137	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
138	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
139	C₂×D₄×C₃₁	—	2	32	no	—
140	C₂×D₄×C₃₁	—	2	8	no	—
141	C₂×D₄×C₃₁	—	2	8	no	—
142	C₂×D₄×C₃₁	—	2	8	no	—
143	C₂×D₄×C₃₁	—	2	8	no	—
144	C₂×D₄×C₃₁	—	2	8	no	—
145	C₂×D₄×C₃₁	—	2	8	no	—
146	C₂×D₄×C₃₁	—	2	8	no	—
147	C₂×D₄×C₃₁	—	2	8	no	—
148	C₂×D₄×C₃₁	—	2	16	no	—
149	C₂×D₄×C₃₁	—	2	4	no	—
150	C₂×D₄×C₃₁	—	2	4	no	—
151	C₂×D₄×C₃₁	—	4	4	no	—
152	C₂×D₄×C₃₁	—	4	8	no	—
153	C₂×D₄×C₃₁	—	4	4	no	—
154	C₂×D₄×C₃₁	—	4	8	no	—
155	C₂×D₄×C₃₁	—	4	32	no	—
156	C₂×D₄×C₃₁	C₂×D₄	1	16	yes	—
157	C₂²×C₄×C₃₁	—	2	2	no	—
158	C₂²×C₄×C₃₁	—	2	2	no	—
159	C₂²×C₄×C₃₁	—	2	2	no	—
160	C₂²×C₄×C₃₁	—	2	4	no	—
161	C₂²×C₄×C₃₁	—	2	2	no	—
162	C₂²×C₄×C₃₁	—	2	8	no	—
163	C₂²×C₄×C₃₁	—	2	4	no	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	2	4	no	—
166	C₂²×C₄×C₃₁	—	2	8	no	—
167	C₂²×C₄×C₃₁	—	2	8	no	—
168	C₂²×C₄×C₃₁	—	2	8	no	—
169	C₂²×C₄×C₃₁	—	2	8	no	—
170	C₂²×C₄×C₃₁	—	4	2	no	—
171	C₂²×C₄×C₃₁	—	4	4	no	—
172	C₂²×C₄×C₃₁	—	4	4	no	—
173	C₂²×C₄×C₃₁	—	4	2	no	—
174	C₂²×C₄×C₃₁	—	4	2	no	—
175	C₂²×C₄×C₃₁	—	4	4	no	—
176	C₂²×C₄×C₃₁	—	4	4	no	—
177	C₂²×C₄×C₃₁	—	4	16	no	—
178	C₂²×C₄×C₃₁	—	4	16	no	—
179	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
180	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
181	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
182	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
183	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
185	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
186	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
187	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
189	C₄⋊C₄×C₃₁	—	2	8	no	—
190	C₄⋊C₄×C₃₁	—	2	16	no	—
191	C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₄⋊C₄×C₃₁	—	2	4	no	—
193	C₄⋊C₄×C₃₁	—	4	4	no	—
194	C₄⋊C₄×C₃₁	—	4	4	no	—
195	C₂²⋊C₄×C₃₁	—	2	8	no	—
196	C₂²⋊C₄×C₃₁	—	2	8	no	—
197	C₂²⋊C₄×C₃₁	—	2	16	no	—
198	C₂²⋊C₄×C₃₁	—	2	8	no	—
199	C₂²⋊C₄×C₃₁	—	2	16	no	—
200	C₂²⋊C₄×C₃₁	—	2	8	no	—
201	C₂²⋊C₄×C₃₁	—	2	8	no	—
202	C₂²⋊C₄×C₃₁	—	2	16	no	—
203	C₂²⋊C₄×C₃₁	—	4	8	no	—
204	C₂²⋊C₄×C₃₁	C₂×D₄	1	16	yes	—
205	C₄²×C₃₁	—	2	8	no	—
206	#34984	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂³×C₃₁	—	2	2	no	—
208	C₂³×C₃₁	—	2	4	no	—
209	C₂³×C₃₁	—	2	2	no	—
210	C₂³×C₃₁	—	2	4	no	—
211	C₂³×C₃₁	—	2	8	no	—
212	C₂³×C₃₁	—	2	2	no	—
213	C₂³×C₃₁	—	2	2	no	—
214	C₂³×C₃₁	—	2	4	no	—
215	C₂³×C₃₁	—	2	4	no	—
216	C₂³×C₃₁	—	2	4	no	—
217	C₂³×C₃₁	—	2	4	no	—
218	C₂³×C₃₁	—	2	4	no	—
219	C₂³×C₃₁	—	2	8	no	—
220	C₂³×C₃₁	—	2	8	no	—
221	C₂³×C₃₁	—	4	4	no	—
222	C₂³×C₃₁	—	4	4	no	—
223	C₂³×C₃₁	—	4	8	no	—
224	C₂³×C₃₁	—	4	8	no	—
225	C₂³×C₃₁	—	4	8	no	—
226	C₂³×C₃₁	—	4	4	no	—
227	C₂³×C₃₁	—	4	8	no	—
228	C₂³×C₃₁	—	4	4	no	—
229	C₂³×C₃₁	—	4	16	no	—
230	C₂³×C₃₁	—	4	8	no	—
231	C₂³×C₃₁	—	8	4	no	—
232	C₂³×C₃₁	—	8	4	no	—
233	C₂³×C₃₁	—	8	8	no	—
234	C₂³×C₃₁	#121	1	16	yes	complemented
235	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
236	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
237	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
238	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
239	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
240	C₂³×C₃₁	C₂×#41	1	2	yes	—
241	C₂³×C₃₁	C₂×#41	1	4	yes	—
242	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
243	D₄×C₃₁	—	2	32	no	—
244	D₄×C₃₁	—	4	16	no	—
245	D₄×C₃₁	—	4	16	no	—
246	D₄×C₃₁	—	8	4	no	—
247	D₄×C₃₁	—	8	4	no	—
248	D₄×C₃₁	—	8	16	no	—
249	C₂×C₄×C₃₁	—	2	8	no	—
250	C₂×C₄×C₃₁	—	2	8	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	2	no	—
253	C₂×C₄×C₃₁	—	2	2	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	4	no	—
257	C₂×C₄×C₃₁	—	2	8	no	—
258	C₂×C₄×C₃₁	—	2	2	no	—
259	C₂×C₄×C₃₁	—	2	4	no	—
260	C₂×C₄×C₃₁	—	2	4	no	—
261	C₂×C₄×C₃₁	—	2	4	no	—
262	C₂×C₄×C₃₁	—	2	8	no	—
263	C₂×C₄×C₃₁	—	2	4	no	—
264	C₂×C₄×C₃₁	—	2	8	no	—
265	C₂×C₄×C₃₁	—	4	4	no	—
266	C₂×C₄×C₃₁	—	4	4	no	—
267	C₂×C₄×C₃₁	—	4	16	no	—
268	C₂×C₄×C₃₁	—	4	16	no	—
269	C₂×C₄×C₃₁	—	4	4	no	—
270	C₂×C₄×C₃₁	—	4	8	no	—
271	C₂×C₄×C₃₁	—	8	8	no	—
272	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
273	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
274	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
275	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
276	C₂×#534	C₂×C₃₁	1	2	yes	complemented
277	C₂×#534	C₂×C₃₁	1	1	yes	characteristic, complemented
278	C₂²×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#2854	C₂×C₃₁	1	2	yes	complemented
280	#2847	C₂×C₃₁	1	2	yes	complemented
281	#2618	C₂×C₃₁	1	2	yes	complemented
282	#2617	C₂×C₃₁	1	2	yes	complemented
283	#2425	C₂×C₃₁	1	8	yes	complemented
284	#2424	C₂×C₃₁	1	8	yes	complemented
285	#2315	C₂×C₃₁	1	1	yes	characteristic, complemented
286	C₂×#386	C₂×C₃₁	1	2	yes	complemented
287	C₂²×C₃₁	—	2	4	no	—
288	C₂²×C₃₁	—	2	8	no	—
289	C₂²×C₃₁	—	2	2	no	—
290	C₂²×C₃₁	—	2	4	no	—
291	C₂²×C₃₁	—	2	2	no	—
292	C₂²×C₃₁	—	2	4	no	—
293	C₂²×C₃₁	—	2	8	no	—
294	C₂²×C₃₁	—	2	4	no	—
295	C₂²×C₃₁	—	2	4	no	—
296	C₂²×C₃₁	—	2	8	no	—
297	C₂²×C₃₁	—	4	16	no	—
298	C₂²×C₃₁	—	4	16	no	—
299	C₂²×C₃₁	—	4	4	no	—
300	C₂²×C₃₁	—	4	2	no	—
301	C₂²×C₃₁	—	4	2	no	—
302	C₂²×C₃₁	—	4	4	no	—
303	C₂²×C₃₁	—	4	8	no	—
304	C₂²×C₃₁	—	4	4	no	—
305	C₂²×C₃₁	—	4	4	no	—
306	C₂²×C₃₁	—	4	8	no	—
307	C₂²×C₃₁	—	8	16	no	—
308	C₂²×C₃₁	C₂×#121	1	4	yes	—
309	C₂²×C₃₁	C₂×#121	1	8	yes	—
310	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
311	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
312	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
313	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
314	C₂²×C₃₁	#585	1	2	yes	—
315	C₄×C₃₁	—	2	4	no	—
316	C₄×C₃₁	—	2	4	no	—
317	C₄×C₃₁	—	2	4	no	—
318	C₄×C₃₁	—	2	4	no	—
319	C₄×C₃₁	—	4	4	no	—
320	C₄×C₃₁	—	4	2	no	—
321	C₄×C₃₁	—	4	8	no	—
322	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
323	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
324	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
325	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
326	#534	—	2	16	no	—
327	#534	—	2	8	no	—
328	C₂×#123	C₂²×C₃₁	1	4	yes	complemented
329	C₂×#123	C₂²×C₃₁	1	4	yes	complemented
330	C₂×#123	C₂²×C₃₁	1	4	yes	complemented
331	C₂×#121	C₂²×C₃₁	1	2	yes	—
332	C₂×#121	C₂²×C₃₁	1	2	yes	—
333	C₂×#121	C₂²×C₃₁	1	8	yes	—
334	C₂×#121	C₂²×C₃₁	1	4	yes	complemented
335	C₂×#120	C₂²×C₃₁	1	4	yes	complemented
336	C₂×#120	C₂²×C₃₁	1	4	yes	complemented
337	C₂×#120	C₂²×C₃₁	1	4	yes	complemented
338	C₂×C₄×D₄	—	2	8	no	—
339	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	complemented
340	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
341	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
342	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
343	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
344	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
345	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic, complemented
346	#398	C₂²×C₃₁	1	4	yes	complemented
347	#397	C₂²×C₃₁	1	4	yes	complemented
348	#396	C₂²×C₃₁	1	8	yes	complemented
349	#394	C₂²×C₃₁	1	8	yes	complemented
350	#394	C₂²×C₃₁	1	4	yes	complemented
351	#392	C₂²×C₃₁	1	4	yes	complemented
352	#390	C₂²×C₃₁	1	4	yes	complemented
353	#389	C₂²×C₃₁	1	4	yes	complemented
354	#388	C₂²×C₃₁	1	8	yes	complemented
355	#386	C₂²×C₃₁	1	4	yes	complemented
356	#386	C₂²×C₃₁	1	8	yes	complemented
357	#386	C₂²×C₃₁	1	4	yes	complemented
358	#386	C₂²×C₃₁	1	8	yes	complemented
359	#385	C₂²×C₃₁	1	8	yes	complemented
360	#380	C₂²×C₃₁	1	2	yes	complemented
361	#380	C₂²×C₃₁	1	2	yes	complemented
362	#380	C₂²×C₃₁	1	4	yes	complemented
363	C₂×#95	C₂²×C₃₁	1	4	yes	complemented
364	C₂×C₃₁	—	2	4	no	—
365	C₂×C₃₁	—	2	8	no	—
366	C₂×C₃₁	—	4	4	no	—
367	C₂×C₃₁	—	8	4	no	—
368	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
369	C₂×C₃₁	#3459	1	2	yes	—
370	C₂×C₃₁	#3484	1	1	yes	characteristic
371	C₂×C₃₁	#3485	1	1	yes	characteristic
372	C₂×C₃₁	#3496	1	2	yes	—
373	C₂⁵	C₂³×C₃₁	1	2	yes	—
374	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
375	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
376	C₂²×D₄	—	2	4	no	—
377	C₂²×D₄	—	2	4	no	—
378	C₂²×D₄	—	2	8	no	—
379	C₂²×D₄	—	2	8	no	—
380	C₂²×D₄	—	2	8	no	—
381	C₂²×D₄	D₄×C₃₁	1	8	yes	complemented
382	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
383	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
384	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
385	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
386	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
387	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
388	C₂³×C₄	—	2	4	no	—
389	C₂³×C₄	—	2	4	no	—
390	C₂³×C₄	—	4	4	no	—
391	C₂³×C₄	—	4	4	no	—
392	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
393	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
394	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
395	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
396	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
397	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
398	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
399	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
400	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
401	#123	—	2	16	no	—
402	#123	—	2	16	no	—
403	#123	—	2	16	no	—
404	#121	—	2	32	no	—
405	#121	—	2	8	no	—
406	#121	—	2	8	no	—
407	#121	—	4	8	no	—
408	#120	—	2	16	no	—
409	#120	—	2	16	no	—
410	#120	—	2	16	no	—
411	C₄×D₄	—	2	32	no	—
412	C₂×C₄⋊C₄	—	2	4	no	—
413	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
414	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
416	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
418	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
419	C₂×C₂²⋊C₄	—	2	8	no	—
420	C₂×C₂²⋊C₄	—	2	4	no	—
421	C₂×C₂²⋊C₄	—	4	8	no	—
422	C₂×C₂²⋊C₄	D₄×C₃₁	1	8	yes	complemented
423	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
433	C₂×C₄²	—	2	4	no	—
434	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
435	#95	C₂³×C₃₁	1	8	yes	—
436	#95	C₂³×C₃₁	1	8	yes	—
437	#95	C₂³×C₃₁	1	4	yes	—
438	#95	C₂³×C₃₁	1	4	yes	—
439	C₃₁	#34984	1	1	yes	characteristic, direct product factor, Sylow
440	C₂⁴	—	2	2	no	—
441	C₂⁴	—	2	2	no	—
442	C₂⁴	—	2	4	no	—
443	C₂⁴	—	2	4	no	—
444	C₂⁴	—	2	4	no	—
445	C₂⁴	—	4	2	no	—
446	C₂⁴	—	4	2	no	—
447	C₂⁴	—	4	2	no	—
448	C₂⁴	—	4	2	no	—
449	C₂⁴	—	4	4	no	—
450	C₂⁴	—	4	4	no	—
451	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
452	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
453	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
454	C₂⁴	#41×C₃₁	1	4	yes	—
455	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
456	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
457	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
458	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
459	C₂×D₄	—	2	32	no	—
460	C₂×D₄	—	2	8	no	—
461	C₂×D₄	—	2	8	no	—
462	C₂×D₄	—	2	8	no	—
463	C₂×D₄	—	2	8	no	—
464	C₂×D₄	—	2	8	no	—
465	C₂×D₄	—	2	8	no	—
466	C₂×D₄	—	2	8	no	—
467	C₂×D₄	—	2	8	no	—
468	C₂×D₄	—	2	16	no	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	4	no	—
471	C₂×D₄	—	4	4	no	—
472	C₂×D₄	—	4	8	no	—
473	C₂×D₄	—	4	4	no	—
474	C₂×D₄	—	4	8	no	—
475	C₂×D₄	—	4	32	no	—
476	C₂×D₄	C₂×D₄×C₃₁	1	16	yes	—
477	C₂²×C₄	—	2	2	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	8	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	4	no	—
486	C₂²×C₄	—	2	8	no	—
487	C₂²×C₄	—	2	8	no	—
488	C₂²×C₄	—	2	8	no	—
489	C₂²×C₄	—	2	8	no	—
490	C₂²×C₄	—	4	2	no	—
491	C₂²×C₄	—	4	4	no	—
492	C₂²×C₄	—	4	4	no	—
493	C₂²×C₄	—	4	2	no	—
494	C₂²×C₄	—	4	2	no	—
495	C₂²×C₄	—	4	4	no	—
496	C₂²×C₄	—	4	4	no	—
497	C₂²×C₄	—	4	16	no	—
498	C₂²×C₄	—	4	16	no	—
499	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
500	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
501	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
502	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
503	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
504	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
505	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
506	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
507	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
508	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
509	C₄⋊C₄	—	2	8	no	—
510	C₄⋊C₄	—	2	16	no	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	4	no	—
513	C₄⋊C₄	—	4	4	no	—
514	C₄⋊C₄	—	4	4	no	—
515	C₂²⋊C₄	—	2	8	no	—
516	C₂²⋊C₄	—	2	8	no	—
517	C₂²⋊C₄	—	2	16	no	—
518	C₂²⋊C₄	—	2	8	no	—
519	C₂²⋊C₄	—	2	16	no	—
520	C₂²⋊C₄	—	2	8	no	—
521	C₂²⋊C₄	—	2	8	no	—
522	C₂²⋊C₄	—	2	16	no	—
523	C₂²⋊C₄	—	4	8	no	—
524	C₂²⋊C₄	C₂×D₄×C₃₁	1	16	yes	—
525	C₄²	—	2	8	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	4	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	4	no	—
530	C₂³	—	2	8	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	2	4	no	—
535	C₂³	—	2	4	no	—
536	C₂³	—	2	4	no	—
537	C₂³	—	2	4	no	—
538	C₂³	—	2	8	no	—
539	C₂³	—	2	8	no	—
540	C₂³	—	4	4	no	—
541	C₂³	—	4	4	no	—
542	C₂³	—	4	8	no	—
543	C₂³	—	4	8	no	—
544	C₂³	—	4	8	no	—
545	C₂³	—	4	4	no	—
546	C₂³	—	4	8	no	—
547	C₂³	—	4	4	no	—
548	C₂³	—	4	16	no	—
549	C₂³	—	4	8	no	—
550	C₂³	—	8	4	no	—
551	C₂³	—	8	4	no	—
552	C₂³	—	8	8	no	—
553	C₂³	C₃₁×#121	1	16	yes	complemented
554	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
555	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
556	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
557	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
558	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
559	C₂³	C₂×#41×C₃₁	1	2	yes	—
560	C₂³	C₂×#41×C₃₁	1	4	yes	—
561	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
562	D₄	—	2	32	no	—
563	D₄	—	4	16	no	—
564	D₄	—	4	16	no	—
565	D₄	—	8	4	no	—
566	D₄	—	8	4	no	—
567	D₄	—	8	16	no	—
568	C₂×C₄	—	2	8	no	—
569	C₂×C₄	—	2	8	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	8	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	8	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	8	no	—
584	C₂×C₄	—	4	4	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	16	no	—
587	C₂×C₄	—	4	16	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	8	no	—
590	C₂×C₄	—	8	8	no	—
591	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
592	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
593	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
594	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
595	C₂²	—	2	4	no	—
596	C₂²	—	2	8	no	—
597	C₂²	—	2	2	no	—
598	C₂²	—	2	4	no	—
599	C₂²	—	2	2	no	—
600	C₂²	—	2	4	no	—
601	C₂²	—	2	8	no	—
602	C₂²	—	2	4	no	—
603	C₂²	—	2	4	no	—
604	C₂²	—	2	8	no	—
605	C₂²	—	4	16	no	—
606	C₂²	—	4	16	no	—
607	C₂²	—	4	4	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	4	no	—
611	C₂²	—	4	8	no	—
612	C₂²	—	4	4	no	—
613	C₂²	—	4	4	no	—
614	C₂²	—	4	8	no	—
615	C₂²	—	8	16	no	—
616	C₂²	C₂×C₃₁×#121	1	4	yes	—
617	C₂²	C₂×C₃₁×#121	1	8	yes	—
618	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
619	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
620	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
621	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
622	C₂²	C₃₁×#585	1	2	yes	—
623	C₄	—	2	4	no	—
624	C₄	—	2	4	no	—
625	C₄	—	2	4	no	—
626	C₄	—	2	4	no	—
627	C₄	—	4	4	no	—
628	C₄	—	4	2	no	—
629	C₄	—	4	8	no	—
630	C₂	—	2	4	no	—
631	C₂	—	2	8	no	—
632	C₂	—	4	4	no	—
633	C₂	—	8	4	no	—
634	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
635	C₂	C₃₁×#3459	1	2	yes	—
636	C₂	C₃₁×#3484	1	1	yes	characteristic
637	C₂	C₃₁×#3485	1	1	yes	characteristic
638	C₂	C₃₁×#3496	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#534	C₂	semidirect
2. C₂×C₃₁×#534	C₂	semidirect
3. C₂²×C₄×D₄×C₃₁	C₂	semidirect
4. C₃₁×#2854	C₂	semidirect
5. C₃₁×#2847	C₂	semidirect
6. C₃₁×#2618	C₂	semidirect
7. C₃₁×#2617	C₂	semidirect
8. C₃₁×#2425	C₂	semidirect
9. C₃₁×#2424	C₂	semidirect
10. C₃₁×#2315	C₂	semidirect
11. C₂×C₃₁×#386	C₂	semidirect
15. C₂⁴×C₄×C₃₁	C₂²	semidirect
18. C₂×C₃₁×#123	C₂²	semidirect
19. C₂×C₃₁×#123	C₂²	semidirect
20. C₂×C₃₁×#123	C₂²	semidirect
24. C₂×C₃₁×#121	C₂²	semidirect
25. C₂×C₃₁×#120	C₂²	semidirect
26. C₂×C₃₁×#120	C₂²	semidirect
27. C₂×C₃₁×#120	C₂²	semidirect
29. C₂×C₄×D₄×C₃₁	C₂²	semidirect
30. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
33. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
34. C₂²×C₂²⋊C₄×C₃₁	C₂²	semidirect
35. C₂²×C₄²×C₃₁	C₂²	semidirect
36. C₃₁×#398	C₂²	semidirect
37. C₃₁×#397	C₂²	semidirect
38. C₃₁×#396	C₂²	semidirect
39. C₃₁×#394	C₂²	semidirect
40. C₃₁×#394	C₂²	semidirect
41. C₃₁×#392	C₂²	semidirect
42. C₃₁×#390	C₂²	semidirect
43. C₃₁×#389	C₂²	semidirect
44. C₃₁×#388	C₂²	semidirect
45. C₃₁×#386	C₂²	semidirect
46. C₃₁×#386	C₂²	semidirect
47. C₃₁×#386	C₂²	semidirect
48. C₃₁×#386	C₂²	semidirect
49. C₃₁×#385	C₂²	semidirect
50. C₃₁×#380	C₂²	semidirect
51. C₃₁×#380	C₂²	semidirect
52. C₃₁×#380	C₂²	semidirect
53. C₂×C₃₁×#95	C₂²	semidirect
62. C₂²×D₄×C₃₁	D₄	semidirect
103. C₂×C₂²⋊C₄×C₃₁	D₄	semidirect
206. #34984	C₃₁	direct (N × H)
234. C₂³×C₃₁	#121	semidirect
276. C₂×#534	C₂×C₃₁	semidirect
277. C₂×#534	C₂×C₃₁	semidirect
278. C₂²×C₄×D₄	C₂×C₃₁	semidirect
279. #2854	C₂×C₃₁	semidirect
280. #2847	C₂×C₃₁	semidirect
281. #2618	C₂×C₃₁	semidirect
282. #2617	C₂×C₃₁	semidirect
283. #2425	C₂×C₃₁	semidirect
284. #2424	C₂×C₃₁	semidirect
285. #2315	C₂×C₃₁	semidirect
286. C₂×#386	C₂×C₃₁	semidirect
325. C₂⁴×C₄	C₂²×C₃₁	semidirect
328. C₂×#123	C₂²×C₃₁	semidirect
329. C₂×#123	C₂²×C₃₁	semidirect
330. C₂×#123	C₂²×C₃₁	semidirect
334. C₂×#121	C₂²×C₃₁	semidirect
335. C₂×#120	C₂²×C₃₁	semidirect
336. C₂×#120	C₂²×C₃₁	semidirect
337. C₂×#120	C₂²×C₃₁	semidirect
339. C₂×C₄×D₄	C₂²×C₃₁	semidirect
340. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
343. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
344. C₂²×C₂²⋊C₄	C₂²×C₃₁	semidirect
345. C₂²×C₄²	C₂²×C₃₁	semidirect
346. #398	C₂²×C₃₁	semidirect
347. #397	C₂²×C₃₁	semidirect
348. #396	C₂²×C₃₁	semidirect
349. #394	C₂²×C₃₁	semidirect
350. #394	C₂²×C₃₁	semidirect
351. #392	C₂²×C₃₁	semidirect
352. #390	C₂²×C₃₁	semidirect
353. #389	C₂²×C₃₁	semidirect
354. #388	C₂²×C₃₁	semidirect
355. #386	C₂²×C₃₁	semidirect
356. #386	C₂²×C₃₁	semidirect
357. #386	C₂²×C₃₁	semidirect
358. #386	C₂²×C₃₁	semidirect
359. #385	C₂²×C₃₁	semidirect
360. #380	C₂²×C₃₁	semidirect
361. #380	C₂²×C₃₁	semidirect
362. #380	C₂²×C₃₁	semidirect
363. C₂×#95	C₂²×C₃₁	semidirect
381. C₂²×D₄	D₄×C₃₁	semidirect
422. C₂×C₂²⋊C₄	D₄×C₃₁	semidirect
439. C₃₁	#34984	direct (N × H)
553. C₂³	C₃₁×#121	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
C₂²×C₄×D₄×C₃₁	2
C₃₁×#2854	2
C₃₁×#2847	2
C₃₁×#2618	2
C₃₁×#2617	2
C₃₁×#2425	2
C₃₁×#2424	2
C₃₁×#2315	2
C₂×C₃₁×#386	2
#34984	31

Subgroups — C31×#34984 #21491743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34984 #21491743