Subgroups — C₃₁×#35081 #21491840

Order 7936.

2334 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2856	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2851	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2847	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2648	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2497	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2490	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2444	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2428	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2424	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2420	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2333	C₂	1	4	yes	maximal, complemented
12	C₄×C₃₁×#121	C₂	1	2	yes	maximal, complemented
13	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃₁×#121	C₂²	1	4	yes	—
18	C₂×C₃₁×#121	C₂²	1	2	yes	—
19	C₂×C₃₁×#121	C₂²	1	2	yes	—
20	C₂×C₃₁×#121	C₂²	1	2	yes	—
21	C₂×C₃₁×#121	C₂²	1	4	yes	—
22	C₂×C₃₁×#120	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#400	C₂²	1	4	yes	complemented
29	C₃₁×#397	C₂²	1	4	yes	complemented
30	C₃₁×#397	C₂²	1	4	yes	complemented
31	C₃₁×#396	C₂²	1	4	yes	—
32	C₃₁×#394	C₂²	1	4	yes	—
33	C₃₁×#394	C₂²	1	4	yes	—
34	C₃₁×#394	C₂²	1	4	yes	complemented
35	C₃₁×#392	C₂²	1	4	yes	—
36	C₃₁×#390	C₂²	1	4	yes	—
37	C₃₁×#390	C₂²	1	4	yes	complemented
38	C₃₁×#390	C₂²	1	2	yes	complemented
39	C₃₁×#390	C₂²	1	2	yes	complemented
40	C₃₁×#389	C₂²	1	2	yes	complemented
41	C₃₁×#389	C₂²	1	2	yes	complemented
42	C₃₁×#388	C₂²	1	4	yes	—
43	C₃₁×#388	C₂²	1	4	yes	complemented
44	C₃₁×#388	C₂²	1	4	yes	complemented
45	C₃₁×#387	C₂²	1	4	yes	complemented
46	C₃₁×#387	C₂²	1	4	yes	complemented
47	C₃₁×#386	C₂²	1	4	yes	—
48	C₃₁×#386	C₂²	1	4	yes	—
49	C₃₁×#386	C₂²	1	4	yes	—
50	C₃₁×#386	C₂²	1	4	yes	—
51	C₃₁×#386	C₂²	1	4	yes	—
52	C₃₁×#386	C₂²	1	4	yes	complemented
53	C₃₁×#385	C₂²	1	4	yes	—
54	C₃₁×#384	C₂²	1	1	yes	characteristic, complemented
55	C₃₁×#382	C₂²	1	1	yes	characteristic, complemented
56	C₃₁×#381	C₂²	1	4	yes	complemented
57	C₃₁×#380	C₂²	1	2	yes	—
58	C₃₁×#380	C₂²	1	2	yes	—
59	C₃₁×#380	C₂²	1	2	yes	—
60	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
61	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
62	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
64	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
65	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
66	C₂×C₃₁×#95	C₂²	1	2	yes	—
67	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
69	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
70	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
71	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
72	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
73	C₂³×C₄×C₃₁	—	2	4	no	—
74	C₂³×C₄×C₃₁	—	4	2	no	—
75	C₂³×C₄×C₃₁	—	4	2	no	—
76	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
77	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
78	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
79	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
81	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
83	C₃₁×#127	—	4	2	no	—
84	C₃₁×#125	—	4	2	no	—
85	C₃₁×#121	—	2	8	no	—
86	C₃₁×#121	—	2	16	no	—
87	C₃₁×#121	—	4	4	no	—
88	C₃₁×#121	—	4	4	no	—
89	C₃₁×#121	—	4	8	no	—
90	C₃₁×#120	—	4	8	no	—
91	C₄×D₄×C₃₁	—	4	4	no	—
92	C₄×D₄×C₃₁	—	4	8	no	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	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	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	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	4	yes	—
111	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
112	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₂×C₄²×C₃₁	—	4	2	no	—
117	C₂×C₄²×C₃₁	—	4	2	no	—
118	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
119	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
123	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
125	C₃₁×#95	C₂³	1	4	yes	—
126	C₃₁×#95	C₂³	1	2	yes	—
127	C₃₁×#95	C₂³	1	4	yes	—
128	C₃₁×#95	C₂³	1	4	yes	—
129	C₃₁×#95	C₂³	1	4	yes	—
130	C₃₁×#95	C₂³	1	2	yes	—
131	C₃₁×#95	C₂³	1	4	yes	—
132	C₃₁×#95	C₂³	1	2	yes	—
133	C₃₁×#95	C₂³	1	2	yes	—
134	C₃₁×#95	C₂³	1	4	yes	—
135	C₃₁×#95	C₂³	1	4	yes	—
136	C₂⁴×C₃₁	—	2	4	no	—
137	C₂⁴×C₃₁	—	2	2	no	—
138	C₂⁴×C₃₁	—	2	4	no	—
139	C₂⁴×C₃₁	—	4	1	no	—
140	C₂⁴×C₃₁	—	4	1	no	—
141	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
142	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
143	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
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	8	no	—
149	C₂×D₄×C₃₁	—	2	4	no	—
150	C₂×D₄×C₃₁	—	2	8	no	—
151	C₂×D₄×C₃₁	—	2	4	no	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	4	4	no	—
154	C₂×D₄×C₃₁	—	4	4	no	—
155	C₂×D₄×C₃₁	—	4	4	no	—
156	C₂×D₄×C₃₁	—	4	4	no	—
157	C₂×D₄×C₃₁	—	4	4	no	—
158	C₂×D₄×C₃₁	—	4	4	no	—
159	C₂²×C₄×C₃₁	—	2	2	no	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	2	no	—
162	C₂²×C₄×C₃₁	—	2	4	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	4	no	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
168	C₂²×C₄×C₃₁	—	2	4	no	—
169	C₂²×C₄×C₃₁	—	2	4	no	—
170	C₂²×C₄×C₃₁	—	2	8	no	—
171	C₂²×C₄×C₃₁	—	4	2	no	—
172	C₂²×C₄×C₃₁	—	4	2	no	—
173	C₂²×C₄×C₃₁	—	4	2	no	—
174	C₂²×C₄×C₃₁	—	4	2	no	—
175	C₂²×C₄×C₃₁	—	4	1	no	—
176	C₂²×C₄×C₃₁	—	4	1	no	—
177	C₂²×C₄×C₃₁	—	4	4	no	—
178	C₂²×C₄×C₃₁	—	4	4	no	—
179	C₂²×C₄×C₃₁	—	4	16	no	—
180	C₂²×C₄×C₃₁	#41	1	4	yes	—
181	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
183	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
187	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
191	C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₄⋊C₄×C₃₁	—	2	8	no	—
193	C₄⋊C₄×C₃₁	—	2	4	no	—
194	C₄⋊C₄×C₃₁	—	2	4	no	—
195	C₄⋊C₄×C₃₁	—	4	4	no	—
196	C₄⋊C₄×C₃₁	—	4	4	no	—
197	C₄⋊C₄×C₃₁	—	4	4	no	—
198	C₂²⋊C₄×C₃₁	—	2	4	no	—
199	C₂²⋊C₄×C₃₁	—	2	8	no	—
200	C₂²⋊C₄×C₃₁	—	2	4	no	—
201	C₂²⋊C₄×C₃₁	—	2	4	no	—
202	C₂²⋊C₄×C₃₁	—	2	8	no	—
203	C₂²⋊C₄×C₃₁	—	2	8	no	—
204	C₂²⋊C₄×C₃₁	—	4	4	no	—
205	C₂²⋊C₄×C₃₁	—	4	4	no	—
206	C₂²⋊C₄×C₃₁	—	4	4	no	—
207	C₂²⋊C₄×C₃₁	—	4	4	no	—
208	C₂²⋊C₄×C₃₁	—	4	4	no	—
209	C₂²⋊C₄×C₃₁	—	4	4	no	—
210	C₂²⋊C₄×C₃₁	—	4	4	no	—
211	C₂²⋊C₄×C₃₁	—	4	4	no	—
212	C₂²⋊C₄×C₃₁	#41	1	8	yes	—
213	C₄²×C₃₁	—	2	4	no	—
214	C₄²×C₃₁	—	2	2	no	—
215	C₄²×C₃₁	—	2	2	no	—
216	C₄²×C₃₁	—	4	1	no	—
217	C₄²×C₃₁	—	4	2	no	—
218	C₄²×C₃₁	—	4	1	no	—
219	C₄²×C₃₁	—	4	4	no	—
220	#35081	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂³×C₃₁	—	2	2	no	—
222	C₂³×C₃₁	—	2	1	no	—
223	C₂³×C₃₁	—	2	1	no	—
224	C₂³×C₃₁	—	2	8	no	—
225	C₂³×C₃₁	—	2	2	no	—
226	C₂³×C₃₁	—	2	2	no	—
227	C₂³×C₃₁	—	2	2	no	—
228	C₂³×C₃₁	—	2	2	no	—
229	C₂³×C₃₁	—	2	2	no	—
230	C₂³×C₃₁	—	2	2	no	—
231	C₂³×C₃₁	—	2	4	no	—
232	C₂³×C₃₁	—	2	4	no	—
233	C₂³×C₃₁	—	2	4	no	—
234	C₂³×C₃₁	—	2	4	no	—
235	C₂³×C₃₁	—	2	4	no	—
236	C₂³×C₃₁	—	2	4	no	—
237	C₂³×C₃₁	—	2	4	no	—
238	C₂³×C₃₁	—	4	8	no	—
239	C₂³×C₃₁	—	4	8	no	—
240	C₂³×C₃₁	—	4	4	no	—
241	C₂³×C₃₁	—	4	4	no	—
242	C₂³×C₃₁	C₂×#41	1	4	yes	—
243	C₂³×C₃₁	C₂×#41	1	2	yes	—
244	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
245	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
246	D₄×C₃₁	—	4	16	no	—
247	D₄×C₃₁	—	4	8	no	—
248	D₄×C₃₁	—	8	8	no	—
249	D₄×C₃₁	—	8	8	no	—
250	D₄×C₃₁	—	8	8	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	2	no	—
253	C₂×C₄×C₃₁	—	2	1	no	—
254	C₂×C₄×C₃₁	—	2	1	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	2	no	—
257	C₂×C₄×C₃₁	—	2	8	no	—
258	C₂×C₄×C₃₁	—	2	2	no	—
259	C₂×C₄×C₃₁	—	2	2	no	—
260	C₂×C₄×C₃₁	—	2	4	no	—
261	C₂×C₄×C₃₁	—	2	4	no	—
262	C₂×C₄×C₃₁	—	2	4	no	—
263	C₂×C₄×C₃₁	—	2	2	no	—
264	C₂×C₄×C₃₁	—	2	2	no	—
265	C₂×C₄×C₃₁	—	2	2	no	—
266	C₂×C₄×C₃₁	—	2	4	no	—
267	C₂×C₄×C₃₁	—	2	4	no	—
268	C₂×C₄×C₃₁	—	2	4	no	—
269	C₂×C₄×C₃₁	—	2	2	no	—
270	C₂×C₄×C₃₁	—	2	4	no	—
271	C₂×C₄×C₃₁	—	2	4	no	—
272	C₂×C₄×C₃₁	—	2	2	no	—
273	C₂×C₄×C₃₁	—	2	2	no	—
274	C₂×C₄×C₃₁	—	2	2	no	—
275	C₂×C₄×C₃₁	—	2	4	no	—
276	C₂×C₄×C₃₁	—	4	8	no	—
277	C₂×C₄×C₃₁	—	4	8	no	—
278	C₂×C₄×C₃₁	—	4	8	no	—
279	C₂×C₄×C₃₁	—	4	4	no	—
280	C₂×C₄×C₃₁	—	4	8	no	—
281	C₂×C₄×C₃₁	—	8	8	no	—
282	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
283	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
284	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
285	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
286	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
287	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
288	#2856	C₂×C₃₁	1	1	yes	characteristic, complemented
289	#2851	C₂×C₃₁	1	1	yes	characteristic, complemented
290	#2847	C₂×C₃₁	1	2	yes	complemented
291	#2648	C₂×C₃₁	1	4	yes	complemented
292	#2497	C₂×C₃₁	1	2	yes	complemented
293	#2490	C₂×C₃₁	1	2	yes	complemented
294	#2444	C₂×C₃₁	1	4	yes	complemented
295	#2428	C₂×C₃₁	1	2	yes	complemented
296	#2424	C₂×C₃₁	1	4	yes	complemented
297	#2420	C₂×C₃₁	1	2	yes	complemented
298	#2333	C₂×C₃₁	1	4	yes	complemented
299	C₄×#121	C₂×C₃₁	1	2	yes	complemented
300	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
301	C₂²×C₃₁	—	2	2	no	—
302	C₂²×C₃₁	—	2	2	no	—
303	C₂²×C₃₁	—	2	4	no	—
304	C₂²×C₃₁	—	2	4	no	—
305	C₂²×C₃₁	—	2	4	no	—
306	C₂²×C₃₁	—	2	4	no	—
307	C₂²×C₃₁	—	2	4	no	—
308	C₂²×C₃₁	—	4	16	no	—
309	C₂²×C₃₁	—	4	2	no	—
310	C₂²×C₃₁	—	4	2	no	—
311	C₂²×C₃₁	—	4	2	no	—
312	C₂²×C₃₁	—	4	2	no	—
313	C₂²×C₃₁	—	4	4	no	—
314	C₂²×C₃₁	—	4	4	no	—
315	C₂²×C₃₁	—	4	4	no	—
316	C₂²×C₃₁	—	4	4	no	—
317	C₂²×C₃₁	—	4	4	no	—
318	C₂²×C₃₁	—	4	4	no	—
319	C₂²×C₃₁	—	4	4	no	—
320	C₂²×C₃₁	#540	1	4	yes	—
321	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
322	C₂²×C₃₁	C₂²×#41	1	2	yes	—
323	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
324	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
325	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
326	C₄×C₃₁	—	2	2	no	—
327	C₄×C₃₁	—	2	4	no	—
328	C₄×C₃₁	—	2	4	no	—
329	C₄×C₃₁	—	4	2	no	—
330	C₄×C₃₁	—	4	4	no	—
331	C₄×C₃₁	—	4	2	no	—
332	C₄×C₃₁	—	4	2	no	—
333	C₄×C₃₁	—	4	4	no	—
334	C₄×C₃₁	—	4	4	no	—
335	C₄×C₃₁	#538	1	4	yes	—
336	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
338	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
339	C₂×#121	C₂²×C₃₁	1	4	yes	—
340	C₂×#121	C₂²×C₃₁	1	2	yes	—
341	C₂×#121	C₂²×C₃₁	1	2	yes	—
342	C₂×#121	C₂²×C₃₁	1	2	yes	—
343	C₂×#121	C₂²×C₃₁	1	4	yes	—
344	C₂×#120	C₂²×C₃₁	1	4	yes	—
345	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
346	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
347	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
348	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
349	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
350	#400	C₂²×C₃₁	1	4	yes	complemented
351	#397	C₂²×C₃₁	1	4	yes	complemented
352	#397	C₂²×C₃₁	1	4	yes	complemented
353	#396	C₂²×C₃₁	1	4	yes	—
354	#394	C₂²×C₃₁	1	4	yes	—
355	#394	C₂²×C₃₁	1	4	yes	—
356	#394	C₂²×C₃₁	1	4	yes	complemented
357	#392	C₂²×C₃₁	1	4	yes	—
358	#390	C₂²×C₃₁	1	4	yes	—
359	#390	C₂²×C₃₁	1	4	yes	complemented
360	#390	C₂²×C₃₁	1	2	yes	complemented
361	#390	C₂²×C₃₁	1	2	yes	complemented
362	#389	C₂²×C₃₁	1	2	yes	complemented
363	#389	C₂²×C₃₁	1	2	yes	complemented
364	#388	C₂²×C₃₁	1	4	yes	—
365	#388	C₂²×C₃₁	1	4	yes	complemented
366	#388	C₂²×C₃₁	1	4	yes	complemented
367	#387	C₂²×C₃₁	1	4	yes	complemented
368	#387	C₂²×C₃₁	1	4	yes	complemented
369	#386	C₂²×C₃₁	1	4	yes	—
370	#386	C₂²×C₃₁	1	4	yes	—
371	#386	C₂²×C₃₁	1	4	yes	—
372	#386	C₂²×C₃₁	1	4	yes	—
373	#386	C₂²×C₃₁	1	4	yes	—
374	#386	C₂²×C₃₁	1	4	yes	complemented
375	#385	C₂²×C₃₁	1	4	yes	—
376	#384	C₂²×C₃₁	1	1	yes	characteristic, complemented
377	#382	C₂²×C₃₁	1	1	yes	characteristic, complemented
378	#381	C₂²×C₃₁	1	4	yes	complemented
379	#380	C₂²×C₃₁	1	2	yes	—
380	#380	C₂²×C₃₁	1	2	yes	—
381	#380	C₂²×C₃₁	1	2	yes	—
382	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
383	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
384	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
385	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
386	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
387	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
388	C₂×#95	C₂²×C₃₁	1	2	yes	—
389	C₂×C₃₁	—	2	4	no	—
390	C₂×C₃₁	—	4	4	no	—
391	C₂×C₃₁	—	8	4	no	—
392	C₂×C₃₁	#3444	1	1	yes	characteristic
393	C₂×C₃₁	C₂×#538	1	1	yes	characteristic
394	C₂×C₃₁	C₂×#540	1	1	yes	characteristic
395	C₂×C₃₁	#3519	1	2	yes	—
396	C₂×C₃₁	#3519	1	2	yes	—
397	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
398	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
399	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
400	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
401	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
402	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
403	C₂³×C₄	—	2	4	no	—
404	C₂³×C₄	—	4	2	no	—
405	C₂³×C₄	—	4	2	no	—
406	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
407	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
408	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
409	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
410	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
411	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
412	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
413	#127	—	4	2	no	—
414	#125	—	4	2	no	—
415	#121	—	2	8	no	—
416	#121	—	2	16	no	—
417	#121	—	4	4	no	—
418	#121	—	4	4	no	—
419	#121	—	4	8	no	—
420	#120	—	4	8	no	—
421	C₄×D₄	—	4	4	no	—
422	C₄×D₄	—	4	8	no	—
423	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
424	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
425	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
426	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
427	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
428	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
429	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
430	C₂×C₂²⋊C₄	—	2	8	no	—
431	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
435	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
436	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
437	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
438	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
439	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
440	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
441	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
442	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
443	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
444	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
445	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
446	C₂×C₄²	—	4	2	no	—
447	C₂×C₄²	—	4	2	no	—
448	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
449	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
450	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
451	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
452	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
453	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
454	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
455	#95	C₂³×C₃₁	1	2	yes	—
456	#95	C₂³×C₃₁	1	2	yes	—
457	#95	C₂³×C₃₁	1	2	yes	—
458	#95	C₂³×C₃₁	1	2	yes	—
459	#95	C₂³×C₃₁	1	4	yes	—
460	#95	C₂³×C₃₁	1	4	yes	—
461	#95	C₂³×C₃₁	1	4	yes	—
462	#95	C₂³×C₃₁	1	4	yes	—
463	#95	C₂³×C₃₁	1	4	yes	—
464	#95	C₂³×C₃₁	1	4	yes	—
465	#95	C₂³×C₃₁	1	4	yes	—
466	C₃₁	#35081	1	1	yes	characteristic, direct product factor, Sylow
467	C₂⁴	—	2	4	no	—
468	C₂⁴	—	2	2	no	—
469	C₂⁴	—	2	4	no	—
470	C₂⁴	—	4	1	no	—
471	C₂⁴	—	4	1	no	—
472	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
473	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
474	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
475	C₂×D₄	—	2	8	no	—
476	C₂×D₄	—	2	8	no	—
477	C₂×D₄	—	2	8	no	—
478	C₂×D₄	—	2	8	no	—
479	C₂×D₄	—	2	8	no	—
480	C₂×D₄	—	2	4	no	—
481	C₂×D₄	—	2	8	no	—
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	4	4	no	—
485	C₂×D₄	—	4	4	no	—
486	C₂×D₄	—	4	4	no	—
487	C₂×D₄	—	4	4	no	—
488	C₂×D₄	—	4	4	no	—
489	C₂×D₄	—	4	4	no	—
490	C₂²×C₄	—	2	2	no	—
491	C₂²×C₄	—	2	2	no	—
492	C₂²×C₄	—	2	2	no	—
493	C₂²×C₄	—	2	4	no	—
494	C₂²×C₄	—	2	4	no	—
495	C₂²×C₄	—	2	4	no	—
496	C₂²×C₄	—	2	4	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	2	4	no	—
499	C₂²×C₄	—	2	4	no	—
500	C₂²×C₄	—	2	4	no	—
501	C₂²×C₄	—	2	8	no	—
502	C₂²×C₄	—	4	2	no	—
503	C₂²×C₄	—	4	2	no	—
504	C₂²×C₄	—	4	2	no	—
505	C₂²×C₄	—	4	2	no	—
506	C₂²×C₄	—	4	1	no	—
507	C₂²×C₄	—	4	1	no	—
508	C₂²×C₄	—	4	4	no	—
509	C₂²×C₄	—	4	4	no	—
510	C₂²×C₄	—	4	16	no	—
511	C₂²×C₄	#41×C₃₁	1	4	yes	—
512	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
513	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
514	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
515	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
516	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
517	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
518	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
519	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
520	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
521	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	8	no	—
524	C₄⋊C₄	—	2	4	no	—
525	C₄⋊C₄	—	2	4	no	—
526	C₄⋊C₄	—	4	4	no	—
527	C₄⋊C₄	—	4	4	no	—
528	C₄⋊C₄	—	4	4	no	—
529	C₂²⋊C₄	—	2	4	no	—
530	C₂²⋊C₄	—	2	8	no	—
531	C₂²⋊C₄	—	2	4	no	—
532	C₂²⋊C₄	—	2	4	no	—
533	C₂²⋊C₄	—	2	8	no	—
534	C₂²⋊C₄	—	2	8	no	—
535	C₂²⋊C₄	—	4	4	no	—
536	C₂²⋊C₄	—	4	4	no	—
537	C₂²⋊C₄	—	4	4	no	—
538	C₂²⋊C₄	—	4	4	no	—
539	C₂²⋊C₄	—	4	4	no	—
540	C₂²⋊C₄	—	4	4	no	—
541	C₂²⋊C₄	—	4	4	no	—
542	C₂²⋊C₄	—	4	4	no	—
543	C₂²⋊C₄	#41×C₃₁	1	8	yes	—
544	C₄²	—	2	4	no	—
545	C₄²	—	2	2	no	—
546	C₄²	—	2	2	no	—
547	C₄²	—	4	1	no	—
548	C₄²	—	4	2	no	—
549	C₄²	—	4	1	no	—
550	C₄²	—	4	4	no	—
551	C₂³	—	2	2	no	—
552	C₂³	—	2	1	no	—
553	C₂³	—	2	1	no	—
554	C₂³	—	2	8	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	2	2	no	—
559	C₂³	—	2	2	no	—
560	C₂³	—	2	2	no	—
561	C₂³	—	2	4	no	—
562	C₂³	—	2	4	no	—
563	C₂³	—	2	4	no	—
564	C₂³	—	2	4	no	—
565	C₂³	—	2	4	no	—
566	C₂³	—	2	4	no	—
567	C₂³	—	2	4	no	—
568	C₂³	—	4	8	no	—
569	C₂³	—	4	8	no	—
570	C₂³	—	4	4	no	—
571	C₂³	—	4	4	no	—
572	C₂³	C₂×#41×C₃₁	1	4	yes	—
573	C₂³	C₂×#41×C₃₁	1	2	yes	—
574	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
575	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
576	D₄	—	4	16	no	—
577	D₄	—	4	8	no	—
578	D₄	—	8	8	no	—
579	D₄	—	8	8	no	—
580	D₄	—	8	8	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	8	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	4	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	4	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	4	no	—
606	C₂×C₄	—	4	8	no	—
607	C₂×C₄	—	4	8	no	—
608	C₂×C₄	—	4	8	no	—
609	C₂×C₄	—	4	4	no	—
610	C₂×C₄	—	4	8	no	—
611	C₂×C₄	—	8	8	no	—
612	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
613	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
614	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
615	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
616	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
617	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	2	no	—
620	C₂²	—	2	4	no	—
621	C₂²	—	2	4	no	—
622	C₂²	—	2	4	no	—
623	C₂²	—	2	4	no	—
624	C₂²	—	2	4	no	—
625	C₂²	—	4	16	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	2	no	—
628	C₂²	—	4	2	no	—
629	C₂²	—	4	2	no	—
630	C₂²	—	4	4	no	—
631	C₂²	—	4	4	no	—
632	C₂²	—	4	4	no	—
633	C₂²	—	4	4	no	—
634	C₂²	—	4	4	no	—
635	C₂²	—	4	4	no	—
636	C₂²	—	4	4	no	—
637	C₂²	C₃₁×#540	1	4	yes	—
638	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
639	C₂²	C₂²×#41×C₃₁	1	2	yes	—
640	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
641	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
642	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
643	C₄	—	2	2	no	—
644	C₄	—	2	4	no	—
645	C₄	—	2	4	no	—
646	C₄	—	4	2	no	—
647	C₄	—	4	4	no	—
648	C₄	—	4	2	no	—
649	C₄	—	4	2	no	—
650	C₄	—	4	4	no	—
651	C₄	—	4	4	no	—
652	C₄	C₃₁×#538	1	4	yes	—
653	C₂	—	2	4	no	—
654	C₂	—	4	4	no	—
655	C₂	—	8	4	no	—
656	C₂	C₃₁×#3444	1	1	yes	characteristic
657	C₂	C₂×C₃₁×#538	1	1	yes	characteristic
658	C₂	C₂×C₃₁×#540	1	1	yes	characteristic
659	C₂	C₃₁×#3519	1	2	yes	—
660	C₂	C₃₁×#3519	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. C₃₁×#2856	C₂	semidirect
2. C₃₁×#2851	C₂	semidirect
3. C₃₁×#2847	C₂	semidirect
4. C₃₁×#2648	C₂	semidirect
5. C₃₁×#2497	C₂	semidirect
6. C₃₁×#2490	C₂	semidirect
7. C₃₁×#2444	C₂	semidirect
8. C₃₁×#2428	C₂	semidirect
9. C₃₁×#2424	C₂	semidirect
10. C₃₁×#2420	C₂	semidirect
11. C₃₁×#2333	C₂	semidirect
12. C₄×C₃₁×#121	C₂	semidirect
13. C₂×C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
15. C₂×C₃₁×#127	C₂²	semidirect
16. C₂×C₃₁×#125	C₂²	semidirect
24. C₂×C₄×D₄×C₃₁	C₂²	semidirect
28. C₃₁×#400	C₂²	semidirect
29. C₃₁×#397	C₂²	semidirect
30. C₃₁×#397	C₂²	semidirect
34. C₃₁×#394	C₂²	semidirect
37. C₃₁×#390	C₂²	semidirect
38. C₃₁×#390	C₂²	semidirect
39. C₃₁×#390	C₂²	semidirect
40. C₃₁×#389	C₂²	semidirect
41. C₃₁×#389	C₂²	semidirect
43. C₃₁×#388	C₂²	semidirect
44. C₃₁×#388	C₂²	semidirect
45. C₃₁×#387	C₂²	semidirect
46. C₃₁×#387	C₂²	semidirect
52. C₃₁×#386	C₂²	semidirect
54. C₃₁×#384	C₂²	semidirect
55. C₃₁×#382	C₂²	semidirect
56. C₃₁×#381	C₂²	semidirect
60. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
65. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
220. #35081	C₃₁	direct (N × H)
288. #2856	C₂×C₃₁	semidirect
289. #2851	C₂×C₃₁	semidirect
290. #2847	C₂×C₃₁	semidirect
291. #2648	C₂×C₃₁	semidirect
292. #2497	C₂×C₃₁	semidirect
293. #2490	C₂×C₃₁	semidirect
294. #2444	C₂×C₃₁	semidirect
295. #2428	C₂×C₃₁	semidirect
296. #2424	C₂×C₃₁	semidirect
297. #2420	C₂×C₃₁	semidirect
298. #2333	C₂×C₃₁	semidirect
299. C₄×#121	C₂×C₃₁	semidirect
300. C₂×C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
337. C₂×#127	C₂²×C₃₁	semidirect
338. C₂×#125	C₂²×C₃₁	semidirect
346. C₂×C₄×D₄	C₂²×C₃₁	semidirect
350. #400	C₂²×C₃₁	semidirect
351. #397	C₂²×C₃₁	semidirect
352. #397	C₂²×C₃₁	semidirect
356. #394	C₂²×C₃₁	semidirect
359. #390	C₂²×C₃₁	semidirect
360. #390	C₂²×C₃₁	semidirect
361. #390	C₂²×C₃₁	semidirect
362. #389	C₂²×C₃₁	semidirect
363. #389	C₂²×C₃₁	semidirect
365. #388	C₂²×C₃₁	semidirect
366. #388	C₂²×C₃₁	semidirect
367. #387	C₂²×C₃₁	semidirect
368. #387	C₂²×C₃₁	semidirect
374. #386	C₂²×C₃₁	semidirect
376. #384	C₂²×C₃₁	semidirect
377. #382	C₂²×C₃₁	semidirect
378. #381	C₂²×C₃₁	semidirect
382. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
387. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
466. C₃₁	#35081	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2856	2
C₃₁×#2851	2
C₃₁×#2847	2
C₃₁×#2648	2
C₃₁×#2497	2
C₃₁×#2490	2
C₃₁×#2444	2
C₃₁×#2428	2
C₃₁×#2424	2
C₃₁×#2420	2
C₃₁×#2333	2
C₄×C₃₁×#121	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
#35081	31

Subgroups — C31×#35081 #21491840

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#35081 #21491840