Subgroups — C₃₁×#34191 #21490950

Order 7936.

2470 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2355	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2352	C₂	1	4	yes	maximal, complemented
4	D₄×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2346	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2343	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2339	C₂	1	2	yes	maximal
8	C₃₁×#2320	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2318	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2283	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₃₁×#382	C₂	1	2	yes	maximal, complemented
14	C₃₁×#2275	C₂	1	2	yes	maximal, complemented
15	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×#41×C₃₁	C₄	1	16	yes	complemented
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
24	C₂×C₃₁×#117	C₂²	1	2	yes	—
25	C₂×C₃₁×#117	C₂²	1	2	yes	—
26	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
28	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
29	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#391	C₂²	1	4	yes	—
31	C₃₁×#389	C₂²	1	4	yes	—
32	C₃₁×#388	C₂²	1	8	yes	—
33	C₃₁×#387	C₂²	1	8	yes	—
34	C₃₁×#387	C₂²	1	8	yes	—
35	C₃₁×#386	C₂²	1	8	yes	—
36	C₃₁×#385	C₂²	1	8	yes	—
37	C₃₁×#384	C₂²	1	2	yes	—
38	C₃₁×#383	C₂²	1	4	yes	—
39	C₃₁×#383	C₂²	1	2	yes	—
40	C₃₁×#382	C₂²	1	4	yes	—
41	C₃₁×#382	C₂²	1	4	yes	—
42	C₃₁×#382	C₂²	1	4	yes	—
43	C₃₁×#382	C₂²	1	1	yes	characteristic
44	C₃₁×#382	C₂²	1	2	yes	—
45	C₃₁×#382	C₂²	1	1	yes	characteristic
46	C₃₁×#381	C₂²	1	2	yes	—
47	C₃₁×#381	C₂²	1	4	yes	—
48	C₃₁×#380	C₂²	1	2	yes	—
49	C₃₁×#380	C₂²	1	2	yes	—
50	C₃₁×#380	C₂²	1	4	yes	—
51	C₃₁×#380	C₂²	1	2	yes	—
52	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
53	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
54	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
55	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
56	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
57	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
58	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
59	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
60	C₃₁×#376	C₂²	1	2	yes	—
61	C₃₁×#376	C₂²	1	4	yes	—
62	C₂×C₃₁×#95	C₂²	1	4	yes	—
63	C₂×C₃₁×#95	C₂²	1	4	yes	—
64	C₂×#41×C₃₁	—	2	8	no	—
65	C₂×#41×C₃₁	C₂×C₄	1	8	yes	—
66	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
70	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
74	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
77	C₄×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
78	C₄×D₄×C₃₁	—	2	16	no	—
79	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
80	C₄×D₄×C₃₁	C₂×C₄	1	8	yes	—
81	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
82	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
83	C₃₁×#117	—	4	4	no	—
84	C₃₁×#117	C₂×C₄	1	16	yes	complemented
85	C₃₁×#117	C₂×C₄	1	8	yes	complemented
86	C₂×C₄⋊C₄×C₃₁	—	4	8	no	—
87	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
98	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
104	C₂×C₄²×C₃₁	C₂×C₄	1	16	yes	complemented
105	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
106	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
107	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
110	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
113	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
115	C₃₁×#95	C₂³	1	2	yes	—
116	C₃₁×#95	C₂³	1	2	yes	—
117	C₃₁×#95	C₂³	1	4	yes	—
118	C₃₁×#95	C₂³	1	4	yes	—
119	C₃₁×#95	C₂³	1	8	yes	—
120	C₃₁×#95	C₂³	1	8	yes	—
121	C₃₁×#95	C₂³	1	2	yes	—
122	C₃₁×#95	C₂³	1	4	yes	—
123	C₃₁×#95	C₂³	1	2	yes	—
124	C₃₁×#95	C₂³	1	8	yes	—
125	C₃₁×#95	C₂³	1	4	yes	—
126	C₃₁×#95	C₂³	1	8	yes	—
127	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
128	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
129	#41×C₃₁	—	2	32	no	—
130	C₂×Q₈×C₃₁	—	2	4	no	—
131	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
132	C₂×D₄×C₃₁	—	2	8	no	—
133	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
134	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
135	C₂×D₄×C₃₁	#41	1	8	yes	—
136	C₂²×C₄×C₃₁	—	2	8	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	—	2	4	no	—
139	C₂²×C₄×C₃₁	—	2	2	no	—
140	C₂²×C₄×C₃₁	—	2	4	no	—
141	C₂²×C₄×C₃₁	—	2	2	no	—
142	C₂²×C₄×C₃₁	—	2	16	no	—
143	C₂²×C₄×C₃₁	—	4	4	no	—
144	C₂²×C₄×C₃₁	—	4	2	no	—
145	C₂²×C₄×C₃₁	—	4	8	no	—
146	C₂²×C₄×C₃₁	—	4	8	no	—
147	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
148	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
149	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
150	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
151	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
152	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
153	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
165	C₄⋊C₄×C₃₁	—	2	8	no	—
166	C₄⋊C₄×C₃₁	—	2	4	no	—
167	C₄⋊C₄×C₃₁	—	2	8	no	—
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	4	8	no	—
170	C₄⋊C₄×C₃₁	—	4	4	no	—
171	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
172	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
173	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
174	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
175	C₂²⋊C₄×C₃₁	—	2	16	no	—
176	C₂²⋊C₄×C₃₁	—	4	4	no	—
177	C₂²⋊C₄×C₃₁	—	4	4	no	—
178	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
179	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
180	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
181	C₄²×C₃₁	—	2	8	no	—
182	C₄²×C₃₁	—	4	8	no	—
183	C₄²×C₃₁	—	4	4	no	—
184	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
185	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
186	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
187	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
188	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
189	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
190	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
191	#34191	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	2	4	no	—
195	C₂³×C₃₁	—	2	2	no	—
196	C₂³×C₃₁	—	2	1	no	—
197	C₂³×C₃₁	—	2	1	no	—
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
200	C₂³×C₃₁	C₂³×C₄	1	4	yes	—
201	C₂³×C₃₁	C₂×#41	1	8	yes	—
202	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
203	Q₈×C₃₁	—	2	8	no	—
204	D₄×C₃₁	—	2	16	no	—
205	D₄×C₃₁	#117	1	16	yes	complemented
206	C₂×C₄×C₃₁	—	2	16	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	8	no	—
217	C₂×C₄×C₃₁	—	2	16	no	—
218	C₂×C₄×C₃₁	—	2	8	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	4	no	—
221	C₂×C₄×C₃₁	—	2	8	no	—
222	C₂×C₄×C₃₁	—	2	8	no	—
223	C₂×C₄×C₃₁	—	2	4	no	—
224	C₂×C₄×C₃₁	—	2	4	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	8	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	4	8	no	—
229	C₂×C₄×C₃₁	—	4	4	no	—
230	C₂×C₄×C₃₁	—	8	16	no	—
231	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
232	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
233	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
234	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
235	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
236	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
237	C₂×C₄×C₃₁	C₂³×C₄	1	4	yes	—
238	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
239	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
240	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
241	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
242	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
243	C₂×C₄×#41	C₂×C₃₁	1	1	yes	characteristic
244	#2355	C₂×C₃₁	1	2	yes	complemented
245	#2352	C₂×C₃₁	1	4	yes	complemented
246	D₄×C₄⋊C₄	C₂×C₃₁	1	2	yes	complemented
247	#2346	C₂×C₃₁	1	2	yes	complemented
248	#2343	C₂×C₃₁	1	4	yes	complemented
249	#2339	C₂×C₃₁	1	2	yes	—
250	#2320	C₂×C₃₁	1	2	yes	complemented
251	#2318	C₂×C₃₁	1	4	yes	complemented
252	#2285	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#2283	C₂×C₃₁	1	2	yes	complemented
255	C₂×#382	C₂×C₃₁	1	2	yes	complemented
256	#2275	C₂×C₃₁	1	2	yes	complemented
257	C₂²×C₃₁	—	2	4	no	—
258	C₂²×C₃₁	—	2	4	no	—
259	C₂²×C₃₁	—	2	8	no	—
260	C₂²×C₃₁	—	2	8	no	—
261	C₂²×C₃₁	—	2	2	no	—
262	C₂²×C₃₁	—	2	2	no	—
263	C₂²×C₃₁	—	2	2	no	—
264	C₂²×C₃₁	—	4	2	no	—
265	C₂²×C₃₁	—	4	2	no	—
266	C₂²×C₃₁	C₂×#117	1	8	yes	—
267	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
268	C₂²×C₃₁	C₂²×#41	1	2	yes	—
269	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
270	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
271	C₂²×C₃₁	#585	1	2	yes	—
272	C₄×C₃₁	—	2	2	no	—
273	C₄×C₃₁	—	2	4	no	—
274	C₄×C₃₁	—	2	2	no	—
275	C₄×C₃₁	—	2	4	no	—
276	C₄×C₃₁	—	2	2	no	—
277	C₄×C₃₁	—	2	4	no	—
278	C₄×C₃₁	—	2	2	no	—
279	C₄×C₃₁	—	4	8	no	—
280	C₄×C₃₁	—	4	4	no	—
281	C₄×C₃₁	—	4	4	no	—
282	C₄×C₃₁	C₂×#117	1	4	yes	—
283	C₄×C₃₁	#520	1	4	yes	—
284	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
285	C₄×#41	C₄×C₃₁	1	16	yes	complemented
286	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
289	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
292	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
293	C₂×#117	C₂²×C₃₁	1	2	yes	—
294	C₂×#117	C₂²×C₃₁	1	2	yes	—
295	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
296	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
297	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
298	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
299	#391	C₂²×C₃₁	1	4	yes	—
300	#389	C₂²×C₃₁	1	4	yes	—
301	#388	C₂²×C₃₁	1	8	yes	—
302	#387	C₂²×C₃₁	1	8	yes	—
303	#387	C₂²×C₃₁	1	8	yes	—
304	#386	C₂²×C₃₁	1	8	yes	—
305	#385	C₂²×C₃₁	1	8	yes	—
306	#384	C₂²×C₃₁	1	2	yes	—
307	#383	C₂²×C₃₁	1	4	yes	—
308	#383	C₂²×C₃₁	1	2	yes	—
309	#382	C₂²×C₃₁	1	4	yes	—
310	#382	C₂²×C₃₁	1	4	yes	—
311	#382	C₂²×C₃₁	1	4	yes	—
312	#382	C₂²×C₃₁	1	1	yes	characteristic
313	#382	C₂²×C₃₁	1	2	yes	—
314	#382	C₂²×C₃₁	1	1	yes	characteristic
315	#381	C₂²×C₃₁	1	2	yes	—
316	#381	C₂²×C₃₁	1	4	yes	—
317	#380	C₂²×C₃₁	1	2	yes	—
318	#380	C₂²×C₃₁	1	2	yes	—
319	#380	C₂²×C₃₁	1	4	yes	—
320	#380	C₂²×C₃₁	1	2	yes	—
321	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
322	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
323	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
324	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
325	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
326	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
327	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
328	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
329	#376	C₂²×C₃₁	1	2	yes	—
330	#376	C₂²×C₃₁	1	4	yes	—
331	C₂×#95	C₂²×C₃₁	1	4	yes	—
332	C₂×#95	C₂²×C₃₁	1	4	yes	—
333	C₂×C₃₁	—	2	8	no	—
334	C₂×C₃₁	—	4	2	no	—
335	C₂×C₃₁	C₂²×#117	1	1	yes	characteristic
336	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
337	C₂×C₃₁	#3428	1	1	yes	characteristic
338	C₂×C₃₁	#3514	1	2	yes	—
339	C₂×C₃₁	#3515	1	2	yes	—
340	C₂×#41	—	2	8	no	—
341	C₂×#41	C₂×C₄×C₃₁	1	8	yes	—
342	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
343	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
344	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
345	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
346	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
347	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
348	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
349	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
350	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
351	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
352	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
353	C₄×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
354	C₄×D₄	—	2	16	no	—
355	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
356	C₄×D₄	C₂×C₄×C₃₁	1	8	yes	—
357	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
358	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
359	#117	—	4	4	no	—
360	#117	C₂×C₄×C₃₁	1	16	yes	complemented
361	#117	C₂×C₄×C₃₁	1	8	yes	complemented
362	C₂×C₄⋊C₄	—	4	8	no	—
363	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
364	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
365	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
367	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
373	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
374	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
379	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
380	C₂×C₄²	C₂×C₄×C₃₁	1	16	yes	complemented
381	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
382	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
383	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
384	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
385	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
388	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
390	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
391	#95	C₂³×C₃₁	1	2	yes	—
392	#95	C₂³×C₃₁	1	2	yes	—
393	#95	C₂³×C₃₁	1	4	yes	—
394	#95	C₂³×C₃₁	1	4	yes	—
395	#95	C₂³×C₃₁	1	8	yes	—
396	#95	C₂³×C₃₁	1	8	yes	—
397	#95	C₂³×C₃₁	1	2	yes	—
398	#95	C₂³×C₃₁	1	4	yes	—
399	#95	C₂³×C₃₁	1	2	yes	—
400	#95	C₂³×C₃₁	1	8	yes	—
401	#95	C₂³×C₃₁	1	4	yes	—
402	#95	C₂³×C₃₁	1	8	yes	—
403	C₃₁	#34191	1	1	yes	characteristic, direct product factor, Sylow
404	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
405	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
406	#41	—	2	32	no	—
407	C₂×Q₈	—	2	4	no	—
408	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
411	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
412	C₂×D₄	#41×C₃₁	1	8	yes	—
413	C₂²×C₄	—	2	8	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	16	no	—
420	C₂²×C₄	—	4	4	no	—
421	C₂²×C₄	—	4	2	no	—
422	C₂²×C₄	—	4	8	no	—
423	C₂²×C₄	—	4	8	no	—
424	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
425	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
426	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
427	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
428	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
429	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
430	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
431	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
432	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
433	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
434	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
436	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
442	C₄⋊C₄	—	2	8	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	8	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	4	8	no	—
447	C₄⋊C₄	—	4	4	no	—
448	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
449	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
450	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
451	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
452	C₂²⋊C₄	—	2	16	no	—
453	C₂²⋊C₄	—	4	4	no	—
454	C₂²⋊C₄	—	4	4	no	—
455	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
456	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
457	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
458	C₄²	—	2	8	no	—
459	C₄²	—	4	8	no	—
460	C₄²	—	4	4	no	—
461	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
462	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
463	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
464	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
465	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
466	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
467	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	4	no	—
471	C₂³	—	2	2	no	—
472	C₂³	—	2	1	no	—
473	C₂³	—	2	1	no	—
474	C₂³	—	2	2	no	—
475	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
476	C₂³	C₂³×C₄×C₃₁	1	4	yes	—
477	C₂³	C₂×#41×C₃₁	1	8	yes	—
478	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
479	Q₈	—	2	8	no	—
480	D₄	—	2	16	no	—
481	D₄	C₃₁×#117	1	16	yes	complemented
482	C₂×C₄	—	2	16	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	4	no	—
485	C₂×C₄	—	2	1	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	8	no	—
493	C₂×C₄	—	2	16	no	—
494	C₂×C₄	—	2	8	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	8	no	—
498	C₂×C₄	—	2	8	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	8	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	4	8	no	—
505	C₂×C₄	—	4	4	no	—
506	C₂×C₄	—	8	16	no	—
507	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
508	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
509	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
510	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
511	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
512	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
513	C₂×C₄	C₂³×C₄×C₃₁	1	4	yes	—
514	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
515	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
516	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
517	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
518	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
519	C₂²	—	2	4	no	—
520	C₂²	—	2	4	no	—
521	C₂²	—	2	8	no	—
522	C₂²	—	2	8	no	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	2	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	4	2	no	—
528	C₂²	C₂×C₃₁×#117	1	8	yes	—
529	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
530	C₂²	C₂²×#41×C₃₁	1	2	yes	—
531	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
532	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
533	C₂²	C₃₁×#585	1	2	yes	—
534	C₄	—	2	2	no	—
535	C₄	—	2	4	no	—
536	C₄	—	2	2	no	—
537	C₄	—	2	4	no	—
538	C₄	—	2	2	no	—
539	C₄	—	2	4	no	—
540	C₄	—	2	2	no	—
541	C₄	—	4	4	no	—
542	C₄	—	4	8	no	—
543	C₄	—	4	4	no	—
544	C₄	C₂×C₃₁×#117	1	4	yes	—
545	C₄	C₃₁×#520	1	4	yes	—
546	C₂	—	2	8	no	—
547	C₂	—	4	2	no	—
548	C₂	C₂²×C₃₁×#117	1	1	yes	characteristic
549	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
550	C₂	C₃₁×#3428	1	1	yes	characteristic
551	C₂	C₃₁×#3514	1	2	yes	—
552	C₂	C₃₁×#3515	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂²×#41×C₃₁	C₂²
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. C₃₁×#2355	C₂	semidirect
3. C₃₁×#2352	C₂	semidirect
4. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
5. C₃₁×#2346	C₂	semidirect
6. C₃₁×#2343	C₂	semidirect
8. C₃₁×#2320	C₂	semidirect
9. C₃₁×#2318	C₂	semidirect
10. C₃₁×#2285	C₂	semidirect
11. C₃₁×#2283	C₂	semidirect
12. C₃₁×#2283	C₂	semidirect
13. C₂×C₃₁×#382	C₂	semidirect
14. C₃₁×#2275	C₂	semidirect
16. C₄×#41×C₃₁	C₄	semidirect
76. C₄×Q₈×C₃₁	C₂×C₄	semidirect
77. C₄×Q₈×C₃₁	C₂×C₄	semidirect
81. C₄×D₄×C₃₁	C₂×C₄	semidirect
82. C₄×D₄×C₃₁	C₂×C₄	semidirect
84. C₃₁×#117	C₂×C₄	semidirect
85. C₃₁×#117	C₂×C₄	semidirect
103. C₂×C₄²×C₃₁	C₂×C₄	semidirect
104. C₂×C₄²×C₃₁	C₂×C₄	semidirect
191. #34191	C₃₁	direct (N × H)
205. D₄×C₃₁	#117	semidirect
244. #2355	C₂×C₃₁	semidirect
245. #2352	C₂×C₃₁	semidirect
246. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
247. #2346	C₂×C₃₁	semidirect
248. #2343	C₂×C₃₁	semidirect
250. #2320	C₂×C₃₁	semidirect
251. #2318	C₂×C₃₁	semidirect
252. #2285	C₂×C₃₁	semidirect
253. #2283	C₂×C₃₁	semidirect
254. #2283	C₂×C₃₁	semidirect
255. C₂×#382	C₂×C₃₁	semidirect
256. #2275	C₂×C₃₁	semidirect
285. C₄×#41	C₄×C₃₁	semidirect
352. C₄×Q₈	C₂×C₄×C₃₁	semidirect
353. C₄×Q₈	C₂×C₄×C₃₁	semidirect
357. C₄×D₄	C₂×C₄×C₃₁	semidirect
358. C₄×D₄	C₂×C₄×C₃₁	semidirect
360. #117	C₂×C₄×C₃₁	semidirect
361. #117	C₂×C₄×C₃₁	semidirect
379. C₂×C₄²	C₂×C₄×C₃₁	semidirect
380. C₂×C₄²	C₂×C₄×C₃₁	semidirect
403. C₃₁	#34191	direct (N × H)
481. D₄	C₃₁×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₃₁	2
C₃₁×#2355	2
C₃₁×#2352	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2346	2
C₃₁×#2343	2
C₃₁×#2339	2
C₃₁×#2320	2
C₃₁×#2318	2
C₃₁×#2285	2
C₃₁×#2283	2
C₂×C₃₁×#382	2
C₃₁×#2275	2
#34191	31

Subgroups — C31×#34191 #21490950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34191 #21490950