Subgroups — C₃₁×#34206 #21490965

Order 7936.

3170 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#520	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2367	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2366	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2365	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2354	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2352	C₂	1	4	yes	maximal, complemented
8	D₄×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2325	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2311	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#121	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#532	C₄	1	16	yes	complemented
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#124	C₂²	1	2	yes	—
19	C₂×C₃₁×#121	C₂²	1	2	yes	—
20	C₂×C₃₁×#121	C₂²	1	2	yes	—
21	C₃₁×#520	—	2	16	no	—
22	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₃₁×#117	C₂²	1	2	yes	—
29	C₂×C₃₁×#117	C₂²	1	4	yes	—
30	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
31	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
32	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
34	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#391	C₂²	1	4	yes	—
36	C₃₁×#390	C₂²	1	4	yes	—
37	C₃₁×#390	C₂²	1	4	yes	—
38	C₃₁×#389	C₂²	1	4	yes	—
39	C₃₁×#388	C₂²	1	8	yes	—
40	C₃₁×#388	C₂²	1	8	yes	—
41	C₃₁×#387	C₂²	1	8	yes	—
42	C₃₁×#386	C₂²	1	8	yes	—
43	C₃₁×#386	C₂²	1	8	yes	—
44	C₃₁×#385	C₂²	1	8	yes	—
45	C₃₁×#384	C₂²	1	1	yes	characteristic
46	C₃₁×#384	C₂²	1	1	yes	characteristic
47	C₃₁×#382	C₂²	1	2	yes	—
48	C₃₁×#380	C₂²	1	4	yes	—
49	C₃₁×#380	C₂²	1	4	yes	—
50	C₃₁×#380	C₂²	1	2	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	2	yes	—
57	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
58	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
59	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
60	C₂×C₃₁×#95	C₂²	1	2	yes	—
61	C₂×#41×C₃₁	—	2	8	no	—
62	C₂×#41×C₃₁	—	2	8	no	—
63	C₂×#41×C₃₁	C₂×C₄	1	16	yes	complemented
64	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
65	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
66	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
67	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
71	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
73	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
77	C₃₁×#127	C₂×C₄	1	8	yes	complemented
78	C₃₁×#124	C₂×C₄	1	16	yes	complemented
79	C₃₁×#121	C₂×C₄	1	16	yes	complemented
80	C₃₁×#121	C₂×C₄	1	16	yes	complemented
81	C₄×Q₈×C₃₁	—	2	16	no	—
82	C₄×D₄×C₃₁	—	2	32	no	—
83	C₄×D₄×C₃₁	—	2	16	no	—
84	C₄×D₄×C₃₁	—	4	8	no	—
85	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
86	C₄×D₄×C₃₁	C₂×C₄	1	16	yes	complemented
87	C₃₁×#117	—	2	8	no	—
88	C₃₁×#117	—	2	16	no	—
89	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
91	C₂×C₄⋊C₄×C₃₁	D₄	1	8	yes	complemented
92	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	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	2	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
107	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
108	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
109	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
111	C₂×C₄²×C₃₁	—	4	4	no	—
112	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
113	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
114	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
115	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
116	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
117	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₃₁×#95	C₂³	1	4	yes	—
122	C₃₁×#95	C₂³	1	4	yes	—
123	C₃₁×#95	C₂³	1	4	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	4	yes	—
128	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
129	#41×C₃₁	—	2	64	no	—
130	C₂×Q₈×C₃₁	—	2	4	no	—
131	C₂×Q₈×C₃₁	—	2	4	no	—
132	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
133	C₂×D₄×C₃₁	—	2	8	no	—
134	C₂×D₄×C₃₁	—	2	8	no	—
135	C₂×D₄×C₃₁	—	2	4	no	—
136	C₂×D₄×C₃₁	—	2	4	no	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	2	4	no	—
139	C₂×D₄×C₃₁	—	4	2	no	—
140	C₂×D₄×C₃₁	—	4	2	no	—
141	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
142	C₂×D₄×C₃₁	C₂²×C₄	1	16	yes	—
143	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
144	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
145	C₂²×C₄×C₃₁	—	2	8	no	—
146	C₂²×C₄×C₃₁	—	2	8	no	—
147	C₂²×C₄×C₃₁	—	2	2	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	16	no	—
151	C₂²×C₄×C₃₁	—	4	2	no	—
152	C₂²×C₄×C₃₁	—	4	2	no	—
153	C₂²×C₄×C₃₁	—	4	1	no	—
154	C₂²×C₄×C₃₁	—	4	1	no	—
155	C₂²×C₄×C₃₁	—	4	2	no	—
156	C₂²×C₄×C₃₁	—	4	2	no	—
157	C₂²×C₄×C₃₁	—	4	8	no	—
158	C₂²×C₄×C₃₁	—	4	8	no	—
159	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
160	C₂²×C₄×C₃₁	C₂²×C₄	1	16	yes	—
161	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
162	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
163	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
164	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
165	C₂²×C₄×C₃₁	#41	1	4	yes	—
166	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₄⋊C₄×C₃₁	—	2	4	no	—
178	C₄⋊C₄×C₃₁	—	2	16	no	—
179	C₄⋊C₄×C₃₁	—	2	4	no	—
180	C₄⋊C₄×C₃₁	—	2	4	no	—
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	—	2	4	no	—
184	C₄⋊C₄×C₃₁	—	4	2	no	—
185	C₄⋊C₄×C₃₁	—	4	2	no	—
186	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
187	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
188	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
189	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
190	C₂²⋊C₄×C₃₁	—	2	8	no	—
191	C₂²⋊C₄×C₃₁	—	2	16	no	—
192	C₂²⋊C₄×C₃₁	—	4	8	no	—
193	C₂²⋊C₄×C₃₁	—	4	8	no	—
194	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
195	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
196	C₄²×C₃₁	—	2	16	no	—
197	C₄²×C₃₁	—	2	8	no	—
198	C₄²×C₃₁	—	2	8	no	—
199	C₄²×C₃₁	—	4	4	no	—
200	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
201	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
202	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
203	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
204	#34206	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂³×C₃₁	—	2	1	no	—
206	C₂³×C₃₁	—	2	1	no	—
207	C₂³×C₃₁	—	2	1	no	—
208	C₂³×C₃₁	—	2	1	no	—
209	C₂³×C₃₁	—	2	4	no	—
210	C₂³×C₃₁	—	2	4	no	—
211	C₂³×C₃₁	—	2	4	no	—
212	C₂³×C₃₁	—	2	4	no	—
213	C₂³×C₃₁	—	2	4	no	—
214	C₂³×C₃₁	—	2	4	no	—
215	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
216	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
217	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
218	C₂³×C₃₁	C₂×#41	1	2	yes	—
219	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
220	Q₈×C₃₁	—	2	16	no	—
221	D₄×C₃₁	—	2	32	no	—
222	D₄×C₃₁	—	2	16	no	—
223	D₄×C₃₁	—	2	16	no	—
224	D₄×C₃₁	—	4	8	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	4	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	—	2	1	no	—
232	C₂×C₄×C₃₁	—	2	4	no	—
233	C₂×C₄×C₃₁	—	2	1	no	—
234	C₂×C₄×C₃₁	—	2	1	no	—
235	C₂×C₄×C₃₁	—	2	32	no	—
236	C₂×C₄×C₃₁	—	2	8	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	4	no	—
242	C₂×C₄×C₃₁	—	2	4	no	—
243	C₂×C₄×C₃₁	—	2	4	no	—
244	C₂×C₄×C₃₁	—	2	4	no	—
245	C₂×C₄×C₃₁	—	2	4	no	—
246	C₂×C₄×C₃₁	—	4	4	no	—
247	C₂×C₄×C₃₁	—	4	8	no	—
248	C₂×C₄×C₃₁	—	4	8	no	—
249	C₂×C₄×C₃₁	—	8	16	no	—
250	C₂×C₄×C₃₁	C₄×D₄	1	16	yes	complemented
251	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
252	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
255	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
256	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
257	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
258	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
259	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
262	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
263	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
264	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
265	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
266	C₂×#520	C₂×C₃₁	1	2	yes	complemented
267	#2367	C₂×C₃₁	1	4	yes	complemented
268	#2366	C₂×C₃₁	1	2	yes	complemented
269	#2365	C₂×C₃₁	1	2	yes	complemented
270	#2354	C₂×C₃₁	1	2	yes	complemented
271	#2352	C₂×C₃₁	1	4	yes	complemented
272	D₄×C₄⋊C₄	C₂×C₃₁	1	2	yes	complemented
273	#2325	C₂×C₃₁	1	4	yes	complemented
274	#2311	C₂×C₃₁	1	2	yes	complemented
275	C₄×#121	C₂×C₃₁	1	4	yes	complemented
276	#2285	C₂×C₃₁	1	1	yes	characteristic, complemented
277	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
278	C₂²×C₃₁	—	2	2	no	—
279	C₂²×C₃₁	—	2	2	no	—
280	C₂²×C₃₁	—	2	2	no	—
281	C₂²×C₃₁	—	2	2	no	—
282	C₂²×C₃₁	—	2	2	no	—
283	C₂²×C₃₁	—	2	2	no	—
284	C₂²×C₃₁	—	2	8	no	—
285	C₂²×C₃₁	—	2	8	no	—
286	C₂²×C₃₁	—	2	8	no	—
287	C₂²×C₃₁	—	4	4	no	—
288	C₂²×C₃₁	—	4	4	no	—
289	C₂²×C₃₁	—	4	4	no	—
290	C₂²×C₃₁	—	4	4	no	—
291	C₂²×C₃₁	C₂×C₄×D₄	1	4	yes	—
292	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
293	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
294	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
295	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
296	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
297	C₂²×C₃₁	#585	1	1	yes	characteristic
298	C₂²×C₃₁	#585	1	1	yes	characteristic
299	C₄×C₃₁	—	2	8	no	—
300	C₄×C₃₁	—	2	2	no	—
301	C₄×C₃₁	—	2	2	no	—
302	C₄×C₃₁	—	2	2	no	—
303	C₄×C₃₁	—	2	8	no	—
304	C₄×C₃₁	—	4	4	no	—
305	C₄×C₃₁	—	4	4	no	—
306	C₄×C₃₁	—	4	4	no	—
307	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
308	C₄×C₃₁	C₂×C₄×D₄	1	4	yes	—
309	C₄×C₃₁	#520	1	4	yes	complemented
310	C₂²×#41	C₂²×C₃₁	1	2	yes	—
311	#532	C₄×C₃₁	1	16	yes	complemented
312	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
313	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×#124	C₂²×C₃₁	1	2	yes	—
315	C₂×#121	C₂²×C₃₁	1	2	yes	—
316	C₂×#121	C₂²×C₃₁	1	2	yes	—
317	#520	—	2	16	no	—
318	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	—
319	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
320	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
321	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
322	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
323	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
324	C₂×#117	C₂²×C₃₁	1	2	yes	—
325	C₂×#117	C₂²×C₃₁	1	4	yes	—
326	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
327	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
328	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
329	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
330	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
331	#391	C₂²×C₃₁	1	4	yes	—
332	#390	C₂²×C₃₁	1	4	yes	—
333	#390	C₂²×C₃₁	1	4	yes	—
334	#389	C₂²×C₃₁	1	4	yes	—
335	#388	C₂²×C₃₁	1	8	yes	—
336	#388	C₂²×C₃₁	1	8	yes	—
337	#387	C₂²×C₃₁	1	8	yes	—
338	#386	C₂²×C₃₁	1	8	yes	—
339	#386	C₂²×C₃₁	1	8	yes	—
340	#385	C₂²×C₃₁	1	8	yes	—
341	#384	C₂²×C₃₁	1	1	yes	characteristic
342	#384	C₂²×C₃₁	1	1	yes	characteristic
343	#382	C₂²×C₃₁	1	2	yes	—
344	#380	C₂²×C₃₁	1	4	yes	—
345	#380	C₂²×C₃₁	1	4	yes	—
346	#380	C₂²×C₃₁	1	2	yes	—
347	#380	C₂²×C₃₁	1	2	yes	—
348	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
349	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
350	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
351	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
352	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
353	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
354	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
355	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
356	C₂×#95	C₂²×C₃₁	1	2	yes	—
357	C₂×C₃₁	—	2	4	no	—
358	C₂×C₃₁	—	4	8	no	—
359	C₂×C₃₁	C₂²×C₄×D₄	1	1	yes	characteristic
360	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
361	C₂×C₃₁	#3428	1	1	yes	characteristic
362	C₂×C₃₁	#3496	1	1	yes	characteristic
363	C₂×C₃₁	#3497	1	1	yes	characteristic
364	C₂×C₃₁	#3506	1	1	yes	characteristic
365	C₂×C₃₁	#3507	1	1	yes	characteristic
366	C₂×#41	—	2	8	no	—
367	C₂×#41	—	2	8	no	—
368	C₂×#41	C₂×C₄×C₃₁	1	16	yes	complemented
369	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
370	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
371	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
372	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
373	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
374	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
375	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
376	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
377	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
378	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
379	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
380	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
381	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
382	#127	C₂×C₄×C₃₁	1	8	yes	complemented
383	#124	C₂×C₄×C₃₁	1	16	yes	complemented
384	#121	C₂×C₄×C₃₁	1	16	yes	complemented
385	#121	C₂×C₄×C₃₁	1	16	yes	complemented
386	C₄×Q₈	—	2	16	no	—
387	C₄×D₄	—	2	32	no	—
388	C₄×D₄	—	2	16	no	—
389	C₄×D₄	—	4	8	no	—
390	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
391	C₄×D₄	C₂×C₄×C₃₁	1	16	yes	complemented
392	#117	—	2	8	no	—
393	#117	—	2	16	no	—
394	C₂×C₄⋊C₄	—	2	4	no	—
395	C₂×C₄⋊C₄	—	2	16	no	—
396	C₂×C₄⋊C₄	D₄×C₃₁	1	8	yes	complemented
397	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
398	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
399	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
401	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
402	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
403	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
416	C₂×C₄²	—	4	4	no	—
417	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
418	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
419	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
420	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
421	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
422	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
424	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
425	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
426	#95	C₂³×C₃₁	1	4	yes	—
427	#95	C₂³×C₃₁	1	4	yes	—
428	#95	C₂³×C₃₁	1	4	yes	—
429	#95	C₂³×C₃₁	1	8	yes	—
430	#95	C₂³×C₃₁	1	4	yes	—
431	#95	C₂³×C₃₁	1	8	yes	—
432	C₃₁	#34206	1	1	yes	characteristic, direct product factor, Sylow
433	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
434	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
435	#41	—	2	64	no	—
436	C₂×Q₈	—	2	4	no	—
437	C₂×Q₈	—	2	4	no	—
438	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
439	C₂×D₄	—	2	8	no	—
440	C₂×D₄	—	2	8	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	4	2	no	—
446	C₂×D₄	—	4	2	no	—
447	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
448	C₂×D₄	C₂²×C₄×C₃₁	1	16	yes	—
449	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
450	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
451	C₂²×C₄	—	2	8	no	—
452	C₂²×C₄	—	2	8	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	16	no	—
457	C₂²×C₄	—	4	2	no	—
458	C₂²×C₄	—	4	2	no	—
459	C₂²×C₄	—	4	1	no	—
460	C₂²×C₄	—	4	1	no	—
461	C₂²×C₄	—	4	2	no	—
462	C₂²×C₄	—	4	2	no	—
463	C₂²×C₄	—	4	8	no	—
464	C₂²×C₄	—	4	8	no	—
465	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
466	C₂²×C₄	C₂²×C₄×C₃₁	1	16	yes	—
467	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
468	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
469	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
470	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
471	C₂²×C₄	#41×C₃₁	1	4	yes	—
472	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
473	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
474	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
475	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
478	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
479	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
480	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
482	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	16	no	—
485	C₄⋊C₄	—	2	4	no	—
486	C₄⋊C₄	—	2	4	no	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	4	2	no	—
491	C₄⋊C₄	—	4	2	no	—
492	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
493	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
494	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
495	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
496	C₂²⋊C₄	—	2	8	no	—
497	C₂²⋊C₄	—	2	16	no	—
498	C₂²⋊C₄	—	4	8	no	—
499	C₂²⋊C₄	—	4	8	no	—
500	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
501	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
502	C₄²	—	2	16	no	—
503	C₄²	—	2	8	no	—
504	C₄²	—	2	8	no	—
505	C₄²	—	4	4	no	—
506	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
507	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
508	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
509	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
510	C₂³	—	2	1	no	—
511	C₂³	—	2	1	no	—
512	C₂³	—	2	1	no	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	4	no	—
515	C₂³	—	2	4	no	—
516	C₂³	—	2	4	no	—
517	C₂³	—	2	4	no	—
518	C₂³	—	2	4	no	—
519	C₂³	—	2	4	no	—
520	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
521	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
522	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
523	C₂³	C₂×#41×C₃₁	1	2	yes	—
524	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
525	Q₈	—	2	16	no	—
526	D₄	—	2	32	no	—
527	D₄	—	2	16	no	—
528	D₄	—	2	16	no	—
529	D₄	—	4	8	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	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	—	2	4	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	32	no	—
541	C₂×C₄	—	2	8	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	4	8	no	—
553	C₂×C₄	—	4	8	no	—
554	C₂×C₄	—	8	16	no	—
555	C₂×C₄	C₄×D₄×C₃₁	1	16	yes	complemented
556	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
557	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
558	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
559	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
560	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
561	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
562	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
563	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
564	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
565	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
566	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
567	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
568	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
569	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	8	no	—
577	C₂²	—	2	8	no	—
578	C₂²	—	2	8	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	4	4	no	—
583	C₂²	C₂×C₄×D₄×C₃₁	1	4	yes	—
584	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
585	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
586	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
587	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
588	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
589	C₂²	C₃₁×#585	1	1	yes	characteristic
590	C₂²	C₃₁×#585	1	1	yes	characteristic
591	C₄	—	2	8	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	2	no	—
595	C₄	—	2	8	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	4	no	—
598	C₄	—	4	4	no	—
599	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
600	C₄	C₂×C₄×D₄×C₃₁	1	4	yes	—
601	C₄	C₃₁×#520	1	4	yes	complemented
602	C₂	—	2	4	no	—
603	C₂	—	4	8	no	—
604	C₂	C₂²×C₄×D₄×C₃₁	1	1	yes	characteristic
605	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
606	C₂	C₃₁×#3428	1	1	yes	characteristic
607	C₂	C₃₁×#3496	1	1	yes	characteristic
608	C₂	C₃₁×#3497	1	1	yes	characteristic
609	C₂	C₃₁×#3506	1	1	yes	characteristic
610	C₂	C₃₁×#3507	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#532	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×#520	C₂	semidirect
3. C₃₁×#2367	C₂	semidirect
4. C₃₁×#2366	C₂	semidirect
5. C₃₁×#2365	C₂	semidirect
6. C₃₁×#2354	C₂	semidirect
7. C₃₁×#2352	C₂	semidirect
8. D₄×C₄⋊C₄×C₃₁	C₂	semidirect
9. C₃₁×#2325	C₂	semidirect
10. C₃₁×#2311	C₂	semidirect
11. C₄×C₃₁×#121	C₂	semidirect
12. C₃₁×#2285	C₂	semidirect
13. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
15. C₃₁×#532	C₄	semidirect
63. C₂×#41×C₃₁	C₂×C₄	semidirect
76. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
77. C₃₁×#127	C₂×C₄	semidirect
78. C₃₁×#124	C₂×C₄	semidirect
79. C₃₁×#121	C₂×C₄	semidirect
80. C₃₁×#121	C₂×C₄	semidirect
85. C₄×D₄×C₃₁	C₂×C₄	semidirect
86. C₄×D₄×C₃₁	C₂×C₄	semidirect
91. C₂×C₄⋊C₄×C₃₁	D₄	semidirect
112. C₂×C₄²×C₃₁	C₂×C₄	semidirect
204. #34206	C₃₁	direct (N × H)
250. C₂×C₄×C₃₁	C₄×D₄	semidirect
266. C₂×#520	C₂×C₃₁	semidirect
267. #2367	C₂×C₃₁	semidirect
268. #2366	C₂×C₃₁	semidirect
269. #2365	C₂×C₃₁	semidirect
270. #2354	C₂×C₃₁	semidirect
271. #2352	C₂×C₃₁	semidirect
272. D₄×C₄⋊C₄	C₂×C₃₁	semidirect
273. #2325	C₂×C₃₁	semidirect
274. #2311	C₂×C₃₁	semidirect
275. C₄×#121	C₂×C₃₁	semidirect
276. #2285	C₂×C₃₁	semidirect
277. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
309. C₄×C₃₁	#520	semidirect
311. #532	C₄×C₃₁	semidirect
368. C₂×#41	C₂×C₄×C₃₁	semidirect
381. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
382. #127	C₂×C₄×C₃₁	semidirect
383. #124	C₂×C₄×C₃₁	semidirect
384. #121	C₂×C₄×C₃₁	semidirect
385. #121	C₂×C₄×C₃₁	semidirect
390. C₄×D₄	C₂×C₄×C₃₁	semidirect
391. C₄×D₄	C₂×C₄×C₃₁	semidirect
396. C₂×C₄⋊C₄	D₄×C₃₁	semidirect
417. C₂×C₄²	C₂×C₄×C₃₁	semidirect
432. C₃₁	#34206	direct (N × H)
555. C₂×C₄	C₄×D₄×C₃₁	semidirect
601. C₄	C₃₁×#520	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#532	2
C₂×C₃₁×#520	2
C₃₁×#2367	2
C₃₁×#2366	2
C₃₁×#2365	2
C₃₁×#2354	2
C₃₁×#2352	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2325	2
C₃₁×#2311	2
C₄×C₃₁×#121	2
C₃₁×#2285	2
C₂×C₄×C₄⋊C₄×C₃₁	2
#34206	31

Subgroups — C31×#34206 #21490965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34206 #21490965