Subgroups — C₃₁×#34255 #21491014

Order 7936.

2682 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#518	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2357	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2354	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2342	C₂	1	8	yes	maximal, complemented
5	C₃₁×#2319	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2317	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2309	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2293	C₂	1	4	yes	maximal, complemented
10	C₂×C₃₁×#390	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#2274	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#518	C₄	1	16	yes	complemented
15	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
16	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#117	C₂²	1	8	yes	complemented
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
24	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
25	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#390	C₂²	1	4	yes	—
27	C₃₁×#390	C₂²	1	2	yes	—
28	C₃₁×#390	C₂²	1	2	yes	—
29	C₃₁×#390	C₂²	1	2	yes	—
30	C₃₁×#390	C₂²	1	4	yes	—
31	C₃₁×#390	C₂²	1	2	yes	—
32	C₃₁×#389	C₂²	1	4	yes	complemented
33	C₃₁×#389	C₂²	1	2	yes	complemented
34	C₃₁×#389	C₂²	1	2	yes	complemented
35	C₃₁×#388	C₂²	1	8	yes	complemented
36	C₃₁×#388	C₂²	1	8	yes	complemented
37	C₃₁×#388	C₂²	1	8	yes	complemented
38	C₃₁×#388	C₂²	1	8	yes	complemented
39	C₃₁×#385	C₂²	1	4	yes	—
40	C₃₁×#385	C₂²	1	4	yes	—
41	C₃₁×#384	C₂²	1	2	yes	complemented
42	C₃₁×#382	C₂²	1	2	yes	complemented
43	C₃₁×#381	C₂²	1	8	yes	complemented
44	C₃₁×#380	C₂²	1	8	yes	complemented
45	C₃₁×#379	C₂²	1	4	yes	—
46	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
47	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
48	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
49	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	complemented
50	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
51	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
52	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
53	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
54	C₃₁×#376	C₂²	1	2	yes	complemented
55	C₄³×C₃₁	C₂²	1	2	yes	complemented
56	C₂⁵×C₃₁	C₂³	1	2	yes	—
57	C₂²×D₄×C₃₁	—	2	4	no	—
58	C₂²×D₄×C₃₁	—	2	4	no	—
59	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	—
60	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
61	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
64	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₄×D₄×C₃₁	C₂×C₄	1	32	yes	complemented
67	C₄×D₄×C₃₁	C₂×C₄	1	32	yes	complemented
68	C₃₁×#117	—	2	4	no	—
69	C₃₁×#117	—	2	4	no	—
70	C₃₁×#117	—	4	16	no	—
71	C₃₁×#117	C₂×C₄	1	8	yes	—
72	C₃₁×#117	C₂×C₄	1	8	yes	—
73	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
84	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
85	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	32	yes	—
86	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
87	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
88	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
89	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
90	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
95	C₂×C₄²×C₃₁	—	2	4	no	—
96	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
99	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
101	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
102	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
103	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
104	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
105	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
106	C₃₁×#95	C₂³	1	4	yes	—
107	C₃₁×#95	C₂³	1	4	yes	—
108	C₃₁×#95	C₂³	1	8	yes	—
109	C₃₁×#95	C₂³	1	4	yes	—
110	C₃₁×#95	C₂³	1	8	yes	—
111	C₃₁×#95	C₂³	1	4	yes	—
112	C₂⁴×C₃₁	—	2	4	no	—
113	C₂⁴×C₃₁	—	2	4	no	—
114	C₂⁴×C₃₁	—	4	4	no	—
115	C₂⁴×C₃₁	—	4	4	no	—
116	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
117	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
118	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
119	C₂×D₄×C₃₁	—	2	32	no	—
120	C₂×D₄×C₃₁	—	2	4	no	—
121	C₂×D₄×C₃₁	—	2	4	no	—
122	C₂×D₄×C₃₁	—	2	4	no	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
125	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
126	C₂²×C₄×C₃₁	—	2	4	no	—
127	C₂²×C₄×C₃₁	—	2	4	no	—
128	C₂²×C₄×C₃₁	—	2	4	no	—
129	C₂²×C₄×C₃₁	—	4	8	no	—
130	C₂²×C₄×C₃₁	—	4	2	no	—
131	C₂²×C₄×C₃₁	—	4	8	no	—
132	C₂²×C₄×C₃₁	—	4	4	no	—
133	C₂²×C₄×C₃₁	C₂²×C₄	1	32	yes	—
134	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
135	C₂²×C₄×C₃₁	#41	1	4	yes	—
136	C₂²×C₄×C₃₁	#41	1	4	yes	—
137	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
140	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
141	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
147	C₄⋊C₄×C₃₁	—	2	4	no	—
148	C₄⋊C₄×C₃₁	—	2	4	no	—
149	C₄⋊C₄×C₃₁	—	2	4	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	8	no	—
153	C₄⋊C₄×C₃₁	—	4	4	no	—
154	C₄⋊C₄×C₃₁	—	4	8	no	—
155	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
156	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
157	C₂²⋊C₄×C₃₁	—	2	4	no	—
158	C₂²⋊C₄×C₃₁	—	2	4	no	—
159	C₂²⋊C₄×C₃₁	—	2	16	no	—
160	C₂²⋊C₄×C₃₁	—	4	8	no	—
161	C₂²⋊C₄×C₃₁	—	4	8	no	—
162	C₂²⋊C₄×C₃₁	—	4	8	no	—
163	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
164	C₂²⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
165	C₂²⋊C₄×C₃₁	C₂²×C₄	1	32	yes	—
166	C₄²×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	2	4	no	—
168	C₄²×C₃₁	—	2	2	no	—
169	C₄²×C₃₁	—	2	4	no	—
170	C₄²×C₃₁	—	2	4	no	—
171	C₄²×C₃₁	—	2	8	no	—
172	C₄²×C₃₁	—	4	4	no	—
173	C₄²×C₃₁	—	4	8	no	—
174	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
175	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
176	#34255	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	—	2	2	no	—
179	C₂³×C₃₁	—	2	4	no	—
180	C₂³×C₃₁	—	2	4	no	—
181	C₂³×C₃₁	—	2	8	no	—
182	C₂³×C₃₁	—	2	8	no	—
183	C₂³×C₃₁	—	2	16	no	—
184	C₂³×C₃₁	—	4	16	no	—
185	C₂³×C₃₁	—	8	8	no	—
186	C₂³×C₃₁	—	8	4	no	—
187	C₂³×C₃₁	—	8	4	no	—
188	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
189	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
190	C₂³×C₃₁	C₂×#41	1	2	yes	—
191	C₂³×C₃₁	C₂×#41	1	2	yes	—
192	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
193	D₄×C₃₁	—	2	16	no	—
194	D₄×C₃₁	—	2	16	no	—
195	C₂×C₄×C₃₁	—	2	2	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	2	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	4	no	—
206	C₂×C₄×C₃₁	—	2	4	no	—
207	C₂×C₄×C₃₁	—	2	4	no	—
208	C₂×C₄×C₃₁	—	2	8	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	4	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	4	32	no	—
214	C₂×C₄×C₃₁	—	4	8	no	—
215	C₂×C₄×C₃₁	—	8	4	no	—
216	C₂×C₄×C₃₁	#117	1	16	yes	complemented
217	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
218	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
219	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
220	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
221	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
222	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
223	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
224	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
225	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
226	C₂×#518	C₂×C₃₁	1	1	yes	characteristic
227	#2357	C₂×C₃₁	1	4	yes	complemented
228	#2354	C₂×C₃₁	1	4	yes	complemented
229	#2342	C₂×C₃₁	1	8	yes	complemented
230	#2319	C₂×C₃₁	1	4	yes	complemented
231	#2317	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#2309	C₂×C₃₁	1	1	yes	characteristic
233	#2296	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#2293	C₂×C₃₁	1	4	yes	complemented
235	C₂×#390	C₂×C₃₁	1	1	yes	characteristic
236	#2274	C₂×C₃₁	1	1	yes	characteristic, complemented
237	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₂²×C₃₁	—	2	4	no	—
239	C₂²×C₃₁	—	2	4	no	—
240	C₂²×C₃₁	—	2	2	no	—
241	C₂²×C₃₁	—	2	2	no	—
242	C₂²×C₃₁	—	2	8	no	—
243	C₂²×C₃₁	—	2	16	no	—
244	C₂²×C₃₁	—	4	8	no	—
245	C₂²×C₃₁	—	4	4	no	—
246	C₂²×C₃₁	—	4	4	no	—
247	C₂²×C₃₁	—	8	16	no	—
248	C₂²×C₃₁	C₂×#117	1	4	yes	—
249	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
250	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
251	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
252	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
253	C₂²×C₃₁	#585	1	2	yes	—
254	C₄×C₃₁	—	2	8	no	—
255	C₄×C₃₁	—	2	8	no	—
256	C₄×C₃₁	—	2	4	no	—
257	C₄×C₃₁	—	4	4	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	4	4	no	—
261	C₄×C₃₁	—	4	2	no	—
262	C₄×C₃₁	C₂×#117	1	4	yes	—
263	C₄×C₃₁	C₂×#117	1	4	yes	—
264	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
265	#518	C₄×C₃₁	1	16	yes	complemented
266	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
267	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
268	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#117	C₂²×C₃₁	1	8	yes	complemented
273	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
274	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
275	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
276	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
277	#390	C₂²×C₃₁	1	4	yes	—
278	#390	C₂²×C₃₁	1	2	yes	—
279	#390	C₂²×C₃₁	1	2	yes	—
280	#390	C₂²×C₃₁	1	2	yes	—
281	#390	C₂²×C₃₁	1	4	yes	—
282	#390	C₂²×C₃₁	1	2	yes	—
283	#389	C₂²×C₃₁	1	4	yes	complemented
284	#389	C₂²×C₃₁	1	2	yes	complemented
285	#389	C₂²×C₃₁	1	2	yes	complemented
286	#388	C₂²×C₃₁	1	8	yes	complemented
287	#388	C₂²×C₃₁	1	8	yes	complemented
288	#388	C₂²×C₃₁	1	8	yes	complemented
289	#388	C₂²×C₃₁	1	8	yes	complemented
290	#385	C₂²×C₃₁	1	4	yes	—
291	#385	C₂²×C₃₁	1	4	yes	—
292	#384	C₂²×C₃₁	1	2	yes	complemented
293	#382	C₂²×C₃₁	1	2	yes	complemented
294	#381	C₂²×C₃₁	1	8	yes	complemented
295	#380	C₂²×C₃₁	1	8	yes	complemented
296	#379	C₂²×C₃₁	1	4	yes	—
297	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
298	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
299	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
300	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	complemented
301	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
302	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
303	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
304	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
305	#376	C₂²×C₃₁	1	2	yes	complemented
306	C₄³	C₂²×C₃₁	1	2	yes	complemented
307	C₂×C₃₁	—	2	4	no	—
308	C₂×C₃₁	—	4	8	no	—
309	C₂×C₃₁	C₂²×#117	1	1	yes	characteristic
310	C₂×C₃₁	C₄×H(5,𝔽₂)	1	2	yes	—
311	C₂×C₃₁	#3494	1	2	yes	—
312	C₂×C₃₁	#3508	1	1	yes	characteristic
313	C₂×C₃₁	#3510	1	1	yes	characteristic
314	C₂⁵	C₂³×C₃₁	1	2	yes	—
315	C₂²×D₄	—	2	4	no	—
316	C₂²×D₄	—	2	4	no	—
317	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	—
318	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
319	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
320	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
321	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
322	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
323	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₄×D₄	C₂×C₄×C₃₁	1	32	yes	complemented
325	C₄×D₄	C₂×C₄×C₃₁	1	32	yes	complemented
326	#117	—	2	4	no	—
327	#117	—	2	4	no	—
328	#117	—	4	16	no	—
329	#117	C₂×C₄×C₃₁	1	8	yes	—
330	#117	C₂×C₄×C₃₁	1	8	yes	—
331	C₂×C₄⋊C₄	—	2	4	no	—
332	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
333	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
335	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
338	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
339	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
340	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
341	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
342	C₂×C₂²⋊C₄	—	4	8	no	—
343	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	32	yes	—
344	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
345	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
353	C₂×C₄²	—	2	4	no	—
354	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
355	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
356	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
357	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
358	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
359	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
360	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
361	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
362	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
363	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
364	#95	C₂³×C₃₁	1	4	yes	—
365	#95	C₂³×C₃₁	1	4	yes	—
366	#95	C₂³×C₃₁	1	8	yes	—
367	#95	C₂³×C₃₁	1	4	yes	—
368	#95	C₂³×C₃₁	1	8	yes	—
369	#95	C₂³×C₃₁	1	4	yes	—
370	C₃₁	#34255	1	1	yes	characteristic, direct product factor, Sylow
371	C₂⁴	—	2	4	no	—
372	C₂⁴	—	2	4	no	—
373	C₂⁴	—	4	4	no	—
374	C₂⁴	—	4	4	no	—
375	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
376	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
377	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
378	C₂×D₄	—	2	32	no	—
379	C₂×D₄	—	2	4	no	—
380	C₂×D₄	—	2	4	no	—
381	C₂×D₄	—	2	4	no	—
382	C₂×D₄	—	2	4	no	—
383	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
384	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	4	8	no	—
389	C₂²×C₄	—	4	2	no	—
390	C₂²×C₄	—	4	8	no	—
391	C₂²×C₄	—	4	4	no	—
392	C₂²×C₄	C₂²×C₄×C₃₁	1	32	yes	—
393	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
394	C₂²×C₄	#41×C₃₁	1	4	yes	—
395	C₂²×C₄	#41×C₃₁	1	4	yes	—
396	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
397	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
398	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
399	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
400	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
401	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
403	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
404	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	8	no	—
412	C₄⋊C₄	—	4	4	no	—
413	C₄⋊C₄	—	4	8	no	—
414	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
415	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	2	16	no	—
419	C₂²⋊C₄	—	4	8	no	—
420	C₂²⋊C₄	—	4	8	no	—
421	C₂²⋊C₄	—	4	8	no	—
422	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
423	C₂²⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
424	C₂²⋊C₄	C₂²×C₄×C₃₁	1	32	yes	—
425	C₄²	—	2	2	no	—
426	C₄²	—	2	4	no	—
427	C₄²	—	2	2	no	—
428	C₄²	—	2	4	no	—
429	C₄²	—	2	4	no	—
430	C₄²	—	2	8	no	—
431	C₄²	—	4	4	no	—
432	C₄²	—	4	8	no	—
433	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
434	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	2	no	—
437	C₂³	—	2	4	no	—
438	C₂³	—	2	4	no	—
439	C₂³	—	2	8	no	—
440	C₂³	—	2	8	no	—
441	C₂³	—	2	16	no	—
442	C₂³	—	4	16	no	—
443	C₂³	—	8	8	no	—
444	C₂³	—	8	4	no	—
445	C₂³	—	8	4	no	—
446	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
447	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
448	C₂³	C₂×#41×C₃₁	1	2	yes	—
449	C₂³	C₂×#41×C₃₁	1	2	yes	—
450	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
451	D₄	—	2	16	no	—
452	D₄	—	2	16	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	4	32	no	—
472	C₂×C₄	—	4	8	no	—
473	C₂×C₄	—	8	4	no	—
474	C₂×C₄	C₃₁×#117	1	16	yes	complemented
475	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
476	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
477	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
478	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
479	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
480	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
481	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
482	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
483	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
484	C₂²	—	2	4	no	—
485	C₂²	—	2	4	no	—
486	C₂²	—	2	2	no	—
487	C₂²	—	2	2	no	—
488	C₂²	—	2	8	no	—
489	C₂²	—	2	16	no	—
490	C₂²	—	4	8	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	8	16	no	—
494	C₂²	C₂×C₃₁×#117	1	4	yes	—
495	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
496	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
497	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
498	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
499	C₂²	C₃₁×#585	1	2	yes	—
500	C₄	—	2	8	no	—
501	C₄	—	2	8	no	—
502	C₄	—	2	4	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	4	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	C₂×C₃₁×#117	1	4	yes	—
509	C₄	C₂×C₃₁×#117	1	4	yes	—
510	C₂	—	2	4	no	—
511	C₂	—	4	8	no	—
512	C₂	C₂²×C₃₁×#117	1	1	yes	characteristic
513	C₂	C₄×C₃₁×H(5,𝔽₂)	1	2	yes	—
514	C₂	C₃₁×#3494	1	2	yes	—
515	C₂	C₃₁×#3508	1	1	yes	characteristic
516	C₂	C₃₁×#3510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂³×D₄×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×#2357	C₂	semidirect
3. C₃₁×#2354	C₂	semidirect
4. C₃₁×#2342	C₂	semidirect
5. C₃₁×#2319	C₂	semidirect
6. C₃₁×#2317	C₂	semidirect
8. C₃₁×#2296	C₂	semidirect
9. C₃₁×#2293	C₂	semidirect
11. C₃₁×#2274	C₂	semidirect
12. C₄×C₃₁×#117	C₂	semidirect
14. C₃₁×#518	C₄	semidirect
21. C₂×C₃₁×#117	C₂²	semidirect
32. C₃₁×#389	C₂²	semidirect
33. C₃₁×#389	C₂²	semidirect
34. C₃₁×#389	C₂²	semidirect
35. C₃₁×#388	C₂²	semidirect
36. C₃₁×#388	C₂²	semidirect
37. C₃₁×#388	C₂²	semidirect
38. C₃₁×#388	C₂²	semidirect
41. C₃₁×#384	C₂²	semidirect
42. C₃₁×#382	C₂²	semidirect
43. C₃₁×#381	C₂²	semidirect
44. C₃₁×#380	C₂²	semidirect
46. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
47. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
48. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
49. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
50. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
53. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
54. C₃₁×#376	C₂²	semidirect
55. C₄³×C₃₁	C₂²	semidirect
66. C₄×D₄×C₃₁	C₂×C₄	semidirect
67. C₄×D₄×C₃₁	C₂×C₄	semidirect
176. #34255	C₃₁	direct (N × H)
216. C₂×C₄×C₃₁	#117	semidirect
227. #2357	C₂×C₃₁	semidirect
228. #2354	C₂×C₃₁	semidirect
229. #2342	C₂×C₃₁	semidirect
230. #2319	C₂×C₃₁	semidirect
231. #2317	C₂×C₃₁	semidirect
233. #2296	C₂×C₃₁	semidirect
234. #2293	C₂×C₃₁	semidirect
236. #2274	C₂×C₃₁	semidirect
237. C₄×#117	C₂×C₃₁	semidirect
265. #518	C₄×C₃₁	semidirect
272. C₂×#117	C₂²×C₃₁	semidirect
283. #389	C₂²×C₃₁	semidirect
284. #389	C₂²×C₃₁	semidirect
285. #389	C₂²×C₃₁	semidirect
286. #388	C₂²×C₃₁	semidirect
287. #388	C₂²×C₃₁	semidirect
288. #388	C₂²×C₃₁	semidirect
289. #388	C₂²×C₃₁	semidirect
292. #384	C₂²×C₃₁	semidirect
293. #382	C₂²×C₃₁	semidirect
294. #381	C₂²×C₃₁	semidirect
295. #380	C₂²×C₃₁	semidirect
297. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
298. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
299. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
300. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
301. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
304. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
305. #376	C₂²×C₃₁	semidirect
306. C₄³	C₂²×C₃₁	semidirect
324. C₄×D₄	C₂×C₄×C₃₁	semidirect
325. C₄×D₄	C₂×C₄×C₃₁	semidirect
370. C₃₁	#34255	direct (N × H)
474. C₂×C₄	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×C₃₁×#518	2
C₃₁×#2357	2
C₃₁×#2354	2
C₃₁×#2342	2
C₃₁×#2319	2
C₃₁×#2317	2
C₃₁×#2309	2
C₃₁×#2296	2
C₃₁×#2293	2
C₂×C₃₁×#390	2
C₃₁×#2274	2
C₄×C₃₁×#117	2
#34255	31

Subgroups — C31×#34255 #21491014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34255 #21491014