Subgroups — C₃₁×#22786 #21479545

Order 7936.

900 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1923	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1917	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1870	C₂	1	2	yes	maximal
5	C₃₁×#1839	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1817	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1753	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1749	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
28	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
30	C₄×C₂²⋊C₄×C₃₁	C₄	1	1	yes	characteristic
31	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₄³×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#336	C₂²	1	1	yes	characteristic
35	C₃₁×#336	C₂²	1	1	yes	characteristic
36	C₃₁×#336	C₂²	1	1	yes	characteristic
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×M₁₆×C₃₁	—	2	4	no	—
51	C₂×M₁₆×C₃₁	—	2	4	no	—
52	C₂×M₁₆×C₃₁	—	4	2	no	—
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₄×D₄×C₃₁	—	2	4	no	—
60	C₄×D₄×C₃₁	—	2	4	no	—
61	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	—	4	2	no	—
71	C₂×C₄²×C₃₁	—	4	2	no	—
72	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×C₄⋊C₈	—	4	2	no	—
86	C₃₁×C₂²⋊C₈	—	2	4	no	—
87	C₃₁×C₂²⋊C₈	—	2	4	no	—
88	C₃₁×C₂²⋊C₈	—	4	2	no	—
89	C₃₁×#97	—	4	2	no	—
90	C₄×C₈×C₃₁	—	2	4	no	—
91	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
92	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
93	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
94	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
95	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
96	C₃₁×#95	C₂×C₄	1	1	yes	characteristic
97	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	4	1	no	—
115	C₂²×C₄×C₃₁	—	4	1	no	—
116	C₂²×C₄×C₃₁	—	4	1	no	—
117	C₂²×C₄×C₃₁	—	4	1	no	—
118	C₂²×C₄×C₃₁	—	4	1	no	—
119	C₂²×C₄×C₃₁	—	4	1	no	—
120	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	#41	1	2	yes	—
133	C₂²×C₄×C₃₁	#41	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
135	M₁₆×C₃₁	—	4	8	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	#41	1	4	yes	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₂²⋊C₄×C₃₁	—	2	2	no	—
152	C₂²⋊C₄×C₃₁	—	2	2	no	—
153	C₂²⋊C₄×C₃₁	—	2	2	no	—
154	C₂²⋊C₄×C₃₁	—	2	2	no	—
155	C₂²⋊C₄×C₃₁	—	2	2	no	—
156	C₂²⋊C₄×C₃₁	M₁₆	1	4	yes	complemented
157	C₄²×C₃₁	—	2	2	no	—
158	C₄²×C₃₁	—	2	2	no	—
159	C₄²×C₃₁	—	2	2	no	—
160	C₄²×C₃₁	—	2	1	no	—
161	C₄²×C₃₁	—	2	1	no	—
162	C₄²×C₃₁	—	2	2	no	—
163	C₄²×C₃₁	—	2	2	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	4	1	no	—
166	C₄²×C₃₁	—	4	1	no	—
167	C₄²×C₃₁	—	4	1	no	—
168	C₄²×C₃₁	—	4	1	no	—
169	C₄²×C₃₁	—	4	1	no	—
170	C₄²×C₃₁	M₁₆	1	4	yes	complemented
171	#22786	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂³×C₃₁	—	2	1	no	—
173	C₂³×C₃₁	—	2	1	no	—
174	C₂³×C₃₁	—	2	1	no	—
175	C₂³×C₃₁	—	2	1	no	—
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	2	1	no	—
178	C₂³×C₃₁	—	2	1	no	—
179	C₂³×C₃₁	—	2	1	no	—
180	C₂³×C₃₁	—	2	1	no	—
181	C₂³×C₃₁	—	2	1	no	—
182	C₂³×C₃₁	—	2	1	no	—
183	C₂³×C₃₁	—	2	1	no	—
184	C₂³×C₃₁	#117	1	2	yes	—
185	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
186	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
187	D₄×C₃₁	—	2	8	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	2	4	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	2	1	no	—
214	C₂×C₄×C₃₁	—	2	1	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	4	4	no	—
220	C₂×C₄×C₃₁	—	4	4	no	—
221	C₂×C₄×C₃₁	—	4	4	no	—
222	C₂×C₄×C₃₁	—	4	4	no	—
223	C₂×C₄×C₃₁	—	4	2	no	—
224	C₂×C₄×C₃₁	—	4	2	no	—
225	C₂×C₄×C₃₁	#117	1	2	yes	—
226	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
227	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
228	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
229	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
230	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
231	C₂×C₄×C₃₁	#131	1	2	yes	—
232	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
234	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
235	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
236	C₈×C₃₁	—	2	4	no	—
237	C₈×C₃₁	—	4	2	no	—
238	C₈×C₃₁	—	4	2	no	—
239	C₈×C₃₁	—	4	2	no	—
240	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
241	#1923	C₂×C₃₁	1	2	yes	complemented
242	#1917	C₂×C₃₁	1	2	yes	complemented
243	#1870	C₂×C₃₁	1	2	yes	—
244	#1839	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#1817	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#1797	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1753	C₂×C₃₁	1	2	yes	complemented
248	#1749	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#1728	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂²×C₃₁	—	2	2	no	—
252	C₂²×C₃₁	—	2	2	no	—
253	C₂²×C₃₁	—	2	2	no	—
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	2	2	no	—
256	C₂²×C₃₁	—	2	2	no	—
257	C₂²×C₃₁	—	4	1	no	—
258	C₂²×C₃₁	—	4	1	no	—
259	C₂²×C₃₁	—	4	1	no	—
260	C₂²×C₃₁	—	4	1	no	—
261	C₂²×C₃₁	—	4	1	no	—
262	C₂²×C₃₁	—	4	1	no	—
263	C₂²×C₃₁	—	4	1	no	—
264	C₂²×C₃₁	—	4	1	no	—
265	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
266	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
267	C₂²×C₃₁	#518	1	1	yes	characteristic
268	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
269	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
270	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
271	C₂²×C₃₁	#568	1	1	yes	characteristic
272	C₄×C₃₁	—	2	2	no	—
273	C₄×C₃₁	—	2	2	no	—
274	C₄×C₃₁	—	2	2	no	—
275	C₄×C₃₁	—	2	2	no	—
276	C₄×C₃₁	—	2	2	no	—
277	C₄×C₃₁	—	2	2	no	—
278	C₄×C₃₁	—	2	2	no	—
279	C₄×C₃₁	—	4	1	no	—
280	C₄×C₃₁	—	4	1	no	—
281	C₄×C₃₁	—	4	1	no	—
282	C₄×C₃₁	—	4	1	no	—
283	C₄×C₃₁	—	4	1	no	—
284	C₄×C₃₁	#551	1	2	yes	—
285	C₄×C₃₁	#553	1	2	yes	—
286	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
287	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
289	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
290	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
291	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
293	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
294	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
295	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
296	C₂×#97	C₂²×C₃₁	1	2	yes	—
297	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
298	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
299	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
300	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
301	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
302	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
303	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
304	C₄×C₂²⋊C₄	C₄×C₃₁	1	1	yes	characteristic
305	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
306	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
307	C₄³	C₂²×C₃₁	1	1	yes	characteristic
308	#336	C₂²×C₃₁	1	1	yes	characteristic
309	#336	C₂²×C₃₁	1	1	yes	characteristic
310	#336	C₂²×C₃₁	1	1	yes	characteristic
311	#336	C₂²×C₃₁	1	2	yes	—
312	#336	C₂²×C₃₁	1	2	yes	—
313	#336	C₂²×C₃₁	1	2	yes	—
314	#336	C₂²×C₃₁	1	2	yes	—
315	#336	C₂²×C₃₁	1	2	yes	—
316	#336	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×C₃₁	—	4	2	no	—
318	C₂×C₃₁	—	4	2	no	—
319	C₂×C₃₁	#2353	1	1	yes	characteristic
320	C₂×C₃₁	#2920	1	1	yes	characteristic
321	C₂×C₃₁	#2925	1	1	yes	characteristic
322	C₂×C₃₁	#2965	1	1	yes	characteristic
323	C₂×C₃₁	#2970	1	1	yes	characteristic
324	C₂×C₃₁	#2974	1	1	yes	characteristic
325	C₂×C₃₁	#2975	1	1	yes	characteristic
326	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
327	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
328	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
329	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
330	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
331	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
332	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
333	C₂×M₁₆	—	2	4	no	—
334	C₂×M₁₆	—	2	4	no	—
335	C₂×M₁₆	—	4	2	no	—
336	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
337	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
338	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
339	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	—	2	2	no	—
342	C₄×D₄	—	2	4	no	—
343	C₄×D₄	—	2	4	no	—
344	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
345	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
346	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
347	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
348	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
349	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
350	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
351	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
352	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
353	C₂×C₄²	—	4	2	no	—
354	C₂×C₄²	—	4	2	no	—
355	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
356	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
357	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
358	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
359	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
360	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
361	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
362	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
363	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
364	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
365	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
366	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
368	C₄⋊C₈	—	4	2	no	—
369	C₂²⋊C₈	—	2	4	no	—
370	C₂²⋊C₈	—	2	4	no	—
371	C₂²⋊C₈	—	4	2	no	—
372	#97	—	4	2	no	—
373	C₄×C₈	—	2	4	no	—
374	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
375	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
376	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
377	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
378	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
379	#95	C₂×C₄×C₃₁	1	1	yes	characteristic
380	C₃₁	#22786	1	1	yes	characteristic, direct product factor, Sylow
381	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
382	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	4	1	no	—
399	C₂²×C₄	—	4	1	no	—
400	C₂²×C₄	—	4	1	no	—
401	C₂²×C₄	—	4	1	no	—
402	C₂²×C₄	—	4	1	no	—
403	C₂²×C₄	—	4	1	no	—
404	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
406	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
407	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
408	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
409	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
410	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
411	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
412	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
413	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
414	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
415	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
416	C₂²×C₄	#41×C₃₁	1	2	yes	—
417	C₂²×C₄	#41×C₃₁	1	2	yes	—
418	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
419	M₁₆	—	4	8	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	2	2	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	#41×C₃₁	1	4	yes	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₂²⋊C₄	—	2	2	no	—
436	C₂²⋊C₄	—	2	2	no	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	—	2	2	no	—
440	C₂²⋊C₄	M₁₆×C₃₁	1	4	yes	complemented
441	C₄²	—	2	2	no	—
442	C₄²	—	2	2	no	—
443	C₄²	—	2	2	no	—
444	C₄²	—	2	1	no	—
445	C₄²	—	2	1	no	—
446	C₄²	—	2	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	2	no	—
449	C₄²	—	4	1	no	—
450	C₄²	—	4	1	no	—
451	C₄²	—	4	1	no	—
452	C₄²	—	4	1	no	—
453	C₄²	—	4	1	no	—
454	C₄²	M₁₆×C₃₁	1	4	yes	complemented
455	C₂³	—	2	1	no	—
456	C₂³	—	2	1	no	—
457	C₂³	—	2	1	no	—
458	C₂³	—	2	1	no	—
459	C₂³	—	2	1	no	—
460	C₂³	—	2	1	no	—
461	C₂³	—	2	1	no	—
462	C₂³	—	2	1	no	—
463	C₂³	—	2	1	no	—
464	C₂³	—	2	1	no	—
465	C₂³	—	2	1	no	—
466	C₂³	—	2	1	no	—
467	C₂³	C₃₁×#117	1	2	yes	—
468	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
469	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
470	D₄	—	2	8	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	1	no	—
485	C₂×C₄	—	2	4	no	—
486	C₂×C₄	—	2	4	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	1	no	—
492	C₂×C₄	—	2	1	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	4	4	no	—
503	C₂×C₄	—	4	4	no	—
504	C₂×C₄	—	4	4	no	—
505	C₂×C₄	—	4	4	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	C₃₁×#117	1	2	yes	—
509	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
510	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
511	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
512	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
513	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
514	C₂×C₄	C₃₁×#131	1	2	yes	—
515	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
516	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
517	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
518	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
519	C₈	—	2	4	no	—
520	C₈	—	4	2	no	—
521	C₈	—	4	2	no	—
522	C₈	—	4	2	no	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	2	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	2	2	no	—
527	C₂²	—	2	2	no	—
528	C₂²	—	2	2	no	—
529	C₂²	—	4	1	no	—
530	C₂²	—	4	1	no	—
531	C₂²	—	4	1	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	1	no	—
535	C₂²	—	4	1	no	—
536	C₂²	—	4	1	no	—
537	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
538	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
539	C₂²	C₃₁×#518	1	1	yes	characteristic
540	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
541	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
542	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
543	C₂²	C₃₁×#568	1	1	yes	characteristic
544	C₄	—	2	2	no	—
545	C₄	—	2	2	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	2	no	—
548	C₄	—	2	2	no	—
549	C₄	—	2	2	no	—
550	C₄	—	2	2	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	1	no	—
555	C₄	—	4	1	no	—
556	C₄	C₃₁×#551	1	2	yes	—
557	C₄	C₃₁×#553	1	2	yes	—
558	C₂	—	4	2	no	—
559	C₂	—	4	2	no	—
560	C₂	C₃₁×#2353	1	1	yes	characteristic
561	C₂	C₃₁×#2920	1	1	yes	characteristic
562	C₂	C₃₁×#2925	1	1	yes	characteristic
563	C₂	C₃₁×#2965	1	1	yes	characteristic
564	C₂	C₃₁×#2970	1	1	yes	characteristic
565	C₂	C₃₁×#2974	1	1	yes	characteristic
566	C₂	C₃₁×#2975	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂²×D₄×C₃₁	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1923	C₂	semidirect
3. C₃₁×#1917	C₂	semidirect
5. C₃₁×#1839	C₂	semidirect
6. C₃₁×#1817	C₂	semidirect
7. C₃₁×#1797	C₂	semidirect
8. C₃₁×#1753	C₂	semidirect
9. C₃₁×#1749	C₂	semidirect
10. C₃₁×#1744	C₂	semidirect
11. C₃₁×#1728	C₂	semidirect
156. C₂²⋊C₄×C₃₁	M₁₆	semidirect
170. C₄²×C₃₁	M₁₆	semidirect
171. #22786	C₃₁	direct (N × H)
241. #1923	C₂×C₃₁	semidirect
242. #1917	C₂×C₃₁	semidirect
244. #1839	C₂×C₃₁	semidirect
245. #1817	C₂×C₃₁	semidirect
246. #1797	C₂×C₃₁	semidirect
247. #1753	C₂×C₃₁	semidirect
248. #1749	C₂×C₃₁	semidirect
249. #1744	C₂×C₃₁	semidirect
250. #1728	C₂×C₃₁	semidirect
380. C₃₁	#22786	direct (N × H)
440. C₂²⋊C₄	M₁₆×C₃₁	semidirect
454. C₄²	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄²×D₄×C₃₁	2
C₃₁×#1923	2
C₃₁×#1917	2
C₃₁×#1870	2
C₃₁×#1839	2
C₃₁×#1817	2
C₃₁×#1797	2
C₃₁×#1753	2
C₃₁×#1749	2
C₃₁×#1744	2
C₃₁×#1728	2
#22786	31

Subgroups — C31×#22786 #21479545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22786 #21479545