Subgroups — C₃₁×#23334 #21480093

Order 7936.

828 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1955	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1924	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1917	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1870	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1870	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1851	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1817	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×#121	C₄	1	1	yes	characteristic
12	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
14	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
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×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×#97	C₂²	1	2	yes	—
24	C₃₁×#390	C₄	1	1	yes	characteristic
25	C₃₁×#389	C₄	1	1	yes	characteristic
26	C₃₁×#388	C₄	1	2	yes	—
27	C₃₁×#386	C₄	1	2	yes	—
28	C₃₁×#380	C₄	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₃₁×#336	C₂²	1	2	yes	—
32	C₃₁×#336	C₂²	1	1	yes	characteristic
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	1	yes	characteristic
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
38	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
43	C₂×M₁₆×C₃₁	—	4	2	no	—
44	C₂×M₁₆×C₃₁	—	4	2	no	—
45	C₂×M₁₆×C₃₁	—	4	2	no	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₃₁×#121	—	2	4	no	—
51	C₄×D₄×C₃₁	—	4	2	no	—
52	C₄×D₄×C₃₁	—	4	2	no	—
53	C₄×D₄×C₃₁	—	4	2	no	—
54	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
59	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
60	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₃₁	—	4	2	no	—
64	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₃₁×C₄⋊C₈	—	4	2	no	—
70	C₃₁×C₄⋊C₈	—	4	2	no	—
71	C₃₁×C₂²⋊C₈	—	4	2	no	—
72	C₃₁×C₂²⋊C₈	—	4	1	no	—
73	C₃₁×C₂²⋊C₈	—	4	2	no	—
74	C₃₁×C₂²⋊C₈	—	4	2	no	—
75	C₃₁×C₂²⋊C₈	—	4	2	no	—
76	C₃₁×C₂²⋊C₈	—	4	1	no	—
77	C₃₁×#97	—	4	2	no	—
78	C₃₁×#95	C₂×C₄	1	2	yes	—
79	C₃₁×#95	C₂×C₄	1	2	yes	—
80	C₃₁×#95	C₂×C₄	1	2	yes	—
81	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
82	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×D₄×C₃₁	—	2	4	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	4	1	no	—
86	C₂×D₄×C₃₁	—	4	2	no	—
87	C₂×D₄×C₃₁	—	4	1	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂²×C₄×C₃₁	—	2	1	no	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	4	1	no	—
100	C₂²×C₄×C₃₁	—	4	1	no	—
101	C₂²×C₄×C₃₁	—	4	1	no	—
102	C₂²×C₄×C₃₁	—	4	1	no	—
103	C₂²×C₄×C₃₁	—	4	1	no	—
104	C₂²×C₄×C₃₁	M₁₆	1	4	yes	complemented
105	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
114	M₁₆×C₃₁	—	4	8	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	4	1	no	—
129	C₄⋊C₄×C₃₁	—	4	1	no	—
130	C₄⋊C₄×C₃₁	—	4	2	no	—
131	C₂²⋊C₄×C₃₁	—	2	4	no	—
132	C₂²⋊C₄×C₃₁	—	4	2	no	—
133	C₂²⋊C₄×C₃₁	—	4	2	no	—
134	C₂²⋊C₄×C₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₃₁	—	4	2	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	2	2	no	—
139	C₄²×C₃₁	—	2	2	no	—
140	C₄²×C₃₁	—	4	1	no	—
141	C₄²×C₃₁	—	4	1	no	—
142	#23334	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	—	2	1	no	—
144	C₂³×C₃₁	—	2	1	no	—
145	C₂³×C₃₁	—	2	1	no	—
146	C₂³×C₃₁	—	2	1	no	—
147	C₂³×C₃₁	—	2	1	no	—
148	C₂³×C₃₁	—	2	1	no	—
149	C₂³×C₃₁	—	2	2	no	—
150	C₂³×C₃₁	—	2	2	no	—
151	C₂³×C₃₁	—	2	2	no	—
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	2	2	no	—
155	C₂³×C₃₁	—	2	2	no	—
156	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
157	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
158	D₄×C₃₁	—	4	4	no	—
159	D₄×C₃₁	—	4	8	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	4	4	no	—
183	C₂×C₄×C₃₁	—	4	4	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
191	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
192	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
193	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
194	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
195	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
196	C₈×C₃₁	—	4	2	no	—
197	C₈×C₃₁	—	4	2	no	—
198	C₈×C₃₁	—	4	2	no	—
199	C₈×C₃₁	—	4	2	no	—
200	C₄×#121	C₂×C₃₁	1	1	yes	characteristic
201	#1955	C₂×C₃₁	1	2	yes	complemented
202	#1924	C₂×C₃₁	1	2	yes	complemented
203	#1917	C₂×C₃₁	1	2	yes	complemented
204	#1870	C₂×C₃₁	1	2	yes	complemented
205	#1870	C₂×C₃₁	1	2	yes	complemented
206	#1851	C₂×C₃₁	1	2	yes	complemented
207	#1817	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1797	C₂×C₃₁	1	1	yes	characteristic, complemented
209	C₂²×C₃₁	—	2	2	no	—
210	C₂²×C₃₁	—	2	2	no	—
211	C₂²×C₃₁	—	2	2	no	—
212	C₂²×C₃₁	—	4	1	no	—
213	C₂²×C₃₁	—	4	1	no	—
214	C₂²×C₃₁	—	4	1	no	—
215	C₂²×C₃₁	—	4	1	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	4	2	no	—
220	C₂²×C₃₁	—	4	2	no	—
221	C₂²×C₃₁	—	4	2	no	—
222	C₂²×C₃₁	—	4	2	no	—
223	C₂²×C₃₁	#518	1	1	yes	characteristic
224	C₂²×C₃₁	#518	1	1	yes	characteristic
225	C₂²×C₃₁	#520	1	1	yes	characteristic
226	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
227	C₂²×C₃₁	#568	1	1	yes	characteristic
228	C₂²×C₃₁	#568	1	1	yes	characteristic
229	C₂²×C₃₁	#568	1	1	yes	characteristic
230	C₄×C₃₁	—	2	2	no	—
231	C₄×C₃₁	—	2	2	no	—
232	C₄×C₃₁	—	4	1	no	—
233	C₄×C₃₁	—	4	2	no	—
234	C₄×C₃₁	—	4	2	no	—
235	C₄×C₃₁	—	4	1	no	—
236	C₄×C₃₁	—	4	1	no	—
237	C₄×C₃₁	—	4	2	no	—
238	C₄×C₃₁	#560	1	2	yes	—
239	C₄×C₃₁	#562	1	2	yes	—
240	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
241	C₂×#121	C₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
244	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
246	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
247	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
250	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
251	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
252	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
253	C₂×#97	C₂²×C₃₁	1	2	yes	—
254	#390	C₄×C₃₁	1	1	yes	characteristic
255	#389	C₄×C₃₁	1	1	yes	characteristic
256	#388	C₄×C₃₁	1	2	yes	—
257	#386	C₄×C₃₁	1	2	yes	—
258	#380	C₄×C₃₁	1	1	yes	characteristic
259	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
260	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
261	#336	C₂²×C₃₁	1	2	yes	—
262	#336	C₂²×C₃₁	1	1	yes	characteristic
263	#336	C₂²×C₃₁	1	2	yes	—
264	#336	C₂²×C₃₁	1	1	yes	characteristic
265	#336	C₂²×C₃₁	1	2	yes	—
266	#336	C₂²×C₃₁	1	2	yes	—
267	C₂×C₃₁	—	4	2	no	—
268	C₂×C₃₁	—	8	2	no	—
269	C₂×C₃₁	#2377	1	1	yes	characteristic
270	C₂×C₃₁	#2973	1	1	yes	characteristic
271	C₂×C₃₁	#2975	1	1	yes	characteristic
272	C₂×C₃₁	#2978	1	1	yes	characteristic
273	C₂×C₃₁	#2980	1	1	yes	characteristic
274	C₂×C₃₁	#2989	1	1	yes	characteristic
275	C₂×C₃₁	#2995	1	1	yes	characteristic
276	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
277	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
278	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
279	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
280	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
281	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
282	C₂×M₁₆	—	4	2	no	—
283	C₂×M₁₆	—	4	2	no	—
284	C₂×M₁₆	—	4	2	no	—
285	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
286	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
287	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
288	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
289	#121	—	2	4	no	—
290	C₄×D₄	—	4	2	no	—
291	C₄×D₄	—	4	2	no	—
292	C₄×D₄	—	4	2	no	—
293	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
296	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
297	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
298	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
299	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
300	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
301	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
302	C₂×C₄²	—	4	2	no	—
303	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
305	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
306	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
307	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
308	C₄⋊C₈	—	4	2	no	—
309	C₄⋊C₈	—	4	2	no	—
310	C₂²⋊C₈	—	4	2	no	—
311	C₂²⋊C₈	—	4	1	no	—
312	C₂²⋊C₈	—	4	2	no	—
313	C₂²⋊C₈	—	4	2	no	—
314	C₂²⋊C₈	—	4	2	no	—
315	C₂²⋊C₈	—	4	1	no	—
316	#97	—	4	2	no	—
317	#95	C₂×C₄×C₃₁	1	2	yes	—
318	#95	C₂×C₄×C₃₁	1	2	yes	—
319	#95	C₂×C₄×C₃₁	1	2	yes	—
320	C₃₁	#23334	1	1	yes	characteristic, direct product factor, Sylow
321	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
322	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
323	C₂×D₄	—	2	4	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	4	1	no	—
326	C₂×D₄	—	4	2	no	—
327	C₂×D₄	—	4	1	no	—
328	C₂×D₄	—	4	2	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	4	1	no	—
340	C₂²×C₄	—	4	1	no	—
341	C₂²×C₄	—	4	1	no	—
342	C₂²×C₄	—	4	1	no	—
343	C₂²×C₄	—	4	1	no	—
344	C₂²×C₄	M₁₆×C₃₁	1	4	yes	complemented
345	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
346	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
347	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
348	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
349	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
350	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
354	M₁₆	—	4	8	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	4	1	no	—
369	C₄⋊C₄	—	4	1	no	—
370	C₄⋊C₄	—	4	2	no	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	4	2	no	—
373	C₂²⋊C₄	—	4	2	no	—
374	C₂²⋊C₄	—	4	2	no	—
375	C₂²⋊C₄	—	4	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	2	2	no	—
380	C₄²	—	4	1	no	—
381	C₄²	—	4	1	no	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	1	no	—
386	C₂³	—	2	1	no	—
387	C₂³	—	2	1	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	2	no	—
394	C₂³	—	2	2	no	—
395	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
396	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
397	D₄	—	4	4	no	—
398	D₄	—	4	8	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	4	4	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
430	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
431	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
432	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
433	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
434	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
435	C₈	—	4	2	no	—
436	C₈	—	4	2	no	—
437	C₈	—	4	2	no	—
438	C₈	—	4	2	no	—
439	C₂²	—	2	2	no	—
440	C₂²	—	2	2	no	—
441	C₂²	—	2	2	no	—
442	C₂²	—	4	1	no	—
443	C₂²	—	4	1	no	—
444	C₂²	—	4	1	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	4	2	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	C₃₁×#518	1	1	yes	characteristic
454	C₂²	C₃₁×#518	1	1	yes	characteristic
455	C₂²	C₃₁×#520	1	1	yes	characteristic
456	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
457	C₂²	C₃₁×#568	1	1	yes	characteristic
458	C₂²	C₃₁×#568	1	1	yes	characteristic
459	C₂²	C₃₁×#568	1	1	yes	characteristic
460	C₄	—	2	2	no	—
461	C₄	—	2	2	no	—
462	C₄	—	4	1	no	—
463	C₄	—	4	2	no	—
464	C₄	—	4	2	no	—
465	C₄	—	4	1	no	—
466	C₄	—	4	1	no	—
467	C₄	—	4	2	no	—
468	C₄	C₃₁×#560	1	2	yes	—
469	C₄	C₃₁×#562	1	2	yes	—
470	C₂	—	4	2	no	—
471	C₂	—	8	2	no	—
472	C₂	C₃₁×#2377	1	1	yes	characteristic
473	C₂	C₃₁×#2973	1	1	yes	characteristic
474	C₂	C₃₁×#2975	1	1	yes	characteristic
475	C₂	C₃₁×#2978	1	1	yes	characteristic
476	C₂	C₃₁×#2980	1	1	yes	characteristic
477	C₂	C₃₁×#2989	1	1	yes	characteristic
478	C₂	C₃₁×#2995	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	C₂×C₃₁×#121	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×#1955	C₂	semidirect
3. C₃₁×#1924	C₂	semidirect
4. C₃₁×#1917	C₂	semidirect
5. C₃₁×#1870	C₂	semidirect
6. C₃₁×#1870	C₂	semidirect
7. C₃₁×#1851	C₂	semidirect
8. C₃₁×#1817	C₂	semidirect
9. C₃₁×#1797	C₂	semidirect
104. C₂²×C₄×C₃₁	M₁₆	semidirect
142. #23334	C₃₁	direct (N × H)
201. #1955	C₂×C₃₁	semidirect
202. #1924	C₂×C₃₁	semidirect
203. #1917	C₂×C₃₁	semidirect
204. #1870	C₂×C₃₁	semidirect
205. #1870	C₂×C₃₁	semidirect
206. #1851	C₂×C₃₁	semidirect
207. #1817	C₂×C₃₁	semidirect
208. #1797	C₂×C₃₁	semidirect
320. C₃₁	#23334	direct (N × H)
344. C₂²×C₄	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#121	2
C₃₁×#1955	2
C₃₁×#1924	2
C₃₁×#1917	2
C₃₁×#1870	2
C₃₁×#1851	2
C₃₁×#1817	2
C₃₁×#1797	2
#23334	31

Subgroups — C31×#23334 #21480093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23334 #21480093