Subgroups — C₃₁×#25032 #21481791

Order 7936.

906 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2420	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2073	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2034	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#121	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#466	—	2	4	no	—
17	C₄×D₈×C₃₁	—	2	4	no	—
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#106	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#397	—	2	2	no	—
29	C₃₁×#394	—	2	2	no	—
30	C₃₁×#390	C₂²	1	2	yes	—
31	C₃₁×#389	C₂²	1	2	yes	—
32	C₃₁×#386	—	2	2	no	—
33	C₃₁×#382	C₂²	1	2	yes	—
34	C₃₁×#381	—	2	2	no	—
35	C₃₁×#380	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₃₁	—	2	2	no	—
42	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
43	C₂³×C₄×C₃₁	—	2	2	no	—
44	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×D₈×C₃₁	—	2	2	no	—
46	C₂×D₈×C₃₁	—	2	2	no	—
47	C₂×D₈×C₃₁	D₄	1	4	yes	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×#121	—	2	4	no	—
56	C₃₁×#121	—	4	2	no	—
57	C₄×D₄×C₃₁	—	2	4	no	—
58	C₄×D₄×C₃₁	—	4	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×#107	—	2	2	no	—
73	C₃₁×#107	—	2	2	no	—
74	C₃₁×#107	—	2	2	no	—
75	C₃₁×#106	—	4	2	no	—
76	C₃₁×#102	—	2	4	no	—
77	C₃₁×#102	—	2	4	no	—
78	C₃₁×#102	—	2	4	no	—
79	C₃₁×C₂²⋊C₈	—	2	2	no	—
80	C₄×C₈×C₃₁	—	2	2	no	—
81	C₄×C₈×C₃₁	—	2	2	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	—	2	2	no	—
86	C₃₁×#95	D₄	1	1	yes	characteristic
87	C₃₁×#95	D₄	1	1	yes	characteristic
88	C₂⁴×C₃₁	—	2	2	no	—
89	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	4	4	no	—
96	C₂×D₄×C₃₁	—	4	4	no	—
97	C₂×D₄×C₃₁	—	8	2	no	—
98	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	1	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	—	4	4	no	—
109	C₂²×C₄×C₃₁	—	4	4	no	—
110	C₂²×C₄×C₃₁	—	8	2	no	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
117	D₈×C₃₁	—	2	8	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	4	4	no	—
127	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₃₁	#41	1	2	yes	—
129	C₂×C₈×C₃₁	#41	1	2	yes	—
130	C₂×C₈×C₃₁	#41	1	2	yes	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	8	2	no	—
138	C₂²⋊C₄×C₃₁	—	2	2	no	—
139	C₂²⋊C₄×C₃₁	—	2	2	no	—
140	C₂²⋊C₄×C₃₁	—	4	4	no	—
141	C₂²⋊C₄×C₃₁	—	4	4	no	—
142	C₂²⋊C₄×C₃₁	—	8	2	no	—
143	C₂²⋊C₄×C₃₁	—	8	2	no	—
144	C₂²⋊C₄×C₃₁	—	8	2	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	#25032	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	—	2	1	no	—
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	2	1	no	—
155	C₂³×C₃₁	—	2	1	no	—
156	C₂³×C₃₁	—	2	4	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	—	4	2	no	—
161	C₂³×C₃₁	—	4	2	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	#135	1	2	yes	—
164	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
165	D₄×C₃₁	—	2	4	no	—
166	D₄×C₃₁	—	2	4	no	—
167	D₄×C₃₁	—	4	2	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	8	8	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	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	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	8	8	no	—
198	C₂×C₄×C₃₁	#135	1	2	yes	—
199	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
203	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
205	C₈×C₃₁	—	2	2	no	—
206	C₈×C₃₁	—	4	1	no	—
207	C₈×C₃₁	—	4	1	no	—
208	C₈×C₃₁	#123	1	4	yes	—
209	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
210	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#2420	C₂×C₃₁	1	2	yes	complemented
212	#2073	C₂×C₃₁	1	2	yes	complemented
213	#2034	C₂×C₃₁	1	2	yes	complemented
214	#1970	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1942	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1864	C₂×C₃₁	1	2	yes	complemented
217	#1846	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#1808	C₂×C₃₁	1	1	yes	characteristic, complemented
219	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
220	C₂²×C₃₁	—	2	2	no	—
221	C₂²×C₃₁	—	2	2	no	—
222	C₂²×C₃₁	—	2	2	no	—
223	C₂²×C₃₁	—	4	1	no	—
224	C₂²×C₃₁	—	4	1	no	—
225	C₂²×C₃₁	—	4	1	no	—
226	C₂²×C₃₁	—	4	1	no	—
227	C₂²×C₃₁	—	4	4	no	—
228	C₂²×C₃₁	—	4	4	no	—
229	C₂²×C₃₁	—	4	4	no	—
230	C₂²×C₃₁	—	8	2	no	—
231	C₂²×C₃₁	—	8	2	no	—
232	C₂²×C₃₁	—	8	2	no	—
233	C₂²×C₃₁	—	8	2	no	—
234	C₂²×C₃₁	#532	1	1	yes	characteristic
235	C₂²×C₃₁	D₄²	1	1	yes	characteristic
236	C₂²×C₃₁	#547	1	1	yes	characteristic
237	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
238	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
239	C₂²×C₃₁	#576	1	1	yes	characteristic
240	C₂²×C₃₁	#576	1	1	yes	characteristic
241	C₄×C₃₁	—	2	2	no	—
242	C₄×C₃₁	—	4	1	no	—
243	C₄×C₃₁	—	4	1	no	—
244	C₄×C₃₁	—	4	1	no	—
245	C₄×C₃₁	—	4	1	no	—
246	C₄×C₃₁	—	8	2	no	—
247	C₄×C₃₁	—	8	2	no	—
248	C₄×C₃₁	—	8	2	no	—
249	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
250	C₄×C₃₁	#542	1	1	yes	characteristic
251	C₄×C₃₁	#547	1	1	yes	characteristic
252	C₄×C₃₁	#553	1	1	yes	characteristic
253	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
254	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×#121	C₂²×C₃₁	1	2	yes	—
256	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
257	#466	—	2	4	no	—
258	C₄×D₈	—	2	4	no	—
259	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
261	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×#106	C₂²×C₃₁	1	2	yes	—
263	C₂×#102	C₂²×C₃₁	1	2	yes	—
264	C₂×#102	C₂²×C₃₁	1	2	yes	—
265	C₂×#102	C₂²×C₃₁	1	2	yes	—
266	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
269	#397	—	2	2	no	—
270	#394	—	2	2	no	—
271	#390	C₂²×C₃₁	1	2	yes	—
272	#389	C₂²×C₃₁	1	2	yes	—
273	#386	—	2	2	no	—
274	#382	C₂²×C₃₁	1	2	yes	—
275	#381	—	2	2	no	—
276	#380	C₂²×C₃₁	1	2	yes	—
277	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
278	#340	C₂²×C₃₁	1	2	yes	—
279	#340	C₂²×C₃₁	1	2	yes	—
280	#336	C₂²×C₃₁	1	1	yes	characteristic
281	#336	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×C₃₁	—	4	2	no	—
283	C₂×C₃₁	—	8	4	no	—
284	C₂×C₃₁	#2422	1	1	yes	characteristic
285	C₂×C₃₁	#3168	1	1	yes	characteristic
286	C₂×C₃₁	#3176	1	1	yes	characteristic
287	C₂×C₃₁	#3280	1	1	yes	characteristic
288	C₂×C₃₁	#3280	1	1	yes	characteristic
289	C₂×C₃₁	#3339	1	1	yes	characteristic
290	C₂×C₃₁	#3341	1	1	yes	characteristic
291	C₂²×D₄	—	2	2	no	—
292	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
293	C₂³×C₄	—	2	2	no	—
294	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
295	C₂×D₈	—	2	2	no	—
296	C₂×D₈	—	2	2	no	—
297	C₂×D₈	D₄×C₃₁	1	4	yes	—
298	C₂²×C₈	—	2	2	no	—
299	C₂²×C₈	—	2	2	no	—
300	C₂²×C₈	—	2	2	no	—
301	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
304	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
305	#121	—	2	4	no	—
306	#121	—	4	2	no	—
307	C₄×D₄	—	2	4	no	—
308	C₄×D₄	—	4	2	no	—
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
311	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
312	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
313	C₂×C₂²⋊C₄	—	2	2	no	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₂²⋊C₄	—	2	2	no	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
319	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
321	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
322	#107	—	2	2	no	—
323	#107	—	2	2	no	—
324	#107	—	2	2	no	—
325	#106	—	4	2	no	—
326	#102	—	2	4	no	—
327	#102	—	2	4	no	—
328	#102	—	2	4	no	—
329	C₂²⋊C₈	—	2	2	no	—
330	C₄×C₈	—	2	2	no	—
331	C₄×C₈	—	2	2	no	—
332	#95	—	2	2	no	—
333	#95	—	2	2	no	—
334	#95	—	2	2	no	—
335	#95	—	2	2	no	—
336	#95	D₄×C₃₁	1	1	yes	characteristic
337	#95	D₄×C₃₁	1	1	yes	characteristic
338	C₃₁	#25032	1	1	yes	characteristic, direct product factor, Sylow
339	C₂⁴	—	2	2	no	—
340	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	4	4	no	—
347	C₂×D₄	—	4	4	no	—
348	C₂×D₄	—	8	2	no	—
349	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	4	4	no	—
360	C₂²×C₄	—	4	4	no	—
361	C₂²×C₄	—	8	2	no	—
362	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
368	D₈	—	2	8	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	4	4	no	—
378	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
379	C₂×C₈	#41×C₃₁	1	2	yes	—
380	C₂×C₈	#41×C₃₁	1	2	yes	—
381	C₂×C₈	#41×C₃₁	1	2	yes	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	8	2	no	—
389	C₂²⋊C₄	—	2	2	no	—
390	C₂²⋊C₄	—	2	2	no	—
391	C₂²⋊C₄	—	4	4	no	—
392	C₂²⋊C₄	—	4	4	no	—
393	C₂²⋊C₄	—	8	2	no	—
394	C₂²⋊C₄	—	8	2	no	—
395	C₂²⋊C₄	—	8	2	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	2	1	no	—
398	C₄²	—	2	1	no	—
399	C₄²	—	2	1	no	—
400	C₂³	—	2	1	no	—
401	C₂³	—	2	1	no	—
402	C₂³	—	2	1	no	—
403	C₂³	—	2	1	no	—
404	C₂³	—	2	1	no	—
405	C₂³	—	2	1	no	—
406	C₂³	—	2	4	no	—
407	C₂³	—	4	2	no	—
408	C₂³	—	4	2	no	—
409	C₂³	—	4	2	no	—
410	C₂³	—	4	2	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	4	2	no	—
413	C₂³	C₃₁×#135	1	2	yes	—
414	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
415	D₄	—	2	4	no	—
416	D₄	—	2	4	no	—
417	D₄	—	4	2	no	—
418	D₄	—	4	2	no	—
419	D₄	—	8	8	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	8	8	no	—
448	C₂×C₄	C₃₁×#135	1	2	yes	—
449	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
450	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
451	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
452	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
453	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
454	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
455	C₈	—	2	2	no	—
456	C₈	—	4	1	no	—
457	C₈	—	4	1	no	—
458	C₈	C₃₁×#123	1	4	yes	—
459	C₂²	—	2	2	no	—
460	C₂²	—	2	2	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	4	1	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	—	4	1	no	—
466	C₂²	—	4	4	no	—
467	C₂²	—	4	4	no	—
468	C₂²	—	4	4	no	—
469	C₂²	—	8	2	no	—
470	C₂²	—	8	2	no	—
471	C₂²	—	8	2	no	—
472	C₂²	—	8	2	no	—
473	C₂²	C₃₁×#532	1	1	yes	characteristic
474	C₂²	D₄²×C₃₁	1	1	yes	characteristic
475	C₂²	C₃₁×#547	1	1	yes	characteristic
476	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
477	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
478	C₂²	C₃₁×#576	1	1	yes	characteristic
479	C₂²	C₃₁×#576	1	1	yes	characteristic
480	C₄	—	2	2	no	—
481	C₄	—	4	1	no	—
482	C₄	—	4	1	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	1	no	—
485	C₄	—	8	2	no	—
486	C₄	—	8	2	no	—
487	C₄	—	8	2	no	—
488	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
489	C₄	C₃₁×#542	1	1	yes	characteristic
490	C₄	C₃₁×#547	1	1	yes	characteristic
491	C₄	C₃₁×#553	1	1	yes	characteristic
492	C₂	—	4	2	no	—
493	C₂	—	8	4	no	—
494	C₂	C₃₁×#2422	1	1	yes	characteristic
495	C₂	C₃₁×#3168	1	1	yes	characteristic
496	C₂	C₃₁×#3176	1	1	yes	characteristic
497	C₂	C₃₁×#3280	1	1	yes	characteristic
498	C₂	C₃₁×#3280	1	1	yes	characteristic
499	C₂	C₃₁×#3339	1	1	yes	characteristic
500	C₂	C₃₁×#3341	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₂²⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₈×C₃₁	C₂	semidirect
3. C₃₁×#2420	C₂	semidirect
4. C₃₁×#2073	C₂	semidirect
5. C₃₁×#2034	C₂	semidirect
6. C₃₁×#1970	C₂	semidirect
7. C₃₁×#1942	C₂	semidirect
8. C₃₁×#1864	C₂	semidirect
9. C₃₁×#1846	C₂	semidirect
10. C₃₁×#1808	C₂	semidirect
11. C₈×C₂²⋊C₄×C₃₁	C₂	semidirect
149. #25032	C₃₁	direct (N × H)
210. C₂×C₄×D₈	C₂×C₃₁	semidirect
211. #2420	C₂×C₃₁	semidirect
212. #2073	C₂×C₃₁	semidirect
213. #2034	C₂×C₃₁	semidirect
214. #1970	C₂×C₃₁	semidirect
215. #1942	C₂×C₃₁	semidirect
216. #1864	C₂×C₃₁	semidirect
217. #1846	C₂×C₃₁	semidirect
218. #1808	C₂×C₃₁	semidirect
219. C₈×C₂²⋊C₄	C₂×C₃₁	semidirect
338. C₃₁	#25032	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#466	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2420	2
C₃₁×#2073	2
C₃₁×#2034	2
C₃₁×#1970	2
C₃₁×#1942	2
C₃₁×#1864	2
C₃₁×#1846	2
C₃₁×#1808	2
C₈×C₂²⋊C₄×C₃₁	2
#25032	31

Subgroups — C31×#25032 #21481791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25032 #21481791