Subgroups — C₃₁×#22461 #21479220

Order 7936.

1316 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3584	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2943	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2901	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1976	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1975	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1906	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1881	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1814	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1794	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1765	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#585	C₂²	1	2	yes	—
13	C₃₁×#578	—	2	1	no	—
14	C₃₁×#578	C₄	1	2	yes	complemented
15	C₃₁×#577	—	2	2	no	—
16	C₃₁×#577	C₄	1	4	yes	complemented
17	C₃₁×#576	—	2	1	no	—
18	C₃₁×#576	C₄	1	2	yes	complemented
19	C₃₁×#575	—	2	2	no	—
20	C₃₁×#575	C₂²	1	2	yes	—
21	C₂×C₃₁×#135	—	2	2	no	—
22	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
24	C₃₁×#568	C₂²	1	2	yes	complemented
25	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
26	C₄×#41×C₃₁	C₂²	1	2	yes	complemented
27	C₃₁×#444	—	2	2	no	—
28	C₃₁×#443	—	2	2	no	—
29	C₃₁×#442	—	2	1	no	—
30	C₃₁×#441	—	2	1	no	—
31	C₃₁×#440	—	2	2	no	—
32	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#427	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#421	—	2	4	no	—
35	C₃₁×#421	C₂²	1	2	yes	—
36	C₃₁×#421	C₂²	1	2	yes	complemented
37	C₃₁×#421	C₂²	1	2	yes	complemented
38	C₂×C₃₁×#104	—	2	4	no	—
39	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#104	C₂²	1	2	yes	complemented
41	C₃₁×#416	C₂²	1	1	yes	characteristic
42	C₃₁×#416	C₂²	1	2	yes	complemented
43	C₃₁×#410	C₂²	1	2	yes	complemented
44	C₃₁×#408	C₂²	1	2	yes	complemented
45	C₃₁×#405	C₂²	1	1	yes	characteristic, complemented
46	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
47	C₃₁×#344	C₂²	1	2	yes	complemented
48	C₃₁×#341	C₂²	1	1	yes	characteristic, complemented
49	C₃₁×#339	C₂²	1	1	yes	characteristic, complemented
50	C₃₁×2₋¹⁺⁴	—	2	2	no	—
51	C₃₁×2₋¹⁺⁴	—	2	2	no	—
52	C₃₁×2₋¹⁺⁴	C₂×C₄	1	4	yes	complemented
53	C₃₁×H(5,𝔽₂)	—	2	2	no	—
54	C₃₁×H(5,𝔽₂)	—	2	2	no	—
55	C₃₁×H(5,𝔽₂)	—	4	2	no	—
56	C₃₁×H(5,𝔽₂)	C₂×C₄	1	4	yes	complemented
57	C₂×#41×C₃₁	—	2	4	no	—
58	C₂×#41×C₃₁	—	2	4	no	—
59	C₂×#41×C₃₁	—	4	2	no	—
60	C₂×#41×C₃₁	C₂³	1	2	yes	—
61	C₂×#41×C₃₁	C₂³	1	2	yes	—
62	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×#137	—	2	2	no	—
64	C₃₁×#137	—	2	4	no	—
65	C₃₁×#137	—	2	4	no	—
66	C₃₁×#137	C₂×C₄	1	4	yes	complemented
67	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
68	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
69	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
70	C₃₁×AGL(1,ℤ/8)	C₂×C₄	1	4	yes	complemented
71	C₃₁×#135	—	2	4	no	—
72	C₃₁×#135	—	2	2	no	—
73	C₃₁×#135	—	2	4	no	—
74	C₃₁×#135	—	2	2	no	—
75	C₃₁×#135	—	2	1	no	—
76	C₃₁×#135	—	2	1	no	—
77	C₃₁×#135	—	2	1	no	—
78	C₃₁×#135	—	2	1	no	—
79	C₃₁×#135	C₂×C₄	1	2	yes	—
80	C₃₁×#135	C₂×C₄	1	2	yes	—
81	C₃₁×#135	D₄	1	2	yes	—
82	C₃₁×#135	D₄	1	2	yes	—
83	C₂×Q₁₆×C₃₁	—	2	2	no	—
84	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
85	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
86	C₂×SD₁₆×C₃₁	—	2	4	no	—
87	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	complemented
88	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
89	C₂×D₈×C₃₁	—	2	2	no	—
90	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
91	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
92	C₃₁×#131	—	2	1	no	—
93	C₃₁×#131	—	2	1	no	—
94	C₃₁×#131	—	2	4	no	—
95	C₃₁×#131	—	4	2	no	—
96	C₃₁×#131	C₂×C₄	1	2	yes	complemented
97	C₃₁×#131	C₂×C₄	1	2	yes	complemented
98	C₂×M₁₆×C₃₁	—	2	1	no	—
99	C₂×M₁₆×C₃₁	—	2	4	no	—
100	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
101	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
103	C₂²×C₈×C₃₁	—	2	1	no	—
104	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
105	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
106	C₄×Q₈×C₃₁	—	2	2	no	—
107	C₄×D₄×C₃₁	—	2	2	no	—
108	C₄×D₄×C₃₁	—	4	2	no	—
109	C₃₁×#117	—	2	2	no	—
110	C₃₁×#117	C₂³	1	1	yes	characteristic
111	C₃₁×#117	C₂³	1	2	yes	—
112	C₂×C₄²×C₃₁	—	2	2	no	—
113	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
114	C₃₁×#108	—	2	1	no	—
115	C₃₁×#108	—	2	1	no	—
116	C₃₁×#107	—	2	1	no	—
117	C₃₁×#106	—	2	1	no	—
118	C₃₁×#104	—	2	4	no	—
119	C₃₁×#104	—	2	4	no	—
120	C₃₁×#104	—	2	4	no	—
121	C₃₁×#104	—	2	2	no	—
122	C₃₁×#104	—	2	2	no	—
123	C₃₁×#104	D₄	1	4	yes	—
124	C₃₁×#103	—	2	2	no	—
125	C₃₁×#103	—	2	1	no	—
126	C₃₁×#102	—	2	2	no	—
127	C₃₁×#102	—	2	1	no	—
128	C₃₁×#99	—	4	2	no	—
129	C₃₁×C₂²⋊C₈	—	4	2	no	—
130	C₃₁×#97	—	2	1	no	—
131	C₃₁×#97	—	2	1	no	—
132	C₃₁×#97	—	2	1	no	—
133	C₄×C₈×C₃₁	—	2	1	no	—
134	#41×C₃₁	—	2	1	no	—
135	#41×C₃₁	—	2	2	no	—
136	#41×C₃₁	—	2	2	no	—
137	#41×C₃₁	—	2	1	no	—
138	#41×C₃₁	—	2	4	no	—
139	#41×C₃₁	—	2	4	no	—
140	#41×C₃₁	—	2	4	no	—
141	#41×C₃₁	—	2	4	no	—
142	#41×C₃₁	—	2	2	no	—
143	#41×C₃₁	—	2	2	no	—
144	#41×C₃₁	—	2	2	no	—
145	#41×C₃₁	—	2	2	no	—
146	#41×C₃₁	—	4	4	no	—
147	#41×C₃₁	—	4	4	no	—
148	#41×C₃₁	—	4	4	no	—
149	#41×C₃₁	—	4	4	no	—
150	#41×C₃₁	—	4	2	no	—
151	#41×C₃₁	—	4	2	no	—
152	#41×C₃₁	C₂²×C₄	1	4	yes	—
153	#41×C₃₁	C₂²×C₄	1	2	yes	—
154	#41×C₃₁	C₂×D₄	1	4	yes	—
155	#41×C₃₁	C₂×D₄	1	2	yes	—
156	C₂×Q₈×C₃₁	—	2	4	no	—
157	C₂×Q₈×C₃₁	—	2	4	no	—
158	C₂×Q₈×C₃₁	—	4	2	no	—
159	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
160	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
161	C₂×Q₈×C₃₁	#41	1	1	yes	characteristic
162	C₂×D₄×C₃₁	—	2	2	no	—
163	C₂×D₄×C₃₁	—	2	1	no	—
164	C₂×D₄×C₃₁	—	2	4	no	—
165	C₂×D₄×C₃₁	—	2	4	no	—
166	C₂×D₄×C₃₁	—	2	2	no	—
167	C₂×D₄×C₃₁	—	4	4	no	—
168	C₂×D₄×C₃₁	—	4	2	no	—
169	C₂×D₄×C₃₁	—	4	4	no	—
170	C₂×D₄×C₃₁	—	4	4	no	—
171	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
172	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₂×D₄×C₃₁	#41	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	—	2	2	no	—
175	C₂²×C₄×C₃₁	—	2	1	no	—
176	C₂²×C₄×C₃₁	—	2	1	no	—
177	C₂²×C₄×C₃₁	—	2	2	no	—
178	C₂²×C₄×C₃₁	—	4	4	no	—
179	C₂²×C₄×C₃₁	—	4	2	no	—
180	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
181	Q₁₆×C₃₁	—	2	4	no	—
182	Q₁₆×C₃₁	—	2	1	no	—
183	Q₁₆×C₃₁	—	2	1	no	—
184	Q₁₆×C₃₁	C₂²⋊C₄	1	2	yes	—
185	Q₁₆×C₃₁	C₂²⋊C₄	1	2	yes	—
186	SD₁₆×C₃₁	—	2	4	no	—
187	SD₁₆×C₃₁	—	2	4	no	—
188	SD₁₆×C₃₁	—	2	2	no	—
189	SD₁₆×C₃₁	—	2	2	no	—
190	SD₁₆×C₃₁	C₂²⋊C₄	1	4	yes	—
191	SD₁₆×C₃₁	C₂²⋊C₄	1	4	yes	—
192	D₈×C₃₁	—	2	4	no	—
193	D₈×C₃₁	—	2	1	no	—
194	D₈×C₃₁	—	2	1	no	—
195	D₈×C₃₁	C₂²⋊C₄	1	2	yes	—
196	D₈×C₃₁	C₂²⋊C₄	1	2	yes	—
197	M₁₆×C₃₁	—	2	1	no	—
198	M₁₆×C₃₁	—	2	1	no	—
199	M₁₆×C₃₁	—	2	2	no	—
200	M₁₆×C₃₁	—	2	2	no	—
201	M₁₆×C₃₁	—	2	2	no	—
202	M₁₆×C₃₁	—	2	2	no	—
203	M₁₆×C₃₁	—	4	4	no	—
204	M₁₆×C₃₁	C₂²×C₄	1	2	yes	—
205	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
206	C₂×C₈×C₃₁	—	2	1	no	—
207	C₂×C₈×C₃₁	—	2	1	no	—
208	C₂×C₈×C₃₁	—	2	2	no	—
209	C₂×C₈×C₃₁	—	2	2	no	—
210	C₂×C₈×C₃₁	—	2	1	no	—
211	C₂×C₈×C₃₁	—	2	1	no	—
212	C₂×C₈×C₃₁	—	2	1	no	—
213	C₂×C₈×C₃₁	—	2	2	no	—
214	C₂×C₈×C₃₁	—	4	4	no	—
215	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
216	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
217	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
218	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
219	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
220	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
221	C₄⋊C₄×C₃₁	—	2	2	no	—
222	C₄⋊C₄×C₃₁	—	2	1	no	—
223	C₄⋊C₄×C₃₁	—	4	2	no	—
224	C₂²⋊C₄×C₃₁	—	2	1	no	—
225	C₂²⋊C₄×C₃₁	—	2	2	no	—
226	C₂²⋊C₄×C₃₁	—	4	2	no	—
227	C₄²×C₃₁	—	2	1	no	—
228	C₄²×C₃₁	—	2	1	no	—
229	C₄²×C₃₁	—	2	2	no	—
230	C₄²×C₃₁	—	4	2	no	—
231	C₄²×C₃₁	C₂×D₄	1	2	yes	—
232	#22461	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
233	C₂³×C₃₁	—	2	2	no	—
234	C₂³×C₃₁	—	2	1	no	—
235	C₂³×C₃₁	—	2	2	no	—
236	C₂³×C₃₁	—	4	4	no	—
237	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
238	Q₈×C₃₁	—	2	1	no	—
239	Q₈×C₃₁	—	2	2	no	—
240	Q₈×C₃₁	—	2	2	no	—
241	Q₈×C₃₁	—	2	2	no	—
242	Q₈×C₃₁	—	4	4	no	—
243	Q₈×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
244	Q₈×C₃₁	C₄×D₄	1	2	yes	—
245	D₄×C₃₁	—	2	2	no	—
246	D₄×C₃₁	—	2	4	no	—
247	D₄×C₃₁	—	2	2	no	—
248	D₄×C₃₁	—	2	4	no	—
249	D₄×C₃₁	—	2	2	no	—
250	D₄×C₃₁	—	2	2	no	—
251	D₄×C₃₁	—	2	1	no	—
252	D₄×C₃₁	—	2	2	no	—
253	D₄×C₃₁	—	4	4	no	—
254	D₄×C₃₁	—	4	4	no	—
255	D₄×C₃₁	—	4	4	no	—
256	D₄×C₃₁	—	4	2	no	—
257	D₄×C₃₁	—	4	2	no	—
258	D₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
259	D₄×C₃₁	C₄×D₄	1	2	yes	—
260	C₂×C₄×C₃₁	—	2	2	no	—
261	C₂×C₄×C₃₁	—	2	1	no	—
262	C₂×C₄×C₃₁	—	2	2	no	—
263	C₂×C₄×C₃₁	—	2	4	no	—
264	C₂×C₄×C₃₁	—	2	4	no	—
265	C₂×C₄×C₃₁	—	2	2	no	—
266	C₂×C₄×C₃₁	—	2	2	no	—
267	C₂×C₄×C₃₁	—	2	2	no	—
268	C₂×C₄×C₃₁	—	4	4	no	—
269	C₂×C₄×C₃₁	—	4	4	no	—
270	C₂×C₄×C₃₁	—	4	4	no	—
271	C₂×C₄×C₃₁	—	4	2	no	—
272	C₂×C₄×C₃₁	—	4	2	no	—
273	C₂×C₄×C₃₁	—	4	2	no	—
274	C₂×C₄×C₃₁	—	4	1	no	—
275	C₂×C₄×C₃₁	—	8	1	no	—
276	C₂×C₄×C₃₁	—	8	4	no	—
277	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
278	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
279	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
282	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
283	C₈×C₃₁	—	2	2	no	—
284	C₈×C₃₁	—	2	2	no	—
285	C₈×C₃₁	—	4	2	no	—
286	C₈×C₃₁	—	4	1	no	—
287	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
288	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
289	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
290	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
291	#3584	C₂×C₃₁	1	1	yes	characteristic
292	#2957	C₂×C₃₁	1	1	yes	characteristic, complemented
293	#2943	C₂×C₃₁	1	1	yes	characteristic, complemented
294	#2901	C₂×C₃₁	1	2	yes	complemented
295	#1976	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#1975	C₂×C₃₁	1	1	yes	characteristic, complemented
297	#1906	C₂×C₃₁	1	2	yes	complemented
298	#1881	C₂×C₃₁	1	2	yes	complemented
299	#1814	C₂×C₃₁	1	1	yes	characteristic, complemented
300	#1794	C₂×C₃₁	1	2	yes	complemented
301	#1765	C₂×C₃₁	1	1	yes	characteristic, complemented
302	C₂²×C₃₁	—	2	2	no	—
303	C₂²×C₃₁	—	2	4	no	—
304	C₂²×C₃₁	—	4	1	no	—
305	C₂²×C₃₁	—	4	2	no	—
306	C₂²×C₃₁	—	8	2	no	—
307	C₂²×C₃₁	—	8	1	no	—
308	C₂²×C₃₁	—	8	2	no	—
309	C₂²×C₃₁	—	16	4	no	—
310	C₂²×C₃₁	#518	1	1	yes	characteristic
311	C₂²×C₃₁	D₄²	1	1	yes	characteristic
312	C₂²×C₃₁	#546	1	1	yes	characteristic
313	C₄×C₃₁	—	2	2	no	—
314	C₄×C₃₁	—	2	4	no	—
315	C₄×C₃₁	—	4	2	no	—
316	C₄×C₃₁	—	4	2	no	—
317	C₄×C₃₁	—	8	1	no	—
318	C₄×C₃₁	—	8	2	no	—
319	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
320	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
321	C₄×C₃₁	D₄²	1	1	yes	characteristic
322	C₄×C₃₁	#546	1	1	yes	characteristic
323	#585	C₂²×C₃₁	1	2	yes	—
324	#578	—	2	1	no	—
325	#578	C₄×C₃₁	1	2	yes	complemented
326	#577	—	2	2	no	—
327	#577	C₄×C₃₁	1	4	yes	complemented
328	#576	—	2	1	no	—
329	#576	C₄×C₃₁	1	2	yes	complemented
330	#575	—	2	2	no	—
331	#575	C₂²×C₃₁	1	2	yes	—
332	C₂×#135	—	2	2	no	—
333	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
334	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
335	#568	C₂²×C₃₁	1	2	yes	complemented
336	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
337	C₄×#41	C₂²×C₃₁	1	2	yes	complemented
338	#444	—	2	2	no	—
339	#443	—	2	2	no	—
340	#442	—	2	1	no	—
341	#441	—	2	1	no	—
342	#440	—	2	2	no	—
343	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic, complemented
344	#427	C₂²×C₃₁	1	1	yes	characteristic, complemented
345	#421	—	2	4	no	—
346	#421	C₂²×C₃₁	1	2	yes	—
347	#421	C₂²×C₃₁	1	2	yes	complemented
348	#421	C₂²×C₃₁	1	2	yes	complemented
349	C₂×#104	—	2	4	no	—
350	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
351	C₂×#104	C₂²×C₃₁	1	2	yes	complemented
352	#416	C₂²×C₃₁	1	1	yes	characteristic
353	#416	C₂²×C₃₁	1	2	yes	complemented
354	#410	C₂²×C₃₁	1	2	yes	complemented
355	#408	C₂²×C₃₁	1	2	yes	complemented
356	#405	C₂²×C₃₁	1	1	yes	characteristic, complemented
357	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
358	#344	C₂²×C₃₁	1	2	yes	complemented
359	#341	C₂²×C₃₁	1	1	yes	characteristic, complemented
360	#339	C₂²×C₃₁	1	1	yes	characteristic, complemented
361	C₂×C₃₁	—	2	1	no	—
362	C₂×C₃₁	—	2	1	no	—
363	C₂×C₃₁	—	2	1	no	—
364	C₂×C₃₁	—	4	4	no	—
365	C₂×C₃₁	—	4	2	no	—
366	C₂×C₃₁	—	8	2	no	—
367	C₂×C₃₁	D₄×C₂²⋊C₄	1	1	yes	characteristic
368	2₋¹⁺⁴	—	2	2	no	—
369	2₋¹⁺⁴	—	2	2	no	—
370	2₋¹⁺⁴	C₂×C₄×C₃₁	1	4	yes	complemented
371	H(5,𝔽₂)	—	2	2	no	—
372	H(5,𝔽₂)	—	2	2	no	—
373	H(5,𝔽₂)	—	4	2	no	—
374	H(5,𝔽₂)	C₂×C₄×C₃₁	1	4	yes	complemented
375	C₂×#41	—	2	4	no	—
376	C₂×#41	—	2	4	no	—
377	C₂×#41	—	4	2	no	—
378	C₂×#41	C₂³×C₃₁	1	2	yes	—
379	C₂×#41	C₂³×C₃₁	1	2	yes	—
380	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
381	#137	—	2	2	no	—
382	#137	—	2	4	no	—
383	#137	—	2	4	no	—
384	#137	C₂×C₄×C₃₁	1	4	yes	complemented
385	AGL(1,ℤ/8)	—	2	2	no	—
386	AGL(1,ℤ/8)	—	2	4	no	—
387	AGL(1,ℤ/8)	—	2	4	no	—
388	AGL(1,ℤ/8)	C₂×C₄×C₃₁	1	4	yes	complemented
389	#135	—	2	4	no	—
390	#135	—	2	2	no	—
391	#135	—	2	4	no	—
392	#135	—	2	2	no	—
393	#135	—	2	1	no	—
394	#135	—	2	1	no	—
395	#135	—	2	1	no	—
396	#135	—	2	1	no	—
397	#135	C₂×C₄×C₃₁	1	2	yes	—
398	#135	C₂×C₄×C₃₁	1	2	yes	—
399	#135	D₄×C₃₁	1	2	yes	—
400	#135	D₄×C₃₁	1	2	yes	—
401	C₂×Q₁₆	—	2	2	no	—
402	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
403	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
404	C₂×SD₁₆	—	2	4	no	—
405	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	complemented
406	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
407	C₂×D₈	—	2	2	no	—
408	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
409	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
410	#131	—	2	1	no	—
411	#131	—	2	1	no	—
412	#131	—	2	4	no	—
413	#131	—	4	2	no	—
414	#131	C₂×C₄×C₃₁	1	2	yes	complemented
415	#131	C₂×C₄×C₃₁	1	2	yes	complemented
416	C₂×M₁₆	—	2	1	no	—
417	C₂×M₁₆	—	2	4	no	—
418	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
419	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
420	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
421	C₂²×C₈	—	2	1	no	—
422	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
423	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
424	C₄×Q₈	—	2	2	no	—
425	C₄×D₄	—	2	2	no	—
426	C₄×D₄	—	4	2	no	—
427	#117	—	2	2	no	—
428	#117	C₂³×C₃₁	1	2	yes	—
429	#117	C₂³×C₃₁	1	1	yes	characteristic
430	C₂×C₄²	—	2	2	no	—
431	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
432	#108	—	2	1	no	—
433	#108	—	2	1	no	—
434	#107	—	2	1	no	—
435	#106	—	2	1	no	—
436	#104	—	2	4	no	—
437	#104	—	2	4	no	—
438	#104	—	2	4	no	—
439	#104	—	2	2	no	—
440	#104	—	2	2	no	—
441	#104	D₄×C₃₁	1	4	yes	—
442	#103	—	2	2	no	—
443	#103	—	2	1	no	—
444	#102	—	2	2	no	—
445	#102	—	2	1	no	—
446	#99	—	4	2	no	—
447	C₂²⋊C₈	—	4	2	no	—
448	#97	—	2	1	no	—
449	#97	—	2	1	no	—
450	#97	—	2	1	no	—
451	C₄×C₈	—	2	1	no	—
452	C₃₁	#22461	1	1	yes	characteristic, direct product factor, Sylow
453	#41	—	2	1	no	—
454	#41	—	2	2	no	—
455	#41	—	2	2	no	—
456	#41	—	2	1	no	—
457	#41	—	2	4	no	—
458	#41	—	2	4	no	—
459	#41	—	2	4	no	—
460	#41	—	2	4	no	—
461	#41	—	2	2	no	—
462	#41	—	2	2	no	—
463	#41	—	2	2	no	—
464	#41	—	2	2	no	—
465	#41	—	4	4	no	—
466	#41	—	4	4	no	—
467	#41	—	4	4	no	—
468	#41	—	4	4	no	—
469	#41	—	4	2	no	—
470	#41	—	4	2	no	—
471	#41	C₂²×C₄×C₃₁	1	4	yes	—
472	#41	C₂²×C₄×C₃₁	1	2	yes	—
473	#41	C₂×D₄×C₃₁	1	4	yes	—
474	#41	C₂×D₄×C₃₁	1	2	yes	—
475	C₂×Q₈	—	2	4	no	—
476	C₂×Q₈	—	2	4	no	—
477	C₂×Q₈	—	4	2	no	—
478	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
479	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
480	C₂×Q₈	#41×C₃₁	1	1	yes	characteristic
481	C₂×D₄	—	2	2	no	—
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	2	4	no	—
485	C₂×D₄	—	2	2	no	—
486	C₂×D₄	—	4	4	no	—
487	C₂×D₄	—	4	2	no	—
488	C₂×D₄	—	4	4	no	—
489	C₂×D₄	—	4	4	no	—
490	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
491	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
492	C₂×D₄	#41×C₃₁	1	1	yes	characteristic
493	C₂²×C₄	—	2	2	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	2	no	—
497	C₂²×C₄	—	4	4	no	—
498	C₂²×C₄	—	4	2	no	—
499	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
500	Q₁₆	—	2	4	no	—
501	Q₁₆	—	2	1	no	—
502	Q₁₆	—	2	1	no	—
503	Q₁₆	C₂²⋊C₄×C₃₁	1	2	yes	—
504	Q₁₆	C₂²⋊C₄×C₃₁	1	2	yes	—
505	SD₁₆	—	2	4	no	—
506	SD₁₆	—	2	4	no	—
507	SD₁₆	—	2	2	no	—
508	SD₁₆	—	2	2	no	—
509	SD₁₆	C₂²⋊C₄×C₃₁	1	4	yes	—
510	SD₁₆	C₂²⋊C₄×C₃₁	1	4	yes	—
511	D₈	—	2	4	no	—
512	D₈	—	2	1	no	—
513	D₈	—	2	1	no	—
514	D₈	C₂²⋊C₄×C₃₁	1	2	yes	—
515	D₈	C₂²⋊C₄×C₃₁	1	2	yes	—
516	M₁₆	—	2	1	no	—
517	M₁₆	—	2	1	no	—
518	M₁₆	—	2	2	no	—
519	M₁₆	—	2	2	no	—
520	M₁₆	—	2	2	no	—
521	M₁₆	—	2	2	no	—
522	M₁₆	—	4	4	no	—
523	M₁₆	C₂²×C₄×C₃₁	1	2	yes	—
524	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
525	C₂×C₈	—	2	1	no	—
526	C₂×C₈	—	2	1	no	—
527	C₂×C₈	—	2	2	no	—
528	C₂×C₈	—	2	2	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	1	no	—
532	C₂×C₈	—	2	2	no	—
533	C₂×C₈	—	4	4	no	—
534	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
535	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
536	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
537	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
538	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
539	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	4	2	no	—
543	C₂²⋊C₄	—	2	1	no	—
544	C₂²⋊C₄	—	2	2	no	—
545	C₂²⋊C₄	—	4	2	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	2	1	no	—
548	C₄²	—	2	2	no	—
549	C₄²	—	4	2	no	—
550	C₄²	C₂×D₄×C₃₁	1	2	yes	—
551	C₂³	—	2	2	no	—
552	C₂³	—	2	1	no	—
553	C₂³	—	2	2	no	—
554	C₂³	—	4	4	no	—
555	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
556	Q₈	—	2	1	no	—
557	Q₈	—	2	2	no	—
558	Q₈	—	2	2	no	—
559	Q₈	—	2	2	no	—
560	Q₈	—	4	4	no	—
561	Q₈	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
562	Q₈	C₄×D₄×C₃₁	1	2	yes	—
563	D₄	—	2	2	no	—
564	D₄	—	2	4	no	—
565	D₄	—	2	2	no	—
566	D₄	—	2	4	no	—
567	D₄	—	2	2	no	—
568	D₄	—	2	2	no	—
569	D₄	—	2	1	no	—
570	D₄	—	2	2	no	—
571	D₄	—	4	4	no	—
572	D₄	—	4	4	no	—
573	D₄	—	4	4	no	—
574	D₄	—	4	2	no	—
575	D₄	—	4	2	no	—
576	D₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
577	D₄	C₄×D₄×C₃₁	1	2	yes	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	4	4	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	2	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	8	1	no	—
594	C₂×C₄	—	8	4	no	—
595	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
596	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
597	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
598	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
599	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
600	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
601	C₈	—	2	2	no	—
602	C₈	—	2	2	no	—
603	C₈	—	4	2	no	—
604	C₈	—	4	1	no	—
605	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
606	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
607	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
608	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
609	C₂²	—	2	2	no	—
610	C₂²	—	2	4	no	—
611	C₂²	—	4	1	no	—
612	C₂²	—	4	2	no	—
613	C₂²	—	8	2	no	—
614	C₂²	—	8	1	no	—
615	C₂²	—	8	2	no	—
616	C₂²	—	16	4	no	—
617	C₂²	C₃₁×#518	1	1	yes	characteristic
618	C₂²	D₄²×C₃₁	1	1	yes	characteristic
619	C₂²	C₃₁×#546	1	1	yes	characteristic
620	C₄	—	2	2	no	—
621	C₄	—	2	4	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	2	no	—
624	C₄	—	8	2	no	—
625	C₄	—	8	1	no	—
626	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
627	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
628	C₄	D₄²×C₃₁	1	1	yes	characteristic
629	C₄	C₃₁×#546	1	1	yes	characteristic
630	C₂	—	2	1	no	—
631	C₂	—	2	1	no	—
632	C₂	—	2	1	no	—
633	C₂	—	4	2	no	—
634	C₂	—	4	4	no	—
635	C₂	—	8	2	no	—
636	C₂	D₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	D₄×C₂²⋊C₄
prime radical	C₃₁×#3584	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2957	C₂	semidirect
3. C₃₁×#2943	C₂	semidirect
4. C₃₁×#2901	C₂	semidirect
5. C₃₁×#1976	C₂	semidirect
6. C₃₁×#1975	C₂	semidirect
7. C₃₁×#1906	C₂	semidirect
8. C₃₁×#1881	C₂	semidirect
9. C₃₁×#1814	C₂	semidirect
10. C₃₁×#1794	C₂	semidirect
11. C₃₁×#1765	C₂	semidirect
14. C₃₁×#578	C₄	semidirect
16. C₃₁×#577	C₄	semidirect
18. C₃₁×#576	C₄	semidirect
24. C₃₁×#568	C₂²	semidirect
26. C₄×#41×C₃₁	C₂²	semidirect
32. C₂×C₃₁×#108	C₂²	semidirect
33. C₃₁×#427	C₂²	semidirect
36. C₃₁×#421	C₂²	semidirect
37. C₃₁×#421	C₂²	semidirect
40. C₂×C₃₁×#104	C₂²	semidirect
42. C₃₁×#416	C₂²	semidirect
43. C₃₁×#410	C₂²	semidirect
44. C₃₁×#408	C₂²	semidirect
45. C₃₁×#405	C₂²	semidirect
46. C₂×C₃₁×#97	C₂²	semidirect
47. C₃₁×#344	C₂²	semidirect
48. C₃₁×#341	C₂²	semidirect
49. C₃₁×#339	C₂²	semidirect
52. C₃₁×2₋¹⁺⁴	C₂×C₄	semidirect
56. C₃₁×H(5,𝔽₂)	C₂×C₄	semidirect
66. C₃₁×#137	C₂×C₄	semidirect
70. C₃₁×AGL(1,ℤ/8)	C₂×C₄	semidirect
84. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
87. C₂×SD₁₆×C₃₁	C₂×C₄	semidirect
90. C₂×D₈×C₃₁	C₂×C₄	semidirect
96. C₃₁×#131	C₂×C₄	semidirect
97. C₃₁×#131	C₂×C₄	semidirect
232. #22461	C₃₁	direct (N × H)
292. #2957	C₂×C₃₁	semidirect
293. #2943	C₂×C₃₁	semidirect
294. #2901	C₂×C₃₁	semidirect
295. #1976	C₂×C₃₁	semidirect
296. #1975	C₂×C₃₁	semidirect
297. #1906	C₂×C₃₁	semidirect
298. #1881	C₂×C₃₁	semidirect
299. #1814	C₂×C₃₁	semidirect
300. #1794	C₂×C₃₁	semidirect
301. #1765	C₂×C₃₁	semidirect
325. #578	C₄×C₃₁	semidirect
327. #577	C₄×C₃₁	semidirect
329. #576	C₄×C₃₁	semidirect
335. #568	C₂²×C₃₁	semidirect
337. C₄×#41	C₂²×C₃₁	semidirect
343. C₂×#108	C₂²×C₃₁	semidirect
344. #427	C₂²×C₃₁	semidirect
347. #421	C₂²×C₃₁	semidirect
348. #421	C₂²×C₃₁	semidirect
351. C₂×#104	C₂²×C₃₁	semidirect
353. #416	C₂²×C₃₁	semidirect
354. #410	C₂²×C₃₁	semidirect
355. #408	C₂²×C₃₁	semidirect
356. #405	C₂²×C₃₁	semidirect
357. C₂×#97	C₂²×C₃₁	semidirect
358. #344	C₂²×C₃₁	semidirect
359. #341	C₂²×C₃₁	semidirect
360. #339	C₂²×C₃₁	semidirect
370. 2₋¹⁺⁴	C₂×C₄×C₃₁	semidirect
374. H(5,𝔽₂)	C₂×C₄×C₃₁	semidirect
384. #137	C₂×C₄×C₃₁	semidirect
388. AGL(1,ℤ/8)	C₂×C₄×C₃₁	semidirect
402. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
405. C₂×SD₁₆	C₂×C₄×C₃₁	semidirect
408. C₂×D₈	C₂×C₄×C₃₁	semidirect
414. #131	C₂×C₄×C₃₁	semidirect
415. #131	C₂×C₄×C₃₁	semidirect
452. C₃₁	#22461	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3584	2
C₃₁×#2957	2
C₃₁×#2943	2
C₃₁×#2901	2
C₃₁×#1976	2
C₃₁×#1975	2
C₃₁×#1906	2
C₃₁×#1881	2
C₃₁×#1814	2
C₃₁×#1794	2
C₃₁×#1765	2
#22461	31

Subgroups — C31×#22461 #21479220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22461 #21479220