Subgroups — C₃₁×#61661 #21518420

Order 7936.

2272 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3549	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3325	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3305	C₂	1	4	yes	maximal, complemented
5	D₄×D₈×C₃₁	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3274	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3205	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3199	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3106	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3008	C₂	1	4	yes	maximal, complemented
11	C₃₁×#2979	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	2	yes	complemented
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
14	C₂²×D₈×C₃₁	C₂²	1	4	yes	complemented
15	C₃₁×H(3,𝔽₄)	—	2	2	no	—
16	C₃₁×#555	—	2	1	no	—
17	C₃₁×#555	C₂²	1	4	yes	complemented
18	C₃₁×#550	C₂²	1	4	yes	complemented
19	D₄²×C₃₁	—	2	1	no	—
20	D₄²×C₃₁	—	2	4	no	—
21	D₄²×C₃₁	C₂²	1	4	yes	complemented
22	D₄²×C₃₁	C₂²	1	4	yes	complemented
23	C₃₁×#535	—	2	2	no	—
24	C₃₁×#535	—	2	4	no	—
25	C₃₁×#535	C₂²	1	2	yes	complemented
26	C₃₁×#535	C₂²	1	4	yes	complemented
27	C₃₁×#532	—	2	1	no	—
28	C₃₁×#532	C₂²	1	1	yes	characteristic
29	C₃₁×#532	C₂²	1	2	yes	complemented
30	C₂×C₃₁×#127	—	2	1	no	—
31	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic, complemented
32	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#493	C₂²	1	4	yes	complemented
34	C₃₁×#486	C₂²	1	4	yes	complemented
35	C₃₁×#482	C₂²	1	4	yes	complemented
36	C₃₁×#482	C₂²	1	8	yes	complemented
37	C₃₁×#477	C₂²	1	2	yes	complemented
38	C₃₁×#476	C₂²	1	2	yes	complemented
39	C₃₁×#469	C₂²	1	8	yes	complemented
40	C₃₁×#466	C₂²	1	8	yes	complemented
41	C₃₁×#460	C₂²	1	2	yes	complemented
42	C₃₁×#459	C₂²	1	2	yes	complemented
43	C₃₁×#459	C₂²	1	4	yes	complemented
44	C₃₁×#459	C₂²	1	4	yes	complemented
45	C₃₁×#449	C₂²	1	4	yes	complemented
46	C₃₁×#449	C₂²	1	8	yes	complemented
47	C₃₁×#447	C₂²	1	8	yes	complemented
48	C₃₁×#447	C₂²	1	8	yes	complemented
49	C₃₁×#447	C₂²	1	8	yes	complemented
50	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
51	C₄×D₈×C₃₁	C₂²	1	4	yes	complemented
52	C₃₁×#436	C₂²	1	4	yes	complemented
53	C₈×D₄×C₃₁	C₂²	1	4	yes	complemented
54	C₃₁×#431	C₂²	1	2	yes	complemented
55	C₃₁×#418	C₂²	1	4	yes	complemented
56	C₃₁×#416	C₂²	1	4	yes	complemented
57	C₃₁×#408	C₂²	1	4	yes	complemented
58	C₃₁×H(5,𝔽₂)	—	4	4	no	—
59	C₃₁×H(5,𝔽₂)	—	4	4	no	—
60	C₂×#41×C₃₁	—	2	2	no	—
61	C₂×#41×C₃₁	—	2	2	no	—
62	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×#41×C₃₁	C₂³	1	4	yes	—
64	C₂×#41×C₃₁	C₂³	1	2	yes	—
65	C₂²×D₄×C₃₁	—	2	2	no	—
66	C₂²×D₄×C₃₁	—	2	4	no	—
67	C₂²×D₄×C₃₁	—	2	4	no	—
68	C₂²×D₄×C₃₁	—	2	4	no	—
69	C₂²×D₄×C₃₁	—	2	2	no	—
70	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
71	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
72	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
73	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
74	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
75	C₂×D₈×C₃₁	—	4	8	no	—
76	C₂×D₈×C₃₁	C₂³	1	4	yes	—
77	C₂×D₈×C₃₁	C₂³	1	8	yes	—
78	C₂×D₈×C₃₁	C₂³	1	8	yes	—
79	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
80	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
81	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
82	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
83	C₃₁×#127	—	2	2	no	—
84	C₃₁×#127	—	2	4	no	—
85	C₃₁×#127	—	2	4	no	—
86	C₃₁×#127	—	2	1	no	—
87	C₃₁×#127	—	2	1	no	—
88	C₃₁×#127	D₄	1	1	yes	characteristic
89	C₃₁×#127	C₂³	1	4	yes	—
90	C₃₁×#127	C₂³	1	2	yes	—
91	C₃₁×#124	—	2	2	no	—
92	C₃₁×#124	—	2	4	no	—
93	C₃₁×#124	—	2	8	no	—
94	C₃₁×#124	—	2	2	no	—
95	C₃₁×#124	D₄	1	2	yes	—
96	C₃₁×#121	—	2	4	no	—
97	C₃₁×#121	—	2	2	no	—
98	C₃₁×#121	—	2	4	no	—
99	C₃₁×#121	—	2	4	no	—
100	C₃₁×#121	—	2	8	no	—
101	C₃₁×#121	D₄	1	4	yes	—
102	C₃₁×#121	C₂³	1	4	yes	—
103	C₃₁×#121	C₂³	1	8	yes	—
104	C₃₁×#121	C₂³	1	8	yes	—
105	C₃₁×#120	—	2	4	no	—
106	C₃₁×#120	—	2	4	no	—
107	C₃₁×#120	—	2	4	no	—
108	C₃₁×#120	—	2	8	no	—
109	C₃₁×#120	—	2	8	no	—
110	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
111	C₄×D₄×C₃₁	—	2	2	no	—
112	C₄×D₄×C₃₁	—	2	4	no	—
113	C₄×D₄×C₃₁	—	2	2	no	—
114	C₄×D₄×C₃₁	C₂³	1	2	yes	—
115	C₄×D₄×C₃₁	C₂³	1	4	yes	—
116	C₄×D₄×C₃₁	C₂³	1	4	yes	—
117	C₃₁×#117	—	2	4	no	—
118	C₃₁×#117	—	2	1	no	—
119	C₃₁×#117	C₂³	1	4	yes	—
120	C₃₁×#117	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	—	2	1	no	—
122	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
123	C₃₁×#107	C₂³	1	4	yes	—
124	C₃₁×#106	C₂³	1	4	yes	—
125	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
126	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
127	C₃₁×#103	C₂³	1	4	yes	—
128	C₃₁×#102	C₂³	1	8	yes	—
129	C₃₁×#102	C₂³	1	4	yes	—
130	C₃₁×#102	C₂³	1	8	yes	—
131	C₃₁×#102	C₂³	1	4	yes	—
132	C₃₁×#102	C₂³	1	4	yes	—
133	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
134	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
135	C₄×C₈×C₃₁	C₂³	1	4	yes	—
136	C₂⁴×C₃₁	—	2	4	no	—
137	C₂⁴×C₃₁	—	2	2	no	—
138	C₂⁴×C₃₁	—	2	4	no	—
139	#41×C₃₁	—	2	2	no	—
140	#41×C₃₁	—	2	2	no	—
141	#41×C₃₁	—	2	8	no	—
142	#41×C₃₁	—	4	2	no	—
143	#41×C₃₁	—	4	2	no	—
144	#41×C₃₁	—	4	4	no	—
145	#41×C₃₁	—	4	8	no	—
146	#41×C₃₁	—	8	4	no	—
147	C₂×Q₈×C₃₁	—	2	2	no	—
148	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
149	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
150	C₂×D₄×C₃₁	—	2	1	no	—
151	C₂×D₄×C₃₁	—	2	8	no	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	2	8	no	—
154	C₂×D₄×C₃₁	—	2	8	no	—
155	C₂×D₄×C₃₁	—	2	4	no	—
156	C₂×D₄×C₃₁	—	2	2	no	—
157	C₂×D₄×C₃₁	—	2	4	no	—
158	C₂×D₄×C₃₁	—	2	4	no	—
159	C₂×D₄×C₃₁	—	2	2	no	—
160	C₂×D₄×C₃₁	—	2	2	no	—
161	C₂×D₄×C₃₁	—	2	4	no	—
162	C₂×D₄×C₃₁	—	2	4	no	—
163	C₂×D₄×C₃₁	—	4	4	no	—
164	C₂×D₄×C₃₁	—	4	2	no	—
165	C₂×D₄×C₃₁	—	4	2	no	—
166	C₂×D₄×C₃₁	—	4	4	no	—
167	C₂×D₄×C₃₁	—	4	4	no	—
168	C₂×D₄×C₃₁	—	4	8	no	—
169	C₂×D₄×C₃₁	—	4	16	no	—
170	C₂×D₄×C₃₁	—	8	4	no	—
171	C₂×D₄×C₃₁	—	8	8	no	—
172	C₂×D₄×C₃₁	—	8	8	no	—
173	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
174	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
175	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
177	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
178	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₃₁	—	2	1	no	—
180	C₂²×C₄×C₃₁	—	2	2	no	—
181	C₂²×C₄×C₃₁	—	2	4	no	—
182	C₂²×C₄×C₃₁	—	2	4	no	—
183	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
184	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
186	SD₁₆×C₃₁	—	2	8	no	—
187	D₈×C₃₁	—	2	8	no	—
188	D₈×C₃₁	—	4	8	no	—
189	D₈×C₃₁	—	4	8	no	—
190	M₁₆×C₃₁	—	4	4	no	—
191	C₂×C₈×C₃₁	—	4	4	no	—
192	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
193	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
194	C₄⋊C₄×C₃₁	—	2	2	no	—
195	C₄⋊C₄×C₃₁	—	2	2	no	—
196	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
197	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
198	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
199	C₂²⋊C₄×C₃₁	—	2	1	no	—
200	C₂²⋊C₄×C₃₁	—	2	4	no	—
201	C₂²⋊C₄×C₃₁	—	2	2	no	—
202	C₂²⋊C₄×C₃₁	—	2	8	no	—
203	C₂²⋊C₄×C₃₁	—	2	8	no	—
204	C₂²⋊C₄×C₃₁	—	2	4	no	—
205	C₂²⋊C₄×C₃₁	—	2	4	no	—
206	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
207	C₄²×C₃₁	—	2	1	no	—
208	C₄²×C₃₁	—	2	2	no	—
209	C₄²×C₃₁	—	2	4	no	—
210	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₃₁	C₂⁴	1	2	yes	—
213	#61661	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂³×C₃₁	—	2	1	no	—
215	C₂³×C₃₁	—	2	4	no	—
216	C₂³×C₃₁	—	2	4	no	—
217	C₂³×C₃₁	—	4	8	no	—
218	C₂³×C₃₁	—	4	4	no	—
219	C₂³×C₃₁	—	8	4	no	—
220	C₂³×C₃₁	—	8	4	no	—
221	C₂³×C₃₁	—	8	8	no	—
222	C₂³×C₃₁	—	8	8	no	—
223	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
224	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	Q₈×C₃₁	—	2	1	no	—
226	Q₈×C₃₁	—	4	2	no	—
227	Q₈×C₃₁	—	8	2	no	—
228	Q₈×C₃₁	H(5,𝔽₂)	1	2	yes	—
229	D₄×C₃₁	—	2	4	no	—
230	D₄×C₃₁	—	2	8	no	—
231	D₄×C₃₁	—	2	1	no	—
232	D₄×C₃₁	—	2	4	no	—
233	D₄×C₃₁	—	2	4	no	—
234	D₄×C₃₁	—	2	4	no	—
235	D₄×C₃₁	—	2	4	no	—
236	D₄×C₃₁	—	4	2	no	—
237	D₄×C₃₁	—	4	4	no	—
238	D₄×C₃₁	—	4	8	no	—
239	D₄×C₃₁	—	4	8	no	—
240	D₄×C₃₁	—	4	4	no	—
241	D₄×C₃₁	—	4	4	no	—
242	D₄×C₃₁	—	4	8	no	—
243	D₄×C₃₁	—	8	4	no	—
244	D₄×C₃₁	—	8	4	no	—
245	D₄×C₃₁	—	8	2	no	—
246	D₄×C₃₁	—	8	8	no	—
247	D₄×C₃₁	—	8	8	no	—
248	D₄×C₃₁	—	8	4	no	—
249	D₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
250	C₂×C₄×C₃₁	—	2	4	no	—
251	C₂×C₄×C₃₁	—	2	1	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	4	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	1	no	—
256	C₂×C₄×C₃₁	—	2	1	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	4	no	—
259	C₂×C₄×C₃₁	—	4	8	no	—
260	C₂×C₄×C₃₁	—	4	4	no	—
261	C₂×C₄×C₃₁	—	4	8	no	—
262	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
264	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
267	C₈×C₃₁	—	2	4	no	—
268	C₈×C₃₁	—	2	4	no	—
269	#3549	C₂×C₃₁	1	1	yes	characteristic, complemented
270	#3498	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#3325	C₂×C₃₁	1	4	yes	complemented
272	#3305	C₂×C₃₁	1	4	yes	complemented
273	D₄×D₈	C₂×C₃₁	1	4	yes	complemented
274	#3274	C₂×C₃₁	1	4	yes	complemented
275	#3205	C₂×C₃₁	1	2	yes	complemented
276	#3199	C₂×C₃₁	1	4	yes	complemented
277	#3106	C₂×C₃₁	1	2	yes	complemented
278	#3008	C₂×C₃₁	1	4	yes	complemented
279	#2979	C₂×C₃₁	1	1	yes	characteristic, complemented
280	C₂²×C₃₁	—	2	1	no	—
281	C₂²×C₃₁	—	2	2	no	—
282	C₂²×C₃₁	—	2	4	no	—
283	C₂²×C₃₁	—	2	2	no	—
284	C₂²×C₃₁	—	2	8	no	—
285	C₂²×C₃₁	—	4	2	no	—
286	C₂²×C₃₁	—	4	4	no	—
287	C₂²×C₃₁	—	4	4	no	—
288	C₂²×C₃₁	—	4	4	no	—
289	C₂²×C₃₁	—	4	8	no	—
290	C₂²×C₃₁	—	4	4	no	—
291	C₂²×C₃₁	—	16	8	no	—
292	C₂²×C₃₁	—	16	4	no	—
293	C₂²×C₃₁	—	16	8	no	—
294	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
295	C₄×C₃₁	—	2	1	no	—
296	C₄×C₃₁	—	2	2	no	—
297	C₄×C₃₁	—	2	4	no	—
298	C₄×C₃₁	—	2	2	no	—
299	C₄×C₃₁	—	4	2	no	—
300	C₄×C₃₁	—	4	4	no	—
301	C₄×C₃₁	#576	1	4	yes	—
302	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
303	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
304	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	2	yes	complemented
305	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	complemented
306	C₂²×D₈	C₂²×C₃₁	1	4	yes	complemented
307	H(3,𝔽₄)	—	2	2	no	—
308	#555	—	2	1	no	—
309	#555	C₂²×C₃₁	1	4	yes	complemented
310	#550	C₂²×C₃₁	1	4	yes	complemented
311	D₄²	—	2	1	no	—
312	D₄²	—	2	4	no	—
313	D₄²	C₂²×C₃₁	1	4	yes	complemented
314	D₄²	C₂²×C₃₁	1	4	yes	complemented
315	#535	—	2	2	no	—
316	#535	—	2	4	no	—
317	#535	C₂²×C₃₁	1	2	yes	complemented
318	#535	C₂²×C₃₁	1	4	yes	complemented
319	#532	—	2	1	no	—
320	#532	C₂²×C₃₁	1	1	yes	characteristic
321	#532	C₂²×C₃₁	1	2	yes	complemented
322	C₂×#127	—	2	1	no	—
323	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
324	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic, complemented
325	#493	C₂²×C₃₁	1	4	yes	complemented
326	#486	C₂²×C₃₁	1	4	yes	complemented
327	#482	C₂²×C₃₁	1	4	yes	complemented
328	#482	C₂²×C₃₁	1	8	yes	complemented
329	#477	C₂²×C₃₁	1	2	yes	complemented
330	#476	C₂²×C₃₁	1	2	yes	complemented
331	#469	C₂²×C₃₁	1	8	yes	complemented
332	#466	C₂²×C₃₁	1	8	yes	complemented
333	#460	C₂²×C₃₁	1	2	yes	complemented
334	#459	C₂²×C₃₁	1	2	yes	complemented
335	#459	C₂²×C₃₁	1	4	yes	complemented
336	#459	C₂²×C₃₁	1	4	yes	complemented
337	#449	C₂²×C₃₁	1	4	yes	complemented
338	#449	C₂²×C₃₁	1	8	yes	complemented
339	#447	C₂²×C₃₁	1	8	yes	complemented
340	#447	C₂²×C₃₁	1	8	yes	complemented
341	#447	C₂²×C₃₁	1	8	yes	complemented
342	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
343	C₄×D₈	C₂²×C₃₁	1	4	yes	complemented
344	#436	C₂²×C₃₁	1	4	yes	complemented
345	C₈×D₄	C₂²×C₃₁	1	4	yes	complemented
346	#431	C₂²×C₃₁	1	2	yes	complemented
347	#418	C₂²×C₃₁	1	4	yes	complemented
348	#416	C₂²×C₃₁	1	4	yes	complemented
349	#408	C₂²×C₃₁	1	4	yes	complemented
350	C₂×C₃₁	—	4	1	no	—
351	C₂×C₃₁	—	4	4	no	—
352	C₂×C₃₁	—	4	2	no	—
353	C₂×C₃₁	—	8	4	no	—
354	C₂×C₃₁	—	8	4	no	—
355	C₂×C₃₁	#3484	1	1	yes	characteristic
356	C₂×C₃₁	C₂×#576	1	2	yes	—
357	H(5,𝔽₂)	—	4	4	no	—
358	H(5,𝔽₂)	—	4	4	no	—
359	C₂×#41	—	2	2	no	—
360	C₂×#41	—	2	2	no	—
361	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
362	C₂×#41	C₂³×C₃₁	1	4	yes	—
363	C₂×#41	C₂³×C₃₁	1	2	yes	—
364	C₂²×D₄	—	2	2	no	—
365	C₂²×D₄	—	2	4	no	—
366	C₂²×D₄	—	2	4	no	—
367	C₂²×D₄	—	2	4	no	—
368	C₂²×D₄	—	2	2	no	—
369	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
370	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
371	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
372	AGL(1,ℤ/8)	—	4	8	no	—
373	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
374	C₂×D₈	—	4	8	no	—
375	C₂×D₈	C₂³×C₃₁	1	4	yes	—
376	C₂×D₈	C₂³×C₃₁	1	8	yes	—
377	C₂×D₈	C₂³×C₃₁	1	8	yes	—
378	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
379	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
380	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
381	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
382	#127	—	2	2	no	—
383	#127	—	2	4	no	—
384	#127	—	2	4	no	—
385	#127	—	2	1	no	—
386	#127	—	2	1	no	—
387	#127	D₄×C₃₁	1	1	yes	characteristic
388	#127	C₂³×C₃₁	1	4	yes	—
389	#127	C₂³×C₃₁	1	2	yes	—
390	#124	—	2	2	no	—
391	#124	—	2	4	no	—
392	#124	—	2	8	no	—
393	#124	—	2	2	no	—
394	#124	D₄×C₃₁	1	2	yes	—
395	#121	—	2	4	no	—
396	#121	—	2	2	no	—
397	#121	—	2	4	no	—
398	#121	—	2	4	no	—
399	#121	—	2	8	no	—
400	#121	D₄×C₃₁	1	4	yes	—
401	#121	C₂³×C₃₁	1	4	yes	—
402	#121	C₂³×C₃₁	1	8	yes	—
403	#121	C₂³×C₃₁	1	8	yes	—
404	#120	—	2	4	no	—
405	#120	—	2	4	no	—
406	#120	—	2	4	no	—
407	#120	—	2	8	no	—
408	#120	—	2	8	no	—
409	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	4	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	C₂³×C₃₁	1	2	yes	—
414	C₄×D₄	C₂³×C₃₁	1	4	yes	—
415	C₄×D₄	C₂³×C₃₁	1	4	yes	—
416	#117	—	2	4	no	—
417	#117	—	2	1	no	—
418	#117	C₂³×C₃₁	1	1	yes	characteristic
419	#117	C₂³×C₃₁	1	4	yes	—
420	C₂×C₄²	—	2	1	no	—
421	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
422	#107	C₂³×C₃₁	1	4	yes	—
423	#106	C₂³×C₃₁	1	4	yes	—
424	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
425	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
426	#103	C₂³×C₃₁	1	4	yes	—
427	#102	C₂³×C₃₁	1	8	yes	—
428	#102	C₂³×C₃₁	1	4	yes	—
429	#102	C₂³×C₃₁	1	8	yes	—
430	#102	C₂³×C₃₁	1	4	yes	—
431	#102	C₂³×C₃₁	1	4	yes	—
432	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
433	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
434	C₄×C₈	C₂³×C₃₁	1	4	yes	—
435	C₃₁	#61661	1	1	yes	characteristic, direct product factor, Sylow
436	C₂⁴	—	2	4	no	—
437	C₂⁴	—	2	2	no	—
438	C₂⁴	—	2	4	no	—
439	#41	—	2	2	no	—
440	#41	—	2	2	no	—
441	#41	—	2	8	no	—
442	#41	—	4	2	no	—
443	#41	—	4	2	no	—
444	#41	—	4	4	no	—
445	#41	—	4	8	no	—
446	#41	—	8	4	no	—
447	C₂×Q₈	—	2	2	no	—
448	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
449	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	2	4	no	—
453	C₂×D₄	—	2	8	no	—
454	C₂×D₄	—	2	8	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	4	no	—
462	C₂×D₄	—	2	4	no	—
463	C₂×D₄	—	4	4	no	—
464	C₂×D₄	—	4	2	no	—
465	C₂×D₄	—	4	2	no	—
466	C₂×D₄	—	4	4	no	—
467	C₂×D₄	—	4	4	no	—
468	C₂×D₄	—	4	8	no	—
469	C₂×D₄	—	4	16	no	—
470	C₂×D₄	—	8	4	no	—
471	C₂×D₄	—	8	8	no	—
472	C₂×D₄	—	8	8	no	—
473	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
474	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
475	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
476	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
477	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
478	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
486	SD₁₆	—	2	8	no	—
487	D₈	—	2	8	no	—
488	D₈	—	4	8	no	—
489	D₈	—	4	8	no	—
490	M₁₆	—	4	4	no	—
491	C₂×C₈	—	4	4	no	—
492	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
493	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
497	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
498	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
499	C₂²⋊C₄	—	2	1	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	2	no	—
502	C₂²⋊C₄	—	2	8	no	—
503	C₂²⋊C₄	—	2	8	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
507	C₄²	—	2	1	no	—
508	C₄²	—	2	2	no	—
509	C₄²	—	2	4	no	—
510	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
511	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
512	C₄²	C₂⁴×C₃₁	1	2	yes	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	4	no	—
515	C₂³	—	2	4	no	—
516	C₂³	—	4	8	no	—
517	C₂³	—	4	4	no	—
518	C₂³	—	8	4	no	—
519	C₂³	—	8	4	no	—
520	C₂³	—	8	8	no	—
521	C₂³	—	8	8	no	—
522	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
523	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
524	Q₈	—	2	1	no	—
525	Q₈	—	4	2	no	—
526	Q₈	—	8	2	no	—
527	Q₈	C₃₁×H(5,𝔽₂)	1	2	yes	—
528	D₄	—	2	4	no	—
529	D₄	—	2	8	no	—
530	D₄	—	2	1	no	—
531	D₄	—	2	4	no	—
532	D₄	—	2	4	no	—
533	D₄	—	2	4	no	—
534	D₄	—	2	4	no	—
535	D₄	—	4	2	no	—
536	D₄	—	4	4	no	—
537	D₄	—	4	8	no	—
538	D₄	—	4	8	no	—
539	D₄	—	4	4	no	—
540	D₄	—	4	4	no	—
541	D₄	—	4	8	no	—
542	D₄	—	8	4	no	—
543	D₄	—	8	4	no	—
544	D₄	—	8	2	no	—
545	D₄	—	8	8	no	—
546	D₄	—	8	8	no	—
547	D₄	—	8	4	no	—
548	D₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	4	8	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	4	8	no	—
561	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
562	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
563	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
564	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
565	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
566	C₈	—	2	4	no	—
567	C₈	—	2	4	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	4	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	8	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	4	4	no	—
575	C₂²	—	4	4	no	—
576	C₂²	—	4	4	no	—
577	C₂²	—	4	8	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	16	8	no	—
580	C₂²	—	16	4	no	—
581	C₂²	—	16	8	no	—
582	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
583	C₄	—	2	1	no	—
584	C₄	—	2	2	no	—
585	C₄	—	2	4	no	—
586	C₄	—	2	2	no	—
587	C₄	—	4	2	no	—
588	C₄	—	4	4	no	—
589	C₄	C₃₁×#576	1	4	yes	—
590	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
591	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
592	C₂	—	4	1	no	—
593	C₂	—	4	4	no	—
594	C₂	—	4	2	no	—
595	C₂	—	8	4	no	—
596	C₂	—	8	4	no	—
597	C₂	C₃₁×#3484	1	1	yes	characteristic
598	C₂	C₂×C₃₁×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#532 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3549	C₂	semidirect
2. C₃₁×#3498	C₂	semidirect
3. C₃₁×#3325	C₂	semidirect
4. C₃₁×#3305	C₂	semidirect
5. D₄×D₈×C₃₁	C₂	semidirect
6. C₃₁×#3274	C₂	semidirect
7. C₃₁×#3205	C₂	semidirect
8. C₃₁×#3199	C₂	semidirect
9. C₃₁×#3106	C₂	semidirect
10. C₃₁×#3008	C₂	semidirect
11. C₃₁×#2979	C₂	semidirect
12. C₂×C₃₁×H(5,𝔽₂)	C₂²	semidirect
13. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
14. C₂²×D₈×C₃₁	C₂²	semidirect
17. C₃₁×#555	C₂²	semidirect
18. C₃₁×#550	C₂²	semidirect
21. D₄²×C₃₁	C₂²	semidirect
22. D₄²×C₃₁	C₂²	semidirect
25. C₃₁×#535	C₂²	semidirect
26. C₃₁×#535	C₂²	semidirect
29. C₃₁×#532	C₂²	semidirect
31. C₂×C₃₁×#127	C₂²	semidirect
32. C₄×#41×C₃₁	C₂²	semidirect
33. C₃₁×#493	C₂²	semidirect
34. C₃₁×#486	C₂²	semidirect
35. C₃₁×#482	C₂²	semidirect
36. C₃₁×#482	C₂²	semidirect
37. C₃₁×#477	C₂²	semidirect
38. C₃₁×#476	C₂²	semidirect
39. C₃₁×#469	C₂²	semidirect
40. C₃₁×#466	C₂²	semidirect
41. C₃₁×#460	C₂²	semidirect
42. C₃₁×#459	C₂²	semidirect
43. C₃₁×#459	C₂²	semidirect
44. C₃₁×#459	C₂²	semidirect
45. C₃₁×#449	C₂²	semidirect
46. C₃₁×#449	C₂²	semidirect
47. C₃₁×#447	C₂²	semidirect
48. C₃₁×#447	C₂²	semidirect
49. C₃₁×#447	C₂²	semidirect
50. C₄×SD₁₆×C₃₁	C₂²	semidirect
51. C₄×D₈×C₃₁	C₂²	semidirect
52. C₃₁×#436	C₂²	semidirect
53. C₈×D₄×C₃₁	C₂²	semidirect
54. C₃₁×#431	C₂²	semidirect
55. C₃₁×#418	C₂²	semidirect
56. C₃₁×#416	C₂²	semidirect
57. C₃₁×#408	C₂²	semidirect
213. #61661	C₃₁	direct (N × H)
269. #3549	C₂×C₃₁	semidirect
270. #3498	C₂×C₃₁	semidirect
271. #3325	C₂×C₃₁	semidirect
272. #3305	C₂×C₃₁	semidirect
273. D₄×D₈	C₂×C₃₁	semidirect
274. #3274	C₂×C₃₁	semidirect
275. #3205	C₂×C₃₁	semidirect
276. #3199	C₂×C₃₁	semidirect
277. #3106	C₂×C₃₁	semidirect
278. #3008	C₂×C₃₁	semidirect
279. #2979	C₂×C₃₁	semidirect
304. C₂×H(5,𝔽₂)	C₂²×C₃₁	semidirect
305. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
306. C₂²×D₈	C₂²×C₃₁	semidirect
309. #555	C₂²×C₃₁	semidirect
310. #550	C₂²×C₃₁	semidirect
313. D₄²	C₂²×C₃₁	semidirect
314. D₄²	C₂²×C₃₁	semidirect
317. #535	C₂²×C₃₁	semidirect
318. #535	C₂²×C₃₁	semidirect
321. #532	C₂²×C₃₁	semidirect
323. C₂×#127	C₂²×C₃₁	semidirect
324. C₄×#41	C₂²×C₃₁	semidirect
325. #493	C₂²×C₃₁	semidirect
326. #486	C₂²×C₃₁	semidirect
327. #482	C₂²×C₃₁	semidirect
328. #482	C₂²×C₃₁	semidirect
329. #477	C₂²×C₃₁	semidirect
330. #476	C₂²×C₃₁	semidirect
331. #469	C₂²×C₃₁	semidirect
332. #466	C₂²×C₃₁	semidirect
333. #460	C₂²×C₃₁	semidirect
334. #459	C₂²×C₃₁	semidirect
335. #459	C₂²×C₃₁	semidirect
336. #459	C₂²×C₃₁	semidirect
337. #449	C₂²×C₃₁	semidirect
338. #449	C₂²×C₃₁	semidirect
339. #447	C₂²×C₃₁	semidirect
340. #447	C₂²×C₃₁	semidirect
341. #447	C₂²×C₃₁	semidirect
342. C₄×SD₁₆	C₂²×C₃₁	semidirect
343. C₄×D₈	C₂²×C₃₁	semidirect
344. #436	C₂²×C₃₁	semidirect
345. C₈×D₄	C₂²×C₃₁	semidirect
346. #431	C₂²×C₃₁	semidirect
347. #418	C₂²×C₃₁	semidirect
348. #416	C₂²×C₃₁	semidirect
349. #408	C₂²×C₃₁	semidirect
435. C₃₁	#61661	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×#3549	2
C₃₁×#3498	2
C₃₁×#3325	2
C₃₁×#3305	2
D₄×D₈×C₃₁	2
C₃₁×#3274	2
C₃₁×#3205	2
C₃₁×#3199	2
C₃₁×#3106	2
C₃₁×#3008	2
C₃₁×#2979	2
#61661	31

Subgroups — C31×#61661 #21518420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61661 #21518420