Subgroups — C₃₁×#62588 #21519347

Order 7936.

2046 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3518	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3362	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3358	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3325	C₂	1	6	yes	maximal, complemented
6	C₃₁×#3324	C₂	1	6	yes	maximal, complemented
7	C₃₁×#3235	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3217	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3171	C₂	1	3	yes	maximal, complemented
10	C₃₁×#3138	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2971	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×H(5,𝔽₂)	—	2	1	no	—
13	Q₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#556	—	2	3	no	—
15	C₃₁×#556	C₂²	1	3	yes	complemented
16	C₃₁×#555	C₂²	1	3	yes	complemented
17	C₃₁×#550	—	2	1	no	—
18	C₃₁×#550	C₂²	1	2	yes	—
19	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
20	C₃₁×#548	—	2	1	no	—
21	D₄²×C₃₁	—	2	3	no	—
22	D₄²×C₃₁	C₂²	1	6	yes	—
23	C₃₁×#542	—	2	3	no	—
24	C₃₁×#535	—	2	3	no	—
25	C₃₁×#533	C₂²	1	3	yes	complemented
26	C₃₁×#532	C₂²	1	3	yes	—
27	C₂×C₃₁×#127	C₂²	1	3	yes	—
28	C₃₁×#529	C₂²	1	3	yes	complemented
29	C₃₁×#519	—	2	1	no	—
30	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#501	C₂²	1	3	yes	complemented
32	C₃₁×#496	C₂²	1	3	yes	complemented
33	C₃₁×#490	C₂²	1	3	yes	complemented
34	C₃₁×#490	C₂²	1	6	yes	complemented
35	C₃₁×#489	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#486	C₂²	1	6	yes	complemented
37	C₃₁×#486	C₂²	1	6	yes	complemented
38	C₃₁×#482	C₂²	1	12	yes	complemented
39	C₃₁×#476	C₂²	1	6	yes	complemented
40	C₃₁×#474	C₂²	1	6	yes	complemented
41	C₃₁×#469	C₂²	1	6	yes	complemented
42	C₃₁×#468	C₂²	1	6	yes	complemented
43	C₃₁×#460	C₂²	1	6	yes	complemented
44	C₃₁×#459	C₂²	1	6	yes	complemented
45	C₃₁×#450	C₂²	1	6	yes	complemented
46	C₃₁×#447	C₂²	1	6	yes	complemented
47	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
48	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	complemented
49	C₄×D₈×C₃₁	C₂²	1	6	yes	complemented
50	C₃₁×#436	C₂²	1	6	yes	complemented
51	C₃₁×#432	C₂²	1	3	yes	complemented
52	C₃₁×#418	C₂²	1	12	yes	complemented
53	C₄×M₁₆×C₃₁	C₂²	1	3	yes	complemented
54	C₃₁×H(5,𝔽₂)	—	4	8	no	—
55	C₂×#41×C₃₁	—	2	2	no	—
56	C₂×#41×C₃₁	—	2	3	no	—
57	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₃₁	—	2	1	no	—
59	C₂²×D₄×C₃₁	—	2	6	no	—
60	C₂²×D₄×C₃₁	—	2	3	no	—
61	C₂²×D₄×C₃₁	—	2	3	no	—
62	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
63	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
64	C₂×D₈×C₃₁	C₂³	1	6	yes	—
65	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
66	C₃₁×C₄⋊Q₈	—	2	3	no	—
67	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
68	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
69	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
70	C₃₁×#127	—	2	3	no	—
71	C₃₁×#127	—	2	3	no	—
72	C₃₁×#127	C₂³	1	6	yes	—
73	C₃₁×#127	C₂³	1	3	yes	—
74	C₃₁×#126	—	2	6	no	—
75	C₃₁×#126	—	2	3	no	—
76	C₃₁×#125	C₂³	1	3	yes	—
77	C₃₁×#124	—	2	3	no	—
78	C₃₁×#124	—	2	6	no	—
79	C₃₁×#124	—	2	3	no	—
80	C₃₁×#124	C₂³	1	1	yes	characteristic
81	C₃₁×#123	—	2	3	no	—
82	C₃₁×C₂²⋊Q₈	—	2	3	no	—
83	C₃₁×C₂²⋊Q₈	—	2	3	no	—
84	C₃₁×C₂²⋊Q₈	—	2	3	no	—
85	C₃₁×C₂²⋊Q₈	C₂³	1	3	yes	—
86	C₃₁×#121	—	2	3	no	—
87	C₃₁×#121	—	2	3	no	—
88	C₃₁×#121	—	2	6	no	—
89	C₃₁×#121	—	2	6	no	—
90	C₃₁×#121	C₂³	1	3	yes	—
91	C₃₁×#121	C₂³	1	6	yes	—
92	C₃₁×#120	—	2	12	no	—
93	C₄×Q₈×C₃₁	—	2	1	no	—
94	C₄×Q₈×C₃₁	—	2	3	no	—
95	C₄×Q₈×C₃₁	—	2	3	no	—
96	C₄×Q₈×C₃₁	—	2	3	no	—
97	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
98	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
99	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×D₄×C₃₁	—	2	3	no	—
101	C₄×D₄×C₃₁	—	2	3	no	—
102	C₄×D₄×C₃₁	D₄	1	3	yes	—
103	C₄×D₄×C₃₁	C₂³	1	3	yes	—
104	C₄×D₄×C₃₁	C₂³	1	6	yes	—
105	C₄×D₄×C₃₁	C₂³	1	3	yes	—
106	C₃₁×#117	—	2	3	no	—
107	C₃₁×#117	—	2	3	no	—
108	C₃₁×#117	D₄	1	3	yes	—
109	C₃₁×#117	C₂³	1	6	yes	—
110	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
111	C₃₁×#107	C₂³	1	6	yes	—
112	C₃₁×#106	C₂³	1	6	yes	—
113	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
114	C₃₁×#103	C₂³	1	2	yes	—
115	C₃₁×#102	C₂³	1	6	yes	—
116	C₃₁×#102	C₂³	1	6	yes	—
117	C₃₁×#102	C₂³	1	12	yes	—
118	C₃₁×#102	C₂³	1	12	yes	—
119	C₃₁×#102	C₂³	1	2	yes	—
120	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
121	C₃₁×#97	C₂³	1	1	yes	characteristic
122	C₃₁×#97	C₂³	1	3	yes	—
123	C₄×C₈×C₃₁	C₂³	1	6	yes	—
124	C₂⁴×C₃₁	—	2	6	no	—
125	#41×C₃₁	—	2	4	no	—
126	#41×C₃₁	—	4	12	no	—
127	#41×C₃₁	—	4	8	no	—
128	C₂×Q₈×C₃₁	—	2	3	no	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	3	no	—
131	C₂×Q₈×C₃₁	—	4	4	no	—
132	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
134	C₂×D₄×C₃₁	—	2	6	no	—
135	C₂×D₄×C₃₁	—	2	3	no	—
136	C₂×D₄×C₃₁	—	2	3	no	—
137	C₂×D₄×C₃₁	—	2	6	no	—
138	C₂×D₄×C₃₁	—	2	3	no	—
139	C₂×D₄×C₃₁	—	2	6	no	—
140	C₂×D₄×C₃₁	—	2	6	no	—
141	C₂×D₄×C₃₁	—	4	12	no	—
142	C₂×D₄×C₃₁	—	4	12	no	—
143	C₂×D₄×C₃₁	—	4	24	no	—
144	C₂×D₄×C₃₁	—	8	6	no	—
145	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
146	C₂×D₄×C₃₁	C₂⁴	1	6	yes	—
147	C₂×D₄×C₃₁	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	—	2	3	no	—
149	C₂²×C₄×C₃₁	—	2	6	no	—
150	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
151	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
152	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
153	SD₁₆×C₃₁	—	4	6	no	—
154	D₈×C₃₁	—	4	6	no	—
155	M₁₆×C₃₁	—	4	6	no	—
156	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
157	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
158	C₄⋊C₄×C₃₁	—	2	3	no	—
159	C₄⋊C₄×C₃₁	—	2	3	no	—
160	C₄⋊C₄×C₃₁	—	2	3	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	3	no	—
163	C₄⋊C₄×C₃₁	—	2	3	no	—
164	C₄⋊C₄×C₃₁	—	2	3	no	—
165	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
166	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
167	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
168	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
169	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
170	C₂²⋊C₄×C₃₁	—	2	6	no	—
171	C₂²⋊C₄×C₃₁	—	2	3	no	—
172	C₂²⋊C₄×C₃₁	—	2	3	no	—
173	C₂²⋊C₄×C₃₁	—	2	1	no	—
174	C₂²⋊C₄×C₃₁	—	2	1	no	—
175	C₂²⋊C₄×C₃₁	—	2	6	no	—
176	C₂²⋊C₄×C₃₁	—	2	6	no	—
177	C₂²⋊C₄×C₃₁	—	2	3	no	—
178	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
179	C₄²×C₃₁	—	2	6	no	—
180	C₄²×C₃₁	—	2	3	no	—
181	C₄²×C₃₁	C₂×D₄	1	3	yes	—
182	C₄²×C₃₁	C₂×D₄	1	3	yes	—
183	C₄²×C₃₁	C₂⁴	1	3	yes	—
184	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
185	#62588	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
186	C₂³×C₃₁	—	2	1	no	—
187	C₂³×C₃₁	—	2	6	no	—
188	C₂³×C₃₁	—	4	12	no	—
189	C₂³×C₃₁	—	8	6	no	—
190	C₂³×C₃₁	—	8	6	no	—
191	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
192	Q₈×C₃₁	—	2	2	no	—
193	Q₈×C₃₁	—	4	2	no	—
194	Q₈×C₃₁	—	4	6	no	—
195	Q₈×C₃₁	—	8	3	no	—
196	Q₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
197	D₄×C₃₁	—	2	6	no	—
198	D₄×C₃₁	—	2	2	no	—
199	D₄×C₃₁	—	2	6	no	—
200	D₄×C₃₁	—	2	6	no	—
201	D₄×C₃₁	—	4	6	no	—
202	D₄×C₃₁	—	4	6	no	—
203	D₄×C₃₁	—	4	12	no	—
204	D₄×C₃₁	—	4	12	no	—
205	D₄×C₃₁	—	8	6	no	—
206	D₄×C₃₁	—	8	6	no	—
207	D₄×C₃₁	—	8	3	no	—
208	C₂×C₄×C₃₁	—	2	6	no	—
209	C₂×C₄×C₃₁	—	2	6	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	3	no	—
212	C₂×C₄×C₃₁	—	2	3	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	4	3	no	—
215	C₂×C₄×C₃₁	—	4	12	no	—
216	C₂×C₄×C₃₁	—	4	6	no	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
218	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
221	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
222	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
223	C₈×C₃₁	—	2	2	no	—
224	C₈×C₃₁	—	2	6	no	—
225	#3518	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#3498	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#3362	C₂×C₃₁	1	2	yes	complemented
228	#3358	C₂×C₃₁	1	2	yes	complemented
229	#3325	C₂×C₃₁	1	6	yes	complemented
230	#3324	C₂×C₃₁	1	6	yes	complemented
231	#3235	C₂×C₃₁	1	3	yes	complemented
232	#3217	C₂×C₃₁	1	3	yes	complemented
233	#3171	C₂×C₃₁	1	3	yes	complemented
234	#3138	C₂×C₃₁	1	3	yes	complemented
235	#2971	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂²×C₃₁	—	2	3	no	—
237	C₂²×C₃₁	—	2	3	no	—
238	C₂²×C₃₁	—	2	3	no	—
239	C₂²×C₃₁	—	4	6	no	—
240	C₂²×C₃₁	—	4	1	no	—
241	C₂²×C₃₁	—	4	6	no	—
242	C₂²×C₃₁	—	4	1	no	—
243	C₂²×C₃₁	—	4	1	no	—
244	C₂²×C₃₁	—	4	6	no	—
245	C₂²×C₃₁	—	16	12	no	—
246	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
247	C₄×C₃₁	—	2	1	no	—
248	C₄×C₃₁	—	2	3	no	—
249	C₄×C₃₁	—	2	3	no	—
250	C₄×C₃₁	—	2	1	no	—
251	C₄×C₃₁	—	4	3	no	—
252	C₄×C₃₁	—	4	1	no	—
253	C₄×C₃₁	—	4	2	no	—
254	C₄×C₃₁	—	4	3	no	—
255	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	6	yes	—
256	C₄×C₃₁	#585	1	2	yes	—
257	C₂×H(5,𝔽₂)	—	2	1	no	—
258	Q₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
259	#556	—	2	3	no	—
260	#556	C₂²×C₃₁	1	3	yes	complemented
261	#555	C₂²×C₃₁	1	3	yes	complemented
262	#550	—	2	1	no	—
263	#550	C₂²×C₃₁	1	2	yes	—
264	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
265	#548	—	2	1	no	—
266	D₄²	—	2	3	no	—
267	D₄²	C₂²×C₃₁	1	6	yes	—
268	#542	—	2	3	no	—
269	#535	—	2	3	no	—
270	#533	C₂²×C₃₁	1	3	yes	complemented
271	#532	C₂²×C₃₁	1	3	yes	—
272	C₂×#127	C₂²×C₃₁	1	3	yes	—
273	#529	C₂²×C₃₁	1	3	yes	complemented
274	#519	—	2	1	no	—
275	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
276	#501	C₂²×C₃₁	1	3	yes	complemented
277	#496	C₂²×C₃₁	1	3	yes	complemented
278	#490	C₂²×C₃₁	1	3	yes	complemented
279	#490	C₂²×C₃₁	1	6	yes	complemented
280	#489	C₂²×C₃₁	1	1	yes	characteristic, complemented
281	#486	C₂²×C₃₁	1	6	yes	complemented
282	#486	C₂²×C₃₁	1	6	yes	complemented
283	#482	C₂²×C₃₁	1	12	yes	complemented
284	#476	C₂²×C₃₁	1	6	yes	complemented
285	#474	C₂²×C₃₁	1	6	yes	complemented
286	#469	C₂²×C₃₁	1	6	yes	complemented
287	#468	C₂²×C₃₁	1	6	yes	complemented
288	#460	C₂²×C₃₁	1	6	yes	complemented
289	#459	C₂²×C₃₁	1	6	yes	complemented
290	#450	C₂²×C₃₁	1	6	yes	complemented
291	#447	C₂²×C₃₁	1	6	yes	complemented
292	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
293	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	complemented
294	C₄×D₈	C₂²×C₃₁	1	6	yes	complemented
295	#436	C₂²×C₃₁	1	6	yes	complemented
296	#432	C₂²×C₃₁	1	3	yes	complemented
297	#418	C₂²×C₃₁	1	12	yes	complemented
298	C₄×M₁₆	C₂²×C₃₁	1	3	yes	complemented
299	C₂×C₃₁	—	4	3	no	—
300	C₂×C₃₁	—	8	1	no	—
301	C₂×C₃₁	—	8	6	no	—
302	C₂×C₃₁	#3500	1	1	yes	characteristic
303	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
304	C₂×C₃₁	#3585	1	1	yes	characteristic
305	H(5,𝔽₂)	—	4	8	no	—
306	C₂×#41	—	2	2	no	—
307	C₂×#41	—	2	3	no	—
308	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
309	C₂²×Q₈	—	2	1	no	—
310	C₂²×D₄	—	2	6	no	—
311	C₂²×D₄	—	2	3	no	—
312	C₂²×D₄	—	2	3	no	—
313	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
314	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
315	C₂×D₈	C₂³×C₃₁	1	6	yes	—
316	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
317	C₄⋊Q₈	—	2	3	no	—
318	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
320	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
321	#127	—	2	3	no	—
322	#127	—	2	3	no	—
323	#127	C₂³×C₃₁	1	3	yes	—
324	#127	C₂³×C₃₁	1	6	yes	—
325	#126	—	2	6	no	—
326	#126	—	2	3	no	—
327	#125	C₂³×C₃₁	1	3	yes	—
328	#124	—	2	3	no	—
329	#124	—	2	6	no	—
330	#124	—	2	3	no	—
331	#124	C₂³×C₃₁	1	1	yes	characteristic
332	#123	—	2	3	no	—
333	C₂²⋊Q₈	—	2	3	no	—
334	C₂²⋊Q₈	—	2	3	no	—
335	C₂²⋊Q₈	—	2	3	no	—
336	C₂²⋊Q₈	C₂³×C₃₁	1	3	yes	—
337	#121	—	2	3	no	—
338	#121	—	2	3	no	—
339	#121	—	2	6	no	—
340	#121	—	2	6	no	—
341	#121	C₂³×C₃₁	1	3	yes	—
342	#121	C₂³×C₃₁	1	6	yes	—
343	#120	—	2	12	no	—
344	C₄×Q₈	—	2	1	no	—
345	C₄×Q₈	—	2	3	no	—
346	C₄×Q₈	—	2	3	no	—
347	C₄×Q₈	—	2	3	no	—
348	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
349	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
350	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
351	C₄×D₄	—	2	3	no	—
352	C₄×D₄	—	2	3	no	—
353	C₄×D₄	D₄×C₃₁	1	3	yes	—
354	C₄×D₄	C₂³×C₃₁	1	3	yes	—
355	C₄×D₄	C₂³×C₃₁	1	6	yes	—
356	C₄×D₄	C₂³×C₃₁	1	3	yes	—
357	#117	—	2	3	no	—
358	#117	—	2	3	no	—
359	#117	D₄×C₃₁	1	3	yes	—
360	#117	C₂³×C₃₁	1	6	yes	—
361	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
362	#107	C₂³×C₃₁	1	6	yes	—
363	#106	C₂³×C₃₁	1	6	yes	—
364	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
365	#103	C₂³×C₃₁	1	2	yes	—
366	#102	C₂³×C₃₁	1	6	yes	—
367	#102	C₂³×C₃₁	1	6	yes	—
368	#102	C₂³×C₃₁	1	12	yes	—
369	#102	C₂³×C₃₁	1	12	yes	—
370	#102	C₂³×C₃₁	1	2	yes	—
371	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
372	#97	C₂³×C₃₁	1	1	yes	characteristic
373	#97	C₂³×C₃₁	1	3	yes	—
374	C₄×C₈	C₂³×C₃₁	1	6	yes	—
375	C₃₁	#62588	1	1	yes	characteristic, direct product factor, Sylow
376	C₂⁴	—	2	6	no	—
377	#41	—	2	4	no	—
378	#41	—	4	12	no	—
379	#41	—	4	8	no	—
380	C₂×Q₈	—	2	3	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	—	2	3	no	—
383	C₂×Q₈	—	4	4	no	—
384	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
386	C₂×D₄	—	2	6	no	—
387	C₂×D₄	—	2	3	no	—
388	C₂×D₄	—	2	3	no	—
389	C₂×D₄	—	2	6	no	—
390	C₂×D₄	—	2	3	no	—
391	C₂×D₄	—	2	6	no	—
392	C₂×D₄	—	2	6	no	—
393	C₂×D₄	—	4	12	no	—
394	C₂×D₄	—	4	12	no	—
395	C₂×D₄	—	4	24	no	—
396	C₂×D₄	—	8	6	no	—
397	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
398	C₂×D₄	C₂⁴×C₃₁	1	6	yes	—
399	C₂×D₄	C₂⁴×C₃₁	1	1	yes	characteristic
400	C₂²×C₄	—	2	3	no	—
401	C₂²×C₄	—	2	6	no	—
402	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
403	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
404	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
405	SD₁₆	—	4	6	no	—
406	D₈	—	4	6	no	—
407	M₁₆	—	4	6	no	—
408	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
409	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
410	C₄⋊C₄	—	2	3	no	—
411	C₄⋊C₄	—	2	3	no	—
412	C₄⋊C₄	—	2	3	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	3	no	—
415	C₄⋊C₄	—	2	3	no	—
416	C₄⋊C₄	—	2	3	no	—
417	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
418	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
419	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
420	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
421	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
422	C₂²⋊C₄	—	2	6	no	—
423	C₂²⋊C₄	—	2	3	no	—
424	C₂²⋊C₄	—	2	3	no	—
425	C₂²⋊C₄	—	2	1	no	—
426	C₂²⋊C₄	—	2	1	no	—
427	C₂²⋊C₄	—	2	6	no	—
428	C₂²⋊C₄	—	2	6	no	—
429	C₂²⋊C₄	—	2	3	no	—
430	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
431	C₄²	—	2	6	no	—
432	C₄²	—	2	3	no	—
433	C₄²	C₂×D₄×C₃₁	1	3	yes	—
434	C₄²	C₂×D₄×C₃₁	1	3	yes	—
435	C₄²	C₂⁴×C₃₁	1	3	yes	—
436	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
437	C₂³	—	2	1	no	—
438	C₂³	—	2	6	no	—
439	C₂³	—	4	12	no	—
440	C₂³	—	8	6	no	—
441	C₂³	—	8	6	no	—
442	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
443	Q₈	—	2	2	no	—
444	Q₈	—	4	2	no	—
445	Q₈	—	4	6	no	—
446	Q₈	—	8	3	no	—
447	Q₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
448	D₄	—	2	6	no	—
449	D₄	—	2	2	no	—
450	D₄	—	2	6	no	—
451	D₄	—	2	6	no	—
452	D₄	—	4	6	no	—
453	D₄	—	4	6	no	—
454	D₄	—	4	12	no	—
455	D₄	—	4	12	no	—
456	D₄	—	8	6	no	—
457	D₄	—	8	6	no	—
458	D₄	—	8	3	no	—
459	C₂×C₄	—	2	6	no	—
460	C₂×C₄	—	2	6	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	3	no	—
463	C₂×C₄	—	2	3	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	4	3	no	—
466	C₂×C₄	—	4	12	no	—
467	C₂×C₄	—	4	6	no	—
468	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
469	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
470	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
472	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
473	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
474	C₈	—	2	2	no	—
475	C₈	—	2	6	no	—
476	C₂²	—	2	3	no	—
477	C₂²	—	2	3	no	—
478	C₂²	—	2	3	no	—
479	C₂²	—	4	6	no	—
480	C₂²	—	4	1	no	—
481	C₂²	—	4	1	no	—
482	C₂²	—	4	6	no	—
483	C₂²	—	4	6	no	—
484	C₂²	—	4	1	no	—
485	C₂²	—	16	12	no	—
486	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
487	C₄	—	2	3	no	—
488	C₄	—	2	1	no	—
489	C₄	—	2	1	no	—
490	C₄	—	2	3	no	—
491	C₄	—	4	1	no	—
492	C₄	—	4	2	no	—
493	C₄	—	4	3	no	—
494	C₄	—	4	3	no	—
495	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	6	yes	—
496	C₄	C₃₁×#585	1	2	yes	—
497	C₂	—	4	3	no	—
498	C₂	—	8	6	no	—
499	C₂	—	8	1	no	—
500	C₂	C₃₁×#3500	1	1	yes	characteristic
501	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
502	C₂	C₃₁×#3585	1	1	yes	characteristic

	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 ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3518	C₂	semidirect
2. C₃₁×#3498	C₂	semidirect
3. C₃₁×#3362	C₂	semidirect
4. C₃₁×#3358	C₂	semidirect
5. C₃₁×#3325	C₂	semidirect
6. C₃₁×#3324	C₂	semidirect
7. C₃₁×#3235	C₂	semidirect
8. C₃₁×#3217	C₂	semidirect
9. C₃₁×#3171	C₂	semidirect
10. C₃₁×#3138	C₂	semidirect
11. C₃₁×#2971	C₂	semidirect
13. Q₈²×C₃₁	C₂²	semidirect
15. C₃₁×#556	C₂²	semidirect
16. C₃₁×#555	C₂²	semidirect
19. D₄×Q₈×C₃₁	C₂²	semidirect
25. C₃₁×#533	C₂²	semidirect
28. C₃₁×#529	C₂²	semidirect
31. C₃₁×#501	C₂²	semidirect
32. C₃₁×#496	C₂²	semidirect
33. C₃₁×#490	C₂²	semidirect
34. C₃₁×#490	C₂²	semidirect
35. C₃₁×#489	C₂²	semidirect
36. C₃₁×#486	C₂²	semidirect
37. C₃₁×#486	C₂²	semidirect
38. C₃₁×#482	C₂²	semidirect
39. C₃₁×#476	C₂²	semidirect
40. C₃₁×#474	C₂²	semidirect
41. C₃₁×#469	C₂²	semidirect
42. C₃₁×#468	C₂²	semidirect
43. C₃₁×#460	C₂²	semidirect
44. C₃₁×#459	C₂²	semidirect
45. C₃₁×#450	C₂²	semidirect
46. C₃₁×#447	C₂²	semidirect
47. C₈×Q₈×C₃₁	C₂²	semidirect
48. C₄×SD₁₆×C₃₁	C₂²	semidirect
49. C₄×D₈×C₃₁	C₂²	semidirect
50. C₃₁×#436	C₂²	semidirect
51. C₃₁×#432	C₂²	semidirect
52. C₃₁×#418	C₂²	semidirect
53. C₄×M₁₆×C₃₁	C₂²	semidirect
185. #62588	C₃₁	direct (N × H)
225. #3518	C₂×C₃₁	semidirect
226. #3498	C₂×C₃₁	semidirect
227. #3362	C₂×C₃₁	semidirect
228. #3358	C₂×C₃₁	semidirect
229. #3325	C₂×C₃₁	semidirect
230. #3324	C₂×C₃₁	semidirect
231. #3235	C₂×C₃₁	semidirect
232. #3217	C₂×C₃₁	semidirect
233. #3171	C₂×C₃₁	semidirect
234. #3138	C₂×C₃₁	semidirect
235. #2971	C₂×C₃₁	semidirect
258. Q₈²	C₂²×C₃₁	semidirect
260. #556	C₂²×C₃₁	semidirect
261. #555	C₂²×C₃₁	semidirect
264. D₄×Q₈	C₂²×C₃₁	semidirect
270. #533	C₂²×C₃₁	semidirect
273. #529	C₂²×C₃₁	semidirect
276. #501	C₂²×C₃₁	semidirect
277. #496	C₂²×C₃₁	semidirect
278. #490	C₂²×C₃₁	semidirect
279. #490	C₂²×C₃₁	semidirect
280. #489	C₂²×C₃₁	semidirect
281. #486	C₂²×C₃₁	semidirect
282. #486	C₂²×C₃₁	semidirect
283. #482	C₂²×C₃₁	semidirect
284. #476	C₂²×C₃₁	semidirect
285. #474	C₂²×C₃₁	semidirect
286. #469	C₂²×C₃₁	semidirect
287. #468	C₂²×C₃₁	semidirect
288. #460	C₂²×C₃₁	semidirect
289. #459	C₂²×C₃₁	semidirect
290. #450	C₂²×C₃₁	semidirect
291. #447	C₂²×C₃₁	semidirect
292. C₈×Q₈	C₂²×C₃₁	semidirect
293. C₄×SD₁₆	C₂²×C₃₁	semidirect
294. C₄×D₈	C₂²×C₃₁	semidirect
295. #436	C₂²×C₃₁	semidirect
296. #432	C₂²×C₃₁	semidirect
297. #418	C₂²×C₃₁	semidirect
298. C₄×M₁₆	C₂²×C₃₁	semidirect
375. C₃₁	#62588	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×#3518	2
C₃₁×#3498	2
C₃₁×#3362	2
C₃₁×#3358	2
C₃₁×#3325	2
C₃₁×#3324	2
C₃₁×#3235	2
C₃₁×#3217	2
C₃₁×#3171	2
C₃₁×#3138	2
C₃₁×#2971	2
#62588	31

Subgroups — C31×#62588 #21519347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62588 #21519347