Subgroups — C₃₁×#24670 #21481429

Order 7936.

838 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2051	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2018	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1978	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1931	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1800	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
14	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
16	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#460	—	2	1	no	—
19	C₃₁×#460	—	2	1	no	—
20	C₃₁×#459	—	2	1	no	—
21	C₃₁×#459	—	2	1	no	—
22	C₃₁×#459	—	2	1	no	—
23	C₃₁×#459	—	2	1	no	—
24	C₃₁×#459	—	2	2	no	—
25	C₃₁×#442	—	2	1	no	—
26	C₃₁×#440	—	2	1	no	—
27	C₄×SD₁₆×C₃₁	—	2	1	no	—
28	C₄×D₈×C₃₁	—	2	1	no	—
29	C₃₁×#433	—	2	2	no	—
30	C₃₁×#432	—	2	1	no	—
31	C₃₁×#431	—	2	1	no	—
32	C₂×C₃₁×#108	C₂²	1	2	yes	—
33	C₃₁×#428	C₂²	1	1	yes	characteristic
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₃₁×#423	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃₁×#418	C₂²	1	1	yes	characteristic
38	C₃₁×#418	C₂²	1	2	yes	—
39	C₃₁×#417	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#102	C₂²	1	2	yes	—
41	C₂×C₃₁×#100	C₂²	1	2	yes	—
42	C₂×C₃₁×#100	C₂²	1	2	yes	—
43	C₂×C₃₁×#100	C₂²	1	2	yes	—
44	C₃₁×#408	C₂²	1	2	yes	—
45	C₃₁×#405	C₂²	1	2	yes	—
46	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
47	C₃₁×#343	C₂²	1	2	yes	—
48	C₃₁×#341	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
53	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	—	2	1	no	—
55	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
56	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
57	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
58	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
59	C₂×SD₁₆×C₃₁	—	2	1	no	—
60	C₂×SD₁₆×C₃₁	—	2	2	no	—
61	C₂×D₈×C₃₁	—	2	1	no	—
62	C₂×D₈×C₃₁	—	2	2	no	—
63	C₂×D₈×C₃₁	—	2	4	no	—
64	C₂×D₈×C₃₁	—	2	2	no	—
65	C₂×D₈×C₃₁	—	2	2	no	—
66	C₂×D₈×C₃₁	—	4	2	no	—
67	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
69	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
70	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
72	C₃₁×#127	—	2	1	no	—
73	C₃₁×#127	—	2	1	no	—
74	C₃₁×#127	—	2	1	no	—
75	C₃₁×#127	—	2	1	no	—
76	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
77	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
78	C₄×D₄×C₃₁	—	2	1	no	—
79	C₄×D₄×C₃₁	—	2	1	no	—
80	C₄×D₄×C₃₁	—	2	1	no	—
81	C₄×D₄×C₃₁	—	2	1	no	—
82	C₄×D₄×C₃₁	—	2	1	no	—
83	C₄×D₄×C₃₁	—	2	1	no	—
84	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
85	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
86	C₃₁×#117	—	2	1	no	—
87	C₃₁×#117	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
90	C₂×C₄²×C₃₁	—	2	1	no	—
91	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
92	C₃₁×#108	—	4	2	no	—
93	C₃₁×#107	—	2	1	no	—
94	C₃₁×#106	—	2	1	no	—
95	C₃₁×C₄⋊C₈	—	2	1	no	—
96	C₃₁×C₄⋊C₈	—	2	1	no	—
97	C₃₁×C₄⋊C₈	—	2	1	no	—
98	C₃₁×C₄⋊C₈	—	2	1	no	—
99	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
100	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
101	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
102	C₃₁×#103	—	2	1	no	—
103	C₃₁×#102	—	2	1	no	—
104	C₃₁×#102	—	2	2	no	—
105	C₃₁×#102	—	2	2	no	—
106	C₃₁×#102	—	2	1	no	—
107	C₃₁×#102	—	2	1	no	—
108	C₃₁×#102	—	2	2	no	—
109	C₃₁×#102	—	2	2	no	—
110	C₃₁×#100	—	2	4	no	—
111	C₃₁×#100	—	4	2	no	—
112	C₃₁×#100	—	4	2	no	—
113	C₃₁×C₂²⋊C₈	—	2	2	no	—
114	C₃₁×C₂²⋊C₈	—	2	2	no	—
115	C₃₁×#97	—	2	2	no	—
116	C₃₁×#97	—	2	1	no	—
117	C₄×C₈×C₃₁	—	2	2	no	—
118	C₄×C₈×C₃₁	—	2	1	no	—
119	C₂⁴×C₃₁	—	2	1	no	—
120	C₂⁴×C₃₁	—	2	2	no	—
121	#41×C₃₁	—	2	1	no	—
122	#41×C₃₁	—	2	1	no	—
123	#41×C₃₁	—	2	1	no	—
124	#41×C₃₁	—	2	1	no	—
125	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂×D₄×C₃₁	—	2	1	no	—
127	C₂×D₄×C₃₁	—	2	1	no	—
128	C₂×D₄×C₃₁	—	2	1	no	—
129	C₂×D₄×C₃₁	—	2	2	no	—
130	C₂×D₄×C₃₁	—	2	2	no	—
131	C₂×D₄×C₃₁	—	2	1	no	—
132	C₂×D₄×C₃₁	—	2	2	no	—
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	2	1	no	—
135	C₂×D₄×C₃₁	—	2	1	no	—
136	C₂×D₄×C₃₁	—	2	2	no	—
137	C₂×D₄×C₃₁	—	2	2	no	—
138	C₂×D₄×C₃₁	—	4	1	no	—
139	C₂×D₄×C₃₁	—	4	1	no	—
140	C₂×D₄×C₃₁	—	4	1	no	—
141	C₂×D₄×C₃₁	—	4	1	no	—
142	C₂×D₄×C₃₁	—	8	2	no	—
143	C₂×D₄×C₃₁	—	8	2	no	—
144	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂×D₄×C₃₁	#41	1	2	yes	—
146	C₂²×C₄×C₃₁	—	2	1	no	—
147	C₂²×C₄×C₃₁	—	2	1	no	—
148	C₂²×C₄×C₃₁	—	2	1	no	—
149	C₂²×C₄×C₃₁	—	2	1	no	—
150	C₂²×C₄×C₃₁	—	4	1	no	—
151	C₂²×C₄×C₃₁	—	4	1	no	—
152	C₂²×C₄×C₃₁	—	8	2	no	—
153	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
156	SD₁₆×C₃₁	—	2	4	no	—
157	SD₁₆×C₃₁	—	4	4	no	—
158	D₈×C₃₁	—	2	4	no	—
159	D₈×C₃₁	—	4	4	no	—
160	D₈×C₃₁	—	4	4	no	—
161	D₈×C₃₁	—	4	4	no	—
162	M₁₆×C₃₁	—	2	1	no	—
163	M₁₆×C₃₁	—	2	2	no	—
164	M₁₆×C₃₁	—	2	1	no	—
165	M₁₆×C₃₁	—	2	2	no	—
166	M₁₆×C₃₁	—	2	2	no	—
167	M₁₆×C₃₁	—	2	1	no	—
168	M₁₆×C₃₁	—	2	2	no	—
169	M₁₆×C₃₁	#41	1	2	yes	—
170	C₂×C₈×C₃₁	—	2	1	no	—
171	C₂×C₈×C₃₁	—	2	2	no	—
172	C₂×C₈×C₃₁	—	2	2	no	—
173	C₂×C₈×C₃₁	—	2	2	no	—
174	C₂×C₈×C₃₁	—	2	1	no	—
175	C₂×C₈×C₃₁	—	2	2	no	—
176	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
177	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
178	C₄⋊C₄×C₃₁	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₂²⋊C₄×C₃₁	—	2	1	no	—
185	C₂²⋊C₄×C₃₁	—	2	1	no	—
186	C₂²⋊C₄×C₃₁	—	2	1	no	—
187	C₂²⋊C₄×C₃₁	—	4	1	no	—
188	C₂²⋊C₄×C₃₁	—	4	1	no	—
189	C₄²×C₃₁	—	2	1	no	—
190	C₄²×C₃₁	—	2	1	no	—
191	C₄²×C₃₁	—	2	1	no	—
192	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
196	#24670	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	1	no	—
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	—	4	1	no	—
200	C₂³×C₃₁	—	4	1	no	—
201	C₂³×C₃₁	—	4	2	no	—
202	C₂³×C₃₁	—	4	2	no	—
203	C₂³×C₃₁	—	4	2	no	—
204	C₂³×C₃₁	—	8	1	no	—
205	C₂³×C₃₁	—	8	1	no	—
206	C₂³×C₃₁	—	8	2	no	—
207	C₂³×C₃₁	—	8	2	no	—
208	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
209	Q₈×C₃₁	—	2	1	no	—
210	Q₈×C₃₁	—	2	1	no	—
211	D₄×C₃₁	—	2	1	no	—
212	D₄×C₃₁	—	2	2	no	—
213	D₄×C₃₁	—	2	1	no	—
214	D₄×C₃₁	—	2	2	no	—
215	D₄×C₃₁	—	4	1	no	—
216	D₄×C₃₁	—	4	1	no	—
217	D₄×C₃₁	—	4	2	no	—
218	D₄×C₃₁	—	4	1	no	—
219	D₄×C₃₁	—	4	2	no	—
220	D₄×C₃₁	—	4	2	no	—
221	D₄×C₃₁	—	4	1	no	—
222	D₄×C₃₁	—	4	2	no	—
223	D₄×C₃₁	—	8	2	no	—
224	D₄×C₃₁	—	8	2	no	—
225	D₄×C₃₁	—	8	2	no	—
226	D₄×C₃₁	—	8	2	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	1	no	—
229	C₂×C₄×C₃₁	—	2	1	no	—
230	C₂×C₄×C₃₁	—	2	1	no	—
231	C₂×C₄×C₃₁	—	4	1	no	—
232	C₂×C₄×C₃₁	—	4	1	no	—
233	C₂×C₄×C₃₁	—	4	1	no	—
234	C₂×C₄×C₃₁	—	4	2	no	—
235	C₂×C₄×C₃₁	—	4	1	no	—
236	C₂×C₄×C₃₁	—	8	1	no	—
237	C₂×C₄×C₃₁	—	8	2	no	—
238	C₂×C₄×C₃₁	—	8	2	no	—
239	C₂×C₄×C₃₁	—	8	1	no	—
240	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
241	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
242	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
243	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
247	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
248	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
249	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
250	C₈×C₃₁	—	2	2	no	—
251	C₈×C₃₁	—	2	2	no	—
252	C₈×C₃₁	—	2	2	no	—
253	C₈×C₃₁	—	4	1	no	—
254	C₈×C₃₁	—	4	2	no	—
255	C₈×C₃₁	—	4	2	no	—
256	#3038	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₂×#459	C₂×C₃₁	1	1	yes	characteristic, complemented
258	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#2925	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#2051	C₂×C₃₁	1	2	yes	complemented
261	#2018	C₂×C₃₁	1	2	yes	complemented
262	#1978	C₂×C₃₁	1	2	yes	complemented
263	#1965	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#1931	C₂×C₃₁	1	2	yes	complemented
266	#1800	C₂×C₃₁	1	1	yes	characteristic, complemented
267	C₂²×C₃₁	—	2	1	no	—
268	C₂²×C₃₁	—	2	1	no	—
269	C₂²×C₃₁	—	4	1	no	—
270	C₂²×C₃₁	—	4	2	no	—
271	C₂²×C₃₁	—	4	1	no	—
272	C₂²×C₃₁	—	4	1	no	—
273	C₂²×C₃₁	—	8	2	no	—
274	C₂²×C₃₁	—	8	2	no	—
275	C₂²×C₃₁	—	8	1	no	—
276	C₂²×C₃₁	—	8	1	no	—
277	C₂²×C₃₁	—	8	2	no	—
278	C₂²×C₃₁	—	8	2	no	—
279	C₂²×C₃₁	—	8	2	no	—
280	C₂²×C₃₁	—	16	2	no	—
281	C₂²×C₃₁	—	16	2	no	—
282	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
283	C₂²×C₃₁	#553	1	1	yes	characteristic
284	C₂²×C₃₁	#555	1	1	yes	characteristic
285	C₄×C₃₁	—	2	1	no	—
286	C₄×C₃₁	—	2	1	no	—
287	C₄×C₃₁	—	2	1	no	—
288	C₄×C₃₁	—	4	1	no	—
289	C₄×C₃₁	—	4	1	no	—
290	C₄×C₃₁	—	4	1	no	—
291	C₄×C₃₁	—	8	1	no	—
292	C₄×C₃₁	#472	1	2	yes	—
293	C₄×C₃₁	#525	1	1	yes	characteristic
294	C₄×C₃₁	#535	1	1	yes	characteristic
295	C₄×C₃₁	#546	1	1	yes	characteristic
296	C₄×C₃₁	#550	1	1	yes	characteristic
297	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
299	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
300	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
301	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
302	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
303	#460	—	2	1	no	—
304	#460	—	2	1	no	—
305	#459	—	2	1	no	—
306	#459	—	2	1	no	—
307	#459	—	2	1	no	—
308	#459	—	2	1	no	—
309	#459	—	2	2	no	—
310	#442	—	2	1	no	—
311	#440	—	2	1	no	—
312	C₄×SD₁₆	—	2	1	no	—
313	C₄×D₈	—	2	1	no	—
314	#433	—	2	2	no	—
315	#432	—	2	1	no	—
316	#431	—	2	1	no	—
317	C₂×#108	C₂²×C₃₁	1	2	yes	—
318	#428	C₂²×C₃₁	1	1	yes	characteristic
319	#423	C₂²×C₃₁	1	1	yes	characteristic
320	#423	C₂²×C₃₁	1	1	yes	characteristic
321	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
322	#418	C₂²×C₃₁	1	1	yes	characteristic
323	#418	C₂²×C₃₁	1	2	yes	—
324	#417	C₂²×C₃₁	1	1	yes	characteristic
325	C₂×#102	C₂²×C₃₁	1	2	yes	—
326	C₂×#100	C₂²×C₃₁	1	2	yes	—
327	C₂×#100	C₂²×C₃₁	1	2	yes	—
328	C₂×#100	C₂²×C₃₁	1	2	yes	—
329	#408	C₂²×C₃₁	1	2	yes	—
330	#405	C₂²×C₃₁	1	2	yes	—
331	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
332	#343	C₂²×C₃₁	1	2	yes	—
333	#341	C₂²×C₃₁	1	2	yes	—
334	C₂×C₃₁	—	2	1	no	—
335	C₂×C₃₁	—	2	1	no	—
336	C₂×C₃₁	—	8	1	no	—
337	C₂×C₃₁	—	8	2	no	—
338	C₂×C₃₁	—	16	2	no	—
339	C₂×C₃₁	#2414	1	1	yes	characteristic
340	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
341	C₂×C₃₁	#3068	1	1	yes	characteristic
342	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
343	C₂²×D₄	—	2	1	no	—
344	C₂²×D₄	—	2	1	no	—
345	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
346	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
347	C₂³×C₄	—	2	1	no	—
348	AGL(1,ℤ/8)	—	2	2	no	—
349	AGL(1,ℤ/8)	—	2	2	no	—
350	AGL(1,ℤ/8)	—	4	2	no	—
351	AGL(1,ℤ/8)	—	4	2	no	—
352	C₂×SD₁₆	—	2	1	no	—
353	C₂×SD₁₆	—	2	2	no	—
354	C₂×D₈	—	2	1	no	—
355	C₂×D₈	—	2	2	no	—
356	C₂×D₈	—	2	4	no	—
357	C₂×D₈	—	2	2	no	—
358	C₂×D₈	—	2	2	no	—
359	C₂×D₈	—	4	2	no	—
360	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
361	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
362	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
363	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
365	#127	—	2	1	no	—
366	#127	—	2	1	no	—
367	#127	—	2	1	no	—
368	#127	—	2	1	no	—
369	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
370	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
371	C₄×D₄	—	2	1	no	—
372	C₄×D₄	—	2	1	no	—
373	C₄×D₄	—	2	1	no	—
374	C₄×D₄	—	2	1	no	—
375	C₄×D₄	—	2	1	no	—
376	C₄×D₄	—	2	1	no	—
377	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
378	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
379	#117	—	2	1	no	—
380	#117	C₂³×C₃₁	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
382	C₂×C₂²⋊C₄	—	2	1	no	—
383	C₂×C₄²	—	2	1	no	—
384	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
385	#108	—	4	2	no	—
386	#107	—	2	1	no	—
387	#106	—	2	1	no	—
388	C₄⋊C₈	—	2	1	no	—
389	C₄⋊C₈	—	2	1	no	—
390	C₄⋊C₈	—	2	1	no	—
391	C₄⋊C₈	—	2	1	no	—
392	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
393	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
394	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
395	#103	—	2	1	no	—
396	#102	—	2	1	no	—
397	#102	—	2	2	no	—
398	#102	—	2	2	no	—
399	#102	—	2	1	no	—
400	#102	—	2	1	no	—
401	#102	—	2	2	no	—
402	#102	—	2	2	no	—
403	#100	—	2	4	no	—
404	#100	—	4	2	no	—
405	#100	—	4	2	no	—
406	C₂²⋊C₈	—	2	2	no	—
407	C₂²⋊C₈	—	2	2	no	—
408	#97	—	2	2	no	—
409	#97	—	2	1	no	—
410	C₄×C₈	—	2	2	no	—
411	C₄×C₈	—	2	1	no	—
412	C₃₁	#24670	1	1	yes	characteristic, direct product factor, Sylow
413	C₂⁴	—	2	1	no	—
414	C₂⁴	—	2	2	no	—
415	#41	—	2	1	no	—
416	#41	—	2	1	no	—
417	#41	—	2	1	no	—
418	#41	—	2	1	no	—
419	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	2	1	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	4	1	no	—
433	C₂×D₄	—	4	1	no	—
434	C₂×D₄	—	4	1	no	—
435	C₂×D₄	—	4	1	no	—
436	C₂×D₄	—	8	2	no	—
437	C₂×D₄	—	8	2	no	—
438	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
439	C₂×D₄	#41×C₃₁	1	2	yes	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	4	1	no	—
445	C₂²×C₄	—	4	1	no	—
446	C₂²×C₄	—	8	2	no	—
447	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
450	SD₁₆	—	2	4	no	—
451	SD₁₆	—	4	4	no	—
452	D₈	—	2	4	no	—
453	D₈	—	4	4	no	—
454	D₈	—	4	4	no	—
455	D₈	—	4	4	no	—
456	M₁₆	—	2	1	no	—
457	M₁₆	—	2	2	no	—
458	M₁₆	—	2	1	no	—
459	M₁₆	—	2	2	no	—
460	M₁₆	—	2	2	no	—
461	M₁₆	—	2	1	no	—
462	M₁₆	—	2	2	no	—
463	M₁₆	#41×C₃₁	1	2	yes	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	2	no	—
466	C₂×C₈	—	2	2	no	—
467	C₂×C₈	—	2	2	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	2	no	—
470	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
471	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₂²⋊C₄	—	2	1	no	—
479	C₂²⋊C₄	—	2	1	no	—
480	C₂²⋊C₄	—	2	1	no	—
481	C₂²⋊C₄	—	4	1	no	—
482	C₂²⋊C₄	—	4	1	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
489	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
490	C₂³	—	2	1	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	4	1	no	—
493	C₂³	—	4	1	no	—
494	C₂³	—	4	2	no	—
495	C₂³	—	4	2	no	—
496	C₂³	—	4	2	no	—
497	C₂³	—	8	1	no	—
498	C₂³	—	8	1	no	—
499	C₂³	—	8	2	no	—
500	C₂³	—	8	2	no	—
501	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
502	Q₈	—	2	1	no	—
503	Q₈	—	2	1	no	—
504	D₄	—	2	1	no	—
505	D₄	—	2	2	no	—
506	D₄	—	2	1	no	—
507	D₄	—	2	2	no	—
508	D₄	—	4	1	no	—
509	D₄	—	4	1	no	—
510	D₄	—	4	2	no	—
511	D₄	—	4	1	no	—
512	D₄	—	4	2	no	—
513	D₄	—	4	2	no	—
514	D₄	—	4	1	no	—
515	D₄	—	4	2	no	—
516	D₄	—	8	2	no	—
517	D₄	—	8	2	no	—
518	D₄	—	8	2	no	—
519	D₄	—	8	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	8	1	no	—
530	C₂×C₄	—	8	2	no	—
531	C₂×C₄	—	8	2	no	—
532	C₂×C₄	—	8	1	no	—
533	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
534	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
535	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
536	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
537	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
538	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
539	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
540	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
541	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
542	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
543	C₈	—	2	2	no	—
544	C₈	—	2	2	no	—
545	C₈	—	2	2	no	—
546	C₈	—	4	1	no	—
547	C₈	—	4	2	no	—
548	C₈	—	4	2	no	—
549	C₂²	—	2	1	no	—
550	C₂²	—	2	1	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	1	no	—
555	C₂²	—	8	2	no	—
556	C₂²	—	8	2	no	—
557	C₂²	—	8	1	no	—
558	C₂²	—	8	1	no	—
559	C₂²	—	8	2	no	—
560	C₂²	—	8	2	no	—
561	C₂²	—	8	2	no	—
562	C₂²	—	16	2	no	—
563	C₂²	—	16	2	no	—
564	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
565	C₂²	C₃₁×#553	1	1	yes	characteristic
566	C₂²	C₃₁×#555	1	1	yes	characteristic
567	C₄	—	2	1	no	—
568	C₄	—	2	1	no	—
569	C₄	—	2	1	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	4	1	no	—
573	C₄	—	8	1	no	—
574	C₄	C₃₁×#472	1	2	yes	—
575	C₄	C₃₁×#525	1	1	yes	characteristic
576	C₄	C₃₁×#535	1	1	yes	characteristic
577	C₄	C₃₁×#546	1	1	yes	characteristic
578	C₄	C₃₁×#550	1	1	yes	characteristic
579	C₂	—	2	1	no	—
580	C₂	—	2	1	no	—
581	C₂	—	8	1	no	—
582	C₂	—	8	2	no	—
583	C₂	—	16	2	no	—
584	C₂	C₃₁×#2414	1	1	yes	characteristic
585	C₂	C₂×C₃₁×#472	1	1	yes	characteristic
586	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3038	C₂	semidirect
2. C₂×C₃₁×#459	C₂	semidirect
3. C₄×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
4. C₃₁×#2925	C₂	semidirect
5. C₃₁×#2051	C₂	semidirect
6. C₃₁×#2018	C₂	semidirect
7. C₃₁×#1978	C₂	semidirect
8. C₃₁×#1965	C₂	semidirect
9. C₃₁×#1950	C₂	semidirect
10. C₃₁×#1931	C₂	semidirect
11. C₃₁×#1800	C₂	semidirect
196. #24670	C₃₁	direct (N × H)
256. #3038	C₂×C₃₁	semidirect
257. C₂×#459	C₂×C₃₁	semidirect
258. C₄×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
259. #2925	C₂×C₃₁	semidirect
260. #2051	C₂×C₃₁	semidirect
261. #2018	C₂×C₃₁	semidirect
262. #1978	C₂×C₃₁	semidirect
263. #1965	C₂×C₃₁	semidirect
264. #1950	C₂×C₃₁	semidirect
265. #1931	C₂×C₃₁	semidirect
266. #1800	C₂×C₃₁	semidirect
412. C₃₁	#24670	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×#3038	2
C₂×C₃₁×#459	2
C₄×C₃₁×AGL(1,ℤ/8)	2
C₃₁×#2925	2
C₃₁×#2051	2
C₃₁×#2018	2
C₃₁×#1978	2
C₃₁×#1965	2
C₃₁×#1950	2
C₃₁×#1931	2
C₃₁×#1800	2
#24670	31

Subgroups — C31×#24670 #21481429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24670 #21481429