Subgroups — C₃₁×#24671 #21481430

Order 7936.

786 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3039	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2052	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2019	C₂	1	2	yes	maximal
7	C₃₁×#1977	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1966	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1930	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1800	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊Q₈	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×#462	—	2	1	no	—
19	C₃₁×#462	—	2	1	no	—
20	C₃₁×#461	—	2	1	no	—
21	C₃₁×#461	—	2	1	no	—
22	C₃₁×#461	—	2	1	no	—
23	C₃₁×#461	—	2	1	no	—
24	C₃₁×#461	—	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×#419	C₂²	1	2	yes	—
38	C₃₁×#418	C₂²	1	1	yes	characteristic
39	C₃₁×#417	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#103	C₂²	1	2	yes	—
41	C₂×C₃₁×#101	C₂²	1	2	yes	—
42	C₂×C₃₁×#101	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×Q₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	—	2	1	no	—
53	C₃₁×#137	—	4	2	no	—
54	C₃₁×#137	—	4	2	no	—
55	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
56	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
57	C₂×Q₁₆×C₃₁	—	2	2	no	—
58	C₂×SD₁₆×C₃₁	—	2	1	no	—
59	C₂×SD₁₆×C₃₁	—	2	2	no	—
60	C₂×SD₁₆×C₃₁	—	2	4	no	—
61	C₂×SD₁₆×C₃₁	—	2	2	no	—
62	C₂×SD₁₆×C₃₁	—	2	2	no	—
63	C₂×SD₁₆×C₃₁	—	4	2	no	—
64	C₂×D₈×C₃₁	—	2	1	no	—
65	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
67	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
68	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
70	C₃₁×C₄⋊Q₈	—	2	1	no	—
71	C₃₁×C₄⋊Q₈	—	2	1	no	—
72	C₃₁×C₄⋊Q₈	—	2	1	no	—
73	C₃₁×C₄⋊Q₈	—	2	1	no	—
74	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
75	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
76	C₄×D₄×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
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	D₄	1	1	yes	characteristic
83	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
84	C₃₁×#117	—	2	1	no	—
85	C₃₁×#117	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
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×#103	—	2	2	no	—
104	C₃₁×#103	—	2	2	no	—
105	C₃₁×#103	—	2	2	no	—
106	C₃₁×#103	—	2	2	no	—
107	C₃₁×#102	—	2	1	no	—
108	C₃₁×#102	—	2	1	no	—
109	C₃₁×#102	—	2	1	no	—
110	C₃₁×#101	—	2	4	no	—
111	C₃₁×#101	—	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	#41×C₃₁	—	2	1	no	—
121	#41×C₃₁	—	2	1	no	—
122	#41×C₃₁	—	2	1	no	—
123	#41×C₃₁	—	2	1	no	—
124	C₂×Q₈×C₃₁	—	2	2	no	—
125	C₂×Q₈×C₃₁	—	2	2	no	—
126	C₂×Q₈×C₃₁	—	2	2	no	—
127	C₂×Q₈×C₃₁	—	2	2	no	—
128	C₂×Q₈×C₃₁	—	2	2	no	—
129	C₂×Q₈×C₃₁	—	2	2	no	—
130	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×D₄×C₃₁	—	2	1	no	—
132	C₂×D₄×C₃₁	—	2	1	no	—
133	C₂×D₄×C₃₁	—	2	1	no	—
134	C₂×D₄×C₃₁	—	2	1	no	—
135	C₂×D₄×C₃₁	—	2	1	no	—
136	C₂×D₄×C₃₁	—	2	1	no	—
137	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂×D₄×C₃₁	#41	1	2	yes	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	1	no	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	2	no	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	—	4	1	no	—
145	C₂²×C₄×C₃₁	—	4	1	no	—
146	C₂²×C₄×C₃₁	—	8	2	no	—
147	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
150	Q₁₆×C₃₁	—	4	4	no	—
151	SD₁₆×C₃₁	—	2	4	no	—
152	SD₁₆×C₃₁	—	4	4	no	—
153	SD₁₆×C₃₁	—	4	4	no	—
154	SD₁₆×C₃₁	—	4	4	no	—
155	D₈×C₃₁	—	2	4	no	—
156	M₁₆×C₃₁	—	2	1	no	—
157	M₁₆×C₃₁	—	2	2	no	—
158	M₁₆×C₃₁	—	2	2	no	—
159	M₁₆×C₃₁	—	2	2	no	—
160	M₁₆×C₃₁	—	2	1	no	—
161	M₁₆×C₃₁	—	2	1	no	—
162	M₁₆×C₃₁	—	2	2	no	—
163	M₁₆×C₃₁	#41	1	2	yes	—
164	C₂×C₈×C₃₁	—	2	1	no	—
165	C₂×C₈×C₃₁	—	2	2	no	—
166	C₂×C₈×C₃₁	—	2	2	no	—
167	C₂×C₈×C₃₁	—	2	2	no	—
168	C₂×C₈×C₃₁	—	2	1	no	—
169	C₂×C₈×C₃₁	—	2	2	no	—
170	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
171	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	4	1	no	—
177	C₄⋊C₄×C₃₁	—	4	1	no	—
178	C₄⋊C₄×C₃₁	—	4	1	no	—
179	C₄⋊C₄×C₃₁	—	4	1	no	—
180	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
182	C₂²⋊C₄×C₃₁	—	2	1	no	—
183	C₂²⋊C₄×C₃₁	—	2	1	no	—
184	C₂²⋊C₄×C₃₁	—	2	1	no	—
185	C₂²⋊C₄×C₃₁	—	4	1	no	—
186	C₂²⋊C₄×C₃₁	—	4	1	no	—
187	C₄²×C₃₁	—	2	1	no	—
188	C₄²×C₃₁	—	2	1	no	—
189	C₄²×C₃₁	—	2	1	no	—
190	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	#24671	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂³×C₃₁	—	2	1	no	—
196	C₂³×C₃₁	—	2	2	no	—
197	C₂³×C₃₁	—	4	1	no	—
198	C₂³×C₃₁	—	4	1	no	—
199	C₂³×C₃₁	—	8	1	no	—
200	C₂³×C₃₁	—	8	1	no	—
201	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
202	Q₈×C₃₁	—	2	1	no	—
203	Q₈×C₃₁	—	2	1	no	—
204	Q₈×C₃₁	—	4	2	no	—
205	Q₈×C₃₁	—	4	2	no	—
206	Q₈×C₃₁	—	4	2	no	—
207	Q₈×C₃₁	—	4	2	no	—
208	D₄×C₃₁	—	2	1	no	—
209	D₄×C₃₁	—	2	2	no	—
210	D₄×C₃₁	—	2	1	no	—
211	D₄×C₃₁	—	2	2	no	—
212	D₄×C₃₁	—	4	1	no	—
213	D₄×C₃₁	—	4	1	no	—
214	D₄×C₃₁	—	4	1	no	—
215	D₄×C₃₁	—	4	1	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	4	1	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	1	no	—
223	C₂×C₄×C₃₁	—	4	2	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	4	2	no	—
227	C₂×C₄×C₃₁	—	4	2	no	—
228	C₂×C₄×C₃₁	—	8	1	no	—
229	C₂×C₄×C₃₁	—	8	2	no	—
230	C₂×C₄×C₃₁	—	8	2	no	—
231	C₂×C₄×C₃₁	—	8	1	no	—
232	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
233	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
234	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
235	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
240	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
241	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
242	C₈×C₃₁	—	2	2	no	—
243	C₈×C₃₁	—	2	2	no	—
244	C₈×C₃₁	—	2	2	no	—
245	C₈×C₃₁	—	4	1	no	—
246	C₈×C₃₁	—	4	2	no	—
247	C₈×C₃₁	—	4	2	no	—
248	#3039	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂×#461	C₂×C₃₁	1	1	yes	characteristic, complemented
250	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic
251	#2925	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#2052	C₂×C₃₁	1	2	yes	complemented
253	#2019	C₂×C₃₁	1	2	yes	—
254	#1977	C₂×C₃₁	1	2	yes	complemented
255	#1966	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1930	C₂×C₃₁	1	2	yes	complemented
258	#1800	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂²×C₃₁	—	2	1	no	—
260	C₂²×C₃₁	—	2	1	no	—
261	C₂²×C₃₁	—	4	1	no	—
262	C₂²×C₃₁	—	4	2	no	—
263	C₂²×C₃₁	—	4	1	no	—
264	C₂²×C₃₁	—	4	1	no	—
265	C₂²×C₃₁	—	8	2	no	—
266	C₂²×C₃₁	—	8	2	no	—
267	C₂²×C₃₁	—	8	1	no	—
268	C₂²×C₃₁	—	8	1	no	—
269	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
270	C₂²×C₃₁	#553	1	1	yes	characteristic
271	C₂²×C₃₁	#555	1	1	yes	characteristic
272	C₄×C₃₁	—	2	1	no	—
273	C₄×C₃₁	—	2	1	no	—
274	C₄×C₃₁	—	2	1	no	—
275	C₄×C₃₁	—	4	1	no	—
276	C₄×C₃₁	—	4	1	no	—
277	C₄×C₃₁	—	4	1	no	—
278	C₄×C₃₁	—	8	2	no	—
279	C₄×C₃₁	—	8	1	no	—
280	C₄×C₃₁	#471	1	2	yes	—
281	C₄×C₃₁	#525	1	1	yes	characteristic
282	C₄×C₃₁	#535	1	1	yes	characteristic
283	C₄×C₃₁	#546	1	1	yes	characteristic
284	C₄×C₃₁	#550	1	1	yes	characteristic
285	C₂×#137	C₂²×C₃₁	1	2	yes	—
286	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
287	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
288	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
289	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
291	#462	—	2	1	no	—
292	#462	—	2	1	no	—
293	#461	—	2	1	no	—
294	#461	—	2	1	no	—
295	#461	—	2	1	no	—
296	#461	—	2	1	no	—
297	#461	—	2	2	no	—
298	#442	—	2	1	no	—
299	#440	—	2	1	no	—
300	C₄×SD₁₆	—	2	1	no	—
301	C₄×D₈	—	2	1	no	—
302	#433	—	2	2	no	—
303	#432	—	2	1	no	—
304	#431	—	2	1	no	—
305	C₂×#108	C₂²×C₃₁	1	2	yes	—
306	#428	C₂²×C₃₁	1	1	yes	characteristic
307	#423	C₂²×C₃₁	1	1	yes	characteristic
308	#423	C₂²×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
310	#419	C₂²×C₃₁	1	2	yes	—
311	#418	C₂²×C₃₁	1	1	yes	characteristic
312	#417	C₂²×C₃₁	1	1	yes	characteristic
313	C₂×#103	C₂²×C₃₁	1	2	yes	—
314	C₂×#101	C₂²×C₃₁	1	2	yes	—
315	C₂×#101	C₂²×C₃₁	1	2	yes	—
316	C₂×#100	C₂²×C₃₁	1	2	yes	—
317	#408	C₂²×C₃₁	1	2	yes	—
318	#405	C₂²×C₃₁	1	2	yes	—
319	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
320	#343	C₂²×C₃₁	1	2	yes	—
321	#341	C₂²×C₃₁	1	2	yes	—
322	C₂×C₃₁	—	2	1	no	—
323	C₂×C₃₁	—	2	1	no	—
324	C₂×C₃₁	—	8	1	no	—
325	C₂×C₃₁	—	8	2	no	—
326	C₂×C₃₁	#2414	1	1	yes	characteristic
327	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
328	C₂×C₃₁	#3069	1	1	yes	characteristic
329	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
330	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
331	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
332	C₂³×C₄	—	2	1	no	—
333	#137	—	4	2	no	—
334	#137	—	4	2	no	—
335	AGL(1,ℤ/8)	—	2	2	no	—
336	AGL(1,ℤ/8)	—	2	2	no	—
337	C₂×Q₁₆	—	2	2	no	—
338	C₂×SD₁₆	—	2	1	no	—
339	C₂×SD₁₆	—	2	2	no	—
340	C₂×SD₁₆	—	2	4	no	—
341	C₂×SD₁₆	—	2	2	no	—
342	C₂×SD₁₆	—	2	2	no	—
343	C₂×SD₁₆	—	4	2	no	—
344	C₂×D₈	—	2	1	no	—
345	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
346	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
347	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
348	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
349	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
350	C₄⋊Q₈	—	2	1	no	—
351	C₄⋊Q₈	—	2	1	no	—
352	C₄⋊Q₈	—	2	1	no	—
353	C₄⋊Q₈	—	2	1	no	—
354	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
355	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
356	C₄×D₄	—	2	1	no	—
357	C₄×D₄	—	2	1	no	—
358	C₄×D₄	—	2	1	no	—
359	C₄×D₄	—	2	1	no	—
360	C₄×D₄	—	2	1	no	—
361	C₄×D₄	—	2	1	no	—
362	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
363	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
364	#117	—	2	1	no	—
365	#117	C₂³×C₃₁	1	1	yes	characteristic
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
369	C₂×C₂²⋊C₄	—	2	1	no	—
370	C₂×C₄²	—	2	1	no	—
371	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
372	#108	—	4	2	no	—
373	#107	—	2	1	no	—
374	#106	—	2	1	no	—
375	C₄⋊C₈	—	2	1	no	—
376	C₄⋊C₈	—	2	1	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	—	2	1	no	—
379	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
380	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
381	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
382	#103	—	2	1	no	—
383	#103	—	2	2	no	—
384	#103	—	2	2	no	—
385	#103	—	2	2	no	—
386	#103	—	2	2	no	—
387	#102	—	2	1	no	—
388	#102	—	2	1	no	—
389	#102	—	2	1	no	—
390	#101	—	2	4	no	—
391	#101	—	4	2	no	—
392	#100	—	4	2	no	—
393	C₂²⋊C₈	—	2	2	no	—
394	C₂²⋊C₈	—	2	2	no	—
395	#97	—	2	2	no	—
396	#97	—	2	1	no	—
397	C₄×C₈	—	2	2	no	—
398	C₄×C₈	—	2	1	no	—
399	C₃₁	#24671	1	1	yes	characteristic, direct product factor, Sylow
400	C₂⁴	—	2	1	no	—
401	#41	—	2	1	no	—
402	#41	—	2	1	no	—
403	#41	—	2	1	no	—
404	#41	—	2	1	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	2	no	—
411	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
412	C₂×D₄	—	2	1	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
419	C₂×D₄	#41×C₃₁	1	2	yes	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	4	1	no	—
426	C₂²×C₄	—	4	1	no	—
427	C₂²×C₄	—	8	2	no	—
428	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
429	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
430	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
431	Q₁₆	—	4	4	no	—
432	SD₁₆	—	2	4	no	—
433	SD₁₆	—	4	4	no	—
434	SD₁₆	—	4	4	no	—
435	SD₁₆	—	4	4	no	—
436	D₈	—	2	4	no	—
437	M₁₆	—	2	1	no	—
438	M₁₆	—	2	2	no	—
439	M₁₆	—	2	2	no	—
440	M₁₆	—	2	2	no	—
441	M₁₆	—	2	1	no	—
442	M₁₆	—	2	1	no	—
443	M₁₆	—	2	2	no	—
444	M₁₆	#41×C₃₁	1	2	yes	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	2	no	—
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
452	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	4	1	no	—
458	C₄⋊C₄	—	4	1	no	—
459	C₄⋊C₄	—	4	1	no	—
460	C₄⋊C₄	—	4	1	no	—
461	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₂²⋊C₄	—	2	1	no	—
464	C₂²⋊C₄	—	2	1	no	—
465	C₂²⋊C₄	—	2	1	no	—
466	C₂²⋊C₄	—	4	1	no	—
467	C₂²⋊C₄	—	4	1	no	—
468	C₄²	—	2	1	no	—
469	C₄²	—	2	1	no	—
470	C₄²	—	2	1	no	—
471	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
472	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
473	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₂³	—	2	1	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	4	1	no	—
478	C₂³	—	4	1	no	—
479	C₂³	—	8	1	no	—
480	C₂³	—	8	1	no	—
481	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
482	Q₈	—	2	1	no	—
483	Q₈	—	2	1	no	—
484	Q₈	—	4	2	no	—
485	Q₈	—	4	2	no	—
486	Q₈	—	4	2	no	—
487	Q₈	—	4	2	no	—
488	D₄	—	2	1	no	—
489	D₄	—	2	2	no	—
490	D₄	—	2	1	no	—
491	D₄	—	2	2	no	—
492	D₄	—	4	1	no	—
493	D₄	—	4	1	no	—
494	D₄	—	4	1	no	—
495	D₄	—	4	1	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	1	no	—
498	C₂×C₄	—	2	1	no	—
499	C₂×C₄	—	2	1	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	8	1	no	—
509	C₂×C₄	—	8	2	no	—
510	C₂×C₄	—	8	2	no	—
511	C₂×C₄	—	8	1	no	—
512	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
513	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
514	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
515	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
516	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
517	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
518	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
519	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
520	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
521	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
522	C₈	—	2	2	no	—
523	C₈	—	2	2	no	—
524	C₈	—	2	2	no	—
525	C₈	—	4	1	no	—
526	C₈	—	4	2	no	—
527	C₈	—	4	2	no	—
528	C₂²	—	2	1	no	—
529	C₂²	—	2	1	no	—
530	C₂²	—	4	1	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	1	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	8	2	no	—
535	C₂²	—	8	2	no	—
536	C₂²	—	8	1	no	—
537	C₂²	—	8	1	no	—
538	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
539	C₂²	C₃₁×#553	1	1	yes	characteristic
540	C₂²	C₃₁×#555	1	1	yes	characteristic
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	1	no	—
544	C₄	—	4	1	no	—
545	C₄	—	4	1	no	—
546	C₄	—	4	1	no	—
547	C₄	—	8	2	no	—
548	C₄	—	8	1	no	—
549	C₄	C₃₁×#471	1	2	yes	—
550	C₄	C₃₁×#525	1	1	yes	characteristic
551	C₄	C₃₁×#535	1	1	yes	characteristic
552	C₄	C₃₁×#546	1	1	yes	characteristic
553	C₄	C₃₁×#550	1	1	yes	characteristic
554	C₂	—	2	1	no	—
555	C₂	—	2	1	no	—
556	C₂	—	8	1	no	—
557	C₂	—	8	2	no	—
558	C₂	C₃₁×#2414	1	1	yes	characteristic
559	C₂	C₂×C₃₁×#471	1	1	yes	characteristic
560	C₂	C₃₁×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×AGL(1,ℤ/8)	C₂²
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×#3039	C₂	semidirect
2. C₂×C₃₁×#461	C₂	semidirect
4. C₃₁×#2925	C₂	semidirect
5. C₃₁×#2052	C₂	semidirect
7. C₃₁×#1977	C₂	semidirect
8. C₃₁×#1966	C₂	semidirect
9. C₃₁×#1950	C₂	semidirect
10. C₃₁×#1930	C₂	semidirect
11. C₃₁×#1800	C₂	semidirect
194. #24671	C₃₁	direct (N × H)
248. #3039	C₂×C₃₁	semidirect
249. C₂×#461	C₂×C₃₁	semidirect
251. #2925	C₂×C₃₁	semidirect
252. #2052	C₂×C₃₁	semidirect
254. #1977	C₂×C₃₁	semidirect
255. #1966	C₂×C₃₁	semidirect
256. #1950	C₂×C₃₁	semidirect
257. #1930	C₂×C₃₁	semidirect
258. #1800	C₂×C₃₁	semidirect
399. C₃₁	#24671	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×#3039	2
C₂×C₃₁×#461	2
C₄×C₃₁×AGL(1,ℤ/8)	2
C₃₁×#2925	2
C₃₁×#2052	2
C₃₁×#2019	2
C₃₁×#1977	2
C₃₁×#1966	2
C₃₁×#1950	2
C₃₁×#1930	2
C₃₁×#1800	2
#24671	31

Subgroups — C31×#24671 #21481430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24671 #21481430