Subgroups — C₃₁×#17765 #21474524

Order 7936.

964 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3511	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2989	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1561	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1557	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1552	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1547	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1547	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1540	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#332	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#556	—	2	1	no	—
14	C₃₁×#555	C₄	1	2	yes	—
15	C₃₁×#554	C₄	1	2	yes	—
16	C₃₁×#553	C₄	1	4	yes	—
17	C₃₁×#548	—	2	2	no	—
18	C₃₁×#547	—	2	1	no	—
19	C₃₁×#542	—	2	2	no	—
20	C₃₁×#537	—	2	1	no	—
21	C₃₁×#533	C₂²	1	2	yes	—
22	C₃₁×#532	C₂²	1	1	yes	characteristic
23	C₃₁×#532	C₂²	1	2	yes	—
24	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃₁×#520	—	2	1	no	—
26	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#446	—	2	1	no	—
28	C₃₁×#436	—	2	1	no	—
29	C₃₁×#435	—	2	2	no	—
30	C₃₁×#432	C₂²	1	2	yes	—
31	C₃₁×#431	C₂²	1	1	yes	characteristic
32	C₃₁×#424	—	2	1	no	—
33	C₃₁×#424	—	2	2	no	—
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₃₁×#423	C₂²	1	2	yes	—
36	C₃₁×#423	C₂²	1	2	yes	—
37	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
40	C₃₁×#332	C₂²	1	2	yes	—
41	C₃₁×#332	C₂²	1	2	yes	—
42	C₃₁×#332	C₂²	1	1	yes	characteristic
43	C₃₁×#332	C₂²	1	1	yes	characteristic
44	C₃₁×#332	C₂²	1	2	yes	—
45	C₃₁×#332	C₂²	1	2	yes	—
46	C₃₁×#332	C₂²	1	2	yes	—
47	C₃₁×#331	C₂²	1	1	yes	characteristic
48	C₃₁×#331	C₂²	1	1	yes	characteristic
49	C₃₁×#331	C₂²	1	2	yes	—
50	C₂×#41×C₃₁	—	2	1	no	—
51	C₂×#41×C₃₁	—	2	2	no	—
52	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
53	C₂²×Q₈×C₃₁	—	2	1	no	—
54	C₃₁×#131	—	4	4	no	—
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	2	1	no	—
57	C₂×M₁₆×C₃₁	—	2	2	no	—
58	C₂²×C₈×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₂²×C₈×C₃₁	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	—	2	1	no	—
63	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
64	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₃₁×#127	C₂³	1	1	yes	characteristic
67	C₃₁×#126	—	2	4	no	—
68	C₃₁×#125	C₂×C₄	1	4	yes	—
69	C₃₁×#124	—	2	4	no	—
70	C₃₁×#124	—	2	1	no	—
71	C₃₁×#124	—	2	2	no	—
72	C₃₁×#123	—	2	4	no	—
73	C₃₁×C₂²⋊Q₈	—	2	2	no	—
74	C₃₁×C₂²⋊Q₈	—	2	2	no	—
75	C₃₁×C₂²⋊Q₈	—	2	2	no	—
76	C₃₁×C₂²⋊Q₈	—	2	4	no	—
77	C₃₁×#121	—	2	2	no	—
78	C₃₁×#121	—	2	4	no	—
79	C₃₁×#121	—	2	2	no	—
80	C₃₁×#121	—	2	1	no	—
81	C₃₁×#121	C₂×C₄	1	2	yes	—
82	C₃₁×#121	C₂×C₄	1	4	yes	—
83	C₄×Q₈×C₃₁	—	2	2	no	—
84	C₄×Q₈×C₃₁	—	2	2	no	—
85	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
87	C₄×D₄×C₃₁	—	2	4	no	—
88	C₄×D₄×C₃₁	—	2	2	no	—
89	C₄×D₄×C₃₁	—	2	1	no	—
90	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
91	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
92	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
93	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₃₁	D₄	1	2	yes	—
95	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
96	C₃₁×#117	—	2	2	no	—
97	C₃₁×#117	C₂×C₄	1	4	yes	—
98	C₃₁×#117	D₄	1	1	yes	characteristic
99	C₃₁×#117	D₄	1	2	yes	—
100	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
101	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
105	C₃₁×C₄⋊C₈	—	2	2	no	—
106	C₃₁×C₄⋊C₈	—	2	1	no	—
107	C₃₁×C₄⋊C₈	—	2	1	no	—
108	C₃₁×C₄⋊C₈	—	2	2	no	—
109	C₃₁×C₄⋊C₈	—	2	1	no	—
110	C₃₁×C₄⋊C₈	—	2	2	no	—
111	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
115	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₃₁×C₂²⋊C₈	—	2	1	no	—
118	C₃₁×C₂²⋊C₈	—	2	2	no	—
119	C₃₁×#97	—	2	2	no	—
120	C₄×C₈×C₃₁	—	2	1	no	—
121	#41×C₃₁	—	2	2	no	—
122	#41×C₃₁	—	2	2	no	—
123	#41×C₃₁	—	4	2	no	—
124	#41×C₃₁	—	4	2	no	—
125	#41×C₃₁	—	8	2	no	—
126	#41×C₃₁	—	8	2	no	—
127	C₂×Q₈×C₃₁	—	2	2	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	—	2	2	no	—
130	C₂×Q₈×C₃₁	—	8	1	no	—
131	C₂×Q₈×C₃₁	—	8	1	no	—
132	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	2	4	no	—
135	C₂×D₄×C₃₁	—	2	1	no	—
136	C₂×D₄×C₃₁	—	2	1	no	—
137	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
138	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
139	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
140	C₂²×C₄×C₃₁	—	2	2	no	—
141	C₂²×C₄×C₃₁	—	2	2	no	—
142	C₂²×C₄×C₃₁	—	2	4	no	—
143	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
149	M₁₆×C₃₁	—	4	4	no	—
150	M₁₆×C₃₁	—	4	2	no	—
151	M₁₆×C₃₁	—	8	2	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	1	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	—	2	2	no	—
157	C₂×C₈×C₃₁	—	2	1	no	—
158	C₂×C₈×C₃₁	—	4	4	no	—
159	C₂×C₈×C₃₁	—	4	2	no	—
160	C₂×C₈×C₃₁	—	8	1	no	—
161	C₄⋊C₄×C₃₁	—	2	2	no	—
162	C₄⋊C₄×C₃₁	—	2	4	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	4	no	—
165	C₄⋊C₄×C₃₁	—	2	2	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
168	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
169	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
170	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
171	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
172	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
173	C₂²⋊C₄×C₃₁	—	2	4	no	—
174	C₂²⋊C₄×C₃₁	—	2	4	no	—
175	C₂²⋊C₄×C₃₁	—	2	2	no	—
176	C₂²⋊C₄×C₃₁	—	2	4	no	—
177	C₂²⋊C₄×C₃₁	—	2	2	no	—
178	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
179	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
180	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
181	C₄²×C₃₁	—	2	4	no	—
182	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
183	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
184	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₃₁	C₂×D₄	1	2	yes	—
186	C₄²×C₃₁	C₂×D₄	1	2	yes	—
187	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
189	#17765	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂³×C₃₁	—	2	2	no	—
191	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
192	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
193	Q₈×C₃₁	—	2	2	no	—
194	Q₈×C₃₁	—	4	1	no	—
195	Q₈×C₃₁	—	4	1	no	—
196	Q₈×C₃₁	—	4	2	no	—
197	Q₈×C₃₁	—	4	2	no	—
198	Q₈×C₃₁	—	8	2	no	—
199	D₄×C₃₁	—	2	4	no	—
200	D₄×C₃₁	—	2	2	no	—
201	D₄×C₃₁	—	4	2	no	—
202	D₄×C₃₁	—	4	4	no	—
203	D₄×C₃₁	—	4	1	no	—
204	D₄×C₃₁	—	4	1	no	—
205	D₄×C₃₁	—	4	1	no	—
206	D₄×C₃₁	—	4	1	no	—
207	D₄×C₃₁	—	8	4	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	4	4	no	—
219	C₂×C₄×C₃₁	—	8	4	no	—
220	C₂×C₄×C₃₁	—	8	2	no	—
221	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
222	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
224	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
225	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
228	C₂×C₄×C₃₁	#137	1	2	yes	—
229	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
230	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₈×C₃₁	—	4	1	no	—
233	C₈×C₃₁	—	4	2	no	—
234	C₈×C₃₁	—	4	1	no	—
235	C₈×C₃₁	—	4	2	no	—
236	C₈×C₃₁	—	4	1	no	—
237	C₈×C₃₁	—	4	1	no	—
238	#3511	C₂×C₃₁	1	1	yes	characteristic
239	#2989	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#2906	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1561	C₂×C₃₁	1	2	yes	complemented
242	#1557	C₂×C₃₁	1	2	yes	complemented
243	#1552	C₂×C₃₁	1	2	yes	complemented
244	#1549	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#1547	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#1547	C₂×C₃₁	1	2	yes	complemented
247	#1540	C₂×C₃₁	1	1	yes	characteristic, complemented
248	C₂×#332	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂²×C₃₁	—	2	1	no	—
250	C₂²×C₃₁	—	2	1	no	—
251	C₂²×C₃₁	—	2	2	no	—
252	C₂²×C₃₁	—	2	2	no	—
253	C₂²×C₃₁	—	2	2	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	—	4	2	no	—
256	C₂²×C₃₁	—	4	2	no	—
257	C₂²×C₃₁	#417	1	2	yes	—
258	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
259	C₄×C₃₁	—	2	1	no	—
260	C₄×C₃₁	—	2	2	no	—
261	C₄×C₃₁	—	2	1	no	—
262	C₄×C₃₁	—	2	2	no	—
263	C₄×C₃₁	—	2	2	no	—
264	C₄×C₃₁	—	2	1	no	—
265	C₄×C₃₁	—	4	2	no	—
266	C₄×C₃₁	—	4	4	no	—
267	C₄×C₃₁	#417	1	2	yes	—
268	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
269	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
270	C₄×C₃₁	#576	1	1	yes	characteristic
271	C₄×C₃₁	#577	1	1	yes	characteristic
272	C₂×#131	—	2	1	no	—
273	#556	—	2	1	no	—
274	#555	C₄×C₃₁	1	2	yes	—
275	#554	C₄×C₃₁	1	2	yes	—
276	#553	C₄×C₃₁	1	4	yes	—
277	#548	—	2	2	no	—
278	#547	—	2	1	no	—
279	#542	—	2	2	no	—
280	#537	—	2	1	no	—
281	#533	C₂²×C₃₁	1	2	yes	—
282	#532	C₂²×C₃₁	1	1	yes	characteristic
283	#532	C₂²×C₃₁	1	2	yes	—
284	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
285	#520	—	2	1	no	—
286	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
287	#446	—	2	1	no	—
288	#436	—	2	1	no	—
289	#435	—	2	2	no	—
290	#432	C₂²×C₃₁	1	2	yes	—
291	#431	C₂²×C₃₁	1	1	yes	characteristic
292	#424	—	2	1	no	—
293	#424	—	2	2	no	—
294	#423	C₂²×C₃₁	1	1	yes	characteristic
295	#423	C₂²×C₃₁	1	2	yes	—
296	#423	C₂²×C₃₁	1	2	yes	—
297	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
299	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
300	#332	C₂²×C₃₁	1	2	yes	—
301	#332	C₂²×C₃₁	1	2	yes	—
302	#332	C₂²×C₃₁	1	1	yes	characteristic
303	#332	C₂²×C₃₁	1	1	yes	characteristic
304	#332	C₂²×C₃₁	1	2	yes	—
305	#332	C₂²×C₃₁	1	2	yes	—
306	#332	C₂²×C₃₁	1	2	yes	—
307	#331	C₂²×C₃₁	1	1	yes	characteristic
308	#331	C₂²×C₃₁	1	1	yes	characteristic
309	#331	C₂²×C₃₁	1	2	yes	—
310	C₂×C₃₁	—	2	2	no	—
311	C₂×C₃₁	—	4	2	no	—
312	C₂×C₃₁	—	8	2	no	—
313	C₂×C₃₁	#2889	1	1	yes	characteristic
314	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
315	C₂×C₃₁	#2897	1	1	yes	characteristic
316	C₂×#41	—	2	1	no	—
317	C₂×#41	—	2	2	no	—
318	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
319	C₂²×Q₈	—	2	1	no	—
320	#131	—	4	4	no	—
321	C₂×M₁₆	—	2	2	no	—
322	C₂×M₁₆	—	2	1	no	—
323	C₂×M₁₆	—	2	2	no	—
324	C₂²×C₈	—	2	2	no	—
325	C₂²×C₈	—	2	1	no	—
326	C₂²×C₈	—	2	1	no	—
327	C₄⋊Q₈	—	2	1	no	—
328	C₄⋊Q₈	—	2	1	no	—
329	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
330	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
331	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
332	#127	C₂³×C₃₁	1	1	yes	characteristic
333	#126	—	2	4	no	—
334	#125	C₂×C₄×C₃₁	1	4	yes	—
335	#124	—	2	4	no	—
336	#124	—	2	1	no	—
337	#124	—	2	2	no	—
338	#123	—	2	4	no	—
339	C₂²⋊Q₈	—	2	2	no	—
340	C₂²⋊Q₈	—	2	2	no	—
341	C₂²⋊Q₈	—	2	2	no	—
342	C₂²⋊Q₈	—	2	4	no	—
343	#121	—	2	2	no	—
344	#121	—	2	4	no	—
345	#121	—	2	2	no	—
346	#121	—	2	1	no	—
347	#121	C₂×C₄×C₃₁	1	2	yes	—
348	#121	C₂×C₄×C₃₁	1	4	yes	—
349	C₄×Q₈	—	2	2	no	—
350	C₄×Q₈	—	2	2	no	—
351	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
352	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
353	C₄×D₄	—	2	4	no	—
354	C₄×D₄	—	2	2	no	—
355	C₄×D₄	—	2	1	no	—
356	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
357	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
358	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
359	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
360	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
361	C₄×D₄	D₄×C₃₁	1	2	yes	—
362	#117	—	2	2	no	—
363	#117	C₂×C₄×C₃₁	1	4	yes	—
364	#117	D₄×C₃₁	1	1	yes	characteristic
365	#117	D₄×C₃₁	1	2	yes	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	2	no	—
368	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
369	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
371	C₄⋊C₈	—	2	2	no	—
372	C₄⋊C₈	—	2	1	no	—
373	C₄⋊C₈	—	2	1	no	—
374	C₄⋊C₈	—	2	2	no	—
375	C₄⋊C₈	—	2	1	no	—
376	C₄⋊C₈	—	2	2	no	—
377	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
378	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
379	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
380	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
381	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
382	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
383	C₂²⋊C₈	—	2	1	no	—
384	C₂²⋊C₈	—	2	2	no	—
385	#97	—	2	2	no	—
386	C₄×C₈	—	2	1	no	—
387	C₃₁	#17765	1	1	yes	characteristic, direct product factor, Sylow
388	#41	—	2	2	no	—
389	#41	—	2	2	no	—
390	#41	—	4	2	no	—
391	#41	—	4	2	no	—
392	#41	—	8	2	no	—
393	#41	—	8	2	no	—
394	C₂×Q₈	—	2	2	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	2	no	—
397	C₂×Q₈	—	8	1	no	—
398	C₂×Q₈	—	8	1	no	—
399	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	1	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
405	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
406	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
407	C₂²×C₄	—	2	2	no	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	4	no	—
410	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
411	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
412	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
414	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
415	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
416	M₁₆	—	4	4	no	—
417	M₁₆	—	4	2	no	—
418	M₁₆	—	8	2	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	1	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	4	4	no	—
426	C₂×C₈	—	4	2	no	—
427	C₂×C₈	—	8	1	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	1	no	—
434	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
435	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
436	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
437	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
438	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
439	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
440	C₂²⋊C₄	—	2	4	no	—
441	C₂²⋊C₄	—	2	4	no	—
442	C₂²⋊C₄	—	2	2	no	—
443	C₂²⋊C₄	—	2	4	no	—
444	C₂²⋊C₄	—	2	2	no	—
445	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
446	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
447	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
448	C₄²	—	2	4	no	—
449	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
450	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
451	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
452	C₄²	C₂×D₄×C₃₁	1	2	yes	—
453	C₄²	C₂×D₄×C₃₁	1	2	yes	—
454	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
456	C₂³	—	2	2	no	—
457	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
458	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
459	Q₈	—	2	2	no	—
460	Q₈	—	4	1	no	—
461	Q₈	—	4	1	no	—
462	Q₈	—	4	2	no	—
463	Q₈	—	4	2	no	—
464	Q₈	—	8	2	no	—
465	D₄	—	2	4	no	—
466	D₄	—	2	2	no	—
467	D₄	—	4	2	no	—
468	D₄	—	4	4	no	—
469	D₄	—	4	1	no	—
470	D₄	—	4	1	no	—
471	D₄	—	4	1	no	—
472	D₄	—	4	1	no	—
473	D₄	—	8	4	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	1	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	4	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	4	no	—
485	C₂×C₄	—	8	4	no	—
486	C₂×C₄	—	8	2	no	—
487	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
488	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
489	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
490	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
491	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
493	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
494	C₂×C₄	C₃₁×#137	1	2	yes	—
495	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
496	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
497	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
498	C₈	—	4	1	no	—
499	C₈	—	4	2	no	—
500	C₈	—	4	1	no	—
501	C₈	—	4	2	no	—
502	C₈	—	4	1	no	—
503	C₈	—	4	1	no	—
504	C₂²	—	2	1	no	—
505	C₂²	—	2	1	no	—
506	C₂²	—	2	2	no	—
507	C₂²	—	2	2	no	—
508	C₂²	—	2	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	2	no	—
512	C₂²	C₃₁×#417	1	2	yes	—
513	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
514	C₄	—	2	1	no	—
515	C₄	—	2	2	no	—
516	C₄	—	2	1	no	—
517	C₄	—	2	2	no	—
518	C₄	—	2	2	no	—
519	C₄	—	2	1	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	4	no	—
522	C₄	C₃₁×#417	1	2	yes	—
523	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
524	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
525	C₄	C₃₁×#576	1	1	yes	characteristic
526	C₄	C₃₁×#577	1	1	yes	characteristic
527	C₂	—	2	2	no	—
528	C₂	—	4	2	no	—
529	C₂	—	8	2	no	—
530	C₂	C₃₁×#2889	1	1	yes	characteristic
531	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
532	C₂	C₃₁×#2897	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3511	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
2. C₃₁×#2989	C₂	semidirect
3. C₃₁×#2906	C₂	semidirect
4. C₃₁×#1561	C₂	semidirect
5. C₃₁×#1557	C₂	semidirect
6. C₃₁×#1552	C₂	semidirect
7. C₃₁×#1549	C₂	semidirect
8. C₃₁×#1547	C₂	semidirect
9. C₃₁×#1547	C₂	semidirect
10. C₃₁×#1540	C₂	semidirect
11. C₂×C₃₁×#332	C₂	semidirect
189. #17765	C₃₁	direct (N × H)
239. #2989	C₂×C₃₁	semidirect
240. #2906	C₂×C₃₁	semidirect
241. #1561	C₂×C₃₁	semidirect
242. #1557	C₂×C₃₁	semidirect
243. #1552	C₂×C₃₁	semidirect
244. #1549	C₂×C₃₁	semidirect
245. #1547	C₂×C₃₁	semidirect
246. #1547	C₂×C₃₁	semidirect
247. #1540	C₂×C₃₁	semidirect
248. C₂×#332	C₂×C₃₁	semidirect
387. C₃₁	#17765	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×#3511	2
C₃₁×#2989	2
C₃₁×#2906	2
C₃₁×#1561	2
C₃₁×#1557	2
C₃₁×#1552	2
C₃₁×#1549	2
C₃₁×#1547	2
C₃₁×#1540	2
C₂×C₃₁×#332	2
#17765	31

Subgroups — C31×#17765 #21474524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17765 #21474524