Subgroups — C₃₁×#22165 #21478924

Order 7936.

764 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3035	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1980	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1974	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1937	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1876	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1866	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1806	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	2	yes	—
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#464	C₄	1	2	yes	complemented
17	C₃₁×#463	C₄	1	2	yes	complemented
18	C₃₁×#462	C₄	1	1	yes	characteristic, complemented
19	C₃₁×#461	C₄	1	1	yes	characteristic, complemented
20	C₃₁×#460	C₄	1	1	yes	characteristic, complemented
21	C₃₁×#459	C₄	1	1	yes	characteristic, complemented
22	C₃₁×#436	—	2	1	no	—
23	C₃₁×#436	—	2	1	no	—
24	C₈×D₄×C₃₁	—	2	1	no	—
25	C₈×D₄×C₃₁	—	2	1	no	—
26	C₃₁×#433	—	2	1	no	—
27	C₃₁×#433	—	2	1	no	—
28	C₃₁×#431	—	2	1	no	—
29	C₃₁×#431	—	2	1	no	—
30	C₂×C₃₁×#108	C₂²	1	2	yes	—
31	C₃₁×#424	C₂²	1	1	yes	characteristic
32	C₃₁×#424	C₂²	1	1	yes	characteristic
33	C₃₁×#423	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#104	C₂²	1	2	yes	—
36	C₂×C₃₁×#104	C₂²	1	2	yes	—
37	C₂×C₃₁×#104	C₂²	1	2	yes	—
38	C₃₁×#416	C₂²	1	2	yes	—
39	C₃₁×#408	C₂²	1	1	yes	characteristic
40	C₃₁×#408	C₂²	1	1	yes	characteristic
41	C₃₁×#408	C₂²	1	2	yes	—
42	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
46	C₃₁×#341	C₂²	1	2	yes	—
47	C₃₁×#339	C₂²	1	2	yes	—
48	C₃₁×#336	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×#41×C₃₁	C₂³	1	2	yes	—
51	C₃₁×#135	—	4	2	no	—
52	C₃₁×#135	—	4	2	no	—
53	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
57	C₃₁×#131	—	4	2	no	—
58	C₃₁×#131	—	4	2	no	—
59	C₂×M₁₆×C₃₁	—	2	2	no	—
60	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
63	C₂²×C₈×C₃₁	—	2	2	no	—
64	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
65	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
68	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
69	C₃₁×#124	C₂×C₄	1	2	yes	—
70	C₃₁×#121	—	2	1	no	—
71	C₃₁×#121	—	2	1	no	—
72	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
73	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₄×D₄×C₃₁	—	2	1	no	—
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
78	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
79	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
80	C₃₁×#117	—	2	1	no	—
81	C₃₁×#117	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×#108	—	4	2	no	—
87	C₃₁×C₄⋊C₈	—	2	1	no	—
88	C₃₁×C₄⋊C₈	—	2	1	no	—
89	C₃₁×C₄⋊C₈	—	2	1	no	—
90	C₃₁×C₄⋊C₈	—	2	1	no	—
91	C₃₁×C₄⋊C₈	—	2	1	no	—
92	C₃₁×C₄⋊C₈	—	2	1	no	—
93	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
94	C₃₁×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
95	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
96	C₃₁×#104	—	4	2	no	—
97	C₃₁×#104	—	4	2	no	—
98	C₃₁×#104	—	4	2	no	—
99	C₃₁×#103	C₂×C₄	1	2	yes	—
100	C₃₁×#103	C₂×C₄	1	2	yes	—
101	C₃₁×#102	C₂×C₄	1	2	yes	—
102	C₃₁×#102	C₂×C₄	1	2	yes	—
103	C₃₁×C₂²⋊C₈	—	2	1	no	—
104	C₃₁×C₂²⋊C₈	—	2	1	no	—
105	C₃₁×C₂²⋊C₈	—	2	1	no	—
106	C₃₁×C₂²⋊C₈	—	2	1	no	—
107	C₃₁×C₂²⋊C₈	—	2	2	no	—
108	C₃₁×C₂²⋊C₈	—	2	2	no	—
109	C₃₁×#97	—	2	1	no	—
110	C₄×C₈×C₃₁	—	2	1	no	—
111	C₄×C₈×C₃₁	—	4	2	no	—
112	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
113	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
114	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
115	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
116	C₃₁×#95	—	2	1	no	—
117	#41×C₃₁	—	2	1	no	—
118	#41×C₃₁	—	2	2	no	—
119	#41×C₃₁	—	2	2	no	—
120	#41×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	2	no	—
125	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
126	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂×D₄×C₃₁	—	2	1	no	—
128	C₂×D₄×C₃₁	—	2	2	no	—
129	C₂×D₄×C₃₁	—	2	1	no	—
130	C₂×D₄×C₃₁	—	2	1	no	—
131	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
132	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	2	1	no	—
135	C₂²×C₄×C₃₁	—	2	1	no	—
136	C₂²×C₄×C₃₁	—	2	1	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
141	Q₁₆×C₃₁	—	4	2	no	—
142	SD₁₆×C₃₁	—	4	2	no	—
143	SD₁₆×C₃₁	—	4	2	no	—
144	D₈×C₃₁	—	4	2	no	—
145	M₁₆×C₃₁	—	2	1	no	—
146	M₁₆×C₃₁	—	2	1	no	—
147	M₁₆×C₃₁	—	2	2	no	—
148	M₁₆×C₃₁	—	2	1	no	—
149	M₁₆×C₃₁	—	2	2	no	—
150	M₁₆×C₃₁	—	2	1	no	—
151	M₁₆×C₃₁	—	4	4	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₂×C₈×C₃₁	—	2	2	no	—
155	C₂×C₈×C₃₁	—	2	1	no	—
156	C₂×C₈×C₃₁	—	2	1	no	—
157	C₂×C₈×C₃₁	—	2	1	no	—
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	2	2	no	—
161	C₂×C₈×C₃₁	—	4	4	no	—
162	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
163	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
164	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₄⋊C₄×C₃₁	—	2	4	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
174	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
175	C₂²⋊C₄×C₃₁	—	2	1	no	—
176	C₂²⋊C₄×C₃₁	—	2	1	no	—
177	C₂²⋊C₄×C₃₁	—	2	1	no	—
178	C₂²⋊C₄×C₃₁	—	2	2	no	—
179	C₂²⋊C₄×C₃₁	—	2	2	no	—
180	C₄²×C₃₁	—	2	1	no	—
181	C₄²×C₃₁	—	2	1	no	—
182	C₄²×C₃₁	—	2	1	no	—
183	C₄²×C₃₁	—	2	2	no	—
184	C₄²×C₃₁	—	2	2	no	—
185	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
186	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
187	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
189	#22165	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂³×C₃₁	—	2	1	no	—
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
193	Q₈×C₃₁	—	2	1	no	—
194	Q₈×C₃₁	—	2	2	no	—
195	Q₈×C₃₁	—	2	2	no	—
196	Q₈×C₃₁	—	2	1	no	—
197	D₄×C₃₁	—	2	1	no	—
198	D₄×C₃₁	—	2	2	no	—
199	D₄×C₃₁	—	2	2	no	—
200	D₄×C₃₁	—	2	1	no	—
201	D₄×C₃₁	—	4	2	no	—
202	D₄×C₃₁	—	4	1	no	—
203	D₄×C₃₁	—	4	2	no	—
204	D₄×C₃₁	—	4	1	no	—
205	D₄×C₃₁	—	4	2	no	—
206	D₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	4	1	no	—
219	C₂×C₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	—	4	2	no	—
222	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
224	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
230	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
231	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
232	C₈×C₃₁	—	2	2	no	—
233	C₈×C₃₁	—	2	2	no	—
234	C₈×C₃₁	—	2	1	no	—
235	C₈×C₃₁	—	2	1	no	—
236	C₈×C₃₁	—	2	1	no	—
237	C₈×C₃₁	—	2	1	no	—
238	C₈×C₃₁	—	4	2	no	—
239	C₈×C₃₁	—	4	1	no	—
240	C₈×C₃₁	—	4	1	no	—
241	#3035	C₂×C₃₁	1	1	yes	characteristic
242	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#1980	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#1974	C₂×C₃₁	1	2	yes	complemented
246	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#1937	C₂×C₃₁	1	2	yes	complemented
248	#1876	C₂×C₃₁	1	2	yes	complemented
249	#1866	C₂×C₃₁	1	2	yes	complemented
250	#1806	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
252	C₂²×C₃₁	—	2	1	no	—
253	C₂²×C₃₁	—	2	1	no	—
254	C₂²×C₃₁	—	4	2	no	—
255	C₂²×C₃₁	—	4	1	no	—
256	C₂²×C₃₁	—	4	1	no	—
257	C₂²×C₃₁	—	4	2	no	—
258	C₂²×C₃₁	—	4	1	no	—
259	C₂²×C₃₁	—	4	2	no	—
260	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
261	C₂²×C₃₁	#540	1	1	yes	characteristic
262	C₂²×C₃₁	#542	1	1	yes	characteristic
263	C₄×C₃₁	—	2	1	no	—
264	C₄×C₃₁	—	2	1	no	—
265	C₄×C₃₁	—	2	1	no	—
266	C₄×C₃₁	—	4	4	no	—
267	C₄×C₃₁	—	4	1	no	—
268	C₄×C₃₁	—	4	2	no	—
269	C₄×C₃₁	—	4	1	no	—
270	C₄×C₃₁	—	4	1	no	—
271	C₄×C₃₁	#443	1	2	yes	—
272	C₄×C₃₁	#518	1	1	yes	characteristic
273	C₄×C₃₁	#520	1	1	yes	characteristic
274	C₄×C₃₁	#535	1	1	yes	characteristic
275	C₄×C₃₁	#537	1	1	yes	characteristic
276	C₂×#135	C₂²×C₃₁	1	2	yes	—
277	C₂×#131	C₂²×C₃₁	1	2	yes	—
278	#532	C₂²×C₃₁	1	1	yes	characteristic
279	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
280	#464	C₄×C₃₁	1	2	yes	complemented
281	#463	C₄×C₃₁	1	2	yes	complemented
282	#462	C₄×C₃₁	1	1	yes	characteristic, complemented
283	#461	C₄×C₃₁	1	1	yes	characteristic, complemented
284	#460	C₄×C₃₁	1	1	yes	characteristic, complemented
285	#459	C₄×C₃₁	1	1	yes	characteristic, complemented
286	#436	—	2	1	no	—
287	#436	—	2	1	no	—
288	C₈×D₄	—	2	1	no	—
289	C₈×D₄	—	2	1	no	—
290	#433	—	2	1	no	—
291	#433	—	2	1	no	—
292	#431	—	2	1	no	—
293	#431	—	2	1	no	—
294	C₂×#108	C₂²×C₃₁	1	2	yes	—
295	#424	C₂²×C₃₁	1	1	yes	characteristic
296	#424	C₂²×C₃₁	1	1	yes	characteristic
297	#423	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
299	C₂×#104	C₂²×C₃₁	1	2	yes	—
300	C₂×#104	C₂²×C₃₁	1	2	yes	—
301	C₂×#104	C₂²×C₃₁	1	2	yes	—
302	#416	C₂²×C₃₁	1	2	yes	—
303	#408	C₂²×C₃₁	1	1	yes	characteristic
304	#408	C₂²×C₃₁	1	1	yes	characteristic
305	#408	C₂²×C₃₁	1	2	yes	—
306	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
308	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
309	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
310	#341	C₂²×C₃₁	1	2	yes	—
311	#339	C₂²×C₃₁	1	2	yes	—
312	#336	C₂²×C₃₁	1	2	yes	—
313	C₂×C₃₁	—	2	1	no	—
314	C₂×C₃₁	—	2	1	no	—
315	C₂×C₃₁	—	8	1	no	—
316	C₂×C₃₁	—	8	2	no	—
317	C₂×C₃₁	#2324	1	1	yes	characteristic
318	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
319	C₂×C₃₁	#2955	1	1	yes	characteristic
320	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
321	C₂×#41	C₂³×C₃₁	1	2	yes	—
322	#135	—	4	2	no	—
323	#135	—	4	2	no	—
324	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
325	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
326	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
327	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
328	#131	—	4	2	no	—
329	#131	—	4	2	no	—
330	C₂×M₁₆	—	2	2	no	—
331	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
332	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
333	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
334	C₂²×C₈	—	2	2	no	—
335	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
336	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
337	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
338	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
339	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
340	#124	C₂×C₄×C₃₁	1	2	yes	—
341	#121	—	2	1	no	—
342	#121	—	2	1	no	—
343	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
344	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
345	C₄×D₄	—	2	1	no	—
346	C₄×D₄	—	2	1	no	—
347	C₄×D₄	—	2	1	no	—
348	C₄×D₄	—	2	1	no	—
349	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
350	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
351	#117	—	2	1	no	—
352	#117	C₂³×C₃₁	1	1	yes	characteristic
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄²	—	2	1	no	—
355	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
356	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
357	#108	—	4	2	no	—
358	C₄⋊C₈	—	2	1	no	—
359	C₄⋊C₈	—	2	1	no	—
360	C₄⋊C₈	—	2	1	no	—
361	C₄⋊C₈	—	2	1	no	—
362	C₄⋊C₈	—	2	1	no	—
363	C₄⋊C₈	—	2	1	no	—
364	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
365	C₄⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
366	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
367	#104	—	4	2	no	—
368	#104	—	4	2	no	—
369	#104	—	4	2	no	—
370	#103	C₂×C₄×C₃₁	1	2	yes	—
371	#103	C₂×C₄×C₃₁	1	2	yes	—
372	#102	C₂×C₄×C₃₁	1	2	yes	—
373	#102	C₂×C₄×C₃₁	1	2	yes	—
374	C₂²⋊C₈	—	2	1	no	—
375	C₂²⋊C₈	—	2	1	no	—
376	C₂²⋊C₈	—	2	1	no	—
377	C₂²⋊C₈	—	2	1	no	—
378	C₂²⋊C₈	—	2	2	no	—
379	C₂²⋊C₈	—	2	2	no	—
380	#97	—	2	1	no	—
381	C₄×C₈	—	2	1	no	—
382	C₄×C₈	—	4	2	no	—
383	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
384	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
385	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
386	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
387	#95	—	2	1	no	—
388	C₃₁	#22165	1	1	yes	characteristic, direct product factor, Sylow
389	#41	—	2	1	no	—
390	#41	—	2	2	no	—
391	#41	—	2	2	no	—
392	#41	—	2	2	no	—
393	#41	—	2	1	no	—
394	#41	—	2	1	no	—
395	#41	—	2	1	no	—
396	#41	—	2	2	no	—
397	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
398	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	1	no	—
403	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
404	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	2	1	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
411	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
412	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
413	Q₁₆	—	4	2	no	—
414	SD₁₆	—	4	2	no	—
415	SD₁₆	—	4	2	no	—
416	D₈	—	4	2	no	—
417	M₁₆	—	2	1	no	—
418	M₁₆	—	2	1	no	—
419	M₁₆	—	2	2	no	—
420	M₁₆	—	2	1	no	—
421	M₁₆	—	2	2	no	—
422	M₁₆	—	2	1	no	—
423	M₁₆	—	4	4	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	1	no	—
428	C₂×C₈	—	2	1	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	1	no	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	2	2	no	—
433	C₂×C₈	—	4	4	no	—
434	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
435	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
436	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
438	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
439	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
440	C₄⋊C₄	—	2	4	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
446	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
447	C₂²⋊C₄	—	2	1	no	—
448	C₂²⋊C₄	—	2	1	no	—
449	C₂²⋊C₄	—	2	1	no	—
450	C₂²⋊C₄	—	2	2	no	—
451	C₂²⋊C₄	—	2	2	no	—
452	C₄²	—	2	1	no	—
453	C₄²	—	2	1	no	—
454	C₄²	—	2	1	no	—
455	C₄²	—	2	2	no	—
456	C₄²	—	2	2	no	—
457	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
458	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
459	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
460	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂³	—	2	1	no	—
462	C₂³	—	2	2	no	—
463	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
464	Q₈	—	2	1	no	—
465	Q₈	—	2	2	no	—
466	Q₈	—	2	2	no	—
467	Q₈	—	2	1	no	—
468	D₄	—	2	1	no	—
469	D₄	—	2	2	no	—
470	D₄	—	2	2	no	—
471	D₄	—	2	1	no	—
472	D₄	—	4	2	no	—
473	D₄	—	4	1	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	1	no	—
476	D₄	—	4	2	no	—
477	D₄	—	4	2	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	4	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
494	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
495	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
496	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
497	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
498	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
499	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
500	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
501	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
502	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
503	C₈	—	2	2	no	—
504	C₈	—	2	2	no	—
505	C₈	—	2	1	no	—
506	C₈	—	2	1	no	—
507	C₈	—	2	1	no	—
508	C₈	—	2	1	no	—
509	C₈	—	4	2	no	—
510	C₈	—	4	1	no	—
511	C₈	—	4	1	no	—
512	C₂²	—	2	1	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	4	2	no	—
515	C₂²	—	4	1	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	1	no	—
519	C₂²	—	4	2	no	—
520	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
521	C₂²	C₃₁×#540	1	1	yes	characteristic
522	C₂²	C₃₁×#542	1	1	yes	characteristic
523	C₄	—	2	1	no	—
524	C₄	—	2	1	no	—
525	C₄	—	2	1	no	—
526	C₄	—	4	4	no	—
527	C₄	—	4	1	no	—
528	C₄	—	4	2	no	—
529	C₄	—	4	1	no	—
530	C₄	—	4	1	no	—
531	C₄	C₃₁×#443	1	2	yes	—
532	C₄	C₃₁×#518	1	1	yes	characteristic
533	C₄	C₃₁×#520	1	1	yes	characteristic
534	C₄	C₃₁×#535	1	1	yes	characteristic
535	C₄	C₃₁×#537	1	1	yes	characteristic
536	C₂	—	2	1	no	—
537	C₂	—	2	1	no	—
538	C₂	—	8	1	no	—
539	C₂	—	8	2	no	—
540	C₂	C₃₁×#2324	1	1	yes	characteristic
541	C₂	C₂×C₃₁×#443	1	1	yes	characteristic
542	C₂	C₃₁×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#3035	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2931	C₂	semidirect
3. C₃₁×#2922	C₂	semidirect
4. C₃₁×#1980	C₂	semidirect
5. C₃₁×#1974	C₂	semidirect
6. C₃₁×#1950	C₂	semidirect
7. C₃₁×#1937	C₂	semidirect
8. C₃₁×#1876	C₂	semidirect
9. C₃₁×#1866	C₂	semidirect
10. C₃₁×#1806	C₂	semidirect
11. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
16. C₃₁×#464	C₄	semidirect
17. C₃₁×#463	C₄	semidirect
18. C₃₁×#462	C₄	semidirect
19. C₃₁×#461	C₄	semidirect
20. C₃₁×#460	C₄	semidirect
21. C₃₁×#459	C₄	semidirect
189. #22165	C₃₁	direct (N × H)
242. #2931	C₂×C₃₁	semidirect
243. #2922	C₂×C₃₁	semidirect
244. #1980	C₂×C₃₁	semidirect
245. #1974	C₂×C₃₁	semidirect
246. #1950	C₂×C₃₁	semidirect
247. #1937	C₂×C₃₁	semidirect
248. #1876	C₂×C₃₁	semidirect
249. #1866	C₂×C₃₁	semidirect
250. #1806	C₂×C₃₁	semidirect
251. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
280. #464	C₄×C₃₁	semidirect
281. #463	C₄×C₃₁	semidirect
282. #462	C₄×C₃₁	semidirect
283. #461	C₄×C₃₁	semidirect
284. #460	C₄×C₃₁	semidirect
285. #459	C₄×C₃₁	semidirect
388. C₃₁	#22165	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×#3035	2
C₃₁×#2931	2
C₃₁×#2922	2
C₃₁×#1980	2
C₃₁×#1974	2
C₃₁×#1950	2
C₃₁×#1937	2
C₃₁×#1876	2
C₃₁×#1866	2
C₃₁×#1806	2
C₈×C₄⋊C₄×C₃₁	2
#22165	31

Subgroups — C31×#22165 #21478924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22165 #21478924