Subgroups — C₃₁×#33192 #21489951

Order 7936.

998 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#473	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#472	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#471	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3040	C₂	1	2	yes	maximal, complemented
6	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1979	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1977	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1967	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
16	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#124	C₂²	1	2	yes	—
20	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#495	—	2	1	no	—
22	C₃₁×#495	—	2	1	no	—
23	C₃₁×#495	—	2	1	no	—
24	C₃₁×#495	—	2	1	no	—
25	C₃₁×#473	—	2	4	no	—
26	C₃₁×#472	—	2	4	no	—
27	C₃₁×#471	—	2	8	no	—
28	C₃₁×#464	—	2	4	no	—
29	C₃₁×#463	—	2	4	no	—
30	C₂×C₃₁×#108	C₂²	1	2	yes	—
31	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
33	C₃₁×#423	C₂²	1	2	yes	—
34	C₃₁×#419	C₂²	1	2	yes	—
35	C₃₁×#418	C₂²	1	2	yes	—
36	C₂×C₃₁×#101	C₂²	1	2	yes	—
37	C₂×C₃₁×#101	C₂²	1	2	yes	—
38	C₂×C₃₁×#100	C₂²	1	2	yes	—
39	C₂×C₃₁×#100	C₂²	1	2	yes	—
40	C₃₁×#405	—	2	1	no	—
41	C₃₁×#405	—	2	1	no	—
42	C₃₁×#405	C₂²	1	1	yes	characteristic
43	C₃₁×#405	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₃₁	—	2	1	no	—
45	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
46	C₂×C₄×C₈×C₃₁	—	2	1	no	—
47	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
48	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
51	C₃₁×#137	—	2	4	no	—
52	C₃₁×#137	—	2	4	no	—
53	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
54	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
55	C₂×Q₁₆×C₃₁	—	2	2	no	—
56	C₂×Q₁₆×C₃₁	—	2	2	no	—
57	C₂×Q₁₆×C₃₁	—	2	1	no	—
58	C₂×Q₁₆×C₃₁	—	2	1	no	—
59	C₂×Q₁₆×C₃₁	—	2	2	no	—
60	C₂×SD₁₆×C₃₁	—	2	2	no	—
61	C₂×SD₁₆×C₃₁	—	2	2	no	—
62	C₂×SD₁₆×C₃₁	—	2	4	no	—
63	C₂×SD₁₆×C₃₁	—	2	2	no	—
64	C₂×SD₁₆×C₃₁	—	2	2	no	—
65	C₂×SD₁₆×C₃₁	—	2	4	no	—
66	C₂×D₈×C₃₁	—	2	2	no	—
67	C₂×D₈×C₃₁	—	2	2	no	—
68	C₂×D₈×C₃₁	—	2	1	no	—
69	C₂×D₈×C₃₁	—	2	1	no	—
70	C₂×D₈×C₃₁	—	2	2	no	—
71	C₃₁×#131	—	2	2	no	—
72	C₃₁×#131	—	2	2	no	—
73	C₃₁×#131	D₄	1	4	yes	—
74	C₃₁×#131	D₄	1	4	yes	—
75	C₂×M₁₆×C₃₁	—	2	1	no	—
76	C₂×M₁₆×C₃₁	—	2	1	no	—
77	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
78	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
79	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₃₁	—	2	1	no	—
82	C₂²×C₈×C₃₁	—	2	1	no	—
83	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₃₁×#124	—	2	4	no	—
86	C₃₁×#124	—	2	2	no	—
87	C₃₁×#124	—	2	2	no	—
88	C₄×Q₈×C₃₁	D₄	1	2	yes	—
89	C₄×D₄×C₃₁	—	2	2	no	—
90	C₄×D₄×C₃₁	D₄	1	2	yes	—
91	C₃₁×#117	—	2	1	no	—
92	C₃₁×#117	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
94	C₂×C₄²×C₃₁	—	2	1	no	—
95	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
96	C₃₁×#108	—	2	4	no	—
97	C₃₁×#108	—	2	2	no	—
98	C₃₁×#108	—	2	2	no	—
99	C₃₁×C₄⋊C₈	—	2	4	no	—
100	C₃₁×#103	—	2	4	no	—
101	C₃₁×#102	—	2	4	no	—
102	C₃₁×#101	—	2	4	no	—
103	C₃₁×#101	—	2	4	no	—
104	C₃₁×#100	—	2	4	no	—
105	C₃₁×#100	—	2	4	no	—
106	C₃₁×#97	—	2	1	no	—
107	C₃₁×#97	—	2	1	no	—
108	C₃₁×#97	—	2	1	no	—
109	C₄×C₈×C₃₁	—	2	1	no	—
110	C₄×C₈×C₃₁	—	2	1	no	—
111	C₄×C₈×C₃₁	—	2	1	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	C₂⁴×C₃₁	—	2	2	no	—
117	#41×C₃₁	—	2	1	no	—
118	#41×C₃₁	—	2	1	no	—
119	#41×C₃₁	C₂×D₄	1	2	yes	—
120	#41×C₃₁	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₃₁	—	2	2	no	—
122	C₂×Q₈×C₃₁	—	2	2	no	—
123	C₂×Q₈×C₃₁	—	2	2	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	C₂×D₄	1	1	yes	characteristic
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	2	no	—
132	C₂×D₄×C₃₁	—	2	2	no	—
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	2	2	no	—
135	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	—	2	1	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	#41	1	2	yes	—
140	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
141	Q₁₆×C₃₁	—	4	4	no	—
142	Q₁₆×C₃₁	—	4	2	no	—
143	Q₁₆×C₃₁	—	4	2	no	—
144	SD₁₆×C₃₁	—	4	4	no	—
145	SD₁₆×C₃₁	—	4	4	no	—
146	SD₁₆×C₃₁	—	4	4	no	—
147	SD₁₆×C₃₁	—	4	4	no	—
148	D₈×C₃₁	—	4	4	no	—
149	D₈×C₃₁	—	4	2	no	—
150	D₈×C₃₁	—	4	2	no	—
151	M₁₆×C₃₁	—	2	1	no	—
152	M₁₆×C₃₁	—	2	1	no	—
153	M₁₆×C₃₁	—	2	4	no	—
154	M₁₆×C₃₁	—	2	4	no	—
155	M₁₆×C₃₁	—	2	2	no	—
156	M₁₆×C₃₁	—	2	2	no	—
157	M₁₆×C₃₁	—	2	2	no	—
158	M₁₆×C₃₁	—	2	2	no	—
159	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
160	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
161	C₂×C₈×C₃₁	—	2	1	no	—
162	C₂×C₈×C₃₁	—	2	1	no	—
163	C₂×C₈×C₃₁	—	2	4	no	—
164	C₂×C₈×C₃₁	—	2	4	no	—
165	C₂×C₈×C₃₁	—	2	2	no	—
166	C₂×C₈×C₃₁	—	2	2	no	—
167	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
168	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
172	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
173	C₂×C₈×C₃₁	#41	1	2	yes	—
174	C₂×C₈×C₃₁	#41	1	2	yes	—
175	C₄⋊C₄×C₃₁	—	2	2	no	—
176	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
177	C₂²⋊C₄×C₃₁	—	2	1	no	—
178	C₂²⋊C₄×C₃₁	—	2	2	no	—
179	C₂²⋊C₄×C₃₁	—	4	4	no	—
180	C₂²⋊C₄×C₃₁	—	4	4	no	—
181	C₄²×C₃₁	—	2	2	no	—
182	C₄²×C₃₁	—	2	1	no	—
183	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
185	C₄²×C₃₁	C₂×D₄	1	2	yes	—
186	#33192	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂³×C₃₁	—	2	1	no	—
188	C₂³×C₃₁	—	4	2	no	—
189	C₂³×C₃₁	—	4	2	no	—
190	C₂³×C₃₁	—	4	2	no	—
191	C₂³×C₃₁	—	8	4	no	—
192	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
193	Q₈×C₃₁	—	2	1	no	—
194	Q₈×C₃₁	—	4	2	no	—
195	Q₈×C₃₁	—	4	2	no	—
196	Q₈×C₃₁	—	4	2	no	—
197	Q₈×C₃₁	—	4	2	no	—
198	Q₈×C₃₁	#127	1	2	yes	—
199	D₄×C₃₁	—	2	2	no	—
200	D₄×C₃₁	—	2	2	no	—
201	D₄×C₃₁	—	2	1	no	—
202	D₄×C₃₁	—	4	2	no	—
203	D₄×C₃₁	—	4	2	no	—
204	D₄×C₃₁	—	4	2	no	—
205	D₄×C₃₁	—	4	2	no	—
206	D₄×C₃₁	#127	1	2	yes	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	—	4	4	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	#121	1	2	yes	—
217	C₂×C₄×C₃₁	#121	1	2	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	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
224	C₈×C₃₁	—	2	2	no	—
225	C₈×C₃₁	—	2	2	no	—
226	C₈×C₃₁	—	4	2	no	—
227	C₈×C₃₁	—	4	2	no	—
228	C₈×C₃₁	#121	1	2	yes	—
229	C₈×C₃₁	#121	1	2	yes	—
230	C₈×C₃₁	#121	1	2	yes	—
231	C₈×C₃₁	#121	1	2	yes	—
232	C₂×#495	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₂×#473	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂×#472	C₂×C₃₁	1	1	yes	characteristic
235	C₂×#471	C₂×C₃₁	1	2	yes	complemented
236	#3040	C₂×C₃₁	1	2	yes	complemented
237	C₄×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#1979	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#1977	C₂×C₃₁	1	2	yes	complemented
241	#1967	C₂×C₃₁	1	2	yes	complemented
242	#1848	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	2	2	no	—
244	C₂²×C₃₁	—	2	2	no	—
245	C₂²×C₃₁	—	4	2	no	—
246	C₂²×C₃₁	—	8	2	no	—
247	C₂²×C₃₁	—	8	4	no	—
248	C₂²×C₃₁	—	16	2	no	—
249	C₂²×C₃₁	—	16	2	no	—
250	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
251	C₂²×C₃₁	#532	1	1	yes	characteristic
252	C₂²×C₃₁	#532	1	1	yes	characteristic
253	C₄×C₃₁	—	2	2	no	—
254	C₄×C₃₁	—	2	2	no	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	8	2	no	—
258	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
259	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
260	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
261	C₄×C₃₁	#532	1	1	yes	characteristic
262	C₂×#137	C₂²×C₃₁	1	2	yes	—
263	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
264	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
265	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
266	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×#124	C₂²×C₃₁	1	2	yes	—
270	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
271	#495	—	2	1	no	—
272	#495	—	2	1	no	—
273	#495	—	2	1	no	—
274	#495	—	2	1	no	—
275	#473	—	2	4	no	—
276	#472	—	2	4	no	—
277	#471	—	2	8	no	—
278	#464	—	2	4	no	—
279	#463	—	2	4	no	—
280	C₂×#108	C₂²×C₃₁	1	2	yes	—
281	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
283	#423	C₂²×C₃₁	1	2	yes	—
284	#419	C₂²×C₃₁	1	2	yes	—
285	#418	C₂²×C₃₁	1	2	yes	—
286	C₂×#101	C₂²×C₃₁	1	2	yes	—
287	C₂×#101	C₂²×C₃₁	1	2	yes	—
288	C₂×#100	C₂²×C₃₁	1	2	yes	—
289	C₂×#100	C₂²×C₃₁	1	2	yes	—
290	#405	—	2	1	no	—
291	#405	—	2	1	no	—
292	#405	C₂²×C₃₁	1	1	yes	characteristic
293	#405	C₂²×C₃₁	1	1	yes	characteristic
294	C₄×M₁₆	—	2	1	no	—
295	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
296	C₂×C₄×C₈	—	2	1	no	—
297	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×C₃₁	—	2	1	no	—
299	C₂×C₃₁	—	2	1	no	—
300	C₂×C₃₁	—	4	2	no	—
301	C₂×C₃₁	—	16	2	no	—
302	C₂×C₃₁	#2856	1	1	yes	characteristic
303	C₂×C₃₁	#3067	1	2	yes	—
304	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
305	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
306	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
307	#137	—	2	4	no	—
308	#137	—	2	4	no	—
309	AGL(1,ℤ/8)	—	2	4	no	—
310	AGL(1,ℤ/8)	—	2	4	no	—
311	C₂×Q₁₆	—	2	2	no	—
312	C₂×Q₁₆	—	2	2	no	—
313	C₂×Q₁₆	—	2	1	no	—
314	C₂×Q₁₆	—	2	1	no	—
315	C₂×Q₁₆	—	2	2	no	—
316	C₂×SD₁₆	—	2	2	no	—
317	C₂×SD₁₆	—	2	2	no	—
318	C₂×SD₁₆	—	2	4	no	—
319	C₂×SD₁₆	—	2	2	no	—
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×SD₁₆	—	2	4	no	—
322	C₂×D₈	—	2	2	no	—
323	C₂×D₈	—	2	2	no	—
324	C₂×D₈	—	2	1	no	—
325	C₂×D₈	—	2	1	no	—
326	C₂×D₈	—	2	2	no	—
327	#131	—	2	2	no	—
328	#131	—	2	2	no	—
329	#131	D₄×C₃₁	1	4	yes	—
330	#131	D₄×C₃₁	1	4	yes	—
331	C₂×M₁₆	—	2	1	no	—
332	C₂×M₁₆	—	2	1	no	—
333	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
334	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
335	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
336	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
337	C₂²×C₈	—	2	1	no	—
338	C₂²×C₈	—	2	1	no	—
339	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
340	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
341	#124	—	2	4	no	—
342	#124	—	2	2	no	—
343	#124	—	2	2	no	—
344	C₄×Q₈	D₄×C₃₁	1	2	yes	—
345	C₄×D₄	—	2	2	no	—
346	C₄×D₄	D₄×C₃₁	1	2	yes	—
347	#117	—	2	1	no	—
348	#117	C₂³×C₃₁	1	1	yes	characteristic
349	C₂×C₂²⋊C₄	—	2	4	no	—
350	C₂×C₄²	—	2	1	no	—
351	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
352	#108	—	2	4	no	—
353	#108	—	2	2	no	—
354	#108	—	2	2	no	—
355	C₄⋊C₈	—	2	4	no	—
356	#103	—	2	4	no	—
357	#102	—	2	4	no	—
358	#101	—	2	4	no	—
359	#101	—	2	4	no	—
360	#100	—	2	4	no	—
361	#100	—	2	4	no	—
362	#97	—	2	1	no	—
363	#97	—	2	1	no	—
364	#97	—	2	1	no	—
365	C₄×C₈	—	2	1	no	—
366	C₄×C₈	—	2	1	no	—
367	C₄×C₈	—	2	1	no	—
368	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
369	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
370	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
371	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
372	C₃₁	#33192	1	1	yes	characteristic, direct product factor, Sylow
373	C₂⁴	—	2	2	no	—
374	#41	—	2	1	no	—
375	#41	—	2	1	no	—
376	#41	C₂×D₄×C₃₁	1	2	yes	—
377	#41	C₂×D₄×C₃₁	1	2	yes	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	2	2	no	—
380	C₂×Q₈	—	2	2	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	2	no	—
384	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	#41×C₃₁	1	2	yes	—
397	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
398	Q₁₆	—	4	4	no	—
399	Q₁₆	—	4	2	no	—
400	Q₁₆	—	4	2	no	—
401	SD₁₆	—	4	4	no	—
402	SD₁₆	—	4	4	no	—
403	SD₁₆	—	4	4	no	—
404	SD₁₆	—	4	4	no	—
405	D₈	—	4	4	no	—
406	D₈	—	4	2	no	—
407	D₈	—	4	2	no	—
408	M₁₆	—	2	1	no	—
409	M₁₆	—	2	1	no	—
410	M₁₆	—	2	4	no	—
411	M₁₆	—	2	4	no	—
412	M₁₆	—	2	2	no	—
413	M₁₆	—	2	2	no	—
414	M₁₆	—	2	2	no	—
415	M₁₆	—	2	2	no	—
416	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
417	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	4	no	—
421	C₂×C₈	—	2	4	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
425	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
426	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
427	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
428	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
429	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
430	C₂×C₈	#41×C₃₁	1	2	yes	—
431	C₂×C₈	#41×C₃₁	1	2	yes	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
434	C₂²⋊C₄	—	2	1	no	—
435	C₂²⋊C₄	—	2	2	no	—
436	C₂²⋊C₄	—	4	4	no	—
437	C₂²⋊C₄	—	4	4	no	—
438	C₄²	—	2	2	no	—
439	C₄²	—	2	1	no	—
440	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₄²	C₂×D₄×C₃₁	1	2	yes	—
443	C₂³	—	2	1	no	—
444	C₂³	—	4	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	8	4	no	—
448	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
449	Q₈	—	2	1	no	—
450	Q₈	—	4	2	no	—
451	Q₈	—	4	2	no	—
452	Q₈	—	4	2	no	—
453	Q₈	—	4	2	no	—
454	Q₈	C₃₁×#127	1	2	yes	—
455	D₄	—	2	2	no	—
456	D₄	—	2	2	no	—
457	D₄	—	2	1	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	2	no	—
460	D₄	—	4	2	no	—
461	D₄	—	4	2	no	—
462	D₄	C₃₁×#127	1	2	yes	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	C₃₁×#121	1	2	yes	—
473	C₂×C₄	C₃₁×#121	1	2	yes	—
474	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
478	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
479	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
480	C₈	—	2	2	no	—
481	C₈	—	2	2	no	—
482	C₈	—	4	2	no	—
483	C₈	—	4	2	no	—
484	C₈	C₃₁×#121	1	2	yes	—
485	C₈	C₃₁×#121	1	2	yes	—
486	C₈	C₃₁×#121	1	2	yes	—
487	C₈	C₃₁×#121	1	2	yes	—
488	C₂²	—	2	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	8	2	no	—
492	C₂²	—	8	4	no	—
493	C₂²	—	16	2	no	—
494	C₂²	—	16	2	no	—
495	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
496	C₂²	C₃₁×#532	1	1	yes	characteristic
497	C₂²	C₃₁×#532	1	1	yes	characteristic
498	C₄	—	2	2	no	—
499	C₄	—	2	2	no	—
500	C₄	—	2	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	8	2	no	—
503	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
504	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
505	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
506	C₄	C₃₁×#532	1	1	yes	characteristic
507	C₂	—	2	1	no	—
508	C₂	—	2	1	no	—
509	C₂	—	4	2	no	—
510	C₂	—	16	2	no	—
511	C₂	C₃₁×#2856	1	1	yes	characteristic
512	C₂	C₃₁×#3067	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#472	C₂
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. C₂×C₃₁×#495	C₂	semidirect
2. C₂×C₃₁×#473	C₂	semidirect
4. C₂×C₃₁×#471	C₂	semidirect
5. C₃₁×#3040	C₂	semidirect
6. C₄×C₃₁×#131	C₂	semidirect
7. C₃₁×#1979	C₂	semidirect
8. C₃₁×#1978	C₂	semidirect
9. C₃₁×#1977	C₂	semidirect
10. C₃₁×#1967	C₂	semidirect
11. C₃₁×#1848	C₂	semidirect
186. #33192	C₃₁	direct (N × H)
232. C₂×#495	C₂×C₃₁	semidirect
233. C₂×#473	C₂×C₃₁	semidirect
235. C₂×#471	C₂×C₃₁	semidirect
236. #3040	C₂×C₃₁	semidirect
237. C₄×#131	C₂×C₃₁	semidirect
238. #1979	C₂×C₃₁	semidirect
239. #1978	C₂×C₃₁	semidirect
240. #1977	C₂×C₃₁	semidirect
241. #1967	C₂×C₃₁	semidirect
242. #1848	C₂×C₃₁	semidirect
372. C₃₁	#33192	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#495	2
C₂×C₃₁×#473	2
C₂×C₃₁×#472	2
C₂×C₃₁×#471	2
C₃₁×#3040	2
C₄×C₃₁×#131	2
C₃₁×#1979	2
C₃₁×#1978	2
C₃₁×#1977	2
C₃₁×#1967	2
C₃₁×#1848	2
#33192	31

Subgroups — C31×#33192 #21489951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33192 #21489951