Subgroups — C₃₁×#33471 #21490230

Order 7936.

668 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#509	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×AGL(1,ℤ/16)	C₂	1	8	yes	maximal, complemented
5	C₃₁×#2155	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2145	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2110	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
10	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#568	—	2	1	no	—
12	C₃₁×#518	—	2	1	no	—
13	C₃₁×#509	C₄	1	4	yes	complemented
14	C₃₁×#509	C₂²	1	4	yes	—
15	C₂×C₃₁×SD₃₂	C₄	1	2	yes	complemented
16	C₂×C₃₁×D₁₆	C₄	1	2	yes	complemented
17	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
19	C₃₁×#444	C₂²	1	4	yes	complemented
20	C₃₁×#443	—	2	2	no	—
21	C₃₁×#442	C₂²	1	4	yes	complemented
22	C₄×D₈×C₃₁	—	2	2	no	—
23	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
24	C₃₁×#427	C₂²	1	1	yes	characteristic
25	C₃₁×#421	—	2	1	no	—
26	C₃₁×#418	—	2	1	no	—
27	C₃₁×#405	C₂²	1	1	yes	characteristic
28	C₃₁×#365	C₂²	1	4	yes	complemented
29	C₃₁×#361	C₂²	1	4	yes	complemented
30	C₃₁×#360	C₂²	1	4	yes	complemented
31	C₃₁×#347	C₂²	1	4	yes	complemented
32	C₂×#41×C₃₁	—	2	1	no	—
33	C₂²×D₄×C₃₁	—	2	1	no	—
34	C₃₁×#135	—	2	1	no	—
35	C₃₁×#135	—	2	1	no	—
36	C₃₁×#135	C₂×C₄	1	2	yes	complemented
37	C₃₁×#135	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×SD₁₆×C₃₁	—	2	1	no	—
40	C₂×D₈×C₃₁	—	2	1	no	—
41	C₂×D₈×C₃₁	—	2	1	no	—
42	C₂×D₈×C₃₁	—	2	1	no	—
43	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
45	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×D₈×C₃₁	C₂³	1	2	yes	—
47	C₃₁×#131	—	2	4	no	—
48	C₃₁×#131	—	4	2	no	—
49	C₂×M₁₆×C₃₁	—	2	1	no	—
50	C₂×M₁₆×C₃₁	—	4	2	no	—
51	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₄×D₄×C₃₁	—	2	4	no	—
54	C₄×D₄×C₃₁	—	4	2	no	—
55	C₃₁×#117	—	2	1	no	—
56	C₃₁×#117	D₄	1	1	yes	characteristic
57	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
58	C₃₁×SD₃₂	C₂×C₄	1	8	yes	complemented
59	C₃₁×D₁₆	C₂×C₄	1	8	yes	complemented
60	C₃₁×M₃₂	—	2	1	no	—
61	C₃₁×M₃₂	C₂×C₄	1	2	yes	complemented
62	C₃₁×M₃₂	C₂³	1	2	yes	—
63	C₃₁×M₃₂	C₂³	1	2	yes	—
64	C₂×C₁₆×C₃₁	—	2	1	no	—
65	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	complemented
66	C₃₁×#108	—	2	1	no	—
67	C₃₁×#108	C₂³	1	2	yes	—
68	C₃₁×#107	—	2	1	no	—
69	C₃₁×#106	C₂³	1	2	yes	—
70	C₃₁×#104	—	2	4	no	—
71	C₃₁×#102	—	2	4	no	—
72	C₃₁×#97	C₂³	1	2	yes	—
73	C₄×C₈×C₃₁	D₄	1	2	yes	—
74	C₂⁴×C₃₁	—	4	1	no	—
75	#41×C₃₁	—	2	2	no	—
76	#41×C₃₁	—	2	2	no	—
77	#41×C₃₁	—	4	1	no	—
78	#41×C₃₁	—	4	1	no	—
79	C₂×Q₈×C₃₁	—	2	1	no	—
80	C₂×D₄×C₃₁	—	2	2	no	—
81	C₂×D₄×C₃₁	—	2	2	no	—
82	C₂×D₄×C₃₁	—	2	1	no	—
83	C₂×D₄×C₃₁	—	2	2	no	—
84	C₂×D₄×C₃₁	—	4	1	no	—
85	C₂×D₄×C₃₁	—	4	1	no	—
86	C₂×D₄×C₃₁	—	4	1	no	—
87	C₂×D₄×C₃₁	—	4	1	no	—
88	C₂²×C₄×C₃₁	—	4	4	no	—
89	C₂²×C₄×C₃₁	—	4	1	no	—
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₃₁	—	2	1	no	—
92	Q₁₆×C₃₁	C₂²×C₄	1	2	yes	—
93	SD₁₆×C₃₁	—	2	2	no	—
94	SD₁₆×C₃₁	—	2	2	no	—
95	D₈×C₃₁	—	2	2	no	—
96	D₈×C₃₁	—	2	2	no	—
97	D₈×C₃₁	—	2	1	no	—
98	D₈×C₃₁	—	2	2	no	—
99	D₈×C₃₁	C₂²×C₄	1	2	yes	—
100	D₈×C₃₁	C₂²×C₄	1	4	yes	—
101	M₁₆×C₃₁	—	2	1	no	—
102	M₁₆×C₃₁	—	2	1	no	—
103	M₁₆×C₃₁	—	2	2	no	—
104	M₁₆×C₃₁	—	4	4	no	—
105	M₁₆×C₃₁	—	4	1	no	—
106	C₂×C₈×C₃₁	—	4	4	no	—
107	C₂×C₈×C₃₁	—	4	1	no	—
108	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
113	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
114	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₃₁	—	2	1	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₂²⋊C₄×C₃₁	—	2	1	no	—
118	C₂²⋊C₄×C₃₁	—	4	4	no	—
119	C₂²⋊C₄×C₃₁	—	4	1	no	—
120	C₄²×C₃₁	—	4	1	no	—
121	C₄²×C₃₁	C₂×D₄	1	2	yes	—
122	C₁₆×C₃₁	—	2	2	no	—
123	C₁₆×C₃₁	C₂²×C₄	1	4	yes	—
124	#33471	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂³×C₃₁	—	4	2	no	—
126	C₂³×C₃₁	—	4	1	no	—
127	C₂³×C₃₁	—	4	2	no	—
128	C₂³×C₃₁	—	4	2	no	—
129	C₂³×C₃₁	—	16	2	no	—
130	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
131	Q₈×C₃₁	—	2	2	no	—
132	Q₈×C₃₁	—	4	1	no	—
133	D₄×C₃₁	—	2	4	no	—
134	D₄×C₃₁	—	2	2	no	—
135	D₄×C₃₁	—	4	2	no	—
136	D₄×C₃₁	—	4	2	no	—
137	D₄×C₃₁	—	4	2	no	—
138	D₄×C₃₁	—	4	2	no	—
139	D₄×C₃₁	—	4	2	no	—
140	D₄×C₃₁	—	4	1	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	4	1	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	16	4	no	—
147	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₃₁	—	2	2	no	—
154	C₈×C₃₁	—	4	2	no	—
155	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
156	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
157	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
158	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
159	C₂×#509	C₂×C₃₁	1	1	yes	characteristic
160	#2957	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#2948	C₂×C₃₁	1	1	yes	characteristic, complemented
162	AGL(1,ℤ/16)	C₂×C₃₁	1	8	yes	complemented
163	#2155	C₂×C₃₁	1	1	yes	characteristic, complemented
164	C₂×#361	C₂×C₃₁	1	1	yes	characteristic, complemented
165	#2145	C₂×C₃₁	1	1	yes	characteristic, complemented
166	#2110	C₂×C₃₁	1	1	yes	characteristic, complemented
167	C₂²×C₃₁	—	4	1	no	—
168	C₂²×C₃₁	—	4	2	no	—
169	C₂²×C₃₁	—	4	4	no	—
170	C₂²×C₃₁	—	8	4	no	—
171	C₂²×C₃₁	—	16	1	no	—
172	C₂²×C₃₁	—	16	2	no	—
173	C₂²×C₃₁	—	16	2	no	—
174	C₂²×C₃₁	#442	1	1	yes	characteristic
175	C₂²×C₃₁	#442	1	1	yes	characteristic
176	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₄×C₃₁	—	4	2	no	—
178	C₄×C₃₁	—	4	2	no	—
179	C₄×C₃₁	—	8	2	no	—
180	C₄×C₃₁	#442	1	1	yes	characteristic
181	C₄×C₃₁	#442	1	1	yes	characteristic
182	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
183	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
185	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
186	#568	—	2	1	no	—
187	#518	—	2	1	no	—
188	#509	C₄×C₃₁	1	4	yes	complemented
189	#509	C₂²×C₃₁	1	4	yes	—
190	C₂×SD₃₂	C₄×C₃₁	1	2	yes	complemented
191	C₂×D₁₆	C₄×C₃₁	1	2	yes	complemented
192	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
193	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
194	#444	C₂²×C₃₁	1	4	yes	complemented
195	#443	—	2	2	no	—
196	#442	C₂²×C₃₁	1	4	yes	complemented
197	C₄×D₈	—	2	2	no	—
198	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
199	#427	C₂²×C₃₁	1	1	yes	characteristic
200	#421	—	2	1	no	—
201	#418	—	2	1	no	—
202	#405	C₂²×C₃₁	1	1	yes	characteristic
203	#365	C₂²×C₃₁	1	4	yes	complemented
204	#361	C₂²×C₃₁	1	4	yes	complemented
205	#360	C₂²×C₃₁	1	4	yes	complemented
206	#347	C₂²×C₃₁	1	4	yes	complemented
207	C₂×C₃₁	—	2	1	no	—
208	C₂×C₃₁	—	2	1	no	—
209	C₂×C₃₁	—	2	1	no	—
210	C₂×C₃₁	—	8	4	no	—
211	C₂×C₃₁	—	8	2	no	—
212	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
213	C₂×#41	—	2	1	no	—
214	C₂²×D₄	—	2	1	no	—
215	#135	—	2	1	no	—
216	#135	—	2	1	no	—
217	#135	C₂×C₄×C₃₁	1	2	yes	complemented
218	#135	C₂³×C₃₁	1	2	yes	—
219	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
220	C₂×SD₁₆	—	2	1	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	—	2	1	no	—
223	C₂×D₈	—	2	1	no	—
224	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
225	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
226	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
227	C₂×D₈	C₂³×C₃₁	1	2	yes	—
228	#131	—	2	4	no	—
229	#131	—	4	2	no	—
230	C₂×M₁₆	—	2	1	no	—
231	C₂×M₁₆	—	4	2	no	—
232	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
234	C₄×D₄	—	2	4	no	—
235	C₄×D₄	—	4	2	no	—
236	#117	—	2	1	no	—
237	#117	D₄×C₃₁	1	1	yes	characteristic
238	C₂×C₂²⋊C₄	—	4	2	no	—
239	SD₃₂	C₂×C₄×C₃₁	1	8	yes	complemented
240	D₁₆	C₂×C₄×C₃₁	1	8	yes	complemented
241	M₃₂	—	2	1	no	—
242	M₃₂	C₂×C₄×C₃₁	1	2	yes	complemented
243	M₃₂	C₂³×C₃₁	1	2	yes	—
244	M₃₂	C₂³×C₃₁	1	2	yes	—
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	complemented
247	#108	—	2	1	no	—
248	#108	C₂³×C₃₁	1	2	yes	—
249	#107	—	2	1	no	—
250	#106	C₂³×C₃₁	1	2	yes	—
251	#104	—	2	4	no	—
252	#102	—	2	4	no	—
253	#97	C₂³×C₃₁	1	2	yes	—
254	C₄×C₈	D₄×C₃₁	1	2	yes	—
255	C₃₁	#33471	1	1	yes	characteristic, direct product factor, Sylow
256	C₂⁴	—	4	1	no	—
257	#41	—	2	2	no	—
258	#41	—	2	2	no	—
259	#41	—	4	1	no	—
260	#41	—	4	1	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂²×C₄	—	4	4	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
273	Q₁₆	—	2	1	no	—
274	Q₁₆	C₂²×C₄×C₃₁	1	2	yes	—
275	SD₁₆	—	2	2	no	—
276	SD₁₆	—	2	2	no	—
277	D₈	—	2	2	no	—
278	D₈	—	2	2	no	—
279	D₈	—	2	1	no	—
280	D₈	—	2	2	no	—
281	D₈	C₂²×C₄×C₃₁	1	2	yes	—
282	D₈	C₂²×C₄×C₃₁	1	4	yes	—
283	M₁₆	—	2	1	no	—
284	M₁₆	—	2	1	no	—
285	M₁₆	—	2	2	no	—
286	M₁₆	—	4	4	no	—
287	M₁₆	—	4	1	no	—
288	C₂×C₈	—	4	4	no	—
289	C₂×C₈	—	4	1	no	—
290	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
291	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
292	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
293	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
294	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
295	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
296	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₂²⋊C₄	—	4	4	no	—
301	C₂²⋊C₄	—	4	1	no	—
302	C₄²	—	4	1	no	—
303	C₄²	C₂×D₄×C₃₁	1	2	yes	—
304	C₁₆	—	2	2	no	—
305	C₁₆	C₂²×C₄×C₃₁	1	4	yes	—
306	C₂³	—	4	2	no	—
307	C₂³	—	4	1	no	—
308	C₂³	—	4	2	no	—
309	C₂³	—	4	2	no	—
310	C₂³	—	16	2	no	—
311	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	4	1	no	—
314	D₄	—	2	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	1	no	—
321	D₄	—	4	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	16	4	no	—
328	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
329	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
330	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
331	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
332	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
333	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	4	2	no	—
336	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
337	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
338	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
339	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
340	C₂²	—	4	1	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	4	no	—
343	C₂²	—	8	4	no	—
344	C₂²	—	16	2	no	—
345	C₂²	—	16	2	no	—
346	C₂²	—	16	1	no	—
347	C₂²	C₃₁×#442	1	1	yes	characteristic
348	C₂²	C₃₁×#442	1	1	yes	characteristic
349	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	2	no	—
353	C₄	C₃₁×#442	1	1	yes	characteristic
354	C₄	C₃₁×#442	1	1	yes	characteristic
355	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
356	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	8	4	no	—
361	C₂	—	8	2	no	—
362	C₂	C₂×C₃₁×#442	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	C₂×#442
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#509	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2957	C₂	semidirect
3. C₃₁×#2948	C₂	semidirect
4. C₃₁×AGL(1,ℤ/16)	C₂	semidirect
5. C₃₁×#2155	C₂	semidirect
6. C₂×C₃₁×#361	C₂	semidirect
7. C₃₁×#2145	C₂	semidirect
8. C₃₁×#2110	C₂	semidirect
13. C₃₁×#509	C₄	semidirect
15. C₂×C₃₁×SD₃₂	C₄	semidirect
16. C₂×C₃₁×D₁₆	C₄	semidirect
19. C₃₁×#444	C₂²	semidirect
21. C₃₁×#442	C₂²	semidirect
28. C₃₁×#365	C₂²	semidirect
29. C₃₁×#361	C₂²	semidirect
30. C₃₁×#360	C₂²	semidirect
31. C₃₁×#347	C₂²	semidirect
36. C₃₁×#135	C₂×C₄	semidirect
38. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
45. C₂×D₈×C₃₁	C₂×C₄	semidirect
58. C₃₁×SD₃₂	C₂×C₄	semidirect
59. C₃₁×D₁₆	C₂×C₄	semidirect
61. C₃₁×M₃₂	C₂×C₄	semidirect
65. C₂×C₁₆×C₃₁	C₂×C₄	semidirect
124. #33471	C₃₁	direct (N × H)
160. #2957	C₂×C₃₁	semidirect
161. #2948	C₂×C₃₁	semidirect
162. AGL(1,ℤ/16)	C₂×C₃₁	semidirect
163. #2155	C₂×C₃₁	semidirect
164. C₂×#361	C₂×C₃₁	semidirect
165. #2145	C₂×C₃₁	semidirect
166. #2110	C₂×C₃₁	semidirect
188. #509	C₄×C₃₁	semidirect
190. C₂×SD₃₂	C₄×C₃₁	semidirect
191. C₂×D₁₆	C₄×C₃₁	semidirect
194. #444	C₂²×C₃₁	semidirect
196. #442	C₂²×C₃₁	semidirect
203. #365	C₂²×C₃₁	semidirect
204. #361	C₂²×C₃₁	semidirect
205. #360	C₂²×C₃₁	semidirect
206. #347	C₂²×C₃₁	semidirect
217. #135	C₂×C₄×C₃₁	semidirect
219. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
226. C₂×D₈	C₂×C₄×C₃₁	semidirect
239. SD₃₂	C₂×C₄×C₃₁	semidirect
240. D₁₆	C₂×C₄×C₃₁	semidirect
242. M₃₂	C₂×C₄×C₃₁	semidirect
246. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
255. C₃₁	#33471	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×#509	2
C₃₁×#2957	2
C₃₁×#2948	2
C₃₁×AGL(1,ℤ/16)	2
C₃₁×#2155	2
C₂×C₃₁×#361	2
C₃₁×#2145	2
C₃₁×#2110	2
#33471	31

Subgroups — C31×#33471 #21490230

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33471 #21490230