Subgroups — C₃₁×#33690 #21490449

Order 7936.

526 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3183	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2956	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2239	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2238	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2231	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2212	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2191	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2190	C₂	1	1	yes	maximal, characteristic
9	C₃₁×AGL(1,ℤ/16)	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2180	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2165	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#575	C₂²	1	1	yes	characteristic
13	C₃₁×#568	—	2	1	no	—
14	C₃₁×#529	—	2	1	no	—
15	C₃₁×#510	C₂²	1	2	yes	—
16	C₃₁×#509	C₂²	1	2	yes	—
17	C₃₁×#499	C₂²	1	1	yes	characteristic, complemented
18	C₃₁×#495	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#488	—	2	1	no	—
20	C₃₁×#470	C₂²	1	1	yes	characteristic
21	C₃₁×#469	C₂²	1	1	yes	characteristic
22	C₃₁×#468	—	2	1	no	—
23	C₃₁×#444	—	2	1	no	—
24	C₃₁×#444	C₂²	1	2	yes	—
25	C₃₁×#443	—	2	1	no	—
26	C₃₁×#443	C₂²	1	2	yes	—
27	C₃₁×#442	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#441	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#440	—	2	1	no	—
30	C₃₁×#430	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#104	—	2	1	no	—
32	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#365	C₂²	1	2	yes	complemented
34	C₃₁×#365	C₂²	1	2	yes	complemented
35	C₃₁×#364	C₂²	1	2	yes	complemented
36	C₃₁×#362	C₂²	1	2	yes	complemented
37	C₃₁×#361	C₂²	1	2	yes	complemented
38	C₃₁×#360	C₂²	1	2	yes	—
39	C₃₁×#360	C₂²	1	2	yes	—
40	C₃₁×#349	C₂²	1	1	yes	characteristic
41	C₃₁×#349	C₂²	1	1	yes	characteristic
42	C₃₁×#347	C₂²	1	2	yes	complemented
43	C₂×#41×C₃₁	—	2	1	no	—
44	C₃₁×#137	—	2	2	no	—
45	C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
46	C₃₁×#135	—	2	1	no	—
47	C₃₁×#135	C₂³	1	2	yes	—
48	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×SD₁₆×C₃₁	—	2	1	no	—
50	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#131	—	2	2	no	—
52	C₃₁×#131	—	2	2	no	—
53	C₃₁×#131	—	4	2	no	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂×M₁₆×C₃₁	—	4	2	no	—
56	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
57	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×#126	—	4	1	no	—
59	C₃₁×#125	—	2	1	no	—
60	C₃₁×#124	—	2	1	no	—
61	C₃₁×#123	—	4	1	no	—
62	C₃₁×C₂²⋊Q₈	—	2	1	no	—
63	C₃₁×#121	—	2	1	no	—
64	C₄×Q₈×C₃₁	—	2	1	no	—
65	C₄×D₄×C₃₁	—	2	1	no	—
66	C₄×D₄×C₃₁	—	4	1	no	—
67	C₃₁×#117	—	4	1	no	—
68	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
69	C₃₁×Q₃₂	—	2	2	no	—
70	C₃₁×SD₃₂	—	2	2	no	—
71	C₃₁×SD₃₂	—	2	2	no	—
72	C₃₁×D₁₆	—	2	2	no	—
73	C₃₁×M₃₂	C₂³	1	2	yes	—
74	C₃₁×M₃₂	C₂³	1	2	yes	—
75	C₃₁×#108	—	2	1	no	—
76	C₃₁×#108	C₂³	1	2	yes	—
77	C₃₁×#107	—	2	1	no	—
78	C₃₁×#106	C₂³	1	1	yes	characteristic
79	C₃₁×#106	C₂³	1	1	yes	characteristic
80	C₃₁×#104	—	2	2	no	—
81	C₃₁×#104	—	2	2	no	—
82	C₃₁×#103	—	2	1	no	—
83	C₃₁×#103	—	2	1	no	—
84	C₃₁×#102	—	2	1	no	—
85	C₃₁×#102	—	2	1	no	—
86	C₃₁×#97	D₄	1	1	yes	characteristic
87	C₃₁×#97	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
89	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
90	#41×C₃₁	—	2	2	no	—
91	#41×C₃₁	—	2	2	no	—
92	#41×C₃₁	—	4	1	no	—
93	#41×C₃₁	—	4	1	no	—
94	C₂×Q₈×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	—	2	1	no	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	4	1	no	—
99	C₂×D₄×C₃₁	—	4	1	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	4	1	no	—
102	C₂²×C₄×C₃₁	—	4	1	no	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	Q₁₆×C₃₁	—	2	1	no	—
105	Q₁₆×C₃₁	—	2	2	no	—
106	Q₁₆×C₃₁	#41	1	2	yes	—
107	SD₁₆×C₃₁	—	2	2	no	—
108	SD₁₆×C₃₁	—	2	2	no	—
109	SD₁₆×C₃₁	—	4	2	no	—
110	D₈×C₃₁	—	2	1	no	—
111	D₈×C₃₁	—	2	2	no	—
112	D₈×C₃₁	#41	1	2	yes	—
113	M₁₆×C₃₁	—	2	2	no	—
114	M₁₆×C₃₁	—	2	2	no	—
115	M₁₆×C₃₁	—	2	2	no	—
116	M₁₆×C₃₁	—	4	2	no	—
117	M₁₆×C₃₁	—	4	1	no	—
118	M₁₆×C₃₁	—	4	2	no	—
119	C₂×C₈×C₃₁	—	4	2	no	—
120	C₂×C₈×C₃₁	—	4	1	no	—
121	C₂×C₈×C₃₁	—	4	2	no	—
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	4	1	no	—
129	C₄⋊C₄×C₃₁	—	4	1	no	—
130	C₄⋊C₄×C₃₁	—	4	1	no	—
131	C₄⋊C₄×C₃₁	—	4	1	no	—
132	C₂²⋊C₄×C₃₁	—	4	1	no	—
133	C₂²⋊C₄×C₃₁	—	4	1	no	—
134	C₂²⋊C₄×C₃₁	—	4	1	no	—
135	C₂²⋊C₄×C₃₁	—	4	1	no	—
136	C₂²⋊C₄×C₃₁	—	4	1	no	—
137	C₄²×C₃₁	—	2	1	no	—
138	C₄²×C₃₁	—	4	1	no	—
139	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₁₆×C₃₁	—	2	2	no	—
142	C₁₆×C₃₁	—	2	2	no	—
143	#33690	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	4	1	no	—
145	C₂³×C₃₁	—	4	1	no	—
146	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
147	Q₈×C₃₁	—	2	2	no	—
148	Q₈×C₃₁	—	4	1	no	—
149	Q₈×C₃₁	—	4	1	no	—
150	Q₈×C₃₁	—	4	1	no	—
151	D₄×C₃₁	—	2	2	no	—
152	D₄×C₃₁	—	4	1	no	—
153	D₄×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	4	2	no	—
155	D₄×C₃₁	—	4	1	no	—
156	D₄×C₃₁	—	4	1	no	—
157	D₄×C₃₁	—	8	2	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	8	1	no	—
169	C₂×C₄×C₃₁	—	8	2	no	—
170	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
172	C₈×C₃₁	—	2	1	no	—
173	C₈×C₃₁	—	2	1	no	—
174	C₈×C₃₁	—	4	2	no	—
175	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
176	C₈×C₃₁	C₂×#41	1	2	yes	—
177	#3183	C₂×C₃₁	1	1	yes	characteristic, complemented
178	#2956	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#2239	C₂×C₃₁	1	1	yes	characteristic, complemented
180	#2238	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#2231	C₂×C₃₁	1	2	yes	complemented
182	#2212	C₂×C₃₁	1	2	yes	complemented
183	#2191	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#2190	C₂×C₃₁	1	1	yes	characteristic
185	AGL(1,ℤ/16)	C₂×C₃₁	1	2	yes	complemented
186	#2180	C₂×C₃₁	1	2	yes	complemented
187	#2165	C₂×C₃₁	1	1	yes	characteristic, complemented
188	C₂²×C₃₁	—	2	1	no	—
189	C₂²×C₃₁	—	4	1	no	—
190	C₂²×C₃₁	—	4	2	no	—
191	C₂²×C₃₁	—	8	1	no	—
192	C₂²×C₃₁	—	8	1	no	—
193	C₂²×C₃₁	—	8	1	no	—
194	C₂²×C₃₁	—	16	1	no	—
195	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
196	C₄×C₃₁	—	2	1	no	—
197	C₄×C₃₁	—	4	1	no	—
198	C₄×C₃₁	—	4	1	no	—
199	C₄×C₃₁	—	4	2	no	—
200	C₄×C₃₁	—	8	1	no	—
201	C₄×C₃₁	—	8	1	no	—
202	C₄×C₃₁	—	8	1	no	—
203	C₄×C₃₁	#525	1	1	yes	characteristic
204	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	#575	C₂²×C₃₁	1	1	yes	characteristic
206	#568	—	2	1	no	—
207	#529	—	2	1	no	—
208	#510	C₂²×C₃₁	1	2	yes	—
209	#509	C₂²×C₃₁	1	2	yes	—
210	#499	C₂²×C₃₁	1	1	yes	characteristic, complemented
211	#495	C₂²×C₃₁	1	1	yes	characteristic, complemented
212	#488	—	2	1	no	—
213	#470	C₂²×C₃₁	1	1	yes	characteristic
214	#469	C₂²×C₃₁	1	1	yes	characteristic
215	#468	—	2	1	no	—
216	#444	—	2	1	no	—
217	#444	C₂²×C₃₁	1	2	yes	—
218	#443	—	2	1	no	—
219	#443	C₂²×C₃₁	1	2	yes	—
220	#442	C₂²×C₃₁	1	1	yes	characteristic, complemented
221	#441	C₂²×C₃₁	1	1	yes	characteristic, complemented
222	#440	—	2	1	no	—
223	#430	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×#104	—	2	1	no	—
225	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
226	#365	C₂²×C₃₁	1	2	yes	complemented
227	#365	C₂²×C₃₁	1	2	yes	complemented
228	#364	C₂²×C₃₁	1	2	yes	complemented
229	#362	C₂²×C₃₁	1	2	yes	complemented
230	#361	C₂²×C₃₁	1	2	yes	complemented
231	#360	C₂²×C₃₁	1	2	yes	—
232	#360	C₂²×C₃₁	1	2	yes	—
233	#349	C₂²×C₃₁	1	1	yes	characteristic
234	#349	C₂²×C₃₁	1	1	yes	characteristic
235	#347	C₂²×C₃₁	1	2	yes	complemented
236	C₂×C₃₁	—	2	1	no	—
237	C₂×C₃₁	—	4	1	no	—
238	C₂×C₃₁	—	8	2	no	—
239	C₂×C₃₁	—	16	1	no	—
240	C₂×C₃₁	#3109	1	1	yes	characteristic
241	C₂×#41	—	2	1	no	—
242	#137	—	2	2	no	—
243	AGL(1,ℤ/8)	—	2	2	no	—
244	#135	—	2	1	no	—
245	#135	C₂³×C₃₁	1	2	yes	—
246	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
247	C₂×SD₁₆	—	2	1	no	—
248	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
249	#131	—	2	2	no	—
250	#131	—	2	2	no	—
251	#131	—	4	2	no	—
252	C₂×M₁₆	—	2	1	no	—
253	C₂×M₁₆	—	4	2	no	—
254	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
255	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
256	#126	—	4	1	no	—
257	#125	—	2	1	no	—
258	#124	—	2	1	no	—
259	#123	—	4	1	no	—
260	C₂²⋊Q₈	—	2	1	no	—
261	#121	—	2	1	no	—
262	C₄×Q₈	—	2	1	no	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	—	4	1	no	—
265	#117	—	4	1	no	—
266	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
267	Q₃₂	—	2	2	no	—
268	SD₃₂	—	2	2	no	—
269	SD₃₂	—	2	2	no	—
270	D₁₆	—	2	2	no	—
271	M₃₂	C₂³×C₃₁	1	2	yes	—
272	M₃₂	C₂³×C₃₁	1	2	yes	—
273	#108	—	2	1	no	—
274	#108	C₂³×C₃₁	1	2	yes	—
275	#107	—	2	1	no	—
276	#106	C₂³×C₃₁	1	1	yes	characteristic
277	#106	C₂³×C₃₁	1	1	yes	characteristic
278	#104	—	2	2	no	—
279	#104	—	2	2	no	—
280	#103	—	2	1	no	—
281	#103	—	2	1	no	—
282	#102	—	2	1	no	—
283	#102	—	2	1	no	—
284	#97	D₄×C₃₁	1	1	yes	characteristic
285	#97	C₂³×C₃₁	1	1	yes	characteristic
286	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
287	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₃₁	#33690	1	1	yes	characteristic, direct product factor, Sylow
289	#41	—	2	2	no	—
290	#41	—	2	2	no	—
291	#41	—	4	1	no	—
292	#41	—	4	1	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×Q₈	—	2	1	no	—
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	—	2	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	4	1	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	4	1	no	—
301	C₂²×C₄	—	4	1	no	—
302	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
303	Q₁₆	—	2	1	no	—
304	Q₁₆	—	2	2	no	—
305	Q₁₆	#41×C₃₁	1	2	yes	—
306	SD₁₆	—	2	2	no	—
307	SD₁₆	—	2	2	no	—
308	SD₁₆	—	4	2	no	—
309	D₈	—	2	1	no	—
310	D₈	—	2	2	no	—
311	D₈	#41×C₃₁	1	2	yes	—
312	M₁₆	—	2	2	no	—
313	M₁₆	—	2	2	no	—
314	M₁₆	—	2	2	no	—
315	M₁₆	—	4	2	no	—
316	M₁₆	—	4	1	no	—
317	M₁₆	—	4	2	no	—
318	C₂×C₈	—	4	2	no	—
319	C₂×C₈	—	4	1	no	—
320	C₂×C₈	—	4	2	no	—
321	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₂²⋊C₄	—	4	1	no	—
332	C₂²⋊C₄	—	4	1	no	—
333	C₂²⋊C₄	—	4	1	no	—
334	C₂²⋊C₄	—	4	1	no	—
335	C₂²⋊C₄	—	4	1	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	4	1	no	—
338	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₁₆	—	2	2	no	—
341	C₁₆	—	2	2	no	—
342	C₂³	—	4	1	no	—
343	C₂³	—	4	1	no	—
344	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
345	Q₈	—	2	2	no	—
346	Q₈	—	4	1	no	—
347	Q₈	—	4	1	no	—
348	Q₈	—	4	1	no	—
349	D₄	—	2	2	no	—
350	D₄	—	4	1	no	—
351	D₄	—	4	2	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	1	no	—
354	D₄	—	4	1	no	—
355	D₄	—	8	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	8	1	no	—
367	C₂×C₄	—	8	2	no	—
368	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
369	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	4	2	no	—
373	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
374	C₈	C₂×#41×C₃₁	1	2	yes	—
375	C₂²	—	2	1	no	—
376	C₂²	—	4	1	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	8	1	no	—
379	C₂²	—	8	1	no	—
380	C₂²	—	8	1	no	—
381	C₂²	—	16	1	no	—
382	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
383	C₄	—	2	1	no	—
384	C₄	—	4	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	8	1	no	—
388	C₄	—	8	1	no	—
389	C₄	—	8	1	no	—
390	C₄	C₃₁×#525	1	1	yes	characteristic
391	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
392	C₂	—	2	1	no	—
393	C₂	—	4	1	no	—
394	C₂	—	8	2	no	—
395	C₂	—	16	1	no	—
396	C₂	C₃₁×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#2190	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂×C₄×C₃₁ ◁ C₄×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3183	C₂	semidirect
2. C₃₁×#2956	C₂	semidirect
3. C₃₁×#2239	C₂	semidirect
4. C₃₁×#2238	C₂	semidirect
5. C₃₁×#2231	C₂	semidirect
6. C₃₁×#2212	C₂	semidirect
7. C₃₁×#2191	C₂	semidirect
9. C₃₁×AGL(1,ℤ/16)	C₂	semidirect
10. C₃₁×#2180	C₂	semidirect
11. C₃₁×#2165	C₂	semidirect
17. C₃₁×#499	C₂²	semidirect
18. C₃₁×#495	C₂²	semidirect
27. C₃₁×#442	C₂²	semidirect
28. C₃₁×#441	C₂²	semidirect
33. C₃₁×#365	C₂²	semidirect
34. C₃₁×#365	C₂²	semidirect
35. C₃₁×#364	C₂²	semidirect
36. C₃₁×#362	C₂²	semidirect
37. C₃₁×#361	C₂²	semidirect
42. C₃₁×#347	C₂²	semidirect
143. #33690	C₃₁	direct (N × H)
177. #3183	C₂×C₃₁	semidirect
178. #2956	C₂×C₃₁	semidirect
179. #2239	C₂×C₃₁	semidirect
180. #2238	C₂×C₃₁	semidirect
181. #2231	C₂×C₃₁	semidirect
182. #2212	C₂×C₃₁	semidirect
183. #2191	C₂×C₃₁	semidirect
185. AGL(1,ℤ/16)	C₂×C₃₁	semidirect
186. #2180	C₂×C₃₁	semidirect
187. #2165	C₂×C₃₁	semidirect
210. #499	C₂²×C₃₁	semidirect
211. #495	C₂²×C₃₁	semidirect
220. #442	C₂²×C₃₁	semidirect
221. #441	C₂²×C₃₁	semidirect
226. #365	C₂²×C₃₁	semidirect
227. #365	C₂²×C₃₁	semidirect
228. #364	C₂²×C₃₁	semidirect
229. #362	C₂²×C₃₁	semidirect
230. #361	C₂²×C₃₁	semidirect
235. #347	C₂²×C₃₁	semidirect
288. C₃₁	#33690	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×#3183	2
C₃₁×#2956	2
C₃₁×#2239	2
C₃₁×#2238	2
C₃₁×#2231	2
C₃₁×#2212	2
C₃₁×#2191	2
C₃₁×#2190	2
C₃₁×AGL(1,ℤ/16)	2
C₃₁×#2180	2
C₃₁×#2165	2
#33690	31

Subgroups — C31×#33690 #21490449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33690 #21490449