Subgroups — C₃₁×#33861 #21490620

Order 7936.

654 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3334	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2232	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2224	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2215	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2184	C₂	1	2	yes	maximal, complemented
9	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#357	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#547	—	2	1	no	—
14	D₄²×C₃₁	—	2	1	no	—
15	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×D₁₆	C₂²	1	2	yes	complemented
17	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	complemented
18	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
19	C₃₁×#493	C₂²	1	1	yes	characteristic
20	C₃₁×#480	—	2	2	no	—
21	C₃₁×#474	—	2	1	no	—
22	C₃₁×#466	C₂²	1	2	yes	—
23	C₃₁×#466	C₂²	1	2	yes	complemented
24	C₃₁×#459	—	2	1	no	—
25	C₃₁×#447	—	2	2	no	—
26	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
28	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
30	C₃₁×#366	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#366	C₂²	1	2	yes	complemented
32	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
33	C₃₁×#357	C₂²	1	1	yes	characteristic
34	C₃₁×#357	C₂²	1	2	yes	—
35	C₃₁×#357	C₂²	1	2	yes	complemented
36	C₃₁×#357	C₂²	1	2	yes	complemented
37	C₃₁×#357	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#357	C₂²	1	2	yes	complemented
39	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
41	C₂×#41×C₃₁	—	4	1	no	—
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	—	4	1	no	—
44	C₂×D₈×C₃₁	—	2	4	no	—
45	C₂×D₈×C₃₁	—	2	2	no	—
46	C₂×D₈×C₃₁	—	2	1	no	—
47	C₂×D₈×C₃₁	—	4	2	no	—
48	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×D₈×C₃₁	C₂³	1	2	yes	—
50	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₃₁×C₄⋊Q₈	—	2	1	no	—
53	C₃₁×#127	—	2	1	no	—
54	C₃₁×#123	—	4	2	no	—
55	C₃₁×C₂²⋊Q₈	—	4	1	no	—
56	C₃₁×#121	—	2	2	no	—
57	C₃₁×#121	—	2	2	no	—
58	C₃₁×#121	—	4	1	no	—
59	C₃₁×#120	—	4	2	no	—
60	C₄×D₄×C₃₁	—	2	1	no	—
61	C₄×D₄×C₃₁	—	2	1	no	—
62	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₃₁×D₁₆	—	2	2	no	—
65	C₃₁×D₁₆	—	4	2	no	—
66	C₂×C₁₆×C₃₁	—	4	2	no	—
67	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
69	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#107	—	2	2	no	—
71	C₃₁×#107	—	2	1	no	—
72	C₃₁×#107	C₂³	1	1	yes	characteristic
73	C₃₁×#107	C₂³	1	2	yes	—
74	C₃₁×#107	C₂³	1	1	yes	characteristic
75	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
76	C₃₁×#102	—	2	1	no	—
77	C₃₁×#102	—	2	1	no	—
78	C₃₁×#102	—	2	2	no	—
79	C₃₁×#102	—	2	2	no	—
80	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂⁴×C₃₁	—	4	2	no	—
83	#41×C₃₁	—	8	4	no	—
84	C₂×Q₈×C₃₁	—	4	1	no	—
85	C₂×D₄×C₃₁	—	2	1	no	—
86	C₂×D₄×C₃₁	—	2	1	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	4	4	no	—
89	C₂×D₄×C₃₁	—	4	2	no	—
90	C₂×D₄×C₃₁	—	4	2	no	—
91	C₂×D₄×C₃₁	—	4	2	no	—
92	C₂×D₄×C₃₁	—	4	1	no	—
93	C₂×D₄×C₃₁	—	4	1	no	—
94	C₂×D₄×C₃₁	—	8	2	no	—
95	C₂×D₄×C₃₁	—	8	4	no	—
96	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	—	4	2	no	—
98	C₂²×C₄×C₃₁	—	4	1	no	—
99	C₂²×C₄×C₃₁	—	4	1	no	—
100	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
101	D₈×C₃₁	—	2	1	no	—
102	D₈×C₃₁	—	2	2	no	—
103	D₈×C₃₁	—	2	1	no	—
104	D₈×C₃₁	—	2	1	no	—
105	D₈×C₃₁	—	2	2	no	—
106	D₈×C₃₁	—	4	4	no	—
107	D₈×C₃₁	—	8	1	no	—
108	D₈×C₃₁	#41	1	2	yes	complemented
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₃₁	—	2	1	no	—
115	C₄⋊C₄×C₃₁	—	2	1	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	4	2	no	—
118	C₄⋊C₄×C₃₁	—	4	1	no	—
119	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
120	C₂²⋊C₄×C₃₁	—	4	1	no	—
121	C₂²⋊C₄×C₃₁	—	4	1	no	—
122	C₂²⋊C₄×C₃₁	—	4	2	no	—
123	C₂²⋊C₄×C₃₁	—	4	2	no	—
124	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
125	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₃₁	—	2	1	no	—
127	C₁₆×C₃₁	—	2	1	no	—
128	C₁₆×C₃₁	—	2	2	no	—
129	#33861	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₃₁	—	4	1	no	—
131	C₂³×C₃₁	—	4	1	no	—
132	C₂³×C₃₁	—	4	2	no	—
133	C₂³×C₃₁	—	8	4	no	—
134	C₂³×C₃₁	—	8	4	no	—
135	C₂³×C₃₁	—	16	2	no	—
136	C₂³×C₃₁	C₂×D₈	1	2	yes	—
137	Q₈×C₃₁	—	8	2	no	—
138	D₄×C₃₁	—	2	2	no	—
139	D₄×C₃₁	—	4	2	no	—
140	D₄×C₃₁	—	4	1	no	—
141	D₄×C₃₁	—	4	1	no	—
142	D₄×C₃₁	—	4	2	no	—
143	D₄×C₃₁	—	4	1	no	—
144	D₄×C₃₁	—	8	4	no	—
145	D₄×C₃₁	—	8	4	no	—
146	D₄×C₃₁	—	8	4	no	—
147	D₄×C₃₁	—	8	2	no	—
148	D₄×C₃₁	—	16	1	no	—
149	D₄×C₃₁	D₁₆	1	4	yes	complemented
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	4	1	no	—
153	C₂×C₄×C₃₁	—	4	1	no	—
154	C₂×C₄×C₃₁	—	4	2	no	—
155	C₂×C₄×C₃₁	—	8	4	no	—
156	C₂×C₄×C₃₁	—	8	2	no	—
157	C₂×C₄×C₃₁	—	8	2	no	—
158	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
160	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₃₁	—	2	2	no	—
163	C₈×C₃₁	—	2	1	no	—
164	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
165	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
166	#3334	C₂×C₃₁	1	1	yes	characteristic, complemented
167	D₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
168	#2240	C₂×C₃₁	1	1	yes	characteristic, complemented
169	#2232	C₂×C₃₁	1	2	yes	complemented
170	#2224	C₂×C₃₁	1	1	yes	characteristic, complemented
171	#2215	C₂×C₃₁	1	2	yes	complemented
172	#2206	C₂×C₃₁	1	1	yes	characteristic, complemented
173	#2184	C₂×C₃₁	1	2	yes	complemented
174	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
175	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
176	C₂×#357	C₂×C₃₁	1	2	yes	complemented
177	C₂²×C₃₁	—	2	2	no	—
178	C₂²×C₃₁	—	2	2	no	—
179	C₂²×C₃₁	—	4	2	no	—
180	C₂²×C₃₁	—	8	1	no	—
181	C₂²×C₃₁	—	8	1	no	—
182	C₂²×C₃₁	—	8	1	no	—
183	C₂²×C₃₁	—	8	2	no	—
184	C₂²×C₃₁	—	8	2	no	—
185	C₂²×C₃₁	—	8	1	no	—
186	C₂²×C₃₁	—	8	2	no	—
187	C₂²×C₃₁	—	8	1	no	—
188	C₂²×C₃₁	—	16	8	no	—
189	C₂²×C₃₁	C₂×D₁₆	1	4	yes	—
190	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
191	C₄×C₃₁	—	2	1	no	—
192	C₄×C₃₁	—	2	2	no	—
193	C₄×C₃₁	—	8	1	no	—
194	C₄×C₃₁	—	8	1	no	—
195	C₄×C₃₁	—	8	2	no	—
196	C₄×C₃₁	C₂×D₁₆	1	2	yes	—
197	C₄×C₃₁	#546	1	1	yes	characteristic
198	C₄×C₃₁	#577	1	1	yes	characteristic
199	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
200	#547	—	2	1	no	—
201	D₄²	—	2	1	no	—
202	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
203	C₂×D₁₆	C₂²×C₃₁	1	2	yes	complemented
204	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	complemented
205	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
206	#493	C₂²×C₃₁	1	1	yes	characteristic
207	#480	—	2	2	no	—
208	#474	—	2	1	no	—
209	#466	C₂²×C₃₁	1	2	yes	—
210	#466	C₂²×C₃₁	1	2	yes	complemented
211	#459	—	2	1	no	—
212	#447	—	2	2	no	—
213	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
214	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
215	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
216	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
217	#366	C₂²×C₃₁	1	1	yes	characteristic, complemented
218	#366	C₂²×C₃₁	1	2	yes	complemented
219	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
220	#357	C₂²×C₃₁	1	1	yes	characteristic
221	#357	C₂²×C₃₁	1	2	yes	—
222	#357	C₂²×C₃₁	1	2	yes	complemented
223	#357	C₂²×C₃₁	1	2	yes	complemented
224	#357	C₂²×C₃₁	1	1	yes	characteristic, complemented
225	#357	C₂²×C₃₁	1	2	yes	complemented
226	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
227	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
228	C₂×C₃₁	—	2	4	no	—
229	C₂×C₃₁	—	8	2	no	—
230	C₂×C₃₁	—	16	1	no	—
231	C₂×C₃₁	—	16	2	no	—
232	C₂×C₃₁	#3294	1	1	yes	characteristic
233	C₂×C₃₁	C₂²×D₁₆	1	1	yes	characteristic
234	C₂×C₃₁	#3416	1	1	yes	characteristic
235	C₂×#41	—	4	1	no	—
236	C₂²×D₄	—	2	2	no	—
237	C₂²×D₄	—	4	1	no	—
238	C₂×D₈	—	2	4	no	—
239	C₂×D₈	—	2	2	no	—
240	C₂×D₈	—	2	1	no	—
241	C₂×D₈	—	4	2	no	—
242	C₂×D₈	C₂³×C₃₁	1	2	yes	—
243	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
244	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
245	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
246	C₄⋊Q₈	—	2	1	no	—
247	#127	—	2	1	no	—
248	#123	—	4	2	no	—
249	C₂²⋊Q₈	—	4	1	no	—
250	#121	—	2	2	no	—
251	#121	—	2	2	no	—
252	#121	—	4	1	no	—
253	#120	—	4	2	no	—
254	C₄×D₄	—	2	1	no	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄⋊C₄	—	2	2	no	—
258	D₁₆	—	2	2	no	—
259	D₁₆	—	4	2	no	—
260	C₂×C₁₆	—	4	2	no	—
261	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
262	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
263	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
264	#107	—	2	2	no	—
265	#107	—	2	1	no	—
266	#107	C₂³×C₃₁	1	1	yes	characteristic
267	#107	C₂³×C₃₁	1	2	yes	—
268	#107	C₂³×C₃₁	1	1	yes	characteristic
269	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
270	#102	—	2	1	no	—
271	#102	—	2	1	no	—
272	#102	—	2	2	no	—
273	#102	—	2	2	no	—
274	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
275	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₃₁	#33861	1	1	yes	characteristic, direct product factor, Sylow
277	C₂⁴	—	4	2	no	—
278	#41	—	8	4	no	—
279	C₂×Q₈	—	4	1	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	4	4	no	—
284	C₂×D₄	—	4	2	no	—
285	C₂×D₄	—	4	2	no	—
286	C₂×D₄	—	4	2	no	—
287	C₂×D₄	—	4	1	no	—
288	C₂×D₄	—	4	1	no	—
289	C₂×D₄	—	8	2	no	—
290	C₂×D₄	—	8	4	no	—
291	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
292	C₂²×C₄	—	4	2	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	—	4	1	no	—
295	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
296	D₈	—	2	1	no	—
297	D₈	—	2	2	no	—
298	D₈	—	2	1	no	—
299	D₈	—	2	1	no	—
300	D₈	—	2	2	no	—
301	D₈	—	4	4	no	—
302	D₈	—	8	1	no	—
303	D₈	#41×C₃₁	1	2	yes	complemented
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
307	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
308	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
315	C₂²⋊C₄	—	4	1	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	4	2	no	—
318	C₂²⋊C₄	—	4	2	no	—
319	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
320	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₁₆	—	2	1	no	—
322	C₁₆	—	2	1	no	—
323	C₁₆	—	2	2	no	—
324	C₂³	—	4	1	no	—
325	C₂³	—	4	1	no	—
326	C₂³	—	4	2	no	—
327	C₂³	—	8	4	no	—
328	C₂³	—	8	4	no	—
329	C₂³	—	16	2	no	—
330	C₂³	C₂×D₈×C₃₁	1	2	yes	—
331	Q₈	—	8	2	no	—
332	D₄	—	2	2	no	—
333	D₄	—	4	2	no	—
334	D₄	—	4	1	no	—
335	D₄	—	4	1	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	1	no	—
338	D₄	—	8	4	no	—
339	D₄	—	8	4	no	—
340	D₄	—	8	4	no	—
341	D₄	—	8	2	no	—
342	D₄	—	16	1	no	—
343	D₄	C₃₁×D₁₆	1	4	yes	complemented
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	4	no	—
350	C₂×C₄	—	8	2	no	—
351	C₂×C₄	—	8	2	no	—
352	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
353	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
354	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	2	1	no	—
358	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
359	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
360	C₂²	—	2	2	no	—
361	C₂²	—	2	2	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	8	1	no	—
369	C₂²	—	8	2	no	—
370	C₂²	—	8	1	no	—
371	C₂²	—	16	8	no	—
372	C₂²	C₂×C₃₁×D₁₆	1	4	yes	—
373	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	1	no	—
378	C₄	—	8	2	no	—
379	C₄	C₂×C₃₁×D₁₆	1	2	yes	—
380	C₄	C₃₁×#546	1	1	yes	characteristic
381	C₄	C₃₁×#577	1	1	yes	characteristic
382	C₂	—	2	4	no	—
383	C₂	—	8	2	no	—
384	C₂	—	16	2	no	—
385	C₂	—	16	1	no	—
386	C₂	C₃₁×#3294	1	1	yes	characteristic
387	C₂	C₂²×C₃₁×D₁₆	1	1	yes	characteristic
388	C₂	C₃₁×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3334	C₂	semidirect
2. D₄×D₈×C₃₁	C₂	semidirect
3. C₃₁×#2240	C₂	semidirect
4. C₃₁×#2232	C₂	semidirect
5. C₃₁×#2224	C₂	semidirect
6. C₃₁×#2215	C₂	semidirect
7. C₃₁×#2206	C₂	semidirect
8. C₃₁×#2184	C₂	semidirect
9. C₄×C₃₁×D₁₆	C₂	semidirect
10. D₄×C₁₆×C₃₁	C₂	semidirect
11. C₂×C₃₁×#357	C₂	semidirect
16. C₂×C₃₁×D₁₆	C₂²	semidirect
17. C₂²×C₁₆×C₃₁	C₂²	semidirect
18. C₃₁×#500	C₂²	semidirect
23. C₃₁×#466	C₂²	semidirect
27. C₄×D₈×C₃₁	C₂²	semidirect
29. C₂×C₃₁×#107	C₂²	semidirect
30. C₃₁×#366	C₂²	semidirect
31. C₃₁×#366	C₂²	semidirect
32. C₃₁×C₄⋊C₁₆	C₂²	semidirect
35. C₃₁×#357	C₂²	semidirect
36. C₃₁×#357	C₂²	semidirect
37. C₃₁×#357	C₂²	semidirect
38. C₃₁×#357	C₂²	semidirect
39. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
40. C₄×C₁₆×C₃₁	C₂²	semidirect
108. D₈×C₃₁	#41	semidirect
129. #33861	C₃₁	direct (N × H)
149. D₄×C₃₁	D₁₆	semidirect
166. #3334	C₂×C₃₁	semidirect
167. D₄×D₈	C₂×C₃₁	semidirect
168. #2240	C₂×C₃₁	semidirect
169. #2232	C₂×C₃₁	semidirect
170. #2224	C₂×C₃₁	semidirect
171. #2215	C₂×C₃₁	semidirect
172. #2206	C₂×C₃₁	semidirect
173. #2184	C₂×C₃₁	semidirect
174. C₄×D₁₆	C₂×C₃₁	semidirect
175. D₄×C₁₆	C₂×C₃₁	semidirect
176. C₂×#357	C₂×C₃₁	semidirect
203. C₂×D₁₆	C₂²×C₃₁	semidirect
204. C₂²×C₁₆	C₂²×C₃₁	semidirect
205. #500	C₂²×C₃₁	semidirect
210. #466	C₂²×C₃₁	semidirect
214. C₄×D₈	C₂²×C₃₁	semidirect
216. C₂×#107	C₂²×C₃₁	semidirect
217. #366	C₂²×C₃₁	semidirect
218. #366	C₂²×C₃₁	semidirect
219. C₄⋊C₁₆	C₂²×C₃₁	semidirect
222. #357	C₂²×C₃₁	semidirect
223. #357	C₂²×C₃₁	semidirect
224. #357	C₂²×C₃₁	semidirect
225. #357	C₂²×C₃₁	semidirect
226. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
227. C₄×C₁₆	C₂²×C₃₁	semidirect
276. C₃₁	#33861	direct (N × H)
303. D₈	#41×C₃₁	semidirect
343. D₄	C₃₁×D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3334	2
D₄×D₈×C₃₁	2
C₃₁×#2240	2
C₃₁×#2232	2
C₃₁×#2224	2
C₃₁×#2215	2
C₃₁×#2206	2
C₃₁×#2184	2
C₄×C₃₁×D₁₆	2
D₄×C₁₆×C₃₁	2
C₂×C₃₁×#357	2
#33861	31

Subgroups — C31×#33861 #21490620

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33861 #21490620