Subgroups — C₃₁×#19989 #21476748

Order 7936.

620 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3341	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3334	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3303	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1683	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1673	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1646	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1635	C₂	1	2	yes	maximal, complemented
10	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#555	C₂²	1	2	yes	—
13	C₃₁×#547	C₂²	1	2	yes	—
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#487	C₂²	1	2	yes	—
17	C₃₁×#487	C₂²	1	2	yes	—
18	C₃₁×#485	—	2	2	no	—
19	C₃₁×#482	—	2	2	no	—
20	C₃₁×#480	—	2	2	no	—
21	C₃₁×#476	C₂²	1	2	yes	—
22	C₃₁×#476	C₂²	1	2	yes	—
23	C₃₁×#474	C₂²	1	2	yes	—
24	C₃₁×#466	—	2	1	no	—
25	C₃₁×#466	—	2	1	no	—
26	C₃₁×#465	—	2	2	no	—
27	C₃₁×#461	C₂²	1	2	yes	—
28	C₃₁×#449	—	2	2	no	—
29	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₈×D₄×C₃₁	C₂²	1	2	yes	—
34	C₃₁×#427	—	2	1	no	—
35	C₂×C₃₁×#107	—	2	1	no	—
36	C₃₁×#416	—	2	2	no	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#335	C₂²	1	1	yes	characteristic
39	C₃₁×#333	C₂²	1	2	yes	—
40	C₃₁×#332	C₂²	1	2	yes	—
41	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
43	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₂×SD₁₆×C₃₁	—	2	2	no	—
47	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
48	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	—	2	1	no	—
50	C₂²×C₈×C₃₁	—	2	1	no	—
51	C₂²×C₈×C₃₁	—	2	2	no	—
52	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
53	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
54	C₃₁×#124	—	4	2	no	—
55	C₃₁×#123	—	4	2	no	—
56	C₃₁×C₂²⋊Q₈	—	2	2	no	—
57	C₃₁×#121	—	2	2	no	—
58	C₃₁×#121	—	2	2	no	—
59	C₃₁×#121	—	2	2	no	—
60	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₄×D₄×C₃₁	C₂³	1	2	yes	—
62	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×#117	—	2	2	no	—
64	C₃₁×#117	—	2	2	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₃₁×#107	—	2	1	no	—
67	C₃₁×#107	—	2	1	no	—
68	C₃₁×#107	—	2	1	no	—
69	C₃₁×#107	—	2	1	no	—
70	C₃₁×#107	—	2	2	no	—
71	C₃₁×#107	D₄	1	1	yes	characteristic
72	C₃₁×#107	D₄	1	1	yes	characteristic
73	C₃₁×#106	—	2	2	no	—
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
76	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
77	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₃₁×#103	—	2	2	no	—
79	C₃₁×#103	—	2	2	no	—
80	C₃₁×#103	—	2	2	no	—
81	C₃₁×#103	—	4	2	no	—
82	C₃₁×#102	—	2	2	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	D₄	1	1	yes	characteristic
86	C₃₁×#102	D₄	1	1	yes	characteristic
87	C₃₁×#102	D₄	1	2	yes	—
88	C₃₁×C₂²⋊C₈	—	2	2	no	—
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	#41×C₃₁	—	4	4	no	—
95	#41×C₃₁	—	8	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	4	2	no	—
99	C₂×D₄×C₃₁	—	4	2	no	—
100	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
101	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	4	2	no	—
107	SD₁₆×C₃₁	—	8	2	no	—
108	D₈×C₃₁	—	2	2	no	—
109	D₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	4	2	no	—
115	C₂×C₈×C₃₁	—	4	2	no	—
116	C₂×C₈×C₃₁	—	8	2	no	—
117	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	4	2	no	—
126	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²⋊C₄×C₃₁	—	2	1	no	—
130	C₂²⋊C₄×C₃₁	—	2	2	no	—
131	C₂²⋊C₄×C₃₁	—	2	1	no	—
132	C₂²⋊C₄×C₃₁	—	4	2	no	—
133	C₂²⋊C₄×C₃₁	—	4	2	no	—
134	C₂²⋊C₄×C₃₁	—	4	2	no	—
135	C₄²×C₃₁	—	4	2	no	—
136	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
137	#19989	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂³×C₃₁	—	2	1	no	—
139	C₂³×C₃₁	—	2	2	no	—
140	C₂³×C₃₁	—	2	1	no	—
141	Q₈×C₃₁	—	4	2	no	—
142	Q₈×C₃₁	—	4	2	no	—
143	Q₈×C₃₁	—	8	2	no	—
144	D₄×C₃₁	—	2	1	no	—
145	D₄×C₃₁	—	2	2	no	—
146	D₄×C₃₁	—	2	1	no	—
147	D₄×C₃₁	—	2	1	no	—
148	D₄×C₃₁	—	2	2	no	—
149	D₄×C₃₁	—	8	4	no	—
150	D₄×C₃₁	—	16	2	no	—
151	D₄×C₃₁	#135	1	2	yes	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	4	1	no	—
156	C₂×C₄×C₃₁	—	4	1	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	1	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	8	4	no	—
164	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
167	C₈×C₃₁	—	2	1	no	—
168	C₈×C₃₁	—	2	1	no	—
169	C₈×C₃₁	—	2	1	no	—
170	C₈×C₃₁	—	4	1	no	—
171	C₈×C₃₁	—	4	1	no	—
172	C₈×C₃₁	—	4	1	no	—
173	C₈×C₃₁	—	4	2	no	—
174	C₈×C₃₁	#135	1	2	yes	—
175	#3341	C₂×C₃₁	1	1	yes	characteristic, complemented
176	#3334	C₂×C₃₁	1	1	yes	characteristic, complemented
177	#3303	C₂×C₃₁	1	2	yes	complemented
178	#1710	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#1683	C₂×C₃₁	1	2	yes	complemented
180	#1675	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#1673	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#1646	C₂×C₃₁	1	2	yes	complemented
183	#1635	C₂×C₃₁	1	2	yes	complemented
184	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
185	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
186	C₂²×C₃₁	—	2	2	no	—
187	C₂²×C₃₁	—	4	1	no	—
188	C₂²×C₃₁	—	4	1	no	—
189	C₂²×C₃₁	—	4	2	no	—
190	C₂²×C₃₁	—	4	1	no	—
191	C₂²×C₃₁	—	4	1	no	—
192	C₂²×C₃₁	—	4	2	no	—
193	C₂²×C₃₁	—	4	1	no	—
194	C₂²×C₃₁	—	4	2	no	—
195	C₂²×C₃₁	D₄²	1	1	yes	characteristic
196	C₄×C₃₁	—	2	1	no	—
197	C₄×C₃₁	—	4	2	no	—
198	C₄×C₃₁	—	4	1	no	—
199	C₄×C₃₁	—	4	1	no	—
200	C₄×C₃₁	—	8	2	no	—
201	C₄×C₃₁	—	8	2	no	—
202	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
203	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
204	C₄×C₃₁	#576	1	1	yes	characteristic
205	C₄×C₃₁	#578	1	1	yes	characteristic
206	#555	C₂²×C₃₁	1	2	yes	—
207	#547	C₂²×C₃₁	1	2	yes	—
208	#500	C₂²×C₃₁	1	1	yes	characteristic
209	#500	C₂²×C₃₁	1	1	yes	characteristic
210	#487	C₂²×C₃₁	1	2	yes	—
211	#487	C₂²×C₃₁	1	2	yes	—
212	#485	—	2	2	no	—
213	#482	—	2	2	no	—
214	#480	—	2	2	no	—
215	#476	C₂²×C₃₁	1	2	yes	—
216	#476	C₂²×C₃₁	1	2	yes	—
217	#474	C₂²×C₃₁	1	2	yes	—
218	#466	—	2	1	no	—
219	#466	—	2	1	no	—
220	#465	—	2	2	no	—
221	#461	C₂²×C₃₁	1	2	yes	—
222	#449	—	2	2	no	—
223	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
224	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
225	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
226	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
227	C₈×D₄	C₂²×C₃₁	1	2	yes	—
228	#427	—	2	1	no	—
229	C₂×#107	—	2	1	no	—
230	#416	—	2	2	no	—
231	#335	C₂²×C₃₁	1	1	yes	characteristic
232	#335	C₂²×C₃₁	1	1	yes	characteristic
233	#333	C₂²×C₃₁	1	2	yes	—
234	#332	C₂²×C₃₁	1	2	yes	—
235	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
236	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
237	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
238	C₈²	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×C₃₁	—	4	2	no	—
240	C₂×C₃₁	—	8	2	no	—
241	C₂×C₃₁	—	8	1	no	—
242	C₂×C₃₁	#3280	1	1	yes	characteristic
243	C₂×C₃₁	#3283	1	1	yes	characteristic
244	C₂×C₃₁	#3290	1	1	yes	characteristic
245	C₂×#41	—	2	2	no	—
246	C₂×SD₁₆	—	2	2	no	—
247	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
248	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
249	C₂²×C₈	—	2	1	no	—
250	C₂²×C₈	—	2	1	no	—
251	C₂²×C₈	—	2	2	no	—
252	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
253	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
254	#124	—	4	2	no	—
255	#123	—	4	2	no	—
256	C₂²⋊Q₈	—	2	2	no	—
257	#121	—	2	2	no	—
258	#121	—	2	2	no	—
259	#121	—	2	2	no	—
260	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
261	C₄×D₄	C₂³×C₃₁	1	2	yes	—
262	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
263	#117	—	2	2	no	—
264	#117	—	2	2	no	—
265	C₂×C₄⋊C₄	—	2	2	no	—
266	#107	—	2	1	no	—
267	#107	—	2	1	no	—
268	#107	—	2	1	no	—
269	#107	—	2	1	no	—
270	#107	—	2	2	no	—
271	#107	D₄×C₃₁	1	1	yes	characteristic
272	#107	D₄×C₃₁	1	1	yes	characteristic
273	#106	—	2	2	no	—
274	#106	—	2	2	no	—
275	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
277	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
278	#103	—	2	2	no	—
279	#103	—	2	2	no	—
280	#103	—	2	2	no	—
281	#103	—	4	2	no	—
282	#102	—	2	2	no	—
283	#102	—	2	2	no	—
284	#102	—	2	2	no	—
285	#102	D₄×C₃₁	1	1	yes	characteristic
286	#102	D₄×C₃₁	1	1	yes	characteristic
287	#102	D₄×C₃₁	1	2	yes	—
288	C₂²⋊C₈	—	2	2	no	—
289	C₂²⋊C₈	—	2	1	no	—
290	C₂²⋊C₈	—	2	1	no	—
291	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
293	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
294	C₃₁	#19989	1	1	yes	characteristic, direct product factor, Sylow
295	#41	—	4	4	no	—
296	#41	—	8	2	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×D₄	—	4	2	no	—
300	C₂×D₄	—	4	2	no	—
301	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
302	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
303	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	—	2	2	no	—
307	C₂²×C₄	—	4	2	no	—
308	SD₁₆	—	8	2	no	—
309	D₈	—	2	2	no	—
310	D₈	—	2	2	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	4	2	no	—
316	C₂×C₈	—	4	2	no	—
317	C₂×C₈	—	8	2	no	—
318	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	2	no	—
327	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
330	C₂²⋊C₄	—	2	1	no	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₂²⋊C₄	—	2	1	no	—
333	C₂²⋊C₄	—	4	2	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₂²⋊C₄	—	4	2	no	—
336	C₄²	—	4	2	no	—
337	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
338	C₂³	—	2	1	no	—
339	C₂³	—	2	2	no	—
340	C₂³	—	2	1	no	—
341	Q₈	—	4	2	no	—
342	Q₈	—	4	2	no	—
343	Q₈	—	8	2	no	—
344	D₄	—	2	1	no	—
345	D₄	—	2	2	no	—
346	D₄	—	2	1	no	—
347	D₄	—	2	1	no	—
348	D₄	—	2	2	no	—
349	D₄	—	8	4	no	—
350	D₄	—	16	2	no	—
351	D₄	C₃₁×#135	1	2	yes	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	8	4	no	—
364	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
365	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
366	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	4	1	no	—
371	C₈	—	4	1	no	—
372	C₈	—	4	1	no	—
373	C₈	—	4	2	no	—
374	C₈	C₃₁×#135	1	2	yes	—
375	C₂²	—	2	2	no	—
376	C₂²	—	4	1	no	—
377	C₂²	—	4	1	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	1	no	—
380	C₂²	—	4	1	no	—
381	C₂²	—	4	2	no	—
382	C₂²	—	4	1	no	—
383	C₂²	—	4	2	no	—
384	C₂²	D₄²×C₃₁	1	1	yes	characteristic
385	C₄	—	2	1	no	—
386	C₄	—	4	2	no	—
387	C₄	—	4	1	no	—
388	C₄	—	4	1	no	—
389	C₄	—	8	2	no	—
390	C₄	—	8	2	no	—
391	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
392	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
393	C₄	C₃₁×#576	1	1	yes	characteristic
394	C₄	C₃₁×#578	1	1	yes	characteristic
395	C₂	—	4	2	no	—
396	C₂	—	8	2	no	—
397	C₂	—	8	1	no	—
398	C₂	C₃₁×#3280	1	1	yes	characteristic
399	C₂	C₃₁×#3283	1	1	yes	characteristic
400	C₂	C₃₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3341	C₂	semidirect
2. C₃₁×#3334	C₂	semidirect
3. C₃₁×#3303	C₂	semidirect
4. C₃₁×#1710	C₂	semidirect
5. C₃₁×#1683	C₂	semidirect
6. C₃₁×#1675	C₂	semidirect
7. C₃₁×#1673	C₂	semidirect
8. C₃₁×#1646	C₂	semidirect
9. C₃₁×#1635	C₂	semidirect
10. C₈×D₈×C₃₁	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
137. #19989	C₃₁	direct (N × H)
175. #3341	C₂×C₃₁	semidirect
176. #3334	C₂×C₃₁	semidirect
177. #3303	C₂×C₃₁	semidirect
178. #1710	C₂×C₃₁	semidirect
179. #1683	C₂×C₃₁	semidirect
180. #1675	C₂×C₃₁	semidirect
181. #1673	C₂×C₃₁	semidirect
182. #1646	C₂×C₃₁	semidirect
183. #1635	C₂×C₃₁	semidirect
184. C₈×D₈	C₂×C₃₁	semidirect
185. C₈×D₈	C₂×C₃₁	semidirect
294. C₃₁	#19989	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×#3341	2
C₃₁×#3334	2
C₃₁×#3303	2
C₃₁×#1710	2
C₃₁×#1683	2
C₃₁×#1675	2
C₃₁×#1673	2
C₃₁×#1646	2
C₃₁×#1635	2
C₈×D₈×C₃₁	2
#19989	31

Subgroups — C31×#19989 #21476748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19989 #21476748