Subgroups — C₃₁×#32836 #21489595

Order 7936.

632 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2860	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2101	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2092	C₂	1	2	yes	maximal
5	C₃₁×#2084	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2083	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2071	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#2071	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#2059	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2025	C₂	1	2	yes	maximal
11	C₃₁×#1839	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#126	—	2	1	no	—
13	C₂×C₃₁×#125	C₂²	1	2	yes	—
14	C₂×C₃₁×#107	C₂²	1	2	yes	—
15	C₂×C₃₁×#106	C₂²	1	2	yes	—
16	C₂×C₃₁×#102	C₂²	1	2	yes	—
17	C₂×C₃₁×#102	C₂²	1	2	yes	—
18	C₃₁×#390	C₂²	1	1	yes	characteristic
19	C₃₁×#390	C₂²	1	1	yes	characteristic
20	C₃₁×#390	C₂²	1	1	yes	characteristic
21	C₃₁×#389	—	2	1	no	—
22	C₃₁×#389	—	2	1	no	—
23	C₃₁×#389	—	2	1	no	—
24	C₃₁×#389	—	2	1	no	—
25	C₃₁×#389	C₂²	1	1	yes	characteristic
26	C₃₁×#389	C₂²	1	1	yes	characteristic
27	C₃₁×#389	C₂²	1	1	yes	characteristic
28	C₃₁×#388	—	2	1	no	—
29	C₃₁×#388	—	2	1	no	—
30	C₃₁×#388	—	2	1	no	—
31	C₃₁×#384	C₂²	1	1	yes	characteristic
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₃₁×#382	C₂²	1	2	yes	—
34	C₄³×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₃₁×#126	—	4	4	no	—
47	C₃₁×#125	—	4	2	no	—
48	C₃₁×#125	—	4	2	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#107	—	4	2	no	—
77	C₃₁×#106	—	4	2	no	—
78	C₃₁×#102	—	4	2	no	—
79	C₃₁×#102	—	4	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₃₁×#95	—	2	1	no	—
83	C₃₁×#95	—	2	1	no	—
84	C₃₁×#95	—	2	1	no	—
85	C₂⁴×C₃₁	—	2	1	no	—
86	C₂×D₄×C₃₁	—	2	1	no	—
87	C₂×D₄×C₃₁	—	2	1	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	1	no	—
91	C₂²×C₄×C₃₁	—	2	1	no	—
92	C₂²×C₄×C₃₁	—	2	1	no	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	2	1	no	—
111	C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₄⋊C₄×C₃₁	—	2	1	no	—
113	C₄⋊C₄×C₃₁	—	2	2	no	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	4	2	no	—
117	C₄⋊C₄×C₃₁	—	4	2	no	—
118	C₄⋊C₄×C₃₁	—	4	4	no	—
119	C₄⋊C₄×C₃₁	—	4	4	no	—
120	C₄⋊C₄×C₃₁	—	4	2	no	—
121	C₄⋊C₄×C₃₁	—	4	2	no	—
122	C₄⋊C₄×C₃₁	—	4	2	no	—
123	C₄⋊C₄×C₃₁	—	4	2	no	—
124	C₄⋊C₄×C₃₁	—	4	2	no	—
125	C₂²⋊C₄×C₃₁	—	4	2	no	—
126	C₂²⋊C₄×C₃₁	—	4	2	no	—
127	C₂²⋊C₄×C₃₁	—	4	2	no	—
128	C₂²⋊C₄×C₃₁	—	4	2	no	—
129	C₂²⋊C₄×C₃₁	—	4	2	no	—
130	C₂²⋊C₄×C₃₁	—	4	2	no	—
131	C₄²×C₃₁	—	2	2	no	—
132	C₄²×C₃₁	—	2	2	no	—
133	C₄²×C₃₁	—	2	1	no	—
134	C₄²×C₃₁	—	2	1	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	4	1	no	—
138	C₄²×C₃₁	—	4	1	no	—
139	#32836	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂³×C₃₁	—	4	1	no	—
141	C₂³×C₃₁	—	4	1	no	—
142	C₂³×C₃₁	—	4	1	no	—
143	C₂³×C₃₁	—	4	1	no	—
144	C₂³×C₃₁	—	4	1	no	—
145	C₂³×C₃₁	—	4	1	no	—
146	C₂³×C₃₁	—	4	1	no	—
147	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
148	D₄×C₃₁	—	4	2	no	—
149	D₄×C₃₁	—	4	2	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	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	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	2	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	#135	1	2	yes	—
182	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
183	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
186	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
187	C₈×C₃₁	—	4	2	no	—
188	C₈×C₃₁	—	4	2	no	—
189	#2860	C₂×C₃₁	1	1	yes	characteristic
190	#2859	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#2101	C₂×C₃₁	1	2	yes	complemented
192	#2092	C₂×C₃₁	1	2	yes	—
193	#2084	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#2083	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#2071	C₂×C₃₁	1	1	yes	characteristic
196	#2071	C₂×C₃₁	1	1	yes	characteristic
197	#2059	C₂×C₃₁	1	2	yes	complemented
198	#2025	C₂×C₃₁	1	2	yes	—
199	#1839	C₂×C₃₁	1	1	yes	characteristic, complemented
200	C₂²×C₃₁	—	8	1	no	—
201	C₂²×C₃₁	—	8	1	no	—
202	C₂²×C₃₁	—	8	1	no	—
203	C₂²×C₃₁	—	8	1	no	—
204	C₂²×C₃₁	—	8	1	no	—
205	C₂²×C₃₁	—	8	1	no	—
206	C₂²×C₃₁	—	8	1	no	—
207	C₂²×C₃₁	#535	1	1	yes	characteristic
208	C₂²×C₃₁	#536	1	1	yes	characteristic
209	C₂²×C₃₁	#536	1	1	yes	characteristic
210	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
211	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
212	C₂²×C₃₁	#575	1	1	yes	characteristic
213	C₂²×C₃₁	#575	1	1	yes	characteristic
214	C₄×C₃₁	—	2	2	no	—
215	C₄×C₃₁	—	2	2	no	—
216	C₄×C₃₁	—	4	1	no	—
217	C₄×C₃₁	—	4	1	no	—
218	C₄×C₃₁	—	4	1	no	—
219	C₄×C₃₁	—	4	1	no	—
220	C₄×C₃₁	—	8	2	no	—
221	C₄×C₃₁	—	8	2	no	—
222	C₄×C₃₁	—	8	1	no	—
223	C₄×C₃₁	—	8	1	no	—
224	C₄×C₃₁	—	8	1	no	—
225	C₄×C₃₁	#563	1	2	yes	—
226	C₄×C₃₁	#563	1	2	yes	—
227	C₂×#126	—	2	1	no	—
228	C₂×#125	C₂²×C₃₁	1	2	yes	—
229	C₂×#107	C₂²×C₃₁	1	2	yes	—
230	C₂×#106	C₂²×C₃₁	1	2	yes	—
231	C₂×#102	C₂²×C₃₁	1	2	yes	—
232	C₂×#102	C₂²×C₃₁	1	2	yes	—
233	#390	C₂²×C₃₁	1	1	yes	characteristic
234	#390	C₂²×C₃₁	1	1	yes	characteristic
235	#390	C₂²×C₃₁	1	1	yes	characteristic
236	#389	—	2	1	no	—
237	#389	—	2	1	no	—
238	#389	—	2	1	no	—
239	#389	—	2	1	no	—
240	#389	C₂²×C₃₁	1	1	yes	characteristic
241	#389	C₂²×C₃₁	1	1	yes	characteristic
242	#389	C₂²×C₃₁	1	1	yes	characteristic
243	#388	—	2	1	no	—
244	#388	—	2	1	no	—
245	#388	—	2	1	no	—
246	#384	C₂²×C₃₁	1	1	yes	characteristic
247	#384	C₂²×C₃₁	1	1	yes	characteristic
248	#382	C₂²×C₃₁	1	2	yes	—
249	C₄³	C₂²×C₃₁	1	1	yes	characteristic
250	#340	C₂²×C₃₁	1	2	yes	—
251	#340	C₂²×C₃₁	1	2	yes	—
252	#340	C₂²×C₃₁	1	2	yes	—
253	#340	C₂²×C₃₁	1	2	yes	—
254	#336	C₂²×C₃₁	1	2	yes	—
255	#336	C₂²×C₃₁	1	1	yes	characteristic
256	#336	C₂²×C₃₁	1	1	yes	characteristic
257	#336	C₂²×C₃₁	1	2	yes	—
258	C₂×C₃₁	—	16	1	no	—
259	C₂×C₃₁	#2821	1	1	yes	characteristic
260	C₂×C₃₁	#3212	1	1	yes	characteristic
261	C₂×C₃₁	#3214	1	1	yes	characteristic
262	C₂×C₃₁	#3228	1	1	yes	characteristic
263	C₂×C₃₁	#3232	1	1	yes	characteristic
264	C₂×C₃₁	#3234	1	1	yes	characteristic
265	C₂×C₃₁	#3240	1	1	yes	characteristic
266	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
267	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
268	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
269	#126	—	4	4	no	—
270	#125	—	4	2	no	—
271	#125	—	4	2	no	—
272	C₂×C₄⋊C₄	—	2	2	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	—	2	1	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
282	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
285	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
286	C₂×C₂²⋊C₄	—	2	1	no	—
287	C₂×C₂²⋊C₄	—	2	1	no	—
288	C₂×C₂²⋊C₄	—	2	1	no	—
289	C₂×C₂²⋊C₄	—	2	1	no	—
290	C₂×C₂²⋊C₄	—	2	1	no	—
291	C₂×C₂²⋊C₄	—	2	1	no	—
292	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
293	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
297	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
298	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
299	#107	—	4	2	no	—
300	#106	—	4	2	no	—
301	#102	—	4	2	no	—
302	#102	—	4	2	no	—
303	#95	—	2	2	no	—
304	#95	—	2	2	no	—
305	#95	—	2	1	no	—
306	#95	—	2	1	no	—
307	#95	—	2	1	no	—
308	C₃₁	#32836	1	1	yes	characteristic, direct product factor, Sylow
309	C₂⁴	—	2	1	no	—
310	C₂×D₄	—	2	1	no	—
311	C₂×D₄	—	2	1	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	4	2	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₄⋊C₄	—	4	4	no	—
343	C₄⋊C₄	—	4	4	no	—
344	C₄⋊C₄	—	4	2	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	—	4	2	no	—
347	C₄⋊C₄	—	4	2	no	—
348	C₄⋊C₄	—	4	2	no	—
349	C₂²⋊C₄	—	4	2	no	—
350	C₂²⋊C₄	—	4	2	no	—
351	C₂²⋊C₄	—	4	2	no	—
352	C₂²⋊C₄	—	4	2	no	—
353	C₂²⋊C₄	—	4	2	no	—
354	C₂²⋊C₄	—	4	2	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	2	1	no	—
358	C₄²	—	2	1	no	—
359	C₄²	—	2	2	no	—
360	C₄²	—	2	2	no	—
361	C₄²	—	4	1	no	—
362	C₄²	—	4	1	no	—
363	C₂³	—	4	1	no	—
364	C₂³	—	4	1	no	—
365	C₂³	—	4	1	no	—
366	C₂³	—	4	1	no	—
367	C₂³	—	4	1	no	—
368	C₂³	—	4	1	no	—
369	C₂³	—	4	1	no	—
370	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	C₃₁×#135	1	2	yes	—
405	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
406	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
407	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
408	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
409	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
410	C₈	—	4	2	no	—
411	C₈	—	4	2	no	—
412	C₂²	—	8	1	no	—
413	C₂²	—	8	1	no	—
414	C₂²	—	8	1	no	—
415	C₂²	—	8	1	no	—
416	C₂²	—	8	1	no	—
417	C₂²	—	8	1	no	—
418	C₂²	—	8	1	no	—
419	C₂²	C₃₁×#535	1	1	yes	characteristic
420	C₂²	C₃₁×#536	1	1	yes	characteristic
421	C₂²	C₃₁×#536	1	1	yes	characteristic
422	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
423	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
424	C₂²	C₃₁×#575	1	1	yes	characteristic
425	C₂²	C₃₁×#575	1	1	yes	characteristic
426	C₄	—	2	2	no	—
427	C₄	—	2	2	no	—
428	C₄	—	4	1	no	—
429	C₄	—	4	1	no	—
430	C₄	—	4	1	no	—
431	C₄	—	4	1	no	—
432	C₄	—	8	2	no	—
433	C₄	—	8	2	no	—
434	C₄	—	8	1	no	—
435	C₄	—	8	1	no	—
436	C₄	—	8	1	no	—
437	C₄	C₃₁×#563	1	2	yes	—
438	C₄	C₃₁×#563	1	2	yes	—
439	C₂	—	16	1	no	—
440	C₂	C₃₁×#2821	1	1	yes	characteristic
441	C₂	C₃₁×#3212	1	1	yes	characteristic
442	C₂	C₃₁×#3214	1	1	yes	characteristic
443	C₂	C₃₁×#3228	1	1	yes	characteristic
444	C₂	C₃₁×#3232	1	1	yes	characteristic
445	C₂	C₃₁×#3234	1	1	yes	characteristic
446	C₂	C₃₁×#3240	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2859	C₂	semidirect
3. C₃₁×#2101	C₂	semidirect
5. C₃₁×#2084	C₂	semidirect
6. C₃₁×#2083	C₂	semidirect
9. C₃₁×#2059	C₂	semidirect
11. C₃₁×#1839	C₂	semidirect
139. #32836	C₃₁	direct (N × H)
190. #2859	C₂×C₃₁	semidirect
191. #2101	C₂×C₃₁	semidirect
193. #2084	C₂×C₃₁	semidirect
194. #2083	C₂×C₃₁	semidirect
197. #2059	C₂×C₃₁	semidirect
199. #1839	C₂×C₃₁	semidirect
308. C₃₁	#32836	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×#2860	2
C₃₁×#2859	2
C₃₁×#2101	2
C₃₁×#2092	2
C₃₁×#2084	2
C₃₁×#2083	2
C₃₁×#2071	2
C₃₁×#2059	2
C₃₁×#2025	2
C₃₁×#1839	2
#32836	31

Subgroups — C31×#32836 #21489595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32836 #21489595