Subgroups — C₃₁×#32647 #21489406

Order 7936.

746 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2567	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2548	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2041	C₂	1	2	yes	maximal
5	C₃₁×#2023	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1936	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1921	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#124	—	2	1	no	—
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#107	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#102	C₂²	1	2	yes	—
17	C₂×C₃₁×#102	C₂²	1	2	yes	—
18	C₂×C₃₁×#102	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
21	C₃₁×#395	—	2	1	no	—
22	C₃₁×#395	—	2	1	no	—
23	C₃₁×#394	—	2	1	no	—
24	C₃₁×#394	—	2	1	no	—
25	C₃₁×#391	—	2	1	no	—
26	C₃₁×#390	—	2	1	no	—
27	C₃₁×#390	C₂²	1	2	yes	—
28	C₃₁×#390	C₂²	1	2	yes	—
29	C₃₁×#389	C₂²	1	2	yes	—
30	C₃₁×#389	C₂²	1	2	yes	—
31	C₃₁×#384	—	2	1	no	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#340	C₂²	1	2	yes	—
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₂²×Q₈×C₃₁	—	2	1	no	—
40	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
41	C₂²×D₄×C₃₁	—	2	1	no	—
42	C₂²×D₄×C₃₁	—	2	1	no	—
43	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₃₁×C₄⋊Q₈	—	4	1	no	—
47	C₃₁×C₄⋊Q₈	—	4	1	no	—
48	C₃₁×#127	—	4	1	no	—
49	C₃₁×#127	—	4	1	no	—
50	C₃₁×#124	—	4	4	no	—
51	C₄×Q₈×C₃₁	—	4	1	no	—
52	C₄×Q₈×C₃₁	—	4	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₄²×C₃₁	—	2	1	no	—
67	C₂×C₄²×C₃₁	—	2	1	no	—
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#107	—	2	4	no	—
72	C₃₁×C₄⋊C₈	—	4	1	no	—
73	C₃₁×#102	—	2	4	no	—
74	C₃₁×#102	—	2	4	no	—
75	C₃₁×#102	—	4	2	no	—
76	C₃₁×#102	—	4	2	no	—
77	C₃₁×#97	—	4	1	no	—
78	C₃₁×#95	—	2	1	no	—
79	C₃₁×#95	—	2	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	1	no	—
82	C₂⁴×C₃₁	—	2	2	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	—	2	2	no	—
85	C₂×Q₈×C₃₁	—	4	1	no	—
86	C₂×Q₈×C₃₁	—	4	2	no	—
87	C₂×Q₈×C₃₁	—	8	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	4	2	no	—
95	C₂×D₄×C₃₁	—	4	2	no	—
96	C₂×D₄×C₃₁	—	4	2	no	—
97	C₂×D₄×C₃₁	—	4	2	no	—
98	C₂×D₄×C₃₁	—	8	2	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	2	no	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	4	1	no	—
121	C₄⋊C₄×C₃₁	—	4	1	no	—
122	C₄⋊C₄×C₃₁	—	4	1	no	—
123	C₄⋊C₄×C₃₁	—	4	2	no	—
124	C₄⋊C₄×C₃₁	—	8	2	no	—
125	C₄⋊C₄×C₃₁	—	8	2	no	—
126	C₄⋊C₄×C₃₁	#41	1	4	yes	—
127	C₂²⋊C₄×C₃₁	—	4	4	no	—
128	C₂²⋊C₄×C₃₁	—	4	4	no	—
129	C₂²⋊C₄×C₃₁	—	8	2	no	—
130	C₂²⋊C₄×C₃₁	—	8	2	no	—
131	C₄²×C₃₁	—	2	1	no	—
132	C₄²×C₃₁	—	2	1	no	—
133	C₄²×C₃₁	—	2	1	no	—
134	C₄²×C₃₁	—	2	1	no	—
135	C₄²×C₃₁	—	2	1	no	—
136	C₄²×C₃₁	—	2	1	no	—
137	C₄²×C₃₁	—	4	2	no	—
138	C₄²×C₃₁	—	4	2	no	—
139	#32647	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂³×C₃₁	—	4	2	no	—
141	C₂³×C₃₁	—	4	2	no	—
142	C₂³×C₃₁	—	4	2	no	—
143	C₂³×C₃₁	—	4	2	no	—
144	C₂³×C₃₁	—	4	2	no	—
145	C₂³×C₃₁	—	4	2	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
148	Q₈×C₃₁	—	4	4	no	—
149	Q₈×C₃₁	—	8	4	no	—
150	D₄×C₃₁	—	4	2	no	—
151	D₄×C₃₁	—	4	2	no	—
152	D₄×C₃₁	—	4	2	no	—
153	D₄×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	8	4	no	—
155	D₄×C₃₁	—	8	4	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	4	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
179	C₂×C₄×C₃₁	#137	1	2	yes	—
180	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
181	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
182	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
184	C₈×C₃₁	—	4	2	no	—
185	C₈×C₃₁	—	4	2	no	—
186	#2567	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#2548	C₂×C₃₁	1	1	yes	characteristic
188	#2067	C₂×C₃₁	1	2	yes	complemented
189	#2041	C₂×C₃₁	1	2	yes	—
190	#2023	C₂×C₃₁	1	2	yes	complemented
191	#1985	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1936	C₂×C₃₁	1	2	yes	complemented
193	#1925	C₂×C₃₁	1	2	yes	complemented
194	#1921	C₂×C₃₁	1	2	yes	complemented
195	C₂²×C₃₁	—	8	2	no	—
196	C₂²×C₃₁	—	8	2	no	—
197	C₂²×C₃₁	—	8	2	no	—
198	C₂²×C₃₁	—	8	2	no	—
199	C₂²×C₃₁	—	8	2	no	—
200	C₂²×C₃₁	—	8	2	no	—
201	C₂²×C₃₁	—	8	2	no	—
202	C₂²×C₃₁	#534	1	1	yes	characteristic
203	C₂²×C₃₁	#548	1	2	yes	—
204	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
206	C₂²×C₃₁	#576	1	2	yes	—
207	C₄×C₃₁	—	2	2	no	—
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	4	4	no	—
210	C₄×C₃₁	—	4	1	no	—
211	C₄×C₃₁	—	4	1	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	—	4	1	no	—
214	C₄×C₃₁	—	8	2	no	—
215	C₄×C₃₁	—	8	2	no	—
216	C₄×C₃₁	#542	1	2	yes	—
217	C₄×C₃₁	#555	1	1	yes	characteristic
218	C₄×C₃₁	#563	1	1	yes	characteristic
219	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
220	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×#124	—	2	1	no	—
222	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×#107	C₂²×C₃₁	1	2	yes	—
224	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
225	C₂×#102	C₂²×C₃₁	1	2	yes	—
226	C₂×#102	C₂²×C₃₁	1	2	yes	—
227	C₂×#102	C₂²×C₃₁	1	2	yes	—
228	C₂×#102	C₂²×C₃₁	1	2	yes	—
229	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
230	#395	—	2	1	no	—
231	#395	—	2	1	no	—
232	#394	—	2	1	no	—
233	#394	—	2	1	no	—
234	#391	—	2	1	no	—
235	#390	—	2	1	no	—
236	#390	C₂²×C₃₁	1	2	yes	—
237	#390	C₂²×C₃₁	1	2	yes	—
238	#389	C₂²×C₃₁	1	2	yes	—
239	#389	C₂²×C₃₁	1	2	yes	—
240	#384	—	2	1	no	—
241	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
242	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
243	#340	C₂²×C₃₁	1	2	yes	—
244	#340	C₂²×C₃₁	1	2	yes	—
245	#340	C₂²×C₃₁	1	2	yes	—
246	#336	C₂²×C₃₁	1	2	yes	—
247	#336	C₂²×C₃₁	1	2	yes	—
248	C₂×C₃₁	—	16	2	no	—
249	C₂×C₃₁	#2731	1	1	yes	characteristic
250	C₂×C₃₁	#3193	1	1	yes	characteristic
251	C₂×C₃₁	#3199	1	1	yes	characteristic
252	C₂×C₃₁	#3366	1	2	yes	—
253	C₂×C₃₁	#3375	1	2	yes	—
254	C₂²×Q₈	—	2	1	no	—
255	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
256	C₂²×D₄	—	2	1	no	—
257	C₂²×D₄	—	2	1	no	—
258	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
259	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
260	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
261	C₄⋊Q₈	—	4	1	no	—
262	C₄⋊Q₈	—	4	1	no	—
263	#127	—	4	1	no	—
264	#127	—	4	1	no	—
265	#124	—	4	4	no	—
266	C₄×Q₈	—	4	1	no	—
267	C₄×Q₈	—	4	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
272	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
273	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
275	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
276	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
277	C₂×C₂²⋊C₄	—	2	2	no	—
278	C₂×C₂²⋊C₄	—	2	2	no	—
279	C₂×C₂²⋊C₄	—	2	2	no	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₄²	—	2	1	no	—
282	C₂×C₄²	—	2	1	no	—
283	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
285	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
286	#107	—	2	4	no	—
287	C₄⋊C₈	—	4	1	no	—
288	#102	—	2	4	no	—
289	#102	—	2	4	no	—
290	#102	—	4	2	no	—
291	#102	—	4	2	no	—
292	#97	—	4	1	no	—
293	#95	—	2	1	no	—
294	#95	—	2	1	no	—
295	#95	—	2	2	no	—
296	#95	—	2	2	no	—
297	C₃₁	#32647	1	1	yes	characteristic, direct product factor, Sylow
298	C₂⁴	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	4	1	no	—
302	C₂×Q₈	—	4	2	no	—
303	C₂×Q₈	—	8	2	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	2	2	no	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	4	2	no	—
311	C₂×D₄	—	4	2	no	—
312	C₂×D₄	—	4	2	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	—	8	2	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
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	C₂⁴×C₃₁	1	1	yes	characteristic
325	C₂×C₈	—	2	2	no	—
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	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	4	1	no	—
337	C₄⋊C₄	—	4	1	no	—
338	C₄⋊C₄	—	4	1	no	—
339	C₄⋊C₄	—	4	2	no	—
340	C₄⋊C₄	—	8	2	no	—
341	C₄⋊C₄	—	8	2	no	—
342	C₄⋊C₄	#41×C₃₁	1	4	yes	—
343	C₂²⋊C₄	—	4	4	no	—
344	C₂²⋊C₄	—	4	4	no	—
345	C₂²⋊C₄	—	8	2	no	—
346	C₂²⋊C₄	—	8	2	no	—
347	C₄²	—	2	1	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	2	1	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	2	1	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	4	2	no	—
354	C₄²	—	4	2	no	—
355	C₂³	—	4	2	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	4	2	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	4	2	no	—
362	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
363	Q₈	—	4	4	no	—
364	Q₈	—	8	4	no	—
365	D₄	—	4	2	no	—
366	D₄	—	4	2	no	—
367	D₄	—	4	2	no	—
368	D₄	—	4	2	no	—
369	D₄	—	8	4	no	—
370	D₄	—	8	4	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	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	4	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
394	C₂×C₄	C₃₁×#137	1	2	yes	—
395	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
396	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
397	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
398	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
399	C₈	—	4	2	no	—
400	C₈	—	4	2	no	—
401	C₂²	—	8	2	no	—
402	C₂²	—	8	2	no	—
403	C₂²	—	8	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	—	8	2	no	—
406	C₂²	—	8	2	no	—
407	C₂²	—	8	2	no	—
408	C₂²	C₃₁×#534	1	1	yes	characteristic
409	C₂²	C₃₁×#548	1	2	yes	—
410	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
411	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
412	C₂²	C₃₁×#576	1	2	yes	—
413	C₄	—	2	2	no	—
414	C₄	—	2	2	no	—
415	C₄	—	4	4	no	—
416	C₄	—	4	1	no	—
417	C₄	—	4	1	no	—
418	C₄	—	4	1	no	—
419	C₄	—	4	1	no	—
420	C₄	—	8	2	no	—
421	C₄	—	8	2	no	—
422	C₄	C₃₁×#542	1	2	yes	—
423	C₄	C₃₁×#555	1	1	yes	characteristic
424	C₄	C₃₁×#563	1	1	yes	characteristic
425	C₂	—	16	2	no	—
426	C₂	C₃₁×#2731	1	1	yes	characteristic
427	C₂	C₃₁×#3193	1	1	yes	characteristic
428	C₂	C₃₁×#3199	1	1	yes	characteristic
429	C₂	C₃₁×#3366	1	2	yes	—
430	C₂	C₃₁×#3375	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2567	C₂	semidirect
3. C₃₁×#2067	C₂	semidirect
5. C₃₁×#2023	C₂	semidirect
6. C₃₁×#1985	C₂	semidirect
7. C₃₁×#1936	C₂	semidirect
8. C₃₁×#1925	C₂	semidirect
9. C₃₁×#1921	C₂	semidirect
139. #32647	C₃₁	direct (N × H)
186. #2567	C₂×C₃₁	semidirect
188. #2067	C₂×C₃₁	semidirect
190. #2023	C₂×C₃₁	semidirect
191. #1985	C₂×C₃₁	semidirect
192. #1936	C₂×C₃₁	semidirect
193. #1925	C₂×C₃₁	semidirect
194. #1921	C₂×C₃₁	semidirect
297. C₃₁	#32647	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×#2567	2
C₃₁×#2548	2
C₃₁×#2067	2
C₃₁×#2041	2
C₃₁×#2023	2
C₃₁×#1985	2
C₃₁×#1936	2
C₃₁×#1925	2
C₃₁×#1921	2
#32647	31

Subgroups — C31×#32647 #21489406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32647 #21489406