Subgroups — C₃₁×#25057 #21481816

Order 7936.

722 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2538	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2355	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2092	C₂	1	2	yes	maximal
4	C₃₁×#2085	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1925	C₂	1	2	yes	maximal
8	C₃₁×#1921	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#107	C₂²	1	2	yes	—
16	C₂×C₃₁×#107	C₂²	1	2	yes	—
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#396	—	2	2	no	—
24	C₃₁×#391	—	2	1	no	—
25	C₃₁×#390	C₂²	1	2	yes	—
26	C₃₁×#389	C₂²	1	2	yes	—
27	C₃₁×#387	—	2	2	no	—
28	C₃₁×#384	C₂²	1	1	yes	characteristic
29	C₃₁×#384	C₂²	1	1	yes	characteristic
30	C₃₁×#382	C₂²	1	2	yes	—
31	C₃₁×#380	—	2	1	no	—
32	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#340	C₂²	1	2	yes	—
36	C₃₁×#340	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	2	yes	—
41	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	—	2	1	no	—
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₂²⋊Q₈	—	4	4	no	—
48	C₄×Q₈×C₃₁	—	2	2	no	—
49	C₄×Q₈×C₃₁	—	2	2	no	—
50	C₄×D₄×C₃₁	—	2	2	no	—
51	C₄×D₄×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	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	D₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	—	2	1	no	—
68	C₂×C₄²×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#107	—	2	4	no	—
72	C₃₁×#107	—	4	2	no	—
73	C₃₁×#106	—	4	2	no	—
74	C₃₁×C₄⋊C₈	—	2	2	no	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	—	4	2	no	—
77	C₃₁×#102	Q₈	1	4	yes	complemented
78	C₄×C₈×C₃₁	—	2	2	no	—
79	C₃₁×#95	—	2	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	1	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₂⁴×C₃₁	—	2	1	no	—
85	C₂×Q₈×C₃₁	—	2	2	no	—
86	C₂×Q₈×C₃₁	—	2	2	no	—
87	C₂×Q₈×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	1	no	—
89	C₂×D₄×C₃₁	—	2	1	no	—
90	C₂×D₄×C₃₁	—	2	1	no	—
91	C₂×D₄×C₃₁	—	2	1	no	—
92	C₂×D₄×C₃₁	—	2	1	no	—
93	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	4	2	no	—
100	C₂²×C₄×C₃₁	—	4	2	no	—
101	C₂²×C₄×C₃₁	—	4	2	no	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₃₁	—	2	1	no	—
108	C₂×C₈×C₃₁	—	2	1	no	—
109	C₂×C₈×C₃₁	—	2	1	no	—
110	C₂×C₈×C₃₁	—	2	1	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	C₂×Q₈	1	2	yes	—
115	C₂×C₈×C₃₁	#41	1	2	yes	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₄⋊C₄×C₃₁	—	2	1	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	1	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	4	4	no	—
128	C₄⋊C₄×C₃₁	—	4	2	no	—
129	C₄⋊C₄×C₃₁	—	4	2	no	—
130	C₄⋊C₄×C₃₁	—	8	1	no	—
131	C₄⋊C₄×C₃₁	—	8	1	no	—
132	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
133	C₂²⋊C₄×C₃₁	—	4	4	no	—
134	C₂²⋊C₄×C₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₃₁	—	8	2	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	4	2	no	—
139	C₄²×C₃₁	—	8	2	no	—
140	C₄²×C₃₁	#41	1	2	yes	—
141	C₄²×C₃₁	#41	1	2	yes	—
142	#25057	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	—	2	2	no	—
144	C₂³×C₃₁	—	4	1	no	—
145	C₂³×C₃₁	—	4	1	no	—
146	C₂³×C₃₁	—	4	1	no	—
147	C₂³×C₃₁	—	4	1	no	—
148	C₂³×C₃₁	—	4	1	no	—
149	C₂³×C₃₁	—	4	1	no	—
150	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₃₁	—	4	4	no	—
152	D₄×C₃₁	—	2	2	no	—
153	D₄×C₃₁	—	2	2	no	—
154	D₄×C₃₁	—	4	1	no	—
155	D₄×C₃₁	—	4	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	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	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	1	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	—	4	1	no	—
179	C₂×C₄×C₃₁	—	8	4	no	—
180	C₂×C₄×C₃₁	#123	1	2	yes	—
181	C₂×C₄×C₃₁	#123	1	2	yes	—
182	C₂×C₄×C₃₁	#135	1	2	yes	—
183	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
184	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
190	C₈×C₃₁	—	2	2	no	—
191	C₈×C₃₁	—	2	2	no	—
192	C₈×C₃₁	—	4	2	no	—
193	#2538	C₂×C₃₁	1	1	yes	characteristic
194	#2355	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#2092	C₂×C₃₁	1	2	yes	—
196	#2085	C₂×C₃₁	1	2	yes	complemented
197	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1925	C₂×C₃₁	1	2	yes	—
200	#1921	C₂×C₃₁	1	2	yes	complemented
201	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1802	C₂×C₃₁	1	1	yes	characteristic
203	C₄×#102	C₂×C₃₁	1	1	yes	characteristic
204	C₂²×C₃₁	—	4	2	no	—
205	C₂²×C₃₁	—	4	2	no	—
206	C₂²×C₃₁	—	4	2	no	—
207	C₂²×C₃₁	—	8	1	no	—
208	C₂²×C₃₁	—	8	1	no	—
209	C₂²×C₃₁	—	8	1	no	—
210	C₂²×C₃₁	—	8	1	no	—
211	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
212	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
213	C₂²×C₃₁	#550	1	1	yes	characteristic
214	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
215	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₂²×C₃₁	#576	1	1	yes	characteristic
217	C₂²×C₃₁	#578	1	1	yes	characteristic
218	C₄×C₃₁	—	2	2	no	—
219	C₄×C₃₁	—	4	2	no	—
220	C₄×C₃₁	—	4	2	no	—
221	C₄×C₃₁	—	4	2	no	—
222	C₄×C₃₁	—	8	1	no	—
223	C₄×C₃₁	—	8	1	no	—
224	C₄×C₃₁	—	8	2	no	—
225	C₄×C₃₁	—	8	2	no	—
226	C₄×C₃₁	#482	1	4	yes	—
227	C₄×C₃₁	#533	1	1	yes	characteristic
228	C₄×C₃₁	#542	1	1	yes	characteristic
229	C₄×C₃₁	#552	1	1	yes	characteristic
230	C₄×C₃₁	#554	1	1	yes	characteristic
231	C₂×C₂²⋊Q₈	—	2	1	no	—
232	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#107	C₂²×C₃₁	1	2	yes	—
235	C₂×#107	C₂²×C₃₁	1	2	yes	—
236	C₂×#106	C₂²×C₃₁	1	2	yes	—
237	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×#102	C₂²×C₃₁	1	2	yes	—
240	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
242	#396	—	2	2	no	—
243	#391	—	2	1	no	—
244	#390	C₂²×C₃₁	1	2	yes	—
245	#389	C₂²×C₃₁	1	2	yes	—
246	#387	—	2	2	no	—
247	#384	C₂²×C₃₁	1	1	yes	characteristic
248	#384	C₂²×C₃₁	1	1	yes	characteristic
249	#382	C₂²×C₃₁	1	2	yes	—
250	#380	—	2	1	no	—
251	C₄×C₄⋊C₄	—	2	1	no	—
252	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
253	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
254	#340	C₂²×C₃₁	1	2	yes	—
255	#340	C₂²×C₃₁	1	1	yes	characteristic
256	#340	C₂²×C₃₁	1	1	yes	characteristic
257	#340	C₂²×C₃₁	1	2	yes	—
258	#336	C₂²×C₃₁	1	2	yes	—
259	#336	C₂²×C₃₁	1	2	yes	—
260	C₂×C₃₁	—	8	2	no	—
261	C₂×C₃₁	#2423	1	1	yes	characteristic
262	C₂×C₃₁	C₂×#482	1	1	yes	characteristic
263	C₂×C₃₁	#3098	1	1	yes	characteristic
264	C₂×C₃₁	#3380	1	1	yes	characteristic
265	C₂×C₃₁	#3386	1	1	yes	characteristic
266	C₂×C₃₁	#3393	1	1	yes	characteristic
267	C₂×C₃₁	#3399	1	1	yes	characteristic
268	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
269	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
270	C₂³×C₄	—	2	1	no	—
271	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
272	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
273	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
274	C₂²⋊Q₈	—	4	4	no	—
275	C₄×Q₈	—	2	2	no	—
276	C₄×Q₈	—	2	2	no	—
277	C₄×D₄	—	2	2	no	—
278	C₄×D₄	—	2	2	no	—
279	C₂×C₄⋊C₄	—	2	1	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
285	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
287	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
288	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
290	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
291	C₂×C₂²⋊C₄	—	2	1	no	—
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₂²⋊C₄	—	2	2	no	—
294	C₂×C₄²	—	2	1	no	—
295	C₂×C₄²	—	2	2	no	—
296	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
297	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
298	#107	—	2	4	no	—
299	#107	—	4	2	no	—
300	#106	—	4	2	no	—
301	C₄⋊C₈	—	2	2	no	—
302	#102	—	2	2	no	—
303	#102	—	4	2	no	—
304	#102	Q₈×C₃₁	1	4	yes	complemented
305	C₄×C₈	—	2	2	no	—
306	#95	—	2	2	no	—
307	#95	—	2	1	no	—
308	#95	—	2	2	no	—
309	#95	—	2	2	no	—
310	#95	—	2	2	no	—
311	C₃₁	#25057	1	1	yes	characteristic, direct product factor, Sylow
312	C₂⁴	—	2	1	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×D₄	—	2	1	no	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	2	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	4	2	no	—
328	C₂²×C₄	—	4	2	no	—
329	C₂²×C₄	—	4	2	no	—
330	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
331	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
332	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
343	C₂×C₈	#41×C₃₁	1	2	yes	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	4	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	4	4	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	—	4	2	no	—
358	C₄⋊C₄	—	8	1	no	—
359	C₄⋊C₄	—	8	1	no	—
360	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
361	C₂²⋊C₄	—	4	4	no	—
362	C₂²⋊C₄	—	4	2	no	—
363	C₂²⋊C₄	—	8	2	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	4	2	no	—
367	C₄²	—	8	2	no	—
368	C₄²	#41×C₃₁	1	2	yes	—
369	C₄²	#41×C₃₁	1	2	yes	—
370	C₂³	—	2	2	no	—
371	C₂³	—	4	1	no	—
372	C₂³	—	4	1	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	4	1	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	1	no	—
377	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
378	Q₈	—	4	4	no	—
379	D₄	—	2	2	no	—
380	D₄	—	2	2	no	—
381	D₄	—	4	1	no	—
382	D₄	—	4	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	2	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	8	4	no	—
407	C₂×C₄	C₃₁×#123	1	2	yes	—
408	C₂×C₄	C₃₁×#123	1	2	yes	—
409	C₂×C₄	C₃₁×#135	1	2	yes	—
410	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
411	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
412	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
415	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
416	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
417	C₈	—	2	2	no	—
418	C₈	—	2	2	no	—
419	C₈	—	4	2	no	—
420	C₂²	—	4	2	no	—
421	C₂²	—	4	2	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	8	1	no	—
424	C₂²	—	8	1	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	1	no	—
427	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
428	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
429	C₂²	C₃₁×#550	1	1	yes	characteristic
430	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
431	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
432	C₂²	C₃₁×#576	1	1	yes	characteristic
433	C₂²	C₃₁×#578	1	1	yes	characteristic
434	C₄	—	2	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	4	2	no	—
437	C₄	—	4	2	no	—
438	C₄	—	8	1	no	—
439	C₄	—	8	1	no	—
440	C₄	—	8	2	no	—
441	C₄	—	8	2	no	—
442	C₄	C₃₁×#482	1	4	yes	—
443	C₄	C₃₁×#533	1	1	yes	characteristic
444	C₄	C₃₁×#542	1	1	yes	characteristic
445	C₄	C₃₁×#552	1	1	yes	characteristic
446	C₄	C₃₁×#554	1	1	yes	characteristic
447	C₂	—	8	2	no	—
448	C₂	C₃₁×#2423	1	1	yes	characteristic
449	C₂	C₂×C₃₁×#482	1	1	yes	characteristic
450	C₂	C₃₁×#3098	1	1	yes	characteristic
451	C₂	C₃₁×#3380	1	1	yes	characteristic
452	C₂	C₃₁×#3386	1	1	yes	characteristic
453	C₂	C₃₁×#3393	1	1	yes	characteristic
454	C₂	C₃₁×#3399	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 ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2355	C₂	semidirect
4. C₃₁×#2085	C₂	semidirect
5. C₃₁×#1984	C₂	semidirect
6. C₃₁×#1948	C₂	semidirect
8. C₃₁×#1921	C₂	semidirect
9. C₃₁×#1845	C₂	semidirect
77. C₃₁×#102	Q₈	semidirect
142. #25057	C₃₁	direct (N × H)
194. #2355	C₂×C₃₁	semidirect
196. #2085	C₂×C₃₁	semidirect
197. #1984	C₂×C₃₁	semidirect
198. #1948	C₂×C₃₁	semidirect
200. #1921	C₂×C₃₁	semidirect
201. #1845	C₂×C₃₁	semidirect
304. #102	Q₈×C₃₁	semidirect
311. C₃₁	#25057	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×#2538	2
C₃₁×#2355	2
C₃₁×#2092	2
C₃₁×#2085	2
C₃₁×#1984	2
C₃₁×#1948	2
C₃₁×#1925	2
C₃₁×#1921	2
C₃₁×#1845	2
C₃₁×#1802	2
C₄×C₃₁×#102	2
#25057	31

Subgroups — C31×#25057 #21481816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25057 #21481816