Subgroups — C₃₁×#24631 #21481390

Order 7936.

974 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2548	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2054	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2012	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1983	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1803	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#124	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#117	—	2	1	no	—
19	C₃₁×#459	—	2	4	no	—
20	C₄×SD₁₆×C₃₁	—	2	4	no	—
21	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#103	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#394	—	2	2	no	—
32	C₃₁×#390	—	2	1	no	—
33	C₃₁×#390	—	2	1	no	—
34	C₃₁×#390	C₂²	1	2	yes	—
35	C₃₁×#390	C₂²	1	2	yes	—
36	C₃₁×#388	—	2	2	no	—
37	C₃₁×#384	C₂²	1	1	yes	characteristic
38	C₃₁×#380	—	2	1	no	—
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
40	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
47	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₃₁	—	2	1	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	—	2	4	no	—
53	C₂×D₈×C₃₁	—	4	2	no	—
54	C₂×D₈×C₃₁	—	4	2	no	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×#127	—	2	2	no	—
59	C₃₁×#127	—	2	2	no	—
60	C₃₁×#124	—	2	4	no	—
61	C₃₁×#124	—	4	2	no	—
62	C₄×Q₈×C₃₁	—	2	2	no	—
63	C₄×Q₈×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	D₄	1	4	yes	—
66	C₃₁×#117	—	4	4	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
70	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
80	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
81	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	—	2	2	no	—
83	C₂×C₄²×C₃₁	—	2	1	no	—
84	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×#106	—	2	2	no	—
87	C₃₁×C₄⋊C₈	—	2	2	no	—
88	C₃₁×C₄⋊C₈	—	4	2	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#103	—	4	2	no	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×#102	—	2	4	no	—
93	C₃₁×#102	—	2	4	no	—
94	C₃₁×#102	—	4	2	no	—
95	C₄×C₈×C₃₁	—	2	2	no	—
96	C₃₁×#95	—	2	2	no	—
97	C₃₁×#95	D₄	1	2	yes	—
98	C₂⁴×C₃₁	—	2	1	no	—
99	C₂⁴×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	2	1	no	—
101	C₂×Q₈×C₃₁	—	2	1	no	—
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	#41	1	2	yes	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	4	2	no	—
118	C₂×D₄×C₃₁	—	4	2	no	—
119	C₂×D₄×C₃₁	—	4	2	no	—
120	C₂×D₄×C₃₁	—	4	2	no	—
121	C₂×D₄×C₃₁	—	8	1	no	—
122	C₂×D₄×C₃₁	—	8	1	no	—
123	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	4	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	4	2	no	—
130	C₂²×C₄×C₃₁	—	4	2	no	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
136	SD₁₆×C₃₁	—	2	8	no	—
137	D₈×C₃₁	—	4	8	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	#41	1	2	yes	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	4	2	no	—
155	C₄⋊C₄×C₃₁	—	4	2	no	—
156	C₄⋊C₄×C₃₁	—	8	1	no	—
157	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
158	C₂²⋊C₄×C₃₁	—	2	4	no	—
159	C₂²⋊C₄×C₃₁	—	4	4	no	—
160	C₂²⋊C₄×C₃₁	—	4	4	no	—
161	C₂²⋊C₄×C₃₁	—	4	4	no	—
162	C₂²⋊C₄×C₃₁	—	8	2	no	—
163	C₂²⋊C₄×C₃₁	—	8	2	no	—
164	C₂²⋊C₄×C₃₁	—	8	2	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	4	4	no	—
168	C₄²×C₃₁	—	4	2	no	—
169	C₄²×C₃₁	C₂×D₄	1	2	yes	—
170	C₄²×C₃₁	#41	1	2	yes	—
171	#24631	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂³×C₃₁	—	2	2	no	—
173	C₂³×C₃₁	—	4	1	no	—
174	C₂³×C₃₁	—	4	1	no	—
175	C₂³×C₃₁	—	4	1	no	—
176	C₂³×C₃₁	—	4	1	no	—
177	C₂³×C₃₁	—	4	1	no	—
178	C₂³×C₃₁	—	4	1	no	—
179	C₂³×C₃₁	—	4	2	no	—
180	C₂³×C₃₁	—	4	2	no	—
181	C₂³×C₃₁	—	4	2	no	—
182	C₂³×C₃₁	—	4	2	no	—
183	C₂³×C₃₁	—	4	2	no	—
184	C₂³×C₃₁	—	4	2	no	—
185	C₂³×C₃₁	—	4	2	no	—
186	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
187	Q₈×C₃₁	—	2	2	no	—
188	Q₈×C₃₁	—	2	2	no	—
189	Q₈×C₃₁	—	4	1	no	—
190	Q₈×C₃₁	—	4	1	no	—
191	D₄×C₃₁	—	2	2	no	—
192	D₄×C₃₁	—	2	2	no	—
193	D₄×C₃₁	—	4	2	no	—
194	D₄×C₃₁	—	4	2	no	—
195	D₄×C₃₁	—	4	1	no	—
196	D₄×C₃₁	—	4	2	no	—
197	D₄×C₃₁	—	4	1	no	—
198	D₄×C₃₁	—	4	2	no	—
199	D₄×C₃₁	—	8	4	no	—
200	D₄×C₃₁	—	8	4	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	2	no	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	8	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	4	2	no	—
219	C₂×C₄×C₃₁	—	4	1	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	#121	1	2	yes	—
223	C₂×C₄×C₃₁	#121	1	2	yes	—
224	C₂×C₄×C₃₁	#135	1	2	yes	—
225	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
229	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
230	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
231	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
232	C₈×C₃₁	—	2	2	no	—
233	C₈×C₃₁	—	4	1	no	—
234	C₈×C₃₁	—	4	2	no	—
235	C₂×#459	C₂×C₃₁	1	1	yes	characteristic
236	C₂×C₄×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#2548	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#2354	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#2054	C₂×C₃₁	1	2	yes	complemented
240	#2012	C₂×C₃₁	1	2	yes	complemented
241	#1983	C₂×C₃₁	1	2	yes	complemented
242	#1957	C₂×C₃₁	1	1	yes	characteristic, complemented
243	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#1926	C₂×C₃₁	1	2	yes	complemented
245	#1803	C₂×C₃₁	1	1	yes	characteristic, complemented
246	C₂²×C₃₁	—	4	2	no	—
247	C₂²×C₃₁	—	4	2	no	—
248	C₂²×C₃₁	—	4	2	no	—
249	C₂²×C₃₁	—	8	1	no	—
250	C₂²×C₃₁	—	8	1	no	—
251	C₂²×C₃₁	—	8	1	no	—
252	C₂²×C₃₁	—	8	1	no	—
253	C₂²×C₃₁	—	8	2	no	—
254	C₂²×C₃₁	—	8	2	no	—
255	C₂²×C₃₁	—	8	2	no	—
256	C₂²×C₃₁	—	8	2	no	—
257	C₂²×C₃₁	—	8	2	no	—
258	C₂²×C₃₁	—	8	2	no	—
259	C₂²×C₃₁	—	8	2	no	—
260	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
261	C₂²×C₃₁	#546	1	1	yes	characteristic
262	C₂²×C₃₁	#550	1	1	yes	characteristic
263	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
264	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
265	C₂²×C₃₁	#576	1	1	yes	characteristic
266	C₂²×C₃₁	#577	1	1	yes	characteristic
267	C₄×C₃₁	—	2	2	no	—
268	C₄×C₃₁	—	4	2	no	—
269	C₄×C₃₁	—	4	2	no	—
270	C₄×C₃₁	—	4	2	no	—
271	C₄×C₃₁	—	4	2	no	—
272	C₄×C₃₁	—	8	2	no	—
273	C₄×C₃₁	—	8	1	no	—
274	C₄×C₃₁	#463	1	4	yes	—
275	C₄×C₃₁	#525	1	1	yes	characteristic
276	C₄×C₃₁	#535	1	1	yes	characteristic
277	C₄×C₃₁	#553	1	1	yes	characteristic
278	C₄×C₃₁	#555	1	1	yes	characteristic
279	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
280	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
281	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×#124	C₂²×C₃₁	1	2	yes	—
283	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
284	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×#117	—	2	1	no	—
286	#459	—	2	4	no	—
287	C₄×SD₁₆	—	2	4	no	—
288	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
290	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
291	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
292	C₂×#103	C₂²×C₃₁	1	2	yes	—
293	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
294	C₂×#102	C₂²×C₃₁	1	2	yes	—
295	C₂×#102	C₂²×C₃₁	1	2	yes	—
296	C₂×#102	C₂²×C₃₁	1	2	yes	—
297	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
298	#394	—	2	2	no	—
299	#390	—	2	1	no	—
300	#390	—	2	1	no	—
301	#390	C₂²×C₃₁	1	2	yes	—
302	#390	C₂²×C₃₁	1	2	yes	—
303	#388	—	2	2	no	—
304	#384	C₂²×C₃₁	1	1	yes	characteristic
305	#380	—	2	1	no	—
306	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
307	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
308	#340	C₂²×C₃₁	1	2	yes	—
309	#336	C₂²×C₃₁	1	2	yes	—
310	C₂×C₃₁	—	8	2	no	—
311	C₂×C₃₁	—	16	2	no	—
312	C₂×C₃₁	#2414	1	1	yes	characteristic
313	C₂×C₃₁	C₂×#463	1	1	yes	characteristic
314	C₂×C₃₁	#3045	1	1	yes	characteristic
315	C₂×C₃₁	#3305	1	1	yes	characteristic
316	C₂×C₃₁	#3332	1	1	yes	characteristic
317	C₂×C₃₁	#3394	1	1	yes	characteristic
318	C₂×C₃₁	#3399	1	1	yes	characteristic
319	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
320	C₂²×D₄	—	2	1	no	—
321	C₂²×D₄	—	2	1	no	—
322	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
323	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₂³×C₄	—	2	1	no	—
325	C₂×SD₁₆	—	2	2	no	—
326	C₂×SD₁₆	—	2	2	no	—
327	C₂×SD₁₆	—	2	2	no	—
328	C₂×D₈	—	2	4	no	—
329	C₂×D₈	—	4	2	no	—
330	C₂×D₈	—	4	2	no	—
331	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
332	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
333	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
334	#127	—	2	2	no	—
335	#127	—	2	2	no	—
336	#124	—	2	4	no	—
337	#124	—	4	2	no	—
338	C₄×Q₈	—	2	2	no	—
339	C₄×Q₈	—	2	2	no	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	D₄×C₃₁	1	4	yes	—
342	#117	—	4	4	no	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	1	no	—
346	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
347	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
348	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
351	C₂×C₂²⋊C₄	—	2	1	no	—
352	C₂×C₂²⋊C₄	—	2	2	no	—
353	C₂×C₂²⋊C₄	—	2	2	no	—
354	C₂×C₂²⋊C₄	—	2	2	no	—
355	C₂×C₂²⋊C₄	—	2	2	no	—
356	C₂×C₂²⋊C₄	—	2	2	no	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₄²	—	2	2	no	—
359	C₂×C₄²	—	2	1	no	—
360	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
361	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
362	#106	—	2	2	no	—
363	C₄⋊C₈	—	2	2	no	—
364	C₄⋊C₈	—	4	2	no	—
365	#103	—	2	2	no	—
366	#103	—	4	2	no	—
367	#102	—	2	2	no	—
368	#102	—	2	4	no	—
369	#102	—	2	4	no	—
370	#102	—	4	2	no	—
371	C₄×C₈	—	2	2	no	—
372	#95	—	2	2	no	—
373	#95	D₄×C₃₁	1	2	yes	—
374	C₃₁	#24631	1	1	yes	characteristic, direct product factor, Sylow
375	C₂⁴	—	2	1	no	—
376	C₂⁴	—	2	2	no	—
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	1	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×Q₈	—	2	1	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	#41×C₃₁	1	2	yes	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	2	2	no	—
394	C₂×D₄	—	4	2	no	—
395	C₂×D₄	—	4	2	no	—
396	C₂×D₄	—	4	2	no	—
397	C₂×D₄	—	4	2	no	—
398	C₂×D₄	—	8	1	no	—
399	C₂×D₄	—	8	1	no	—
400	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
401	C₂²×C₄	—	2	1	no	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	4	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	1	no	—
406	C₂²×C₄	—	4	2	no	—
407	C₂²×C₄	—	4	2	no	—
408	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
409	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
410	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
411	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
412	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
413	SD₁₆	—	2	8	no	—
414	D₈	—	4	8	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	#41×C₃₁	1	2	yes	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	4	2	no	—
432	C₄⋊C₄	—	4	2	no	—
433	C₄⋊C₄	—	8	1	no	—
434	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	4	4	no	—
437	C₂²⋊C₄	—	4	4	no	—
438	C₂²⋊C₄	—	4	4	no	—
439	C₂²⋊C₄	—	8	2	no	—
440	C₂²⋊C₄	—	8	2	no	—
441	C₂²⋊C₄	—	8	2	no	—
442	C₄²	—	2	2	no	—
443	C₄²	—	2	2	no	—
444	C₄²	—	4	4	no	—
445	C₄²	—	4	2	no	—
446	C₄²	C₂×D₄×C₃₁	1	2	yes	—
447	C₄²	#41×C₃₁	1	2	yes	—
448	C₂³	—	2	2	no	—
449	C₂³	—	4	1	no	—
450	C₂³	—	4	1	no	—
451	C₂³	—	4	1	no	—
452	C₂³	—	4	1	no	—
453	C₂³	—	4	1	no	—
454	C₂³	—	4	1	no	—
455	C₂³	—	4	2	no	—
456	C₂³	—	4	2	no	—
457	C₂³	—	4	2	no	—
458	C₂³	—	4	2	no	—
459	C₂³	—	4	2	no	—
460	C₂³	—	4	2	no	—
461	C₂³	—	4	2	no	—
462	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
463	Q₈	—	2	2	no	—
464	Q₈	—	2	2	no	—
465	Q₈	—	4	1	no	—
466	Q₈	—	4	1	no	—
467	D₄	—	2	2	no	—
468	D₄	—	2	2	no	—
469	D₄	—	4	2	no	—
470	D₄	—	4	2	no	—
471	D₄	—	4	1	no	—
472	D₄	—	4	2	no	—
473	D₄	—	4	1	no	—
474	D₄	—	4	2	no	—
475	D₄	—	8	4	no	—
476	D₄	—	8	4	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	1	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	2	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	8	no	—
492	C₂×C₄	—	4	1	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	4	1	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	C₃₁×#121	1	2	yes	—
499	C₂×C₄	C₃₁×#121	1	2	yes	—
500	C₂×C₄	C₃₁×#135	1	2	yes	—
501	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
502	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
503	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
504	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
505	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
506	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
507	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
508	C₈	—	2	2	no	—
509	C₈	—	4	1	no	—
510	C₈	—	4	2	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	4	2	no	—
514	C₂²	—	8	1	no	—
515	C₂²	—	8	1	no	—
516	C₂²	—	8	1	no	—
517	C₂²	—	8	1	no	—
518	C₂²	—	8	2	no	—
519	C₂²	—	8	2	no	—
520	C₂²	—	8	2	no	—
521	C₂²	—	8	2	no	—
522	C₂²	—	8	2	no	—
523	C₂²	—	8	2	no	—
524	C₂²	—	8	2	no	—
525	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
526	C₂²	C₃₁×#546	1	1	yes	characteristic
527	C₂²	C₃₁×#550	1	1	yes	characteristic
528	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
529	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
530	C₂²	C₃₁×#576	1	1	yes	characteristic
531	C₂²	C₃₁×#577	1	1	yes	characteristic
532	C₄	—	2	2	no	—
533	C₄	—	4	2	no	—
534	C₄	—	4	2	no	—
535	C₄	—	4	2	no	—
536	C₄	—	4	2	no	—
537	C₄	—	8	2	no	—
538	C₄	—	8	1	no	—
539	C₄	C₃₁×#463	1	4	yes	—
540	C₄	C₃₁×#525	1	1	yes	characteristic
541	C₄	C₃₁×#535	1	1	yes	characteristic
542	C₄	C₃₁×#553	1	1	yes	characteristic
543	C₄	C₃₁×#555	1	1	yes	characteristic
544	C₂	—	8	2	no	—
545	C₂	—	16	2	no	—
546	C₂	C₃₁×#2414	1	1	yes	characteristic
547	C₂	C₂×C₃₁×#463	1	1	yes	characteristic
548	C₂	C₃₁×#3045	1	1	yes	characteristic
549	C₂	C₃₁×#3305	1	1	yes	characteristic
550	C₂	C₃₁×#3332	1	1	yes	characteristic
551	C₂	C₃₁×#3394	1	1	yes	characteristic
552	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	C₂×C₃₁×#459	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×SD₁₆×C₃₁	C₂	semidirect
3. C₃₁×#2548	C₂	semidirect
4. C₃₁×#2354	C₂	semidirect
5. C₃₁×#2054	C₂	semidirect
6. C₃₁×#2012	C₂	semidirect
7. C₃₁×#1983	C₂	semidirect
8. C₃₁×#1957	C₂	semidirect
9. C₃₁×#1939	C₂	semidirect
10. C₃₁×#1926	C₂	semidirect
11. C₃₁×#1803	C₂	semidirect
171. #24631	C₃₁	direct (N × H)
236. C₂×C₄×SD₁₆	C₂×C₃₁	semidirect
237. #2548	C₂×C₃₁	semidirect
238. #2354	C₂×C₃₁	semidirect
239. #2054	C₂×C₃₁	semidirect
240. #2012	C₂×C₃₁	semidirect
241. #1983	C₂×C₃₁	semidirect
242. #1957	C₂×C₃₁	semidirect
243. #1939	C₂×C₃₁	semidirect
244. #1926	C₂×C₃₁	semidirect
245. #1803	C₂×C₃₁	semidirect
374. C₃₁	#24631	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#459	2
C₂×C₄×SD₁₆×C₃₁	2
C₃₁×#2548	2
C₃₁×#2354	2
C₃₁×#2054	2
C₃₁×#2012	2
C₃₁×#1983	2
C₃₁×#1957	2
C₃₁×#1939	2
C₃₁×#1926	2
C₃₁×#1803	2
#24631	31

Subgroups — C31×#24631 #21481390

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24631 #21481390