Subgroups — C₃₁×#60704 #21517463

Order 7936.

1974 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3455	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#540	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3179	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3176	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3174	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3166	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3142	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3055	C₂	1	4	yes	maximal, complemented
11	C₄×C₃₁×AGL(1,ℤ/8)	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2873	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	—
14	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#542	C₂²	1	4	yes	complemented
16	C₃₁×#541	C₂²	1	2	yes	complemented
17	C₃₁×#540	—	2	4	no	—
18	C₃₁×#540	C₂²	1	4	yes	—
19	C₃₁×#540	C₂²	1	4	yes	—
20	C₃₁×#540	C₂²	1	2	yes	complemented
21	C₃₁×#539	—	2	2	no	—
22	C₃₁×#538	—	2	2	no	—
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₃₁×#532	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
26	C₃₁×#529	—	2	4	no	—
27	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
28	C₃₁×#525	C₂²	1	2	yes	—
29	C₂×C₃₁×#123	—	2	2	no	—
30	C₂×C₃₁×#121	—	2	2	no	—
31	C₂×C₃₁×#121	C₂²	1	2	yes	—
32	C₄×#41×C₃₁	C₂²	1	2	yes	—
33	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
34	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
35	C₃₁×#501	C₂²	1	4	yes	complemented
36	C₃₁×#499	C₂²	1	4	yes	complemented
37	C₃₁×#496	C₂²	1	4	yes	complemented
38	C₃₁×#493	C₂²	1	2	yes	complemented
39	C₃₁×#492	C₂²	1	2	yes	complemented
40	C₃₁×#491	C₂²	1	2	yes	complemented
41	C₃₁×#490	C₂²	1	4	yes	complemented
42	C₃₁×#490	C₂²	1	2	yes	complemented
43	C₃₁×#488	C₂²	1	4	yes	complemented
44	C₃₁×#486	C₂²	1	4	yes	complemented
45	C₃₁×#469	C₂²	1	8	yes	complemented
46	C₃₁×#468	C₂²	1	8	yes	complemented
47	C₃₁×#466	C₂²	1	8	yes	complemented
48	C₃₁×#465	C₂²	1	8	yes	complemented
49	C₃₁×#442	C₂²	1	8	yes	complemented
50	C₃₁×#440	C₂²	1	8	yes	complemented
51	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
52	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
53	C₃₁×#428	C₂²	1	4	yes	—
54	C₃₁×#418	C₂²	1	4	yes	—
55	C₃₁×#417	C₂²	1	4	yes	—
56	C₃₁×#405	C₂²	1	4	yes	complemented
57	C₃₁×#405	C₂²	1	4	yes	complemented
58	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
59	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
60	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
61	C₂×#41×C₃₁	C₂³	1	2	yes	—
62	C₂²×D₄×C₃₁	—	2	2	no	—
63	C₂²×D₄×C₃₁	—	2	2	no	—
64	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
65	C₂³×C₄×C₃₁	—	2	2	no	—
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×AGL(1,ℤ/8)	—	2	16	no	—
68	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
69	C₂×D₈×C₃₁	C₂³	1	8	yes	—
70	C₂×M₁₆×C₃₁	—	2	8	no	—
71	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
72	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
73	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
74	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
75	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
76	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
77	C₃₁×#127	—	2	2	no	—
78	C₃₁×#127	C₂³	1	2	yes	—
79	C₃₁×#127	C₂³	1	2	yes	—
80	C₃₁×#127	C₂³	1	1	yes	characteristic
81	C₃₁×#126	—	2	4	no	—
82	C₃₁×#126	—	2	4	no	—
83	C₃₁×#125	—	2	4	no	—
84	C₃₁×#125	—	2	2	no	—
85	C₃₁×#125	C₂³	1	2	yes	—
86	C₃₁×#125	C₂³	1	2	yes	—
87	C₃₁×#125	C₂³	1	2	yes	—
88	C₃₁×#124	—	2	4	no	—
89	C₃₁×#124	C₂³	1	2	yes	—
90	C₃₁×#123	—	2	2	no	—
91	C₃₁×#123	—	2	8	no	—
92	C₃₁×#123	—	2	8	no	—
93	C₃₁×#123	—	2	4	no	—
94	C₃₁×#123	—	2	4	no	—
95	C₃₁×C₂²⋊Q₈	—	2	2	no	—
96	C₃₁×C₂²⋊Q₈	—	2	4	no	—
97	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
98	C₃₁×#121	—	2	4	no	—
99	C₃₁×#121	—	2	2	no	—
100	C₃₁×#121	—	2	8	no	—
101	C₃₁×#121	—	2	8	no	—
102	C₃₁×#121	—	2	4	no	—
103	C₃₁×#121	C₂³	1	8	yes	—
104	C₃₁×#121	C₂³	1	4	yes	—
105	C₃₁×#120	—	2	2	no	—
106	C₄×Q₈×C₃₁	—	2	2	no	—
107	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
108	C₄×D₄×C₃₁	—	2	4	no	—
109	C₄×D₄×C₃₁	—	2	4	no	—
110	C₄×D₄×C₃₁	—	2	4	no	—
111	C₄×D₄×C₃₁	—	2	2	no	—
112	C₄×D₄×C₃₁	C₂³	1	4	yes	—
113	C₄×D₄×C₃₁	C₂³	1	8	yes	—
114	C₃₁×#117	—	2	2	no	—
115	C₃₁×#117	D₄	1	4	yes	—
116	C₃₁×#117	C₂³	1	2	yes	—
117	C₃₁×#117	C₂³	1	2	yes	—
118	C₃₁×#117	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
123	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
124	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
125	C₂×C₄²×C₃₁	—	2	2	no	—
126	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
128	C₃₁×#107	C₂³	1	8	yes	—
129	C₃₁×#106	C₂³	1	8	yes	—
130	C₃₁×#103	C₂³	1	8	yes	—
131	C₃₁×#102	C₂³	1	8	yes	—
132	C₃₁×#102	C₂³	1	8	yes	—
133	C₃₁×#102	C₂³	1	8	yes	—
134	C₃₁×#97	C₂³	1	4	yes	—
135	C₃₁×#97	C₂³	1	2	yes	—
136	C₃₁×#97	C₂³	1	2	yes	—
137	C₄×C₈×C₃₁	C₂³	1	4	yes	—
138	C₄×C₈×C₃₁	C₂³	1	2	yes	—
139	C₄×C₈×C₃₁	C₂³	1	2	yes	—
140	C₂⁴×C₃₁	—	2	2	no	—
141	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
142	#41×C₃₁	—	2	8	no	—
143	C₂×Q₈×C₃₁	—	2	2	no	—
144	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
145	C₂×D₄×C₃₁	—	2	2	no	—
146	C₂×D₄×C₃₁	—	2	4	no	—
147	C₂×D₄×C₃₁	—	2	8	no	—
148	C₂×D₄×C₃₁	—	2	2	no	—
149	C₂×D₄×C₃₁	—	2	2	no	—
150	C₂×D₄×C₃₁	—	2	4	no	—
151	C₂×D₄×C₃₁	—	2	4	no	—
152	C₂×D₄×C₃₁	—	2	8	no	—
153	C₂×D₄×C₃₁	—	8	4	no	—
154	C₂×D₄×C₃₁	—	8	4	no	—
155	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
156	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₃₁	—	2	4	no	—
158	C₂²×C₄×C₃₁	—	2	2	no	—
159	C₂²×C₄×C₃₁	—	2	2	no	—
160	C₂²×C₄×C₃₁	—	2	4	no	—
161	C₂²×C₄×C₃₁	—	2	8	no	—
162	C₂²×C₄×C₃₁	—	8	4	no	—
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
169	SD₁₆×C₃₁	—	2	16	no	—
170	D₈×C₃₁	—	2	16	no	—
171	M₁₆×C₃₁	—	2	4	no	—
172	M₁₆×C₃₁	—	2	8	no	—
173	M₁₆×C₃₁	#41	1	8	yes	—
174	C₂×C₈×C₃₁	—	2	8	no	—
175	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
176	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
177	C₄⋊C₄×C₃₁	—	2	4	no	—
178	C₄⋊C₄×C₃₁	—	2	4	no	—
179	C₄⋊C₄×C₃₁	—	2	2	no	—
180	C₄⋊C₄×C₃₁	—	2	4	no	—
181	C₄⋊C₄×C₃₁	—	2	2	no	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	—	2	4	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
186	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
187	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
189	C₂²⋊C₄×C₃₁	—	2	2	no	—
190	C₂²⋊C₄×C₃₁	—	2	4	no	—
191	C₂²⋊C₄×C₃₁	—	2	8	no	—
192	C₂²⋊C₄×C₃₁	—	2	8	no	—
193	C₂²⋊C₄×C₃₁	—	2	4	no	—
194	C₂²⋊C₄×C₃₁	—	2	4	no	—
195	C₂²⋊C₄×C₃₁	—	2	4	no	—
196	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
197	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
198	C₄²×C₃₁	—	2	2	no	—
199	C₄²×C₃₁	—	2	4	no	—
200	C₄²×C₃₁	C₂×D₄	1	2	yes	—
201	C₄²×C₃₁	C₂×D₄	1	2	yes	—
202	C₄²×C₃₁	C₂⁴	1	2	yes	—
203	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
204	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
205	#60704	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂³×C₃₁	—	2	2	no	—
207	C₂³×C₃₁	—	2	2	no	—
208	C₂³×C₃₁	—	2	4	no	—
209	C₂³×C₃₁	—	4	2	no	—
210	C₂³×C₃₁	—	4	4	no	—
211	C₂³×C₃₁	—	4	4	no	—
212	C₂³×C₃₁	—	4	4	no	—
213	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
214	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	Q₈×C₃₁	—	2	4	no	—
216	Q₈×C₃₁	—	8	2	no	—
217	D₄×C₃₁	—	2	4	no	—
218	D₄×C₃₁	—	2	8	no	—
219	D₄×C₃₁	—	4	4	no	—
220	D₄×C₃₁	—	4	4	no	—
221	D₄×C₃₁	—	8	2	no	—
222	D₄×C₃₁	—	8	4	no	—
223	D₄×C₃₁	—	8	4	no	—
224	D₄×C₃₁	—	8	4	no	—
225	D₄×C₃₁	—	8	8	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	2	4	no	—
233	C₂×C₄×C₃₁	—	4	2	no	—
234	C₂×C₄×C₃₁	—	4	2	no	—
235	C₂×C₄×C₃₁	—	4	4	no	—
236	C₂×C₄×C₃₁	—	4	4	no	—
237	C₂×C₄×C₃₁	—	8	4	no	—
238	C₂×C₄×C₃₁	—	8	8	no	—
239	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
240	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
245	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
246	C₈×C₃₁	—	2	4	no	—
247	C₈×C₃₁	C₂×#41	1	8	yes	—
248	#3455	C₂×C₃₁	1	1	yes	characteristic, complemented
249	C₂×#540	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#3179	C₂×C₃₁	1	8	yes	complemented
251	#3176	C₂×C₃₁	1	4	yes	complemented
252	#3174	C₂×C₃₁	1	4	yes	complemented
253	#3166	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#3151	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#3142	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#3138	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#3055	C₂×C₃₁	1	4	yes	complemented
258	C₄×AGL(1,ℤ/8)	C₂×C₃₁	1	4	yes	complemented
259	#2873	C₂×C₃₁	1	1	yes	characteristic, complemented
260	C₂²×C₃₁	—	2	2	no	—
261	C₂²×C₃₁	—	2	4	no	—
262	C₂²×C₃₁	—	2	2	no	—
263	C₂²×C₃₁	—	4	4	no	—
264	C₂²×C₃₁	—	4	4	no	—
265	C₂²×C₃₁	—	4	4	no	—
266	C₂²×C₃₁	—	4	2	no	—
267	C₂²×C₃₁	—	4	4	no	—
268	C₂²×C₃₁	—	8	2	no	—
269	C₂²×C₃₁	—	8	8	no	—
270	C₂²×C₃₁	#532	1	2	yes	—
271	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
272	C₄×C₃₁	—	2	4	no	—
273	C₄×C₃₁	—	2	2	no	—
274	C₄×C₃₁	—	2	2	no	—
275	C₄×C₃₁	—	2	2	no	—
276	C₄×C₃₁	—	4	4	no	—
277	C₄×C₃₁	—	4	4	no	—
278	C₄×C₃₁	—	4	2	no	—
279	C₄×C₃₁	#532	1	2	yes	—
280	C₄×C₃₁	C₂²×#41	1	2	yes	—
281	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	—
282	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
283	#542	C₂²×C₃₁	1	4	yes	complemented
284	#541	C₂²×C₃₁	1	2	yes	complemented
285	#540	—	2	4	no	—
286	#540	C₂²×C₃₁	1	4	yes	—
287	#540	C₂²×C₃₁	1	4	yes	—
288	#540	C₂²×C₃₁	1	2	yes	complemented
289	#539	—	2	2	no	—
290	#538	—	2	2	no	—
291	#533	C₂²×C₃₁	1	1	yes	characteristic
292	#532	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
294	#529	—	2	4	no	—
295	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
296	#525	C₂²×C₃₁	1	2	yes	—
297	C₂×#123	—	2	2	no	—
298	C₂×#121	—	2	2	no	—
299	C₂×#121	C₂²×C₃₁	1	2	yes	—
300	C₄×#41	C₂²×C₃₁	1	2	yes	—
301	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
302	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
303	#501	C₂²×C₃₁	1	4	yes	complemented
304	#499	C₂²×C₃₁	1	4	yes	complemented
305	#496	C₂²×C₃₁	1	4	yes	complemented
306	#493	C₂²×C₃₁	1	2	yes	complemented
307	#492	C₂²×C₃₁	1	2	yes	complemented
308	#491	C₂²×C₃₁	1	2	yes	complemented
309	#490	C₂²×C₃₁	1	4	yes	complemented
310	#490	C₂²×C₃₁	1	2	yes	complemented
311	#488	C₂²×C₃₁	1	4	yes	complemented
312	#486	C₂²×C₃₁	1	4	yes	complemented
313	#469	C₂²×C₃₁	1	8	yes	complemented
314	#468	C₂²×C₃₁	1	8	yes	complemented
315	#466	C₂²×C₃₁	1	8	yes	complemented
316	#465	C₂²×C₃₁	1	8	yes	complemented
317	#442	C₂²×C₃₁	1	8	yes	complemented
318	#440	C₂²×C₃₁	1	8	yes	complemented
319	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
320	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
321	#428	C₂²×C₃₁	1	4	yes	—
322	#418	C₂²×C₃₁	1	4	yes	—
323	#417	C₂²×C₃₁	1	4	yes	—
324	#405	C₂²×C₃₁	1	4	yes	complemented
325	#405	C₂²×C₃₁	1	4	yes	complemented
326	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
327	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
328	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
329	C₂×C₃₁	—	2	2	no	—
330	C₂×C₃₁	—	4	2	no	—
331	C₂×C₃₁	—	8	2	no	—
332	C₂×C₃₁	—	8	4	no	—
333	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
334	C₂×C₃₁	#3585	1	2	yes	—
335	C₂×#41	C₂³×C₃₁	1	2	yes	—
336	C₂²×D₄	—	2	2	no	—
337	C₂²×D₄	—	2	2	no	—
338	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
339	C₂³×C₄	—	2	2	no	—
340	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
341	AGL(1,ℤ/8)	—	2	16	no	—
342	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
343	C₂×D₈	C₂³×C₃₁	1	8	yes	—
344	C₂×M₁₆	—	2	8	no	—
345	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
346	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
347	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
348	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
349	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
350	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
351	#127	—	2	2	no	—
352	#127	C₂³×C₃₁	1	2	yes	—
353	#127	C₂³×C₃₁	1	2	yes	—
354	#127	C₂³×C₃₁	1	1	yes	characteristic
355	#126	—	2	4	no	—
356	#126	—	2	4	no	—
357	#125	—	2	4	no	—
358	#125	—	2	2	no	—
359	#125	C₂³×C₃₁	1	2	yes	—
360	#125	C₂³×C₃₁	1	2	yes	—
361	#125	C₂³×C₃₁	1	2	yes	—
362	#124	—	2	4	no	—
363	#124	C₂³×C₃₁	1	2	yes	—
364	#123	—	2	2	no	—
365	#123	—	2	8	no	—
366	#123	—	2	8	no	—
367	#123	—	2	4	no	—
368	#123	—	2	4	no	—
369	C₂²⋊Q₈	—	2	2	no	—
370	C₂²⋊Q₈	—	2	4	no	—
371	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
372	#121	—	2	4	no	—
373	#121	—	2	2	no	—
374	#121	—	2	8	no	—
375	#121	—	2	8	no	—
376	#121	—	2	4	no	—
377	#121	C₂³×C₃₁	1	8	yes	—
378	#121	C₂³×C₃₁	1	4	yes	—
379	#120	—	2	2	no	—
380	C₄×Q₈	—	2	2	no	—
381	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
382	C₄×D₄	—	2	4	no	—
383	C₄×D₄	—	2	4	no	—
384	C₄×D₄	—	2	4	no	—
385	C₄×D₄	—	2	2	no	—
386	C₄×D₄	C₂³×C₃₁	1	8	yes	—
387	C₄×D₄	C₂³×C₃₁	1	4	yes	—
388	#117	—	2	2	no	—
389	#117	D₄×C₃₁	1	4	yes	—
390	#117	C₂³×C₃₁	1	2	yes	—
391	#117	C₂³×C₃₁	1	2	yes	—
392	#117	C₂³×C₃₁	1	2	yes	—
393	C₂×C₄⋊C₄	—	2	2	no	—
394	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
395	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
396	C₂×C₂²⋊C₄	—	2	2	no	—
397	C₂×C₂²⋊C₄	—	2	2	no	—
398	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
399	C₂×C₄²	—	2	2	no	—
400	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
402	#107	C₂³×C₃₁	1	8	yes	—
403	#106	C₂³×C₃₁	1	8	yes	—
404	#103	C₂³×C₃₁	1	8	yes	—
405	#102	C₂³×C₃₁	1	8	yes	—
406	#102	C₂³×C₃₁	1	8	yes	—
407	#102	C₂³×C₃₁	1	8	yes	—
408	#97	C₂³×C₃₁	1	4	yes	—
409	#97	C₂³×C₃₁	1	2	yes	—
410	#97	C₂³×C₃₁	1	2	yes	—
411	C₄×C₈	C₂³×C₃₁	1	4	yes	—
412	C₄×C₈	C₂³×C₃₁	1	2	yes	—
413	C₄×C₈	C₂³×C₃₁	1	2	yes	—
414	C₃₁	#60704	1	1	yes	characteristic, direct product factor, Sylow
415	C₂⁴	—	2	2	no	—
416	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
417	#41	—	2	8	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	8	no	—
428	C₂×D₄	—	8	4	no	—
429	C₂×D₄	—	8	4	no	—
430	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
431	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	8	no	—
437	C₂²×C₄	—	8	4	no	—
438	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
440	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
441	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
442	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
444	SD₁₆	—	2	16	no	—
445	D₈	—	2	16	no	—
446	M₁₆	—	2	4	no	—
447	M₁₆	—	2	8	no	—
448	M₁₆	#41×C₃₁	1	8	yes	—
449	C₂×C₈	—	2	8	no	—
450	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
451	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	2	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
461	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
462	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
463	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
464	C₂²⋊C₄	—	2	2	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	8	no	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	4	no	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
472	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
473	C₄²	—	2	2	no	—
474	C₄²	—	2	4	no	—
475	C₄²	C₂×D₄×C₃₁	1	2	yes	—
476	C₄²	C₂×D₄×C₃₁	1	2	yes	—
477	C₄²	C₂⁴×C₃₁	1	2	yes	—
478	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
479	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	4	no	—
483	C₂³	—	4	2	no	—
484	C₂³	—	4	4	no	—
485	C₂³	—	4	4	no	—
486	C₂³	—	4	4	no	—
487	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
488	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
489	Q₈	—	2	4	no	—
490	Q₈	—	8	2	no	—
491	D₄	—	2	4	no	—
492	D₄	—	2	8	no	—
493	D₄	—	4	4	no	—
494	D₄	—	4	4	no	—
495	D₄	—	8	2	no	—
496	D₄	—	8	4	no	—
497	D₄	—	8	4	no	—
498	D₄	—	8	4	no	—
499	D₄	—	8	8	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	4	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	4	4	no	—
510	C₂×C₄	—	4	4	no	—
511	C₂×C₄	—	8	4	no	—
512	C₂×C₄	—	8	8	no	—
513	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
514	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
515	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
516	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
517	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
518	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
519	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
520	C₈	—	2	4	no	—
521	C₈	C₂×#41×C₃₁	1	8	yes	—
522	C₂²	—	2	2	no	—
523	C₂²	—	2	2	no	—
524	C₂²	—	2	4	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	4	4	no	—
528	C₂²	—	4	4	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	8	2	no	—
531	C₂²	—	8	8	no	—
532	C₂²	C₃₁×#532	1	2	yes	—
533	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
534	C₄	—	2	2	no	—
535	C₄	—	2	4	no	—
536	C₄	—	2	2	no	—
537	C₄	—	2	2	no	—
538	C₄	—	4	2	no	—
539	C₄	—	4	4	no	—
540	C₄	—	4	4	no	—
541	C₄	C₃₁×#532	1	2	yes	—
542	C₄	C₂²×#41×C₃₁	1	2	yes	—
543	C₂	—	2	2	no	—
544	C₂	—	4	2	no	—
545	C₂	—	8	4	no	—
546	C₂	—	8	2	no	—
547	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
548	C₂	C₃₁×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3455	C₂	semidirect
2. C₂×C₃₁×#540	C₂	semidirect
3. C₃₁×#3179	C₂	semidirect
4. C₃₁×#3176	C₂	semidirect
5. C₃₁×#3174	C₂	semidirect
6. C₃₁×#3166	C₂	semidirect
7. C₃₁×#3151	C₂	semidirect
8. C₃₁×#3142	C₂	semidirect
9. C₃₁×#3138	C₂	semidirect
10. C₃₁×#3055	C₂	semidirect
11. C₄×C₃₁×AGL(1,ℤ/8)	C₂	semidirect
12. C₃₁×#2873	C₂	semidirect
15. C₃₁×#542	C₂²	semidirect
16. C₃₁×#541	C₂²	semidirect
20. C₃₁×#540	C₂²	semidirect
35. C₃₁×#501	C₂²	semidirect
36. C₃₁×#499	C₂²	semidirect
37. C₃₁×#496	C₂²	semidirect
38. C₃₁×#493	C₂²	semidirect
39. C₃₁×#492	C₂²	semidirect
40. C₃₁×#491	C₂²	semidirect
41. C₃₁×#490	C₂²	semidirect
42. C₃₁×#490	C₂²	semidirect
43. C₃₁×#488	C₂²	semidirect
44. C₃₁×#486	C₂²	semidirect
45. C₃₁×#469	C₂²	semidirect
46. C₃₁×#468	C₂²	semidirect
47. C₃₁×#466	C₂²	semidirect
48. C₃₁×#465	C₂²	semidirect
49. C₃₁×#442	C₂²	semidirect
50. C₃₁×#440	C₂²	semidirect
51. C₄×SD₁₆×C₃₁	C₂²	semidirect
52. C₄×D₈×C₃₁	C₂²	semidirect
56. C₃₁×#405	C₂²	semidirect
57. C₃₁×#405	C₂²	semidirect
205. #60704	C₃₁	direct (N × H)
248. #3455	C₂×C₃₁	semidirect
249. C₂×#540	C₂×C₃₁	semidirect
250. #3179	C₂×C₃₁	semidirect
251. #3176	C₂×C₃₁	semidirect
252. #3174	C₂×C₃₁	semidirect
253. #3166	C₂×C₃₁	semidirect
254. #3151	C₂×C₃₁	semidirect
255. #3142	C₂×C₃₁	semidirect
256. #3138	C₂×C₃₁	semidirect
257. #3055	C₂×C₃₁	semidirect
258. C₄×AGL(1,ℤ/8)	C₂×C₃₁	semidirect
259. #2873	C₂×C₃₁	semidirect
283. #542	C₂²×C₃₁	semidirect
284. #541	C₂²×C₃₁	semidirect
288. #540	C₂²×C₃₁	semidirect
303. #501	C₂²×C₃₁	semidirect
304. #499	C₂²×C₃₁	semidirect
305. #496	C₂²×C₃₁	semidirect
306. #493	C₂²×C₃₁	semidirect
307. #492	C₂²×C₃₁	semidirect
308. #491	C₂²×C₃₁	semidirect
309. #490	C₂²×C₃₁	semidirect
310. #490	C₂²×C₃₁	semidirect
311. #488	C₂²×C₃₁	semidirect
312. #486	C₂²×C₃₁	semidirect
313. #469	C₂²×C₃₁	semidirect
314. #468	C₂²×C₃₁	semidirect
315. #466	C₂²×C₃₁	semidirect
316. #465	C₂²×C₃₁	semidirect
317. #442	C₂²×C₃₁	semidirect
318. #440	C₂²×C₃₁	semidirect
319. C₄×SD₁₆	C₂²×C₃₁	semidirect
320. C₄×D₈	C₂²×C₃₁	semidirect
324. #405	C₂²×C₃₁	semidirect
325. #405	C₂²×C₃₁	semidirect
414. C₃₁	#60704	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×#3455	2
C₂×C₃₁×#540	2
C₃₁×#3179	2
C₃₁×#3176	2
C₃₁×#3174	2
C₃₁×#3166	2
C₃₁×#3151	2
C₃₁×#3142	2
C₃₁×#3138	2
C₃₁×#3055	2
C₄×C₃₁×AGL(1,ℤ/8)	2
C₃₁×#2873	2
#60704	31

Subgroups — C31×#60704 #21517463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60704 #21517463