Subgroups — C₃₁×#17337 #21474096

Order 7936.

798 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1455	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1453	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1451	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
9	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	C₄	1	2	yes	—
13	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#446	C₄	1	2	yes	—
15	C₃₁×#446	C₄	1	2	yes	—
16	C₃₁×#436	C₄	1	2	yes	—
17	C₃₁×#436	C₄	1	2	yes	—
18	C₃₁×#435	C₄	1	4	yes	—
19	C₃₁×#435	C₄	1	4	yes	—
20	C₃₁×#431	C₂²	1	1	yes	characteristic
21	C₃₁×#431	C₂²	1	1	yes	characteristic
22	C₃₁×#431	C₂²	1	2	yes	—
23	C₃₁×#431	C₂²	1	2	yes	—
24	C₃₁×#405	C₄	1	4	yes	—
25	C₃₁×#405	C₄	1	2	yes	—
26	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
28	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
29	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#322	C₂²	1	1	yes	characteristic
35	C₃₁×#322	C₂²	1	1	yes	characteristic
36	C₃₁×#322	C₂²	1	1	yes	characteristic
37	C₃₁×#322	C₂²	1	1	yes	characteristic
38	C₃₁×#322	C₂²	1	2	yes	—
39	C₃₁×#322	C₂²	1	2	yes	—
40	C₃₁×#322	C₂²	1	2	yes	—
41	C₃₁×#322	C₂²	1	2	yes	—
42	C₈²×C₃₁	C₂²	1	2	yes	—
43	C₈²×C₃₁	C₂²	1	2	yes	—
44	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₃₁×#131	—	2	8	no	—
46	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
47	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
48	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
49	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
54	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
57	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₃₁×#117	C₂×C₄	1	2	yes	—
61	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
64	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
65	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
66	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
67	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
68	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
69	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
70	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
71	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
72	C₃₁×#97	C₂×C₄	1	4	yes	—
73	C₃₁×#97	C₂×C₄	1	2	yes	—
74	C₃₁×#97	C₂×C₄	1	4	yes	—
75	C₃₁×#97	C₂×C₄	1	4	yes	—
76	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
78	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
80	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₃₁	C₂³	1	2	yes	—
82	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₃₁	C₂³	1	2	yes	—
88	C₄×C₈×C₃₁	C₂³	1	2	yes	—
89	#41×C₃₁	—	2	4	no	—
90	C₂×Q₈×C₃₁	C₄²	1	1	yes	characteristic
91	C₂×D₄×C₃₁	C₄²	1	2	yes	—
92	C₂×D₄×C₃₁	C₄²	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	M₁₆×C₃₁	—	2	8	no	—
100	M₁₆×C₃₁	—	2	8	no	—
101	M₁₆×C₃₁	—	4	4	no	—
102	M₁₆×C₃₁	M₁₆	1	8	yes	complemented
103	C₂×C₈×C₃₁	—	2	4	no	—
104	C₂×C₈×C₃₁	—	2	8	no	—
105	C₂×C₈×C₃₁	—	2	4	no	—
106	C₂×C₈×C₃₁	—	4	2	no	—
107	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
108	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
109	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
113	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
114	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
115	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
116	C₄⋊C₄×C₃₁	C₄²	1	2	yes	—
117	C₄⋊C₄×C₃₁	C₄²	1	2	yes	—
118	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
119	C₄⋊C₄×C₃₁	C₄²	1	1	yes	characteristic
120	C₂²⋊C₄×C₃₁	C₄²	1	2	yes	—
121	C₂²⋊C₄×C₃₁	C₄²	1	2	yes	—
122	C₂²⋊C₄×C₃₁	C₄²	1	2	yes	—
123	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
125	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
128	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
129	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
130	#17337	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
132	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
133	Q₈×C₃₁	—	2	2	no	—
134	D₄×C₃₁	—	2	4	no	—
135	D₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	4	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	2	4	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	4	no	—
141	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
142	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
143	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
145	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
146	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
148	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
151	C₈×C₃₁	—	2	2	no	—
152	C₈×C₃₁	—	2	2	no	—
153	C₈×C₃₁	—	2	2	no	—
154	C₈×C₃₁	—	2	4	no	—
155	C₈×C₃₁	—	2	2	no	—
156	C₈×C₃₁	—	2	2	no	—
157	C₈×C₃₁	—	2	4	no	—
158	C₈×C₃₁	C₂×M₁₆	1	8	yes	—
159	C₈×C₃₁	#131	1	4	yes	complemented
160	#2965	C₂×C₃₁	1	1	yes	characteristic
161	#2965	C₂×C₃₁	1	1	yes	characteristic
162	C₄×#131	C₂×C₃₁	1	1	yes	characteristic
163	#1455	C₂×C₃₁	1	2	yes	complemented
164	#1453	C₂×C₃₁	1	2	yes	complemented
165	#1452	C₂×C₃₁	1	1	yes	characteristic, complemented
166	#1451	C₂×C₃₁	1	2	yes	complemented
167	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
168	C₈×M₁₆	C₂×C₃₁	1	2	yes	complemented
169	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
170	C₂×#322	C₂×C₃₁	1	1	yes	characteristic, complemented
171	C₂²×C₃₁	—	2	1	no	—
172	C₂²×C₃₁	—	2	1	no	—
173	C₂²×C₃₁	—	2	2	no	—
174	C₂²×C₃₁	—	2	2	no	—
175	C₂²×C₃₁	—	2	2	no	—
176	C₂²×C₃₁	C₄×M₁₆	1	2	yes	—
177	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
178	C₄×C₃₁	—	2	2	no	—
179	C₄×C₃₁	—	2	1	no	—
180	C₄×C₃₁	—	2	2	no	—
181	C₄×C₃₁	—	2	1	no	—
182	C₄×C₃₁	—	2	2	no	—
183	C₄×C₃₁	C₄×M₁₆	1	2	yes	—
184	C₄×C₃₁	C₂²×M₁₆	1	2	yes	—
185	C₄×C₃₁	C₂×#131	1	2	yes	—
186	C₄×C₃₁	#568	1	2	yes	—
187	C₂×#131	C₄×C₃₁	1	2	yes	—
188	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
189	#446	C₄×C₃₁	1	2	yes	—
190	#446	C₄×C₃₁	1	2	yes	—
191	#436	C₄×C₃₁	1	2	yes	—
192	#436	C₄×C₃₁	1	2	yes	—
193	#435	C₄×C₃₁	1	4	yes	—
194	#435	C₄×C₃₁	1	4	yes	—
195	#431	C₂²×C₃₁	1	1	yes	characteristic
196	#431	C₂²×C₃₁	1	1	yes	characteristic
197	#431	C₂²×C₃₁	1	2	yes	—
198	#431	C₂²×C₃₁	1	2	yes	—
199	#405	C₄×C₃₁	1	4	yes	—
200	#405	C₄×C₃₁	1	2	yes	—
201	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
202	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
203	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
204	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
205	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
206	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
207	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
208	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
209	#322	C₂²×C₃₁	1	1	yes	characteristic
210	#322	C₂²×C₃₁	1	1	yes	characteristic
211	#322	C₂²×C₃₁	1	1	yes	characteristic
212	#322	C₂²×C₃₁	1	1	yes	characteristic
213	#322	C₂²×C₃₁	1	2	yes	—
214	#322	C₂²×C₃₁	1	2	yes	—
215	#322	C₂²×C₃₁	1	2	yes	—
216	#322	C₂²×C₃₁	1	2	yes	—
217	C₈²	C₂²×C₃₁	1	2	yes	—
218	C₈²	C₂²×C₃₁	1	2	yes	—
219	C₂×C₃₁	—	2	2	no	—
220	C₂×C₃₁	—	4	2	no	—
221	C₂×C₃₁	C₂×C₄×M₁₆	1	1	yes	characteristic
222	C₂×C₃₁	C₄×#131	1	1	yes	characteristic
223	C₂×C₃₁	#2875	1	1	yes	characteristic
224	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
225	#131	—	2	8	no	—
226	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
227	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
228	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
229	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
230	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
231	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
232	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
233	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
234	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
235	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
236	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
237	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
238	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
239	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
240	#117	C₂×C₄×C₃₁	1	2	yes	—
241	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
243	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
244	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
245	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
246	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
247	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
248	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
249	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
250	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
251	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
252	#97	C₂×C₄×C₃₁	1	4	yes	—
253	#97	C₂×C₄×C₃₁	1	2	yes	—
254	#97	C₂×C₄×C₃₁	1	4	yes	—
255	#97	C₂×C₄×C₃₁	1	4	yes	—
256	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
257	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
258	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
259	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
260	C₄×C₈	C₂³×C₃₁	1	2	yes	—
261	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
263	C₄×C₈	C₂³×C₃₁	1	2	yes	—
264	C₄×C₈	C₂³×C₃₁	1	2	yes	—
265	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
266	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
267	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
268	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
269	C₃₁	#17337	1	1	yes	characteristic, direct product factor, Sylow
270	#41	—	2	4	no	—
271	C₂×Q₈	C₄²×C₃₁	1	1	yes	characteristic
272	C₂×D₄	C₄²×C₃₁	1	2	yes	—
273	C₂×D₄	C₄²×C₃₁	1	1	yes	characteristic
274	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
275	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
276	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
277	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
278	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
279	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
280	M₁₆	—	2	8	no	—
281	M₁₆	—	2	8	no	—
282	M₁₆	—	4	4	no	—
283	M₁₆	M₁₆×C₃₁	1	8	yes	complemented
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	2	8	no	—
286	C₂×C₈	—	2	4	no	—
287	C₂×C₈	—	4	2	no	—
288	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
289	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
290	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
291	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
292	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
293	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
294	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
295	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
296	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
297	C₄⋊C₄	C₄²×C₃₁	1	2	yes	—
298	C₄⋊C₄	C₄²×C₃₁	1	2	yes	—
299	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
300	C₄⋊C₄	C₄²×C₃₁	1	1	yes	characteristic
301	C₂²⋊C₄	C₄²×C₃₁	1	2	yes	—
302	C₂²⋊C₄	C₄²×C₃₁	1	2	yes	—
303	C₂²⋊C₄	C₄²×C₃₁	1	2	yes	—
304	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
305	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
306	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
307	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
308	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
309	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
310	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
311	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
312	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	D₄	—	2	4	no	—
315	D₄	—	2	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
322	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
323	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
324	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
325	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
326	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
327	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
328	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
329	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
330	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	2	no	—
333	C₈	—	2	2	no	—
334	C₈	—	2	4	no	—
335	C₈	—	2	2	no	—
336	C₈	—	2	2	no	—
337	C₈	—	2	4	no	—
338	C₈	C₂×M₁₆×C₃₁	1	8	yes	—
339	C₈	C₃₁×#131	1	4	yes	complemented
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	C₄×M₁₆×C₃₁	1	2	yes	—
346	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	C₄×M₁₆×C₃₁	1	2	yes	—
353	C₄	C₂²×M₁₆×C₃₁	1	2	yes	—
354	C₄	C₂×C₃₁×#131	1	2	yes	—
355	C₄	C₃₁×#568	1	2	yes	—
356	C₂	—	2	2	no	—
357	C₂	—	4	2	no	—
358	C₂	C₂×C₄×M₁₆×C₃₁	1	1	yes	characteristic
359	C₂	C₄×C₃₁×#131	1	1	yes	characteristic
360	C₂	C₃₁×#2875	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₃₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄²
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1455	C₂	semidirect
5. C₃₁×#1453	C₂	semidirect
6. C₃₁×#1452	C₂	semidirect
7. C₃₁×#1451	C₂	semidirect
8. C₈×M₁₆×C₃₁	C₂	semidirect
9. C₈×M₁₆×C₃₁	C₂	semidirect
10. C₈×M₁₆×C₃₁	C₂	semidirect
11. C₂×C₃₁×#322	C₂	semidirect
102. M₁₆×C₃₁	M₁₆	semidirect
130. #17337	C₃₁	direct (N × H)
159. C₈×C₃₁	#131	semidirect
163. #1455	C₂×C₃₁	semidirect
164. #1453	C₂×C₃₁	semidirect
165. #1452	C₂×C₃₁	semidirect
166. #1451	C₂×C₃₁	semidirect
167. C₈×M₁₆	C₂×C₃₁	semidirect
168. C₈×M₁₆	C₂×C₃₁	semidirect
169. C₈×M₁₆	C₂×C₃₁	semidirect
170. C₂×#322	C₂×C₃₁	semidirect
269. C₃₁	#17337	direct (N × H)
283. M₁₆	M₁₆×C₃₁	semidirect
339. C₈	C₃₁×#131	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2965	2
C₄×C₃₁×#131	2
C₃₁×#1455	2
C₃₁×#1453	2
C₃₁×#1452	2
C₃₁×#1451	2
C₈×M₁₆×C₃₁	2
C₂×C₃₁×#322	2
#17337	31

Subgroups — C31×#17337 #21474096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17337 #21474096