Subgroups — C₃₁×#22210 #21478969

Order 7936.

866 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#436	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1954	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1923	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1851	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×#124	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₃₁	—	2	4	no	—
15	C₃₁×#436	—	2	4	no	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#391	C₄	1	1	yes	characteristic
26	C₃₁×#390	C₄	1	1	yes	characteristic
27	C₃₁×#388	C₄	1	4	yes	—
28	C₃₁×#382	C₄	1	1	yes	characteristic
29	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
30	C₄³×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#336	C₂²	1	2	yes	—
32	C₃₁×#336	C₂²	1	4	yes	—
33	C₃₁×#336	C₂²	1	2	yes	—
34	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
37	C₂×M₁₆×C₃₁	—	2	4	no	—
38	C₂×M₁₆×C₃₁	—	4	4	no	—
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×#124	—	2	4	no	—
45	C₄×Q₈×C₃₁	—	2	2	no	—
46	C₄×Q₈×C₃₁	—	2	2	no	—
47	C₄×D₄×C₃₁	—	2	2	no	—
48	C₄×D₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
54	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
55	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
59	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊C₈	—	2	4	no	—
62	C₃₁×C₄⋊C₈	—	2	2	no	—
63	C₃₁×C₄⋊C₈	—	2	2	no	—
64	C₃₁×C₄⋊C₈	—	4	4	no	—
65	C₃₁×C₂²⋊C₈	—	2	4	no	—
66	C₃₁×C₂²⋊C₈	—	4	4	no	—
67	C₄×C₈×C₃₁	—	2	4	no	—
68	C₄×C₈×C₃₁	—	2	2	no	—
69	C₄×C₈×C₃₁	D₄	1	4	yes	—
70	C₃₁×#95	C₂×C₄	1	4	yes	—
71	C₃₁×#95	C₂×C₄	1	4	yes	—
72	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
73	C₂×Q₈×C₃₁	—	2	2	no	—
74	C₂×Q₈×C₃₁	—	2	2	no	—
75	C₂×Q₈×C₃₁	M₁₆	1	4	yes	complemented
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	2	4	no	—
82	C₂²×C₄×C₃₁	—	2	4	no	—
83	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
91	M₁₆×C₃₁	—	4	8	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	2	2	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	4	no	—
99	C₂×C₈×C₃₁	—	2	4	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	2	4	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₂²⋊C₄×C₃₁	—	2	4	no	—
107	C₂²⋊C₄×C₃₁	—	2	8	no	—
108	C₂²⋊C₄×C₃₁	—	2	8	no	—
109	C₄²×C₃₁	—	2	2	no	—
110	C₄²×C₃₁	—	2	2	no	—
111	C₄²×C₃₁	—	2	4	no	—
112	C₄²×C₃₁	—	2	2	no	—
113	C₄²×C₃₁	—	2	2	no	—
114	C₄²×C₃₁	—	2	2	no	—
115	C₄²×C₃₁	—	2	2	no	—
116	C₄²×C₃₁	C₂×D₄	1	2	yes	—
117	C₄²×C₃₁	#41	1	2	yes	—
118	#22210	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
119	C₂³×C₃₁	—	2	2	no	—
120	C₂³×C₃₁	—	2	4	no	—
121	C₂³×C₃₁	—	2	2	no	—
122	C₂³×C₃₁	—	2	4	no	—
123	C₂³×C₃₁	—	2	2	no	—
124	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
125	Q₈×C₃₁	—	2	8	no	—
126	D₄×C₃₁	—	4	4	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	4	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	2	4	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	4	4	no	—
142	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
145	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
146	C₂×C₄×C₃₁	#131	1	4	yes	—
147	C₂×C₄×C₃₁	#131	1	2	yes	—
148	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
151	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
152	C₈×C₃₁	—	2	2	no	—
153	C₈×C₃₁	—	2	2	no	—
154	C₈×C₃₁	—	4	2	no	—
155	C₈×C₃₁	—	4	2	no	—
156	C₈×C₃₁	—	4	2	no	—
157	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂×#436	C₂×C₃₁	1	1	yes	characteristic
159	C₄×#124	C₂×C₃₁	1	1	yes	characteristic
160	#1984	C₂×C₃₁	1	1	yes	characteristic, complemented
161	#1954	C₂×C₃₁	1	1	yes	characteristic
162	#1923	C₂×C₃₁	1	4	yes	complemented
163	#1851	C₂×C₃₁	1	4	yes	complemented
164	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
165	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
166	C₂²×C₃₁	—	4	2	no	—
167	C₂²×C₃₁	—	4	2	no	—
168	C₂²×C₃₁	—	4	2	no	—
169	C₂²×C₃₁	—	4	4	no	—
170	C₂²×C₃₁	—	4	4	no	—
171	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₃₁	#520	1	2	yes	—
173	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
174	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
175	C₂²×C₃₁	C₂×#131	1	2	yes	—
176	C₄×C₃₁	—	2	4	no	—
177	C₄×C₃₁	—	2	2	no	—
178	C₄×C₃₁	—	2	4	no	—
179	C₄×C₃₁	—	4	2	no	—
180	C₄×C₃₁	—	4	2	no	—
181	C₄×C₃₁	—	4	2	no	—
182	C₄×C₃₁	#435	1	4	yes	—
183	C₄×C₃₁	#535	1	1	yes	characteristic
184	C₄×C₃₁	#536	1	1	yes	characteristic
185	C₄×C₃₁	#542	1	2	yes	—
186	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
187	C₂×#124	C₄×C₃₁	1	1	yes	characteristic
188	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
190	C₈×Q₈	—	2	4	no	—
191	#436	—	2	4	no	—
192	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
193	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
194	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
195	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
196	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
197	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
198	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
199	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
200	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
201	#391	C₄×C₃₁	1	1	yes	characteristic
202	#390	C₄×C₃₁	1	1	yes	characteristic
203	#388	C₄×C₃₁	1	4	yes	—
204	#382	C₄×C₃₁	1	1	yes	characteristic
205	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
206	C₄³	C₂²×C₃₁	1	1	yes	characteristic
207	#336	C₂²×C₃₁	1	2	yes	—
208	#336	C₂²×C₃₁	1	4	yes	—
209	#336	C₂²×C₃₁	1	2	yes	—
210	C₂×C₃₁	—	8	2	no	—
211	C₂×C₃₁	#2326	1	1	yes	characteristic
212	C₂×C₃₁	C₂×#435	1	1	yes	characteristic
213	C₂×C₃₁	#2931	1	1	yes	characteristic
214	C₂×C₃₁	#2991	1	2	yes	—
215	C₂×C₃₁	#2994	1	2	yes	—
216	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
217	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
218	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
219	C₂×M₁₆	—	2	4	no	—
220	C₂×M₁₆	—	4	4	no	—
221	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
222	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
223	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
224	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
225	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
226	#124	—	2	4	no	—
227	C₄×Q₈	—	2	2	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	2	2	no	—
231	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
234	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
235	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
236	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
237	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
239	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
241	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
242	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
243	C₄⋊C₈	—	2	4	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	4	4	no	—
247	C₂²⋊C₈	—	2	4	no	—
248	C₂²⋊C₈	—	4	4	no	—
249	C₄×C₈	—	2	4	no	—
250	C₄×C₈	—	2	2	no	—
251	C₄×C₈	D₄×C₃₁	1	4	yes	—
252	#95	C₂×C₄×C₃₁	1	4	yes	—
253	#95	C₂×C₄×C₃₁	1	4	yes	—
254	C₃₁	#22210	1	1	yes	characteristic, direct product factor, Sylow
255	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	M₁₆×C₃₁	1	4	yes	complemented
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	2	4	no	—
266	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
267	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
268	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
269	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
270	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
271	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
272	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
273	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
274	M₁₆	—	4	8	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	4	no	—
282	C₂×C₈	—	2	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	2	4	no	—
290	C₂²⋊C₄	—	2	8	no	—
291	C₂²⋊C₄	—	2	8	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	4	no	—
295	C₄²	—	2	2	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	C₂×D₄×C₃₁	1	2	yes	—
300	C₄²	#41×C₃₁	1	2	yes	—
301	C₂³	—	2	2	no	—
302	C₂³	—	2	4	no	—
303	C₂³	—	2	2	no	—
304	C₂³	—	2	4	no	—
305	C₂³	—	2	2	no	—
306	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
307	Q₈	—	2	8	no	—
308	D₄	—	4	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
325	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
326	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
327	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
328	C₂×C₄	C₃₁×#131	1	4	yes	—
329	C₂×C₄	C₃₁×#131	1	2	yes	—
330	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
332	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
333	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
334	C₈	—	2	2	no	—
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	—	4	2	no	—
339	C₂²	—	4	2	no	—
340	C₂²	—	4	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	4	4	no	—
343	C₂²	—	4	4	no	—
344	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
345	C₂²	C₃₁×#520	1	2	yes	—
346	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
347	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
348	C₂²	C₂×C₃₁×#131	1	2	yes	—
349	C₄	—	2	4	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	4	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	C₃₁×#435	1	4	yes	—
356	C₄	C₃₁×#535	1	1	yes	characteristic
357	C₄	C₃₁×#536	1	1	yes	characteristic
358	C₄	C₃₁×#542	1	2	yes	—
359	C₂	—	8	2	no	—
360	C₂	C₃₁×#2326	1	1	yes	characteristic
361	C₂	C₂×C₃₁×#435	1	1	yes	characteristic
362	C₂	C₃₁×#2931	1	1	yes	characteristic
363	C₂	C₃₁×#2991	1	2	yes	—
364	C₂	C₃₁×#2994	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×Q₈×C₃₁	C₂	semidirect
4. C₃₁×#1984	C₂	semidirect
6. C₃₁×#1923	C₂	semidirect
7. C₃₁×#1851	C₂	semidirect
8. C₃₁×#1842	C₂	semidirect
9. C₃₁×#1744	C₂	semidirect
75. C₂×Q₈×C₃₁	M₁₆	semidirect
118. #22210	C₃₁	direct (N × H)
157. C₂×C₈×Q₈	C₂×C₃₁	semidirect
160. #1984	C₂×C₃₁	semidirect
162. #1923	C₂×C₃₁	semidirect
163. #1851	C₂×C₃₁	semidirect
164. #1842	C₂×C₃₁	semidirect
165. #1744	C₂×C₃₁	semidirect
254. C₃₁	#22210	direct (N × H)
258. C₂×Q₈	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₃₁	2
C₂×C₃₁×#436	2
C₄×C₃₁×#124	2
C₃₁×#1984	2
C₃₁×#1954	2
C₃₁×#1923	2
C₃₁×#1851	2
C₃₁×#1842	2
C₃₁×#1744	2
#22210	31

Subgroups — C31×#22210 #21478969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22210 #21478969