Subgroups — C₃₁×#19191 #21475950

Order 7936.

662 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3227	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3206	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1691	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1671	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#131	—	2	1	no	—
11	C₃₁×#532	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₃₁×#498	—	2	1	no	—
15	C₃₁×#498	—	2	1	no	—
16	C₃₁×#498	—	2	1	no	—
17	C₃₁×#498	C₂²	1	2	yes	—
18	C₃₁×#498	C₂²	1	2	yes	—
19	C₃₁×#492	—	2	1	no	—
20	C₃₁×#487	—	2	1	no	—
21	C₃₁×#486	—	2	1	no	—
22	C₃₁×#481	—	2	2	no	—
23	C₃₁×#476	C₂²	1	4	yes	—
24	C₃₁×#461	C₂²	1	4	yes	—
25	C₃₁×#448	—	2	2	no	—
26	C₃₁×#436	—	2	1	no	—
27	C₃₁×#436	C₂²	1	2	yes	—
28	C₈×D₄×C₃₁	—	2	1	no	—
29	C₈×D₄×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#431	C₂²	1	1	yes	characteristic
31	C₃₁×#431	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#106	—	2	1	no	—
33	C₂×C₃₁×#106	—	2	1	no	—
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₃₁×#408	—	2	1	no	—
36	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#335	C₂²	1	2	yes	—
38	C₃₁×#334	C₂²	1	2	yes	—
39	C₃₁×#333	C₂²	1	4	yes	—
40	C₃₁×#332	C₂²	1	4	yes	—
41	C₂×#41×C₃₁	—	2	1	no	—
42	C₂²×Q₈×C₃₁	—	2	1	no	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂×SD₁₆×C₃₁	—	2	4	no	—
45	C₃₁×#131	—	4	4	no	—
46	C₂×M₁₆×C₃₁	—	2	1	no	—
47	C₂×M₁₆×C₃₁	—	2	2	no	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
51	C₃₁×C₄⋊Q₈	—	2	1	no	—
52	C₃₁×C₄⋊Q₈	—	2	1	no	—
53	C₃₁×C₄⋊Q₈	—	2	1	no	—
54	C₃₁×C₄⋊Q₈	—	2	1	no	—
55	C₃₁×C₄⋊Q₈	—	2	1	no	—
56	C₃₁×C₄⋊Q₈	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
59	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
60	C₃₁×#127	D₄	1	1	yes	characteristic
61	C₃₁×#124	—	2	1	no	—
62	C₃₁×#121	—	2	1	no	—
63	C₃₁×#121	D₄	1	2	yes	—
64	C₄×D₄×C₃₁	—	2	1	no	—
65	C₄×D₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	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×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#106	—	2	2	no	—
72	C₃₁×#106	—	2	2	no	—
73	C₃₁×#106	—	2	2	no	—
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×#106	—	2	4	no	—
76	C₃₁×C₄⋊C₈	—	2	1	no	—
77	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
78	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
79	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
80	C₃₁×#103	—	2	4	no	—
81	C₃₁×#102	—	2	4	no	—
82	C₃₁×C₂²⋊C₈	—	2	1	no	—
83	C₃₁×C₂²⋊C₈	—	2	1	no	—
84	C₃₁×C₂²⋊C₈	—	2	2	no	—
85	C₃₁×C₂²⋊C₈	—	2	2	no	—
86	C₄×C₈×C₃₁	—	2	1	no	—
87	C₄×C₈×C₃₁	—	2	1	no	—
88	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
89	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
90	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	#41×C₃₁	—	4	2	no	—
93	#41×C₃₁	—	4	2	no	—
94	C₂×Q₈×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	4	1	no	—
98	C₂×Q₈×C₃₁	—	4	1	no	—
99	C₂×Q₈×C₃₁	—	8	2	no	—
100	C₂×Q₈×C₃₁	—	8	2	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	1	no	—
103	C₂×D₄×C₃₁	—	2	1	no	—
104	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	4	1	no	—
107	C₂²×C₄×C₃₁	—	4	1	no	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
111	SD₁₆×C₃₁	—	8	4	no	—
112	M₁₆×C₃₁	—	4	2	no	—
113	M₁₆×C₃₁	—	4	4	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	4	4	no	—
119	C₂×C₈×C₃₁	SD₁₆	1	4	yes	complemented
120	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄×C₃₁	—	2	1	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	4	1	no	—
130	C₄⋊C₄×C₃₁	—	4	1	no	—
131	C₄⋊C₄×C₃₁	—	4	1	no	—
132	C₄⋊C₄×C₃₁	—	4	1	no	—
133	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₂²⋊C₄×C₃₁	—	2	2	no	—
135	C₂²⋊C₄×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₃₁	#41	1	1	yes	characteristic
139	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
140	#19191	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	#121	1	1	yes	characteristic
143	Q₈×C₃₁	—	4	2	no	—
144	Q₈×C₃₁	—	4	4	no	—
145	Q₈×C₃₁	—	8	2	no	—
146	Q₈×C₃₁	—	8	2	no	—
147	Q₈×C₃₁	—	8	2	no	—
148	D₄×C₃₁	—	2	4	no	—
149	D₄×C₃₁	—	4	4	no	—
150	D₄×C₃₁	—	4	2	no	—
151	D₄×C₃₁	—	4	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	4	2	no	—
156	C₂×C₄×C₃₁	—	4	2	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	8	2	no	—
160	C₂×C₄×C₃₁	—	8	2	no	—
161	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
162	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
163	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
164	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
165	C₂×C₄×C₃₁	#137	1	2	yes	—
166	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
169	C₈×C₃₁	—	2	1	no	—
170	C₈×C₃₁	—	2	1	no	—
171	C₈×C₃₁	—	4	2	no	—
172	C₈×C₃₁	—	4	2	no	—
173	C₈×C₃₁	—	4	2	no	—
174	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
175	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
176	#3227	C₂×C₃₁	1	1	yes	characteristic
177	#3206	C₂×C₃₁	1	1	yes	characteristic
178	C₂×#498	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#2931	C₂×C₃₁	1	1	yes	characteristic
180	#1691	C₂×C₃₁	1	4	yes	complemented
181	#1671	C₂×C₃₁	1	4	yes	complemented
182	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#1549	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1548	C₂×C₃₁	1	1	yes	characteristic, complemented
185	C₂²×C₃₁	—	2	1	no	—
186	C₂²×C₃₁	—	2	1	no	—
187	C₂²×C₃₁	—	4	2	no	—
188	C₂²×C₃₁	—	4	2	no	—
189	C₂²×C₃₁	—	4	2	no	—
190	C₂²×C₃₁	#461	1	2	yes	—
191	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
192	C₄×C₃₁	—	2	1	no	—
193	C₄×C₃₁	—	2	1	no	—
194	C₄×C₃₁	—	4	2	no	—
195	C₄×C₃₁	—	8	2	no	—
196	C₄×C₃₁	—	8	2	no	—
197	C₄×C₃₁	#461	1	2	yes	—
198	C₄×C₃₁	#471	1	2	yes	—
199	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
200	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
201	C₄×C₃₁	#577	1	2	yes	—
202	C₂×#131	—	2	1	no	—
203	#532	C₂²×C₃₁	1	1	yes	characteristic
204	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
205	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
206	#498	—	2	1	no	—
207	#498	—	2	1	no	—
208	#498	—	2	1	no	—
209	#498	C₂²×C₃₁	1	2	yes	—
210	#498	C₂²×C₃₁	1	2	yes	—
211	#492	—	2	1	no	—
212	#487	—	2	1	no	—
213	#486	—	2	1	no	—
214	#481	—	2	2	no	—
215	#476	C₂²×C₃₁	1	4	yes	—
216	#461	C₂²×C₃₁	1	4	yes	—
217	#448	—	2	2	no	—
218	#436	—	2	1	no	—
219	#436	C₂²×C₃₁	1	2	yes	—
220	C₈×D₄	—	2	1	no	—
221	C₈×D₄	C₂²×C₃₁	1	2	yes	—
222	#431	C₂²×C₃₁	1	1	yes	characteristic
223	#431	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×#106	—	2	1	no	—
225	C₂×#106	—	2	1	no	—
226	#423	C₂²×C₃₁	1	1	yes	characteristic
227	#408	—	2	1	no	—
228	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
229	#335	C₂²×C₃₁	1	2	yes	—
230	#334	C₂²×C₃₁	1	2	yes	—
231	#333	C₂²×C₃₁	1	4	yes	—
232	#332	C₂²×C₃₁	1	4	yes	—
233	C₂×C₃₁	—	2	2	no	—
234	C₂×C₃₁	—	8	2	no	—
235	C₂×C₃₁	C₂×#461	1	1	yes	characteristic
236	C₂×C₃₁	#3057	1	1	yes	characteristic
237	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
238	C₂×#41	—	2	1	no	—
239	C₂²×Q₈	—	2	1	no	—
240	C₂²×Q₈	—	2	1	no	—
241	C₂×SD₁₆	—	2	4	no	—
242	#131	—	4	4	no	—
243	C₂×M₁₆	—	2	1	no	—
244	C₂×M₁₆	—	2	2	no	—
245	C₂²×C₈	—	2	2	no	—
246	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
247	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	2	1	no	—
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
255	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
256	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
257	#127	D₄×C₃₁	1	1	yes	characteristic
258	#124	—	2	1	no	—
259	#121	—	2	1	no	—
260	#121	D₄×C₃₁	1	2	yes	—
261	C₄×D₄	—	2	1	no	—
262	C₄×D₄	C₂³×C₃₁	1	2	yes	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	2	1	no	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
268	#106	—	2	2	no	—
269	#106	—	2	2	no	—
270	#106	—	2	2	no	—
271	#106	—	2	2	no	—
272	#106	—	2	4	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
275	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
276	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
277	#103	—	2	4	no	—
278	#102	—	2	4	no	—
279	C₂²⋊C₈	—	2	1	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	2	2	no	—
282	C₂²⋊C₈	—	2	2	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	—	2	1	no	—
285	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
286	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
287	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₃₁	#19191	1	1	yes	characteristic, direct product factor, Sylow
290	#41	—	4	2	no	—
291	#41	—	4	2	no	—
292	C₂×Q₈	—	2	1	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	4	1	no	—
296	C₂×Q₈	—	4	1	no	—
297	C₂×Q₈	—	8	2	no	—
298	C₂×Q₈	—	8	2	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	1	no	—
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	4	1	no	—
305	C₂²×C₄	—	4	1	no	—
306	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
308	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
309	SD₁₆	—	8	4	no	—
310	M₁₆	—	4	2	no	—
311	M₁₆	—	4	4	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	4	4	no	—
317	C₂×C₈	SD₁₆×C₃₁	1	4	yes	complemented
318	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
332	C₂²⋊C₄	—	2	2	no	—
333	C₂²⋊C₄	—	2	2	no	—
334	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₄²	#41×C₃₁	1	1	yes	characteristic
337	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
338	C₂³	—	2	2	no	—
339	C₂³	C₃₁×#121	1	1	yes	characteristic
340	Q₈	—	4	2	no	—
341	Q₈	—	4	4	no	—
342	Q₈	—	8	2	no	—
343	Q₈	—	8	2	no	—
344	Q₈	—	8	2	no	—
345	D₄	—	2	4	no	—
346	D₄	—	4	4	no	—
347	D₄	—	4	2	no	—
348	D₄	—	4	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	8	2	no	—
357	C₂×C₄	—	8	2	no	—
358	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
359	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
360	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
361	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
362	C₂×C₄	C₃₁×#137	1	2	yes	—
363	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
364	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
365	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	—	4	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	4	2	no	—
371	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
372	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	4	2	no	—
376	C₂²	—	4	2	no	—
377	C₂²	—	4	2	no	—
378	C₂²	C₃₁×#461	1	2	yes	—
379	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	4	2	no	—
383	C₄	—	8	2	no	—
384	C₄	—	8	2	no	—
385	C₄	C₃₁×#461	1	2	yes	—
386	C₄	C₃₁×#471	1	2	yes	—
387	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
388	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
389	C₄	C₃₁×#577	1	2	yes	—
390	C₂	—	2	2	no	—
391	C₂	—	8	2	no	—
392	C₂	C₂×C₃₁×#461	1	1	yes	characteristic
393	C₂	C₃₁×#3057	1	1	yes	characteristic
394	C₂	C₂×C₃₁×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₁×#498	C₂	semidirect
5. C₃₁×#1691	C₂	semidirect
6. C₃₁×#1671	C₂	semidirect
7. C₃₁×#1564	C₂	semidirect
8. C₃₁×#1549	C₂	semidirect
9. C₃₁×#1548	C₂	semidirect
119. C₂×C₈×C₃₁	SD₁₆	semidirect
140. #19191	C₃₁	direct (N × H)
178. C₂×#498	C₂×C₃₁	semidirect
180. #1691	C₂×C₃₁	semidirect
181. #1671	C₂×C₃₁	semidirect
182. #1564	C₂×C₃₁	semidirect
183. #1549	C₂×C₃₁	semidirect
184. #1548	C₂×C₃₁	semidirect
289. C₃₁	#19191	direct (N × H)
317. C₂×C₈	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3227	2
C₃₁×#3206	2
C₂×C₃₁×#498	2
C₃₁×#2931	2
C₃₁×#1691	2
C₃₁×#1671	2
C₃₁×#1564	2
C₃₁×#1549	2
C₃₁×#1548	2
#19191	31

Subgroups — C31×#19191 #21475950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19191 #21475950