Subgroups — C₃₁×#22235 #21478994

Order 7936.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3135	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1995	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1963	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1929	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1866	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1849	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#131	C₂²	1	2	yes	—
11	C₃₁×#533	C₂²	1	1	yes	characteristic
12	C₃₁×#532	C₂²	1	1	yes	characteristic
13	C₃₁×#491	C₄	1	2	yes	complemented
14	C₃₁×#490	C₄	1	2	yes	complemented
15	C₃₁×#489	C₄	1	2	yes	complemented
16	C₃₁×#489	C₄	1	2	yes	complemented
17	C₃₁×#446	—	2	1	no	—
18	C₃₁×#446	—	2	1	no	—
19	C₈×Q₈×C₃₁	—	2	1	no	—
20	C₈×Q₈×C₃₁	—	2	1	no	—
21	C₃₁×#436	—	2	1	no	—
22	C₃₁×#436	—	2	1	no	—
23	C₈×D₄×C₃₁	—	2	1	no	—
24	C₈×D₄×C₃₁	—	2	1	no	—
25	C₂×C₃₁×#108	C₂²	1	4	yes	—
26	C₃₁×#424	C₂²	1	2	yes	—
27	C₃₁×#423	C₂²	1	1	yes	characteristic
28	C₃₁×#423	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#104	C₂²	1	4	yes	—
30	C₃₁×#416	C₂²	1	4	yes	—
31	C₃₁×#408	C₂²	1	2	yes	—
32	C₃₁×#405	C₂²	1	2	yes	—
33	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#376	C₂²	1	1	yes	characteristic
36	C₃₁×#339	C₂²	1	4	yes	—
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₂×#41×C₃₁	C₂³	1	2	yes	—
40	C₃₁×#131	—	4	4	no	—
41	C₂×M₁₆×C₃₁	—	2	2	no	—
42	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
43	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	—	2	2	no	—
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
49	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
50	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
51	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
52	C₃₁×#125	C₂×C₄	1	2	yes	—
53	C₃₁×#124	C₂×C₄	1	2	yes	—
54	C₃₁×C₂²⋊Q₈	—	2	1	no	—
55	C₃₁×C₂²⋊Q₈	—	2	1	no	—
56	C₃₁×#121	—	2	1	no	—
57	C₃₁×#121	—	2	1	no	—
58	C₄×Q₈×C₃₁	—	2	1	no	—
59	C₄×Q₈×C₃₁	—	2	1	no	—
60	C₄×D₄×C₃₁	—	2	1	no	—
61	C₄×D₄×C₃₁	—	2	1	no	—
62	C₃₁×#117	C₂³	1	2	yes	—
63	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#108	—	4	4	no	—
66	C₃₁×C₄⋊C₈	—	2	2	no	—
67	C₃₁×C₄⋊C₈	—	2	2	no	—
68	C₃₁×C₄⋊C₈	—	2	1	no	—
69	C₃₁×C₄⋊C₈	—	2	1	no	—
70	C₃₁×C₄⋊C₈	—	2	1	no	—
71	C₃₁×C₄⋊C₈	—	2	1	no	—
72	C₃₁×#104	—	4	4	no	—
73	C₃₁×#103	C₂×C₄	1	4	yes	—
74	C₃₁×#103	C₂×C₄	1	4	yes	—
75	C₃₁×#102	C₂×C₄	1	4	yes	—
76	C₃₁×#102	C₂×C₄	1	4	yes	—
77	C₃₁×C₂²⋊C₈	—	2	2	no	—
78	C₃₁×C₂²⋊C₈	—	2	2	no	—
79	C₃₁×#97	—	2	2	no	—
80	C₃₁×#97	C₂×C₄	1	2	yes	—
81	C₃₁×#97	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₃₁	—	2	2	no	—
83	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
84	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
85	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
86	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
87	C₃₁×#95	—	2	1	no	—
88	C₃₁×#95	—	2	1	no	—
89	#41×C₃₁	—	2	4	no	—
90	#41×C₃₁	—	2	4	no	—
91	C₂×Q₈×C₃₁	—	2	1	no	—
92	C₂×Q₈×C₃₁	—	2	1	no	—
93	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	—	2	1	no	—
97	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
103	M₁₆×C₃₁	—	2	4	no	—
104	M₁₆×C₃₁	—	2	4	no	—
105	M₁₆×C₃₁	—	4	4	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	4	4	no	—
113	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
114	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄⋊C₄×C₃₁	—	2	1	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	1	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
125	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
126	C₂²⋊C₄×C₃₁	—	2	2	no	—
127	C₂²⋊C₄×C₃₁	—	2	2	no	—
128	C₂²⋊C₄×C₃₁	—	2	2	no	—
129	C₄²×C₃₁	—	2	2	no	—
130	C₄²×C₃₁	—	2	2	no	—
131	C₄²×C₃₁	—	2	2	no	—
132	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
134	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
136	#22235	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₃₁	—	2	2	no	—
138	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
139	Q₈×C₃₁	—	2	4	no	—
140	Q₈×C₃₁	—	4	2	no	—
141	Q₈×C₃₁	—	4	2	no	—
142	D₄×C₃₁	—	2	4	no	—
143	D₄×C₃₁	—	4	2	no	—
144	D₄×C₃₁	—	4	2	no	—
145	D₄×C₃₁	—	4	4	no	—
146	C₂×C₄×C₃₁	—	2	2	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	2	no	—
149	C₂×C₄×C₃₁	—	4	1	no	—
150	C₂×C₄×C₃₁	—	4	1	no	—
151	C₂×C₄×C₃₁	—	4	2	no	—
152	C₂×C₄×C₃₁	—	4	2	no	—
153	C₂×C₄×C₃₁	—	4	2	no	—
154	C₂×C₄×C₃₁	—	4	2	no	—
155	C₂×C₄×C₃₁	—	4	2	no	—
156	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
160	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
163	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
164	C₈×C₃₁	—	2	4	no	—
165	C₈×C₃₁	—	2	1	no	—
166	C₈×C₃₁	—	2	1	no	—
167	C₈×C₃₁	—	2	1	no	—
168	C₈×C₃₁	—	2	1	no	—
169	C₈×C₃₁	—	4	2	no	—
170	C₈×C₃₁	—	4	2	no	—
171	#3135	C₂×C₃₁	1	1	yes	characteristic
172	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
173	#2931	C₂×C₃₁	1	1	yes	characteristic
174	#1995	C₂×C₃₁	1	1	yes	characteristic, complemented
175	#1963	C₂×C₃₁	1	1	yes	characteristic
176	#1929	C₂×C₃₁	1	4	yes	complemented
177	#1866	C₂×C₃₁	1	4	yes	complemented
178	#1849	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
180	C₂²×C₃₁	—	2	1	no	—
181	C₂²×C₃₁	—	2	1	no	—
182	C₂²×C₃₁	—	4	2	no	—
183	C₂²×C₃₁	—	4	2	no	—
184	C₂²×C₃₁	—	4	2	no	—
185	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
186	C₂²×C₃₁	#542	1	2	yes	—
187	C₄×C₃₁	—	2	1	no	—
188	C₄×C₃₁	—	2	1	no	—
189	C₄×C₃₁	—	2	1	no	—
190	C₄×C₃₁	—	2	1	no	—
191	C₄×C₃₁	—	4	2	no	—
192	C₄×C₃₁	—	4	2	no	—
193	C₄×C₃₁	—	4	2	no	—
194	C₄×C₃₁	—	8	2	no	—
195	C₄×C₃₁	#520	1	2	yes	—
196	C₄×C₃₁	#535	1	1	yes	characteristic
197	C₄×C₃₁	#536	1	1	yes	characteristic
198	C₂×#131	C₂²×C₃₁	1	2	yes	—
199	#533	C₂²×C₃₁	1	1	yes	characteristic
200	#532	C₂²×C₃₁	1	1	yes	characteristic
201	#491	C₄×C₃₁	1	2	yes	complemented
202	#490	C₄×C₃₁	1	2	yes	complemented
203	#489	C₄×C₃₁	1	2	yes	complemented
204	#489	C₄×C₃₁	1	2	yes	complemented
205	#446	—	2	1	no	—
206	#446	—	2	1	no	—
207	C₈×Q₈	—	2	1	no	—
208	C₈×Q₈	—	2	1	no	—
209	#436	—	2	1	no	—
210	#436	—	2	1	no	—
211	C₈×D₄	—	2	1	no	—
212	C₈×D₄	—	2	1	no	—
213	C₂×#108	C₂²×C₃₁	1	4	yes	—
214	#424	C₂²×C₃₁	1	2	yes	—
215	#423	C₂²×C₃₁	1	1	yes	characteristic
216	#423	C₂²×C₃₁	1	1	yes	characteristic
217	C₂×#104	C₂²×C₃₁	1	4	yes	—
218	#416	C₂²×C₃₁	1	4	yes	—
219	#408	C₂²×C₃₁	1	2	yes	—
220	#405	C₂²×C₃₁	1	2	yes	—
221	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
223	#376	C₂²×C₃₁	1	1	yes	characteristic
224	#339	C₂²×C₃₁	1	4	yes	—
225	#336	C₂²×C₃₁	1	2	yes	—
226	#336	C₂²×C₃₁	1	2	yes	—
227	C₂×C₃₁	—	2	2	no	—
228	C₂×C₃₁	—	8	2	no	—
229	C₂×C₃₁	#2326	1	1	yes	characteristic
230	C₂×C₃₁	#2955	1	1	yes	characteristic
231	C₂×C₃₁	#2955	1	1	yes	characteristic
232	C₂×#41	C₂³×C₃₁	1	2	yes	—
233	#131	—	4	4	no	—
234	C₂×M₁₆	—	2	2	no	—
235	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
236	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
237	C₂²×C₈	—	2	2	no	—
238	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
239	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
240	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
241	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
242	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
243	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
244	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
245	#125	C₂×C₄×C₃₁	1	2	yes	—
246	#124	C₂×C₄×C₃₁	1	2	yes	—
247	C₂²⋊Q₈	—	2	1	no	—
248	C₂²⋊Q₈	—	2	1	no	—
249	#121	—	2	1	no	—
250	#121	—	2	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	2	1	no	—
253	C₄×D₄	—	2	1	no	—
254	C₄×D₄	—	2	1	no	—
255	#117	C₂³×C₃₁	1	2	yes	—
256	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
257	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
258	#108	—	4	4	no	—
259	C₄⋊C₈	—	2	2	no	—
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	2	1	no	—
264	C₄⋊C₈	—	2	1	no	—
265	#104	—	4	4	no	—
266	#103	C₂×C₄×C₃₁	1	4	yes	—
267	#103	C₂×C₄×C₃₁	1	4	yes	—
268	#102	C₂×C₄×C₃₁	1	4	yes	—
269	#102	C₂×C₄×C₃₁	1	4	yes	—
270	C₂²⋊C₈	—	2	2	no	—
271	C₂²⋊C₈	—	2	2	no	—
272	#97	—	2	2	no	—
273	#97	C₂×C₄×C₃₁	1	2	yes	—
274	#97	C₂×C₄×C₃₁	1	2	yes	—
275	C₄×C₈	—	2	2	no	—
276	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
277	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
278	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
279	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
280	#95	—	2	1	no	—
281	#95	—	2	1	no	—
282	C₃₁	#22235	1	1	yes	characteristic, direct product factor, Sylow
283	#41	—	2	4	no	—
284	#41	—	2	4	no	—
285	C₂×Q₈	—	2	1	no	—
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	2	1	no	—
291	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
295	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
296	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
297	M₁₆	—	2	4	no	—
298	M₁₆	—	2	4	no	—
299	M₁₆	—	4	4	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	4	4	no	—
307	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
308	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
310	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	1	no	—
317	C₄⋊C₄	—	2	1	no	—
318	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
319	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	2	2	no	—
322	C₂²⋊C₄	—	2	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	2	2	no	—
326	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
327	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
328	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂³	—	2	2	no	—
331	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
332	Q₈	—	2	4	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	2	no	—
335	D₄	—	2	4	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	4	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
350	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
351	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
352	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
353	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
355	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
356	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
357	C₈	—	2	4	no	—
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	4	2	no	—
363	C₈	—	4	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
370	C₂²	C₃₁×#542	1	2	yes	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	8	2	no	—
379	C₄	C₃₁×#520	1	2	yes	—
380	C₄	C₃₁×#535	1	1	yes	characteristic
381	C₄	C₃₁×#536	1	1	yes	characteristic
382	C₂	—	2	2	no	—
383	C₂	—	8	2	no	—
384	C₂	C₃₁×#2326	1	1	yes	characteristic
385	C₂	C₃₁×#2955	1	1	yes	characteristic
386	C₂	C₃₁×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2960	C₂	semidirect
4. C₃₁×#1995	C₂	semidirect
6. C₃₁×#1929	C₂	semidirect
7. C₃₁×#1866	C₂	semidirect
8. C₃₁×#1849	C₂	semidirect
9. C₃₁×#1746	C₂	semidirect
13. C₃₁×#491	C₄	semidirect
14. C₃₁×#490	C₄	semidirect
15. C₃₁×#489	C₄	semidirect
16. C₃₁×#489	C₄	semidirect
136. #22235	C₃₁	direct (N × H)
172. #2960	C₂×C₃₁	semidirect
174. #1995	C₂×C₃₁	semidirect
176. #1929	C₂×C₃₁	semidirect
177. #1866	C₂×C₃₁	semidirect
178. #1849	C₂×C₃₁	semidirect
179. #1746	C₂×C₃₁	semidirect
201. #491	C₄×C₃₁	semidirect
202. #490	C₄×C₃₁	semidirect
203. #489	C₄×C₃₁	semidirect
204. #489	C₄×C₃₁	semidirect
282. C₃₁	#22235	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×#3135	2
C₃₁×#2960	2
C₃₁×#2931	2
C₃₁×#1995	2
C₃₁×#1963	2
C₃₁×#1929	2
C₃₁×#1866	2
C₃₁×#1849	2
C₃₁×#1746	2
#22235	31

Subgroups — C31×#22235 #21478994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22235 #21478994