Subgroups — C₃₁×#19203 #21475962

Order 7936.

774 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3208	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1688	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1668	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1545	C₂	1	2	yes	maximal, complemented
8	C₂²×D₈×C₃₁	—	2	1	no	—
9	C₂²×D₈×C₃₁	—	2	1	no	—
10	C₃₁×#533	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	2	yes	—
12	C₃₁×#493	—	2	1	no	—
13	C₃₁×#493	—	2	1	no	—
14	C₃₁×#493	C₂²	1	2	yes	—
15	C₃₁×#493	C₂²	1	2	yes	—
16	C₃₁×#492	—	2	2	no	—
17	C₃₁×#486	—	2	2	no	—
18	C₃₁×#460	C₂²	1	8	yes	complemented
19	C₃₁×#450	—	2	4	no	—
20	C₃₁×#446	—	2	1	no	—
21	C₃₁×#446	C₂²	1	2	yes	complemented
22	C₈×Q₈×C₃₁	—	2	1	no	—
23	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
24	C₃₁×#431	C₂²	1	2	yes	—
25	C₃₁×#424	—	2	1	no	—
26	C₃₁×#423	C₂²	1	1	yes	characteristic
27	C₃₁×#405	—	2	1	no	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#335	C₂²	1	2	yes	complemented
30	C₃₁×#334	C₂²	1	2	yes	complemented
31	C₃₁×#331	C₂²	1	8	yes	complemented
32	C₂²×D₄×C₃₁	—	2	2	no	—
33	C₂²×D₄×C₃₁	—	2	2	no	—
34	C₂²×D₄×C₃₁	—	2	2	no	—
35	C₂×SD₁₆×C₃₁	—	2	8	no	—
36	C₂×D₈×C₃₁	—	2	2	no	—
37	C₂×D₈×C₃₁	—	2	2	no	—
38	C₂×D₈×C₃₁	—	2	2	no	—
39	C₂×D₈×C₃₁	—	2	2	no	—
40	C₂×D₈×C₃₁	—	4	4	no	—
41	C₂×D₈×C₃₁	—	4	4	no	—
42	C₂×M₁₆×C₃₁	—	2	1	no	—
43	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₃₁×#127	—	2	2	no	—
48	C₃₁×#127	—	2	2	no	—
49	C₃₁×#127	—	2	2	no	—
50	C₃₁×#127	C₂³	1	4	yes	—
51	C₃₁×#125	—	2	1	no	—
52	C₃₁×C₂²⋊Q₈	—	2	1	no	—
53	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
54	C₄×Q₈×C₃₁	—	2	1	no	—
55	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
56	C₃₁×#117	—	2	1	no	—
57	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×C₄⋊C₈	—	2	2	no	—
59	C₃₁×C₄⋊C₈	—	2	2	no	—
60	C₃₁×C₄⋊C₈	—	2	1	no	—
61	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
62	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
63	C₃₁×#102	—	2	8	no	—
64	C₃₁×C₂²⋊C₈	—	2	2	no	—
65	C₃₁×#97	—	2	2	no	—
66	C₄×C₈×C₃₁	—	2	2	no	—
67	C₄×C₈×C₃₁	—	2	2	no	—
68	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂⁴×C₃₁	—	4	2	no	—
73	C₂×Q₈×C₃₁	—	2	1	no	—
74	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
75	C₂×D₄×C₃₁	—	2	4	no	—
76	C₂×D₄×C₃₁	—	2	4	no	—
77	C₂×D₄×C₃₁	—	2	4	no	—
78	C₂×D₄×C₃₁	—	4	2	no	—
79	C₂×D₄×C₃₁	—	4	2	no	—
80	C₂×D₄×C₃₁	—	4	2	no	—
81	C₂×D₄×C₃₁	—	4	4	no	—
82	C₂×D₄×C₃₁	—	4	4	no	—
83	C₂×D₄×C₃₁	—	8	4	no	—
84	C₂×D₄×C₃₁	—	8	4	no	—
85	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
88	SD₁₆×C₃₁	—	8	8	no	—
89	D₈×C₃₁	—	4	4	no	—
90	D₈×C₃₁	—	4	4	no	—
91	D₈×C₃₁	—	4	4	no	—
92	D₈×C₃₁	—	4	4	no	—
93	M₁₆×C₃₁	—	4	2	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	2	4	no	—
97	C₂×C₈×C₃₁	SD₁₆	1	4	yes	complemented
98	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₄⋊C₄×C₃₁	—	2	1	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
108	C₂²⋊C₄×C₃₁	—	2	2	no	—
109	C₄²×C₃₁	—	2	2	no	—
110	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₃₁	#41	1	1	yes	characteristic
113	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
114	#19203	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	—	4	4	no	—
116	C₂³×C₃₁	—	8	4	no	—
117	C₂³×C₃₁	—	8	4	no	—
118	C₂³×C₃₁	—	8	4	no	—
119	C₂³×C₃₁	#121	1	1	yes	characteristic
120	Q₈×C₃₁	—	2	4	no	—
121	Q₈×C₃₁	—	4	2	no	—
122	D₄×C₃₁	—	4	4	no	—
123	D₄×C₃₁	—	4	4	no	—
124	D₄×C₃₁	—	4	8	no	—
125	D₄×C₃₁	—	8	4	no	—
126	D₄×C₃₁	—	8	4	no	—
127	D₄×C₃₁	—	8	4	no	—
128	D₄×C₃₁	—	8	4	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	#121	1	1	yes	characteristic
134	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
137	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
138	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
141	C₈×C₃₁	—	2	1	no	—
142	C₈×C₃₁	—	2	1	no	—
143	C₈×C₃₁	—	4	2	no	—
144	C₈×C₃₁	—	4	4	no	—
145	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
146	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
147	#3208	C₂×C₃₁	1	2	yes	complemented
148	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
149	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#1688	C₂×C₃₁	1	4	yes	complemented
151	#1668	C₂×C₃₁	1	4	yes	complemented
152	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#1545	C₂×C₃₁	1	2	yes	complemented
154	C₂²×C₃₁	—	2	1	no	—
155	C₂²×C₃₁	—	2	1	no	—
156	C₂²×C₃₁	—	8	4	no	—
157	C₂²×C₃₁	—	8	4	no	—
158	C₂²×C₃₁	—	8	4	no	—
159	C₂²×C₃₁	—	16	8	no	—
160	C₂²×C₃₁	#460	1	2	yes	—
161	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
162	C₄×C₃₁	—	2	1	no	—
163	C₄×C₃₁	—	2	1	no	—
164	C₄×C₃₁	—	4	2	no	—
165	C₄×C₃₁	—	4	2	no	—
166	C₄×C₃₁	#460	1	2	yes	—
167	C₄×C₃₁	#472	1	2	yes	—
168	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
169	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₄×C₃₁	#576	1	2	yes	—
171	C₂²×D₈	—	2	1	no	—
172	C₂²×D₈	—	2	1	no	—
173	#533	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×#127	C₂²×C₃₁	1	2	yes	—
175	#493	—	2	1	no	—
176	#493	—	2	1	no	—
177	#493	C₂²×C₃₁	1	2	yes	—
178	#493	C₂²×C₃₁	1	2	yes	—
179	#492	—	2	2	no	—
180	#486	—	2	2	no	—
181	#460	C₂²×C₃₁	1	8	yes	complemented
182	#450	—	2	4	no	—
183	#446	—	2	1	no	—
184	#446	C₂²×C₃₁	1	2	yes	complemented
185	C₈×Q₈	—	2	1	no	—
186	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
187	#431	C₂²×C₃₁	1	2	yes	—
188	#424	—	2	1	no	—
189	#423	C₂²×C₃₁	1	1	yes	characteristic
190	#405	—	2	1	no	—
191	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
192	#335	C₂²×C₃₁	1	2	yes	complemented
193	#334	C₂²×C₃₁	1	2	yes	complemented
194	#331	C₂²×C₃₁	1	8	yes	complemented
195	C₂×C₃₁	—	2	2	no	—
196	C₂×C₃₁	—	16	4	no	—
197	C₂×C₃₁	C₂×#460	1	1	yes	characteristic
198	C₂×C₃₁	#3058	1	1	yes	characteristic
199	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
200	C₂²×D₄	—	2	2	no	—
201	C₂²×D₄	—	2	2	no	—
202	C₂²×D₄	—	2	2	no	—
203	C₂×SD₁₆	—	2	8	no	—
204	C₂×D₈	—	2	2	no	—
205	C₂×D₈	—	2	2	no	—
206	C₂×D₈	—	2	2	no	—
207	C₂×D₈	—	2	2	no	—
208	C₂×D₈	—	4	4	no	—
209	C₂×D₈	—	4	4	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
213	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
214	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
215	#127	—	2	2	no	—
216	#127	—	2	2	no	—
217	#127	—	2	2	no	—
218	#127	C₂³×C₃₁	1	4	yes	—
219	#125	—	2	1	no	—
220	C₂²⋊Q₈	—	2	1	no	—
221	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
224	#117	—	2	1	no	—
225	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
230	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
231	#102	—	2	8	no	—
232	C₂²⋊C₈	—	2	2	no	—
233	#97	—	2	2	no	—
234	C₄×C₈	—	2	2	no	—
235	C₄×C₈	—	2	2	no	—
236	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
237	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
240	C₃₁	#19203	1	1	yes	characteristic, direct product factor, Sylow
241	C₂⁴	—	4	2	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	4	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂×D₄	—	8	4	no	—
253	C₂×D₄	—	8	4	no	—
254	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
257	SD₁₆	—	8	8	no	—
258	D₈	—	4	4	no	—
259	D₈	—	4	4	no	—
260	D₈	—	4	4	no	—
261	D₈	—	4	4	no	—
262	M₁₆	—	4	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	SD₁₆×C₃₁	1	4	yes	complemented
267	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
281	C₄²	#41×C₃₁	1	1	yes	characteristic
282	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
283	C₂³	—	4	4	no	—
284	C₂³	—	8	4	no	—
285	C₂³	—	8	4	no	—
286	C₂³	—	8	4	no	—
287	C₂³	C₃₁×#121	1	1	yes	characteristic
288	Q₈	—	2	4	no	—
289	Q₈	—	4	2	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	4	8	no	—
293	D₄	—	8	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	8	4	no	—
296	D₄	—	8	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
302	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
303	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
304	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
305	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
306	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
308	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	2	no	—
312	C₈	—	4	4	no	—
313	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
314	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	8	4	no	—
318	C₂²	—	8	4	no	—
319	C₂²	—	8	4	no	—
320	C₂²	—	16	8	no	—
321	C₂²	C₃₁×#460	1	2	yes	—
322	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	C₃₁×#460	1	2	yes	—
328	C₄	C₃₁×#472	1	2	yes	—
329	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
330	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
331	C₄	C₃₁×#576	1	2	yes	—
332	C₂	—	2	2	no	—
333	C₂	—	16	4	no	—
334	C₂	C₂×C₃₁×#460	1	1	yes	characteristic
335	C₂	C₃₁×#3058	1	1	yes	characteristic
336	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#493	C₂
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
1. C₃₁×#3208	C₂	semidirect
3. C₃₁×#2960	C₂	semidirect
4. C₃₁×#1688	C₂	semidirect
5. C₃₁×#1668	C₂	semidirect
6. C₃₁×#1564	C₂	semidirect
7. C₃₁×#1545	C₂	semidirect
18. C₃₁×#460	C₂²	semidirect
21. C₃₁×#446	C₂²	semidirect
23. C₈×Q₈×C₃₁	C₂²	semidirect
29. C₃₁×#335	C₂²	semidirect
30. C₃₁×#334	C₂²	semidirect
31. C₃₁×#331	C₂²	semidirect
97. C₂×C₈×C₃₁	SD₁₆	semidirect
114. #19203	C₃₁	direct (N × H)
147. #3208	C₂×C₃₁	semidirect
149. #2960	C₂×C₃₁	semidirect
150. #1688	C₂×C₃₁	semidirect
151. #1668	C₂×C₃₁	semidirect
152. #1564	C₂×C₃₁	semidirect
153. #1545	C₂×C₃₁	semidirect
181. #460	C₂²×C₃₁	semidirect
184. #446	C₂²×C₃₁	semidirect
186. C₈×Q₈	C₂²×C₃₁	semidirect
192. #335	C₂²×C₃₁	semidirect
193. #334	C₂²×C₃₁	semidirect
194. #331	C₂²×C₃₁	semidirect
240. C₃₁	#19203	direct (N × H)
266. 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×#3208	2
C₂×C₃₁×#493	2
C₃₁×#2960	2
C₃₁×#1688	2
C₃₁×#1668	2
C₃₁×#1564	2
C₃₁×#1545	2
#19203	31

Subgroups — C31×#19203 #21475962

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19203 #21475962