Subgroups — C₃₁×#19192 #21475951

Order 7936.

674 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3206	C₂	1	2	yes	maximal
2	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1693	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1673	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1549	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#131	—	2	1	no	—
9	C₃₁×#532	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃₁×#500	—	2	1	no	—
12	C₃₁×#500	—	2	1	no	—
13	C₃₁×#500	C₂²	1	2	yes	—
14	C₃₁×#500	C₂²	1	2	yes	—
15	C₃₁×#492	—	2	2	no	—
16	C₃₁×#487	—	2	2	no	—
17	C₃₁×#461	C₂²	1	8	yes	—
18	C₃₁×#448	—	2	4	no	—
19	C₃₁×#436	—	2	1	no	—
20	C₃₁×#436	C₂²	1	2	yes	—
21	C₈×D₄×C₃₁	—	2	1	no	—
22	C₈×D₄×C₃₁	C₂²	1	2	yes	—
23	C₃₁×#431	C₂²	1	2	yes	—
24	C₂×C₃₁×#107	—	2	1	no	—
25	C₂×C₃₁×#107	—	2	1	no	—
26	C₃₁×#423	C₂²	1	1	yes	characteristic
27	C₃₁×#408	—	2	1	no	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#335	C₂²	1	2	yes	—
30	C₃₁×#334	C₂²	1	2	yes	—
31	C₃₁×#332	C₂²	1	8	yes	—
32	C₂×#41×C₃₁	—	2	1	no	—
33	C₂²×Q₈×C₃₁	—	2	2	no	—
34	C₂×SD₁₆×C₃₁	—	2	8	no	—
35	C₃₁×#131	—	4	4	no	—
36	C₂×M₁₆×C₃₁	—	2	1	no	—
37	C₂×M₁₆×C₃₁	—	2	2	no	—
38	C₂²×C₈×C₃₁	—	2	2	no	—
39	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₃₁×C₄⋊Q₈	—	2	2	no	—
42	C₃₁×C₄⋊Q₈	—	2	2	no	—
43	C₃₁×C₄⋊Q₈	—	2	2	no	—
44	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
45	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
46	C₃₁×#127	D₄	1	1	yes	characteristic
47	C₃₁×#124	—	2	1	no	—
48	C₃₁×#121	—	2	1	no	—
49	C₃₁×#121	D₄	1	2	yes	—
50	C₄×D₄×C₃₁	—	2	1	no	—
51	C₄×D₄×C₃₁	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×#107	—	2	2	no	—
56	C₃₁×#107	—	2	2	no	—
57	C₃₁×#107	—	2	2	no	—
58	C₃₁×#107	—	2	2	no	—
59	C₃₁×C₄⋊C₈	—	2	1	no	—
60	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
61	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
62	C₃₁×#103	—	2	8	no	—
63	C₃₁×C₂²⋊C₈	—	2	2	no	—
64	C₃₁×C₂²⋊C₈	—	2	2	no	—
65	C₃₁×C₂²⋊C₈	—	2	2	no	—
66	C₄×C₈×C₃₁	—	2	2	no	—
67	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	#41×C₃₁	—	4	2	no	—
72	#41×C₃₁	—	4	2	no	—
73	C₂×Q₈×C₃₁	—	2	1	no	—
74	C₂×Q₈×C₃₁	—	2	4	no	—
75	C₂×Q₈×C₃₁	—	4	2	no	—
76	C₂×Q₈×C₃₁	—	8	4	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	2	1	no	—
79	C₂×D₄×C₃₁	—	2	1	no	—
80	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	4	2	no	—
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
86	SD₁₆×C₃₁	—	8	8	no	—
87	M₁₆×C₃₁	—	4	2	no	—
88	M₁₆×C₃₁	—	4	4	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	4	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	—	4	4	no	—
93	C₂×C₈×C₃₁	SD₁₆	1	4	yes	complemented
94	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₄⋊C₄×C₃₁	—	2	4	no	—
100	C₄⋊C₄×C₃₁	—	2	4	no	—
101	C₄⋊C₄×C₃₁	—	4	2	no	—
102	C₄⋊C₄×C₃₁	—	4	2	no	—
103	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²⋊C₄×C₃₁	—	2	2	no	—
105	C₂²⋊C₄×C₃₁	—	2	2	no	—
106	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₃₁	#41	1	1	yes	characteristic
109	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
110	#19192	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₃₁	—	2	2	no	—
112	C₂³×C₃₁	#121	1	1	yes	characteristic
113	Q₈×C₃₁	—	4	2	no	—
114	Q₈×C₃₁	—	4	8	no	—
115	Q₈×C₃₁	—	8	4	no	—
116	D₄×C₃₁	—	2	4	no	—
117	D₄×C₃₁	—	4	4	no	—
118	D₄×C₃₁	—	4	2	no	—
119	D₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	4	no	—
126	C₂×C₄×C₃₁	—	8	4	no	—
127	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
128	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
129	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
131	C₂×C₄×C₃₁	#137	1	2	yes	—
132	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
135	C₈×C₃₁	—	2	1	no	—
136	C₈×C₃₁	—	2	1	no	—
137	C₈×C₃₁	—	4	2	no	—
138	C₈×C₃₁	—	4	4	no	—
139	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
140	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
141	#3206	C₂×C₃₁	1	2	yes	—
142	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#2931	C₂×C₃₁	1	1	yes	characteristic
144	#1693	C₂×C₃₁	1	4	yes	complemented
145	#1673	C₂×C₃₁	1	4	yes	complemented
146	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#1549	C₂×C₃₁	1	2	yes	complemented
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	2	1	no	—
150	C₂²×C₃₁	—	4	2	no	—
151	C₂²×C₃₁	—	4	2	no	—
152	C₂²×C₃₁	—	4	2	no	—
153	C₂²×C₃₁	#461	1	2	yes	—
154	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
155	C₄×C₃₁	—	2	1	no	—
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	4	2	no	—
158	C₄×C₃₁	—	8	4	no	—
159	C₄×C₃₁	#461	1	2	yes	—
160	C₄×C₃₁	#473	1	2	yes	—
161	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
162	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
163	C₄×C₃₁	#576	1	2	yes	—
164	C₂×#131	—	2	1	no	—
165	#532	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
167	#500	—	2	1	no	—
168	#500	—	2	1	no	—
169	#500	C₂²×C₃₁	1	2	yes	—
170	#500	C₂²×C₃₁	1	2	yes	—
171	#492	—	2	2	no	—
172	#487	—	2	2	no	—
173	#461	C₂²×C₃₁	1	8	yes	—
174	#448	—	2	4	no	—
175	#436	—	2	1	no	—
176	#436	C₂²×C₃₁	1	2	yes	—
177	C₈×D₄	—	2	1	no	—
178	C₈×D₄	C₂²×C₃₁	1	2	yes	—
179	#431	C₂²×C₃₁	1	2	yes	—
180	C₂×#107	—	2	1	no	—
181	C₂×#107	—	2	1	no	—
182	#423	C₂²×C₃₁	1	1	yes	characteristic
183	#408	—	2	1	no	—
184	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
185	#335	C₂²×C₃₁	1	2	yes	—
186	#334	C₂²×C₃₁	1	2	yes	—
187	#332	C₂²×C₃₁	1	8	yes	—
188	C₂×C₃₁	—	2	2	no	—
189	C₂×C₃₁	—	8	2	no	—
190	C₂×C₃₁	C₂×#461	1	1	yes	characteristic
191	C₂×C₃₁	#3058	1	1	yes	characteristic
192	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
193	C₂×#41	—	2	1	no	—
194	C₂²×Q₈	—	2	2	no	—
195	C₂×SD₁₆	—	2	8	no	—
196	#131	—	4	4	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
201	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
206	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
207	#127	D₄×C₃₁	1	1	yes	characteristic
208	#124	—	2	1	no	—
209	#121	—	2	1	no	—
210	#121	D₄×C₃₁	1	2	yes	—
211	C₄×D₄	—	2	1	no	—
212	C₄×D₄	C₂³×C₃₁	1	2	yes	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
216	#107	—	2	2	no	—
217	#107	—	2	2	no	—
218	#107	—	2	2	no	—
219	#107	—	2	2	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
222	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
223	#103	—	2	8	no	—
224	C₂²⋊C₈	—	2	2	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	2	2	no	—
227	C₄×C₈	—	2	2	no	—
228	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
229	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
232	C₃₁	#19192	1	1	yes	characteristic, direct product factor, Sylow
233	#41	—	4	2	no	—
234	#41	—	4	2	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	4	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	8	4	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
247	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
248	SD₁₆	—	8	8	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	4	no	—
255	C₂×C₈	SD₁₆×C₃₁	1	4	yes	complemented
256	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
270	C₄²	#41×C₃₁	1	1	yes	characteristic
271	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
272	C₂³	—	2	2	no	—
273	C₂³	C₃₁×#121	1	1	yes	characteristic
274	Q₈	—	4	2	no	—
275	Q₈	—	4	8	no	—
276	Q₈	—	8	4	no	—
277	D₄	—	2	4	no	—
278	D₄	—	4	4	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	8	4	no	—
288	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
289	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
290	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
291	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
292	C₂×C₄	C₃₁×#137	1	2	yes	—
293	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	4	2	no	—
299	C₈	—	4	4	no	—
300	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
301	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	C₃₁×#461	1	2	yes	—
308	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	4	no	—
313	C₄	C₃₁×#461	1	2	yes	—
314	C₄	C₃₁×#473	1	2	yes	—
315	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
316	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
317	C₄	C₃₁×#576	1	2	yes	—
318	C₂	—	2	2	no	—
319	C₂	—	8	2	no	—
320	C₂	C₂×C₃₁×#461	1	1	yes	characteristic
321	C₂	C₃₁×#3058	1	1	yes	characteristic
322	C₂	C₂×C₃₁×#473	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
2. C₂×C₃₁×#500	C₂	semidirect
4. C₃₁×#1693	C₂	semidirect
5. C₃₁×#1673	C₂	semidirect
6. C₃₁×#1564	C₂	semidirect
7. C₃₁×#1549	C₂	semidirect
93. C₂×C₈×C₃₁	SD₁₆	semidirect
110. #19192	C₃₁	direct (N × H)
142. C₂×#500	C₂×C₃₁	semidirect
144. #1693	C₂×C₃₁	semidirect
145. #1673	C₂×C₃₁	semidirect
146. #1564	C₂×C₃₁	semidirect
147. #1549	C₂×C₃₁	semidirect
232. C₃₁	#19192	direct (N × H)
255. 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×#3206	2
C₂×C₃₁×#500	2
C₃₁×#2931	2
C₃₁×#1693	2
C₃₁×#1673	2
C₃₁×#1564	2
C₃₁×#1549	2
#19192	31

Subgroups — C31×#19192 #21475951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19192 #21475951