Subgroups — C₃₁×#19296 #21476055

Order 7936.

680 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3039	C₂	1	2	yes	maximal
3	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1694	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1693	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1546	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#137	—	2	2	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#500	—	2	1	no	—
12	C₃₁×#500	—	2	1	no	—
13	C₃₁×#500	C₂²	1	4	yes	—
14	C₃₁×#462	—	2	2	no	—
15	C₃₁×#462	C₂²	1	4	yes	—
16	C₃₁×#461	—	2	2	no	—
17	C₃₁×#461	C₂²	1	4	yes	—
18	C₃₁×#446	C₂²	1	2	yes	—
19	C₃₁×#436	C₂²	1	2	yes	—
20	C₃₁×#435	—	2	2	no	—
21	C₃₁×#431	—	2	1	no	—
22	C₃₁×#431	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#107	—	2	2	no	—
24	C₃₁×#423	C₂²	1	2	yes	—
25	C₃₁×#419	—	2	2	no	—
26	C₄×M₁₆×C₃₁	—	2	1	no	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#334	C₂²	1	4	yes	—
29	C₃₁×#332	C₂²	1	8	yes	—
30	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×Q₈×C₃₁	—	2	2	no	—
32	C₃₁×#137	—	4	8	no	—
33	C₂×Q₁₆×C₃₁	—	2	4	no	—
34	C₂×SD₁₆×C₃₁	—	2	4	no	—
35	C₂×M₁₆×C₃₁	—	2	2	no	—
36	C₂×M₁₆×C₃₁	—	2	2	no	—
37	C₂²×C₈×C₃₁	D₄	1	2	yes	—
38	C₃₁×C₄⋊Q₈	—	2	2	no	—
39	C₃₁×C₄⋊Q₈	—	2	2	no	—
40	C₃₁×C₄⋊Q₈	—	2	2	no	—
41	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
42	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
43	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₄×D₄×C₃₁	—	2	1	no	—
45	C₄×D₄×C₃₁	—	2	1	no	—
46	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
47	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#117	—	2	1	no	—
49	C₃₁×#117	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	—	2	1	no	—
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#107	—	2	4	no	—
55	C₃₁×#107	—	2	4	no	—
56	C₃₁×C₄⋊C₈	—	2	2	no	—
57	C₃₁×C₄⋊C₈	—	2	2	no	—
58	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
59	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
60	C₃₁×#103	—	2	4	no	—
61	C₃₁×#103	—	2	4	no	—
62	C₃₁×C₂²⋊C₈	—	2	1	no	—
63	C₃₁×C₂²⋊C₈	—	2	2	no	—
64	C₃₁×#97	—	2	2	no	—
65	C₄×C₈×C₃₁	—	2	1	no	—
66	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₃₁	C₂³	1	2	yes	—
69	#41×C₃₁	—	2	4	no	—
70	C₂×Q₈×C₃₁	—	2	4	no	—
71	C₂×Q₈×C₃₁	—	4	2	no	—
72	C₂×Q₈×C₃₁	—	4	4	no	—
73	C₂×Q₈×C₃₁	—	4	4	no	—
74	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂×D₄×C₃₁	—	2	1	no	—
76	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	—	2	1	no	—
78	C₂²×C₄×C₃₁	—	2	1	no	—
79	C₂²×C₄×C₃₁	—	2	1	no	—
80	C₂²×C₄×C₃₁	—	4	2	no	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
84	Q₁₆×C₃₁	—	4	8	no	—
85	SD₁₆×C₃₁	—	4	8	no	—
86	M₁₆×C₃₁	—	4	4	no	—
87	M₁₆×C₃₁	—	4	4	no	—
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	4	no	—
91	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
93	C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₄⋊C₄×C₃₁	—	2	1	no	—
95	C₄⋊C₄×C₃₁	—	2	4	no	—
96	C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₄⋊C₄×C₃₁	—	4	2	no	—
98	C₄⋊C₄×C₃₁	—	4	2	no	—
99	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²⋊C₄×C₃₁	—	2	1	no	—
102	C₂²⋊C₄×C₃₁	—	2	1	no	—
103	C₂²⋊C₄×C₃₁	—	2	1	no	—
104	C₄²×C₃₁	—	2	1	no	—
105	C₄²×C₃₁	—	2	1	no	—
106	C₄²×C₃₁	—	2	1	no	—
107	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₃₁	#41	1	1	yes	characteristic
110	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
111	#19296	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂³×C₃₁	—	2	1	no	—
113	C₂³×C₃₁	#121	1	1	yes	characteristic
114	Q₈×C₃₁	—	2	2	no	—
115	Q₈×C₃₁	—	4	4	no	—
116	Q₈×C₃₁	—	4	4	no	—
117	Q₈×C₃₁	—	8	4	no	—
118	D₄×C₃₁	—	2	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	1	no	—
123	C₂×C₄×C₃₁	—	2	1	no	—
124	C₂×C₄×C₃₁	—	2	1	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	—	4	2	no	—
128	C₂×C₄×C₃₁	—	4	4	no	—
129	C₂×C₄×C₃₁	—	8	4	no	—
130	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
131	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
132	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
134	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
137	C₈×C₃₁	—	2	2	no	—
138	C₈×C₃₁	—	4	4	no	—
139	C₈×C₃₁	—	4	2	no	—
140	C₈×C₃₁	#137	1	4	yes	complemented
141	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
142	#3039	C₂×C₃₁	1	2	yes	—
143	#2965	C₂×C₃₁	1	1	yes	characteristic
144	#1694	C₂×C₃₁	1	4	yes	complemented
145	#1693	C₂×C₃₁	1	4	yes	complemented
146	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#1546	C₂×C₃₁	1	2	yes	complemented
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	2	1	no	—
150	C₂²×C₃₁	—	4	1	no	—
151	C₂²×C₃₁	—	4	1	no	—
152	C₂²×C₃₁	—	4	1	no	—
153	C₂²×C₃₁	#469	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	1	no	—
158	C₄×C₃₁	—	4	1	no	—
159	C₄×C₃₁	—	4	1	no	—
160	C₄×C₃₁	—	8	4	no	—
161	C₄×C₃₁	#469	1	2	yes	—
162	C₄×C₃₁	#473	1	2	yes	—
163	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
164	C₄×C₃₁	C₂×#137	1	2	yes	—
165	C₂×#137	—	2	2	no	—
166	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
167	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
168	#500	—	2	1	no	—
169	#500	—	2	1	no	—
170	#500	C₂²×C₃₁	1	4	yes	—
171	#462	—	2	2	no	—
172	#462	C₂²×C₃₁	1	4	yes	—
173	#461	—	2	2	no	—
174	#461	C₂²×C₃₁	1	4	yes	—
175	#446	C₂²×C₃₁	1	2	yes	—
176	#436	C₂²×C₃₁	1	2	yes	—
177	#435	—	2	2	no	—
178	#431	—	2	1	no	—
179	#431	C₂²×C₃₁	1	1	yes	characteristic
180	C₂×#107	—	2	2	no	—
181	#423	C₂²×C₃₁	1	2	yes	—
182	#419	—	2	2	no	—
183	C₄×M₁₆	—	2	1	no	—
184	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
185	#334	C₂²×C₃₁	1	4	yes	—
186	#332	C₂²×C₃₁	1	8	yes	—
187	C₂×C₃₁	—	2	2	no	—
188	C₂×C₃₁	—	8	1	no	—
189	C₂×C₃₁	#3039	1	1	yes	characteristic
190	C₂×C₃₁	C₂×#469	1	1	yes	characteristic
191	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
192	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
193	C₂²×Q₈	—	2	2	no	—
194	#137	—	4	8	no	—
195	C₂×Q₁₆	—	2	4	no	—
196	C₂×SD₁₆	—	2	4	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂²×C₈	D₄×C₃₁	1	2	yes	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	2	2	no	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
204	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
205	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
206	C₄×D₄	—	2	1	no	—
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
209	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
210	#117	—	2	1	no	—
211	#117	D₄×C₃₁	1	1	yes	characteristic
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
216	#107	—	2	4	no	—
217	#107	—	2	4	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
221	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
222	#103	—	2	4	no	—
223	#103	—	2	4	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	#97	—	2	2	no	—
227	C₄×C₈	—	2	1	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	2	yes	—
231	C₃₁	#19296	1	1	yes	characteristic, direct product factor, Sylow
232	#41	—	2	4	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	4	no	—
236	C₂×Q₈	—	4	4	no	—
237	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
246	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
247	Q₁₆	—	4	8	no	—
248	SD₁₆	—	4	8	no	—
249	M₁₆	—	4	4	no	—
250	M₁₆	—	4	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
255	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
272	C₄²	#41×C₃₁	1	1	yes	characteristic
273	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
274	C₂³	—	2	1	no	—
275	C₂³	C₃₁×#121	1	1	yes	characteristic
276	Q₈	—	2	2	no	—
277	Q₈	—	4	4	no	—
278	Q₈	—	4	4	no	—
279	Q₈	—	8	4	no	—
280	D₄	—	2	2	no	—
281	D₄	—	4	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	8	4	no	—
292	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
293	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
294	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
295	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
296	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
298	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	4	4	no	—
301	C₈	—	4	2	no	—
302	C₈	C₃₁×#137	1	4	yes	complemented
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	C₃₁×#469	1	2	yes	—
309	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	8	4	no	—
316	C₄	C₃₁×#469	1	2	yes	—
317	C₄	C₃₁×#473	1	2	yes	—
318	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
319	C₄	C₂×C₃₁×#137	1	2	yes	—
320	C₂	—	2	2	no	—
321	C₂	—	8	1	no	—
322	C₂	C₃₁×#3039	1	1	yes	characteristic
323	C₂	C₂×C₃₁×#469	1	1	yes	characteristic
324	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	C₂×#41×C₃₁	D₄
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. C₂×C₃₁×#500	C₂	semidirect
4. C₃₁×#1694	C₂	semidirect
5. C₃₁×#1693	C₂	semidirect
6. C₃₁×#1566	C₂	semidirect
7. C₃₁×#1546	C₂	semidirect
111. #19296	C₃₁	direct (N × H)
140. C₈×C₃₁	#137	semidirect
141. C₂×#500	C₂×C₃₁	semidirect
144. #1694	C₂×C₃₁	semidirect
145. #1693	C₂×C₃₁	semidirect
146. #1566	C₂×C₃₁	semidirect
147. #1546	C₂×C₃₁	semidirect
231. C₃₁	#19296	direct (N × H)
302. C₈	C₃₁×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#500	2
C₃₁×#3039	2
C₃₁×#2965	2
C₃₁×#1694	2
C₃₁×#1693	2
C₃₁×#1566	2
C₃₁×#1546	2
#19296	31

Subgroups — C31×#19296 #21476055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19296 #21476055