Subgroups — C₃₁×#33064 #21489823

Order 7936.

798 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3080	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2875	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1998	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1931	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1907	C₂	1	4	yes	maximal, complemented
9	C₃₁×#1838	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
11	C₂×C₃₁×#131	C₂²	1	2	yes	—
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
14	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#490	—	2	1	no	—
16	C₃₁×#490	—	2	1	no	—
17	C₃₁×#490	—	2	2	no	—
18	C₃₁×#479	—	2	2	no	—
19	C₃₁×#478	—	2	2	no	—
20	C₃₁×#460	—	2	2	no	—
21	C₃₁×#459	—	2	2	no	—
22	C₃₁×#428	C₂²	1	2	yes	—
23	C₃₁×#423	C₂²	1	1	yes	characteristic
24	C₃₁×#423	C₂²	1	1	yes	characteristic
25	C₃₁×#418	C₂²	1	2	yes	—
26	C₃₁×#417	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	4	yes	—
28	C₂×C₃₁×#100	C₂²	1	4	yes	—
29	C₃₁×#405	—	2	2	no	—
30	C₃₁×#405	C₂²	1	2	yes	—
31	C₄×M₁₆×C₃₁	—	2	1	no	—
32	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×#97	—	2	1	no	—
34	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
35	C₃₁×#343	C₂²	1	4	yes	—
36	C₃₁×#341	C₂²	1	4	yes	—
37	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×D₄×C₃₁	—	2	2	no	—
39	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
40	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
41	C₂×SD₁₆×C₃₁	—	2	2	no	—
42	C₂×D₈×C₃₁	—	2	2	no	—
43	C₃₁×#131	—	2	4	no	—
44	C₃₁×#131	—	2	4	no	—
45	C₂×M₁₆×C₃₁	—	2	2	no	—
46	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
47	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
48	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₃₁×#127	—	2	2	no	—
52	C₃₁×#127	—	2	1	no	—
53	C₃₁×#127	—	2	1	no	—
54	C₃₁×#125	—	2	2	no	—
55	C₃₁×#125	—	2	1	no	—
56	C₃₁×#125	—	2	1	no	—
57	C₄×Q₈×C₃₁	D₄	1	2	yes	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₄×D₄×C₃₁	D₄	1	2	yes	—
60	C₃₁×#117	—	2	1	no	—
61	C₃₁×#117	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄²×C₃₁	—	2	1	no	—
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×#107	—	2	2	no	—
67	C₃₁×#106	—	2	2	no	—
68	C₃₁×C₄⋊C₈	—	2	2	no	—
69	C₃₁×C₄⋊C₈	—	2	2	no	—
70	C₃₁×#103	—	2	2	no	—
71	C₃₁×#102	—	2	2	no	—
72	C₃₁×#102	—	2	4	no	—
73	C₃₁×#102	—	2	4	no	—
74	C₃₁×#102	—	2	4	no	—
75	C₃₁×#100	—	2	8	no	—
76	C₃₁×#97	—	2	2	no	—
77	C₃₁×#97	—	2	1	no	—
78	C₃₁×#97	D₄	1	1	yes	characteristic
79	C₃₁×#97	D₄	1	1	yes	characteristic
80	C₃₁×#97	D₄	1	2	yes	—
81	C₄×C₈×C₃₁	—	2	2	no	—
82	C₄×C₈×C₃₁	—	2	1	no	—
83	C₂⁴×C₃₁	—	2	2	no	—
84	#41×C₃₁	—	2	2	no	—
85	#41×C₃₁	#41	1	4	yes	—
86	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂×D₄×C₃₁	—	2	1	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	2	4	no	—
91	C₂×D₄×C₃₁	—	2	4	no	—
92	C₂×D₄×C₃₁	—	4	2	no	—
93	C₂×D₄×C₃₁	—	4	2	no	—
94	C₂×D₄×C₃₁	—	8	4	no	—
95	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	#41	1	2	yes	—
100	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
101	SD₁₆×C₃₁	—	4	4	no	—
102	D₈×C₃₁	—	4	4	no	—
103	M₁₆×C₃₁	—	2	8	no	—
104	M₁₆×C₃₁	—	2	2	no	—
105	M₁₆×C₃₁	—	2	4	no	—
106	M₁₆×C₃₁	—	2	4	no	—
107	M₁₆×C₃₁	#41	1	4	yes	—
108	C₂×C₈×C₃₁	—	2	8	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	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
116	C₄⋊C₄×C₃₁	—	2	2	no	—
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	4	2	no	—
120	C₄⋊C₄×C₃₁	—	4	2	no	—
121	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²⋊C₄×C₃₁	—	2	1	no	—
123	C₂²⋊C₄×C₃₁	—	2	2	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	1	no	—
126	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₃₁	C₂×D₄	1	2	yes	—
129	#33064	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂³×C₃₁	—	2	1	no	—
131	C₂³×C₃₁	—	4	2	no	—
132	C₂³×C₃₁	—	4	4	no	—
133	C₂³×C₃₁	—	8	4	no	—
134	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
135	Q₈×C₃₁	—	2	1	no	—
136	Q₈×C₃₁	#124	1	2	yes	—
137	D₄×C₃₁	—	2	4	no	—
138	D₄×C₃₁	—	2	1	no	—
139	D₄×C₃₁	—	4	4	no	—
140	D₄×C₃₁	—	4	2	no	—
141	D₄×C₃₁	—	4	2	no	—
142	D₄×C₃₁	—	8	4	no	—
143	D₄×C₃₁	—	8	4	no	—
144	D₄×C₃₁	#124	1	2	yes	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	2	4	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	4	2	no	—
149	C₂×C₄×C₃₁	—	4	2	no	—
150	C₂×C₄×C₃₁	—	8	2	no	—
151	C₂×C₄×C₃₁	—	8	2	no	—
152	C₂×C₄×C₃₁	#121	1	2	yes	—
153	C₂×C₄×C₃₁	#121	1	2	yes	—
154	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
157	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
158	C₈×C₃₁	—	2	2	no	—
159	C₈×C₃₁	—	2	4	no	—
160	C₈×C₃₁	—	2	2	no	—
161	C₈×C₃₁	—	4	2	no	—
162	C₈×C₃₁	—	4	2	no	—
163	C₂×#490	C₂×C₃₁	1	1	yes	characteristic, complemented
164	#3080	C₂×C₃₁	1	1	yes	characteristic, complemented
165	#3038	C₂×C₃₁	1	1	yes	characteristic
166	#2875	C₂×C₃₁	1	1	yes	characteristic, complemented
167	#1998	C₂×C₃₁	1	1	yes	characteristic, complemented
168	#1965	C₂×C₃₁	1	1	yes	characteristic, complemented
169	#1931	C₂×C₃₁	1	4	yes	complemented
170	#1907	C₂×C₃₁	1	4	yes	complemented
171	#1838	C₂×C₃₁	1	1	yes	characteristic, complemented
172	C₂²×C₃₁	—	2	2	no	—
173	C₂²×C₃₁	—	2	2	no	—
174	C₂²×C₃₁	—	4	2	no	—
175	C₂²×C₃₁	—	8	2	no	—
176	C₂²×C₃₁	—	8	4	no	—
177	C₂²×C₃₁	—	16	4	no	—
178	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
179	C₂²×C₃₁	#529	1	2	yes	—
180	C₄×C₃₁	—	2	2	no	—
181	C₄×C₃₁	—	2	2	no	—
182	C₄×C₃₁	—	2	2	no	—
183	C₄×C₃₁	—	4	2	no	—
184	C₄×C₃₁	—	8	2	no	—
185	C₄×C₃₁	#525	1	2	yes	—
186	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
187	C₄×C₃₁	#532	1	1	yes	characteristic
188	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
189	C₂×#131	C₂²×C₃₁	1	2	yes	—
190	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
191	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
192	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
193	#490	—	2	1	no	—
194	#490	—	2	1	no	—
195	#490	—	2	2	no	—
196	#479	—	2	2	no	—
197	#478	—	2	2	no	—
198	#460	—	2	2	no	—
199	#459	—	2	2	no	—
200	#428	C₂²×C₃₁	1	2	yes	—
201	#423	C₂²×C₃₁	1	1	yes	characteristic
202	#423	C₂²×C₃₁	1	1	yes	characteristic
203	#418	C₂²×C₃₁	1	2	yes	—
204	#417	C₂²×C₃₁	1	2	yes	—
205	C₂×#102	C₂²×C₃₁	1	4	yes	—
206	C₂×#100	C₂²×C₃₁	1	4	yes	—
207	#405	—	2	2	no	—
208	#405	C₂²×C₃₁	1	2	yes	—
209	C₄×M₁₆	—	2	1	no	—
210	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
211	C₂×#97	—	2	1	no	—
212	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
213	#343	C₂²×C₃₁	1	4	yes	—
214	#341	C₂²×C₃₁	1	4	yes	—
215	C₂×C₃₁	—	2	2	no	—
216	C₂×C₃₁	—	4	2	no	—
217	C₂×C₃₁	—	16	2	no	—
218	C₂×C₃₁	#2850	1	1	yes	characteristic
219	C₂×C₃₁	#3068	1	2	yes	—
220	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
221	C₂²×D₄	—	2	2	no	—
222	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
223	AGL(1,ℤ/8)	—	2	8	no	—
224	C₂×SD₁₆	—	2	2	no	—
225	C₂×D₈	—	2	2	no	—
226	#131	—	2	4	no	—
227	#131	—	2	4	no	—
228	C₂×M₁₆	—	2	2	no	—
229	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
230	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
231	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
232	C₂²×C₈	—	2	2	no	—
233	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
234	#127	—	2	2	no	—
235	#127	—	2	1	no	—
236	#127	—	2	1	no	—
237	#125	—	2	2	no	—
238	#125	—	2	1	no	—
239	#125	—	2	1	no	—
240	C₄×Q₈	D₄×C₃₁	1	2	yes	—
241	C₄×D₄	—	2	2	no	—
242	C₄×D₄	D₄×C₃₁	1	2	yes	—
243	#117	—	2	1	no	—
244	#117	C₂³×C₃₁	1	1	yes	characteristic
245	C₂×C₄⋊C₄	—	2	2	no	—
246	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
247	C₂×C₄²	—	2	1	no	—
248	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
249	#107	—	2	2	no	—
250	#106	—	2	2	no	—
251	C₄⋊C₈	—	2	2	no	—
252	C₄⋊C₈	—	2	2	no	—
253	#103	—	2	2	no	—
254	#102	—	2	2	no	—
255	#102	—	2	4	no	—
256	#102	—	2	4	no	—
257	#102	—	2	4	no	—
258	#100	—	2	8	no	—
259	#97	—	2	2	no	—
260	#97	—	2	1	no	—
261	#97	D₄×C₃₁	1	1	yes	characteristic
262	#97	D₄×C₃₁	1	1	yes	characteristic
263	#97	D₄×C₃₁	1	2	yes	—
264	C₄×C₈	—	2	2	no	—
265	C₄×C₈	—	2	1	no	—
266	C₃₁	#33064	1	1	yes	characteristic, direct product factor, Sylow
267	C₂⁴	—	2	2	no	—
268	#41	—	2	2	no	—
269	#41	#41×C₃₁	1	4	yes	—
270	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	4	no	—
275	C₂×D₄	—	2	4	no	—
276	C₂×D₄	—	4	2	no	—
277	C₂×D₄	—	4	2	no	—
278	C₂×D₄	—	8	4	no	—
279	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	2	2	no	—
283	C₂²×C₄	#41×C₃₁	1	2	yes	—
284	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
285	SD₁₆	—	4	4	no	—
286	D₈	—	4	4	no	—
287	M₁₆	—	2	8	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	4	no	—
290	M₁₆	—	2	4	no	—
291	M₁₆	#41×C₃₁	1	4	yes	—
292	C₂×C₈	—	2	8	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
299	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
306	C₂²⋊C₄	—	2	1	no	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	1	no	—
310	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
312	C₄²	C₂×D₄×C₃₁	1	2	yes	—
313	C₂³	—	2	1	no	—
314	C₂³	—	4	2	no	—
315	C₂³	—	4	4	no	—
316	C₂³	—	8	4	no	—
317	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
318	Q₈	—	2	1	no	—
319	Q₈	C₃₁×#124	1	2	yes	—
320	D₄	—	2	4	no	—
321	D₄	—	2	1	no	—
322	D₄	—	4	4	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	8	4	no	—
326	D₄	—	8	4	no	—
327	D₄	C₃₁×#124	1	2	yes	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	2	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	C₃₁×#121	1	2	yes	—
336	C₂×C₄	C₃₁×#121	1	2	yes	—
337	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
338	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
339	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
340	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
341	C₈	—	2	2	no	—
342	C₈	—	2	4	no	—
343	C₈	—	2	2	no	—
344	C₈	—	4	2	no	—
345	C₈	—	4	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	8	2	no	—
350	C₂²	—	8	4	no	—
351	C₂²	—	16	4	no	—
352	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
353	C₂²	C₃₁×#529	1	2	yes	—
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	8	2	no	—
359	C₄	C₃₁×#525	1	2	yes	—
360	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
361	C₄	C₃₁×#532	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	4	2	no	—
364	C₂	—	16	2	no	—
365	C₂	C₃₁×#2850	1	1	yes	characteristic
366	C₂	C₃₁×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3038	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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×#490	C₂	semidirect
2. C₃₁×#3080	C₂	semidirect
4. C₃₁×#2875	C₂	semidirect
5. C₃₁×#1998	C₂	semidirect
6. C₃₁×#1965	C₂	semidirect
7. C₃₁×#1931	C₂	semidirect
8. C₃₁×#1907	C₂	semidirect
9. C₃₁×#1838	C₂	semidirect
129. #33064	C₃₁	direct (N × H)
163. C₂×#490	C₂×C₃₁	semidirect
164. #3080	C₂×C₃₁	semidirect
166. #2875	C₂×C₃₁	semidirect
167. #1998	C₂×C₃₁	semidirect
168. #1965	C₂×C₃₁	semidirect
169. #1931	C₂×C₃₁	semidirect
170. #1907	C₂×C₃₁	semidirect
171. #1838	C₂×C₃₁	semidirect
266. C₃₁	#33064	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#490	2
C₃₁×#3080	2
C₃₁×#3038	2
C₃₁×#2875	2
C₃₁×#1998	2
C₃₁×#1965	2
C₃₁×#1931	2
C₃₁×#1907	2
C₃₁×#1838	2
#33064	31

Subgroups — C31×#33064 #21489823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33064 #21489823