Subgroups — C₃₁×#33239 #21489998

Order 7936.

1050 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	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2862	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2858	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1968	C₂	1	8	yes	maximal, complemented
8	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#124	—	2	2	no	—
15	C₃₁×#498	—	2	4	no	—
16	C₃₁×#492	—	2	4	no	—
17	C₃₁×#487	—	2	4	no	—
18	C₃₁×#486	—	2	4	no	—
19	C₂×C₃₁×#106	C₂²	1	4	yes	—
20	C₂×C₃₁×#103	C₂²	1	4	yes	—
21	C₂×C₃₁×#102	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#391	—	2	2	no	—
26	C₃₁×#391	C₂²	1	4	yes	—
27	C₃₁×#390	—	2	2	no	—
28	C₃₁×#390	C₂²	1	4	yes	—
29	C₃₁×#382	—	2	2	no	—
30	C₄³×C₃₁	C₂²	1	1	yes	characteristic
31	C₂²×Q₈×C₃₁	—	2	1	no	—
32	C₂²×Q₈×C₃₁	—	2	1	no	—
33	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
34	C₂²×D₄×C₃₁	—	2	1	no	—
35	C₂²×D₄×C₃₁	—	2	1	no	—
36	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
37	C₂×SD₁₆×C₃₁	—	2	8	no	—
38	C₂×SD₁₆×C₃₁	—	2	16	no	—
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
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	—	2	2	no	—
45	C₃₁×C₄⋊Q₈	—	2	2	no	—
46	C₃₁×C₄⋊Q₈	—	2	2	no	—
47	C₃₁×#127	—	2	2	no	—
48	C₃₁×#127	—	2	2	no	—
49	C₃₁×#124	—	4	8	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
57	C₂×C₄²×C₃₁	D₄	1	2	yes	—
58	C₂×C₄²×C₃₁	D₄	1	2	yes	—
59	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×#106	—	2	8	no	—
62	C₃₁×#103	—	2	8	no	—
63	C₃₁×#102	—	2	8	no	—
64	C₄×C₈×C₃₁	—	2	8	no	—
65	C₄×C₈×C₃₁	D₄	1	4	yes	—
66	C₄×C₈×C₃₁	D₄	1	4	yes	—
67	C₃₁×#95	—	2	8	no	—
68	C₂⁴×C₃₁	—	2	2	no	—
69	C₂×Q₈×C₃₁	—	2	2	no	—
70	C₂×Q₈×C₃₁	—	2	2	no	—
71	C₂×Q₈×C₃₁	—	2	8	no	—
72	C₂×Q₈×C₃₁	—	4	2	no	—
73	C₂×Q₈×C₃₁	—	4	2	no	—
74	C₂×Q₈×C₃₁	—	4	4	no	—
75	C₂×Q₈×C₃₁	—	4	4	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	2	8	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	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
90	SD₁₆×C₃₁	—	4	16	no	—
91	C₂×C₈×C₃₁	—	2	4	no	—
92	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₃₁	#41	1	8	yes	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	2	no	—
99	C₄⋊C₄×C₃₁	—	4	2	no	—
100	C₄⋊C₄×C₃₁	—	4	2	no	—
101	C₄⋊C₄×C₃₁	—	4	2	no	—
102	C₄⋊C₄×C₃₁	—	4	2	no	—
103	C₂²⋊C₄×C₃₁	—	4	16	no	—
104	C₄²×C₃₁	—	2	4	no	—
105	C₄²×C₃₁	—	2	4	no	—
106	C₄²×C₃₁	SD₁₆	1	8	yes	complemented
107	C₄²×C₃₁	C₂×D₄	1	2	yes	—
108	C₄²×C₃₁	C₂×D₄	1	2	yes	—
109	#33239	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₃₁	—	4	2	no	—
111	C₂³×C₃₁	—	4	4	no	—
112	C₂³×C₃₁	—	4	4	no	—
113	C₂³×C₃₁	—	4	4	no	—
114	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₃₁	—	4	4	no	—
116	Q₈×C₃₁	—	4	4	no	—
117	Q₈×C₃₁	—	8	4	no	—
118	Q₈×C₃₁	—	8	4	no	—
119	D₄×C₃₁	—	4	4	no	—
120	D₄×C₃₁	—	4	4	no	—
121	D₄×C₃₁	—	8	4	no	—
122	D₄×C₃₁	—	8	4	no	—
123	C₂×C₄×C₃₁	—	2	4	no	—
124	C₂×C₄×C₃₁	—	2	4	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	—	4	4	no	—
130	C₂×C₄×C₃₁	—	4	4	no	—
131	C₂×C₄×C₃₁	#127	1	2	yes	—
132	C₂×C₄×C₃₁	#127	1	2	yes	—
133	C₂×C₄×C₃₁	C₂×SD₁₆	1	8	yes	—
134	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
135	C₂×C₄×C₃₁	#135	1	4	yes	—
136	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
139	C₈×C₃₁	—	2	4	no	—
140	C₈×C₃₁	#124	1	8	yes	complemented
141	C₂×#498	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂×#492	C₂×C₃₁	1	1	yes	characteristic
143	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
144	C₂×#486	C₂×C₃₁	1	1	yes	characteristic
145	#2862	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#2858	C₂×C₃₁	1	1	yes	characteristic
147	#1968	C₂×C₃₁	1	8	yes	complemented
148	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
149	C₂²×C₃₁	—	8	2	no	—
150	C₂²×C₃₁	—	8	2	no	—
151	C₂²×C₃₁	—	8	2	no	—
152	C₂²×C₃₁	—	8	8	no	—
153	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
154	C₂²×C₃₁	#532	1	2	yes	—
155	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
156	C₂²×C₃₁	C₂×#135	1	2	yes	—
157	C₄×C₃₁	—	2	8	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	—	8	2	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	#492	1	4	yes	—
162	C₄×C₃₁	#495	1	4	yes	—
163	C₄×C₃₁	C₂×#124	1	2	yes	—
164	C₄×C₃₁	#532	1	2	yes	—
165	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	—
166	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
168	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×#124	—	2	2	no	—
171	#498	—	2	4	no	—
172	#492	—	2	4	no	—
173	#487	—	2	4	no	—
174	#486	—	2	4	no	—
175	C₂×#106	C₂²×C₃₁	1	4	yes	—
176	C₂×#103	C₂²×C₃₁	1	4	yes	—
177	C₂×#102	C₂²×C₃₁	1	4	yes	—
178	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
179	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
181	#391	—	2	2	no	—
182	#391	C₂²×C₃₁	1	4	yes	—
183	#390	—	2	2	no	—
184	#390	C₂²×C₃₁	1	4	yes	—
185	#382	—	2	2	no	—
186	C₄³	C₂²×C₃₁	1	1	yes	characteristic
187	C₂×C₃₁	—	16	2	no	—
188	C₂×C₃₁	#2858	1	1	yes	characteristic
189	C₂×C₃₁	C₂×#492	1	1	yes	characteristic
190	C₂×C₃₁	C₂×#495	1	1	yes	characteristic
191	C₂×C₃₁	#3167	1	4	yes	—
192	C₂²×Q₈	—	2	1	no	—
193	C₂²×Q₈	—	2	1	no	—
194	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
195	C₂²×D₄	—	2	1	no	—
196	C₂²×D₄	—	2	1	no	—
197	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
198	C₂×SD₁₆	—	2	8	no	—
199	C₂×SD₁₆	—	2	16	no	—
200	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
201	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
202	C₄⋊Q₈	—	2	2	no	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	2	no	—
205	C₄⋊Q₈	—	2	2	no	—
206	C₄⋊Q₈	—	2	2	no	—
207	C₄⋊Q₈	—	2	2	no	—
208	#127	—	2	2	no	—
209	#127	—	2	2	no	—
210	#124	—	4	8	no	—
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
217	C₂×C₂²⋊C₄	—	2	8	no	—
218	C₂×C₄²	D₄×C₃₁	1	2	yes	—
219	C₂×C₄²	D₄×C₃₁	1	2	yes	—
220	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
221	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
222	#106	—	2	8	no	—
223	#103	—	2	8	no	—
224	#102	—	2	8	no	—
225	C₄×C₈	—	2	8	no	—
226	C₄×C₈	D₄×C₃₁	1	4	yes	—
227	C₄×C₈	D₄×C₃₁	1	4	yes	—
228	#95	—	2	8	no	—
229	C₃₁	#33239	1	1	yes	characteristic, direct product factor, Sylow
230	C₂⁴	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	8	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×Q₈	—	4	4	no	—
237	C₂×Q₈	—	4	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	8	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	4	4	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
249	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
252	SD₁₆	—	4	16	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
255	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
256	C₂×C₈	#41×C₃₁	1	8	yes	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	16	no	—
266	C₄²	—	2	4	no	—
267	C₄²	—	2	4	no	—
268	C₄²	SD₁₆×C₃₁	1	8	yes	complemented
269	C₄²	C₂×D₄×C₃₁	1	2	yes	—
270	C₄²	C₂×D₄×C₃₁	1	2	yes	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	4	no	—
273	C₂³	—	4	4	no	—
274	C₂³	—	4	4	no	—
275	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
276	Q₈	—	4	4	no	—
277	Q₈	—	4	4	no	—
278	Q₈	—	8	4	no	—
279	Q₈	—	8	4	no	—
280	D₄	—	4	4	no	—
281	D₄	—	4	4	no	—
282	D₄	—	8	4	no	—
283	D₄	—	8	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	C₃₁×#127	1	2	yes	—
293	C₂×C₄	C₃₁×#127	1	2	yes	—
294	C₂×C₄	C₂×SD₁₆×C₃₁	1	8	yes	—
295	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
296	C₂×C₄	C₃₁×#135	1	4	yes	—
297	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
299	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
300	C₈	—	2	4	no	—
301	C₈	C₃₁×#124	1	8	yes	complemented
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	8	no	—
306	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
307	C₂²	C₃₁×#532	1	2	yes	—
308	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
309	C₂²	C₂×C₃₁×#135	1	2	yes	—
310	C₄	—	2	8	no	—
311	C₄	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	2	no	—
314	C₄	C₃₁×#492	1	4	yes	—
315	C₄	C₃₁×#495	1	4	yes	—
316	C₄	C₂×C₃₁×#124	1	2	yes	—
317	C₄	C₃₁×#532	1	2	yes	—
318	C₂	—	16	2	no	—
319	C₂	C₃₁×#2858	1	1	yes	characteristic
320	C₂	C₂×C₃₁×#492	1	1	yes	characteristic
321	C₂	C₂×C₃₁×#495	1	1	yes	characteristic
322	C₂	C₃₁×#3167	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#498	C₂	semidirect
3. C₂×C₃₁×#487	C₂	semidirect
5. C₃₁×#2862	C₂	semidirect
7. C₃₁×#1968	C₂	semidirect
8. C₄²×C₈×C₃₁	C₂	semidirect
106. C₄²×C₃₁	SD₁₆	semidirect
109. #33239	C₃₁	direct (N × H)
140. C₈×C₃₁	#124	semidirect
141. C₂×#498	C₂×C₃₁	semidirect
143. C₂×#487	C₂×C₃₁	semidirect
145. #2862	C₂×C₃₁	semidirect
147. #1968	C₂×C₃₁	semidirect
148. C₄²×C₈	C₂×C₃₁	semidirect
229. C₃₁	#33239	direct (N × H)
268. C₄²	SD₁₆×C₃₁	semidirect
301. C₈	C₃₁×#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#498	2
C₂×C₃₁×#492	2
C₂×C₃₁×#487	2
C₂×C₃₁×#486	2
C₃₁×#2862	2
C₃₁×#2858	2
C₃₁×#1968	2
C₄²×C₈×C₃₁	2
#33239	31

Subgroups — C31×#33239 #21489998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33239 #21489998