Subgroups — C₃₁×#13237 #21469996

Order 7936.

474 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#367	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#358	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#357	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×SD₃₂	—	2	2	no	—
11	C₂×C₃₁×SD₃₂	—	2	2	no	—
12	C₂×C₃₁×SD₃₂	C₄	1	4	yes	complemented
13	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#103	—	2	1	no	—
18	C₂×C₃₁×#102	—	2	1	no	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#391	—	2	1	no	—
21	C₃₁×#390	—	2	1	no	—
22	C₃₁×#367	—	2	2	no	—
23	C₃₁×#358	—	2	2	no	—
24	C₃₁×#357	—	2	2	no	—
25	C₄×C₁₆×C₃₁	—	2	2	no	—
26	C₂²×Q₈×C₃₁	—	2	1	no	—
27	C₂²×D₄×C₃₁	—	2	1	no	—
28	C₂×Q₁₆×C₃₁	—	2	1	no	—
29	C₂×Q₁₆×C₃₁	—	2	1	no	—
30	C₂×Q₁₆×C₃₁	—	2	1	no	—
31	C₂×Q₁₆×C₃₁	—	2	1	no	—
32	C₂×Q₁₆×C₃₁	—	2	1	no	—
33	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
34	C₂×D₈×C₃₁	—	2	1	no	—
35	C₂×D₈×C₃₁	—	2	1	no	—
36	C₂×D₈×C₃₁	—	2	1	no	—
37	C₂×D₈×C₃₁	—	2	1	no	—
38	C₂×D₈×C₃₁	—	2	1	no	—
39	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
44	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
45	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
46	C₃₁×SD₃₂	—	2	8	no	—
47	C₂×C₁₆×C₃₁	—	2	1	no	—
48	C₂×C₁₆×C₃₁	—	2	1	no	—
49	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
51	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
52	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
53	C₃₁×#107	—	2	1	no	—
54	C₃₁×#107	—	2	1	no	—
55	C₃₁×#107	—	2	1	no	—
56	C₃₁×#107	—	2	1	no	—
57	C₃₁×#103	—	4	2	no	—
58	C₃₁×#102	—	4	2	no	—
59	C₄×C₈×C₃₁	—	2	1	no	—
60	C₄×C₈×C₃₁	—	2	1	no	—
61	C₃₁×#95	—	4	2	no	—
62	C₂⁴×C₃₁	—	4	1	no	—
63	C₂×Q₈×C₃₁	—	2	2	no	—
64	C₂×Q₈×C₃₁	—	4	1	no	—
65	C₂×Q₈×C₃₁	—	4	1	no	—
66	C₂×Q₈×C₃₁	—	4	1	no	—
67	C₂×Q₈×C₃₁	—	4	1	no	—
68	C₂×Q₈×C₃₁	—	4	1	no	—
69	C₂×D₄×C₃₁	—	2	2	no	—
70	C₂×D₄×C₃₁	—	4	1	no	—
71	C₂×D₄×C₃₁	—	4	1	no	—
72	C₂×D₄×C₃₁	—	4	1	no	—
73	C₂×D₄×C₃₁	—	4	1	no	—
74	C₂×D₄×C₃₁	—	4	1	no	—
75	C₂²×C₄×C₃₁	—	4	1	no	—
76	C₂²×C₄×C₃₁	—	4	1	no	—
77	C₂²×C₄×C₃₁	—	4	1	no	—
78	C₂²×C₄×C₃₁	D₈	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	Q₁₆×C₃₁	—	2	2	no	—
81	Q₁₆×C₃₁	—	2	2	no	—
82	Q₁₆×C₃₁	—	4	1	no	—
83	Q₁₆×C₃₁	—	4	1	no	—
84	D₈×C₃₁	—	2	2	no	—
85	D₈×C₃₁	—	2	2	no	—
86	D₈×C₃₁	—	4	1	no	—
87	D₈×C₃₁	—	4	1	no	—
88	C₂×C₈×C₃₁	—	2	1	no	—
89	C₂×C₈×C₃₁	—	2	1	no	—
90	C₂×C₈×C₃₁	D₈	1	2	yes	—
91	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
96	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
97	C₄⋊C₄×C₃₁	—	4	1	no	—
98	C₄⋊C₄×C₃₁	—	4	1	no	—
99	C₄⋊C₄×C₃₁	—	4	1	no	—
100	C₄⋊C₄×C₃₁	—	4	1	no	—
101	C₂²⋊C₄×C₃₁	—	8	4	no	—
102	C₄²×C₃₁	—	2	1	no	—
103	C₄²×C₃₁	—	2	1	no	—
104	C₁₆×C₃₁	—	2	2	no	—
105	C₁₆×C₃₁	C₂²⋊C₄	1	4	yes	complemented
106	#13237	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₃₁	—	4	2	no	—
108	C₂³×C₃₁	—	8	1	no	—
109	C₂³×C₃₁	—	8	1	no	—
110	C₂³×C₃₁	—	8	2	no	—
111	C₂³×C₃₁	—	8	2	no	—
112	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
113	Q₈×C₃₁	—	4	2	no	—
114	Q₈×C₃₁	—	4	2	no	—
115	Q₈×C₃₁	—	8	1	no	—
116	Q₈×C₃₁	—	8	1	no	—
117	D₄×C₃₁	—	4	2	no	—
118	D₄×C₃₁	—	4	2	no	—
119	D₄×C₃₁	—	8	1	no	—
120	D₄×C₃₁	—	8	1	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	—	8	1	no	—
125	C₂×C₄×C₃₁	—	8	1	no	—
126	C₂×C₄×C₃₁	—	8	1	no	—
127	C₂×C₄×C₃₁	—	8	1	no	—
128	C₂×C₄×C₃₁	—	8	2	no	—
129	C₂×C₄×C₃₁	—	8	2	no	—
130	C₂×C₄×C₃₁	SD₃₂	1	4	yes	complemented
131	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
134	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
136	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
137	C₈×C₃₁	—	2	2	no	—
138	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
140	C₈×C₃₁	#121	1	1	yes	characteristic
141	C₈×C₃₁	#124	1	1	yes	characteristic
142	C₂²×SD₃₂	C₂×C₃₁	1	1	yes	characteristic
143	C₂×#367	C₂×C₃₁	1	1	yes	characteristic, complemented
144	C₂×#358	C₂×C₃₁	1	1	yes	characteristic, complemented
145	C₂×#357	C₂×C₃₁	1	1	yes	characteristic
146	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#1970	C₂×C₃₁	1	1	yes	characteristic
148	#1969	C₂×C₃₁	1	1	yes	characteristic, complemented
149	C₂²×C₃₁	—	8	4	no	—
150	C₂²×C₃₁	—	8	2	no	—
151	C₂²×C₃₁	—	16	2	no	—
152	C₂²×C₃₁	—	16	1	no	—
153	C₂²×C₃₁	—	16	1	no	—
154	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
155	C₂²×C₃₁	#466	1	1	yes	characteristic
156	C₂²×C₃₁	#493	1	1	yes	characteristic
157	C₂²×C₃₁	C₂×SD₃₂	1	2	yes	—
158	C₂²×C₃₁	#508	1	2	yes	—
159	C₄×C₃₁	—	2	4	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	—	16	1	no	—
162	C₄×C₃₁	—	16	1	no	—
163	C₄×C₃₁	#390	1	1	yes	characteristic
164	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
165	C₄×C₃₁	#466	1	1	yes	characteristic
166	C₄×C₃₁	#495	1	1	yes	characteristic
167	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
168	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×SD₃₂	—	2	2	no	—
170	C₂×SD₃₂	—	2	2	no	—
171	C₂×SD₃₂	C₄×C₃₁	1	4	yes	complemented
172	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
173	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
176	C₂×#103	—	2	1	no	—
177	C₂×#102	—	2	1	no	—
178	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
179	#391	—	2	1	no	—
180	#390	—	2	1	no	—
181	#367	—	2	2	no	—
182	#358	—	2	2	no	—
183	#357	—	2	2	no	—
184	C₄×C₁₆	—	2	2	no	—
185	C₂×C₃₁	—	16	2	no	—
186	C₂×C₃₁	#1970	1	1	yes	characteristic
187	C₂×C₃₁	C₄×SD₃₂	1	2	yes	—
188	C₂×C₃₁	#2217	1	2	yes	—
189	C₂×C₃₁	#2248	1	1	yes	characteristic
190	C₂×C₃₁	#2249	1	1	yes	characteristic
191	C₂²×Q₈	—	2	1	no	—
192	C₂²×D₄	—	2	1	no	—
193	C₂×Q₁₆	—	2	1	no	—
194	C₂×Q₁₆	—	2	1	no	—
195	C₂×Q₁₆	—	2	1	no	—
196	C₂×Q₁₆	—	2	1	no	—
197	C₂×Q₁₆	—	2	1	no	—
198	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
205	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
206	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₂²⋊C₄	—	4	2	no	—
210	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
211	SD₃₂	—	2	8	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
215	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
216	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
217	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	—	2	1	no	—
222	#103	—	4	2	no	—
223	#102	—	4	2	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	#95	—	4	2	no	—
227	C₃₁	#13237	1	1	yes	characteristic, direct product factor, Sylow
228	C₂⁴	—	4	1	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	1	no	—
233	C₂×Q₈	—	4	1	no	—
234	C₂×Q₈	—	4	1	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	D₈×C₃₁	1	2	yes	—
245	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
246	Q₁₆	—	2	2	no	—
247	Q₁₆	—	2	2	no	—
248	Q₁₆	—	4	1	no	—
249	Q₁₆	—	4	1	no	—
250	D₈	—	2	2	no	—
251	D₈	—	2	2	no	—
252	D₈	—	4	1	no	—
253	D₈	—	4	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	D₈×C₃₁	1	2	yes	—
257	C₂×C₈	C₂²×C₄×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	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
261	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
262	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	8	4	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	C₂²⋊C₄×C₃₁	1	4	yes	complemented
272	C₂³	—	4	2	no	—
273	C₂³	—	8	1	no	—
274	C₂³	—	8	1	no	—
275	C₂³	—	8	2	no	—
276	C₂³	—	8	2	no	—
277	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	8	1	no	—
281	Q₈	—	8	1	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	D₄	—	8	1	no	—
285	D₄	—	8	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	8	2	no	—
295	C₂×C₄	C₃₁×SD₃₂	1	4	yes	complemented
296	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
297	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
298	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
299	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
300	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
301	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
304	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
305	C₈	C₃₁×#121	1	1	yes	characteristic
306	C₈	C₃₁×#124	1	1	yes	characteristic
307	C₂²	—	8	4	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	16	2	no	—
310	C₂²	—	16	1	no	—
311	C₂²	—	16	1	no	—
312	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
313	C₂²	C₃₁×#466	1	1	yes	characteristic
314	C₂²	C₃₁×#493	1	1	yes	characteristic
315	C₂²	C₂×C₃₁×SD₃₂	1	2	yes	—
316	C₂²	C₃₁×#508	1	2	yes	—
317	C₄	—	2	4	no	—
318	C₄	—	8	2	no	—
319	C₄	—	16	1	no	—
320	C₄	—	16	1	no	—
321	C₄	C₃₁×#390	1	1	yes	characteristic
322	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
323	C₄	C₃₁×#466	1	1	yes	characteristic
324	C₄	C₃₁×#495	1	1	yes	characteristic
325	C₂	—	16	2	no	—
326	C₂	C₃₁×#1970	1	1	yes	characteristic
327	C₂	C₄×C₃₁×SD₃₂	1	2	yes	—
328	C₂	C₃₁×#2217	1	2	yes	—
329	C₂	C₃₁×#2248	1	1	yes	characteristic
330	C₂	C₃₁×#2249	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#367	C₂	semidirect
3. C₂×C₃₁×#358	C₂	semidirect
5. C₂×C₄×C₁₆×C₃₁	C₂	semidirect
7. C₃₁×#1969	C₂	semidirect
12. C₂×C₃₁×SD₃₂	C₄	semidirect
105. C₁₆×C₃₁	C₂²⋊C₄	semidirect
106. #13237	C₃₁	direct (N × H)
130. C₂×C₄×C₃₁	SD₃₂	semidirect
143. C₂×#367	C₂×C₃₁	semidirect
144. C₂×#358	C₂×C₃₁	semidirect
146. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
148. #1969	C₂×C₃₁	semidirect
171. C₂×SD₃₂	C₄×C₃₁	semidirect
227. C₃₁	#13237	direct (N × H)
271. C₁₆	C₂²⋊C₄×C₃₁	semidirect
295. C₂×C₄	C₃₁×SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×SD₃₂	2
C₂×C₃₁×#367	2
C₂×C₃₁×#358	2
C₂×C₃₁×#357	2
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1970	2
C₃₁×#1969	2
#13237	31

Subgroups — C31×#13237 #21469996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13237 #21469996