Subgroups — C₃₁×#11376 #21468135

Order 7936.

654 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#341	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#339	C₂	1	2	yes	maximal
4	C₃₁×#1729	C₂	1	2	yes	maximal
5	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#427	—	2	2	no	—
12	C₃₁×#427	—	2	2	no	—
13	C₂×C₃₁×#107	C₄	1	1	yes	characteristic, complemented
14	C₂×C₃₁×#107	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃₁×#106	C₄	1	2	yes	complemented
16	C₃₁×#341	—	2	2	no	—
17	C₃₁×#339	—	2	4	no	—
18	C₃₁×#336	—	2	2	no	—
19	C₃₁×#336	—	2	4	no	—
20	C₂³×C₄×C₃₁	—	2	1	no	—
21	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	Q₈	1	4	yes	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	—	2	1	no	—
31	C₂²×C₈×C₃₁	—	2	2	no	—
32	C₂²×C₈×C₃₁	—	2	1	no	—
33	C₂²×C₈×C₃₁	—	2	2	no	—
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	D₄	1	2	yes	—
38	C₂²×C₈×C₃₁	D₄	1	2	yes	—
39	C₂²×C₈×C₃₁	D₄	1	2	yes	—
40	C₃₁×#117	—	2	2	no	—
41	C₃₁×#117	—	2	2	no	—
42	C₃₁×#117	—	2	2	no	—
43	C₃₁×#117	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄²×C₃₁	—	2	2	no	—
48	C₂×C₄²×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	—	2	1	no	—
50	C₂×C₄²×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	D₄	1	2	yes	—
52	C₃₁×#107	—	2	2	no	—
53	C₃₁×#107	—	2	2	no	—
54	C₃₁×#106	—	2	4	no	—
55	C₂⁴×C₃₁	#41	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	—	2	4	no	—
57	C₂²×C₄×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	—	2	2	no	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	4	2	no	—
61	C₂²×C₄×C₃₁	—	4	2	no	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₃₁	#41	1	2	yes	—
69	C₂²×C₄×C₃₁	#41	1	2	yes	—
70	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	#41	1	2	yes	—
72	M₁₆×C₃₁	—	2	4	no	—
73	M₁₆×C₃₁	—	2	4	no	—
74	M₁₆×C₃₁	—	4	2	no	—
75	M₁₆×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	4	no	—
83	C₂×C₈×C₃₁	—	2	4	no	—
84	C₂×C₈×C₃₁	—	4	1	no	—
85	C₂×C₈×C₃₁	—	4	2	no	—
86	C₂×C₈×C₃₁	—	4	1	no	—
87	C₂×C₈×C₃₁	—	4	2	no	—
88	C₂×C₈×C₃₁	—	4	2	no	—
89	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₂²⋊C₄×C₃₁	—	4	2	no	—
96	C₂²⋊C₄×C₃₁	—	4	2	no	—
97	C₄²×C₃₁	—	2	4	no	—
98	C₄²×C₃₁	—	4	2	no	—
99	C₄²×C₃₁	—	4	1	no	—
100	C₄²×C₃₁	—	4	1	no	—
101	C₄²×C₃₁	—	4	2	no	—
102	#11376	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	—	2	2	no	—
104	C₂³×C₃₁	—	2	2	no	—
105	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
106	C₂³×C₃₁	#121	1	1	yes	characteristic
107	C₂³×C₃₁	#124	1	1	yes	characteristic
108	C₂³×C₃₁	#125	1	2	yes	—
109	C₂³×C₃₁	#126	1	2	yes	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	4	8	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₃₁	#117	1	2	yes	—
120	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
122	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
127	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
128	C₂×C₄×C₃₁	#123	1	2	yes	—
129	C₂×C₄×C₃₁	#123	1	2	yes	—
130	C₂×C₄×C₃₁	#123	1	2	yes	—
131	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
132	C₂×C₄×C₃₁	#125	1	2	yes	—
133	C₂×C₄×C₃₁	#126	1	2	yes	—
134	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
136	C₈×C₃₁	—	2	2	no	—
137	C₈×C₃₁	—	2	2	no	—
138	C₈×C₃₁	—	4	2	no	—
139	C₈×C₃₁	—	4	4	no	—
140	C₂×#427	C₂×C₃₁	1	1	yes	characteristic
141	C₂×#341	C₂×C₃₁	1	1	yes	characteristic
142	C₂×#339	C₂×C₃₁	1	2	yes	—
143	#1729	C₂×C₃₁	1	2	yes	—
144	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
145	C₂²×C₃₁	—	2	4	no	—
146	C₂²×C₃₁	—	2	4	no	—
147	C₂²×C₃₁	—	2	1	no	—
148	C₂²×C₃₁	—	2	1	no	—
149	C₂²×C₃₁	—	2	1	no	—
150	C₂²×C₃₁	—	2	1	no	—
151	C₂²×C₃₁	#387	1	2	yes	—
152	C₂²×C₃₁	#390	1	1	yes	characteristic
153	C₂²×C₃₁	#397	1	1	yes	characteristic
154	C₂²×C₃₁	#398	1	1	yes	characteristic
155	C₂²×C₃₁	#400	1	2	yes	—
156	C₂²×C₃₁	#443	1	2	yes	—
157	C₂²×C₃₁	#444	1	2	yes	—
158	C₄×C₃₁	—	4	4	no	—
159	C₄×C₃₁	—	8	2	no	—
160	C₄×C₃₁	—	8	2	no	—
161	C₄×C₃₁	#381	1	1	yes	characteristic
162	C₄×C₃₁	#387	1	2	yes	—
163	C₄×C₃₁	#391	1	1	yes	characteristic
164	C₄×C₃₁	#397	1	1	yes	characteristic
165	C₄×C₃₁	#398	1	1	yes	characteristic
166	C₄×C₃₁	#399	1	2	yes	—
167	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
168	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
169	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×#117	C₂²×C₃₁	1	2	yes	—
171	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
172	#427	—	2	2	no	—
173	#427	—	2	2	no	—
174	C₂×#107	C₄×C₃₁	1	1	yes	characteristic, complemented
175	C₂×#107	C₄×C₃₁	1	1	yes	characteristic, complemented
176	C₂×#106	C₄×C₃₁	1	2	yes	complemented
177	#341	—	2	2	no	—
178	#339	—	2	4	no	—
179	#336	—	2	2	no	—
180	#336	—	2	4	no	—
181	C₂×C₃₁	—	2	1	no	—
182	C₂×C₃₁	—	2	1	no	—
183	C₂×C₃₁	—	2	2	no	—
184	C₂×C₃₁	#1445	1	1	yes	characteristic
185	C₂×C₃₁	#1922	1	4	yes	—
186	C₂×C₃₁	#1974	1	1	yes	characteristic
187	C₂×C₃₁	#1975	1	1	yes	characteristic
188	C₂³×C₄	—	2	1	no	—
189	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	Q₈×C₃₁	1	4	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
205	C₂²×C₈	D₄×C₃₁	1	2	yes	—
206	C₂²×C₈	D₄×C₃₁	1	2	yes	—
207	C₂²×C₈	D₄×C₃₁	1	2	yes	—
208	#117	—	2	2	no	—
209	#117	—	2	2	no	—
210	#117	—	2	2	no	—
211	#117	—	2	2	no	—
212	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
213	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	D₄×C₃₁	1	2	yes	—
220	#107	—	2	2	no	—
221	#107	—	2	2	no	—
222	#106	—	2	4	no	—
223	C₃₁	#11376	1	1	yes	characteristic, direct product factor, Sylow
224	C₂⁴	#41×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
237	C₂²×C₄	#41×C₃₁	1	2	yes	—
238	C₂²×C₄	#41×C₃₁	1	2	yes	—
239	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	#41×C₃₁	1	2	yes	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	4	2	no	—
244	M₁₆	—	4	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
259	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	2	4	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
274	C₂³	C₃₁×#121	1	1	yes	characteristic
275	C₂³	C₃₁×#124	1	1	yes	characteristic
276	C₂³	C₃₁×#125	1	2	yes	—
277	C₂³	C₃₁×#126	1	2	yes	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₃₁×#117	1	2	yes	—
288	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
290	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
291	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
292	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
293	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
294	C₂×C₄	C₃₁×#123	1	2	yes	—
295	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
296	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
297	C₂×C₄	C₃₁×#123	1	2	yes	—
298	C₂×C₄	C₃₁×#123	1	2	yes	—
299	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
300	C₂×C₄	C₃₁×#125	1	2	yes	—
301	C₂×C₄	C₃₁×#126	1	2	yes	—
302	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
303	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	4	4	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	C₃₁×#387	1	2	yes	—
315	C₂²	C₃₁×#390	1	1	yes	characteristic
316	C₂²	C₃₁×#397	1	1	yes	characteristic
317	C₂²	C₃₁×#398	1	1	yes	characteristic
318	C₂²	C₃₁×#400	1	2	yes	—
319	C₂²	C₃₁×#443	1	2	yes	—
320	C₂²	C₃₁×#444	1	2	yes	—
321	C₄	—	4	4	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	C₃₁×#381	1	1	yes	characteristic
325	C₄	C₃₁×#387	1	2	yes	—
326	C₄	C₃₁×#391	1	1	yes	characteristic
327	C₄	C₃₁×#397	1	1	yes	characteristic
328	C₄	C₃₁×#398	1	1	yes	characteristic
329	C₄	C₃₁×#399	1	2	yes	—
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	2	no	—
333	C₂	C₃₁×#1445	1	1	yes	characteristic
334	C₂	C₃₁×#1922	1	4	yes	—
335	C₂	C₃₁×#1974	1	1	yes	characteristic
336	C₂	C₃₁×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂³×C₃₁	C₄×D₄
prime radical	C₂⁴×C₃₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×C₃₁×#107	C₄	semidirect
14. C₂×C₃₁×#107	C₄	semidirect
15. C₂×C₃₁×#106	C₄	semidirect
102. #11376	C₃₁	direct (N × H)
174. C₂×#107	C₄×C₃₁	semidirect
175. C₂×#107	C₄×C₃₁	semidirect
176. C₂×#106	C₄×C₃₁	semidirect
223. C₃₁	#11376	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×#427	2
C₂×C₃₁×#341	2
C₂×C₃₁×#339	2
C₃₁×#1729	2
C₂×C₃₁×#336	2
#11376	31

Subgroups — C31×#11376 #21468135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11376 #21468135