Subgroups — C₃₁×#29360 #21486119

Order 7936.

718 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2641	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2330	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2071	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2025	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1989	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1982	C₂	1	2	yes	maximal
7	C₃₁×#1936	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1934	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1768	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×#102	C₂²	1	2	yes	—
15	C₂×C₃₁×#102	C₂²	1	2	yes	—
16	C₂×C₃₁×#102	C₂²	1	2	yes	—
17	C₂×C₃₁×#102	C₂²	1	2	yes	—
18	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#394	—	2	2	no	—
21	C₃₁×#394	—	2	2	no	—
22	C₃₁×#390	C₂²	1	2	yes	—
23	C₃₁×#390	C₂²	1	2	yes	—
24	C₃₁×#389	—	2	2	no	—
25	C₃₁×#389	C₂²	1	2	yes	—
26	C₃₁×#387	—	2	2	no	—
27	C₃₁×#384	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#376	C₂²	1	1	yes	characteristic
30	C₃₁×#340	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₂²×D₄×C₃₁	—	2	2	no	—
36	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₃₁×#127	—	2	4	no	—
40	C₃₁×#125	—	2	4	no	—
41	C₃₁×#125	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₄²×C₃₁	—	2	2	no	—
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×C₄⋊C₈	—	2	4	no	—
58	C₃₁×#102	—	2	4	no	—
59	C₃₁×#102	—	2	4	no	—
60	C₃₁×#102	—	4	2	no	—
61	C₃₁×#102	—	4	2	no	—
62	C₃₁×#97	—	4	1	no	—
63	C₄×C₈×C₃₁	—	4	1	no	—
64	C₃₁×#95	—	2	2	no	—
65	C₃₁×#95	—	2	2	no	—
66	C₃₁×#95	D₄	1	2	yes	—
67	C₃₁×#95	D₄	1	2	yes	—
68	C₂⁴×C₃₁	—	2	2	no	—
69	C₂×D₄×C₃₁	—	2	2	no	—
70	C₂×D₄×C₃₁	—	2	2	no	—
71	C₂×D₄×C₃₁	—	2	2	no	—
72	C₂×D₄×C₃₁	—	2	2	no	—
73	C₂×D₄×C₃₁	—	2	2	no	—
74	C₂×D₄×C₃₁	—	2	2	no	—
75	C₂×D₄×C₃₁	—	4	4	no	—
76	C₂×D₄×C₃₁	—	8	2	no	—
77	C₂×D₄×C₃₁	—	8	2	no	—
78	C₂²×C₄×C₃₁	—	2	2	no	—
79	C₂²×C₄×C₃₁	—	2	2	no	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	2	2	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	8	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	4	4	no	—
98	C₄⋊C₄×C₃₁	—	8	2	no	—
99	C₄⋊C₄×C₃₁	—	8	2	no	—
100	C₄⋊C₄×C₃₁	#41	1	4	yes	—
101	C₂²⋊C₄×C₃₁	—	8	2	no	—
102	C₂²⋊C₄×C₃₁	—	8	2	no	—
103	C₂²⋊C₄×C₃₁	—	8	2	no	—
104	C₂²⋊C₄×C₃₁	—	8	2	no	—
105	C₄²×C₃₁	—	2	2	no	—
106	C₄²×C₃₁	—	2	1	no	—
107	C₄²×C₃₁	—	2	1	no	—
108	C₄²×C₃₁	—	8	2	no	—
109	C₄²×C₃₁	#41	1	4	yes	—
110	#29360	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂³×C₃₁	—	4	2	no	—
112	C₂³×C₃₁	—	4	2	no	—
113	C₂³×C₃₁	—	4	2	no	—
114	C₂³×C₃₁	—	4	2	no	—
115	C₂³×C₃₁	—	4	2	no	—
116	C₂³×C₃₁	—	4	2	no	—
117	C₂³×C₃₁	—	4	2	no	—
118	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
119	D₄×C₃₁	—	4	2	no	—
120	D₄×C₃₁	—	4	2	no	—
121	D₄×C₃₁	—	4	2	no	—
122	D₄×C₃₁	—	4	2	no	—
123	D₄×C₃₁	—	8	8	no	—
124	C₂×C₄×C₃₁	—	2	2	no	—
125	C₂×C₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	2	4	no	—
128	C₂×C₄×C₃₁	—	2	1	no	—
129	C₂×C₄×C₃₁	—	2	1	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	2	2	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	—	4	2	no	—
136	C₂×C₄×C₃₁	—	4	2	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	4	2	no	—
139	C₂×C₄×C₃₁	—	4	2	no	—
140	C₂×C₄×C₃₁	—	4	2	no	—
141	C₂×C₄×C₃₁	—	4	2	no	—
142	C₂×C₄×C₃₁	#124	1	4	yes	—
143	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
144	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
145	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
147	C₈×C₃₁	—	4	2	no	—
148	C₈×C₃₁	—	4	2	no	—
149	#2641	C₂×C₃₁	1	1	yes	characteristic
150	#2330	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#2071	C₂×C₃₁	1	2	yes	complemented
152	#2025	C₂×C₃₁	1	2	yes	complemented
153	#1989	C₂×C₃₁	1	2	yes	complemented
154	#1982	C₂×C₃₁	1	2	yes	—
155	#1936	C₂×C₃₁	1	2	yes	complemented
156	#1934	C₂×C₃₁	1	2	yes	complemented
157	#1768	C₂×C₃₁	1	1	yes	characteristic, complemented
158	C₂²×C₃₁	—	8	2	no	—
159	C₂²×C₃₁	—	8	2	no	—
160	C₂²×C₃₁	—	8	2	no	—
161	C₂²×C₃₁	—	8	2	no	—
162	C₂²×C₃₁	—	8	2	no	—
163	C₂²×C₃₁	—	8	2	no	—
164	C₂²×C₃₁	—	8	2	no	—
165	C₂²×C₃₁	#525	1	1	yes	characteristic
166	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
167	C₂²×C₃₁	#537	1	1	yes	characteristic
168	C₂²×C₃₁	#576	1	2	yes	—
169	C₂²×C₃₁	#577	1	2	yes	—
170	C₄×C₃₁	—	2	4	no	—
171	C₄×C₃₁	—	4	2	no	—
172	C₄×C₃₁	—	4	4	no	—
173	C₄×C₃₁	—	4	1	no	—
174	C₄×C₃₁	—	4	1	no	—
175	C₄×C₃₁	—	8	2	no	—
176	C₄×C₃₁	—	8	2	no	—
177	C₄×C₃₁	#556	1	2	yes	—
178	C₄×C₃₁	#559	1	2	yes	—
179	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
180	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
181	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
183	C₂×#102	C₂²×C₃₁	1	2	yes	—
184	C₂×#102	C₂²×C₃₁	1	2	yes	—
185	C₂×#102	C₂²×C₃₁	1	2	yes	—
186	C₂×#102	C₂²×C₃₁	1	2	yes	—
187	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
189	#394	—	2	2	no	—
190	#394	—	2	2	no	—
191	#390	C₂²×C₃₁	1	2	yes	—
192	#390	C₂²×C₃₁	1	2	yes	—
193	#389	—	2	2	no	—
194	#389	C₂²×C₃₁	1	2	yes	—
195	#387	—	2	2	no	—
196	#384	C₂²×C₃₁	1	1	yes	characteristic
197	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
198	#376	C₂²×C₃₁	1	1	yes	characteristic
199	#340	C₂²×C₃₁	1	2	yes	—
200	#340	C₂²×C₃₁	1	2	yes	—
201	#340	C₂²×C₃₁	1	2	yes	—
202	#340	C₂²×C₃₁	1	2	yes	—
203	#336	C₂²×C₃₁	1	2	yes	—
204	C₂×C₃₁	—	16	2	no	—
205	C₂×C₃₁	#2592	1	1	yes	characteristic
206	C₂×C₃₁	#3119	1	2	yes	—
207	C₂×C₃₁	#3136	1	2	yes	—
208	C₂×C₃₁	#3259	1	1	yes	characteristic
209	C₂×C₃₁	#3260	1	1	yes	characteristic
210	C₂²×D₄	—	2	2	no	—
211	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
212	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
213	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
214	#127	—	2	4	no	—
215	#125	—	2	4	no	—
216	#125	—	2	4	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
222	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₄²	—	2	2	no	—
229	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
230	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
231	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
232	C₄⋊C₈	—	2	4	no	—
233	#102	—	2	4	no	—
234	#102	—	2	4	no	—
235	#102	—	4	2	no	—
236	#102	—	4	2	no	—
237	#97	—	4	1	no	—
238	C₄×C₈	—	4	1	no	—
239	#95	—	2	2	no	—
240	#95	—	2	2	no	—
241	#95	D₄×C₃₁	1	2	yes	—
242	#95	D₄×C₃₁	1	2	yes	—
243	C₃₁	#29360	1	1	yes	characteristic, direct product factor, Sylow
244	C₂⁴	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	—	2	2	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	4	no	—
252	C₂×D₄	—	8	2	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
259	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
261	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₄⋊C₄	—	2	8	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	4	no	—
274	C₄⋊C₄	—	8	2	no	—
275	C₄⋊C₄	—	8	2	no	—
276	C₄⋊C₄	#41×C₃₁	1	4	yes	—
277	C₂²⋊C₄	—	8	2	no	—
278	C₂²⋊C₄	—	8	2	no	—
279	C₂²⋊C₄	—	8	2	no	—
280	C₂²⋊C₄	—	8	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	8	2	no	—
285	C₄²	#41×C₃₁	1	4	yes	—
286	C₂³	—	4	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	4	2	no	—
289	C₂³	—	4	2	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	8	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	C₃₁×#124	1	4	yes	—
318	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
319	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
320	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
321	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
322	C₈	—	4	2	no	—
323	C₈	—	4	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	C₃₁×#525	1	1	yes	characteristic
332	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
333	C₂²	C₃₁×#537	1	1	yes	characteristic
334	C₂²	C₃₁×#576	1	2	yes	—
335	C₂²	C₃₁×#577	1	2	yes	—
336	C₄	—	2	4	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	4	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₃₁×#556	1	2	yes	—
344	C₄	C₃₁×#559	1	2	yes	—
345	C₂	—	16	2	no	—
346	C₂	C₃₁×#2592	1	1	yes	characteristic
347	C₂	C₃₁×#3119	1	2	yes	—
348	C₂	C₃₁×#3136	1	2	yes	—
349	C₂	C₃₁×#3259	1	1	yes	characteristic
350	C₂	C₃₁×#3260	1	1	yes	characteristic

	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 ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2330	C₂	semidirect
3. C₃₁×#2071	C₂	semidirect
4. C₃₁×#2025	C₂	semidirect
5. C₃₁×#1989	C₂	semidirect
7. C₃₁×#1936	C₂	semidirect
8. C₃₁×#1934	C₂	semidirect
9. C₃₁×#1768	C₂	semidirect
110. #29360	C₃₁	direct (N × H)
150. #2330	C₂×C₃₁	semidirect
151. #2071	C₂×C₃₁	semidirect
152. #2025	C₂×C₃₁	semidirect
153. #1989	C₂×C₃₁	semidirect
155. #1936	C₂×C₃₁	semidirect
156. #1934	C₂×C₃₁	semidirect
157. #1768	C₂×C₃₁	semidirect
243. C₃₁	#29360	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2641	2
C₃₁×#2330	2
C₃₁×#2071	2
C₃₁×#2025	2
C₃₁×#1989	2
C₃₁×#1982	2
C₃₁×#1936	2
C₃₁×#1934	2
C₃₁×#1768	2
#29360	31

Subgroups — C31×#29360 #21486119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29360 #21486119