Subgroups — C₃₁×#29358 #21486117

Order 7936.

726 conjugacy classes of subgroups.

The subgroup lattice has 360 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×#2331	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2073	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2023	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×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	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×#391	—	2	2	no	—
23	C₃₁×#390	C₂²	1	2	yes	—
24	C₃₁×#390	C₂²	1	2	yes	—
25	C₃₁×#389	C₂²	1	2	yes	—
26	C₃₁×#387	—	2	2	no	—
27	C₃₁×#382	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×Q₈×C₃₁	—	2	1	no	—
36	C₂²×D₄×C₃₁	—	2	2	no	—
37	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₃₁×C₄⋊Q₈	—	2	4	no	—
41	C₃₁×#127	—	2	4	no	—
42	C₃₁×#125	—	2	4	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	1	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	1	no	—
66	C₃₁×#95	—	2	1	no	—
67	C₃₁×#95	—	2	2	no	—
68	C₃₁×#95	D₄	1	2	yes	—
69	C₃₁×#95	D₄	1	2	yes	—
70	C₂⁴×C₃₁	—	2	2	no	—
71	C₂×Q₈×C₃₁	—	2	4	no	—
72	C₂×Q₈×C₃₁	—	8	2	no	—
73	C₂×D₄×C₃₁	—	2	2	no	—
74	C₂×D₄×C₃₁	—	2	2	no	—
75	C₂×D₄×C₃₁	—	2	2	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	4	4	no	—
80	C₂×D₄×C₃₁	—	8	2	no	—
81	C₂×D₄×C₃₁	—	8	2	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	2	2	no	—
94	C₂×C₈×C₃₁	—	2	2	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	4	no	—
99	C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	4	4	no	—
102	C₄⋊C₄×C₃₁	—	8	2	no	—
103	C₄⋊C₄×C₃₁	#41	1	4	yes	—
104	C₂²⋊C₄×C₃₁	—	8	2	no	—
105	C₂²⋊C₄×C₃₁	—	8	2	no	—
106	C₂²⋊C₄×C₃₁	—	8	2	no	—
107	C₂²⋊C₄×C₃₁	—	8	2	no	—
108	C₄²×C₃₁	—	2	2	no	—
109	C₄²×C₃₁	—	2	1	no	—
110	C₄²×C₃₁	—	2	1	no	—
111	C₄²×C₃₁	—	8	2	no	—
112	C₄²×C₃₁	#41	1	4	yes	—
113	#29358	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	—	4	2	no	—
119	C₂³×C₃₁	—	4	2	no	—
120	C₂³×C₃₁	—	4	2	no	—
121	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
122	Q₈×C₃₁	—	8	4	no	—
123	D₄×C₃₁	—	4	2	no	—
124	D₄×C₃₁	—	4	2	no	—
125	D₄×C₃₁	—	4	2	no	—
126	D₄×C₃₁	—	4	2	no	—
127	D₄×C₃₁	—	8	8	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	1	no	—
131	C₂×C₄×C₃₁	—	2	1	no	—
132	C₂×C₄×C₃₁	—	2	4	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	2	no	—
136	C₂×C₄×C₃₁	—	2	2	no	—
137	C₂×C₄×C₃₁	—	2	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	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	#124	1	4	yes	—
147	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
148	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
149	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
151	C₈×C₃₁	—	4	2	no	—
152	C₈×C₃₁	—	4	2	no	—
153	#2641	C₂×C₃₁	1	1	yes	characteristic
154	#2331	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#2073	C₂×C₃₁	1	2	yes	complemented
156	#2023	C₂×C₃₁	1	2	yes	complemented
157	#1989	C₂×C₃₁	1	2	yes	complemented
158	#1982	C₂×C₃₁	1	2	yes	—
159	#1936	C₂×C₃₁	1	2	yes	complemented
160	#1934	C₂×C₃₁	1	2	yes	complemented
161	#1768	C₂×C₃₁	1	1	yes	characteristic, complemented
162	C₂²×C₃₁	—	8	2	no	—
163	C₂²×C₃₁	—	8	2	no	—
164	C₂²×C₃₁	—	8	2	no	—
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	8	2	no	—
167	C₂²×C₃₁	—	8	2	no	—
168	C₂²×C₃₁	—	8	2	no	—
169	C₂²×C₃₁	#525	1	1	yes	characteristic
170	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
171	C₂²×C₃₁	#537	1	1	yes	characteristic
172	C₂²×C₃₁	#576	1	2	yes	—
173	C₂²×C₃₁	#577	1	2	yes	—
174	C₄×C₃₁	—	2	4	no	—
175	C₄×C₃₁	—	4	1	no	—
176	C₄×C₃₁	—	4	4	no	—
177	C₄×C₃₁	—	4	1	no	—
178	C₄×C₃₁	—	4	2	no	—
179	C₄×C₃₁	—	8	2	no	—
180	C₄×C₃₁	—	8	2	no	—
181	C₄×C₃₁	#556	1	2	yes	—
182	C₄×C₃₁	#559	1	2	yes	—
183	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
184	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
185	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
187	C₂×#102	C₂²×C₃₁	1	2	yes	—
188	C₂×#102	C₂²×C₃₁	1	2	yes	—
189	C₂×#102	C₂²×C₃₁	1	2	yes	—
190	C₂×#102	C₂²×C₃₁	1	2	yes	—
191	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
192	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
193	#394	—	2	2	no	—
194	#394	—	2	2	no	—
195	#391	—	2	2	no	—
196	#390	C₂²×C₃₁	1	2	yes	—
197	#390	C₂²×C₃₁	1	2	yes	—
198	#389	C₂²×C₃₁	1	2	yes	—
199	#387	—	2	2	no	—
200	#382	C₂²×C₃₁	1	1	yes	characteristic
201	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
202	#376	C₂²×C₃₁	1	1	yes	characteristic
203	#340	C₂²×C₃₁	1	2	yes	—
204	#340	C₂²×C₃₁	1	2	yes	—
205	#340	C₂²×C₃₁	1	2	yes	—
206	#340	C₂²×C₃₁	1	2	yes	—
207	#336	C₂²×C₃₁	1	2	yes	—
208	C₂×C₃₁	—	16	2	no	—
209	C₂×C₃₁	#2592	1	1	yes	characteristic
210	C₂×C₃₁	#3122	1	1	yes	characteristic
211	C₂×C₃₁	#3126	1	1	yes	characteristic
212	C₂×C₃₁	#3136	1	2	yes	—
213	C₂×C₃₁	#3257	1	2	yes	—
214	C₂²×Q₈	—	2	1	no	—
215	C₂²×D₄	—	2	2	no	—
216	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
217	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
218	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
219	C₄⋊Q₈	—	2	4	no	—
220	#127	—	2	4	no	—
221	#125	—	2	4	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
227	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
236	C₄⋊C₈	—	2	4	no	—
237	#102	—	2	4	no	—
238	#102	—	2	4	no	—
239	#102	—	4	2	no	—
240	#102	—	4	2	no	—
241	#97	—	4	1	no	—
242	C₄×C₈	—	4	1	no	—
243	#95	—	2	2	no	—
244	#95	—	2	1	no	—
245	#95	—	2	1	no	—
246	#95	—	2	2	no	—
247	#95	D₄×C₃₁	1	2	yes	—
248	#95	D₄×C₃₁	1	2	yes	—
249	C₃₁	#29358	1	1	yes	characteristic, direct product factor, Sylow
250	C₂⁴	—	2	2	no	—
251	C₂×Q₈	—	2	4	no	—
252	C₂×Q₈	—	8	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	4	no	—
260	C₂×D₄	—	8	2	no	—
261	C₂×D₄	—	8	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
269	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	4	no	—
282	C₄⋊C₄	—	8	2	no	—
283	C₄⋊C₄	#41×C₃₁	1	4	yes	—
284	C₂²⋊C₄	—	8	2	no	—
285	C₂²⋊C₄	—	8	2	no	—
286	C₂²⋊C₄	—	8	2	no	—
287	C₂²⋊C₄	—	8	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	8	2	no	—
292	C₄²	#41×C₃₁	1	4	yes	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
301	Q₈	—	8	4	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	8	8	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	C₃₁×#124	1	4	yes	—
326	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
327	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
328	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
329	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
330	C₈	—	4	2	no	—
331	C₈	—	4	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	C₃₁×#525	1	1	yes	characteristic
340	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
341	C₂²	C₃₁×#537	1	1	yes	characteristic
342	C₂²	C₃₁×#576	1	2	yes	—
343	C₂²	C₃₁×#577	1	2	yes	—
344	C₄	—	2	4	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	4	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	8	2	no	—
351	C₄	C₃₁×#556	1	2	yes	—
352	C₄	C₃₁×#559	1	2	yes	—
353	C₂	—	16	2	no	—
354	C₂	C₃₁×#2592	1	1	yes	characteristic
355	C₂	C₃₁×#3122	1	1	yes	characteristic
356	C₂	C₃₁×#3126	1	1	yes	characteristic
357	C₂	C₃₁×#3136	1	2	yes	—
358	C₂	C₃₁×#3257	1	2	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 ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2331	C₂	semidirect
3. C₃₁×#2073	C₂	semidirect
4. C₃₁×#2023	C₂	semidirect
5. C₃₁×#1989	C₂	semidirect
7. C₃₁×#1936	C₂	semidirect
8. C₃₁×#1934	C₂	semidirect
9. C₃₁×#1768	C₂	semidirect
113. #29358	C₃₁	direct (N × H)
154. #2331	C₂×C₃₁	semidirect
155. #2073	C₂×C₃₁	semidirect
156. #2023	C₂×C₃₁	semidirect
157. #1989	C₂×C₃₁	semidirect
159. #1936	C₂×C₃₁	semidirect
160. #1934	C₂×C₃₁	semidirect
161. #1768	C₂×C₃₁	semidirect
249. C₃₁	#29358	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×#2331	2
C₃₁×#2073	2
C₃₁×#2023	2
C₃₁×#1989	2
C₃₁×#1982	2
C₃₁×#1936	2
C₃₁×#1934	2
C₃₁×#1768	2
#29358	31

Subgroups — C31×#29358 #21486117

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29358 #21486117