Subgroups — C₃₁×#32827 #21489586

Order 7936.

654 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2647	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2641	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2092	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2071	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2065	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2059	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2041	C₂	1	2	yes	maximal
9	C₃₁×#1832	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×#107	C₂²	1	2	yes	—
13	C₂×C₃₁×#106	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#102	C₂²	1	2	yes	—
17	C₂×C₃₁×#102	C₂²	1	2	yes	—
18	C₂×C₃₁×#102	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₃₁×#400	—	2	2	no	—
21	C₃₁×#398	—	2	2	no	—
22	C₃₁×#394	—	2	2	no	—
23	C₃₁×#394	—	2	2	no	—
24	C₃₁×#390	C₂²	1	2	yes	—
25	C₃₁×#390	C₂²	1	2	yes	—
26	C₃₁×#389	C₂²	1	2	yes	—
27	C₃₁×#389	C₂²	1	2	yes	—
28	C₃₁×#384	C₂²	1	1	yes	characteristic
29	C₃₁×#382	C₂²	1	1	yes	characteristic
30	C₃₁×#376	C₂²	1	1	yes	characteristic
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#336	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	—	4	1	no	—
40	C₃₁×#127	—	4	1	no	—
41	C₃₁×#125	—	4	1	no	—
42	C₃₁×#125	—	4	1	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	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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×C₂²⋊C₄×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×#107	—	4	2	no	—
58	C₃₁×#106	—	4	2	no	—
59	C₃₁×C₄⋊C₈	—	4	1	no	—
60	C₃₁×C₄⋊C₈	—	4	1	no	—
61	C₃₁×#102	—	4	2	no	—
62	C₃₁×#102	—	4	2	no	—
63	C₃₁×#102	—	4	2	no	—
64	C₃₁×#102	—	4	2	no	—
65	C₃₁×#95	—	2	2	no	—
66	C₃₁×#95	—	2	2	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×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	—	2	2	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	4	4	no	—
78	C₂×D₄×C₃₁	—	8	2	no	—
79	C₂×D₄×C₃₁	—	8	2	no	—
80	C₂²×C₄×C₃₁	—	2	2	no	—
81	C₂²×C₄×C₃₁	—	2	2	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
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	C₂×C₈×C₃₁	—	2	2	no	—
93	C₂×C₈×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	—	2	2	no	—
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	—	2	2	no	—
100	C₄⋊C₄×C₃₁	—	4	4	no	—
101	C₄⋊C₄×C₃₁	—	8	2	no	—
102	C₄⋊C₄×C₃₁	—	8	2	no	—
103	C₄⋊C₄×C₃₁	—	8	2	no	—
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	1	no	—
109	C₄²×C₃₁	—	2	1	no	—
110	C₄²×C₃₁	—	2	1	no	—
111	C₄²×C₃₁	—	2	1	no	—
112	C₄²×C₃₁	—	2	1	no	—
113	C₄²×C₃₁	—	2	1	no	—
114	#32827	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	—	4	2	no	—
122	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
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	2	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	—	4	2	no	—
147	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
151	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
153	C₈×C₃₁	—	4	2	no	—
154	C₈×C₃₁	—	4	2	no	—
155	#2647	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#2641	C₂×C₃₁	1	1	yes	characteristic
157	#2092	C₂×C₃₁	1	2	yes	complemented
158	#2071	C₂×C₃₁	1	2	yes	complemented
159	#2067	C₂×C₃₁	1	2	yes	complemented
160	#2065	C₂×C₃₁	1	2	yes	complemented
161	#2059	C₂×C₃₁	1	2	yes	complemented
162	#2041	C₂×C₃₁	1	2	yes	—
163	#1832	C₂×C₃₁	1	1	yes	characteristic, complemented
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	—	8	2	no	—
170	C₂²×C₃₁	—	8	2	no	—
171	C₂²×C₃₁	#535	1	1	yes	characteristic
172	C₂²×C₃₁	#536	1	1	yes	characteristic
173	C₂²×C₃₁	#537	1	1	yes	characteristic
174	C₂²×C₃₁	#576	1	2	yes	—
175	C₂²×C₃₁	#577	1	2	yes	—
176	C₄×C₃₁	—	4	2	no	—
177	C₄×C₃₁	—	4	2	no	—
178	C₄×C₃₁	—	4	1	no	—
179	C₄×C₃₁	—	4	1	no	—
180	C₄×C₃₁	—	8	2	no	—
181	C₄×C₃₁	—	8	2	no	—
182	C₄×C₃₁	—	8	2	no	—
183	C₄×C₃₁	#562	1	1	yes	characteristic
184	C₄×C₃₁	#562	1	1	yes	characteristic
185	C₄×C₃₁	#563	1	1	yes	characteristic
186	C₄×C₃₁	#563	1	1	yes	characteristic
187	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
188	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×#107	C₂²×C₃₁	1	2	yes	—
190	C₂×#106	C₂²×C₃₁	1	2	yes	—
191	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
192	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
193	C₂×#102	C₂²×C₃₁	1	2	yes	—
194	C₂×#102	C₂²×C₃₁	1	2	yes	—
195	C₂×#102	C₂²×C₃₁	1	2	yes	—
196	C₂×#102	C₂²×C₃₁	1	2	yes	—
197	#400	—	2	2	no	—
198	#398	—	2	2	no	—
199	#394	—	2	2	no	—
200	#394	—	2	2	no	—
201	#390	C₂²×C₃₁	1	2	yes	—
202	#390	C₂²×C₃₁	1	2	yes	—
203	#389	C₂²×C₃₁	1	2	yes	—
204	#389	C₂²×C₃₁	1	2	yes	—
205	#384	C₂²×C₃₁	1	1	yes	characteristic
206	#382	C₂²×C₃₁	1	1	yes	characteristic
207	#376	C₂²×C₃₁	1	1	yes	characteristic
208	#340	C₂²×C₃₁	1	2	yes	—
209	#340	C₂²×C₃₁	1	2	yes	—
210	#336	C₂²×C₃₁	1	2	yes	—
211	#336	C₂²×C₃₁	1	2	yes	—
212	C₂×C₃₁	—	16	2	no	—
213	C₂×C₃₁	#2820	1	1	yes	characteristic
214	C₂×C₃₁	#3224	1	2	yes	—
215	C₂×C₃₁	#3241	1	2	yes	—
216	C₂×C₃₁	#3259	1	1	yes	characteristic
217	C₂×C₃₁	#3260	1	1	yes	characteristic
218	C₂²×D₄	—	2	2	no	—
219	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
220	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
221	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
222	#127	—	4	1	no	—
223	#127	—	4	1	no	—
224	#125	—	4	1	no	—
225	#125	—	4	1	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
231	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
232	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
233	C₂×C₂²⋊C₄	—	2	2	no	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
238	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
239	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
240	#107	—	4	2	no	—
241	#106	—	4	2	no	—
242	C₄⋊C₈	—	4	1	no	—
243	C₄⋊C₈	—	4	1	no	—
244	#102	—	4	2	no	—
245	#102	—	4	2	no	—
246	#102	—	4	2	no	—
247	#102	—	4	2	no	—
248	#95	—	2	2	no	—
249	#95	—	2	2	no	—
250	#95	—	2	2	no	—
251	#95	D₄×C₃₁	1	2	yes	—
252	#95	D₄×C₃₁	1	2	yes	—
253	C₃₁	#32827	1	1	yes	characteristic, direct product factor, Sylow
254	C₂⁴	—	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	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	4	4	no	—
262	C₂×D₄	—	8	2	no	—
263	C₂×D₄	—	8	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
270	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
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	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	4	no	—
285	C₄⋊C₄	—	8	2	no	—
286	C₄⋊C₄	—	8	2	no	—
287	C₄⋊C₄	—	8	2	no	—
288	C₂²⋊C₄	—	8	2	no	—
289	C₂²⋊C₄	—	8	2	no	—
290	C₂²⋊C₄	—	8	2	no	—
291	C₂²⋊C₄	—	8	2	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₄²	—	2	1	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	—	4	2	no	—
305	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	8	8	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	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	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
331	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
332	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
333	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
334	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
335	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	C₃₁×#535	1	1	yes	characteristic
346	C₂²	C₃₁×#536	1	1	yes	characteristic
347	C₂²	C₃₁×#537	1	1	yes	characteristic
348	C₂²	C₃₁×#576	1	2	yes	—
349	C₂²	C₃₁×#577	1	2	yes	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	2	no	—
357	C₄	C₃₁×#562	1	1	yes	characteristic
358	C₄	C₃₁×#562	1	1	yes	characteristic
359	C₄	C₃₁×#563	1	1	yes	characteristic
360	C₄	C₃₁×#563	1	1	yes	characteristic
361	C₂	—	16	2	no	—
362	C₂	C₃₁×#2820	1	1	yes	characteristic
363	C₂	C₃₁×#3224	1	2	yes	—
364	C₂	C₃₁×#3241	1	2	yes	—
365	C₂	C₃₁×#3259	1	1	yes	characteristic
366	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
1. C₃₁×#2647	C₂	semidirect
3. C₃₁×#2092	C₂	semidirect
4. C₃₁×#2071	C₂	semidirect
5. C₃₁×#2067	C₂	semidirect
6. C₃₁×#2065	C₂	semidirect
7. C₃₁×#2059	C₂	semidirect
9. C₃₁×#1832	C₂	semidirect
114. #32827	C₃₁	direct (N × H)
155. #2647	C₂×C₃₁	semidirect
157. #2092	C₂×C₃₁	semidirect
158. #2071	C₂×C₃₁	semidirect
159. #2067	C₂×C₃₁	semidirect
160. #2065	C₂×C₃₁	semidirect
161. #2059	C₂×C₃₁	semidirect
163. #1832	C₂×C₃₁	semidirect
253. C₃₁	#32827	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×#2647	2
C₃₁×#2641	2
C₃₁×#2092	2
C₃₁×#2071	2
C₃₁×#2067	2
C₃₁×#2065	2
C₃₁×#2059	2
C₃₁×#2041	2
C₃₁×#1832	2
#32827	31

Subgroups — C31×#32827 #21489586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32827 #21489586