Subgroups — C₃₁×#9197 #21465956

Order 7936.

624 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2905	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2619	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1545	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1458	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1286	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₃₁	—	2	1	no	—
7	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#121	—	2	1	no	—
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₃₁×#431	C₂²	1	2	yes	—
11	C₃₁×#424	—	2	4	no	—
12	C₃₁×#423	—	2	1	no	—
13	C₃₁×#423	C₂²	1	2	yes	—
14	C₃₁×#394	—	2	1	no	—
15	C₃₁×#394	C₄	1	2	yes	—
16	C₃₁×#392	—	2	1	no	—
17	C₃₁×#392	C₄	1	2	yes	—
18	C₃₁×#390	—	2	1	no	—
19	C₃₁×#382	C₂²	1	1	yes	characteristic
20	C₃₁×#331	—	2	4	no	—
21	C₃₁×#324	—	2	1	no	—
22	C₃₁×#323	—	2	1	no	—
23	C₂²×D₄×C₃₁	—	2	2	no	—
24	C₂²×D₄×C₃₁	—	2	1	no	—
25	C₂²×D₄×C₃₁	—	2	1	no	—
26	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂×M₁₆×C₃₁	—	4	2	no	—
31	C₂×M₁₆×C₃₁	—	4	2	no	—
32	C₂²×C₈×C₃₁	—	2	2	no	—
33	C₃₁×#127	—	2	2	no	—
34	C₃₁×#127	—	2	2	no	—
35	C₃₁×#121	—	4	2	no	—
36	C₃₁×#121	—	4	2	no	—
37	C₃₁×#117	—	2	2	no	—
38	C₃₁×#117	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊C₈	—	2	4	no	—
50	C₃₁×C₄⋊C₈	—	2	2	no	—
51	C₃₁×C₄⋊C₈	—	2	2	no	—
52	C₃₁×C₄⋊C₈	D₄	1	4	yes	complemented
53	C₃₁×C₂²⋊C₈	—	2	2	no	—
54	C₄×C₈×C₃₁	—	2	2	no	—
55	C₃₁×#95	—	2	1	no	—
56	C₃₁×#95	C₂×C₄	1	2	yes	—
57	C₂⁴×C₃₁	—	2	2	no	—
58	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₄×C₃₁	—	2	4	no	—
60	C₂×D₄×C₃₁	—	4	4	no	—
61	C₂×D₄×C₃₁	—	4	1	no	—
62	C₂×D₄×C₃₁	—	4	2	no	—
63	C₂×D₄×C₃₁	—	4	1	no	—
64	C₂×D₄×C₃₁	—	4	1	no	—
65	C₂×D₄×C₃₁	—	4	1	no	—
66	C₂×D₄×C₃₁	—	4	1	no	—
67	C₂×D₄×C₃₁	—	4	1	no	—
68	C₂×D₄×C₃₁	—	8	2	no	—
69	C₂×D₄×C₃₁	—	8	2	no	—
70	C₂×D₄×C₃₁	—	8	2	no	—
71	C₂×D₄×C₃₁	—	8	2	no	—
72	C₂²×C₄×C₃₁	—	2	1	no	—
73	C₂²×C₄×C₃₁	—	2	1	no	—
74	C₂²×C₄×C₃₁	—	2	1	no	—
75	C₂²×C₄×C₃₁	—	2	1	no	—
76	C₂²×C₄×C₃₁	—	2	2	no	—
77	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	D₈	1	2	yes	—
81	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
82	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	M₁₆×C₃₁	—	4	4	no	—
86	M₁₆×C₃₁	—	4	2	no	—
87	M₁₆×C₃₁	—	4	2	no	—
88	C₂×C₈×C₃₁	—	2	4	no	—
89	C₂×C₈×C₃₁	—	4	4	no	—
90	C₂×C₈×C₃₁	—	4	2	no	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	4	1	no	—
94	C₄⋊C₄×C₃₁	—	4	1	no	—
95	C₂²⋊C₄×C₃₁	—	2	2	no	—
96	C₂²⋊C₄×C₃₁	—	2	2	no	—
97	C₂²⋊C₄×C₃₁	—	4	4	no	—
98	C₂²⋊C₄×C₃₁	—	8	2	no	—
99	C₂²⋊C₄×C₃₁	—	8	2	no	—
100	C₄²×C₃₁	—	2	1	no	—
101	C₄²×C₃₁	—	2	1	no	—
102	C₄²×C₃₁	C₂×D₄	1	2	yes	—
103	C₄²×C₃₁	#41	1	2	yes	—
104	#9197	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₃₁	—	2	1	no	—
106	C₂³×C₃₁	—	2	1	no	—
107	C₂³×C₃₁	—	4	1	no	—
108	C₂³×C₃₁	—	4	1	no	—
109	C₂³×C₃₁	—	4	2	no	—
110	C₂³×C₃₁	—	4	2	no	—
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	#102	1	2	yes	—
117	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	D₄×C₃₁	—	8	4	no	—
119	D₄×C₃₁	—	8	4	no	—
120	D₄×C₃₁	—	8	8	no	—
121	D₄×C₃₁	—	8	1	no	—
122	D₄×C₃₁	—	8	1	no	—
123	D₄×C₃₁	—	8	1	no	—
124	D₄×C₃₁	—	8	1	no	—
125	C₂×C₄×C₃₁	—	2	1	no	—
126	C₂×C₄×C₃₁	—	2	1	no	—
127	C₂×C₄×C₃₁	—	2	2	no	—
128	C₂×C₄×C₃₁	—	2	2	no	—
129	C₂×C₄×C₃₁	—	2	2	no	—
130	C₂×C₄×C₃₁	—	2	2	no	—
131	C₂×C₄×C₃₁	—	2	1	no	—
132	C₂×C₄×C₃₁	—	2	1	no	—
133	C₂×C₄×C₃₁	—	4	1	no	—
134	C₂×C₄×C₃₁	—	4	1	no	—
135	C₂×C₄×C₃₁	—	4	1	no	—
136	C₂×C₄×C₃₁	—	4	1	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	#102	1	2	yes	—
141	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
143	C₂×C₄×C₃₁	#121	1	2	yes	—
144	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
145	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
149	C₈×C₃₁	—	4	4	no	—
150	C₈×C₃₁	—	8	2	no	—
151	#2905	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#2619	C₂×C₃₁	1	1	yes	characteristic
153	#1545	C₂×C₃₁	1	2	yes	complemented
154	#1458	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#1286	C₂×C₃₁	1	2	yes	complemented
156	C₂²×C₃₁	—	2	1	no	—
157	C₂²×C₃₁	—	2	1	no	—
158	C₂²×C₃₁	—	2	2	no	—
159	C₂²×C₃₁	—	4	2	no	—
160	C₂²×C₃₁	—	4	1	no	—
161	C₂²×C₃₁	—	4	1	no	—
162	C₂²×C₃₁	—	8	1	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	—	8	2	no	—
170	C₂²×C₃₁	#386	1	1	yes	characteristic
171	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
172	C₂²×C₃₁	#417	1	1	yes	characteristic
173	C₄×C₃₁	—	2	2	no	—
174	C₄×C₃₁	—	2	2	no	—
175	C₄×C₃₁	—	2	2	no	—
176	C₄×C₃₁	—	4	1	no	—
177	C₄×C₃₁	—	4	1	no	—
178	C₄×C₃₁	—	8	2	no	—
179	C₄×C₃₁	#447	1	1	yes	characteristic
180	C₄×C₃₁	#448	1	1	yes	characteristic
181	C₄×C₃₁	#480	1	1	yes	characteristic
182	C₄×C₃₁	#481	1	1	yes	characteristic
183	C₂²×M₁₆	—	2	1	no	—
184	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
185	C₂×#121	—	2	1	no	—
186	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
187	#431	C₂²×C₃₁	1	2	yes	—
188	#424	—	2	4	no	—
189	#423	—	2	1	no	—
190	#423	C₂²×C₃₁	1	2	yes	—
191	#394	—	2	1	no	—
192	#394	C₄×C₃₁	1	2	yes	—
193	#392	—	2	1	no	—
194	#392	C₄×C₃₁	1	2	yes	—
195	#390	—	2	1	no	—
196	#382	C₂²×C₃₁	1	1	yes	characteristic
197	#331	—	2	4	no	—
198	#324	—	2	1	no	—
199	#323	—	2	1	no	—
200	C₂×C₃₁	—	2	1	no	—
201	C₂×C₃₁	—	2	1	no	—
202	C₂×C₃₁	—	4	2	no	—
203	C₂×C₃₁	—	16	2	no	—
204	C₂×C₃₁	#1874	1	1	yes	characteristic
205	C₂×C₃₁	#1910	1	1	yes	characteristic
206	C₂×C₃₁	#1914	1	1	yes	characteristic
207	C₂²×D₄	—	2	2	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×D₄	—	2	1	no	—
210	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
211	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
212	C₂×M₁₆	—	2	2	no	—
213	C₂×M₁₆	—	2	2	no	—
214	C₂×M₁₆	—	4	2	no	—
215	C₂×M₁₆	—	4	2	no	—
216	C₂²×C₈	—	2	2	no	—
217	#127	—	2	2	no	—
218	#127	—	2	2	no	—
219	#121	—	4	2	no	—
220	#121	—	4	2	no	—
221	#117	—	2	2	no	—
222	#117	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
225	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
230	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
231	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
233	C₄⋊C₈	—	2	4	no	—
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	D₄×C₃₁	1	4	yes	complemented
237	C₂²⋊C₈	—	2	2	no	—
238	C₄×C₈	—	2	2	no	—
239	#95	—	2	1	no	—
240	#95	C₂×C₄×C₃₁	1	2	yes	—
241	C₃₁	#9197	1	1	yes	characteristic, direct product factor, Sylow
242	C₂⁴	—	2	2	no	—
243	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂×D₄	—	8	2	no	—
255	C₂×D₄	—	8	2	no	—
256	C₂×D₄	—	8	2	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
263	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
264	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
265	C₂²×C₄	D₈×C₃₁	1	2	yes	—
266	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
267	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
270	M₁₆	—	4	4	no	—
271	M₁₆	—	4	2	no	—
272	M₁₆	—	4	2	no	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	4	4	no	—
275	C₂×C₈	—	4	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	8	2	no	—
284	C₂²⋊C₄	—	8	2	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	C₂×D₄×C₃₁	1	2	yes	—
288	C₄²	#41×C₃₁	1	2	yes	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	4	1	no	—
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	C₃₁×#102	1	2	yes	—
301	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
302	D₄	—	8	4	no	—
303	D₄	—	8	4	no	—
304	D₄	—	8	8	no	—
305	D₄	—	8	1	no	—
306	D₄	—	8	1	no	—
307	D₄	—	8	1	no	—
308	D₄	—	8	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	C₃₁×#102	1	2	yes	—
325	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
326	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
327	C₂×C₄	C₃₁×#121	1	2	yes	—
328	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
329	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
330	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
331	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
332	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
333	C₈	—	4	4	no	—
334	C₈	—	8	2	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	C₃₁×#386	1	1	yes	characteristic
350	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
351	C₂²	C₃₁×#417	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	8	2	no	—
358	C₄	C₃₁×#447	1	1	yes	characteristic
359	C₄	C₃₁×#448	1	1	yes	characteristic
360	C₄	C₃₁×#480	1	1	yes	characteristic
361	C₄	C₃₁×#481	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	4	2	no	—
365	C₂	—	16	2	no	—
366	C₂	C₃₁×#1874	1	1	yes	characteristic
367	C₂	C₃₁×#1910	1	1	yes	characteristic
368	C₂	C₃₁×#1914	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#386
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₃₁×#2619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2905	C₂	semidirect
3. C₃₁×#1545	C₂	semidirect
4. C₃₁×#1458	C₂	semidirect
5. C₃₁×#1286	C₂	semidirect
52. C₃₁×C₄⋊C₈	D₄	semidirect
104. #9197	C₃₁	direct (N × H)
151. #2905	C₂×C₃₁	semidirect
153. #1545	C₂×C₃₁	semidirect
154. #1458	C₂×C₃₁	semidirect
155. #1286	C₂×C₃₁	semidirect
236. C₄⋊C₈	D₄×C₃₁	semidirect
241. C₃₁	#9197	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×#2905	2
C₃₁×#2619	2
C₃₁×#1545	2
C₃₁×#1458	2
C₃₁×#1286	2
#9197	31

Subgroups — C31×#9197 #21465956

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9197 #21465956