Subgroups — C₃₁×#19292 #21476051

Order 7936.

848 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3038	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1688	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1687	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1545	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
9	C₂²×D₈×C₃₁	—	2	2	no	—
10	C₂×C₃₁×#127	C₂²	1	2	yes	—
11	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#493	—	2	1	no	—
13	C₃₁×#493	—	2	1	no	—
14	C₃₁×#493	C₂²	1	4	yes	—
15	C₃₁×#460	—	2	2	no	—
16	C₃₁×#460	C₂²	1	4	yes	complemented
17	C₃₁×#459	—	2	2	no	—
18	C₃₁×#459	C₂²	1	4	yes	complemented
19	C₃₁×#446	C₂²	1	2	yes	complemented
20	C₃₁×#436	C₂²	1	2	yes	complemented
21	C₃₁×#435	—	2	2	no	—
22	C₃₁×#431	—	2	1	no	—
23	C₃₁×#431	C₂²	1	1	yes	characteristic
24	C₃₁×#423	C₂²	1	2	yes	—
25	C₃₁×#418	—	2	2	no	—
26	C₄×M₁₆×C₃₁	—	2	1	no	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#334	C₂²	1	4	yes	complemented
29	C₃₁×#331	C₂²	1	8	yes	complemented
30	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×D₄×C₃₁	—	2	2	no	—
32	C₂²×D₄×C₃₁	—	2	2	no	—
33	C₂²×D₄×C₃₁	—	2	2	no	—
34	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
35	C₂×SD₁₆×C₃₁	—	2	4	no	—
36	C₂×D₈×C₃₁	—	2	4	no	—
37	C₂×D₈×C₃₁	—	2	4	no	—
38	C₂×D₈×C₃₁	—	2	4	no	—
39	C₂×D₈×C₃₁	—	4	8	no	—
40	C₂×M₁₆×C₃₁	—	2	2	no	—
41	C₂×M₁₆×C₃₁	—	2	2	no	—
42	C₂²×C₈×C₃₁	D₄	1	2	yes	—
43	C₃₁×#127	—	2	2	no	—
44	C₃₁×#127	—	2	2	no	—
45	C₃₁×#127	—	2	2	no	—
46	C₃₁×#127	C₂³	1	4	yes	—
47	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
48	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₄×D₄×C₃₁	—	2	1	no	—
50	C₄×D₄×C₃₁	—	2	1	no	—
51	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
52	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#117	—	2	1	no	—
54	C₃₁×#117	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	—	2	1	no	—
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×C₄⋊C₈	—	2	2	no	—
58	C₃₁×C₄⋊C₈	—	2	2	no	—
59	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
60	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
61	C₃₁×#102	—	2	4	no	—
62	C₃₁×#102	—	2	4	no	—
63	C₃₁×C₂²⋊C₈	—	2	1	no	—
64	C₃₁×C₂²⋊C₈	—	2	2	no	—
65	C₃₁×#97	—	2	2	no	—
66	C₄×C₈×C₃₁	—	2	1	no	—
67	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
68	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
69	C₄×C₈×C₃₁	C₂³	1	2	yes	—
70	C₂⁴×C₃₁	—	4	2	no	—
71	#41×C₃₁	—	2	4	no	—
72	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂×D₄×C₃₁	—	2	1	no	—
74	C₂×D₄×C₃₁	—	2	4	no	—
75	C₂×D₄×C₃₁	—	2	4	no	—
76	C₂×D₄×C₃₁	—	2	4	no	—
77	C₂×D₄×C₃₁	—	4	8	no	—
78	C₂×D₄×C₃₁	—	4	2	no	—
79	C₂×D₄×C₃₁	—	4	2	no	—
80	C₂×D₄×C₃₁	—	4	2	no	—
81	C₂×D₄×C₃₁	—	4	4	no	—
82	C₂×D₄×C₃₁	—	4	4	no	—
83	C₂×D₄×C₃₁	—	8	4	no	—
84	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₃₁	—	2	1	no	—
86	C₂²×C₄×C₃₁	—	2	1	no	—
87	C₂²×C₄×C₃₁	—	2	1	no	—
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
91	SD₁₆×C₃₁	—	4	8	no	—
92	D₈×C₃₁	—	4	8	no	—
93	D₈×C₃₁	—	4	8	no	—
94	D₈×C₃₁	—	4	8	no	—
95	M₁₆×C₃₁	—	4	4	no	—
96	M₁₆×C₃₁	—	4	4	no	—
97	C₂×C₈×C₃₁	—	2	4	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	2	4	no	—
100	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²⋊C₄×C₃₁	—	2	1	no	—
107	C₂²⋊C₄×C₃₁	—	2	1	no	—
108	C₂²⋊C₄×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	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₃₁	#41	1	1	yes	characteristic
115	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
116	#19292	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₃₁	—	2	1	no	—
118	C₂³×C₃₁	—	4	4	no	—
119	C₂³×C₃₁	—	8	4	no	—
120	C₂³×C₃₁	—	8	4	no	—
121	C₂³×C₃₁	—	8	4	no	—
122	C₂³×C₃₁	#121	1	1	yes	characteristic
123	Q₈×C₃₁	—	2	2	no	—
124	D₄×C₃₁	—	2	2	no	—
125	D₄×C₃₁	—	4	2	no	—
126	D₄×C₃₁	—	4	8	no	—
127	D₄×C₃₁	—	4	4	no	—
128	D₄×C₃₁	—	4	4	no	—
129	D₄×C₃₁	—	8	4	no	—
130	D₄×C₃₁	—	8	4	no	—
131	D₄×C₃₁	—	8	4	no	—
132	D₄×C₃₁	—	8	4	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	2	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	1	no	—
137	C₂×C₄×C₃₁	—	2	1	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	#121	1	1	yes	characteristic
142	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
143	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
144	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
145	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
148	C₈×C₃₁	—	2	2	no	—
149	C₈×C₃₁	—	4	2	no	—
150	C₈×C₃₁	—	4	4	no	—
151	C₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	complemented
152	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#3038	C₂×C₃₁	1	2	yes	complemented
154	#2965	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#1688	C₂×C₃₁	1	4	yes	complemented
156	#1687	C₂×C₃₁	1	4	yes	complemented
157	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#1545	C₂×C₃₁	1	2	yes	complemented
159	C₂²×C₃₁	—	2	1	no	—
160	C₂²×C₃₁	—	2	1	no	—
161	C₂²×C₃₁	—	4	1	no	—
162	C₂²×C₃₁	—	4	1	no	—
163	C₂²×C₃₁	—	4	1	no	—
164	C₂²×C₃₁	—	8	4	no	—
165	C₂²×C₃₁	—	8	4	no	—
166	C₂²×C₃₁	—	8	4	no	—
167	C₂²×C₃₁	—	16	8	no	—
168	C₂²×C₃₁	#469	1	2	yes	—
169	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
170	C₄×C₃₁	—	2	1	no	—
171	C₄×C₃₁	—	2	1	no	—
172	C₄×C₃₁	—	4	1	no	—
173	C₄×C₃₁	—	4	1	no	—
174	C₄×C₃₁	—	4	1	no	—
175	C₄×C₃₁	#469	1	2	yes	—
176	C₄×C₃₁	#472	1	2	yes	—
177	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
178	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
179	C₂×AGL(1,ℤ/8)	—	2	2	no	—
180	C₂²×D₈	—	2	2	no	—
181	C₂×#127	C₂²×C₃₁	1	2	yes	—
182	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
183	#493	—	2	1	no	—
184	#493	—	2	1	no	—
185	#493	C₂²×C₃₁	1	4	yes	—
186	#460	—	2	2	no	—
187	#460	C₂²×C₃₁	1	4	yes	complemented
188	#459	—	2	2	no	—
189	#459	C₂²×C₃₁	1	4	yes	complemented
190	#446	C₂²×C₃₁	1	2	yes	complemented
191	#436	C₂²×C₃₁	1	2	yes	complemented
192	#435	—	2	2	no	—
193	#431	—	2	1	no	—
194	#431	C₂²×C₃₁	1	1	yes	characteristic
195	#423	C₂²×C₃₁	1	2	yes	—
196	#418	—	2	2	no	—
197	C₄×M₁₆	—	2	1	no	—
198	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
199	#334	C₂²×C₃₁	1	4	yes	complemented
200	#331	C₂²×C₃₁	1	8	yes	complemented
201	C₂×C₃₁	—	2	2	no	—
202	C₂×C₃₁	—	8	1	no	—
203	C₂×C₃₁	—	16	4	no	—
204	C₂×C₃₁	#3038	1	1	yes	characteristic
205	C₂×C₃₁	C₂×#469	1	1	yes	characteristic
206	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
207	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
208	C₂²×D₄	—	2	2	no	—
209	C₂²×D₄	—	2	2	no	—
210	C₂²×D₄	—	2	2	no	—
211	AGL(1,ℤ/8)	—	4	8	no	—
212	C₂×SD₁₆	—	2	4	no	—
213	C₂×D₈	—	2	4	no	—
214	C₂×D₈	—	2	4	no	—
215	C₂×D₈	—	2	4	no	—
216	C₂×D₈	—	4	8	no	—
217	C₂×M₁₆	—	2	2	no	—
218	C₂×M₁₆	—	2	2	no	—
219	C₂²×C₈	D₄×C₃₁	1	2	yes	—
220	#127	—	2	2	no	—
221	#127	—	2	2	no	—
222	#127	—	2	2	no	—
223	#127	C₂³×C₃₁	1	4	yes	—
224	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
225	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
226	C₄×D₄	—	2	1	no	—
227	C₄×D₄	—	2	1	no	—
228	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
229	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
230	#117	—	2	1	no	—
231	#117	D₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄²	—	2	1	no	—
233	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
237	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
238	#102	—	2	4	no	—
239	#102	—	2	4	no	—
240	C₂²⋊C₈	—	2	1	no	—
241	C₂²⋊C₈	—	2	2	no	—
242	#97	—	2	2	no	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
245	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₃₁	1	2	yes	—
247	C₃₁	#19292	1	1	yes	characteristic, direct product factor, Sylow
248	C₂⁴	—	4	2	no	—
249	#41	—	2	4	no	—
250	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	4	no	—
253	C₂×D₄	—	2	4	no	—
254	C₂×D₄	—	2	4	no	—
255	C₂×D₄	—	4	8	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	2	no	—
259	C₂×D₄	—	4	4	no	—
260	C₂×D₄	—	4	4	no	—
261	C₂×D₄	—	8	4	no	—
262	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
268	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
269	SD₁₆	—	4	8	no	—
270	D₈	—	4	8	no	—
271	D₈	—	4	8	no	—
272	D₈	—	4	8	no	—
273	M₁₆	—	4	4	no	—
274	M₁₆	—	4	4	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
279	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
283	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	2	1	no	—
286	C₂²⋊C₄	—	2	1	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	2	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
291	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
292	C₄²	#41×C₃₁	1	1	yes	characteristic
293	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
294	C₂³	—	2	1	no	—
295	C₂³	—	4	4	no	—
296	C₂³	—	8	4	no	—
297	C₂³	—	8	4	no	—
298	C₂³	—	8	4	no	—
299	C₂³	C₃₁×#121	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	D₄	—	2	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	8	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	4	no	—
306	D₄	—	8	4	no	—
307	D₄	—	8	4	no	—
308	D₄	—	8	4	no	—
309	D₄	—	8	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
319	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
320	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
321	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
322	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	4	2	no	—
327	C₈	—	4	4	no	—
328	C₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	complemented
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	8	4	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	8	4	no	—
337	C₂²	—	16	8	no	—
338	C₂²	C₃₁×#469	1	2	yes	—
339	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	C₃₁×#469	1	2	yes	—
346	C₄	C₃₁×#472	1	2	yes	—
347	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
348	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
349	C₂	—	2	2	no	—
350	C₂	—	8	1	no	—
351	C₂	—	16	4	no	—
352	C₂	C₃₁×#3038	1	1	yes	characteristic
353	C₂	C₂×C₃₁×#469	1	1	yes	characteristic
354	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ 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. C₂×C₃₁×#493	C₂	semidirect
2. C₃₁×#3038	C₂	semidirect
3. C₃₁×#2965	C₂	semidirect
4. C₃₁×#1688	C₂	semidirect
5. C₃₁×#1687	C₂	semidirect
6. C₃₁×#1566	C₂	semidirect
7. C₃₁×#1545	C₂	semidirect
16. C₃₁×#460	C₂²	semidirect
18. C₃₁×#459	C₂²	semidirect
19. C₃₁×#446	C₂²	semidirect
20. C₃₁×#436	C₂²	semidirect
28. C₃₁×#334	C₂²	semidirect
29. C₃₁×#331	C₂²	semidirect
116. #19292	C₃₁	direct (N × H)
151. C₈×C₃₁	AGL(1,ℤ/8)	semidirect
152. C₂×#493	C₂×C₃₁	semidirect
153. #3038	C₂×C₃₁	semidirect
154. #2965	C₂×C₃₁	semidirect
155. #1688	C₂×C₃₁	semidirect
156. #1687	C₂×C₃₁	semidirect
157. #1566	C₂×C₃₁	semidirect
158. #1545	C₂×C₃₁	semidirect
187. #460	C₂²×C₃₁	semidirect
189. #459	C₂²×C₃₁	semidirect
190. #446	C₂²×C₃₁	semidirect
191. #436	C₂²×C₃₁	semidirect
199. #334	C₂²×C₃₁	semidirect
200. #331	C₂²×C₃₁	semidirect
247. C₃₁	#19292	direct (N × H)
328. C₈	C₃₁×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#493	2
C₃₁×#3038	2
C₃₁×#2965	2
C₃₁×#1688	2
C₃₁×#1687	2
C₃₁×#1566	2
C₃₁×#1545	2
#19292	31

Subgroups — C31×#19292 #21476051

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19292 #21476051