Subgroups — C₃₁×#13414 #21470173

Order 7936.

456 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#362	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2141	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2140	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1945	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×SD₁₆×C₃₁	—	2	1	no	—
10	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#124	—	2	1	no	—
12	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
14	C₃₁×#495	—	2	2	no	—
15	C₃₁×#495	—	2	2	no	—
16	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
17	C₃₁×#427	C₂²	1	1	yes	characteristic
18	C₃₁×#424	—	2	1	no	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#362	—	2	2	no	—
21	C₃₁×#361	—	2	2	no	—
22	C₃₁×#358	—	2	2	no	—
23	C₃₁×#357	—	2	2	no	—
24	C₃₁×#339	—	2	1	no	—
25	C₂²×Q₈×C₃₁	—	2	1	no	—
26	C₂²×D₄×C₃₁	—	2	1	no	—
27	C₂×Q₁₆×C₃₁	—	2	1	no	—
28	C₂×Q₁₆×C₃₁	—	2	1	no	—
29	C₂×Q₁₆×C₃₁	—	2	1	no	—
30	C₂×Q₁₆×C₃₁	—	2	1	no	—
31	C₂×Q₁₆×C₃₁	—	2	1	no	—
32	C₂×Q₁₆×C₃₁	D₄	1	2	yes	—
33	C₂×SD₁₆×C₃₁	—	2	4	no	—
34	C₂×SD₁₆×C₃₁	—	4	2	no	—
35	C₂×SD₁₆×C₃₁	—	4	2	no	—
36	C₂×D₈×C₃₁	—	2	1	no	—
37	C₂×D₈×C₃₁	—	2	1	no	—
38	C₂×D₈×C₃₁	—	2	1	no	—
39	C₂×D₈×C₃₁	—	2	1	no	—
40	C₂×D₈×C₃₁	—	2	1	no	—
41	C₂×D₈×C₃₁	D₄	1	2	yes	—
42	C₂×M₁₆×C₃₁	—	2	1	no	—
43	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×#124	—	2	4	no	—
46	C₃₁×#124	—	4	2	no	—
47	C₃₁×#117	—	2	1	no	—
48	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
49	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
50	C₃₁×M₃₂	—	2	1	no	—
51	C₃₁×M₃₂	—	2	1	no	—
52	C₃₁×M₃₂	—	2	1	no	—
53	C₃₁×M₃₂	—	2	1	no	—
54	C₂×C₁₆×C₃₁	—	2	1	no	—
55	C₂×C₁₆×C₃₁	—	2	1	no	—
56	C₃₁×#108	—	2	1	no	—
57	C₃₁×#108	—	2	1	no	—
58	C₃₁×#107	D₄	1	2	yes	—
59	C₃₁×#106	—	2	1	no	—
60	C₃₁×C₄⋊C₈	—	4	2	no	—
61	C₄×C₈×C₃₁	—	2	1	no	—
62	C₄×C₈×C₃₁	—	2	1	no	—
63	C₂⁴×C₃₁	—	4	1	no	—
64	C₂×Q₈×C₃₁	—	2	2	no	—
65	C₂×Q₈×C₃₁	—	4	1	no	—
66	C₂×Q₈×C₃₁	—	4	1	no	—
67	C₂×Q₈×C₃₁	—	4	1	no	—
68	C₂×Q₈×C₃₁	—	4	1	no	—
69	C₂×Q₈×C₃₁	—	4	1	no	—
70	C₂×D₄×C₃₁	—	2	2	no	—
71	C₂×D₄×C₃₁	—	4	1	no	—
72	C₂×D₄×C₃₁	—	4	1	no	—
73	C₂×D₄×C₃₁	—	4	1	no	—
74	C₂×D₄×C₃₁	—	4	1	no	—
75	C₂×D₄×C₃₁	—	4	1	no	—
76	C₂²×C₄×C₃₁	—	2	1	no	—
77	C₂²×C₄×C₃₁	—	4	1	no	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	Q₁₆×C₃₁	—	2	2	no	—
80	Q₁₆×C₃₁	—	2	2	no	—
81	Q₁₆×C₃₁	—	4	1	no	—
82	Q₁₆×C₃₁	—	4	1	no	—
83	SD₁₆×C₃₁	—	4	8	no	—
84	D₈×C₃₁	—	2	2	no	—
85	D₈×C₃₁	—	2	2	no	—
86	D₈×C₃₁	—	4	1	no	—
87	D₈×C₃₁	—	4	1	no	—
88	M₁₆×C₃₁	—	4	1	no	—
89	M₁₆×C₃₁	—	4	1	no	—
90	C₂×C₈×C₃₁	—	2	1	no	—
91	C₂×C₈×C₃₁	—	2	1	no	—
92	C₂×C₈×C₃₁	—	4	1	no	—
93	C₂×C₈×C₃₁	D₈	1	2	yes	—
94	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
98	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
99	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
100	C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₂²⋊C₄×C₃₁	—	4	8	no	—
102	C₂²⋊C₄×C₃₁	—	4	1	no	—
103	C₄²×C₃₁	—	2	1	no	—
104	C₄²×C₃₁	—	4	1	no	—
105	C₄²×C₃₁	SD₁₆	1	2	yes	—
106	C₁₆×C₃₁	—	4	1	no	—
107	C₁₆×C₃₁	—	4	1	no	—
108	#13414	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₃₁	—	4	2	no	—
110	C₂³×C₃₁	—	8	4	no	—
111	C₂³×C₃₁	—	8	1	no	—
112	C₂³×C₃₁	—	8	1	no	—
113	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
114	Q₈×C₃₁	—	4	2	no	—
115	Q₈×C₃₁	—	4	2	no	—
116	Q₈×C₃₁	—	8	1	no	—
117	Q₈×C₃₁	—	8	1	no	—
118	D₄×C₃₁	—	4	2	no	—
119	D₄×C₃₁	—	4	2	no	—
120	D₄×C₃₁	—	8	1	no	—
121	D₄×C₃₁	—	8	1	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	2	no	—
126	C₂×C₄×C₃₁	—	8	1	no	—
127	C₂×C₄×C₃₁	—	8	1	no	—
128	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
129	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
130	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₈×C₃₁	—	2	2	no	—
135	C₈×C₃₁	—	8	1	no	—
136	C₈×C₃₁	#121	1	1	yes	characteristic
137	C₈×C₃₁	#121	1	1	yes	characteristic
138	C₈×C₃₁	#123	1	1	yes	characteristic
139	C₈×C₃₁	#123	1	1	yes	characteristic
140	C₂×#495	C₂×C₃₁	1	1	yes	characteristic
141	C₂×#362	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂×#361	C₂×C₃₁	1	1	yes	characteristic
143	#2141	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#2140	C₂×C₃₁	1	1	yes	characteristic
145	#1945	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#1376	C₂×C₃₁	1	1	yes	characteristic, complemented
147	C₂²×C₃₁	—	2	2	no	—
148	C₂²×C₃₁	—	8	4	no	—
149	C₂²×C₃₁	—	8	2	no	—
150	C₂²×C₃₁	—	16	2	no	—
151	C₂²×C₃₁	—	16	2	no	—
152	C₂²×C₃₁	#450	1	1	yes	characteristic
153	C₂²×C₃₁	#459	1	1	yes	characteristic
154	C₂²×C₃₁	#490	1	1	yes	characteristic
155	C₄×C₃₁	—	4	2	no	—
156	C₄×C₃₁	—	8	2	no	—
157	C₄×C₃₁	—	8	2	no	—
158	C₄×C₃₁	#394	1	1	yes	characteristic
159	C₄×C₃₁	#447	1	1	yes	characteristic
160	C₄×C₃₁	#460	1	1	yes	characteristic
161	C₄×C₃₁	#486	1	1	yes	characteristic
162	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
163	C₂²×SD₁₆	—	2	1	no	—
164	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×#124	—	2	1	no	—
166	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
167	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
168	#495	—	2	2	no	—
169	#495	—	2	2	no	—
170	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
171	#427	C₂²×C₃₁	1	1	yes	characteristic
172	#424	—	2	1	no	—
173	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
174	#362	—	2	2	no	—
175	#361	—	2	2	no	—
176	#358	—	2	2	no	—
177	#357	—	2	2	no	—
178	#339	—	2	1	no	—
179	C₂×C₃₁	—	2	1	no	—
180	C₂×C₃₁	—	2	1	no	—
181	C₂×C₃₁	—	16	2	no	—
182	C₂×C₃₁	#2041	1	1	yes	characteristic
183	C₂×C₃₁	#2212	1	2	yes	—
184	C₂²×Q₈	—	2	1	no	—
185	C₂²×D₄	—	2	1	no	—
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	D₄×C₃₁	1	2	yes	—
192	C₂×SD₁₆	—	2	4	no	—
193	C₂×SD₁₆	—	4	2	no	—
194	C₂×SD₁₆	—	4	2	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	—	2	1	no	—
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	D₄×C₃₁	1	2	yes	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
203	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
204	#124	—	2	4	no	—
205	#124	—	4	2	no	—
206	#117	—	2	1	no	—
207	C₂×C₂²⋊C₄	—	4	2	no	—
208	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
209	M₃₂	—	2	1	no	—
210	M₃₂	—	2	1	no	—
211	M₃₂	—	2	1	no	—
212	M₃₂	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	#108	—	2	1	no	—
216	#108	—	2	1	no	—
217	#107	D₄×C₃₁	1	2	yes	—
218	#106	—	2	1	no	—
219	C₄⋊C₈	—	4	2	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₃₁	#13414	1	1	yes	characteristic, direct product factor, Sylow
223	C₂⁴	—	4	1	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂×Q₈	—	4	1	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	1	no	—
233	C₂×D₄	—	4	1	no	—
234	C₂×D₄	—	4	1	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	4	1	no	—
242	Q₁₆	—	4	1	no	—
243	SD₁₆	—	4	8	no	—
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	D₈	—	4	1	no	—
247	D₈	—	4	1	no	—
248	M₁₆	—	4	1	no	—
249	M₁₆	—	4	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	D₈×C₃₁	1	2	yes	—
254	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
255	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
258	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
259	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₂²⋊C₄	—	4	8	no	—
262	C₂²⋊C₄	—	4	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	4	1	no	—
265	C₄²	SD₁₆×C₃₁	1	2	yes	—
266	C₁₆	—	4	1	no	—
267	C₁₆	—	4	1	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	8	4	no	—
270	C₂³	—	8	1	no	—
271	C₂³	—	8	1	no	—
272	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	8	1	no	—
276	Q₈	—	8	1	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	8	1	no	—
280	D₄	—	8	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
288	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
289	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
290	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
292	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	8	1	no	—
295	C₈	C₃₁×#121	1	1	yes	characteristic
296	C₈	C₃₁×#121	1	1	yes	characteristic
297	C₈	C₃₁×#123	1	1	yes	characteristic
298	C₈	C₃₁×#123	1	1	yes	characteristic
299	C₂²	—	2	2	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	16	2	no	—
303	C₂²	—	16	2	no	—
304	C₂²	C₃₁×#450	1	1	yes	characteristic
305	C₂²	C₃₁×#459	1	1	yes	characteristic
306	C₂²	C₃₁×#490	1	1	yes	characteristic
307	C₄	—	4	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₃₁×#394	1	1	yes	characteristic
311	C₄	C₃₁×#447	1	1	yes	characteristic
312	C₄	C₃₁×#460	1	1	yes	characteristic
313	C₄	C₃₁×#486	1	1	yes	characteristic
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	16	2	no	—
317	C₂	C₃₁×#2041	1	1	yes	characteristic
318	C₂	C₃₁×#2212	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#362	C₂	semidirect
4. C₃₁×#2141	C₂	semidirect
6. C₃₁×#1945	C₂	semidirect
7. C₃₁×#1376	C₂	semidirect
108. #13414	C₃₁	direct (N × H)
141. C₂×#362	C₂×C₃₁	semidirect
143. #2141	C₂×C₃₁	semidirect
145. #1945	C₂×C₃₁	semidirect
146. #1376	C₂×C₃₁	semidirect
222. C₃₁	#13414	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#495	2
C₂×C₃₁×#362	2
C₂×C₃₁×#361	2
C₃₁×#2141	2
C₃₁×#2140	2
C₃₁×#1945	2
C₃₁×#1376	2
#13414	31

Subgroups — C31×#13414 #21470173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13414 #21470173