Subgroups — C₃₁×#13440 #21470199

Order 7936.

398 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×SD₃₂	—	2	2	no	—
11	C₂×C₃₁×SD₃₂	—	2	2	no	—
12	C₂×C₃₁×SD₃₂	—	2	2	no	—
13	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
17	C₃₁×#419	—	2	1	no	—
18	C₃₁×#418	—	2	1	no	—
19	C₂×C₃₁×#101	—	2	1	no	—
20	C₂×C₃₁×#100	—	2	1	no	—
21	C₃₁×#405	C₂²	1	1	yes	characteristic
22	C₃₁×C₄⋊C₁₆	—	2	2	no	—
23	C₃₁×#362	—	2	2	no	—
24	C₃₁×#361	—	2	2	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×D₈×C₃₁	—	2	1	no	—
34	C₂×D₈×C₃₁	—	2	1	no	—
35	C₂×D₈×C₃₁	—	2	1	no	—
36	C₂×D₈×C₃₁	—	2	1	no	—
37	C₂×D₈×C₃₁	—	2	1	no	—
38	C₂×D₈×C₃₁	D₄	1	2	yes	—
39	C₂×M₁₆×C₃₁	—	2	1	no	—
40	C₂×M₁₆×C₃₁	—	2	1	no	—
41	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#117	D₄	1	1	yes	characteristic
44	C₃₁×SD₃₂	—	2	8	no	—
45	C₃₁×M₃₂	—	2	1	no	—
46	C₃₁×M₃₂	—	2	1	no	—
47	C₂×C₁₆×C₃₁	—	2	1	no	—
48	C₂×C₁₆×C₃₁	—	2	1	no	—
49	C₂×C₁₆×C₃₁	—	2	1	no	—
50	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
51	C₃₁×#108	—	2	1	no	—
52	C₃₁×#108	—	2	1	no	—
53	C₃₁×#108	—	2	1	no	—
54	C₃₁×#108	—	2	1	no	—
55	C₃₁×#103	—	4	2	no	—
56	C₃₁×#102	—	4	2	no	—
57	C₃₁×#101	—	4	2	no	—
58	C₃₁×#100	—	4	2	no	—
59	C₃₁×#97	—	2	1	no	—
60	C₄×C₈×C₃₁	—	2	1	no	—
61	C₂⁴×C₃₁	—	4	1	no	—
62	C₂×Q₈×C₃₁	—	2	2	no	—
63	C₂×Q₈×C₃₁	—	4	1	no	—
64	C₂×Q₈×C₃₁	—	4	1	no	—
65	C₂×Q₈×C₃₁	—	4	1	no	—
66	C₂×Q₈×C₃₁	—	4	1	no	—
67	C₂×Q₈×C₃₁	—	4	1	no	—
68	C₂×D₄×C₃₁	—	2	2	no	—
69	C₂×D₄×C₃₁	—	4	1	no	—
70	C₂×D₄×C₃₁	—	4	1	no	—
71	C₂×D₄×C₃₁	—	4	1	no	—
72	C₂×D₄×C₃₁	—	4	1	no	—
73	C₂×D₄×C₃₁	—	4	1	no	—
74	C₂²×C₄×C₃₁	—	4	1	no	—
75	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	Q₁₆×C₃₁	—	2	2	no	—
77	Q₁₆×C₃₁	—	2	2	no	—
78	Q₁₆×C₃₁	—	4	1	no	—
79	Q₁₆×C₃₁	—	4	1	no	—
80	D₈×C₃₁	—	2	2	no	—
81	D₈×C₃₁	—	2	2	no	—
82	D₈×C₃₁	—	4	1	no	—
83	D₈×C₃₁	—	4	1	no	—
84	M₁₆×C₃₁	—	2	1	no	—
85	M₁₆×C₃₁	—	2	1	no	—
86	M₁₆×C₃₁	—	4	1	no	—
87	M₁₆×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	—	4	1	no	—
92	C₂×C₈×C₃₁	—	4	1	no	—
93	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
97	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
98	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
99	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
100	C₂²⋊C₄×C₃₁	—	2	1	no	—
101	C₄²×C₃₁	—	2	1	no	—
102	C₁₆×C₃₁	—	2	2	no	—
103	C₁₆×C₃₁	—	4	1	no	—
104	#13440	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₃₁	—	4	2	no	—
106	C₂³×C₃₁	—	8	1	no	—
107	C₂³×C₃₁	—	8	1	no	—
108	C₂³×C₃₁	—	16	2	no	—
109	C₂³×C₃₁	#135	1	1	yes	characteristic
110	Q₈×C₃₁	—	4	2	no	—
111	Q₈×C₃₁	—	4	2	no	—
112	Q₈×C₃₁	—	8	1	no	—
113	Q₈×C₃₁	—	8	1	no	—
114	D₄×C₃₁	—	4	2	no	—
115	D₄×C₃₁	—	4	2	no	—
116	D₄×C₃₁	—	8	1	no	—
117	D₄×C₃₁	—	8	1	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	8	1	no	—
121	C₂×C₄×C₃₁	—	8	1	no	—
122	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
123	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
124	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₈×C₃₁	—	4	1	no	—
129	C₈×C₃₁	—	8	1	no	—
130	C₈×C₃₁	—	8	1	no	—
131	C₈×C₃₁	#120	1	1	yes	characteristic
132	C₈×C₃₁	#123	1	1	yes	characteristic
133	C₈×C₃₁	#123	1	1	yes	characteristic
134	C₈×C₃₁	#124	1	1	yes	characteristic
135	C₂²×SD₃₂	C₂×C₃₁	1	1	yes	characteristic
136	#2151	C₂×C₃₁	1	1	yes	characteristic, complemented
137	C₂×#362	C₂×C₃₁	1	1	yes	characteristic, complemented
138	C₂×#361	C₂×C₃₁	1	1	yes	characteristic
139	#1979	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#1978	C₂×C₃₁	1	1	yes	characteristic
141	#1381	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂²×C₃₁	—	2	2	no	—
143	C₂²×C₃₁	—	8	2	no	—
144	C₂²×C₃₁	—	16	2	no	—
145	C₂²×C₃₁	—	16	2	no	—
146	C₂²×C₃₁	—	16	2	no	—
147	C₂²×C₃₁	#463	1	1	yes	characteristic
148	C₂²×C₃₁	#463	1	1	yes	characteristic
149	C₂²×C₃₁	#466	1	1	yes	characteristic
150	C₄×C₃₁	—	4	2	no	—
151	C₄×C₃₁	—	8	2	no	—
152	C₄×C₃₁	#394	1	1	yes	characteristic
153	C₄×C₃₁	#459	1	1	yes	characteristic
154	C₄×C₃₁	#459	1	1	yes	characteristic
155	C₄×C₃₁	#469	1	1	yes	characteristic
156	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
157	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
158	C₂×SD₃₂	—	2	2	no	—
159	C₂×SD₃₂	—	2	2	no	—
160	C₂×SD₃₂	—	2	2	no	—
161	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
162	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
165	#419	—	2	1	no	—
166	#418	—	2	1	no	—
167	C₂×#101	—	2	1	no	—
168	C₂×#100	—	2	1	no	—
169	#405	C₂²×C₃₁	1	1	yes	characteristic
170	C₄⋊C₁₆	—	2	2	no	—
171	#362	—	2	2	no	—
172	#361	—	2	2	no	—
173	C₂×C₃₁	—	2	1	no	—
174	C₂×C₃₁	—	2	1	no	—
175	C₂×C₃₁	—	16	2	no	—
176	C₂×C₃₁	#2054	1	1	yes	characteristic
177	C₂×C₃₁	#2223	1	2	yes	—
178	C₂²×Q₈	—	2	1	no	—
179	C₂²×D₄	—	2	1	no	—
180	C₂×Q₁₆	—	2	1	no	—
181	C₂×Q₁₆	—	2	1	no	—
182	C₂×Q₁₆	—	2	1	no	—
183	C₂×Q₁₆	—	2	1	no	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	D₄×C₃₁	1	2	yes	—
186	C₂×D₈	—	2	1	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	—	2	1	no	—
189	C₂×D₈	—	2	1	no	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	D₄×C₃₁	1	2	yes	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
195	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
196	#117	D₄×C₃₁	1	1	yes	characteristic
197	SD₃₂	—	2	8	no	—
198	M₃₂	—	2	1	no	—
199	M₃₂	—	2	1	no	—
200	C₂×C₁₆	—	2	1	no	—
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	—	2	1	no	—
203	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
204	#108	—	2	1	no	—
205	#108	—	2	1	no	—
206	#108	—	2	1	no	—
207	#108	—	2	1	no	—
208	#103	—	4	2	no	—
209	#102	—	4	2	no	—
210	#101	—	4	2	no	—
211	#100	—	4	2	no	—
212	#97	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₃₁	#13440	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	—	4	1	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂×Q₈	—	4	1	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	1	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂×D₄	—	4	1	no	—
227	C₂×D₄	—	4	1	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
230	Q₁₆	—	2	2	no	—
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	4	1	no	—
233	Q₁₆	—	4	1	no	—
234	D₈	—	2	2	no	—
235	D₈	—	2	2	no	—
236	D₈	—	4	1	no	—
237	D₈	—	4	1	no	—
238	M₁₆	—	2	1	no	—
239	M₁₆	—	2	1	no	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	4	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
250	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
251	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
252	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
253	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₁₆	—	2	2	no	—
257	C₁₆	—	4	1	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	8	1	no	—
260	C₂³	—	8	1	no	—
261	C₂³	—	16	2	no	—
262	C₂³	C₃₁×#135	1	1	yes	characteristic
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	8	1	no	—
266	Q₈	—	8	1	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	8	1	no	—
270	D₄	—	8	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
276	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
277	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
278	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
281	C₈	—	4	1	no	—
282	C₈	—	8	1	no	—
283	C₈	—	8	1	no	—
284	C₈	C₃₁×#120	1	1	yes	characteristic
285	C₈	C₃₁×#123	1	1	yes	characteristic
286	C₈	C₃₁×#123	1	1	yes	characteristic
287	C₈	C₃₁×#124	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	16	2	no	—
291	C₂²	—	16	2	no	—
292	C₂²	—	16	2	no	—
293	C₂²	C₃₁×#463	1	1	yes	characteristic
294	C₂²	C₃₁×#463	1	1	yes	characteristic
295	C₂²	C₃₁×#466	1	1	yes	characteristic
296	C₄	—	4	2	no	—
297	C₄	—	8	2	no	—
298	C₄	C₃₁×#394	1	1	yes	characteristic
299	C₄	C₃₁×#459	1	1	yes	characteristic
300	C₄	C₃₁×#459	1	1	yes	characteristic
301	C₄	C₃₁×#469	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	16	2	no	—
305	C₂	C₃₁×#2054	1	1	yes	characteristic
306	C₂	C₃₁×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	#466
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 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2151	C₂	semidirect
3. C₂×C₃₁×#362	C₂	semidirect
5. C₃₁×#1979	C₂	semidirect
7. C₃₁×#1381	C₂	semidirect
104. #13440	C₃₁	direct (N × H)
136. #2151	C₂×C₃₁	semidirect
137. C₂×#362	C₂×C₃₁	semidirect
139. #1979	C₂×C₃₁	semidirect
141. #1381	C₂×C₃₁	semidirect
214. C₃₁	#13440	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×SD₃₂	2
C₃₁×#2151	2
C₂×C₃₁×#362	2
C₂×C₃₁×#361	2
C₃₁×#1979	2
C₃₁×#1978	2
C₃₁×#1381	2
#13440	31

Subgroups — C31×#13440 #21470199

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13440 #21470199