Subgroups — C₃₁×#33720 #21490479

Order 7936.

858 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#509	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3155	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2250	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2249	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2246	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
10	C₂×C₃₁×#135	C₂²	1	2	yes	—
11	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#535	—	2	2	no	—
13	C₃₁×#509	—	2	8	no	—
14	C₃₁×#508	—	2	4	no	—
15	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	complemented
17	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	complemented
18	C₂×C₃₁×D₁₆	—	2	4	no	—
19	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
20	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
21	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃₁×D₁₆	C₂²	1	4	yes	complemented
23	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#496	—	2	4	no	—
26	C₃₁×#495	C₂²	1	4	yes	complemented
27	C₃₁×#493	C₂²	1	4	yes	complemented
28	C₃₁×#405	C₂²	1	1	yes	characteristic
29	C₃₁×#346	C₂²	1	2	yes	complemented
30	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
31	C₂×#41×C₃₁	—	2	2	no	—
32	C₂²×D₄×C₃₁	—	2	2	no	—
33	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
34	C₃₁×#135	—	2	4	no	—
35	C₃₁×#135	—	2	4	no	—
36	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
37	C₂×SD₁₆×C₃₁	—	2	4	no	—
38	C₂×SD₁₆×C₃₁	—	2	2	no	—
39	C₂×D₈×C₃₁	—	2	4	no	—
40	C₂×D₈×C₃₁	—	2	4	no	—
41	C₂×D₈×C₃₁	—	2	4	no	—
42	C₂×D₈×C₃₁	—	2	2	no	—
43	C₂×D₈×C₃₁	C₂³	1	2	yes	—
44	C₂×D₈×C₃₁	C₂³	1	4	yes	—
45	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#127	—	2	4	no	—
48	C₃₁×#124	—	2	4	no	—
49	C₃₁×#121	—	4	4	no	—
50	C₃₁×#120	—	4	4	no	—
51	C₃₁×#117	D₄	1	1	yes	characteristic
52	C₃₁×Q₃₂	—	2	2	no	—
53	C₃₁×SD₃₂	—	2	4	no	—
54	C₃₁×SD₃₂	—	2	8	no	—
55	C₃₁×D₁₆	—	2	4	no	—
56	C₃₁×D₁₆	—	2	4	no	—
57	C₃₁×D₁₆	—	2	2	no	—
58	C₃₁×D₁₆	—	2	8	no	—
59	C₃₁×M₃₂	D₄	1	4	yes	complemented
60	C₂×C₁₆×C₃₁	D₄	1	4	yes	complemented
61	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
64	C₃₁×#97	D₄	1	2	yes	—
65	C₄×C₈×C₃₁	C₂³	1	2	yes	—
66	C₂⁴×C₃₁	—	4	2	no	—
67	#41×C₃₁	—	4	4	no	—
68	#41×C₃₁	—	4	4	no	—
69	C₂×Q₈×C₃₁	—	2	2	no	—
70	C₂×D₄×C₃₁	—	2	4	no	—
71	C₂×D₄×C₃₁	—	2	2	no	—
72	C₂×D₄×C₃₁	—	4	2	no	—
73	C₂×D₄×C₃₁	—	4	4	no	—
74	C₂×D₄×C₃₁	—	4	4	no	—
75	C₂×D₄×C₃₁	—	4	2	no	—
76	C₂×D₄×C₃₁	—	4	8	no	—
77	C₂²×C₄×C₃₁	—	4	2	no	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	Q₁₆×C₃₁	—	2	2	no	—
80	Q₁₆×C₃₁	—	2	2	no	—
81	SD₁₆×C₃₁	—	4	4	no	—
82	SD₁₆×C₃₁	—	4	8	no	—
83	D₈×C₃₁	—	2	4	no	—
84	D₈×C₃₁	—	2	2	no	—
85	D₈×C₃₁	—	2	4	no	—
86	D₈×C₃₁	—	2	2	no	—
87	D₈×C₃₁	—	4	4	no	—
88	D₈×C₃₁	—	4	8	no	—
89	M₁₆×C₃₁	D₈	1	4	yes	—
90	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
96	C₂²⋊C₄×C₃₁	—	4	8	no	—
97	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
98	C₄²×C₃₁	C₂×D₄	1	2	yes	—
99	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
100	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
101	C₁₆×C₃₁	C₂×D₄	1	4	yes	—
102	#33720	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	—	4	4	no	—
104	C₂³×C₃₁	—	4	2	no	—
105	C₂³×C₃₁	—	8	4	no	—
106	C₂³×C₃₁	—	16	4	no	—
107	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
108	Q₈×C₃₁	—	4	2	no	—
109	Q₈×C₃₁	—	4	2	no	—
110	D₄×C₃₁	—	4	4	no	—
111	D₄×C₃₁	—	4	2	no	—
112	D₄×C₃₁	—	4	2	no	—
113	D₄×C₃₁	—	4	4	no	—
114	D₄×C₃₁	—	8	4	no	—
115	D₄×C₃₁	—	8	4	no	—
116	D₄×C₃₁	—	16	8	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	8	4	no	—
119	C₂×C₄×C₃₁	#127	1	2	yes	—
120	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
121	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₃₁	#127	1	2	yes	—
125	C₈×C₃₁	C₂×D₈	1	4	yes	—
126	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×#509	C₂×C₃₁	1	2	yes	complemented
129	C₂×#508	C₂×C₃₁	1	1	yes	characteristic, complemented
130	C₂²×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#3155	C₂×C₃₁	1	2	yes	complemented
132	#2250	C₂×C₃₁	1	4	yes	complemented
133	#2249	C₂×C₃₁	1	2	yes	complemented
134	#2246	C₂×C₃₁	1	2	yes	complemented
135	#2108	C₂×C₃₁	1	1	yes	characteristic, complemented
136	C₂²×C₃₁	—	2	2	no	—
137	C₂²×C₃₁	—	8	2	no	—
138	C₂²×C₃₁	—	8	4	no	—
139	C₂²×C₃₁	—	8	4	no	—
140	C₂²×C₃₁	—	8	8	no	—
141	C₂²×C₃₁	—	16	8	no	—
142	C₂²×C₃₁	#493	1	2	yes	—
143	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₃₁	—	2	4	no	—
145	C₄×C₃₁	—	8	2	no	—
146	C₄×C₃₁	#493	1	2	yes	—
147	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
148	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
149	C₂×AGL(1,ℤ/8)	—	2	2	no	—
150	C₂×#135	C₂²×C₃₁	1	2	yes	—
151	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
152	#535	—	2	2	no	—
153	#509	—	2	8	no	—
154	#508	—	2	4	no	—
155	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic, complemented
156	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	complemented
157	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	complemented
158	C₂×D₁₆	—	2	4	no	—
159	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
160	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
161	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
162	C₂×D₁₆	C₂²×C₃₁	1	4	yes	complemented
163	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
164	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
165	#496	—	2	4	no	—
166	#495	C₂²×C₃₁	1	4	yes	complemented
167	#493	C₂²×C₃₁	1	4	yes	complemented
168	#405	C₂²×C₃₁	1	1	yes	characteristic
169	#346	C₂²×C₃₁	1	2	yes	complemented
170	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
171	C₂×C₃₁	—	2	2	no	—
172	C₂×C₃₁	—	16	2	no	—
173	C₂×C₃₁	—	16	4	no	—
174	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
175	C₂×C₃₁	#3415	1	2	yes	—
176	C₂×#41	—	2	2	no	—
177	C₂²×D₄	—	2	2	no	—
178	AGL(1,ℤ/8)	—	4	8	no	—
179	#135	—	2	4	no	—
180	#135	—	2	4	no	—
181	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
182	C₂×SD₁₆	—	2	2	no	—
183	C₂×SD₁₆	—	2	4	no	—
184	C₂×D₈	—	2	4	no	—
185	C₂×D₈	—	2	4	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	2	4	no	—
188	C₂×D₈	C₂³×C₃₁	1	4	yes	—
189	C₂×D₈	C₂³×C₃₁	1	2	yes	—
190	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
192	#127	—	2	4	no	—
193	#124	—	2	4	no	—
194	#121	—	4	4	no	—
195	#120	—	4	4	no	—
196	#117	D₄×C₃₁	1	1	yes	characteristic
197	Q₃₂	—	2	2	no	—
198	SD₃₂	—	2	4	no	—
199	SD₃₂	—	2	8	no	—
200	D₁₆	—	2	4	no	—
201	D₁₆	—	2	4	no	—
202	D₁₆	—	2	2	no	—
203	D₁₆	—	2	8	no	—
204	M₃₂	D₄×C₃₁	1	4	yes	complemented
205	C₂×C₁₆	D₄×C₃₁	1	4	yes	complemented
206	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
207	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
208	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
209	#97	D₄×C₃₁	1	2	yes	—
210	C₄×C₈	C₂³×C₃₁	1	2	yes	—
211	C₃₁	#33720	1	1	yes	characteristic, direct product factor, Sylow
212	C₂⁴	—	4	2	no	—
213	#41	—	4	4	no	—
214	#41	—	4	4	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	4	no	—
220	C₂×D₄	—	4	4	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	8	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	2	2	no	—
227	SD₁₆	—	4	4	no	—
228	SD₁₆	—	4	8	no	—
229	D₈	—	2	4	no	—
230	D₈	—	2	2	no	—
231	D₈	—	2	4	no	—
232	D₈	—	2	2	no	—
233	D₈	—	4	4	no	—
234	D₈	—	4	8	no	—
235	M₁₆	D₈×C₃₁	1	4	yes	—
236	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
237	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
238	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
239	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
240	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
241	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
242	C₂²⋊C₄	—	4	8	no	—
243	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
244	C₄²	C₂×D₄×C₃₁	1	2	yes	—
245	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
246	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
247	C₁₆	C₂×D₄×C₃₁	1	4	yes	—
248	C₂³	—	4	4	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	8	4	no	—
251	C₂³	—	16	4	no	—
252	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	4	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	2	no	—
258	D₄	—	4	4	no	—
259	D₄	—	8	4	no	—
260	D₄	—	8	4	no	—
261	D₄	—	16	8	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	C₃₁×#127	1	2	yes	—
265	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
266	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
269	C₈	C₃₁×#127	1	2	yes	—
270	C₈	C₂×D₈×C₃₁	1	4	yes	—
271	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
272	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
273	C₂²	—	2	2	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	8	4	no	—
277	C₂²	—	8	8	no	—
278	C₂²	—	16	8	no	—
279	C₂²	C₃₁×#493	1	2	yes	—
280	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	8	2	no	—
283	C₄	C₃₁×#493	1	2	yes	—
284	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
285	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	16	4	no	—
288	C₂	—	16	2	no	—
289	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
290	C₂	C₃₁×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#509	C₂	semidirect
2. C₂×C₃₁×#508	C₂	semidirect
3. C₂²×C₃₁×D₁₆	C₂	semidirect
4. C₃₁×#3155	C₂	semidirect
5. C₃₁×#2250	C₂	semidirect
6. C₃₁×#2249	C₂	semidirect
7. C₃₁×#2246	C₂	semidirect
8. C₃₁×#2108	C₂	semidirect
15. C₂×C₃₁×Q₃₂	C₂²	semidirect
16. C₂×C₃₁×SD₃₂	C₂²	semidirect
17. C₂×C₃₁×SD₃₂	C₂²	semidirect
21. C₂×C₃₁×D₁₆	C₂²	semidirect
22. C₂×C₃₁×D₁₆	C₂²	semidirect
26. C₃₁×#495	C₂²	semidirect
27. C₃₁×#493	C₂²	semidirect
29. C₃₁×#346	C₂²	semidirect
30. C₄×C₁₆×C₃₁	C₂²	semidirect
59. C₃₁×M₃₂	D₄	semidirect
60. C₂×C₁₆×C₃₁	D₄	semidirect
102. #33720	C₃₁	direct (N × H)
128. C₂×#509	C₂×C₃₁	semidirect
129. C₂×#508	C₂×C₃₁	semidirect
130. C₂²×D₁₆	C₂×C₃₁	semidirect
131. #3155	C₂×C₃₁	semidirect
132. #2250	C₂×C₃₁	semidirect
133. #2249	C₂×C₃₁	semidirect
134. #2246	C₂×C₃₁	semidirect
135. #2108	C₂×C₃₁	semidirect
155. C₂×Q₃₂	C₂²×C₃₁	semidirect
156. C₂×SD₃₂	C₂²×C₃₁	semidirect
157. C₂×SD₃₂	C₂²×C₃₁	semidirect
161. C₂×D₁₆	C₂²×C₃₁	semidirect
162. C₂×D₁₆	C₂²×C₃₁	semidirect
166. #495	C₂²×C₃₁	semidirect
167. #493	C₂²×C₃₁	semidirect
169. #346	C₂²×C₃₁	semidirect
170. C₄×C₁₆	C₂²×C₃₁	semidirect
204. M₃₂	D₄×C₃₁	semidirect
205. C₂×C₁₆	D₄×C₃₁	semidirect
211. C₃₁	#33720	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×#509	2
C₂×C₃₁×#508	2
C₂²×C₃₁×D₁₆	2
C₃₁×#3155	2
C₃₁×#2250	2
C₃₁×#2249	2
C₃₁×#2246	2
C₃₁×#2108	2
#33720	31

Subgroups — C31×#33720 #21490479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33720 #21490479