Subgroups — C₃₁×#33582 #21490341

Order 7936.

806 conjugacy classes of subgroups.

The subgroup lattice has 312 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×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2207	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2206	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2150	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2140	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₃₁	—	2	1	no	—
10	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×#127	—	2	1	no	—
12	C₄×#41×C₃₁	—	2	1	no	—
13	C₃₁×#509	—	2	8	no	—
14	C₂×C₃₁×SD₃₂	C₂²	1	4	yes	complemented
15	C₂×C₃₁×D₁₆	C₂²	1	4	yes	complemented
16	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
17	C₃₁×#493	—	2	2	no	—
18	C₃₁×#493	C₂²	1	4	yes	—
19	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
20	C₄×SD₁₆×C₃₁	—	2	2	no	—
21	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
22	C₃₁×#427	C₂²	1	1	yes	characteristic
23	C₃₁×#416	—	2	1	no	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×C₄⋊C₁₆	C₂²	1	4	yes	complemented
26	C₃₁×#357	C₂²	1	8	yes	complemented
27	C₂×#41×C₃₁	—	2	1	no	—
28	C₂²×D₄×C₃₁	—	2	2	no	—
29	C₂²×D₄×C₃₁	—	4	2	no	—
30	C₃₁×#135	—	2	2	no	—
31	C₃₁×#135	D₄	1	4	yes	complemented
32	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
33	C₂×SD₁₆×C₃₁	—	2	1	no	—
34	C₂×D₈×C₃₁	—	2	4	no	—
35	C₂×D₈×C₃₁	—	2	2	no	—
36	C₂×D₈×C₃₁	—	2	4	no	—
37	C₂×D₈×C₃₁	—	2	4	no	—
38	C₂×D₈×C₃₁	—	4	4	no	—
39	C₂×D₈×C₃₁	C₂³	1	4	yes	—
40	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#127	—	2	4	no	—
44	C₃₁×#127	—	4	1	no	—
45	C₃₁×#127	—	4	1	no	—
46	C₄×Q₈×C₃₁	—	2	2	no	—
47	C₄×D₄×C₃₁	—	2	2	no	—
48	C₄×D₄×C₃₁	—	4	2	no	—
49	C₃₁×#117	—	2	1	no	—
50	C₃₁×#117	—	4	1	no	—
51	C₂×C₄²×C₃₁	—	4	1	no	—
52	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
53	C₃₁×SD₃₂	—	2	8	no	—
54	C₃₁×D₁₆	—	2	8	no	—
55	C₃₁×M₃₂	—	2	4	no	—
56	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
57	C₃₁×#107	C₂³	1	2	yes	—
58	C₃₁×#106	—	2	1	no	—
59	C₃₁×#103	—	2	2	no	—
60	C₃₁×#102	—	2	2	no	—
61	C₄×C₈×C₃₁	D₄	1	2	yes	—
62	C₄×C₈×C₃₁	C₂³	1	2	yes	—
63	C₂⁴×C₃₁	—	4	2	no	—
64	#41×C₃₁	—	2	4	no	—
65	#41×C₃₁	—	4	2	no	—
66	C₂×Q₈×C₃₁	—	2	1	no	—
67	C₂×D₄×C₃₁	—	2	1	no	—
68	C₂×D₄×C₃₁	—	2	4	no	—
69	C₂×D₄×C₃₁	—	4	1	no	—
70	C₂×D₄×C₃₁	—	4	4	no	—
71	C₂×D₄×C₃₁	—	4	4	no	—
72	C₂×D₄×C₃₁	—	4	4	no	—
73	C₂×D₄×C₃₁	—	4	2	no	—
74	C₂×D₄×C₃₁	—	4	4	no	—
75	C₂×D₄×C₃₁	—	8	8	no	—
76	C₂²×C₄×C₃₁	—	4	1	no	—
77	C₂²×C₄×C₃₁	—	4	2	no	—
78	C₂²×C₄×C₃₁	D₈	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	Q₁₆×C₃₁	—	2	1	no	—
81	Q₁₆×C₃₁	C₂×D₄	1	2	yes	complemented
82	SD₁₆×C₃₁	—	2	4	no	—
83	D₈×C₃₁	—	2	4	no	—
84	D₈×C₃₁	—	2	1	no	—
85	D₈×C₃₁	—	2	4	no	—
86	D₈×C₃₁	—	4	8	no	—
87	D₈×C₃₁	—	4	4	no	—
88	D₈×C₃₁	C₂×D₄	1	2	yes	complemented
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₃₁	#41	1	2	yes	—
94	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₃₁	—	2	2	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₂²⋊C₄×C₃₁	—	4	1	no	—
98	C₂²⋊C₄×C₃₁	—	4	2	no	—
99	C₄²×C₃₁	—	4	1	no	—
100	C₄²×C₃₁	—	4	2	no	—
101	C₄²×C₃₁	D₈	1	1	yes	characteristic
102	C₄²×C₃₁	D₈	1	1	yes	characteristic
103	C₄²×C₃₁	C₂×D₄	1	2	yes	—
104	C₁₆×C₃₁	—	2	4	no	—
105	#33582	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₃₁	—	4	1	no	—
107	C₂³×C₃₁	—	4	4	no	—
108	C₂³×C₃₁	—	8	8	no	—
109	C₂³×C₃₁	—	8	4	no	—
110	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
111	Q₈×C₃₁	—	2	2	no	—
112	Q₈×C₃₁	—	4	1	no	—
113	D₄×C₃₁	—	2	2	no	—
114	D₄×C₃₁	—	4	4	no	—
115	D₄×C₃₁	—	4	4	no	—
116	D₄×C₃₁	—	4	1	no	—
117	D₄×C₃₁	—	4	4	no	—
118	D₄×C₃₁	—	8	8	no	—
119	D₄×C₃₁	—	8	4	no	—
120	D₄×C₃₁	—	8	8	no	—
121	C₂×C₄×C₃₁	—	2	4	no	—
122	C₂×C₄×C₃₁	—	4	1	no	—
123	C₂×C₄×C₃₁	—	4	4	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	8	4	no	—
126	C₂×C₄×C₃₁	#121	1	2	yes	—
127	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₃₁	—	2	2	no	—
133	C₈×C₃₁	#121	1	2	yes	—
134	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
136	C₂×#509	C₂×C₃₁	1	2	yes	complemented
137	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
138	C₄×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#2207	C₂×C₃₁	1	4	yes	complemented
140	#2206	C₂×C₃₁	1	4	yes	complemented
141	#2150	C₂×C₃₁	1	1	yes	characteristic, complemented
142	#2140	C₂×C₃₁	1	2	yes	complemented
143	C₂²×C₃₁	—	2	2	no	—
144	C₂²×C₃₁	—	4	2	no	—
145	C₂²×C₃₁	—	8	2	no	—
146	C₂²×C₃₁	—	8	8	no	—
147	C₂²×C₃₁	—	8	4	no	—
148	C₂²×C₃₁	—	16	8	no	—
149	C₂²×C₃₁	#459	1	2	yes	—
150	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
151	C₄×C₃₁	—	2	2	no	—
152	C₄×C₃₁	—	4	2	no	—
153	C₄×C₃₁	—	8	2	no	—
154	C₄×C₃₁	#459	1	2	yes	—
155	C₄×C₃₁	#509	1	4	yes	complemented
156	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
157	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
159	C₂²×D₈	—	2	1	no	—
160	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
161	C₂×#127	—	2	1	no	—
162	C₄×#41	—	2	1	no	—
163	#509	—	2	8	no	—
164	C₂×SD₃₂	C₂²×C₃₁	1	4	yes	complemented
165	C₂×D₁₆	C₂²×C₃₁	1	4	yes	complemented
166	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
167	#493	—	2	2	no	—
168	#493	C₂²×C₃₁	1	4	yes	—
169	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
170	C₄×SD₁₆	—	2	2	no	—
171	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
172	#427	C₂²×C₃₁	1	1	yes	characteristic
173	#416	—	2	1	no	—
174	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
175	C₄⋊C₁₆	C₂²×C₃₁	1	4	yes	complemented
176	#357	C₂²×C₃₁	1	8	yes	complemented
177	C₂×C₃₁	—	2	2	no	—
178	C₂×C₃₁	—	8	2	no	—
179	C₂×C₃₁	—	16	4	no	—
180	C₂×C₃₁	C₂×#459	1	1	yes	characteristic
181	C₂×C₃₁	C₂×#509	1	2	yes	—
182	C₂×#41	—	2	1	no	—
183	C₂²×D₄	—	2	2	no	—
184	C₂²×D₄	—	4	2	no	—
185	#135	—	2	2	no	—
186	#135	D₄×C₃₁	1	4	yes	complemented
187	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
188	C₂×SD₁₆	—	2	1	no	—
189	C₂×D₈	—	2	4	no	—
190	C₂×D₈	—	2	4	no	—
191	C₂×D₈	—	2	2	no	—
192	C₂×D₈	—	2	4	no	—
193	C₂×D₈	—	4	4	no	—
194	C₂×D₈	C₂³×C₃₁	1	4	yes	—
195	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
196	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
198	#127	—	2	4	no	—
199	#127	—	4	1	no	—
200	#127	—	4	1	no	—
201	C₄×Q₈	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	—	4	2	no	—
204	#117	—	2	1	no	—
205	#117	—	4	1	no	—
206	C₂×C₄²	—	4	1	no	—
207	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
208	SD₃₂	—	2	8	no	—
209	D₁₆	—	2	8	no	—
210	M₃₂	—	2	4	no	—
211	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
212	#107	C₂³×C₃₁	1	2	yes	—
213	#106	—	2	1	no	—
214	#103	—	2	2	no	—
215	#102	—	2	2	no	—
216	C₄×C₈	D₄×C₃₁	1	2	yes	—
217	C₄×C₈	C₂³×C₃₁	1	2	yes	—
218	C₃₁	#33582	1	1	yes	characteristic, direct product factor, Sylow
219	C₂⁴	—	4	2	no	—
220	#41	—	2	4	no	—
221	#41	—	4	2	no	—
222	C₂×Q₈	—	2	1	no	—
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	8	8	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	D₈×C₃₁	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
236	Q₁₆	—	2	1	no	—
237	Q₁₆	C₂×D₄×C₃₁	1	2	yes	complemented
238	SD₁₆	—	2	4	no	—
239	D₈	—	2	4	no	—
240	D₈	—	2	1	no	—
241	D₈	—	2	4	no	—
242	D₈	—	4	8	no	—
243	D₈	—	4	4	no	—
244	D₈	C₂×D₄×C₃₁	1	2	yes	complemented
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
247	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
249	C₂×C₈	#41×C₃₁	1	2	yes	—
250	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	1	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₄²	—	4	1	no	—
256	C₄²	—	4	2	no	—
257	C₄²	D₈×C₃₁	1	1	yes	characteristic
258	C₄²	D₈×C₃₁	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₃₁	1	2	yes	—
260	C₁₆	—	2	4	no	—
261	C₂³	—	4	1	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	8	8	no	—
264	C₂³	—	8	4	no	—
265	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	—	4	1	no	—
268	D₄	—	2	2	no	—
269	D₄	—	4	4	no	—
270	D₄	—	4	4	no	—
271	D₄	—	4	1	no	—
272	D₄	—	4	4	no	—
273	D₄	—	8	8	no	—
274	D₄	—	8	4	no	—
275	D₄	—	8	8	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	4	no	—
281	C₂×C₄	C₃₁×#121	1	2	yes	—
282	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
283	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
284	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
285	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
286	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	C₃₁×#121	1	2	yes	—
289	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
290	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
291	C₂²	—	2	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	8	no	—
295	C₂²	—	8	4	no	—
296	C₂²	—	16	8	no	—
297	C₂²	C₃₁×#459	1	2	yes	—
298	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	8	2	no	—
302	C₄	C₃₁×#459	1	2	yes	—
303	C₄	C₃₁×#509	1	4	yes	complemented
304	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
305	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
306	C₂	—	2	2	no	—
307	C₂	—	8	2	no	—
308	C₂	—	16	4	no	—
309	C₂	C₂×C₃₁×#459	1	1	yes	characteristic
310	C₂	C₂×C₃₁×#509	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 ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ 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×#493	C₂	semidirect
3. C₄×C₃₁×#135	C₂	semidirect
4. C₃₁×#2207	C₂	semidirect
5. C₃₁×#2206	C₂	semidirect
6. C₃₁×#2150	C₂	semidirect
7. C₃₁×#2140	C₂	semidirect
14. C₂×C₃₁×SD₃₂	C₂²	semidirect
15. C₂×C₃₁×D₁₆	C₂²	semidirect
19. C₄×Q₁₆×C₃₁	C₂²	semidirect
21. C₄×D₈×C₃₁	C₂²	semidirect
25. C₃₁×C₄⋊C₁₆	C₂²	semidirect
26. C₃₁×#357	C₂²	semidirect
31. C₃₁×#135	D₄	semidirect
81. Q₁₆×C₃₁	C₂×D₄	semidirect
88. D₈×C₃₁	C₂×D₄	semidirect
105. #33582	C₃₁	direct (N × H)
136. C₂×#509	C₂×C₃₁	semidirect
137. C₂×#493	C₂×C₃₁	semidirect
138. C₄×#135	C₂×C₃₁	semidirect
139. #2207	C₂×C₃₁	semidirect
140. #2206	C₂×C₃₁	semidirect
141. #2150	C₂×C₃₁	semidirect
142. #2140	C₂×C₃₁	semidirect
155. C₄×C₃₁	#509	semidirect
164. C₂×SD₃₂	C₂²×C₃₁	semidirect
165. C₂×D₁₆	C₂²×C₃₁	semidirect
169. C₄×Q₁₆	C₂²×C₃₁	semidirect
171. C₄×D₈	C₂²×C₃₁	semidirect
175. C₄⋊C₁₆	C₂²×C₃₁	semidirect
176. #357	C₂²×C₃₁	semidirect
186. #135	D₄×C₃₁	semidirect
218. C₃₁	#33582	direct (N × H)
237. Q₁₆	C₂×D₄×C₃₁	semidirect
244. D₈	C₂×D₄×C₃₁	semidirect
303. C₄	C₃₁×#509	semidirect

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×#493	2
C₄×C₃₁×#135	2
C₃₁×#2207	2
C₃₁×#2206	2
C₃₁×#2150	2
C₃₁×#2140	2
#33582	31

Subgroups — C31×#33582 #21490341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33582 #21490341