Subgroups — C₃₁×#20824 #21477583

Order 7936.

930 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2285	C₂	1	2	yes	maximal
3	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1834	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1809	C₂	1	4	yes	maximal
7	C₃₁×#1776	C₂	1	4	yes	maximal, complemented
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×#117	—	2	2	no	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#107	C₂²	1	4	yes	—
12	C₂×C₃₁×#106	C₂²	1	4	yes	—
13	C₄×M₁₆×C₃₁	C₄	1	8	yes	complemented
14	C₂×C₃₁×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#384	C₂²	1	2	yes	—
18	C₃₁×#384	C₂²	1	2	yes	—
19	C₃₁×#382	—	2	2	no	—
20	C₃₁×#380	—	2	2	no	—
21	C₃₁×#380	C₂²	1	4	yes	—
22	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
23	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
24	C₃₁×#340	C₂²	1	8	yes	—
25	C₂³×C₄×C₃₁	D₄	1	2	yes	—
26	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂×M₁₆×C₃₁	—	2	4	no	—
28	C₂×M₁₆×C₃₁	—	2	4	no	—
29	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
30	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
32	C₃₁×#117	—	4	8	no	—
33	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₄²×C₃₁	—	2	4	no	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
42	C₂×C₄²×C₃₁	D₄	1	2	yes	—
43	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×#107	—	2	8	no	—
47	C₃₁×#106	—	2	8	no	—
48	C₃₁×#97	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
51	C₃₁×#95	—	2	4	no	—
52	C₃₁×#95	—	2	4	no	—
53	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	—	2	4	no	—
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
61	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
67	M₁₆×C₃₁	—	2	16	no	—
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₃₁	#41	1	4	yes	—
73	C₂×C₈×C₃₁	#41	1	4	yes	—
74	C₄⋊C₄×C₃₁	—	2	8	no	—
75	C₄⋊C₄×C₃₁	—	2	8	no	—
76	C₄⋊C₄×C₃₁	—	4	4	no	—
77	C₄⋊C₄×C₃₁	—	4	4	no	—
78	C₄⋊C₄×C₃₁	—	4	4	no	—
79	C₄⋊C₄×C₃₁	—	4	4	no	—
80	C₂²⋊C₄×C₃₁	—	4	8	no	—
81	C₄²×C₃₁	—	4	8	no	—
82	C₄²×C₃₁	C₄⋊C₄	1	8	yes	—
83	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
84	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
85	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
86	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
87	#20824	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	2	1	no	—
89	C₂³×C₃₁	—	2	1	no	—
90	C₂³×C₃₁	—	2	2	no	—
91	C₂³×C₃₁	—	2	2	no	—
92	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	4	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	2	16	no	—
101	C₂×C₄×C₃₁	—	4	4	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	4	4	no	—
105	C₂×C₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	—	4	4	no	—
107	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
111	C₂×C₄×C₃₁	#137	1	4	yes	—
112	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
115	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
116	C₈×C₃₁	—	2	4	no	—
117	C₈×C₃₁	#117	1	8	yes	complemented
118	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
119	#2285	C₂×C₃₁	1	2	yes	—
120	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
121	#1844	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#1834	C₂×C₃₁	1	2	yes	complemented
123	#1809	C₂×C₃₁	1	4	yes	—
124	#1776	C₂×C₃₁	1	4	yes	complemented
125	C₂²×C₃₁	—	2	4	no	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	4	2	no	—
128	C₂²×C₃₁	—	4	1	no	—
129	C₂²×C₃₁	—	4	1	no	—
130	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₃₁	#532	1	1	yes	characteristic
132	C₂²×C₃₁	#532	1	1	yes	characteristic
133	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
134	C₂²×C₃₁	C₂×#137	1	2	yes	—
135	C₄×C₃₁	—	2	4	no	—
136	C₄×C₃₁	—	2	2	no	—
137	C₄×C₃₁	—	8	4	no	—
138	C₄×C₃₁	—	8	4	no	—
139	C₄×C₃₁	#428	1	8	yes	—
140	C₄×C₃₁	C₂×#117	1	2	yes	—
141	C₄×C₃₁	#533	1	2	yes	—
142	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
143	C₂×#117	—	2	2	no	—
144	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
145	C₂×#107	C₂²×C₃₁	1	4	yes	—
146	C₂×#106	C₂²×C₃₁	1	4	yes	—
147	C₄×M₁₆	C₄×C₃₁	1	8	yes	complemented
148	C₂×#97	C₂²×C₃₁	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
151	#384	C₂²×C₃₁	1	2	yes	—
152	#384	C₂²×C₃₁	1	2	yes	—
153	#382	—	2	2	no	—
154	#380	—	2	2	no	—
155	#380	C₂²×C₃₁	1	4	yes	—
156	C₄×C₄⋊C₄	—	2	2	no	—
157	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
158	#340	C₂²×C₃₁	1	8	yes	—
159	C₂×C₃₁	—	4	2	no	—
160	C₂×C₃₁	#2285	1	1	yes	characteristic
161	C₂×C₃₁	C₂×#428	1	2	yes	—
162	C₂×C₃₁	#3182	1	2	yes	—
163	C₂×C₃₁	#3184	1	1	yes	characteristic
164	C₂×C₃₁	#3185	1	1	yes	characteristic
165	C₂³×C₄	D₄×C₃₁	1	2	yes	—
166	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	—	2	4	no	—
169	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
170	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
171	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
172	#117	—	4	8	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
178	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
179	C₂×C₂²⋊C₄	—	2	4	no	—
180	C₂×C₄²	—	2	4	no	—
181	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
182	C₂×C₄²	D₄×C₃₁	1	2	yes	—
183	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
184	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
185	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
186	#107	—	2	8	no	—
187	#106	—	2	8	no	—
188	#97	C₂×C₄×C₃₁	1	8	yes	—
189	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
190	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
191	#95	—	2	4	no	—
192	#95	—	2	4	no	—
193	C₃₁	#20824	1	1	yes	characteristic, direct product factor, Sylow
194	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
202	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
206	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
207	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
208	M₁₆	—	2	16	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
212	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
213	C₂×C₈	#41×C₃₁	1	4	yes	—
214	C₂×C₈	#41×C₃₁	1	4	yes	—
215	C₄⋊C₄	—	2	8	no	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₂²⋊C₄	—	4	8	no	—
222	C₄²	—	4	8	no	—
223	C₄²	C₄⋊C₄×C₃₁	1	8	yes	—
224	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
225	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
226	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
227	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
233	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	16	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
248	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
249	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
250	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
251	C₂×C₄	C₃₁×#137	1	4	yes	—
252	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
253	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
254	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
255	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	C₃₁×#117	1	8	yes	complemented
258	C₂²	—	2	4	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	1	no	—
263	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
264	C₂²	C₃₁×#532	1	1	yes	characteristic
265	C₂²	C₃₁×#532	1	1	yes	characteristic
266	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
267	C₂²	C₂×C₃₁×#137	1	2	yes	—
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	8	4	no	—
271	C₄	—	8	4	no	—
272	C₄	C₃₁×#428	1	8	yes	—
273	C₄	C₂×C₃₁×#117	1	2	yes	—
274	C₄	C₃₁×#533	1	2	yes	—
275	C₂	—	4	2	no	—
276	C₂	C₃₁×#2285	1	1	yes	characteristic
277	C₂	C₂×C₃₁×#428	1	2	yes	—
278	C₂	C₃₁×#3182	1	2	yes	—
279	C₂	C₃₁×#3184	1	1	yes	characteristic
280	C₂	C₃₁×#3185	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1845	C₂	semidirect
4. C₃₁×#1844	C₂	semidirect
5. C₃₁×#1834	C₂	semidirect
7. C₃₁×#1776	C₂	semidirect
13. C₄×M₁₆×C₃₁	C₄	semidirect
87. #20824	C₃₁	direct (N × H)
117. C₈×C₃₁	#117	semidirect
120. #1845	C₂×C₃₁	semidirect
121. #1844	C₂×C₃₁	semidirect
122. #1834	C₂×C₃₁	semidirect
124. #1776	C₂×C₃₁	semidirect
147. C₄×M₁₆	C₄×C₃₁	semidirect
193. C₃₁	#20824	direct (N × H)
257. C₈	C₃₁×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2285	2
C₃₁×#1845	2
C₃₁×#1844	2
C₃₁×#1834	2
C₃₁×#1809	2
C₃₁×#1776	2
#20824	31

Subgroups — C31×#20824 #21477583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20824 #21477583