Subgroups — C₃₁×#19661 #21476420

Order 7936.

990 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3205	C₂	1	4	yes	maximal, complemented
2	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1721	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1712	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1709	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	—	2	2	no	—
9	C₂²×D₈×C₃₁	—	2	2	no	—
10	C₃₁×#532	C₂²	1	2	yes	—
11	C₂×C₃₁×#127	C₂²	1	2	yes	—
12	C₃₁×#495	—	2	2	no	—
13	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
14	C₃₁×#493	—	2	2	no	—
15	C₃₁×#493	C₂²	1	2	yes	—
16	C₃₁×#493	C₂²	1	2	yes	—
17	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
18	C₃₁×#493	C₂²	1	8	yes	complemented
19	C₃₁×#492	C₂²	1	2	yes	complemented
20	C₃₁×#486	C₂²	1	8	yes	complemented
21	C₃₁×#459	—	2	8	no	—
22	C₃₁×#447	—	2	8	no	—
23	C₃₁×#431	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#322	C₂²	1	2	yes	complemented
26	C₈²×C₃₁	C₂²	1	2	yes	complemented
27	C₂×#41×C₃₁	—	2	2	no	—
28	C₂²×D₄×C₃₁	—	2	2	no	—
29	C₂²×D₄×C₃₁	—	2	4	no	—
30	C₃₁×#135	—	4	8	no	—
31	C₂×Q₁₆×C₃₁	—	2	2	no	—
32	C₂×SD₁₆×C₃₁	—	2	4	no	—
33	C₂×D₈×C₃₁	—	2	4	no	—
34	C₂×D₈×C₃₁	—	2	4	no	—
35	C₂×D₈×C₃₁	—	2	2	no	—
36	C₂×D₈×C₃₁	—	2	16	no	—
37	C₂×D₈×C₃₁	—	4	8	no	—
38	C₂²×C₈×C₃₁	D₄	1	2	yes	—
39	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
40	C₃₁×#127	—	2	4	no	—
41	C₃₁×#127	—	2	2	no	—
42	C₃₁×#127	C₂³	1	4	yes	—
43	C₃₁×#127	C₂³	1	2	yes	—
44	C₃₁×#124	—	2	2	no	—
45	C₃₁×#121	—	2	4	no	—
46	C₄×D₄×C₃₁	—	2	4	no	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
49	C₃₁×#102	—	2	16	no	—
50	C₃₁×C₂²⋊C₈	D₄	1	4	yes	—
51	C₄×C₈×C₃₁	D₄	1	2	yes	—
52	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₃₁	C₂³	1	4	yes	—
55	C₂⁴×C₃₁	—	4	2	no	—
56	#41×C₃₁	—	4	8	no	—
57	C₂×Q₈×C₃₁	—	2	2	no	—
58	C₂×D₄×C₃₁	—	2	8	no	—
59	C₂×D₄×C₃₁	—	2	4	no	—
60	C₂×D₄×C₃₁	—	2	2	no	—
61	C₂×D₄×C₃₁	—	2	4	no	—
62	C₂×D₄×C₃₁	—	4	2	no	—
63	C₂×D₄×C₃₁	—	4	4	no	—
64	C₂×D₄×C₃₁	—	4	2	no	—
65	C₂×D₄×C₃₁	—	4	8	no	—
66	C₂×D₄×C₃₁	—	8	8	no	—
67	C₂²×C₄×C₃₁	—	4	2	no	—
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	Q₁₆×C₃₁	—	4	4	no	—
71	SD₁₆×C₃₁	—	4	8	no	—
72	D₈×C₃₁	—	4	8	no	—
73	D₈×C₃₁	—	4	8	no	—
74	D₈×C₃₁	—	4	4	no	—
75	D₈×C₃₁	—	8	16	no	—
76	C₂×C₈×C₃₁	D₈	1	8	yes	complemented
77	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
78	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₂²⋊C₄×C₃₁	—	4	4	no	—
82	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₃₁	C₂×D₄	1	2	yes	—
84	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
85	#19661	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₃₁	—	4	2	no	—
87	C₂³×C₃₁	—	4	4	no	—
88	C₂³×C₃₁	—	8	8	no	—
89	C₂³×C₃₁	—	8	4	no	—
90	C₂³×C₃₁	#127	1	1	yes	characteristic
91	Q₈×C₃₁	—	4	4	no	—
92	D₄×C₃₁	—	4	8	no	—
93	D₄×C₃₁	—	4	4	no	—
94	D₄×C₃₁	—	4	8	no	—
95	D₄×C₃₁	—	4	16	no	—
96	D₄×C₃₁	—	8	8	no	—
97	D₄×C₃₁	—	8	4	no	—
98	D₄×C₃₁	—	8	4	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	8	4	no	—
102	C₂×C₄×C₃₁	#127	1	2	yes	—
103	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
105	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
107	C₈×C₃₁	—	2	8	no	—
108	C₈×C₃₁	C₂×D₈	1	4	yes	—
109	C₈×C₃₁	C₂×D₈	1	4	yes	—
110	#3205	C₂×C₃₁	1	4	yes	complemented
111	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
112	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
113	#1721	C₂×C₃₁	1	4	yes	complemented
114	#1712	C₂×C₃₁	1	2	yes	complemented
115	#1709	C₂×C₃₁	1	2	yes	complemented
116	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
117	C₂²×C₃₁	—	2	2	no	—
118	C₂²×C₃₁	—	8	2	no	—
119	C₂²×C₃₁	—	8	4	no	—
120	C₂²×C₃₁	—	8	8	no	—
121	C₂²×C₃₁	—	8	4	no	—
122	C₂²×C₃₁	—	16	8	no	—
123	C₂²×C₃₁	#493	1	2	yes	—
124	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
125	C₄×C₃₁	—	2	2	no	—
126	C₄×C₃₁	—	8	2	no	—
127	C₄×C₃₁	#493	1	2	yes	—
128	C₄×C₃₁	C₂²×D₈	1	2	yes	—
129	C₄×C₃₁	#576	1	4	yes	—
130	C₂×#135	—	2	2	no	—
131	C₂²×D₈	—	2	2	no	—
132	#532	C₂²×C₃₁	1	2	yes	—
133	C₂×#127	C₂²×C₃₁	1	2	yes	—
134	#495	—	2	2	no	—
135	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
136	#493	—	2	2	no	—
137	#493	C₂²×C₃₁	1	2	yes	—
138	#493	C₂²×C₃₁	1	2	yes	—
139	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
140	#493	C₂²×C₃₁	1	8	yes	complemented
141	#492	C₂²×C₃₁	1	2	yes	complemented
142	#486	C₂²×C₃₁	1	8	yes	complemented
143	#459	—	2	8	no	—
144	#447	—	2	8	no	—
145	#431	C₂²×C₃₁	1	2	yes	—
146	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
147	#322	C₂²×C₃₁	1	2	yes	complemented
148	C₈²	C₂²×C₃₁	1	2	yes	complemented
149	C₂×C₃₁	—	2	2	no	—
150	C₂×C₃₁	—	16	2	no	—
151	C₂×C₃₁	—	16	4	no	—
152	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
153	C₂×C₃₁	#3155	1	2	yes	—
154	C₂×#41	—	2	2	no	—
155	C₂²×D₄	—	2	2	no	—
156	C₂²×D₄	—	2	4	no	—
157	#135	—	4	8	no	—
158	C₂×Q₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	4	no	—
160	C₂×D₈	—	2	4	no	—
161	C₂×D₈	—	2	4	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂×D₈	—	2	16	no	—
164	C₂×D₈	—	4	8	no	—
165	C₂²×C₈	D₄×C₃₁	1	2	yes	—
166	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
167	#127	—	2	4	no	—
168	#127	—	2	2	no	—
169	#127	C₂³×C₃₁	1	4	yes	—
170	#127	C₂³×C₃₁	1	2	yes	—
171	#124	—	2	2	no	—
172	#121	—	2	4	no	—
173	C₄×D₄	—	2	4	no	—
174	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
175	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
176	#102	—	2	16	no	—
177	C₂²⋊C₈	D₄×C₃₁	1	4	yes	—
178	C₄×C₈	D₄×C₃₁	1	2	yes	—
179	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
180	C₄×C₈	C₂³×C₃₁	1	4	yes	—
181	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
182	C₃₁	#19661	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	—	4	2	no	—
184	#41	—	4	8	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×D₄	—	2	8	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂×D₄	—	4	8	no	—
194	C₂×D₄	—	8	8	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
198	Q₁₆	—	4	4	no	—
199	SD₁₆	—	4	8	no	—
200	D₈	—	4	8	no	—
201	D₈	—	4	8	no	—
202	D₈	—	4	4	no	—
203	D₈	—	8	16	no	—
204	C₂×C₈	D₈×C₃₁	1	8	yes	complemented
205	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
206	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
207	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
208	C₄⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
211	C₄²	C₂×D₄×C₃₁	1	2	yes	—
212	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
213	C₂³	—	4	2	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	8	8	no	—
216	C₂³	—	8	4	no	—
217	C₂³	C₃₁×#127	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	D₄	—	4	8	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	8	no	—
222	D₄	—	4	16	no	—
223	D₄	—	8	8	no	—
224	D₄	—	8	4	no	—
225	D₄	—	8	4	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	8	4	no	—
229	C₂×C₄	C₃₁×#127	1	2	yes	—
230	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
231	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
232	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
233	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
234	C₈	—	2	8	no	—
235	C₈	C₂×D₈×C₃₁	1	4	yes	—
236	C₈	C₂×D₈×C₃₁	1	4	yes	—
237	C₂²	—	2	2	no	—
238	C₂²	—	8	2	no	—
239	C₂²	—	8	4	no	—
240	C₂²	—	8	8	no	—
241	C₂²	—	8	4	no	—
242	C₂²	—	16	8	no	—
243	C₂²	C₃₁×#493	1	2	yes	—
244	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	8	2	no	—
247	C₄	C₃₁×#493	1	2	yes	—
248	C₄	C₂²×D₈×C₃₁	1	2	yes	—
249	C₄	C₃₁×#576	1	4	yes	—
250	C₂	—	2	2	no	—
251	C₂	—	16	4	no	—
252	C₂	—	16	2	no	—
253	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
254	C₂	C₃₁×#3155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3205	C₂	semidirect
2. C₃₁×#3147	C₂	semidirect
3. C₂×C₃₁×#493	C₂	semidirect
4. C₃₁×#1721	C₂	semidirect
5. C₃₁×#1712	C₂	semidirect
6. C₃₁×#1709	C₂	semidirect
7. C₃₁×#1450	C₂	semidirect
13. C₃₁×#494	C₂²	semidirect
17. C₃₁×#493	C₂²	semidirect
18. C₃₁×#493	C₂²	semidirect
19. C₃₁×#492	C₂²	semidirect
20. C₃₁×#486	C₂²	semidirect
25. C₃₁×#322	C₂²	semidirect
26. C₈²×C₃₁	C₂²	semidirect
76. C₂×C₈×C₃₁	D₈	semidirect
85. #19661	C₃₁	direct (N × H)
110. #3205	C₂×C₃₁	semidirect
111. #3147	C₂×C₃₁	semidirect
112. C₂×#493	C₂×C₃₁	semidirect
113. #1721	C₂×C₃₁	semidirect
114. #1712	C₂×C₃₁	semidirect
115. #1709	C₂×C₃₁	semidirect
116. #1450	C₂×C₃₁	semidirect
135. #494	C₂²×C₃₁	semidirect
139. #493	C₂²×C₃₁	semidirect
140. #493	C₂²×C₃₁	semidirect
141. #492	C₂²×C₃₁	semidirect
142. #486	C₂²×C₃₁	semidirect
147. #322	C₂²×C₃₁	semidirect
148. C₈²	C₂²×C₃₁	semidirect
182. C₃₁	#19661	direct (N × H)
204. C₂×C₈	D₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3205	2
C₃₁×#3147	2
C₂×C₃₁×#493	2
C₃₁×#1721	2
C₃₁×#1712	2
C₃₁×#1709	2
C₃₁×#1450	2
#19661	31

Subgroups — C31×#19661 #21476420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19661 #21476420