Subgroups — C₃₁×#20806 #21477565

Order 7936.

994 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#430	C₂	1	4	yes	maximal
2	C₂×C₃₁×#423	C₂	1	2	yes	maximal
3	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1849	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1848	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#430	—	2	16	no	—
10	C₂×C₃₁×#108	C₂²	1	16	yes	—
11	C₃₁×#423	—	2	4	no	—
12	C₃₁×#423	C₂²	1	4	yes	—
13	C₃₁×#423	C₂²	1	4	yes	—
14	C₂×C₃₁×C₄⋊C₈	—	2	4	no	—
15	C₄×M₁₆×C₃₁	C₄	1	4	yes	—
16	C₄×M₁₆×C₃₁	C₄	1	4	yes	—
17	C₂×C₃₁×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
19	C₂³×C₄×C₃₁	Q₈	1	2	yes	—
20	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₃₁	—	2	4	no	—
22	C₂×M₁₆×C₃₁	—	2	16	no	—
23	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
24	C₂×M₁₆×C₃₁	D₄	1	8	yes	—
25	C₂×M₁₆×C₃₁	Q₈	1	8	yes	—
26	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
27	C₂²×C₈×C₃₁	—	2	4	no	—
28	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₃₁	D₄	1	4	yes	—
32	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×#108	—	2	32	no	—
36	C₃₁×C₄⋊C₈	—	2	8	no	—
37	C₃₁×C₄⋊C₈	—	2	8	no	—
38	C₃₁×#97	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
41	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	—	4	4	no	—
43	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
47	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
53	M₁₆×C₃₁	—	2	16	no	—
54	M₁₆×C₃₁	—	4	8	no	—
55	M₁₆×C₃₁	C₄⋊C₄	1	32	yes	—
56	C₂×C₈×C₃₁	—	2	8	no	—
57	C₂×C₈×C₃₁	—	4	8	no	—
58	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
59	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
60	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
61	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
62	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
63	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
66	C₄²×C₃₁	C₂×D₄	1	2	yes	—
67	C₄²×C₃₁	C₂×D₄	1	2	yes	—
68	#20806	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₃₁	—	4	2	no	—
70	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₃₁	C₄⋊Q₈	1	4	yes	—
72	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₃₁	—	2	8	no	—
74	C₂×C₄×C₃₁	—	2	4	no	—
75	C₂×C₄×C₃₁	—	4	4	no	—
76	C₂×C₄×C₃₁	—	4	2	no	—
77	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
79	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
80	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
82	C₂×C₄×C₃₁	#127	1	4	yes	—
83	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
84	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
85	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
87	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
89	C₈×C₃₁	—	4	8	no	—
90	C₈×C₃₁	C₂×C₄⋊C₄	1	16	yes	—
91	C₂×#430	C₂×C₃₁	1	4	yes	—
92	C₂×#423	C₂×C₃₁	1	2	yes	—
93	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
94	#1849	C₂×C₃₁	1	4	yes	complemented
95	#1848	C₂×C₃₁	1	4	yes	complemented
96	C₂²×C₃₁	—	2	4	no	—
97	C₂²×C₃₁	—	2	2	no	—
98	C₂²×C₃₁	—	4	4	no	—
99	C₂²×C₃₁	#384	1	4	yes	—
100	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₃₁	C₂×C₄⋊Q₈	1	2	yes	—
102	C₄×C₃₁	—	2	4	no	—
103	C₄×C₃₁	—	2	2	no	—
104	C₄×C₃₁	#384	1	4	yes	—
105	C₄×C₃₁	#430	1	4	yes	complemented
106	C₄×C₃₁	C₂²×C₄⋊C₄	1	2	yes	—
107	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
108	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
109	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
110	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
111	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
112	#430	—	2	16	no	—
113	C₂×#108	C₂²×C₃₁	1	16	yes	—
114	#423	—	2	4	no	—
115	#423	C₂²×C₃₁	1	4	yes	—
116	#423	C₂²×C₃₁	1	4	yes	—
117	C₂×C₄⋊C₈	—	2	4	no	—
118	C₄×M₁₆	C₄×C₃₁	1	4	yes	—
119	C₄×M₁₆	C₄×C₃₁	1	4	yes	—
120	C₂×#97	C₂²×C₃₁	1	2	yes	—
121	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
122	C₂×C₃₁	—	2	2	no	—
123	C₂×C₃₁	—	2	4	no	—
124	C₂×C₃₁	C₂×#384	1	1	yes	characteristic
125	C₂×C₃₁	C₂×#430	1	2	yes	—
126	C₂³×C₄	Q₈×C₃₁	1	2	yes	—
127	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
128	C₂×M₁₆	—	2	4	no	—
129	C₂×M₁₆	—	2	16	no	—
130	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
131	C₂×M₁₆	D₄×C₃₁	1	8	yes	—
132	C₂×M₁₆	Q₈×C₃₁	1	8	yes	—
133	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
134	C₂²×C₈	—	2	4	no	—
135	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
136	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
137	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
138	C₂×C₄²	D₄×C₃₁	1	4	yes	—
139	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
140	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
141	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
142	#108	—	2	32	no	—
143	C₄⋊C₈	—	2	8	no	—
144	C₄⋊C₈	—	2	8	no	—
145	#97	C₂×C₄×C₃₁	1	8	yes	—
146	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
147	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
148	C₃₁	#20806	1	1	yes	characteristic, direct product factor, Sylow
149	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
150	C₂²×C₄	—	4	4	no	—
151	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
152	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
153	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
154	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
155	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
156	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
157	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
158	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
159	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
160	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
161	M₁₆	—	2	16	no	—
162	M₁₆	—	4	8	no	—
163	M₁₆	C₄⋊C₄×C₃₁	1	32	yes	—
164	C₂×C₈	—	2	8	no	—
165	C₂×C₈	—	4	8	no	—
166	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
167	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
168	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
169	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
170	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
171	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
172	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
173	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
174	C₄²	C₂×D₄×C₃₁	1	2	yes	—
175	C₄²	C₂×D₄×C₃₁	1	2	yes	—
176	C₂³	—	4	2	no	—
177	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
178	C₂³	C₃₁×C₄⋊Q₈	1	4	yes	—
179	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
185	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
186	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
187	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
188	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
189	C₂×C₄	C₃₁×#127	1	4	yes	—
190	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
191	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
192	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
193	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
194	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
196	C₈	—	4	8	no	—
197	C₈	C₂×C₄⋊C₄×C₃₁	1	16	yes	—
198	C₂²	—	2	4	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	4	no	—
201	C₂²	C₃₁×#384	1	4	yes	—
202	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂²	C₂×C₃₁×C₄⋊Q₈	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	C₃₁×#384	1	4	yes	—
207	C₄	C₃₁×#430	1	4	yes	complemented
208	C₄	C₂²×C₄⋊C₄×C₃₁	1	2	yes	—
209	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
210	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
211	C₂	—	2	2	no	—
212	C₂	—	2	4	no	—
213	C₂	C₂×C₃₁×#384	1	1	yes	characteristic
214	C₂	C₂×C₃₁×#430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1849	C₂	semidirect
5. C₃₁×#1848	C₂	semidirect
68. #20806	C₃₁	direct (N × H)
94. #1849	C₂×C₃₁	semidirect
95. #1848	C₂×C₃₁	semidirect
105. C₄×C₃₁	#430	semidirect
148. C₃₁	#20806	direct (N × H)
207. C₄	C₃₁×#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#430	2
C₂×C₃₁×#423	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#1849	2
C₃₁×#1848	2
#20806	31

Subgroups — C31×#20806 #21477565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20806 #21477565