Subgroups — C₃₁×#20539 #21477298

Order 7936.

866 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#432	C₂	1	2	yes	maximal
2	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1771	C₂	1	8	yes	maximal, complemented
4	C₃₁×#1750	C₂	1	4	yes	maximal
5	C₂×C₃₁×#117	C₂²	1	2	yes	—
6	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#432	C₄	1	16	yes	complemented
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
9	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
10	C₂×C₃₁×#97	C₂²	1	4	yes	—
11	C₂×C₃₁×#97	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
13	C₃₁×#384	C₄	1	2	yes	—
14	C₃₁×#382	C₄	1	2	yes	—
15	C₃₁×#380	C₄	1	4	yes	—
16	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
17	C₃₁×#336	C₂²	1	8	yes	—
18	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
21	C₃₁×#117	—	4	2	no	—
22	C₃₁×#117	—	4	2	no	—
23	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
25	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
26	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	—
29	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
30	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
31	C₃₁×C₄⋊C₈	C₂×C₄	1	16	yes	—
32	C₃₁×C₂²⋊C₈	C₂×C₄	1	16	yes	—
33	C₃₁×#97	—	4	4	no	—
34	C₃₁×#97	C₂×C₄	1	16	yes	—
35	C₄×C₈×C₃₁	—	4	4	no	—
36	C₃₁×#95	C₂×C₄	1	8	yes	—
37	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	—	2	2	no	—
39	C₂²×C₄×C₃₁	—	2	2	no	—
40	C₂²×C₄×C₃₁	—	4	4	no	—
41	C₂²×C₄×C₃₁	C₄²	1	8	yes	—
42	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
43	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
47	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
48	C₂×C₈×C₃₁	—	2	8	no	—
49	C₂×C₈×C₃₁	—	2	8	no	—
50	C₂×C₈×C₃₁	C₂²×C₄	1	16	yes	—
51	C₄⋊C₄×C₃₁	—	4	8	no	—
52	C₄⋊C₄×C₃₁	—	4	4	no	—
53	C₂²⋊C₄×C₃₁	—	4	4	no	—
54	C₄²×C₃₁	—	2	4	no	—
55	C₄²×C₃₁	—	2	4	no	—
56	C₄²×C₃₁	C₄²	1	4	yes	—
57	C₄²×C₃₁	C₄²	1	4	yes	—
58	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
59	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
60	#20539	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₃₁	—	2	1	no	—
62	C₂³×C₃₁	—	2	1	no	—
63	C₂³×C₃₁	—	2	4	no	—
64	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
65	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
66	C₂×C₄×C₃₁	—	2	1	no	—
67	C₂×C₄×C₃₁	—	2	1	no	—
68	C₂×C₄×C₃₁	—	2	4	no	—
69	C₂×C₄×C₃₁	—	2	4	no	—
70	C₂×C₄×C₃₁	—	2	8	no	—
71	C₂×C₄×C₃₁	—	2	8	no	—
72	C₂×C₄×C₃₁	—	2	16	no	—
73	C₂×C₄×C₃₁	—	4	8	no	—
74	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
75	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
76	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
77	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
78	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
79	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
80	C₈×C₃₁	—	2	16	no	—
81	C₂×#432	C₂×C₃₁	1	2	yes	—
82	#2285	C₂×C₃₁	1	1	yes	characteristic
83	#1771	C₂×C₃₁	1	8	yes	complemented
84	#1750	C₂×C₃₁	1	4	yes	—
85	C₂²×C₃₁	—	2	4	no	—
86	C₂²×C₃₁	—	2	2	no	—
87	C₂²×C₃₁	—	4	2	no	—
88	C₂²×C₃₁	—	4	2	no	—
89	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
90	C₂²×C₃₁	#518	1	1	yes	characteristic
91	C₂²×C₃₁	#519	1	1	yes	characteristic
92	C₂²×C₃₁	#568	1	4	yes	—
93	C₄×C₃₁	—	2	4	no	—
94	C₄×C₃₁	—	2	4	no	—
95	C₄×C₃₁	—	2	2	no	—
96	C₄×C₃₁	—	4	8	no	—
97	C₄×C₃₁	#520	1	4	yes	—
98	C₂×#117	C₂²×C₃₁	1	2	yes	—
99	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
100	#432	C₄×C₃₁	1	16	yes	complemented
101	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
102	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
103	C₂×#97	C₂²×C₃₁	1	4	yes	—
104	C₂×#97	C₂²×C₃₁	1	4	yes	—
105	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
106	#384	C₄×C₃₁	1	2	yes	—
107	#382	C₄×C₃₁	1	2	yes	—
108	#380	C₄×C₃₁	1	4	yes	—
109	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
110	#336	C₂²×C₃₁	1	8	yes	—
111	C₂×C₃₁	—	4	2	no	—
112	C₂×C₃₁	#2274	1	1	yes	characteristic
113	C₂×C₃₁	#2873	1	2	yes	—
114	C₂×C₃₁	#2980	1	2	yes	—
115	C₂×C₃₁	#2987	1	2	yes	—
116	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
117	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
119	#117	—	4	2	no	—
120	#117	—	4	2	no	—
121	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
122	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
123	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
124	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
125	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
126	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	—
127	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
128	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
129	C₄⋊C₈	C₂×C₄×C₃₁	1	16	yes	—
130	C₂²⋊C₈	C₂×C₄×C₃₁	1	16	yes	—
131	#97	—	4	4	no	—
132	#97	C₂×C₄×C₃₁	1	16	yes	—
133	C₄×C₈	—	4	4	no	—
134	#95	C₂×C₄×C₃₁	1	8	yes	—
135	C₃₁	#20539	1	1	yes	characteristic, direct product factor, Sylow
136	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	4	4	no	—
140	C₂²×C₄	C₄²×C₃₁	1	8	yes	—
141	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
142	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
143	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
144	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
145	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
146	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
147	C₂×C₈	—	2	8	no	—
148	C₂×C₈	—	2	8	no	—
149	C₂×C₈	C₂²×C₄×C₃₁	1	16	yes	—
150	C₄⋊C₄	—	4	8	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₂²⋊C₄	—	4	4	no	—
153	C₄²	—	2	4	no	—
154	C₄²	—	2	4	no	—
155	C₄²	C₄²×C₃₁	1	4	yes	—
156	C₄²	C₄²×C₃₁	1	4	yes	—
157	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
158	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
159	C₂³	—	2	1	no	—
160	C₂³	—	2	1	no	—
161	C₂³	—	2	4	no	—
162	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
163	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	—	2	16	no	—
171	C₂×C₄	—	4	8	no	—
172	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
173	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
174	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
175	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
176	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
177	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
178	C₈	—	2	16	no	—
179	C₂²	—	2	4	no	—
180	C₂²	—	2	2	no	—
181	C₂²	—	4	2	no	—
182	C₂²	—	4	2	no	—
183	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
184	C₂²	C₃₁×#518	1	1	yes	characteristic
185	C₂²	C₃₁×#519	1	1	yes	characteristic
186	C₂²	C₃₁×#568	1	4	yes	—
187	C₄	—	2	4	no	—
188	C₄	—	2	4	no	—
189	C₄	—	2	2	no	—
190	C₄	—	4	8	no	—
191	C₄	C₃₁×#520	1	4	yes	—
192	C₂	—	4	2	no	—
193	C₂	C₃₁×#2274	1	1	yes	characteristic
194	C₂	C₃₁×#2873	1	2	yes	—
195	C₂	C₃₁×#2980	1	2	yes	—
196	C₂	C₃₁×#2987	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1771	C₂	semidirect
7. C₃₁×#432	C₄	semidirect
60. #20539	C₃₁	direct (N × H)
83. #1771	C₂×C₃₁	semidirect
100. #432	C₄×C₃₁	semidirect
135. C₃₁	#20539	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×#432	2
C₃₁×#2285	2
C₃₁×#1771	2
C₃₁×#1750	2
#20539	31

Subgroups — C31×#20539 #21477298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20539 #21477298