Subgroups — C₃₁×#21965 #21478724

Order 7936.

786 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2283	C₂	1	2	yes	maximal
3	C₃₁×#1949	C₂	1	8	yes	maximal, complemented
4	C₃₁×#1809	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
6	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#107	C₂²	1	4	yes	—
8	C₂×C₃₁×#106	C₂²	1	4	yes	—
9	C₃₁×#423	C₄	1	8	yes	complemented
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
11	C₃₁×#383	—	2	2	no	—
12	C₃₁×#382	C₂²	1	4	yes	—
13	C₃₁×#381	—	2	2	no	—
14	C₃₁×#380	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
16	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
17	C₃₁×#376	—	2	2	no	—
18	C₃₁×#340	C₂²	1	8	yes	—
19	C₂³×C₄×C₃₁	D₄	1	2	yes	—
20	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₃₁	—	4	2	no	—
22	C₂×M₁₆×C₃₁	—	4	2	no	—
23	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
25	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
26	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
27	C₂×C₄²×C₃₁	—	2	4	no	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
32	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
33	C₃₁×#107	—	4	4	no	—
34	C₃₁×#106	—	4	4	no	—
35	C₃₁×C₄⋊C₈	C₂×C₄	1	16	yes	—
36	C₃₁×#95	—	2	4	no	—
37	C₃₁×#95	—	2	4	no	—
38	C₃₁×#95	—	2	4	no	—
39	C₃₁×#95	—	2	4	no	—
40	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	—	2	2	no	—
42	C₂²×C₄×C₃₁	—	2	2	no	—
43	C₂²×C₄×C₃₁	—	2	8	no	—
44	C₂²×C₄×C₃₁	—	4	4	no	—
45	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
46	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
52	M₁₆×C₃₁	—	4	8	no	—
53	C₂×C₈×C₃₁	—	2	4	no	—
54	C₂×C₈×C₃₁	—	2	4	no	—
55	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
56	C₄⋊C₄×C₃₁	—	2	8	no	—
57	C₄⋊C₄×C₃₁	—	2	8	no	—
58	C₂²⋊C₄×C₃₁	—	4	8	no	—
59	C₄²×C₃₁	—	4	8	no	—
60	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
61	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
62	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
63	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
64	#21965	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₃₁	—	2	2	no	—
66	C₂³×C₃₁	—	2	4	no	—
67	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
68	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	—	2	2	no	—
70	C₂×C₄×C₃₁	—	2	4	no	—
71	C₂×C₄×C₃₁	—	2	4	no	—
72	C₂×C₄×C₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	—	2	4	no	—
75	C₂×C₄×C₃₁	—	4	8	no	—
76	C₂×C₄×C₃₁	—	4	8	no	—
77	C₂×C₄×C₃₁	—	4	8	no	—
78	C₂×C₄×C₃₁	—	4	8	no	—
79	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
81	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
82	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
85	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
86	C₈×C₃₁	—	2	8	no	—
87	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
88	#2283	C₂×C₃₁	1	2	yes	—
89	#1949	C₂×C₃₁	1	8	yes	complemented
90	#1809	C₂×C₃₁	1	4	yes	—
91	C₂²×C₃₁	—	2	2	no	—
92	C₂²×C₃₁	—	2	4	no	—
93	C₂²×C₃₁	—	4	4	no	—
94	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₃₁	#537	1	2	yes	—
96	C₂²×C₃₁	#575	1	4	yes	—
97	C₄×C₃₁	—	2	2	no	—
98	C₄×C₃₁	—	2	4	no	—
99	C₄×C₃₁	—	2	4	no	—
100	C₄×C₃₁	—	8	8	no	—
101	C₄×C₃₁	#518	1	1	yes	characteristic
102	C₄×C₃₁	#519	1	1	yes	characteristic
103	C₄×C₃₁	#544	1	2	yes	—
104	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
105	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
106	C₂×#107	C₂²×C₃₁	1	4	yes	—
107	C₂×#106	C₂²×C₃₁	1	4	yes	—
108	#423	C₄×C₃₁	1	8	yes	complemented
109	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
110	#383	—	2	2	no	—
111	#382	C₂²×C₃₁	1	4	yes	—
112	#381	—	2	2	no	—
113	#380	C₂²×C₃₁	1	4	yes	—
114	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
115	C₄×C₂²⋊C₄	—	2	2	no	—
116	#376	—	2	2	no	—
117	#340	C₂²×C₃₁	1	8	yes	—
118	C₂×C₃₁	—	4	2	no	—
119	C₂×C₃₁	#2316	1	1	yes	characteristic
120	C₂×C₃₁	#2911	1	2	yes	—
121	C₂×C₃₁	#3256	1	4	yes	—
122	C₂³×C₄	D₄×C₃₁	1	2	yes	—
123	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
124	C₂×M₁₆	—	4	2	no	—
125	C₂×M₁₆	—	4	2	no	—
126	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
127	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
128	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
129	C₂×C₂²⋊C₄	—	2	4	no	—
130	C₂×C₄²	—	2	4	no	—
131	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
132	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
133	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
134	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
135	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
136	#107	—	4	4	no	—
137	#106	—	4	4	no	—
138	C₄⋊C₈	C₂×C₄×C₃₁	1	16	yes	—
139	#95	—	2	4	no	—
140	#95	—	2	4	no	—
141	#95	—	2	4	no	—
142	#95	—	2	4	no	—
143	C₃₁	#21965	1	1	yes	characteristic, direct product factor, Sylow
144	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	8	no	—
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
150	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
151	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
153	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
154	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
155	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
156	M₁₆	—	4	8	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
160	C₄⋊C₄	—	2	8	no	—
161	C₄⋊C₄	—	2	8	no	—
162	C₂²⋊C₄	—	4	8	no	—
163	C₄²	—	4	8	no	—
164	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
165	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
166	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
167	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	4	no	—
170	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
171	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	8	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	—	4	8	no	—
181	C₂×C₄	—	4	8	no	—
182	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
183	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
185	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
186	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
187	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
188	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
189	C₈	—	2	8	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
194	C₂²	C₃₁×#537	1	2	yes	—
195	C₂²	C₃₁×#575	1	4	yes	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	—	2	4	no	—
199	C₄	—	8	8	no	—
200	C₄	C₃₁×#518	1	1	yes	characteristic
201	C₄	C₃₁×#519	1	1	yes	characteristic
202	C₄	C₃₁×#544	1	2	yes	—
203	C₂	—	4	2	no	—
204	C₂	C₃₁×#2316	1	1	yes	characteristic
205	C₂	C₃₁×#2911	1	2	yes	—
206	C₂	C₃₁×#3256	1	4	yes	—

	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×#1949	C₂	semidirect
9. C₃₁×#423	C₄	semidirect
64. #21965	C₃₁	direct (N × H)
89. #1949	C₂×C₃₁	semidirect
108. #423	C₄×C₃₁	semidirect
143. C₃₁	#21965	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×#423	2
C₃₁×#2283	2
C₃₁×#1949	2
C₃₁×#1809	2
#21965	31

Subgroups — C31×#21965 #21478724

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21965 #21478724