Subgroups — C₃₁×#13289 #21470048

Order 7936.

342 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#499	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1387	C₂	1	4	yes	maximal
5	C₂×C₃₁×#125	—	2	1	no	—
6	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
7	C₃₁×#499	—	2	1	no	—
8	C₃₁×#499	—	2	1	no	—
9	C₃₁×#499	C₄	1	2	yes	—
10	C₃₁×#499	C₄	1	2	yes	—
11	C₂×C₃₁×#108	—	2	1	no	—
12	C₂×C₃₁×#108	C₂²	1	2	yes	—
13	C₂×C₃₁×#107	C₂²	1	2	yes	—
14	C₂×C₃₁×#106	—	2	1	no	—
15	C₃₁×#423	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#346	—	2	1	no	—
18	C₄×C₁₆×C₃₁	—	2	1	no	—
19	C₂×M₁₆×C₃₁	—	2	2	no	—
20	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
22	C₃₁×#125	—	2	2	no	—
23	C₃₁×#125	—	2	2	no	—
24	C₃₁×#125	—	4	2	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
27	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
28	C₃₁×M₃₂	—	2	2	no	—
29	C₃₁×M₃₂	Q₈	1	4	yes	—
30	C₂×C₁₆×C₃₁	—	2	2	no	—
31	C₃₁×#108	—	2	4	no	—
32	C₃₁×#108	—	4	2	no	—
33	C₃₁×#107	—	2	2	no	—
34	C₃₁×#107	C₂×C₄	1	4	yes	—
35	C₃₁×#106	—	2	2	no	—
36	C₃₁×#106	—	2	2	no	—
37	C₃₁×C₄⋊C₈	—	4	2	no	—
38	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	—	4	2	no	—
43	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	M₁₆×C₃₁	—	4	2	no	—
46	M₁₆×C₃₁	—	4	2	no	—
47	C₂×C₈×C₃₁	—	4	2	no	—
48	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
51	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
52	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₃₁	#41	1	2	yes	—
56	C₄⋊C₄×C₃₁	—	2	4	no	—
57	C₄⋊C₄×C₃₁	—	4	4	no	—
58	C₄⋊C₄×C₃₁	—	4	4	no	—
59	C₄⋊C₄×C₃₁	—	4	2	no	—
60	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₃₁	D₈	1	2	yes	—
63	C₁₆×C₃₁	—	2	4	no	—
64	#13289	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₃₁	—	4	2	no	—
67	C₂×C₄×C₃₁	—	4	4	no	—
68	C₂×C₄×C₃₁	—	8	4	no	—
69	C₂×C₄×C₃₁	#102	1	2	yes	—
70	C₂×C₄×C₃₁	#102	1	2	yes	—
71	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
77	C₈×C₃₁	—	8	2	no	—
78	C₈×C₃₁	#103	1	2	yes	—
79	C₈×C₃₁	#103	1	2	yes	—
80	C₈×C₃₁	C₄×Q₈	1	2	yes	—
81	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
82	C₂×#499	C₂×C₃₁	1	1	yes	characteristic
83	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic
84	#1848	C₂×C₃₁	1	1	yes	characteristic
85	#1387	C₂×C₃₁	1	4	yes	—
86	C₂²×C₃₁	—	2	2	no	—
87	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
88	C₂²×C₃₁	#487	1	1	yes	characteristic
89	C₂²×C₃₁	#498	1	1	yes	characteristic
90	C₄×C₃₁	—	2	2	no	—
91	C₄×C₃₁	—	2	2	no	—
92	C₄×C₃₁	—	8	4	no	—
93	C₄×C₃₁	#391	1	1	yes	characteristic
94	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
95	C₄×C₃₁	#486	1	1	yes	characteristic
96	C₄×C₃₁	#500	1	1	yes	characteristic
97	C₂×#125	—	2	1	no	—
98	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
99	#499	—	2	1	no	—
100	#499	—	2	1	no	—
101	#499	C₄×C₃₁	1	2	yes	—
102	#499	C₄×C₃₁	1	2	yes	—
103	C₂×#108	—	2	1	no	—
104	C₂×#108	C₂²×C₃₁	1	2	yes	—
105	C₂×#107	C₂²×C₃₁	1	2	yes	—
106	C₂×#106	—	2	1	no	—
107	#423	—	2	1	no	—
108	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
109	#346	—	2	1	no	—
110	C₄×C₁₆	—	2	1	no	—
111	C₂×C₃₁	—	2	1	no	—
112	C₂×C₃₁	—	2	1	no	—
113	C₂×C₃₁	#1990	1	1	yes	characteristic
114	C₂×C₃₁	#2148	1	2	yes	—
115	C₂×M₁₆	—	2	2	no	—
116	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
117	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
118	#125	—	2	2	no	—
119	#125	—	2	2	no	—
120	#125	—	4	2	no	—
121	C₂×C₄⋊C₄	—	2	2	no	—
122	C₂×C₄⋊C₄	—	4	2	no	—
123	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
124	M₃₂	—	2	2	no	—
125	M₃₂	Q₈×C₃₁	1	4	yes	—
126	C₂×C₁₆	—	2	2	no	—
127	#108	—	2	4	no	—
128	#108	—	4	2	no	—
129	#107	—	2	2	no	—
130	#107	C₂×C₄×C₃₁	1	4	yes	—
131	#106	—	2	2	no	—
132	#106	—	2	2	no	—
133	C₄⋊C₈	—	4	2	no	—
134	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
135	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
136	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
137	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
138	C₃₁	#13289	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
141	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
142	M₁₆	—	4	2	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
146	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
147	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
148	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
149	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
150	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
152	C₂×C₈	#41×C₃₁	1	2	yes	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
158	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
159	C₄²	D₈×C₃₁	1	2	yes	—
160	C₁₆	—	2	4	no	—
161	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	—	8	4	no	—
165	C₂×C₄	C₃₁×#102	1	2	yes	—
166	C₂×C₄	C₃₁×#102	1	2	yes	—
167	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
168	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
169	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
170	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
171	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
172	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
173	C₈	—	8	2	no	—
174	C₈	C₃₁×#103	1	2	yes	—
175	C₈	C₃₁×#103	1	2	yes	—
176	C₈	C₄×Q₈×C₃₁	1	2	yes	—
177	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
180	C₂²	C₃₁×#487	1	1	yes	characteristic
181	C₂²	C₃₁×#498	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	C₃₁×#391	1	1	yes	characteristic
186	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
187	C₄	C₃₁×#486	1	1	yes	characteristic
188	C₄	C₃₁×#500	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	1	no	—
191	C₂	C₃₁×#1990	1	1	yes	characteristic
192	C₂	C₃₁×#2148	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂³×C₃₁	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
64. #13289	C₃₁	direct (N × H)
138. C₃₁	#13289	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×#499	2
C₄×C₃₁×M₃₂	2
C₃₁×#1848	2
C₃₁×#1387	2
#13289	31

Subgroups — C31×#13289 #21470048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13289 #21470048