Subgroups — C₃₁×#13273 #21470032

Order 7936.

334 conjugacy classes of subgroups.

The subgroup lattice has 202 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	C₂×C₃₁×#364	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1772	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1386	C₂	1	2	yes	maximal
5	C₃₁×#1383	C₂	1	2	yes	maximal
6	C₂×C₃₁×#125	—	2	1	no	—
7	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
8	C₃₁×#499	—	2	2	no	—
9	C₃₁×#499	C₄	1	4	yes	complemented
10	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
11	C₃₁×#427	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#107	—	2	1	no	—
13	C₂×C₃₁×#106	C₂²	1	2	yes	—
14	C₃₁×#405	—	2	1	no	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#364	—	2	2	no	—
17	C₃₁×#339	—	2	1	no	—
18	C₂×M₁₆×C₃₁	—	2	1	no	—
19	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
21	C₃₁×#125	—	2	4	no	—
22	C₃₁×#125	—	4	2	no	—
23	C₃₁×#117	—	2	1	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
26	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
27	C₃₁×M₃₂	—	2	2	no	—
28	C₃₁×M₃₂	—	2	2	no	—
29	C₂×C₁₆×C₃₁	—	2	2	no	—
30	C₃₁×#108	—	2	1	no	—
31	C₃₁×#108	Q₈	1	2	yes	—
32	C₃₁×#107	—	2	4	no	—
33	C₃₁×#107	Q₈	1	2	yes	—
34	C₃₁×#106	—	2	2	no	—
35	C₃₁×#106	—	2	1	no	—
36	C₃₁×#106	C₂×C₄	1	4	yes	—
37	C₃₁×#97	—	4	1	no	—
38	C₄×C₈×C₃₁	—	4	1	no	—
39	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₃₁	D₄	1	2	yes	—
41	C₂²×C₄×C₃₁	—	2	1	no	—
42	C₂²×C₄×C₃₁	—	4	2	no	—
43	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₃₁	—	2	2	no	—
45	M₁₆×C₃₁	—	4	1	no	—
46	C₂×C₈×C₃₁	—	4	1	no	—
47	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
51	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
56	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
57	C₄⋊C₄×C₃₁	—	2	4	no	—
58	C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₄⋊C₄×C₃₁	—	4	8	no	—
60	C₄⋊C₄×C₃₁	—	4	2	no	—
61	C₂²⋊C₄×C₃₁	—	4	1	no	—
62	C₄²×C₃₁	—	2	1	no	—
63	C₄²×C₃₁	—	4	1	no	—
64	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
65	C₁₆×C₃₁	—	4	2	no	—
66	#13273	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂×C₄×C₃₁	—	2	2	no	—
69	C₂×C₄×C₃₁	—	4	2	no	—
70	C₂×C₄×C₃₁	—	4	4	no	—
71	C₂×C₄×C₃₁	—	4	2	no	—
72	C₂×C₄×C₃₁	—	8	4	no	—
73	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
78	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₈×C₃₁	—	4	2	no	—
80	C₈×C₃₁	#103	1	2	yes	—
81	C₈×C₃₁	#103	1	2	yes	—
82	C₈×C₃₁	C₄×Q₈	1	1	yes	characteristic
83	C₈×C₃₁	C₄×Q₈	1	1	yes	characteristic
84	C₈×C₃₁	#124	1	1	yes	characteristic
85	C₈×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×#499	C₂×C₃₁	1	1	yes	characteristic
87	C₂×#364	C₂×C₃₁	1	1	yes	characteristic
88	#1772	C₂×C₃₁	1	1	yes	characteristic
89	#1386	C₂×C₃₁	1	2	yes	—
90	#1383	C₂×C₃₁	1	2	yes	—
91	C₂²×C₃₁	—	2	2	no	—
92	C₂²×C₃₁	#418	1	1	yes	characteristic
93	C₂²×C₃₁	#476	1	1	yes	characteristic
94	C₂²×C₃₁	#477	1	1	yes	characteristic
95	C₄×C₃₁	—	4	2	no	—
96	C₄×C₃₁	—	8	4	no	—
97	C₄×C₃₁	—	8	2	no	—
98	C₄×C₃₁	#391	1	1	yes	characteristic
99	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
100	C₄×C₃₁	#476	1	1	yes	characteristic
101	C₄×C₃₁	#477	1	1	yes	characteristic
102	C₂×#125	—	2	1	no	—
103	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
104	#499	—	2	2	no	—
105	#499	C₄×C₃₁	1	4	yes	complemented
106	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
107	#427	C₂²×C₃₁	1	1	yes	characteristic
108	C₂×#107	—	2	1	no	—
109	C₂×#106	C₂²×C₃₁	1	2	yes	—
110	#405	—	2	1	no	—
111	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
112	#364	—	2	2	no	—
113	#339	—	2	1	no	—
114	C₂×C₃₁	—	2	1	no	—
115	C₂×C₃₁	—	2	1	no	—
116	C₂×C₃₁	#1987	1	1	yes	characteristic
117	C₂×C₃₁	#2231	1	2	yes	—
118	C₂×M₁₆	—	2	1	no	—
119	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
120	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
121	#125	—	2	4	no	—
122	#125	—	4	2	no	—
123	#117	—	2	1	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	4	2	no	—
126	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
127	M₃₂	—	2	2	no	—
128	M₃₂	—	2	2	no	—
129	C₂×C₁₆	—	2	2	no	—
130	#108	—	2	1	no	—
131	#108	Q₈×C₃₁	1	2	yes	—
132	#107	—	2	4	no	—
133	#107	Q₈×C₃₁	1	2	yes	—
134	#106	—	2	2	no	—
135	#106	—	2	1	no	—
136	#106	C₂×C₄×C₃₁	1	4	yes	—
137	#97	—	4	1	no	—
138	C₄×C₈	—	4	1	no	—
139	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
140	C₄×C₈	D₄×C₃₁	1	2	yes	—
141	C₃₁	#13273	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	4	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
149	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
150	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
151	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
152	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
153	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
154	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
155	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
156	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
157	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	4	8	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
166	C₁₆	—	4	2	no	—
167	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	8	4	no	—
173	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
174	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
175	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
176	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
177	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
178	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₈	—	4	2	no	—
180	C₈	C₃₁×#103	1	2	yes	—
181	C₈	C₃₁×#103	1	2	yes	—
182	C₈	C₄×Q₈×C₃₁	1	1	yes	characteristic
183	C₈	C₄×Q₈×C₃₁	1	1	yes	characteristic
184	C₈	C₃₁×#124	1	1	yes	characteristic
185	C₈	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	C₃₁×#418	1	1	yes	characteristic
188	C₂²	C₃₁×#476	1	1	yes	characteristic
189	C₂²	C₃₁×#477	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	8	4	no	—
192	C₄	—	8	2	no	—
193	C₄	C₃₁×#391	1	1	yes	characteristic
194	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
195	C₄	C₃₁×#476	1	1	yes	characteristic
196	C₄	C₃₁×#477	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	C₃₁×#1987	1	1	yes	characteristic
200	C₂	C₃₁×#2231	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	#476
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂³×C₃₁	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₃₁×#499	C₄	semidirect
66. #13273	C₃₁	direct (N × H)
105. #499	C₄×C₃₁	semidirect
141. C₃₁	#13273	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
C₂×C₃₁×#364	2
C₃₁×#1772	2
C₃₁×#1386	2
C₃₁×#1383	2
#13273	31

Subgroups — C31×#13273 #21470032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13273 #21470032