Subgroups — C₃₁×#13272 #21470031

Order 7936.

366 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#498	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×C₄⋊Q₈	—	2	1	no	—
7	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
8	C₃₁×#498	—	2	2	no	—
9	C₃₁×#498	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×#106	—	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	C₂²×Q₈×C₃₁	—	4	1	no	—
19	C₂×M₁₆×C₃₁	—	2	1	no	—
20	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
22	C₃₁×C₄⋊Q₈	—	2	4	no	—
23	C₃₁×C₄⋊Q₈	—	4	2	no	—
24	C₃₁×#117	—	2	1	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
27	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
28	C₃₁×M₃₂	—	2	2	no	—
29	C₃₁×M₃₂	—	2	2	no	—
30	C₂×C₁₆×C₃₁	—	2	2	no	—
31	C₃₁×#108	—	2	1	no	—
32	C₃₁×#108	Q₈	1	2	yes	complemented
33	C₃₁×#107	Q₈	1	2	yes	complemented
34	C₃₁×#106	—	2	4	no	—
35	C₃₁×#106	—	2	2	no	—
36	C₃₁×#106	—	2	1	no	—
37	C₃₁×#106	C₂×C₄	1	4	yes	—
38	C₃₁×#97	—	4	1	no	—
39	C₄×C₈×C₃₁	—	4	1	no	—
40	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₃₁	D₄	1	2	yes	—
42	C₂×Q₈×C₃₁	—	4	4	no	—
43	C₂×Q₈×C₃₁	—	8	4	no	—
44	C₂²×C₄×C₃₁	—	2	1	no	—
45	C₂²×C₄×C₃₁	—	4	2	no	—
46	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	M₁₆×C₃₁	—	2	2	no	—
48	M₁₆×C₃₁	—	4	1	no	—
49	C₂×C₈×C₃₁	—	4	1	no	—
50	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
51	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₃₁	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
54	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
59	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
60	C₄⋊C₄×C₃₁	—	2	4	no	—
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	—	4	4	no	—
63	C₄⋊C₄×C₃₁	—	4	2	no	—
64	C₂²⋊C₄×C₃₁	—	4	1	no	—
65	C₄²×C₃₁	—	2	1	no	—
66	C₄²×C₃₁	—	4	1	no	—
67	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
68	C₁₆×C₃₁	—	4	2	no	—
69	#13272	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
71	Q₈×C₃₁	—	8	8	no	—
72	C₂×C₄×C₃₁	—	2	4	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	—	4	4	no	—
75	C₂×C₄×C₃₁	—	4	2	no	—
76	C₂×C₄×C₃₁	—	8	4	no	—
77	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₈×C₃₁	—	4	2	no	—
84	C₈×C₃₁	#103	1	2	yes	—
85	C₈×C₃₁	#103	1	2	yes	—
86	C₈×C₃₁	C₄×Q₈	1	1	yes	characteristic
87	C₈×C₃₁	C₄×Q₈	1	1	yes	characteristic
88	C₈×C₃₁	#124	1	1	yes	characteristic
89	C₈×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
90	C₂×#498	C₂×C₃₁	1	1	yes	characteristic
91	C₂×#364	C₂×C₃₁	1	1	yes	characteristic
92	#1772	C₂×C₃₁	1	1	yes	characteristic
93	#1386	C₂×C₃₁	1	2	yes	—
94	#1383	C₂×C₃₁	1	2	yes	—
95	C₂²×C₃₁	—	2	2	no	—
96	C₂²×C₃₁	#418	1	1	yes	characteristic
97	C₂²×C₃₁	#476	1	1	yes	characteristic
98	C₂²×C₃₁	#477	1	1	yes	characteristic
99	C₄×C₃₁	—	2	4	no	—
100	C₄×C₃₁	—	8	4	no	—
101	C₄×C₃₁	—	8	2	no	—
102	C₄×C₃₁	#391	1	1	yes	characteristic
103	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
104	C₄×C₃₁	#476	1	1	yes	characteristic
105	C₄×C₃₁	#477	1	1	yes	characteristic
106	C₂×C₄⋊Q₈	—	2	1	no	—
107	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
108	#498	—	2	2	no	—
109	#498	C₄×C₃₁	1	4	yes	complemented
110	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
111	#427	C₂²×C₃₁	1	1	yes	characteristic
112	C₂×#106	—	2	1	no	—
113	C₂×#106	C₂²×C₃₁	1	2	yes	—
114	#405	—	2	1	no	—
115	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
116	#364	—	2	2	no	—
117	#339	—	2	1	no	—
118	C₂×C₃₁	—	2	1	no	—
119	C₂×C₃₁	—	2	1	no	—
120	C₂×C₃₁	#1987	1	1	yes	characteristic
121	C₂×C₃₁	#2230	1	2	yes	—
122	C₂²×Q₈	—	4	1	no	—
123	C₂×M₁₆	—	2	1	no	—
124	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
126	C₄⋊Q₈	—	2	4	no	—
127	C₄⋊Q₈	—	4	2	no	—
128	#117	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
132	M₃₂	—	2	2	no	—
133	M₃₂	—	2	2	no	—
134	C₂×C₁₆	—	2	2	no	—
135	#108	—	2	1	no	—
136	#108	Q₈×C₃₁	1	2	yes	complemented
137	#107	Q₈×C₃₁	1	2	yes	complemented
138	#106	—	2	4	no	—
139	#106	—	2	2	no	—
140	#106	—	2	1	no	—
141	#106	C₂×C₄×C₃₁	1	4	yes	—
142	#97	—	4	1	no	—
143	C₄×C₈	—	4	1	no	—
144	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
145	C₄×C₈	D₄×C₃₁	1	2	yes	—
146	C₃₁	#13272	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×Q₈	—	4	4	no	—
148	C₂×Q₈	—	8	4	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	4	1	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
156	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
157	C₂×C₈	SD₁₆×C₃₁	1	1	yes	characteristic
158	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
159	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
160	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
161	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
162	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
163	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
164	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
173	C₁₆	—	4	2	no	—
174	C₂³	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
175	Q₈	—	8	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	8	4	no	—
181	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
182	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
183	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
185	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
186	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₈	—	4	2	no	—
188	C₈	C₃₁×#103	1	2	yes	—
189	C₈	C₃₁×#103	1	2	yes	—
190	C₈	C₄×Q₈×C₃₁	1	1	yes	characteristic
191	C₈	C₄×Q₈×C₃₁	1	1	yes	characteristic
192	C₈	C₃₁×#124	1	1	yes	characteristic
193	C₈	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
194	C₂²	—	2	2	no	—
195	C₂²	C₃₁×#418	1	1	yes	characteristic
196	C₂²	C₃₁×#476	1	1	yes	characteristic
197	C₂²	C₃₁×#477	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	8	4	no	—
200	C₄	—	8	2	no	—
201	C₄	C₃₁×#391	1	1	yes	characteristic
202	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
203	C₄	C₃₁×#476	1	1	yes	characteristic
204	C₄	C₃₁×#477	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	C₃₁×#1987	1	1	yes	characteristic
208	C₂	C₃₁×#2230	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×#498	C₄	semidirect
32. C₃₁×#108	Q₈	semidirect
33. C₃₁×#107	Q₈	semidirect
69. #13272	C₃₁	direct (N × H)
109. #498	C₄×C₃₁	semidirect
136. #108	Q₈×C₃₁	semidirect
137. #107	Q₈×C₃₁	semidirect
146. C₃₁	#13272	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×#498	2
C₂×C₃₁×#364	2
C₃₁×#1772	2
C₃₁×#1386	2
C₃₁×#1383	2
#13272	31

Subgroups — C31×#13272 #21470031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13272 #21470031