Subgroups — C₃₁×#8285 #21465044

Order 7936.

430 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊C₄×C₃₁	C₂	1	2	yes	maximal
3	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1279	C₂	1	2	yes	maximal
6	C₂×C₃₁×#106	C₄	1	2	yes	—
7	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₃₁	—	2	2	no	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#340	C₄	1	2	yes	—
14	C₃₁×#336	—	2	2	no	—
15	C₃₁×#335	C₄	1	4	yes	complemented
16	C₃₁×#322	C₄	1	4	yes	complemented
17	C₂²×C₈×C₃₁	—	2	2	no	—
18	C₂²×C₈×C₃₁	—	2	1	no	—
19	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	D₄	1	2	yes	—
22	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₃₁	—	2	2	no	—
24	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
25	C₃₁×#106	C₈	1	8	yes	complemented
26	C₃₁×C₄⋊C₈	—	2	2	no	—
27	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₃₁	—	2	1	no	—
29	C₄×C₈×C₃₁	—	4	4	no	—
30	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₃₁	Q₈	1	2	yes	—
35	C₃₁×#95	—	2	2	no	—
36	C₂²×C₄×C₃₁	—	2	4	no	—
37	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
40	C₂×C₈×C₃₁	—	2	2	no	—
41	C₂×C₈×C₃₁	—	2	2	no	—
42	C₂×C₈×C₃₁	—	2	2	no	—
43	C₂×C₈×C₃₁	—	2	2	no	—
44	C₂×C₈×C₃₁	—	2	4	no	—
45	C₂×C₈×C₃₁	—	4	2	no	—
46	C₂×C₈×C₃₁	—	4	2	no	—
47	C₂×C₈×C₃₁	—	4	2	no	—
48	C₂×C₈×C₃₁	C₄²	1	2	yes	—
49	C₂×C₈×C₃₁	C₄²	1	2	yes	—
50	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
51	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
52	C₄⋊C₄×C₃₁	—	2	4	no	—
53	C₄⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
54	C₄²×C₃₁	—	4	2	no	—
55	C₄²×C₃₁	—	4	2	no	—
56	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₃₁	#41	1	1	yes	characteristic
60	#8285	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	2	8	no	—
64	C₂×C₄×C₃₁	—	4	4	no	—
65	C₂×C₄×C₃₁	—	4	4	no	—
66	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
73	C₂×C₄×C₃₁	#131	1	2	yes	—
74	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
75	C₂×C₄×C₃₁	#137	1	2	yes	—
76	C₈×C₃₁	—	2	2	no	—
77	C₈×C₃₁	—	2	4	no	—
78	C₈×C₃₁	—	4	2	no	—
79	C₈×C₃₁	—	4	4	no	—
80	C₈×C₃₁	C₄×C₈	1	4	yes	complemented
81	C₄×#106	C₂×C₃₁	1	1	yes	characteristic
82	C₈×C₄⋊C₄	C₂×C₃₁	1	2	yes	—
83	C₂×#335	C₂×C₃₁	1	1	yes	characteristic
84	C₂×#322	C₂×C₃₁	1	1	yes	characteristic
85	#1279	C₂×C₃₁	1	2	yes	—
86	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₃₁	C₈×D₄	1	2	yes	—
88	C₂²×C₃₁	#440	1	2	yes	—
89	C₂²×C₃₁	#444	1	2	yes	—
90	C₄×C₃₁	—	2	1	no	—
91	C₄×C₃₁	—	2	1	no	—
92	C₄×C₃₁	—	4	8	no	—
93	C₄×C₃₁	C₂×C₄×C₈	1	1	yes	characteristic
94	C₄×C₃₁	#405	1	1	yes	characteristic
95	C₄×C₃₁	#428	1	2	yes	—
96	C₄×C₃₁	#441	1	1	yes	characteristic
97	C₄×C₃₁	#442	1	1	yes	characteristic
98	C₄×C₃₁	C₈×Q₈	1	2	yes	—
99	C₂×#106	C₄×C₃₁	1	2	yes	—
100	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
101	C₂×C₄×C₈	—	2	2	no	—
102	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
104	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
105	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
106	#340	C₄×C₃₁	1	2	yes	—
107	#336	—	2	2	no	—
108	#335	C₄×C₃₁	1	4	yes	complemented
109	#322	C₄×C₃₁	1	4	yes	complemented
110	C₂×C₃₁	SD₁₆⋊C₈	1	4	yes	—
111	C₂×C₃₁	C₈×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₃₁	#1773	1	1	yes	characteristic
113	C₂×C₃₁	#1776	1	1	yes	characteristic
114	C₂²×C₈	—	2	2	no	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
118	C₂²×C₈	D₄×C₃₁	1	2	yes	—
119	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
120	C₂×C₄²	—	2	2	no	—
121	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
122	#106	C₈×C₃₁	1	8	yes	complemented
123	C₄⋊C₈	—	2	2	no	—
124	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	4	4	no	—
127	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
128	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
129	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
130	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
131	C₄×C₈	Q₈×C₃₁	1	2	yes	—
132	#95	—	2	2	no	—
133	C₃₁	#8285	1	1	yes	characteristic, direct product factor, Sylow
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
137	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	4	no	—
143	C₂×C₈	—	4	2	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	C₄²×C₃₁	1	2	yes	—
147	C₂×C₈	C₄²×C₃₁	1	2	yes	—
148	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
149	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
152	C₄²	—	4	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
155	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
156	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
157	C₄²	#41×C₃₁	1	1	yes	characteristic
158	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
166	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
167	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
168	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
170	C₂×C₄	C₃₁×#131	1	2	yes	—
171	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
172	C₂×C₄	C₃₁×#137	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	2	4	no	—
175	C₈	—	4	2	no	—
176	C₈	—	4	4	no	—
177	C₈	C₄×C₈×C₃₁	1	4	yes	complemented
178	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
179	C₂²	C₈×D₄×C₃₁	1	2	yes	—
180	C₂²	C₃₁×#440	1	2	yes	—
181	C₂²	C₃₁×#444	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	8	no	—
185	C₄	C₂×C₄×C₈×C₃₁	1	1	yes	characteristic
186	C₄	C₃₁×#405	1	1	yes	characteristic
187	C₄	C₃₁×#428	1	2	yes	—
188	C₄	C₃₁×#441	1	1	yes	characteristic
189	C₄	C₃₁×#442	1	1	yes	characteristic
190	C₄	C₈×Q₈×C₃₁	1	2	yes	—
191	C₂	C₃₁×SD₁₆⋊C₈	1	4	yes	—
192	C₂	C₈×C₄⋊C₄×C₃₁	1	1	yes	characteristic
193	C₂	C₃₁×#1773	1	1	yes	characteristic
194	C₂	C₃₁×#1776	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₄×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₃₁×#335	C₄	semidirect
16. C₃₁×#322	C₄	semidirect
25. C₃₁×#106	C₈	semidirect
60. #8285	C₃₁	direct (N × H)
80. C₈×C₃₁	C₄×C₈	semidirect
108. #335	C₄×C₃₁	semidirect
109. #322	C₄×C₃₁	semidirect
122. #106	C₈×C₃₁	semidirect
133. C₃₁	#8285	direct (N × H)
177. C₈	C₄×C₈×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₃₁×#106	2
C₈×C₄⋊C₄×C₃₁	2
C₂×C₃₁×#335	2
C₂×C₃₁×#322	2
C₃₁×#1279	2
#8285	31

Subgroups — C31×#8285 #21465044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8285 #21465044