Subgroups — C₃₁×#13288 #21470047

Order 7936.

354 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1386	C₂	1	4	yes	maximal
5	C₃₁×#533	—	2	1	no	—
6	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
7	C₃₁×#500	C₄	1	2	yes	complemented
8	C₃₁×#499	—	2	1	no	—
9	C₃₁×#499	C₄	1	2	yes	complemented
10	C₃₁×#498	—	2	1	no	—
11	C₃₁×#427	—	2	1	no	—
12	C₃₁×#427	C₂²	1	2	yes	—
13	C₂×C₃₁×#107	—	2	1	no	—
14	C₂×C₃₁×#106	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#382	—	2	1	no	—
17	C₃₁×#346	—	2	1	no	—
18	C₄×C₁₆×C₃₁	—	2	1	no	—
19	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
21	C₃₁×C₄⋊Q₈	—	2	2	no	—
22	C₃₁×#125	—	2	2	no	—
23	C₃₁×C₂²⋊Q₈	—	4	2	no	—
24	C₄×Q₈×C₃₁	—	4	2	no	—
25	C₃₁×#117	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
28	C₃₁×M₃₂	—	2	2	no	—
29	C₃₁×M₃₂	Q₈	1	4	yes	complemented
30	C₂×C₁₆×C₃₁	—	2	2	no	—
31	C₃₁×#107	—	2	2	no	—
32	C₃₁×#107	—	4	2	no	—
33	C₃₁×#107	C₂×C₄	1	4	yes	—
34	C₃₁×#106	—	2	2	no	—
35	C₃₁×#106	—	2	2	no	—
36	C₃₁×#106	—	2	4	no	—
37	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₃₁×#95	—	4	2	no	—
42	C₂×Q₈×C₃₁	—	4	2	no	—
43	C₂²×C₄×C₃₁	—	4	2	no	—
44	C₂²×C₄×C₃₁	D₈	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
49	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
50	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₃₁	#41	1	2	yes	—
54	C₄⋊C₄×C₃₁	—	2	4	no	—
55	C₄⋊C₄×C₃₁	—	4	4	no	—
56	C₄⋊C₄×C₃₁	—	4	2	no	—
57	C₄⋊C₄×C₃₁	—	4	2	no	—
58	C₄⋊C₄×C₃₁	—	4	2	no	—
59	C₂²⋊C₄×C₃₁	—	4	2	no	—
60	C₄²×C₃₁	—	4	2	no	—
61	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₃₁	SD₁₆	1	2	yes	—
64	C₁₆×C₃₁	—	2	4	no	—
65	#13288	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₃₁	—	8	4	no	—
68	C₂×C₄×C₃₁	—	4	2	no	—
69	C₂×C₄×C₃₁	—	4	4	no	—
70	C₂×C₄×C₃₁	—	8	4	no	—
71	C₂×C₄×C₃₁	—	8	2	no	—
72	C₂×C₄×C₃₁	#102	1	2	yes	—
73	C₂×C₄×C₃₁	#102	1	2	yes	—
74	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
76	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
80	C₈×C₃₁	#103	1	2	yes	—
81	C₈×C₃₁	#103	1	2	yes	—
82	C₈×C₃₁	C₄×Q₈	1	2	yes	—
83	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
84	#3160	C₂×C₃₁	1	1	yes	characteristic
85	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic
86	#1846	C₂×C₃₁	1	1	yes	characteristic
87	#1386	C₂×C₃₁	1	4	yes	—
88	C₂²×C₃₁	—	2	2	no	—
89	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
90	C₂²×C₃₁	#486	1	1	yes	characteristic
91	C₂²×C₃₁	#500	1	1	yes	characteristic
92	C₄×C₃₁	—	2	2	no	—
93	C₄×C₃₁	—	2	2	no	—
94	C₄×C₃₁	—	8	4	no	—
95	C₄×C₃₁	—	16	2	no	—
96	C₄×C₃₁	#391	1	1	yes	characteristic
97	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
98	C₄×C₃₁	#487	1	1	yes	characteristic
99	C₄×C₃₁	#498	1	1	yes	characteristic
100	#533	—	2	1	no	—
101	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
102	#500	C₄×C₃₁	1	2	yes	complemented
103	#499	—	2	1	no	—
104	#499	C₄×C₃₁	1	2	yes	complemented
105	#498	—	2	1	no	—
106	#427	—	2	1	no	—
107	#427	C₂²×C₃₁	1	2	yes	—
108	C₂×#107	—	2	1	no	—
109	C₂×#106	C₂²×C₃₁	1	2	yes	—
110	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
111	#382	—	2	1	no	—
112	#346	—	2	1	no	—
113	C₄×C₁₆	—	2	1	no	—
114	C₂×C₃₁	—	2	1	no	—
115	C₂×C₃₁	—	2	1	no	—
116	C₂×C₃₁	#1990	1	1	yes	characteristic
117	C₂×C₃₁	#2145	1	2	yes	—
118	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
119	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
120	C₄⋊Q₈	—	2	2	no	—
121	#125	—	2	2	no	—
122	C₂²⋊Q₈	—	4	2	no	—
123	C₄×Q₈	—	4	2	no	—
124	#117	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
127	M₃₂	—	2	2	no	—
128	M₃₂	Q₈×C₃₁	1	4	yes	complemented
129	C₂×C₁₆	—	2	2	no	—
130	#107	—	2	2	no	—
131	#107	—	4	2	no	—
132	#107	C₂×C₄×C₃₁	1	4	yes	—
133	#106	—	2	2	no	—
134	#106	—	2	2	no	—
135	#106	—	2	4	no	—
136	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
137	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
138	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
139	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
140	#95	—	4	2	no	—
141	C₃₁	#13288	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×Q₈	—	4	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	D₈×C₃₁	1	2	yes	—
145	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
146	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
147	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
148	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
149	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
150	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
151	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
152	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
153	C₂×C₈	#41×C₃₁	1	2	yes	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	4	2	no	—
160	C₄²	—	4	2	no	—
161	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
162	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
163	C₄²	SD₁₆×C₃₁	1	2	yes	—
164	C₁₆	—	2	4	no	—
165	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
166	Q₈	—	8	4	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	C₃₁×#102	1	2	yes	—
172	C₂×C₄	C₃₁×#102	1	2	yes	—
173	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
174	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
175	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
176	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
177	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
178	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
179	C₈	C₃₁×#103	1	2	yes	—
180	C₈	C₃₁×#103	1	2	yes	—
181	C₈	C₄×Q₈×C₃₁	1	2	yes	—
182	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
185	C₂²	C₃₁×#486	1	1	yes	characteristic
186	C₂²	C₃₁×#500	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	2	2	no	—
189	C₄	—	8	4	no	—
190	C₄	—	16	2	no	—
191	C₄	C₃₁×#391	1	1	yes	characteristic
192	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
193	C₄	C₃₁×#487	1	1	yes	characteristic
194	C₄	C₃₁×#498	1	1	yes	characteristic
195	C₂	—	2	1	no	—
196	C₂	—	2	1	no	—
197	C₂	C₃₁×#1990	1	1	yes	characteristic
198	C₂	C₃₁×#2145	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×D₈
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
7. C₃₁×#500	C₄	semidirect
9. C₃₁×#499	C₄	semidirect
29. C₃₁×M₃₂	Q₈	semidirect
65. #13288	C₃₁	direct (N × H)
102. #500	C₄×C₃₁	semidirect
104. #499	C₄×C₃₁	semidirect
128. M₃₂	Q₈×C₃₁	semidirect
141. C₃₁	#13288	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3160	2
C₄×C₃₁×M₃₂	2
C₃₁×#1846	2
C₃₁×#1386	2
#13288	31

Subgroups — C31×#13288 #21470047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13288 #21470047