Subgroups — C₃₁×#13165 #21469924

Order 7936.

326 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×#366	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1942	C₂	1	2	yes	maximal
5	C₃₁×#1380	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#107	C₂²	1	2	yes	—
9	C₂×C₃₁×#107	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#389	—	2	2	no	—
12	C₃₁×#366	—	2	2	no	—
13	C₃₁×#366	C₄	1	4	yes	complemented
14	C₄×C₁₆×C₃₁	—	2	2	no	—
15	C₃₁×#340	—	2	2	no	—
16	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
19	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
20	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
21	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₃₁	—	2	1	no	—
23	C₂×C₁₆×C₃₁	—	2	1	no	—
24	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
26	C₂×C₁₆×C₃₁	Q₈	1	2	yes	—
27	C₂×C₁₆×C₃₁	Q₈	1	2	yes	—
28	C₃₁×#107	—	2	2	no	—
29	C₃₁×#107	—	2	2	no	—
30	C₃₁×#107	—	2	2	no	—
31	C₃₁×#107	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₃₁	—	2	1	no	—
33	C₄×C₈×C₃₁	—	2	1	no	—
34	C₃₁×#95	—	4	2	no	—
35	C₂²×C₄×C₃₁	—	4	2	no	—
36	C₂²×C₄×C₃₁	—	4	2	no	—
37	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₃₁	—	2	1	no	—
41	C₂×C₈×C₃₁	—	2	1	no	—
42	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
43	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
48	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
49	C₄⋊C₄×C₃₁	—	2	4	no	—
50	C₄⋊C₄×C₃₁	—	4	2	no	—
51	C₄⋊C₄×C₃₁	—	4	2	no	—
52	C₄⋊C₄×C₃₁	—	4	2	no	—
53	C₄⋊C₄×C₃₁	—	8	4	no	—
54	C₄²×C₃₁	—	2	1	no	—
55	C₄²×C₃₁	—	2	1	no	—
56	C₁₆×C₃₁	—	2	2	no	—
57	C₁₆×C₃₁	C₄⋊C₄	1	4	yes	complemented
58	#13165	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₃₁	—	2	1	no	—
61	C₂×C₄×C₃₁	—	2	1	no	—
62	C₂×C₄×C₃₁	—	2	1	no	—
63	C₂×C₄×C₃₁	—	2	1	no	—
64	C₂×C₄×C₃₁	—	4	4	no	—
65	C₂×C₄×C₃₁	—	8	2	no	—
66	C₂×C₄×C₃₁	—	8	2	no	—
67	C₂×C₄×C₃₁	—	8	2	no	—
68	C₂×C₄×C₃₁	—	8	2	no	—
69	C₂×C₄×C₃₁	—	8	2	no	—
70	C₂×C₄×C₃₁	D₁₆	1	2	yes	—
71	C₂×C₄×C₃₁	Q₃₂	1	2	yes	—
72	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
77	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
78	C₈×C₃₁	—	2	2	no	—
79	C₈×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₈×C₃₁	C₄×Q₈	1	1	yes	characteristic
81	C₈×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
82	C₈×C₃₁	#125	1	1	yes	characteristic
83	C₂×#366	C₂×C₃₁	1	1	yes	characteristic
84	C₂×#366	C₂×C₃₁	1	1	yes	characteristic
85	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
86	#1942	C₂×C₃₁	1	2	yes	—
87	#1380	C₂×C₃₁	1	2	yes	—
88	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
89	C₂²×C₃₁	#466	1	1	yes	characteristic
90	C₂²×C₃₁	#500	1	1	yes	characteristic
91	C₂²×C₃₁	C₂×D₁₆	1	1	yes	characteristic
92	C₂²×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
93	C₂²×C₃₁	#508	1	1	yes	characteristic
94	C₂²×C₃₁	#508	1	1	yes	characteristic
95	C₄×C₃₁	—	2	2	no	—
96	C₄×C₃₁	—	2	2	no	—
97	C₄×C₃₁	—	8	4	no	—
98	C₄×C₃₁	—	16	2	no	—
99	C₄×C₃₁	#389	1	1	yes	characteristic
100	C₄×C₃₁	C₄×Q₁₆	1	1	yes	characteristic
101	C₄×C₃₁	#467	1	1	yes	characteristic
102	C₄×C₃₁	#499	1	1	yes	characteristic
103	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
104	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
105	C₂×#107	C₂²×C₃₁	1	2	yes	—
106	C₂×#107	C₂²×C₃₁	1	2	yes	—
107	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
108	#389	—	2	2	no	—
109	#366	—	2	2	no	—
110	#366	C₄×C₃₁	1	4	yes	complemented
111	C₄×C₁₆	—	2	2	no	—
112	#340	—	2	2	no	—
113	C₂×C₃₁	#1942	1	1	yes	characteristic
114	C₂×C₃₁	C₄×D₁₆	1	1	yes	characteristic
115	C₂×C₃₁	C₄×Q₃₂	1	1	yes	characteristic
116	C₂×C₃₁	#2215	1	1	yes	characteristic
117	C₂×C₃₁	#2216	1	1	yes	characteristic
118	C₂×C₃₁	#2252	1	1	yes	characteristic
119	C₂×C₃₁	#2253	1	1	yes	characteristic
120	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	4	2	no	—
125	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	—	2	1	no	—
128	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
129	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
130	C₂×C₁₆	Q₈×C₃₁	1	2	yes	—
131	C₂×C₁₆	Q₈×C₃₁	1	2	yes	—
132	#107	—	2	2	no	—
133	#107	—	2	2	no	—
134	#107	—	2	2	no	—
135	#107	C₂×C₄×C₃₁	1	4	yes	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	—	2	1	no	—
138	#95	—	4	2	no	—
139	C₃₁	#13165	1	1	yes	characteristic, direct product factor, Sylow
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
143	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
148	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
149	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
150	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
152	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
153	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₄⋊C₄	—	8	4	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	2	1	no	—
161	C₁₆	—	2	2	no	—
162	C₁₆	C₄⋊C₄×C₃₁	1	4	yes	complemented
163	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	2	no	—
170	C₂×C₄	—	8	2	no	—
171	C₂×C₄	—	8	2	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	C₃₁×D₁₆	1	2	yes	—
175	C₂×C₄	C₃₁×Q₃₂	1	2	yes	—
176	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
177	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
178	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
179	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
180	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
181	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
184	C₈	C₄×Q₈×C₃₁	1	1	yes	characteristic
185	C₈	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
186	C₈	C₃₁×#125	1	1	yes	characteristic
187	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
188	C₂²	C₃₁×#466	1	1	yes	characteristic
189	C₂²	C₃₁×#500	1	1	yes	characteristic
190	C₂²	C₂×C₃₁×D₁₆	1	1	yes	characteristic
191	C₂²	C₂×C₃₁×Q₃₂	1	1	yes	characteristic
192	C₂²	C₃₁×#508	1	1	yes	characteristic
193	C₂²	C₃₁×#508	1	1	yes	characteristic
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	8	4	no	—
197	C₄	—	16	2	no	—
198	C₄	C₃₁×#389	1	1	yes	characteristic
199	C₄	C₄×Q₁₆×C₃₁	1	1	yes	characteristic
200	C₄	C₃₁×#467	1	1	yes	characteristic
201	C₄	C₃₁×#499	1	1	yes	characteristic
202	C₂	C₃₁×#1942	1	1	yes	characteristic
203	C₂	C₄×C₃₁×D₁₆	1	1	yes	characteristic
204	C₂	C₄×C₃₁×Q₃₂	1	1	yes	characteristic
205	C₂	C₃₁×#2215	1	1	yes	characteristic
206	C₂	C₃₁×#2216	1	1	yes	characteristic
207	C₂	C₃₁×#2252	1	1	yes	characteristic
208	C₂	C₃₁×#2253	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
13. C₃₁×#366	C₄	semidirect
57. C₁₆×C₃₁	C₄⋊C₄	semidirect
58. #13165	C₃₁	direct (N × H)
110. #366	C₄×C₃₁	semidirect
139. C₃₁	#13165	direct (N × H)
162. C₁₆	C₄⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#366	2
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1942	2
C₃₁×#1380	2
#13165	31

Subgroups — C31×#13165 #21469924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13165 #21469924