Subgroups — C₃₁×#13252 #21470011

Order 7936.

598 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#357	C₂	1	4	yes	maximal, complemented
4	C₄×C₃₁×#107	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₃₁	—	2	1	no	—
6	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×#127	—	2	1	no	—
8	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#493	—	2	2	no	—
10	C₃₁×#493	C₄	1	4	yes	complemented
11	C₂×C₃₁×#107	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
14	C₃₁×C₄⋊C₁₆	—	2	2	no	—
15	C₃₁×#357	—	2	8	no	—
16	C₃₁×#340	—	2	1	no	—
17	C₂²×D₄×C₃₁	—	2	2	no	—
18	C₂²×D₄×C₃₁	—	4	1	no	—
19	C₂²×D₄×C₃₁	—	4	1	no	—
20	C₂×D₈×C₃₁	—	2	4	no	—
21	C₂×D₈×C₃₁	—	2	4	no	—
22	C₂×D₈×C₃₁	—	2	2	no	—
23	C₂×D₈×C₃₁	—	2	4	no	—
24	C₂×D₈×C₃₁	—	4	4	no	—
25	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₃₁×#127	—	2	4	no	—
29	C₃₁×#127	—	4	2	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄²×C₃₁	—	4	1	no	—
32	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₃₁	—	2	4	no	—
34	C₂×C₁₆×C₃₁	—	2	2	no	—
35	C₃₁×#107	—	2	2	no	—
36	C₃₁×#107	D₄	1	4	yes	complemented
37	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₃₁	D₄	1	2	yes	—
39	C₃₁×#95	—	4	1	no	—
40	C₂⁴×C₃₁	—	4	2	no	—
41	C₂×D₄×C₃₁	—	2	4	no	—
42	C₂×D₄×C₃₁	—	4	4	no	—
43	C₂×D₄×C₃₁	—	4	4	no	—
44	C₂×D₄×C₃₁	—	4	4	no	—
45	C₂×D₄×C₃₁	—	4	2	no	—
46	C₂×D₄×C₃₁	—	4	4	no	—
47	C₂×D₄×C₃₁	—	8	4	no	—
48	C₂×D₄×C₃₁	—	8	2	no	—
49	C₂×D₄×C₃₁	—	8	2	no	—
50	C₂²×C₄×C₃₁	—	4	2	no	—
51	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	D₈×C₃₁	—	2	8	no	—
55	D₈×C₃₁	—	4	8	no	—
56	D₈×C₃₁	—	4	2	no	—
57	D₈×C₃₁	—	4	2	no	—
58	C₂×C₈×C₃₁	—	2	2	no	—
59	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₃₁	#41	1	2	yes	—
64	C₄⋊C₄×C₃₁	—	2	4	no	—
65	C₄⋊C₄×C₃₁	—	4	2	no	—
66	C₄²×C₃₁	—	8	2	no	—
67	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₃₁	SD₁₆	1	2	yes	—
69	C₁₆×C₃₁	—	4	2	no	—
70	#13252	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₃₁	—	4	4	no	—
72	C₂³×C₃₁	—	8	4	no	—
73	C₂³×C₃₁	—	8	2	no	—
74	C₂³×C₃₁	—	8	2	no	—
75	C₂³×C₃₁	—	8	4	no	—
76	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
77	D₄×C₃₁	—	4	4	no	—
78	D₄×C₃₁	—	4	4	no	—
79	D₄×C₃₁	—	8	8	no	—
80	D₄×C₃₁	—	8	8	no	—
81	D₄×C₃₁	—	8	4	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	4	4	no	—
84	C₂×C₄×C₃₁	—	8	2	no	—
85	C₂×C₄×C₃₁	—	8	2	no	—
86	C₂×C₄×C₃₁	#102	1	2	yes	—
87	C₂×C₄×C₃₁	#102	1	2	yes	—
88	C₂×C₄×C₃₁	D₁₆	1	2	yes	—
89	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
90	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	#121	1	2	yes	—
92	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
94	C₈×C₃₁	—	2	2	no	—
95	C₈×C₃₁	C₄×D₄	1	2	yes	—
96	C₈×C₃₁	#124	1	1	yes	characteristic
97	C₈×C₃₁	#127	1	1	yes	characteristic
98	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
99	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
100	C₂×#357	C₂×C₃₁	1	4	yes	complemented
101	C₄×#107	C₂×C₃₁	1	1	yes	characteristic, complemented
102	C₂²×C₃₁	—	8	4	no	—
103	C₂²×C₃₁	—	8	4	no	—
104	C₂²×C₃₁	—	8	4	no	—
105	C₂²×C₃₁	—	16	4	no	—
106	C₂²×C₃₁	—	16	4	no	—
107	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
108	C₂²×C₃₁	#459	1	2	yes	—
109	C₂²×C₃₁	C₂×D₁₆	1	1	yes	characteristic
110	C₂²×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
111	C₂²×C₃₁	#509	1	2	yes	—
112	C₄×C₃₁	—	2	2	no	—
113	C₄×C₃₁	—	8	4	no	—
114	C₄×C₃₁	#357	1	4	yes	complemented
115	C₄×C₃₁	#390	1	1	yes	characteristic
116	C₄×C₃₁	#418	1	1	yes	characteristic
117	C₄×C₃₁	#460	1	2	yes	—
118	C₂²×D₈	—	2	1	no	—
119	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
120	C₂×#127	—	2	1	no	—
121	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
122	#493	—	2	2	no	—
123	#493	C₄×C₃₁	1	4	yes	complemented
124	C₂×#107	C₂²×C₃₁	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
126	C₄×C₄⋊C₄	—	2	1	no	—
127	C₄⋊C₁₆	—	2	2	no	—
128	#357	—	2	8	no	—
129	#340	—	2	1	no	—
130	C₂×C₃₁	—	16	4	no	—
131	C₂×C₃₁	#1982	1	1	yes	characteristic
132	C₂×C₃₁	C₂×#357	1	1	yes	characteristic
133	C₂×C₃₁	#2140	1	1	yes	characteristic
134	C₂×C₃₁	#2206	1	2	yes	—
135	C₂×C₃₁	#2207	1	2	yes	—
136	C₂²×D₄	—	2	2	no	—
137	C₂²×D₄	—	4	1	no	—
138	C₂²×D₄	—	4	1	no	—
139	C₂×D₈	—	2	4	no	—
140	C₂×D₈	—	2	4	no	—
141	C₂×D₈	—	2	2	no	—
142	C₂×D₈	—	2	4	no	—
143	C₂×D₈	—	4	4	no	—
144	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
145	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
147	#127	—	2	4	no	—
148	#127	—	4	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄²	—	4	1	no	—
151	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
152	C₂×C₁₆	—	2	4	no	—
153	C₂×C₁₆	—	2	2	no	—
154	#107	—	2	2	no	—
155	#107	D₄×C₃₁	1	4	yes	complemented
156	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
157	C₄×C₈	D₄×C₃₁	1	2	yes	—
158	#95	—	4	1	no	—
159	C₃₁	#13252	1	1	yes	characteristic, direct product factor, Sylow
160	C₂⁴	—	4	2	no	—
161	C₂×D₄	—	2	4	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂×D₄	—	4	4	no	—
164	C₂×D₄	—	4	4	no	—
165	C₂×D₄	—	4	2	no	—
166	C₂×D₄	—	4	4	no	—
167	C₂×D₄	—	8	4	no	—
168	C₂×D₄	—	8	2	no	—
169	C₂×D₄	—	8	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
172	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
174	D₈	—	2	8	no	—
175	D₈	—	4	8	no	—
176	D₈	—	4	2	no	—
177	D₈	—	4	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
180	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
182	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
183	C₂×C₈	#41×C₃₁	1	2	yes	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄²	—	8	2	no	—
187	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
188	C₄²	SD₁₆×C₃₁	1	2	yes	—
189	C₁₆	—	4	2	no	—
190	C₂³	—	4	4	no	—
191	C₂³	—	8	4	no	—
192	C₂³	—	8	2	no	—
193	C₂³	—	8	2	no	—
194	C₂³	—	8	4	no	—
195	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
196	D₄	—	4	4	no	—
197	D₄	—	4	4	no	—
198	D₄	—	8	8	no	—
199	D₄	—	8	4	no	—
200	D₄	—	8	8	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	8	2	no	—
205	C₂×C₄	C₃₁×#102	1	2	yes	—
206	C₂×C₄	C₃₁×#102	1	2	yes	—
207	C₂×C₄	C₃₁×D₁₆	1	2	yes	—
208	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₃₁×#121	1	2	yes	—
211	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
212	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	C₄×D₄×C₃₁	1	2	yes	—
215	C₈	C₃₁×#124	1	1	yes	characteristic
216	C₈	C₃₁×#127	1	1	yes	characteristic
217	C₂²	—	8	4	no	—
218	C₂²	—	8	4	no	—
219	C₂²	—	8	4	no	—
220	C₂²	—	16	4	no	—
221	C₂²	—	16	4	no	—
222	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
223	C₂²	C₃₁×#459	1	2	yes	—
224	C₂²	C₂×C₃₁×D₁₆	1	1	yes	characteristic
225	C₂²	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
226	C₂²	C₃₁×#509	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	8	4	no	—
229	C₄	C₃₁×#357	1	4	yes	complemented
230	C₄	C₃₁×#390	1	1	yes	characteristic
231	C₄	C₃₁×#418	1	1	yes	characteristic
232	C₄	C₃₁×#460	1	2	yes	—
233	C₂	—	16	4	no	—
234	C₂	C₃₁×#1982	1	1	yes	characteristic
235	C₂	C₂×C₃₁×#357	1	1	yes	characteristic
236	C₂	C₃₁×#2140	1	1	yes	characteristic
237	C₂	C₃₁×#2206	1	2	yes	—
238	C₂	C₃₁×#2207	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#493	C₂
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
2. C₂×C₃₁×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₃₁×#357	C₂	semidirect
4. C₄×C₃₁×#107	C₂	semidirect
10. C₃₁×#493	C₄	semidirect
36. C₃₁×#107	D₄	semidirect
70. #13252	C₃₁	direct (N × H)
99. C₂×C₄⋊C₁₆	C₂×C₃₁	semidirect
100. C₂×#357	C₂×C₃₁	semidirect
101. C₄×#107	C₂×C₃₁	semidirect
114. C₄×C₃₁	#357	semidirect
123. #493	C₄×C₃₁	semidirect
155. #107	D₄×C₃₁	semidirect
159. C₃₁	#13252	direct (N × H)
229. C₄	C₃₁×#357	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#493	2
C₂×C₃₁×C₄⋊C₁₆	2
C₂×C₃₁×#357	2
C₄×C₃₁×#107	2
#13252	31

Subgroups — C31×#13252 #21470011

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13252 #21470011