Subgroups — C₃₁×#8243 #21465002

Order 7936.

436 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1543	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1525	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1451	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1299	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#131	C₄	1	2	yes	—
7	C₃₁×#532	C₂²	1	1	yes	characteristic
8	C₃₁×#436	—	2	2	no	—
9	C₈×D₄×C₃₁	—	2	2	no	—
10	C₃₁×#431	C₂²	1	2	yes	—
11	C₃₁×#423	C₂²	1	1	yes	characteristic
12	C₃₁×#408	C₄	1	2	yes	—
13	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#333	C₄	1	2	yes	—
16	C₃₁×#331	C₄	1	2	yes	—
17	C₃₁×#329	C₄	1	4	yes	—
18	C₃₁×#322	—	2	2	no	—
19	C₃₁×#322	—	2	2	no	—
20	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
21	C₃₁×#131	—	4	4	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
29	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
30	C₃₁×#124	C₂×C₄	1	2	yes	—
31	C₃₁×#121	—	2	2	no	—
32	C₄×D₄×C₃₁	—	2	2	no	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊C₈	—	2	2	no	—
35	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
36	C₃₁×C₂²⋊C₈	—	2	2	no	—
37	C₃₁×C₂²⋊C₈	—	2	4	no	—
38	C₃₁×#97	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₃₁	—	2	2	no	—
40	C₄×C₈×C₃₁	—	2	2	no	—
41	C₄×C₈×C₃₁	D₄	1	2	yes	—
42	C₄×C₈×C₃₁	D₄	1	2	yes	—
43	#41×C₃₁	—	4	2	no	—
44	#41×C₃₁	—	4	2	no	—
45	C₂×Q₈×C₃₁	C₄²	1	2	yes	—
46	C₂×D₄×C₃₁	—	2	2	no	—
47	C₂×D₄×C₃₁	—	2	2	no	—
48	C₂×D₄×C₃₁	C₄²	1	2	yes	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	#41	1	2	yes	—
52	M₁₆×C₃₁	—	4	4	no	—
53	M₁₆×C₃₁	—	4	4	no	—
54	M₁₆×C₃₁	—	4	2	no	—
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	—	2	4	no	—
58	C₂×C₈×C₃₁	—	2	4	no	—
59	C₂×C₈×C₃₁	—	4	1	no	—
60	C₂×C₈×C₃₁	—	4	4	no	—
61	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₂²⋊C₄×C₃₁	—	2	4	no	—
66	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₃₁	C₂×D₄	1	2	yes	—
69	#8243	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₃₁	—	2	2	no	—
71	C₂³×C₃₁	#117	1	1	yes	characteristic
72	Q₈×C₃₁	—	4	2	no	—
73	D₄×C₃₁	—	4	2	no	—
74	D₄×C₃₁	—	4	4	no	—
75	D₄×C₃₁	—	4	4	no	—
76	C₂×C₄×C₃₁	—	2	1	no	—
77	C₂×C₄×C₃₁	—	2	1	no	—
78	C₂×C₄×C₃₁	—	2	4	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	4	2	no	—
81	C₂×C₄×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
86	C₈×C₃₁	—	2	1	no	—
87	C₈×C₃₁	—	2	2	no	—
88	C₈×C₃₁	—	4	4	no	—
89	C₈×C₃₁	—	4	2	no	—
90	C₈×C₃₁	—	4	4	no	—
91	C₈×C₃₁	#100	1	2	yes	—
92	#2931	C₂×C₃₁	1	1	yes	characteristic
93	#1543	C₂×C₃₁	1	1	yes	characteristic
94	#1525	C₂×C₃₁	1	1	yes	characteristic
95	#1451	C₂×C₃₁	1	2	yes	complemented
96	#1299	C₂×C₃₁	1	2	yes	complemented
97	C₂²×C₃₁	—	2	1	no	—
98	C₂²×C₃₁	—	2	2	no	—
99	C₂²×C₃₁	—	4	2	no	—
100	C₂²×C₃₁	—	4	4	no	—
101	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
102	C₄×C₃₁	—	2	1	no	—
103	C₄×C₃₁	—	2	2	no	—
104	C₄×C₃₁	—	4	2	no	—
105	C₄×C₃₁	C₂×#100	1	1	yes	characteristic
106	C₄×C₃₁	#413	1	1	yes	characteristic
107	C₄×C₃₁	#444	1	4	yes	—
108	C₂×#131	C₄×C₃₁	1	2	yes	—
109	#532	C₂²×C₃₁	1	1	yes	characteristic
110	#436	—	2	2	no	—
111	C₈×D₄	—	2	2	no	—
112	#431	C₂²×C₃₁	1	2	yes	—
113	#423	C₂²×C₃₁	1	1	yes	characteristic
114	#408	C₄×C₃₁	1	2	yes	—
115	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
117	#333	C₄×C₃₁	1	2	yes	—
118	#331	C₄×C₃₁	1	2	yes	—
119	#329	C₄×C₃₁	1	4	yes	—
120	#322	—	2	2	no	—
121	#322	—	2	2	no	—
122	C₂×C₃₁	—	4	1	no	—
123	C₂×C₃₁	—	8	2	no	—
124	C₂×C₃₁	C₄×#100	1	1	yes	characteristic
125	C₂×C₃₁	#1765	1	2	yes	—
126	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
127	#131	—	4	4	no	—
128	C₂×M₁₆	—	2	2	no	—
129	C₂×M₁₆	—	2	2	no	—
130	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
133	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
134	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
135	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
136	#124	C₂×C₄×C₃₁	1	2	yes	—
137	#121	—	2	2	no	—
138	C₄×D₄	—	2	2	no	—
139	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
142	C₂²⋊C₈	—	2	2	no	—
143	C₂²⋊C₈	—	2	4	no	—
144	#97	C₂×C₄×C₃₁	1	4	yes	—
145	C₄×C₈	—	2	2	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	D₄×C₃₁	1	2	yes	—
148	C₄×C₈	D₄×C₃₁	1	2	yes	—
149	C₃₁	#8243	1	1	yes	characteristic, direct product factor, Sylow
150	#41	—	4	2	no	—
151	#41	—	4	2	no	—
152	C₂×Q₈	C₄²×C₃₁	1	2	yes	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	C₄²×C₃₁	1	2	yes	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
158	C₂²×C₄	#41×C₃₁	1	2	yes	—
159	M₁₆	—	4	4	no	—
160	M₁₆	—	4	4	no	—
161	M₁₆	—	4	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	4	1	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
169	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
170	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
174	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
175	C₄²	C₂×D₄×C₃₁	1	2	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	C₃₁×#117	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	4	no	—
181	D₄	—	4	4	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
189	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
190	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
191	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
192	C₈	—	2	1	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	4	no	—
195	C₈	—	4	2	no	—
196	C₈	—	4	4	no	—
197	C₈	C₃₁×#100	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	4	no	—
202	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	4	2	no	—
206	C₄	C₂×C₃₁×#100	1	1	yes	characteristic
207	C₄	C₃₁×#413	1	1	yes	characteristic
208	C₄	C₃₁×#444	1	4	yes	—
209	C₂	—	4	1	no	—
210	C₂	—	8	2	no	—
211	C₂	C₄×C₃₁×#100	1	1	yes	characteristic
212	C₂	C₃₁×#1765	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×C₂²⋊C₄
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1451	C₂	semidirect
5. C₃₁×#1299	C₂	semidirect
69. #8243	C₃₁	direct (N × H)
95. #1451	C₂×C₃₁	semidirect
96. #1299	C₂×C₃₁	semidirect
149. C₃₁	#8243	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×#2931	2
C₃₁×#1543	2
C₃₁×#1525	2
C₃₁×#1451	2
C₃₁×#1299	2
#8243	31

Subgroups — C31×#8243 #21465002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8243 #21465002