Subgroups — C₃₁×#22265 #21479024

Order 7936.

962 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#436	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2397	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1955	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1870	C₂	1	8	yes	maximal, complemented
5	C₃₁×#1832	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
8	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
9	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
10	C₃₁×#436	—	2	8	no	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₃₁×C₂²⋊C₈	C₂²	1	8	yes	—
14	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#399	C₄	1	4	yes	—
17	C₃₁×#392	C₄	1	4	yes	—
18	C₃₁×#376	C₂²	1	1	yes	characteristic
19	C₃₁×#336	C₂²	1	4	yes	—
20	C₃₁×#336	C₂²	1	4	yes	—
21	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×D₄×C₃₁	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
24	C₂×M₁₆×C₃₁	—	2	8	no	—
25	C₂×M₁₆×C₃₁	—	4	8	no	—
26	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
27	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
28	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
29	C₄×D₄×C₃₁	—	2	4	no	—
30	C₄×D₄×C₃₁	—	2	4	no	—
31	C₄×D₄×C₃₁	—	4	8	no	—
32	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
37	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×C₄⋊C₈	—	2	4	no	—
39	C₃₁×C₂²⋊C₈	—	2	8	no	—
40	C₃₁×C₂²⋊C₈	—	4	8	no	—
41	C₃₁×#97	—	2	2	no	—
42	C₄×C₈×C₃₁	D₄	1	4	yes	complemented
43	C₃₁×#95	C₂×C₄	1	4	yes	—
44	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
45	C₂×D₄×C₃₁	—	2	4	no	—
46	C₂×D₄×C₃₁	—	4	4	no	—
47	C₂×D₄×C₃₁	—	4	4	no	—
48	C₂×D₄×C₃₁	—	4	8	no	—
49	C₂²×C₄×C₃₁	—	2	4	no	—
50	C₂²×C₄×C₃₁	—	2	4	no	—
51	C₂²×C₄×C₃₁	—	2	8	no	—
52	C₂²×C₄×C₃₁	—	2	8	no	—
53	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
57	M₁₆×C₃₁	—	4	16	no	—
58	C₂×C₈×C₃₁	—	2	4	no	—
59	C₂×C₈×C₃₁	—	2	4	no	—
60	C₂×C₈×C₃₁	—	2	2	no	—
61	C₂×C₈×C₃₁	—	2	4	no	—
62	C₂×C₈×C₃₁	—	2	8	no	—
63	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
64	C₄⋊C₄×C₃₁	—	2	4	no	—
65	C₄⋊C₄×C₃₁	—	4	4	no	—
66	C₂²⋊C₄×C₃₁	—	2	8	no	—
67	C₂²⋊C₄×C₃₁	—	4	8	no	—
68	C₄²×C₃₁	—	2	4	no	—
69	C₄²×C₃₁	C₂×D₄	1	2	yes	—
70	C₄²×C₃₁	#41	1	2	yes	—
71	#22265	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₃₁	—	2	4	no	—
73	C₂³×C₃₁	—	2	8	no	—
74	C₂³×C₃₁	—	2	4	no	—
75	C₂³×C₃₁	—	2	4	no	—
76	C₂³×C₃₁	—	2	8	no	—
77	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
78	D₄×C₃₁	—	4	8	no	—
79	D₄×C₃₁	—	4	16	no	—
80	C₂×C₄×C₃₁	—	2	4	no	—
81	C₂×C₄×C₃₁	—	2	4	no	—
82	C₂×C₄×C₃₁	—	2	4	no	—
83	C₂×C₄×C₃₁	—	2	8	no	—
84	C₂×C₄×C₃₁	—	2	8	no	—
85	C₂×C₄×C₃₁	—	4	4	no	—
86	C₂×C₄×C₃₁	—	4	4	no	—
87	C₂×C₄×C₃₁	—	4	8	no	—
88	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
89	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
91	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
92	C₈×C₃₁	—	2	4	no	—
93	C₈×C₃₁	—	4	4	no	—
94	C₈×C₃₁	—	4	2	no	—
95	C₂×#436	C₂×C₃₁	1	2	yes	complemented
96	#2397	C₂×C₃₁	1	1	yes	characteristic
97	#1955	C₂×C₃₁	1	2	yes	complemented
98	#1870	C₂×C₃₁	1	8	yes	complemented
99	#1832	C₂×C₃₁	1	1	yes	characteristic, complemented
100	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
101	C₂²×C₃₁	—	4	4	no	—
102	C₂²×C₃₁	—	4	4	no	—
103	C₂²×C₃₁	—	4	4	no	—
104	C₂²×C₃₁	—	4	8	no	—
105	C₂²×C₃₁	—	4	8	no	—
106	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
107	C₂²×C₃₁	#518	1	2	yes	—
108	C₂²×C₃₁	#568	1	4	yes	—
109	C₄×C₃₁	—	2	4	no	—
110	C₄×C₃₁	—	4	4	no	—
111	C₄×C₃₁	—	4	4	no	—
112	C₄×C₃₁	#535	1	2	yes	—
113	C₄×C₃₁	#540	1	2	yes	—
114	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
115	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
116	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
117	#436	—	2	8	no	—
118	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
119	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
120	C₂×C₂²⋊C₈	C₂²×C₃₁	1	8	yes	—
121	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
123	#399	C₄×C₃₁	1	4	yes	—
124	#392	C₄×C₃₁	1	4	yes	—
125	#376	C₂²×C₃₁	1	1	yes	characteristic
126	#336	C₂²×C₃₁	1	4	yes	—
127	#336	C₂²×C₃₁	1	4	yes	—
128	C₂×C₃₁	—	8	4	no	—
129	C₂×C₃₁	#2328	1	1	yes	characteristic
130	C₂×C₃₁	#2930	1	2	yes	—
131	C₂×C₃₁	#2978	1	4	yes	—
132	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
133	C₂²×D₄	C₂×C₄×C₃₁	1	4	yes	—
134	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
135	C₂×M₁₆	—	2	8	no	—
136	C₂×M₁₆	—	4	8	no	—
137	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
138	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
139	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
140	C₄×D₄	—	2	4	no	—
141	C₄×D₄	—	2	4	no	—
142	C₄×D₄	—	4	8	no	—
143	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
144	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
145	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
146	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
147	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
148	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
149	C₄⋊C₈	—	2	4	no	—
150	C₂²⋊C₈	—	2	8	no	—
151	C₂²⋊C₈	—	4	8	no	—
152	#97	—	2	2	no	—
153	C₄×C₈	D₄×C₃₁	1	4	yes	complemented
154	#95	C₂×C₄×C₃₁	1	4	yes	—
155	C₃₁	#22265	1	1	yes	characteristic, direct product factor, Sylow
156	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
157	C₂×D₄	—	2	4	no	—
158	C₂×D₄	—	4	4	no	—
159	C₂×D₄	—	4	4	no	—
160	C₂×D₄	—	4	8	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
166	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
167	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
168	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
169	M₁₆	—	4	16	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₂²⋊C₄	—	2	8	no	—
179	C₂²⋊C₄	—	4	8	no	—
180	C₄²	—	2	4	no	—
181	C₄²	C₂×D₄×C₃₁	1	2	yes	—
182	C₄²	#41×C₃₁	1	2	yes	—
183	C₂³	—	2	4	no	—
184	C₂³	—	2	8	no	—
185	C₂³	—	2	4	no	—
186	C₂³	—	2	4	no	—
187	C₂³	—	2	8	no	—
188	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
189	D₄	—	4	8	no	—
190	D₄	—	4	16	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	8	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	8	no	—
199	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
200	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
201	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
202	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
203	C₈	—	2	4	no	—
204	C₈	—	4	4	no	—
205	C₈	—	4	2	no	—
206	C₂²	—	4	4	no	—
207	C₂²	—	4	4	no	—
208	C₂²	—	4	4	no	—
209	C₂²	—	4	8	no	—
210	C₂²	—	4	8	no	—
211	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
212	C₂²	C₃₁×#518	1	2	yes	—
213	C₂²	C₃₁×#568	1	4	yes	—
214	C₄	—	2	4	no	—
215	C₄	—	4	4	no	—
216	C₄	—	4	4	no	—
217	C₄	C₃₁×#535	1	2	yes	—
218	C₄	C₃₁×#540	1	2	yes	—
219	C₂	—	8	4	no	—
220	C₂	C₃₁×#2328	1	1	yes	characteristic
221	C₂	C₃₁×#2930	1	2	yes	—
222	C₂	C₃₁×#2978	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₃₁×#2397	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#436	C₂	semidirect
3. C₃₁×#1955	C₂	semidirect
4. C₃₁×#1870	C₂	semidirect
5. C₃₁×#1832	C₂	semidirect
6. C₃₁×#1746	C₂	semidirect
42. C₄×C₈×C₃₁	D₄	semidirect
71. #22265	C₃₁	direct (N × H)
95. C₂×#436	C₂×C₃₁	semidirect
97. #1955	C₂×C₃₁	semidirect
98. #1870	C₂×C₃₁	semidirect
99. #1832	C₂×C₃₁	semidirect
100. #1746	C₂×C₃₁	semidirect
153. C₄×C₈	D₄×C₃₁	semidirect
155. C₃₁	#22265	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#436	2
C₃₁×#2397	2
C₃₁×#1955	2
C₃₁×#1870	2
C₃₁×#1832	2
C₃₁×#1746	2
#22265	31

Subgroups — C31×#22265 #21479024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22265 #21479024