Subgroups — C₃₁×#23372 #21480131

Order 7936.

584 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1940	C₂	1	4	yes	maximal
3	C₃₁×#1917	C₂	1	4	yes	maximal
4	C₃₁×#1917	C₂	1	4	yes	maximal
5	C₃₁×#1831	C₂	1	2	yes	maximal
6	C₂×C₃₁×#125	C₄	1	1	yes	characteristic
7	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₃₁×#97	C₂²	1	4	yes	—
11	C₃₁×#389	C₄	1	2	yes	—
12	C₃₁×#387	C₄	1	4	yes	—
13	C₃₁×#382	C₄	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
16	C₄³×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#336	C₂²	1	4	yes	—
18	C₃₁×#336	C₂²	1	4	yes	—
19	C₃₁×#336	C₂²	1	4	yes	—
20	C₃₁×#336	C₂²	1	4	yes	—
21	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
22	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
23	C₃₁×#125	—	2	4	no	—
24	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
32	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊C₈	—	4	4	no	—
35	C₃₁×C₄⋊C₈	—	4	2	no	—
36	C₃₁×C₄⋊C₈	—	4	2	no	—
37	C₃₁×#97	—	4	4	no	—
38	C₃₁×#95	C₂×C₄	1	4	yes	—
39	C₃₁×#95	C₂×C₄	1	4	yes	—
40	C₃₁×#95	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
43	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
48	C₂×C₈×C₃₁	—	2	4	no	—
49	C₂×C₈×C₃₁	—	2	4	no	—
50	C₂×C₈×C₃₁	—	2	4	no	—
51	C₂×C₈×C₃₁	—	2	4	no	—
52	C₂×C₈×C₃₁	—	2	4	no	—
53	C₂×C₈×C₃₁	—	2	4	no	—
54	C₄⋊C₄×C₃₁	—	2	8	no	—
55	C₄⋊C₄×C₃₁	—	2	4	no	—
56	C₄⋊C₄×C₃₁	—	4	2	no	—
57	C₄⋊C₄×C₃₁	—	4	4	no	—
58	C₄²×C₃₁	—	2	4	no	—
59	C₄²×C₃₁	—	2	2	no	—
60	C₄²×C₃₁	—	2	1	no	—
61	C₄²×C₃₁	—	2	1	no	—
62	C₄²×C₃₁	—	2	2	no	—
63	C₄²×C₃₁	—	2	4	no	—
64	C₄²×C₃₁	—	4	1	no	—
65	C₄²×C₃₁	—	4	2	no	—
66	C₄²×C₃₁	M₁₆	1	4	yes	—
67	#23372	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	—	2	1	no	—
70	C₂×C₄×C₃₁	—	2	2	no	—
71	C₂×C₄×C₃₁	—	2	1	no	—
72	C₂×C₄×C₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	1	no	—
74	C₂×C₄×C₃₁	—	2	4	no	—
75	C₂×C₄×C₃₁	—	2	4	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	4	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	4	no	—
82	C₂×C₄×C₃₁	—	2	4	no	—
83	C₂×C₄×C₃₁	—	2	4	no	—
84	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
85	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
86	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
87	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
88	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
89	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
90	C₈×C₃₁	—	4	4	no	—
91	C₈×C₃₁	—	4	4	no	—
92	C₄×#125	C₂×C₃₁	1	1	yes	characteristic
93	#1940	C₂×C₃₁	1	4	yes	—
94	#1917	C₂×C₃₁	1	4	yes	—
95	#1917	C₂×C₃₁	1	4	yes	—
96	#1831	C₂×C₃₁	1	2	yes	—
97	C₂²×C₃₁	#519	1	1	yes	characteristic
98	C₂²×C₃₁	#520	1	2	yes	—
99	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
100	C₂²×C₃₁	#568	1	1	yes	characteristic
101	C₂²×C₃₁	#568	1	2	yes	—
102	C₄×C₃₁	—	2	2	no	—
103	C₄×C₃₁	—	2	4	no	—
104	C₄×C₃₁	—	4	1	no	—
105	C₄×C₃₁	—	4	4	no	—
106	C₄×C₃₁	—	4	2	no	—
107	C₄×C₃₁	—	4	4	no	—
108	C₄×C₃₁	#562	1	1	yes	characteristic
109	C₄×C₃₁	#563	1	2	yes	—
110	C₄×C₃₁	#564	1	1	yes	characteristic
111	C₂×#125	C₄×C₃₁	1	1	yes	characteristic
112	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
114	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
115	C₂×#97	C₂²×C₃₁	1	4	yes	—
116	#389	C₄×C₃₁	1	2	yes	—
117	#387	C₄×C₃₁	1	4	yes	—
118	#382	C₄×C₃₁	1	1	yes	characteristic
119	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
120	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
121	C₄³	C₂²×C₃₁	1	1	yes	characteristic
122	#336	C₂²×C₃₁	1	4	yes	—
123	#336	C₂²×C₃₁	1	4	yes	—
124	#336	C₂²×C₃₁	1	4	yes	—
125	#336	C₂²×C₃₁	1	4	yes	—
126	C₂×C₃₁	#2382	1	1	yes	characteristic
127	C₂×C₃₁	#2983	1	1	yes	characteristic
128	C₂×C₃₁	#2987	1	1	yes	characteristic
129	C₂×C₃₁	#2989	1	2	yes	—
130	C₂×C₃₁	#2995	1	2	yes	—
131	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
132	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
133	#125	—	2	4	no	—
134	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
135	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
136	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
137	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
138	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
139	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
140	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
141	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
142	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
143	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
144	C₄⋊C₈	—	4	4	no	—
145	C₄⋊C₈	—	4	2	no	—
146	C₄⋊C₈	—	4	2	no	—
147	#97	—	4	4	no	—
148	#95	C₂×C₄×C₃₁	1	4	yes	—
149	#95	C₂×C₄×C₃₁	1	4	yes	—
150	#95	C₂×C₄×C₃₁	1	4	yes	—
151	C₃₁	#23372	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
153	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
154	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
155	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
156	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
157	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
158	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	2	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₄⋊C₄	—	2	8	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄²	—	2	4	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	—	2	2	no	—
174	C₄²	—	2	4	no	—
175	C₄²	—	4	1	no	—
176	C₄²	—	4	2	no	—
177	C₄²	M₁₆×C₃₁	1	4	yes	—
178	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
195	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
196	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
197	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
198	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
199	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
200	C₈	—	4	4	no	—
201	C₈	—	4	4	no	—
202	C₂²	C₃₁×#519	1	1	yes	characteristic
203	C₂²	C₃₁×#520	1	2	yes	—
204	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
205	C₂²	C₃₁×#568	1	1	yes	characteristic
206	C₂²	C₃₁×#568	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	4	1	no	—
210	C₄	—	4	4	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	4	no	—
213	C₄	C₃₁×#562	1	1	yes	characteristic
214	C₄	C₃₁×#563	1	2	yes	—
215	C₄	C₃₁×#564	1	1	yes	characteristic
216	C₂	C₃₁×#2382	1	1	yes	characteristic
217	C₂	C₃₁×#2983	1	1	yes	characteristic
218	C₂	C₃₁×#2987	1	1	yes	characteristic
219	C₂	C₃₁×#2989	1	2	yes	—
220	C₂	C₃₁×#2995	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle, prime radical	C₂³×C₃₁	C₂³×C₄
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
67. #23372	C₃₁	direct (N × H)
151. C₃₁	#23372	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#125	2
C₃₁×#1940	2
C₃₁×#1917	2
C₃₁×#1831	2
#23372	31

Subgroups — C31×#23372 #21480131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23372 #21480131