Subgroups — C₃₁×#23378 #21480137

Order 7936.

642 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#126	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1940	C₂	1	3	yes	maximal, complemented
3	C₃₁×#1924	C₂	1	3	yes	maximal
4	C₃₁×#1917	C₂	1	3	yes	maximal, complemented
5	C₃₁×#1851	C₂	1	3	yes	maximal
6	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1839	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#126	C₄	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	—
13	C₂×C₃₁×C₂²⋊C₈	C₂²	1	3	yes	—
14	C₂×C₃₁×#97	C₂²	1	3	yes	—
15	C₃₁×#388	C₄	1	3	yes	—
16	C₃₁×#387	C₄	1	3	yes	—
17	C₃₁×#383	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
19	C₄×C₂²⋊C₄×C₃₁	C₂²	1	3	yes	—
20	C₄³×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#336	C₂²	1	3	yes	—
22	C₃₁×#336	C₂²	1	3	yes	—
23	C₃₁×#336	C₂²	1	3	yes	—
24	C₃₁×#336	C₂²	1	3	yes	—
25	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₃₁	—	4	3	no	—
27	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
28	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
29	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
31	C₃₁×#126	—	2	4	no	—
32	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
33	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
34	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
35	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
39	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
40	C₃₁×C₄⋊C₈	—	4	3	no	—
41	C₃₁×C₄⋊C₈	—	4	3	no	—
42	C₃₁×C₄⋊C₈	—	4	3	no	—
43	C₃₁×C₂²⋊C₈	—	4	3	no	—
44	C₃₁×#97	—	4	3	no	—
45	C₃₁×#95	C₂×C₄	1	3	yes	—
46	C₃₁×#95	C₂×C₄	1	3	yes	—
47	C₃₁×#95	C₂×C₄	1	3	yes	—
48	C₃₁×#95	C₂×C₄	1	3	yes	—
49	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	—	2	3	no	—
51	C₂²×C₄×C₃₁	—	2	3	no	—
52	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
58	M₁₆×C₃₁	—	4	4	no	—
59	C₂×C₈×C₃₁	—	2	3	no	—
60	C₂×C₈×C₃₁	—	2	3	no	—
61	C₂×C₈×C₃₁	—	2	3	no	—
62	C₂×C₈×C₃₁	—	2	3	no	—
63	C₂×C₈×C₃₁	—	2	3	no	—
64	C₂×C₈×C₃₁	—	2	3	no	—
65	C₂×C₈×C₃₁	—	2	6	no	—
66	C₄⋊C₄×C₃₁	—	2	6	no	—
67	C₄⋊C₄×C₃₁	—	4	3	no	—
68	C₂²⋊C₄×C₃₁	—	2	6	no	—
69	C₂²⋊C₄×C₃₁	—	4	3	no	—
70	C₄²×C₃₁	—	2	3	no	—
71	C₄²×C₃₁	—	2	3	no	—
72	C₄²×C₃₁	—	2	3	no	—
73	C₄²×C₃₁	—	2	3	no	—
74	C₄²×C₃₁	—	4	3	no	—
75	C₄²×C₃₁	M₁₆	1	4	yes	complemented
76	#23378	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	—	2	3	no	—
78	C₂³×C₃₁	—	2	1	no	—
79	C₂³×C₃₁	—	2	3	no	—
80	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	—	2	3	no	—
82	C₂×C₄×C₃₁	—	2	3	no	—
83	C₂×C₄×C₃₁	—	2	3	no	—
84	C₂×C₄×C₃₁	—	2	3	no	—
85	C₂×C₄×C₃₁	—	2	3	no	—
86	C₂×C₄×C₃₁	—	2	3	no	—
87	C₂×C₄×C₃₁	—	2	3	no	—
88	C₂×C₄×C₃₁	—	2	3	no	—
89	C₂×C₄×C₃₁	—	2	3	no	—
90	C₂×C₄×C₃₁	—	2	3	no	—
91	C₂×C₄×C₃₁	—	2	3	no	—
92	C₂×C₄×C₃₁	—	2	3	no	—
93	C₂×C₄×C₃₁	—	4	1	no	—
94	C₂×C₄×C₃₁	—	4	3	no	—
95	C₂×C₄×C₃₁	C₂×M₁₆	1	6	yes	—
96	C₂×C₄×C₃₁	H(5,𝔽₂)	1	3	yes	—
97	C₂×C₄×C₃₁	2₋¹⁺⁴	1	3	yes	—
98	C₈×C₃₁	—	4	3	no	—
99	C₈×C₃₁	—	4	1	no	—
100	C₈×C₃₁	—	4	1	no	—
101	C₈×C₃₁	—	4	3	no	—
102	C₄×#126	C₂×C₃₁	1	1	yes	characteristic
103	#1940	C₂×C₃₁	1	3	yes	complemented
104	#1924	C₂×C₃₁	1	3	yes	—
105	#1917	C₂×C₃₁	1	3	yes	complemented
106	#1851	C₂×C₃₁	1	3	yes	—
107	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
108	#1839	C₂×C₃₁	1	1	yes	characteristic, complemented
109	C₂²×C₃₁	—	4	1	no	—
110	C₂²×C₃₁	—	4	3	no	—
111	C₂²×C₃₁	—	4	3	no	—
112	C₂²×C₃₁	#520	1	3	yes	—
113	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
114	C₂²×C₃₁	#568	1	3	yes	—
115	C₄×C₃₁	—	2	6	no	—
116	C₄×C₃₁	—	4	1	no	—
117	C₄×C₃₁	—	4	3	no	—
118	C₄×C₃₁	—	4	3	no	—
119	C₄×C₃₁	—	4	3	no	—
120	C₄×C₃₁	#560	1	1	yes	characteristic
121	C₄×C₃₁	#563	1	3	yes	—
122	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
123	C₂×#126	C₄×C₃₁	1	1	yes	characteristic
124	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
125	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
126	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	—
127	C₂×C₂²⋊C₈	C₂²×C₃₁	1	3	yes	—
128	C₂×#97	C₂²×C₃₁	1	3	yes	—
129	#388	C₄×C₃₁	1	3	yes	—
130	#387	C₄×C₃₁	1	3	yes	—
131	#383	C₄×C₃₁	1	1	yes	characteristic
132	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
133	C₄×C₂²⋊C₄	C₂²×C₃₁	1	3	yes	—
134	C₄³	C₂²×C₃₁	1	1	yes	characteristic
135	#336	C₂²×C₃₁	1	3	yes	—
136	#336	C₂²×C₃₁	1	3	yes	—
137	#336	C₂²×C₃₁	1	3	yes	—
138	#336	C₂²×C₃₁	1	3	yes	—
139	C₂×C₃₁	—	8	1	no	—
140	C₂×C₃₁	#2383	1	1	yes	characteristic
141	C₂×C₃₁	#2989	1	3	yes	—
142	C₂×C₃₁	#2995	1	3	yes	—
143	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
144	C₂×M₁₆	—	4	3	no	—
145	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
146	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
148	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
149	#126	—	2	4	no	—
150	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
151	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
152	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
153	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
154	C₂×C₄²	C₂×C₄×C₃₁	1	3	yes	—
155	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
156	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
157	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
158	C₄⋊C₈	—	4	3	no	—
159	C₄⋊C₈	—	4	3	no	—
160	C₄⋊C₈	—	4	3	no	—
161	C₂²⋊C₈	—	4	3	no	—
162	#97	—	4	3	no	—
163	#95	C₂×C₄×C₃₁	1	3	yes	—
164	#95	C₂×C₄×C₃₁	1	3	yes	—
165	#95	C₂×C₄×C₃₁	1	3	yes	—
166	#95	C₂×C₄×C₃₁	1	3	yes	—
167	C₃₁	#23378	1	1	yes	characteristic, direct product factor, Sylow
168	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
169	C₂²×C₄	—	2	3	no	—
170	C₂²×C₄	—	2	3	no	—
171	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
172	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
173	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
174	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
175	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
176	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
177	M₁₆	—	4	4	no	—
178	C₂×C₈	—	2	3	no	—
179	C₂×C₈	—	2	3	no	—
180	C₂×C₈	—	2	3	no	—
181	C₂×C₈	—	2	3	no	—
182	C₂×C₈	—	2	3	no	—
183	C₂×C₈	—	2	3	no	—
184	C₂×C₈	—	2	6	no	—
185	C₄⋊C₄	—	2	6	no	—
186	C₄⋊C₄	—	4	3	no	—
187	C₂²⋊C₄	—	2	6	no	—
188	C₂²⋊C₄	—	4	3	no	—
189	C₄²	—	2	3	no	—
190	C₄²	—	2	3	no	—
191	C₄²	—	2	3	no	—
192	C₄²	—	2	3	no	—
193	C₄²	—	4	3	no	—
194	C₄²	M₁₆×C₃₁	1	4	yes	complemented
195	C₂³	—	2	3	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	3	no	—
198	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄	—	2	3	no	—
200	C₂×C₄	—	2	3	no	—
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	2	3	no	—
207	C₂×C₄	—	2	3	no	—
208	C₂×C₄	—	2	3	no	—
209	C₂×C₄	—	2	3	no	—
210	C₂×C₄	—	2	3	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	3	no	—
213	C₂×C₄	C₂×M₁₆×C₃₁	1	6	yes	—
214	C₂×C₄	C₃₁×H(5,𝔽₂)	1	3	yes	—
215	C₂×C₄	C₃₁×2₋¹⁺⁴	1	3	yes	—
216	C₈	—	4	3	no	—
217	C₈	—	4	1	no	—
218	C₈	—	4	1	no	—
219	C₈	—	4	3	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	3	no	—
222	C₂²	—	4	3	no	—
223	C₂²	C₃₁×#520	1	3	yes	—
224	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
225	C₂²	C₃₁×#568	1	3	yes	—
226	C₄	—	2	6	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	3	no	—
229	C₄	—	4	3	no	—
230	C₄	—	4	3	no	—
231	C₄	C₃₁×#560	1	1	yes	characteristic
232	C₄	C₃₁×#563	1	3	yes	—
233	C₂	—	8	1	no	—
234	C₂	C₃₁×#2383	1	1	yes	characteristic
235	C₂	C₃₁×#2989	1	3	yes	—
236	C₂	C₃₁×#2995	1	3	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	C₂⁴×C₃₁	C₂²×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
2. C₃₁×#1940	C₂	semidirect
4. C₃₁×#1917	C₂	semidirect
6. C₃₁×#1842	C₂	semidirect
7. C₃₁×#1839	C₂	semidirect
75. C₄²×C₃₁	M₁₆	semidirect
76. #23378	C₃₁	direct (N × H)
103. #1940	C₂×C₃₁	semidirect
105. #1917	C₂×C₃₁	semidirect
107. #1842	C₂×C₃₁	semidirect
108. #1839	C₂×C₃₁	semidirect
167. C₃₁	#23378	direct (N × H)
194. C₄²	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#126	2
C₃₁×#1940	2
C₃₁×#1924	2
C₃₁×#1917	2
C₃₁×#1851	2
C₃₁×#1842	2
C₃₁×#1839	2
#23378	31

Subgroups — C31×#23378 #21480137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23378 #21480137