Subgroups — C₃₁×#2500 #16842433

Order 3968.

568 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×#121	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#394	C₂	1	4	yes	maximal, complemented
4	C₃₁×#390	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#390	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#388	C₂	1	4	yes	maximal, complemented
7	C₃₁×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
12	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#121	—	2	4	no	—
14	C₃₁×#117	—	2	4	no	—
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
23	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
24	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
25	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#95	C₂²	1	2	yes	—
28	C₃₁×#95	C₂²	1	4	yes	—
29	C₂⁴×C₃₁	C₂³	1	2	yes	—
30	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×D₄×C₃₁	—	2	4	no	—
32	C₂×D₄×C₃₁	—	2	2	no	—
33	C₂×D₄×C₃₁	—	2	2	no	—
34	C₂×D₄×C₃₁	—	2	2	no	—
35	C₂×D₄×C₃₁	—	4	4	no	—
36	C₂²×C₄×C₃₁	—	2	2	no	—
37	C₂²×C₄×C₃₁	—	2	2	no	—
38	C₂²×C₄×C₃₁	D₄	1	4	yes	—
39	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₄⋊C₄×C₃₁	—	4	4	no	—
50	C₂²⋊C₄×C₃₁	—	2	4	no	—
51	C₂²⋊C₄×C₃₁	—	2	4	no	—
52	C₂²⋊C₄×C₃₁	—	2	4	no	—
53	C₂²⋊C₄×C₃₁	—	2	8	no	—
54	C₂²⋊C₄×C₃₁	—	4	4	no	—
55	C₄²×C₃₁	—	2	4	no	—
56	C₄²×C₃₁	—	2	2	no	—
57	C₄²×C₃₁	—	2	2	no	—
58	C₂³×C₃₁	—	2	2	no	—
59	C₂³×C₃₁	—	2	1	no	—
60	C₂³×C₃₁	—	2	1	no	—
61	C₂³×C₃₁	—	2	2	no	—
62	C₂³×C₃₁	—	2	2	no	—
63	C₂³×C₃₁	—	2	4	no	—
64	C₂³×C₃₁	—	2	2	no	—
65	C₂³×C₃₁	—	2	2	no	—
66	C₂³×C₃₁	—	2	4	no	—
67	C₂³×C₃₁	C₂×D₄	1	2	yes	—
68	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
69	D₄×C₃₁	—	4	8	no	—
70	D₄×C₃₁	—	4	4	no	—
71	C₂×C₄×C₃₁	—	2	2	no	—
72	C₂×C₄×C₃₁	—	2	1	no	—
73	C₂×C₄×C₃₁	—	2	1	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	8	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	4	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	4	no	—
84	C₂×C₄×C₃₁	—	2	4	no	—
85	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₃₁	#41	1	4	yes	—
88	C₂×C₄×C₃₁	#41	1	2	yes	—
89	C₂×C₄×C₃₁	#41	1	2	yes	—
90	#2500	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₃₁	—	2	2	no	—
92	C₂²×C₃₁	—	2	2	no	—
93	C₂²×C₃₁	—	2	2	no	—
94	C₂²×C₃₁	—	4	2	no	—
95	C₂²×C₃₁	—	4	2	no	—
96	C₂²×C₃₁	—	4	2	no	—
97	C₂²×C₃₁	—	4	2	no	—
98	C₂²×C₃₁	—	4	2	no	—
99	C₂²×C₃₁	—	4	4	no	—
100	C₂²×C₃₁	—	4	4	no	—
101	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
102	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
103	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
104	C₂²×C₃₁	C₂×#41	1	2	yes	—
105	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
106	C₄×C₃₁	—	2	2	no	—
107	C₄×C₃₁	—	2	4	no	—
108	C₄×C₃₁	—	4	2	no	—
109	C₄×C₃₁	—	4	2	no	—
110	C₄×C₃₁	—	4	2	no	—
111	C₄×C₃₁	—	4	2	no	—
112	C₄×C₃₁	#123	1	4	yes	—
113	C₂×#121	C₂×C₃₁	1	1	yes	characteristic
114	C₂×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
115	#394	C₂×C₃₁	1	4	yes	complemented
116	#390	C₂×C₃₁	1	1	yes	characteristic, complemented
117	#390	C₂×C₃₁	1	1	yes	characteristic, complemented
118	#388	C₂×C₃₁	1	4	yes	complemented
119	#384	C₂×C₃₁	1	1	yes	characteristic, complemented
120	#382	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
122	C₂×C₃₁	—	4	2	no	—
123	C₂×C₃₁	—	8	2	no	—
124	C₂×C₃₁	C₂×#123	1	1	yes	characteristic
125	C₂×C₃₁	#532	1	1	yes	characteristic
126	C₂×C₃₁	#535	1	1	yes	characteristic
127	C₂×C₃₁	#553	1	2	yes	—
128	C₂×C₃₁	#555	1	2	yes	—
129	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
130	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
131	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
132	#121	—	2	4	no	—
133	#117	—	2	4	no	—
134	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
135	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
136	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
137	C₂×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
139	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
140	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
143	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
144	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
145	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
146	#95	C₂²×C₃₁	1	2	yes	—
147	#95	C₂²×C₃₁	1	4	yes	—
148	C₃₁	#2500	1	1	yes	characteristic, direct product factor, Sylow
149	C₂⁴	C₂³×C₃₁	1	2	yes	—
150	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
151	C₂×D₄	—	2	4	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	4	4	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	D₄×C₃₁	1	4	yes	—
159	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
160	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
161	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
162	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
163	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
164	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
165	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₂²⋊C₄	—	2	4	no	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₂²⋊C₄	—	2	4	no	—
173	C₂²⋊C₄	—	2	8	no	—
174	C₂²⋊C₄	—	4	4	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	1	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	4	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	4	no	—
187	C₂³	C₂×D₄×C₃₁	1	2	yes	—
188	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
189	D₄	—	4	8	no	—
190	D₄	—	4	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	8	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
206	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
207	C₂×C₄	#41×C₃₁	1	4	yes	—
208	C₂×C₄	#41×C₃₁	1	2	yes	—
209	C₂×C₄	#41×C₃₁	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	4	no	—
219	C₂²	—	4	4	no	—
220	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
221	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
222	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
223	C₂²	C₂×#41×C₃₁	1	2	yes	—
224	C₂²	C₃₁×H(5,𝔽₂)	1	2	yes	—
225	C₄	—	2	2	no	—
226	C₄	—	2	4	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	C₃₁×#123	1	4	yes	—
232	C₂	—	4	2	no	—
233	C₂	—	8	2	no	—
234	C₂	C₂×C₃₁×#123	1	1	yes	characteristic
235	C₂	C₃₁×#532	1	1	yes	characteristic
236	C₂	C₃₁×#535	1	1	yes	characteristic
237	C₂	C₃₁×#553	1	2	yes	—
238	C₂	C₃₁×#555	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#117	C₂	semidirect
3. C₃₁×#394	C₂	semidirect
4. C₃₁×#390	C₂	semidirect
5. C₃₁×#390	C₂	semidirect
6. C₃₁×#388	C₂	semidirect
7. C₃₁×#384	C₂	semidirect
8. C₃₁×#382	C₂	semidirect
9. C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
90. #2500	C₃₁	direct (N × H)
114. C₂×#117	C₂×C₃₁	semidirect
115. #394	C₂×C₃₁	semidirect
116. #390	C₂×C₃₁	semidirect
117. #390	C₂×C₃₁	semidirect
118. #388	C₂×C₃₁	semidirect
119. #384	C₂×C₃₁	semidirect
120. #382	C₂×C₃₁	semidirect
121. C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
148. C₃₁	#2500	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×#121	2
C₂×C₃₁×#117	2
C₃₁×#394	2
C₃₁×#390	2
C₃₁×#388	2
C₃₁×#384	2
C₃₁×#382	2
C₄×C₂²⋊C₄×C₃₁	2
#2500	31

Subgroups — C31×#2500 #16842433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2500 #16842433