Subgroups — C₃₁×#2637 #16842570

Order 3968.

450 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#126	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#398	C₂	1	2	yes	maximal, complemented
4	C₃₁×#398	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#397	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#396	C₂	1	2	yes	maximal
7	C₃₁×#395	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#393	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#388	C₂	1	2	yes	maximal
10	C₃₁×#387	C₂	1	2	yes	maximal, complemented
11	C₃₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#126	—	2	4	no	—
14	C₄×Q₈×C₃₁	—	2	4	no	—
15	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#95	C₂²	1	1	yes	characteristic
31	C₃₁×#95	C₂²	1	2	yes	—
32	C₃₁×#95	C₂²	1	2	yes	—
33	C₃₁×#95	C₂²	1	2	yes	—
34	C₃₁×#95	C₂²	1	1	yes	characteristic
35	C₃₁×#95	C₂²	1	2	yes	—
36	C₃₁×#95	C₂²	1	2	yes	—
37	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₃₁	—	2	2	no	—
39	C₂×Q₈×C₃₁	—	2	4	no	—
40	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
50	C₄⋊C₄×C₃₁	—	2	4	no	—
51	C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₄⋊C₄×C₃₁	—	2	2	no	—
53	C₄⋊C₄×C₃₁	—	2	2	no	—
54	C₄⋊C₄×C₃₁	—	2	2	no	—
55	C₄⋊C₄×C₃₁	—	2	4	no	—
56	C₄⋊C₄×C₃₁	—	4	2	no	—
57	C₄⋊C₄×C₃₁	—	4	2	no	—
58	C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂²⋊C₄×C₃₁	—	2	2	no	—
60	C₂²⋊C₄×C₃₁	—	2	2	no	—
61	C₂²⋊C₄×C₃₁	—	2	4	no	—
62	C₂²⋊C₄×C₃₁	—	4	2	no	—
63	C₄²×C₃₁	—	2	4	no	—
64	C₄²×C₃₁	D₄	1	4	yes	—
65	C₂³×C₃₁	—	2	1	no	—
66	C₂³×C₃₁	—	2	2	no	—
67	C₂³×C₃₁	—	2	1	no	—
68	C₂³×C₃₁	—	2	2	no	—
69	C₂³×C₃₁	—	2	1	no	—
70	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
71	Q₈×C₃₁	—	2	8	no	—
72	C₂×C₄×C₃₁	—	2	2	no	—
73	C₂×C₄×C₃₁	—	2	2	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	1	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	1	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	1	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₃₁	#41	1	4	yes	—
98	#2637	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₃₁	—	4	1	no	—
100	C₂²×C₃₁	—	4	1	no	—
101	C₂²×C₃₁	—	4	1	no	—
102	C₂²×C₃₁	—	4	2	no	—
103	C₂²×C₃₁	—	4	2	no	—
104	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
105	C₂²×C₃₁	C₂×#41	1	2	yes	—
106	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂²×C₃₁	2₋¹⁺⁴	1	2	yes	—
108	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
109	C₄×C₃₁	—	2	2	no	—
110	C₄×C₃₁	—	2	4	no	—
111	C₄×C₃₁	—	2	2	no	—
112	C₄×C₃₁	—	2	2	no	—
113	C₄×C₃₁	—	4	1	no	—
114	C₄×C₃₁	—	4	1	no	—
115	C₄×C₃₁	—	4	2	no	—
116	C₄×C₃₁	—	4	1	no	—
117	C₄×C₃₁	—	4	2	no	—
118	C₄×C₃₁	—	4	2	no	—
119	C₂×#126	C₂×C₃₁	1	1	yes	characteristic
120	C₂×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
121	#398	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#398	C₂×C₃₁	1	2	yes	complemented
123	#397	C₂×C₃₁	1	1	yes	characteristic
124	#396	C₂×C₃₁	1	2	yes	—
125	#395	C₂×C₃₁	1	1	yes	characteristic, complemented
126	#393	C₂×C₃₁	1	1	yes	characteristic
127	#388	C₂×C₃₁	1	2	yes	—
128	#387	C₂×C₃₁	1	2	yes	complemented
129	#382	C₂×C₃₁	1	1	yes	characteristic, complemented
130	C₂×C₃₁	—	8	1	no	—
131	C₂×C₃₁	#525	1	1	yes	characteristic
132	C₂×C₃₁	#536	1	1	yes	characteristic
133	C₂×C₃₁	#537	1	1	yes	characteristic
134	C₂×C₃₁	#541	1	2	yes	—
135	C₂×C₃₁	#542	1	2	yes	—
136	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
137	#126	—	2	4	no	—
138	C₄×Q₈	—	2	4	no	—
139	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
146	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
152	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
153	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
154	#95	C₂²×C₃₁	1	2	yes	—
155	#95	C₂²×C₃₁	1	1	yes	characteristic
156	#95	C₂²×C₃₁	1	1	yes	characteristic
157	#95	C₂²×C₃₁	1	2	yes	—
158	#95	C₂²×C₃₁	1	2	yes	—
159	#95	C₂²×C₃₁	1	2	yes	—
160	#95	C₂²×C₃₁	1	2	yes	—
161	C₃₁	#2637	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
163	C₂×Q₈	—	2	2	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
167	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
168	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
169	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
174	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₄²	—	2	4	no	—
189	C₄²	D₄×C₃₁	1	4	yes	—
190	C₂³	—	2	1	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	2	no	—
194	C₂³	—	2	1	no	—
195	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
196	Q₈	—	2	8	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
220	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
221	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
222	C₂×C₄	#41×C₃₁	1	4	yes	—
223	C₂²	—	4	1	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
229	C₂²	C₂×#41×C₃₁	1	2	yes	—
230	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
231	C₂²	C₃₁×2₋¹⁺⁴	1	2	yes	—
232	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	4	1	no	—
238	C₄	—	4	1	no	—
239	C₄	—	4	2	no	—
240	C₄	—	4	1	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	2	no	—
243	C₂	—	8	1	no	—
244	C₂	C₃₁×#525	1	1	yes	characteristic
245	C₂	C₃₁×#536	1	1	yes	characteristic
246	C₂	C₃₁×#537	1	1	yes	characteristic
247	C₂	C₃₁×#541	1	2	yes	—
248	C₂	C₃₁×#542	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂³
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×C₄×Q₈×C₃₁	C₂	semidirect
3. C₃₁×#398	C₂	semidirect
4. C₃₁×#398	C₂	semidirect
7. C₃₁×#395	C₂	semidirect
10. C₃₁×#387	C₂	semidirect
11. C₃₁×#382	C₂	semidirect
98. #2637	C₃₁	direct (N × H)
120. C₂×C₄×Q₈	C₂×C₃₁	semidirect
121. #398	C₂×C₃₁	semidirect
122. #398	C₂×C₃₁	semidirect
125. #395	C₂×C₃₁	semidirect
128. #387	C₂×C₃₁	semidirect
129. #382	C₂×C₃₁	semidirect
161. C₃₁	#2637	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×#126	2
C₂×C₄×Q₈×C₃₁	2
C₃₁×#398	2
C₃₁×#397	2
C₃₁×#396	2
C₃₁×#395	2
C₃₁×#393	2
C₃₁×#388	2
C₃₁×#387	2
C₃₁×#382	2
#2637	31

Subgroups — C31×#2637 #16842570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2637 #16842570