Subgroups — C₃₁×#3040 #16842973

Order 3968.

500 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#464	C₂	1	4	yes	maximal, complemented
6	C₃₁×#463	C₂	1	4	yes	maximal, complemented
7	C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#419	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#418	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#137	—	2	4	no	—
14	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
15	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
16	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
17	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
18	C₂×D₈×C₃₁	C₂²	1	2	yes	—
19	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#124	—	2	1	no	—
22	C₃₁×#124	—	2	1	no	—
23	C₃₁×#124	C₂²	1	4	yes	—
24	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
25	C₄×D₄×C₃₁	—	2	2	no	—
26	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
27	C₃₁×#117	—	2	1	no	—
28	C₃₁×#117	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
30	C₂×C₄²×C₃₁	—	2	1	no	—
31	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₈	C₂²	1	4	yes	complemented
33	C₃₁×#103	C₂²	1	4	yes	complemented
34	C₃₁×#102	C₂²	1	4	yes	—
35	C₂⁴×C₃₁	—	2	1	no	—
36	#41×C₃₁	—	2	2	no	—
37	#41×C₃₁	D₄	1	4	yes	—
38	C₂×Q₈×C₃₁	—	2	1	no	—
39	C₂×Q₈×C₃₁	—	2	2	no	—
40	C₂×Q₈×C₃₁	—	2	2	no	—
41	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
42	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×D₄×C₃₁	—	2	1	no	—
44	C₂×D₄×C₃₁	—	2	2	no	—
45	C₂×D₄×C₃₁	—	2	1	no	—
46	C₂×D₄×C₃₁	—	2	2	no	—
47	C₂×D₄×C₃₁	C₂³	1	2	yes	—
48	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	—	2	1	no	—
50	C₂²×C₄×C₃₁	—	2	1	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	D₄	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	Q₁₆×C₃₁	—	2	4	no	—
55	SD₁₆×C₃₁	—	2	4	no	—
56	SD₁₆×C₃₁	—	2	4	no	—
57	D₈×C₃₁	—	2	4	no	—
58	M₁₆×C₃₁	—	2	2	no	—
59	M₁₆×C₃₁	—	2	2	no	—
60	C₂×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂×C₈×C₃₁	C₂³	1	2	yes	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂²⋊C₄×C₃₁	—	2	1	no	—
65	C₂²⋊C₄×C₃₁	—	2	4	no	—
66	C₂²⋊C₄×C₃₁	—	2	4	no	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	—	2	1	no	—
69	C₄²×C₃₁	—	2	2	no	—
70	C₄²×C₃₁	D₄	1	1	yes	characteristic
71	C₄²×C₃₁	D₄	1	1	yes	characteristic
72	C₄²×C₃₁	C₂³	1	2	yes	—
73	C₂³×C₃₁	—	2	1	no	—
74	C₂³×C₃₁	—	2	2	no	—
75	C₂³×C₃₁	—	4	4	no	—
76	C₂³×C₃₁	—	4	2	no	—
77	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
78	Q₈×C₃₁	—	2	1	no	—
79	Q₈×C₃₁	—	2	2	no	—
80	Q₈×C₃₁	—	2	2	no	—
81	Q₈×C₃₁	—	4	2	no	—
82	Q₈×C₃₁	C₂×D₄	1	2	yes	—
83	D₄×C₃₁	—	2	4	no	—
84	D₄×C₃₁	—	2	2	no	—
85	D₄×C₃₁	—	2	1	no	—
86	D₄×C₃₁	—	2	2	no	—
87	D₄×C₃₁	—	4	2	no	—
88	D₄×C₃₁	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	2	4	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	4	2	no	—
94	C₂×C₄×C₃₁	—	4	4	no	—
95	C₂×C₄×C₃₁	—	4	2	no	—
96	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₃₁	#41	1	2	yes	—
101	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
102	C₈×C₃₁	—	2	2	no	—
103	C₈×C₃₁	—	2	2	no	—
104	#3040	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₃₁	—	2	2	no	—
106	C₂²×C₃₁	—	2	2	no	—
107	C₂²×C₃₁	—	4	2	no	—
108	C₂²×C₃₁	—	4	2	no	—
109	C₂²×C₃₁	—	4	4	no	—
110	C₂²×C₃₁	—	8	4	no	—
111	C₂²×C₃₁	#121	1	2	yes	—
112	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₃₁	—	2	2	no	—
114	C₄×C₃₁	—	2	2	no	—
115	C₄×C₃₁	—	2	2	no	—
116	C₄×C₃₁	—	4	2	no	—
117	C₄×C₃₁	—	4	2	no	—
118	C₄×C₃₁	#121	1	2	yes	—
119	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
120	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
121	C₂×#137	C₂×C₃₁	1	1	yes	characteristic, complemented
122	C₂×AGL(1,ℤ/8)	C₂×C₃₁	1	1	yes	characteristic
123	C₂×#124	C₂×C₃₁	1	1	yes	characteristic, complemented
124	C₄×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
125	#464	C₂×C₃₁	1	4	yes	complemented
126	#463	C₂×C₃₁	1	4	yes	complemented
127	#423	C₂×C₃₁	1	1	yes	characteristic, complemented
128	#419	C₂×C₃₁	1	1	yes	characteristic, complemented
129	#418	C₂×C₃₁	1	1	yes	characteristic, complemented
130	C₂×C₃₁	—	2	2	no	—
131	C₂×C₃₁	—	4	2	no	—
132	C₂×C₃₁	—	8	2	no	—
133	C₂×C₃₁	C₂×#121	1	1	yes	characteristic
134	C₂×C₃₁	#575	1	2	yes	—
135	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
136	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
137	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
138	#137	—	2	4	no	—
139	AGL(1,ℤ/8)	—	2	4	no	—
140	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
141	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
142	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
143	C₂×D₈	C₂²×C₃₁	1	2	yes	—
144	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
145	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
146	#124	—	2	1	no	—
147	#124	—	2	1	no	—
148	#124	C₂²×C₃₁	1	4	yes	—
149	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
150	C₄×D₄	—	2	2	no	—
151	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
152	#117	—	2	1	no	—
153	#117	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×C₂²⋊C₄	—	2	2	no	—
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
157	C₄⋊C₈	C₂²×C₃₁	1	4	yes	complemented
158	#103	C₂²×C₃₁	1	4	yes	complemented
159	#102	C₂²×C₃₁	1	4	yes	—
160	C₃₁	#3040	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	—	2	1	no	—
162	#41	—	2	2	no	—
163	#41	D₄×C₃₁	1	4	yes	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
168	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
169	C₂×D₄	—	2	1	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	—	2	1	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	C₂³×C₃₁	1	2	yes	—
174	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	D₄×C₃₁	1	2	yes	—
179	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
180	Q₁₆	—	2	4	no	—
181	SD₁₆	—	2	4	no	—
182	SD₁₆	—	2	4	no	—
183	D₈	—	2	4	no	—
184	M₁₆	—	2	2	no	—
185	M₁₆	—	2	2	no	—
186	C₂×C₈	C₂³×C₃₁	1	2	yes	—
187	C₂×C₈	C₂³×C₃₁	1	2	yes	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	—	2	4	no	—
192	C₂²⋊C₄	—	2	4	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	2	no	—
196	C₄²	D₄×C₃₁	1	1	yes	characteristic
197	C₄²	D₄×C₃₁	1	1	yes	characteristic
198	C₄²	C₂³×C₃₁	1	2	yes	—
199	C₂³	—	2	1	no	—
200	C₂³	—	2	2	no	—
201	C₂³	—	4	4	no	—
202	C₂³	—	4	2	no	—
203	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
204	Q₈	—	2	1	no	—
205	Q₈	—	2	2	no	—
206	Q₈	—	2	2	no	—
207	Q₈	—	4	2	no	—
208	Q₈	C₂×D₄×C₃₁	1	2	yes	—
209	D₄	—	2	4	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	1	no	—
212	D₄	—	2	2	no	—
213	D₄	—	4	2	no	—
214	D₄	C₂×D₄×C₃₁	1	2	yes	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
223	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
224	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
225	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
226	C₂×C₄	#41×C₃₁	1	2	yes	—
227	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	2	no	—
233	C₂²	—	4	2	no	—
234	C₂²	—	4	4	no	—
235	C₂²	—	8	4	no	—
236	C₂²	C₃₁×#121	1	2	yes	—
237	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	2	no	—
243	C₄	C₃₁×#121	1	2	yes	—
244	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
245	C₄	C₂×#41×C₃₁	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	4	2	no	—
248	C₂	—	8	2	no	—
249	C₂	C₂×C₃₁×#121	1	1	yes	characteristic
250	C₂	C₃₁×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#137	C₂	semidirect
3. C₂×C₃₁×#124	C₂	semidirect
4. C₄×#41×C₃₁	C₂	semidirect
5. C₃₁×#464	C₂	semidirect
6. C₃₁×#463	C₂	semidirect
7. C₃₁×#423	C₂	semidirect
8. C₃₁×#419	C₂	semidirect
9. C₃₁×#418	C₂	semidirect
15. C₂×Q₁₆×C₃₁	C₂²	semidirect
17. C₂×SD₁₆×C₃₁	C₂²	semidirect
24. C₄×Q₈×C₃₁	C₂²	semidirect
26. C₄×D₄×C₃₁	C₂²	semidirect
32. C₃₁×C₄⋊C₈	C₂²	semidirect
33. C₃₁×#103	C₂²	semidirect
104. #3040	C₃₁	direct (N × H)
121. C₂×#137	C₂×C₃₁	semidirect
123. C₂×#124	C₂×C₃₁	semidirect
124. C₄×#41	C₂×C₃₁	semidirect
125. #464	C₂×C₃₁	semidirect
126. #463	C₂×C₃₁	semidirect
127. #423	C₂×C₃₁	semidirect
128. #419	C₂×C₃₁	semidirect
129. #418	C₂×C₃₁	semidirect
140. C₂×Q₁₆	C₂²×C₃₁	semidirect
142. C₂×SD₁₆	C₂²×C₃₁	semidirect
149. C₄×Q₈	C₂²×C₃₁	semidirect
151. C₄×D₄	C₂²×C₃₁	semidirect
157. C₄⋊C₈	C₂²×C₃₁	semidirect
158. #103	C₂²×C₃₁	semidirect
160. C₃₁	#3040	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×#137	2
C₂×C₃₁×AGL(1,ℤ/8)	2
C₂×C₃₁×#124	2
C₄×#41×C₃₁	2
C₃₁×#464	2
C₃₁×#463	2
C₃₁×#423	2
C₃₁×#419	2
C₃₁×#418	2
#3040	31

Subgroups — C31×#3040 #16842973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3040 #16842973