Subgroups — C₃₁×#17981 #21474740

Order 7936.

726 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2931	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1586	C₂	1	8	yes	maximal, complemented
4	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1492	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#131	—	2	2	no	—
8	C₃₁×#532	C₂²	1	2	yes	—
9	C₃₁×#500	C₄	1	1	yes	characteristic
10	C₃₁×#493	C₄	1	1	yes	characteristic
11	C₃₁×#486	C₄	1	2	yes	—
12	C₃₁×#466	C₄	1	4	yes	—
13	C₄×D₈×C₃₁	C₂²	1	4	yes	—
14	C₃₁×#436	—	2	2	no	—
15	C₃₁×#436	C₂²	1	4	yes	—
16	C₈×D₄×C₃₁	—	2	2	no	—
17	C₈×D₄×C₃₁	C₂²	1	4	yes	—
18	C₃₁×#423	C₂²	1	2	yes	—
19	C₃₁×#408	—	2	2	no	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#334	C₂²	1	4	yes	—
24	C₃₁×D₄⋊C₈	C₂²	1	8	yes	—
25	C₃₁×#322	C₂²	1	4	yes	—
26	C₂×#41×C₃₁	—	2	2	no	—
27	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
28	C₃₁×#131	—	4	8	no	—
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂×M₁₆×C₃₁	—	2	4	no	—
31	C₂²×C₈×C₃₁	—	2	1	no	—
32	C₂²×C₈×C₃₁	—	2	4	no	—
33	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₃₁	D₄	1	2	yes	—
35	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₃₁×#127	C₂×C₄	1	2	yes	—
37	C₃₁×#124	—	2	2	no	—
38	C₃₁×#121	—	2	2	no	—
39	C₃₁×#121	C₂×C₄	1	4	yes	—
40	C₄×D₄×C₃₁	—	2	2	no	—
41	C₄×D₄×C₃₁	C₂³	1	4	yes	—
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#107	C₂×C₄	1	4	yes	—
44	C₃₁×C₄⋊C₈	—	2	2	no	—
45	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
46	C₃₁×#102	C₂×C₄	1	8	yes	—
47	C₃₁×C₂²⋊C₈	—	2	4	no	—
48	C₃₁×C₂²⋊C₈	—	2	4	no	—
49	C₄×C₈×C₃₁	—	2	1	no	—
50	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
53	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₃₁	C₂³	1	2	yes	—
55	C₄×C₈×C₃₁	C₂³	1	2	yes	—
56	C₄×C₈×C₃₁	C₂³	1	2	yes	—
57	#41×C₃₁	—	4	4	no	—
58	#41×C₃₁	—	4	4	no	—
59	C₂×Q₈×C₃₁	—	2	2	no	—
60	C₂×D₄×C₃₁	—	2	4	no	—
61	C₂×D₄×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	—	2	2	no	—
63	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₃₁	—	2	4	no	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
68	D₈×C₃₁	—	4	4	no	—
69	M₁₆×C₃₁	—	4	8	no	—
70	M₁₆×C₃₁	—	4	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	4	8	no	—
77	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄×C₃₁	—	2	4	no	—
84	C₂²⋊C₄×C₃₁	—	2	4	no	—
85	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₃₁	#41	1	1	yes	characteristic
88	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
89	#17981	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	4	no	—
91	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₃₁	—	4	4	no	—
93	D₄×C₃₁	—	2	8	no	—
94	D₄×C₃₁	—	4	8	no	—
95	D₄×C₃₁	—	4	4	no	—
96	D₄×C₃₁	—	4	4	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	—	4	4	no	—
102	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
106	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
109	C₈×C₃₁	—	2	4	no	—
110	C₈×C₃₁	—	2	2	no	—
111	C₈×C₃₁	—	2	2	no	—
112	C₈×C₃₁	—	2	2	no	—
113	C₈×C₃₁	—	2	2	no	—
114	C₈×C₃₁	—	4	4	no	—
115	#3169	C₂×C₃₁	1	1	yes	characteristic
116	#2931	C₂×C₃₁	1	2	yes	complemented
117	#1586	C₂×C₃₁	1	8	yes	complemented
118	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#1492	C₂×C₃₁	1	2	yes	complemented
120	C₂×#322	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₂²×C₃₁	—	2	1	no	—
122	C₂²×C₃₁	—	2	1	no	—
123	C₂²×C₃₁	—	4	4	no	—
124	C₂²×C₃₁	—	4	4	no	—
125	C₂²×C₃₁	—	4	4	no	—
126	C₂²×C₃₁	#442	1	2	yes	—
127	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	4	4	no	—
131	C₄×C₃₁	#442	1	2	yes	—
132	C₄×C₃₁	#443	1	2	yes	—
133	C₄×C₃₁	#568	1	2	yes	—
134	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
135	C₂×#131	—	2	2	no	—
136	#532	C₂²×C₃₁	1	2	yes	—
137	#500	C₄×C₃₁	1	1	yes	characteristic
138	#493	C₄×C₃₁	1	1	yes	characteristic
139	#486	C₄×C₃₁	1	2	yes	—
140	#466	C₄×C₃₁	1	4	yes	—
141	C₄×D₈	C₂²×C₃₁	1	4	yes	—
142	#436	—	2	2	no	—
143	#436	C₂²×C₃₁	1	4	yes	—
144	C₈×D₄	—	2	2	no	—
145	C₈×D₄	C₂²×C₃₁	1	4	yes	—
146	#423	C₂²×C₃₁	1	2	yes	—
147	#408	—	2	2	no	—
148	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
151	#334	C₂²×C₃₁	1	4	yes	—
152	D₄⋊C₈	C₂²×C₃₁	1	8	yes	—
153	#322	C₂²×C₃₁	1	4	yes	—
154	C₂×C₃₁	—	2	2	no	—
155	C₂×C₃₁	—	8	4	no	—
156	C₂×C₃₁	#2932	1	1	yes	characteristic
157	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
158	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
159	C₂×#41	—	2	2	no	—
160	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
161	#131	—	4	8	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	4	no	—
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	—	2	4	no	—
166	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
167	C₂²×C₈	D₄×C₃₁	1	2	yes	—
168	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
169	#127	C₂×C₄×C₃₁	1	2	yes	—
170	#124	—	2	2	no	—
171	#121	—	2	2	no	—
172	#121	C₂×C₄×C₃₁	1	4	yes	—
173	C₄×D₄	—	2	2	no	—
174	C₄×D₄	C₂³×C₃₁	1	4	yes	—
175	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
176	#107	C₂×C₄×C₃₁	1	4	yes	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
179	#102	C₂×C₄×C₃₁	1	8	yes	—
180	C₂²⋊C₈	—	2	4	no	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
184	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
186	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
187	C₄×C₈	C₂³×C₃₁	1	2	yes	—
188	C₄×C₈	C₂³×C₃₁	1	2	yes	—
189	C₄×C₈	C₂³×C₃₁	1	2	yes	—
190	C₃₁	#17981	1	1	yes	characteristic, direct product factor, Sylow
191	#41	—	4	4	no	—
192	#41	—	4	4	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
202	D₈	—	4	4	no	—
203	M₁₆	—	4	8	no	—
204	M₁₆	—	4	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	8	no	—
211	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
214	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
215	C₄⋊C₄	—	2	2	no	—
216	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	—	2	4	no	—
219	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₄²	#41×C₃₁	1	1	yes	characteristic
222	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
223	C₂³	—	2	4	no	—
224	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
225	Q₈	—	4	4	no	—
226	D₄	—	2	8	no	—
227	D₄	—	4	8	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
236	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
239	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₈	—	2	2	no	—
247	C₈	—	4	4	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	C₃₁×#442	1	2	yes	—
254	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	C₃₁×#442	1	2	yes	—
259	C₄	C₃₁×#443	1	2	yes	—
260	C₄	C₃₁×#568	1	2	yes	—
261	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	8	4	no	—
264	C₂	C₃₁×#2932	1	1	yes	characteristic
265	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
266	C₂	C₂×C₃₁×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2931	C₂	semidirect
3. C₃₁×#1586	C₂	semidirect
4. C₃₁×#1566	C₂	semidirect
5. C₃₁×#1492	C₂	semidirect
6. C₂×C₃₁×#322	C₂	semidirect
89. #17981	C₃₁	direct (N × H)
116. #2931	C₂×C₃₁	semidirect
117. #1586	C₂×C₃₁	semidirect
118. #1566	C₂×C₃₁	semidirect
119. #1492	C₂×C₃₁	semidirect
120. C₂×#322	C₂×C₃₁	semidirect
190. C₃₁	#17981	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3169	2
C₃₁×#2931	2
C₃₁×#1586	2
C₃₁×#1566	2
C₃₁×#1492	2
C₂×C₃₁×#322	2
#17981	31

Subgroups — C31×#17981 #21474740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17981 #21474740