Subgroups — C₃₁×#17962 #21474721

Order 7936.

734 conjugacy classes of subgroups.

The subgroup lattice has 272 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×#1594	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1580	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1492	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#131	—	2	2	no	—
9	C₃₁×#532	C₂²	1	2	yes	—
10	C₃₁×#500	C₄	1	1	yes	characteristic
11	C₃₁×#493	C₄	1	1	yes	characteristic
12	C₃₁×#486	C₄	1	2	yes	—
13	C₃₁×#466	C₄	1	4	yes	—
14	C₄×D₈×C₃₁	C₂²	1	4	yes	—
15	C₃₁×#436	—	2	2	no	—
16	C₃₁×#436	C₂²	1	4	yes	—
17	C₈×D₄×C₃₁	—	2	2	no	—
18	C₈×D₄×C₃₁	C₂²	1	4	yes	—
19	C₃₁×#423	C₂²	1	2	yes	—
20	C₃₁×#408	—	2	2	no	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#334	C₂²	1	4	yes	—
25	C₃₁×D₄⋊C₈	C₂²	1	8	yes	—
26	C₃₁×#322	C₂²	1	2	yes	—
27	C₃₁×#322	C₂²	1	2	yes	—
28	C₂×#41×C₃₁	—	2	2	no	—
29	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
30	C₃₁×#131	—	4	8	no	—
31	C₂×M₁₆×C₃₁	—	2	2	no	—
32	C₂×M₁₆×C₃₁	—	2	4	no	—
33	C₂²×C₈×C₃₁	—	2	1	no	—
34	C₂²×C₈×C₃₁	—	2	4	no	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₃₁	D₄	1	2	yes	—
37	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₃₁×#127	C₂×C₄	1	2	yes	—
39	C₃₁×#124	—	2	2	no	—
40	C₃₁×#121	—	2	2	no	—
41	C₃₁×#121	C₂×C₄	1	4	yes	—
42	C₄×D₄×C₃₁	—	2	2	no	—
43	C₄×D₄×C₃₁	C₂³	1	4	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₃₁×#107	C₂×C₄	1	4	yes	—
46	C₃₁×C₄⋊C₈	—	2	2	no	—
47	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
48	C₃₁×#102	C₂×C₄	1	8	yes	—
49	C₃₁×C₂²⋊C₈	—	2	4	no	—
50	C₃₁×C₂²⋊C₈	—	2	4	no	—
51	C₄×C₈×C₃₁	—	2	1	no	—
52	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₃₁	D₄	1	2	yes	—
55	C₄×C₈×C₃₁	C₂³	1	2	yes	—
56	C₄×C₈×C₃₁	C₂³	1	2	yes	—
57	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	#41×C₃₁	—	4	4	no	—
60	#41×C₃₁	—	4	4	no	—
61	C₂×Q₈×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	—	2	4	no	—
63	C₂×D₄×C₃₁	—	2	2	no	—
64	C₂×D₄×C₃₁	—	2	2	no	—
65	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₃₁	—	2	4	no	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
70	D₈×C₃₁	—	2	8	no	—
71	M₁₆×C₃₁	—	4	8	no	—
72	M₁₆×C₃₁	—	4	4	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	4	8	no	—
78	C₂×C₈×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
83	C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
85	C₂²⋊C₄×C₃₁	—	2	4	no	—
86	C₂²⋊C₄×C₃₁	—	2	4	no	—
87	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₃₁	#41	1	1	yes	characteristic
90	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
91	#17962	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	2	4	no	—
93	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
94	Q₈×C₃₁	—	4	4	no	—
95	D₄×C₃₁	—	2	8	no	—
96	D₄×C₃₁	—	4	8	no	—
97	D₄×C₃₁	—	4	4	no	—
98	D₄×C₃₁	—	4	4	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
108	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
111	C₈×C₃₁	—	2	2	no	—
112	C₈×C₃₁	—	2	2	no	—
113	C₈×C₃₁	—	2	2	no	—
114	C₈×C₃₁	—	4	2	no	—
115	C₈×C₃₁	—	4	4	no	—
116	C₈×C₃₁	#131	1	4	yes	complemented
117	#3169	C₂×C₃₁	1	1	yes	characteristic
118	#2931	C₂×C₃₁	1	2	yes	complemented
119	#1594	C₂×C₃₁	1	4	yes	complemented
120	#1580	C₂×C₃₁	1	4	yes	complemented
121	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#1492	C₂×C₃₁	1	2	yes	complemented
123	C₂×#322	C₂×C₃₁	1	1	yes	characteristic, complemented
124	C₂²×C₃₁	—	2	1	no	—
125	C₂²×C₃₁	—	2	1	no	—
126	C₂²×C₃₁	—	4	4	no	—
127	C₂²×C₃₁	—	4	4	no	—
128	C₂²×C₃₁	—	4	4	no	—
129	C₂²×C₃₁	#442	1	2	yes	—
130	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
131	C₄×C₃₁	—	2	1	no	—
132	C₄×C₃₁	—	2	1	no	—
133	C₄×C₃₁	—	4	4	no	—
134	C₄×C₃₁	#442	1	2	yes	—
135	C₄×C₃₁	#444	1	2	yes	—
136	C₄×C₃₁	C₂×#131	1	2	yes	—
137	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
138	C₂×#131	—	2	2	no	—
139	#532	C₂²×C₃₁	1	2	yes	—
140	#500	C₄×C₃₁	1	1	yes	characteristic
141	#493	C₄×C₃₁	1	1	yes	characteristic
142	#486	C₄×C₃₁	1	2	yes	—
143	#466	C₄×C₃₁	1	4	yes	—
144	C₄×D₈	C₂²×C₃₁	1	4	yes	—
145	#436	—	2	2	no	—
146	#436	C₂²×C₃₁	1	4	yes	—
147	C₈×D₄	—	2	2	no	—
148	C₈×D₄	C₂²×C₃₁	1	4	yes	—
149	#423	C₂²×C₃₁	1	2	yes	—
150	#408	—	2	2	no	—
151	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
154	#334	C₂²×C₃₁	1	4	yes	—
155	D₄⋊C₈	C₂²×C₃₁	1	8	yes	—
156	#322	C₂²×C₃₁	1	2	yes	—
157	#322	C₂²×C₃₁	1	2	yes	—
158	C₂×C₃₁	—	2	2	no	—
159	C₂×C₃₁	—	8	4	no	—
160	C₂×C₃₁	#2931	1	1	yes	characteristic
161	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
162	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
163	C₂×#41	—	2	2	no	—
164	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
165	#131	—	4	8	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	4	no	—
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
171	C₂²×C₈	D₄×C₃₁	1	2	yes	—
172	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
173	#127	C₂×C₄×C₃₁	1	2	yes	—
174	#124	—	2	2	no	—
175	#121	—	2	2	no	—
176	#121	C₂×C₄×C₃₁	1	4	yes	—
177	C₄×D₄	—	2	2	no	—
178	C₄×D₄	C₂³×C₃₁	1	4	yes	—
179	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
180	#107	C₂×C₄×C₃₁	1	4	yes	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
183	#102	C₂×C₄×C₃₁	1	8	yes	—
184	C₂²⋊C₈	—	2	4	no	—
185	C₂²⋊C₈	—	2	4	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
188	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
189	C₄×C₈	D₄×C₃₁	1	2	yes	—
190	C₄×C₈	C₂³×C₃₁	1	2	yes	—
191	C₄×C₈	C₂³×C₃₁	1	2	yes	—
192	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
194	C₃₁	#17962	1	1	yes	characteristic, direct product factor, Sylow
195	#41	—	4	4	no	—
196	#41	—	4	4	no	—
197	C₂×Q₈	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
206	D₈	—	2	8	no	—
207	M₁₆	—	4	8	no	—
208	M₁₆	—	4	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	4	8	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
224	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
225	C₄²	#41×C₃₁	1	1	yes	characteristic
226	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
227	C₂³	—	2	4	no	—
228	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
229	Q₈	—	4	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
243	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	2	2	no	—
249	C₈	—	4	2	no	—
250	C₈	—	4	4	no	—
251	C₈	C₃₁×#131	1	4	yes	complemented
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₃₁×#442	1	2	yes	—
258	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	4	no	—
262	C₄	C₃₁×#442	1	2	yes	—
263	C₄	C₃₁×#444	1	2	yes	—
264	C₄	C₂×C₃₁×#131	1	2	yes	—
265	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	8	4	no	—
268	C₂	C₃₁×#2931	1	1	yes	characteristic
269	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
270	C₂	C₂×C₃₁×#444	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×#1594	C₂	semidirect
4. C₃₁×#1580	C₂	semidirect
5. C₃₁×#1566	C₂	semidirect
6. C₃₁×#1492	C₂	semidirect
7. C₂×C₃₁×#322	C₂	semidirect
91. #17962	C₃₁	direct (N × H)
116. C₈×C₃₁	#131	semidirect
118. #2931	C₂×C₃₁	semidirect
119. #1594	C₂×C₃₁	semidirect
120. #1580	C₂×C₃₁	semidirect
121. #1566	C₂×C₃₁	semidirect
122. #1492	C₂×C₃₁	semidirect
123. C₂×#322	C₂×C₃₁	semidirect
194. C₃₁	#17962	direct (N × H)
251. C₈	C₃₁×#131	semidirect

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×#1594	2
C₃₁×#1580	2
C₃₁×#1566	2
C₃₁×#1492	2
C₂×C₃₁×#322	2
#17962	31

Subgroups — C31×#17962 #21474721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17962 #21474721