Subgroups — C₃₁×#3180 #16843113

Order 3968.

410 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#496	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#466	C₂	1	2	yes	maximal, complemented
8	C₃₁×#465	C₂	1	2	yes	maximal, complemented
9	C₃₁×#442	C₂	1	2	yes	maximal, complemented
10	C₃₁×#440	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₃₁	—	2	1	no	—
13	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
14	C₂×D₈×C₃₁	C₂²	1	2	yes	—
15	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₃₁×#127	C₂²	1	1	yes	characteristic
18	C₃₁×#126	—	2	1	no	—
19	C₃₁×#125	C₂²	1	1	yes	characteristic
20	C₃₁×#124	—	2	1	no	—
21	C₃₁×#124	C₂²	1	1	yes	characteristic
22	C₃₁×#123	—	2	1	no	—
23	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₃₁×#121	—	2	1	no	—
25	C₃₁×#121	C₂²	1	2	yes	—
26	C₃₁×#121	C₂²	1	1	yes	characteristic
27	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₃₁	—	2	1	no	—
29	C₄×D₄×C₃₁	—	2	1	no	—
30	C₄×D₄×C₃₁	C₂²	1	2	yes	—
31	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#117	—	2	1	no	—
33	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#107	C₂²	1	2	yes	—
35	C₃₁×#106	C₂²	1	2	yes	—
36	C₃₁×#103	C₂²	1	2	yes	—
37	C₃₁×#102	C₂²	1	2	yes	—
38	C₃₁×#102	C₂²	1	2	yes	—
39	C₃₁×#102	C₂²	1	2	yes	—
40	C₃₁×#97	C₂²	1	2	yes	—
41	C₃₁×#97	C₂²	1	1	yes	characteristic
42	C₃₁×#97	C₂²	1	1	yes	characteristic
43	#41×C₃₁	—	4	4	no	—
44	C₂×Q₈×C₃₁	—	2	1	no	—
45	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₃₁	—	2	1	no	—
47	C₂×D₄×C₃₁	—	2	2	no	—
48	C₂×D₄×C₃₁	—	2	1	no	—
49	C₂×D₄×C₃₁	—	2	1	no	—
50	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₄×C₃₁	—	2	1	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
57	SD₁₆×C₃₁	—	2	4	no	—
58	D₈×C₃₁	—	2	4	no	—
59	C₂×C₈×C₃₁	—	2	4	no	—
60	C₂×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂×C₈×C₃₁	C₂³	1	2	yes	—
62	C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
70	C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂²⋊C₄×C₃₁	—	2	1	no	—
77	C₄²×C₃₁	—	2	1	no	—
78	C₄²×C₃₁	D₄	1	1	yes	characteristic
79	C₄²×C₃₁	D₄	1	1	yes	characteristic
80	C₄²×C₃₁	C₂³	1	1	yes	characteristic
81	C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₂³×C₃₁	—	2	1	no	—
83	C₂³×C₃₁	—	2	2	no	—
84	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
85	Q₈×C₃₁	—	2	2	no	—
86	Q₈×C₃₁	—	4	2	no	—
87	D₄×C₃₁	—	2	2	no	—
88	D₄×C₃₁	—	2	2	no	—
89	D₄×C₃₁	—	2	2	no	—
90	D₄×C₃₁	—	4	2	no	—
91	D₄×C₃₁	—	4	2	no	—
92	D₄×C₃₁	—	4	2	no	—
93	D₄×C₃₁	—	4	4	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	1	no	—
98	C₂×C₄×C₃₁	—	2	1	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	4	1	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
109	C₈×C₃₁	—	2	2	no	—
110	C₈×C₃₁	#41	1	4	yes	complemented
111	#3180	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂²×C₃₁	—	2	1	no	—
113	C₂²×C₃₁	—	2	1	no	—
114	C₂²×C₃₁	—	4	1	no	—
115	C₂²×C₃₁	—	4	2	no	—
116	C₂²×C₃₁	—	4	2	no	—
117	C₂²×C₃₁	—	4	1	no	—
118	C₂²×C₃₁	—	4	1	no	—
119	C₂²×C₃₁	—	4	2	no	—
120	C₂²×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
121	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
122	C₄×C₃₁	—	2	1	no	—
123	C₄×C₃₁	—	2	1	no	—
124	C₄×C₃₁	—	2	1	no	—
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	4	1	no	—
128	C₄×C₃₁	—	4	2	no	—
129	C₄×C₃₁	—	4	1	no	—
130	C₄×C₃₁	—	4	1	no	—
131	C₄×C₃₁	C₂×#41	1	2	yes	—
132	#532	C₂×C₃₁	1	1	yes	characteristic, complemented
133	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
134	#501	C₂×C₃₁	1	1	yes	characteristic, complemented
135	#496	C₂×C₃₁	1	1	yes	characteristic, complemented
136	#490	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#489	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#466	C₂×C₃₁	1	2	yes	complemented
139	#465	C₂×C₃₁	1	2	yes	complemented
140	#442	C₂×C₃₁	1	2	yes	complemented
141	#440	C₂×C₃₁	1	2	yes	complemented
142	C₂×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
143	C₂×C₃₁	—	2	2	no	—
144	C₂×C₃₁	—	8	2	no	—
145	C₂×C₃₁	—	8	1	no	—
146	C₂×C₃₁	#532	1	1	yes	characteristic
147	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₃₁	#575	1	1	yes	characteristic
149	C₂×#41	—	2	1	no	—
150	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
151	C₂×D₈	C₂²×C₃₁	1	2	yes	—
152	C₂²×C₈	C₂²×C₃₁	1	2	yes	—
153	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
154	#127	C₂²×C₃₁	1	1	yes	characteristic
155	#126	—	2	1	no	—
156	#125	C₂²×C₃₁	1	1	yes	characteristic
157	#124	—	2	1	no	—
158	#124	C₂²×C₃₁	1	1	yes	characteristic
159	#123	—	2	1	no	—
160	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
161	#121	—	2	1	no	—
162	#121	C₂²×C₃₁	1	1	yes	characteristic
163	#121	C₂²×C₃₁	1	2	yes	—
164	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
168	C₄×D₄	C₂²×C₃₁	1	2	yes	—
169	#117	—	2	1	no	—
170	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
171	#107	C₂²×C₃₁	1	2	yes	—
172	#106	C₂²×C₃₁	1	2	yes	—
173	#103	C₂²×C₃₁	1	2	yes	—
174	#102	C₂²×C₃₁	1	2	yes	—
175	#102	C₂²×C₃₁	1	2	yes	—
176	#102	C₂²×C₃₁	1	2	yes	—
177	#97	C₂²×C₃₁	1	2	yes	—
178	#97	C₂²×C₃₁	1	1	yes	characteristic
179	#97	C₂²×C₃₁	1	1	yes	characteristic
180	C₃₁	#3180	1	1	yes	characteristic, direct product factor, Sylow
181	#41	—	4	4	no	—
182	C₂×Q₈	—	2	1	no	—
183	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
189	C₂×D₄	C₂³×C₃₁	1	2	yes	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
193	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
195	SD₁₆	—	2	4	no	—
196	D₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	C₂³×C₃₁	1	2	yes	—
199	C₂×C₈	C₂³×C₃₁	1	2	yes	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
206	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
207	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₄²	—	2	1	no	—
216	C₄²	D₄×C₃₁	1	1	yes	characteristic
217	C₄²	D₄×C₃₁	1	1	yes	characteristic
218	C₄²	C₂³×C₃₁	1	1	yes	characteristic
219	C₄²	C₂³×C₃₁	1	1	yes	characteristic
220	C₂³	—	2	1	no	—
221	C₂³	—	2	2	no	—
222	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	Q₈	—	4	2	no	—
225	D₄	—	2	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
246	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	#41×C₃₁	1	4	yes	complemented
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	1	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	2	no	—
257	C₂²	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
258	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	1	no	—
267	C₄	—	4	1	no	—
268	C₄	C₂×#41×C₃₁	1	2	yes	—
269	C₂	—	2	2	no	—
270	C₂	—	8	2	no	—
271	C₂	—	8	1	no	—
272	C₂	C₃₁×#532	1	1	yes	characteristic
273	C₂	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₂	C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
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. C₃₁×#532	C₂	semidirect
2. C₃₁×#529	C₂	semidirect
3. C₃₁×#501	C₂	semidirect
4. C₃₁×#496	C₂	semidirect
5. C₃₁×#490	C₂	semidirect
6. C₃₁×#489	C₂	semidirect
7. C₃₁×#466	C₂	semidirect
8. C₃₁×#465	C₂	semidirect
9. C₃₁×#442	C₂	semidirect
10. C₃₁×#440	C₂	semidirect
11. C₂×C₃₁×#97	C₂	semidirect
110. C₈×C₃₁	#41	semidirect
111. #3180	C₃₁	direct (N × H)
132. #532	C₂×C₃₁	semidirect
133. #529	C₂×C₃₁	semidirect
134. #501	C₂×C₃₁	semidirect
135. #496	C₂×C₃₁	semidirect
136. #490	C₂×C₃₁	semidirect
137. #489	C₂×C₃₁	semidirect
138. #466	C₂×C₃₁	semidirect
139. #465	C₂×C₃₁	semidirect
140. #442	C₂×C₃₁	semidirect
141. #440	C₂×C₃₁	semidirect
142. C₂×#97	C₂×C₃₁	semidirect
180. C₃₁	#3180	direct (N × H)
248. C₈	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#532	2
C₃₁×#529	2
C₃₁×#501	2
C₃₁×#496	2
C₃₁×#490	2
C₃₁×#489	2
C₃₁×#466	2
C₃₁×#465	2
C₃₁×#442	2
C₃₁×#440	2
C₂×C₃₁×#97	2
#3180	31

Subgroups — C31×#3180 #16843113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3180 #16843113