Subgroups — C₃₁×#3011 #16842944

Order 3968.

700 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#137	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#536	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#452	C₂	1	4	yes	maximal, complemented
6	C₃₁×#451	C₂	1	4	yes	maximal, complemented
7	C₃₁×#419	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×2₋¹⁺⁴	—	2	4	no	—
11	C₃₁×2₋¹⁺⁴	—	2	4	no	—
12	C₂×#41×C₃₁	—	2	2	no	—
13	C₂×#41×C₃₁	—	2	1	no	—
14	C₂×#41×C₃₁	C₂²	1	2	yes	—
15	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×Q₈×C₃₁	—	2	1	no	—
18	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
19	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#137	—	2	4	no	—
22	C₂×Q₁₆×C₃₁	—	2	4	no	—
23	C₂×Q₁₆×C₃₁	C₂²	1	2	yes	—
24	C₂×Q₁₆×C₃₁	C₂²	1	4	yes	—
25	C₂×SD₁₆×C₃₁	C₂²	1	2	yes	—
26	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×C₄⋊Q₈	—	2	1	no	—
29	C₃₁×#124	—	2	1	no	—
30	C₃₁×#123	—	2	2	no	—
31	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
32	C₃₁×#117	C₂²	1	1	yes	characteristic
33	C₃₁×#103	C₂²	1	2	yes	—
34	C₃₁×#103	C₂²	1	4	yes	—
35	C₃₁×#102	C₂²	1	2	yes	—
36	C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
37	#41×C₃₁	—	2	16	no	—
38	#41×C₃₁	—	2	8	no	—
39	#41×C₃₁	—	2	4	no	—
40	#41×C₃₁	—	2	4	no	—
41	#41×C₃₁	—	2	2	no	—
42	#41×C₃₁	—	2	2	no	—
43	#41×C₃₁	—	2	2	no	—
44	#41×C₃₁	D₄	1	4	yes	—
45	C₂×Q₈×C₃₁	—	2	8	no	—
46	C₂×Q₈×C₃₁	—	2	4	no	—
47	C₂×Q₈×C₃₁	—	2	4	no	—
48	C₂×Q₈×C₃₁	—	2	2	no	—
49	C₂×Q₈×C₃₁	—	2	1	no	—
50	C₂×Q₈×C₃₁	—	2	2	no	—
51	C₂×Q₈×C₃₁	—	2	1	no	—
52	C₂×Q₈×C₃₁	—	2	2	no	—
53	C₂×Q₈×C₃₁	—	2	2	no	—
54	C₂×Q₈×C₃₁	D₄	1	4	yes	—
55	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
56	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
57	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
59	C₂×D₄×C₃₁	—	2	2	no	—
60	C₂×D₄×C₃₁	—	2	1	no	—
61	C₂×D₄×C₃₁	D₄	1	2	yes	—
62	C₂×D₄×C₃₁	D₄	1	1	yes	characteristic
63	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	—	2	1	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	1	no	—
68	C₂²×C₄×C₃₁	—	2	1	no	—
69	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	Q₁₆×C₃₁	—	2	8	no	—
72	Q₁₆×C₃₁	—	2	4	no	—
73	SD₁₆×C₃₁	—	2	4	no	—
74	M₁₆×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	C₂³	1	2	yes	—
77	C₂×C₈×C₃₁	C₂³	1	2	yes	—
78	C₄⋊C₄×C₃₁	—	2	4	no	—
79	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₃₁	—	2	1	no	—
81	C₂²⋊C₄×C₃₁	—	2	4	no	—
82	C₄²×C₃₁	—	2	1	no	—
83	C₂³×C₃₁	—	2	1	no	—
84	C₂³×C₃₁	C₂×D₄	1	2	yes	—
85	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₃₁	—	2	8	no	—
87	Q₈×C₃₁	—	2	4	no	—
88	Q₈×C₃₁	—	2	2	no	—
89	Q₈×C₃₁	—	2	1	no	—
90	Q₈×C₃₁	—	2	2	no	—
91	Q₈×C₃₁	—	2	2	no	—
92	Q₈×C₃₁	—	2	2	no	—
93	Q₈×C₃₁	—	4	2	no	—
94	Q₈×C₃₁	C₂×D₄	1	2	yes	—
95	Q₈×C₃₁	C₂×D₄	1	4	yes	—
96	D₄×C₃₁	—	2	8	no	—
97	D₄×C₃₁	—	2	4	no	—
98	D₄×C₃₁	—	2	4	no	—
99	D₄×C₃₁	—	2	2	no	—
100	D₄×C₃₁	—	2	1	no	—
101	D₄×C₃₁	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₃₁	—	2	8	no	—
103	C₂×C₄×C₃₁	—	2	8	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
114	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
118	C₈×C₃₁	—	2	2	no	—
119	C₈×C₃₁	—	2	2	no	—
120	#3011	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂²×C₃₁	—	2	2	no	—
122	C₂²×C₃₁	—	2	2	no	—
123	C₂²×C₃₁	—	2	2	no	—
124	C₂²×C₃₁	—	2	4	no	—
125	C₂²×C₃₁	—	4	2	no	—
126	C₂²×C₃₁	#120	1	2	yes	—
127	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₃₁	—	2	4	no	—
129	C₄×C₃₁	—	2	2	no	—
130	C₄×C₃₁	—	2	2	no	—
131	C₄×C₃₁	—	4	2	no	—
132	C₄×C₃₁	—	4	2	no	—
133	C₄×C₃₁	#120	1	2	yes	—
134	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
136	C₂×2₋¹⁺⁴	C₂×C₃₁	1	1	yes	characteristic
137	C₂×#137	C₂×C₃₁	1	1	yes	characteristic, complemented
138	C₂²×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
139	#536	C₂×C₃₁	1	1	yes	characteristic, complemented
140	#452	C₂×C₃₁	1	4	yes	complemented
141	#451	C₂×C₃₁	1	4	yes	complemented
142	#419	C₂×C₃₁	1	1	yes	characteristic, complemented
143	#416	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#408	C₂×C₃₁	1	1	yes	characteristic, complemented
145	C₂×C₃₁	—	2	2	no	—
146	C₂×C₃₁	—	4	2	no	—
147	C₂×C₃₁	—	4	2	no	—
148	C₂×C₃₁	C₂×#120	1	1	yes	characteristic
149	C₂×C₃₁	#578	1	2	yes	—
150	2₋¹⁺⁴	—	2	4	no	—
151	2₋¹⁺⁴	—	2	4	no	—
152	C₂×#41	—	2	2	no	—
153	C₂×#41	—	2	1	no	—
154	C₂×#41	C₂²×C₃₁	1	2	yes	—
155	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
157	C₂²×Q₈	—	2	1	no	—
158	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₂²×Q₈	C₂²×C₃₁	1	2	yes	—
160	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
161	#137	—	2	4	no	—
162	C₂×Q₁₆	—	2	4	no	—
163	C₂×Q₁₆	C₂²×C₃₁	1	4	yes	—
164	C₂×Q₁₆	C₂²×C₃₁	1	2	yes	—
165	C₂×SD₁₆	C₂²×C₃₁	1	2	yes	—
166	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
167	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
168	C₄⋊Q₈	—	2	1	no	—
169	#124	—	2	1	no	—
170	#123	—	2	2	no	—
171	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
172	#117	C₂²×C₃₁	1	1	yes	characteristic
173	#103	C₂²×C₃₁	1	2	yes	—
174	#103	C₂²×C₃₁	1	4	yes	—
175	#102	C₂²×C₃₁	1	2	yes	—
176	C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
177	C₃₁	#3011	1	1	yes	characteristic, direct product factor, Sylow
178	#41	—	2	16	no	—
179	#41	—	2	8	no	—
180	#41	—	2	4	no	—
181	#41	—	2	4	no	—
182	#41	—	2	2	no	—
183	#41	—	2	2	no	—
184	#41	—	2	2	no	—
185	#41	D₄×C₃₁	1	4	yes	—
186	C₂×Q₈	—	2	8	no	—
187	C₂×Q₈	—	2	4	no	—
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	2	1	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	D₄×C₃₁	1	4	yes	—
196	C₂×Q₈	D₄×C₃₁	1	1	yes	characteristic
197	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
198	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
199	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	D₄×C₃₁	1	2	yes	—
203	C₂×D₄	D₄×C₃₁	1	1	yes	characteristic
204	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
211	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
212	Q₁₆	—	2	8	no	—
213	Q₁₆	—	2	4	no	—
214	SD₁₆	—	2	4	no	—
215	M₁₆	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	C₂³×C₃₁	1	2	yes	—
218	C₂×C₈	C₂³×C₃₁	1	2	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	4	no	—
223	C₄²	—	2	1	no	—
224	C₂³	—	2	1	no	—
225	C₂³	C₂×D₄×C₃₁	1	2	yes	—
226	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
227	Q₈	—	2	8	no	—
228	Q₈	—	2	4	no	—
229	Q₈	—	2	2	no	—
230	Q₈	—	2	1	no	—
231	Q₈	—	2	2	no	—
232	Q₈	—	2	2	no	—
233	Q₈	—	2	2	no	—
234	Q₈	—	4	2	no	—
235	Q₈	C₂×D₄×C₃₁	1	2	yes	—
236	Q₈	C₂×D₄×C₃₁	1	4	yes	—
237	D₄	—	2	8	no	—
238	D₄	—	2	4	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	2	no	—
241	D₄	—	2	1	no	—
242	D₄	C₂×D₄×C₃₁	1	2	yes	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
255	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
257	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
258	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₃₁×#120	1	2	yes	—
267	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	2	no	—
273	C₄	C₃₁×#120	1	2	yes	—
274	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
275	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
276	C₂	—	2	2	no	—
277	C₂	—	4	2	no	—
278	C₂	—	4	2	no	—
279	C₂	C₂×C₃₁×#120	1	1	yes	characteristic
280	C₂	C₃₁×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×2₋¹⁺⁴	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#137	C₂	semidirect
3. C₂²×Q₁₆×C₃₁	C₂	semidirect
4. C₃₁×#536	C₂	semidirect
5. C₃₁×#452	C₂	semidirect
6. C₃₁×#451	C₂	semidirect
7. C₃₁×#419	C₂	semidirect
8. C₃₁×#416	C₂	semidirect
9. C₃₁×#408	C₂	semidirect
120. #3011	C₃₁	direct (N × H)
137. C₂×#137	C₂×C₃₁	semidirect
138. C₂²×Q₁₆	C₂×C₃₁	semidirect
139. #536	C₂×C₃₁	semidirect
140. #452	C₂×C₃₁	semidirect
141. #451	C₂×C₃₁	semidirect
142. #419	C₂×C₃₁	semidirect
143. #416	C₂×C₃₁	semidirect
144. #408	C₂×C₃₁	semidirect
177. C₃₁	#3011	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×C₃₁×2₋¹⁺⁴	2
C₂×C₃₁×#137	2
C₂²×Q₁₆×C₃₁	2
C₃₁×#536	2
C₃₁×#452	2
C₃₁×#451	2
C₃₁×#419	2
C₃₁×#416	2
C₃₁×#408	2
#3011	31

Subgroups — C31×#3011 #16842944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3011 #16842944