Subgroups — C₃₁×#2976 #16842909

Order 3968.

412 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×#525	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#436	C₂	1	2	yes	maximal, complemented
3	C₃₁×#435	C₂	1	2	yes	maximal, complemented
4	C₃₁×#435	C₂	1	2	yes	maximal, complemented
5	C₈×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
6	C₃₁×#433	C₂	1	2	yes	maximal, complemented
7	C₃₁×#408	C₂	1	2	yes	maximal, complemented
8	C₃₁×#407	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₃₁	C₂²	1	2	yes	complemented
15	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₃₁	C₂²	1	2	yes	complemented
18	C₃₁×#123	C₄	1	2	yes	—
19	C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
20	C₃₁×C₂²⋊Q₈	C₄	1	1	yes	characteristic
21	C₃₁×#121	C₄	1	1	yes	characteristic
22	C₃₁×#121	C₄	1	1	yes	characteristic
23	C₃₁×#120	C₄	1	2	yes	—
24	C₄×D₄×C₃₁	C₂²	1	2	yes	—
25	C₄×D₄×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#117	C₂²	1	1	yes	characteristic
27	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
28	C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
29	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
30	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
31	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
32	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
34	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
35	C₃₁×#97	C₂²	1	2	yes	complemented
36	C₄×C₈×C₃₁	C₂²	1	2	yes	complemented
37	C₂⁴×C₃₁	C₂×C₄	1	1	yes	characteristic
38	#41×C₃₁	—	4	1	no	—
39	#41×C₃₁	—	4	1	no	—
40	C₂×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	—	4	1	no	—
46	C₂²×C₄×C₃₁	—	4	1	no	—
47	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
54	M₁₆×C₃₁	—	4	1	no	—
55	M₁₆×C₃₁	—	4	1	no	—
56	M₁₆×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	—	4	1	no	—
58	C₂×C₈×C₃₁	—	4	1	no	—
59	C₂×C₈×C₃₁	—	4	2	no	—
60	C₂×C₈×C₃₁	C₂³	1	2	yes	—
61	C₂×C₈×C₃₁	C₂³	1	2	yes	—
62	C₂×C₈×C₃₁	C₂³	1	2	yes	—
63	C₂×C₈×C₃₁	C₂³	1	2	yes	—
64	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
67	C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
68	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
69	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
70	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
71	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
72	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
73	C₄²×C₃₁	C₂³	1	2	yes	—
74	C₂³×C₃₁	—	2	2	no	—
75	C₂³×C₃₁	—	4	1	no	—
76	C₂³×C₃₁	—	4	1	no	—
77	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
80	C₂³×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	Q₈×C₃₁	—	4	1	no	—
82	D₄×C₃₁	—	2	4	no	—
83	D₄×C₃₁	—	4	2	no	—
84	D₄×C₃₁	—	4	1	no	—
85	D₄×C₃₁	—	4	2	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	1	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	4	1	no	—
95	C₂×C₄×C₃₁	—	4	1	no	—
96	C₂×C₄×C₃₁	—	4	1	no	—
97	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
99	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
104	C₈×C₃₁	—	2	2	no	—
105	C₈×C₃₁	—	2	2	no	—
106	C₈×C₃₁	—	2	2	no	—
107	C₈×C₃₁	—	2	2	no	—
108	#2976	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂²×C₃₁	—	2	1	no	—
110	C₂²×C₃₁	—	2	1	no	—
111	C₂²×C₃₁	—	2	1	no	—
112	C₂²×C₃₁	—	2	1	no	—
113	C₂²×C₃₁	—	2	2	no	—
114	C₂²×C₃₁	—	2	2	no	—
115	C₂²×C₃₁	—	2	1	no	—
116	C₂²×C₃₁	—	2	1	no	—
117	C₂²×C₃₁	—	2	1	no	—
118	C₂²×C₃₁	—	2	2	no	—
119	C₂²×C₃₁	—	2	1	no	—
120	C₂²×C₃₁	—	4	4	no	—
121	C₂²×C₃₁	#131	1	2	yes	—
122	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₃₁	—	2	1	no	—
124	C₄×C₃₁	—	2	1	no	—
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	2	no	—
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	2	2	no	—
129	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
130	C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
131	#525	C₂×C₃₁	1	1	yes	characteristic
132	#436	C₂×C₃₁	1	2	yes	complemented
133	#435	C₂×C₃₁	1	2	yes	complemented
134	#435	C₂×C₃₁	1	2	yes	complemented
135	C₈×D₄	C₂×C₃₁	1	2	yes	complemented
136	#433	C₂×C₃₁	1	2	yes	complemented
137	#408	C₂×C₃₁	1	2	yes	complemented
138	#407	C₂×C₃₁	1	1	yes	characteristic, complemented
139	C₂×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
140	C₂×C₃₁	—	2	2	no	—
141	C₂×C₃₁	—	4	2	no	—
142	C₂×C₃₁	—	4	1	no	—
143	C₂×C₃₁	—	4	1	no	—
144	C₂×C₃₁	#518	1	1	yes	characteristic
145	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
146	C₂×C₃₁	#568	1	1	yes	characteristic
147	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
148	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
151	C₂×M₁₆	C₂²×C₃₁	1	2	yes	complemented
152	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
153	C₂²×C₈	C₂²×C₃₁	1	1	yes	characteristic
154	C₂²×C₈	C₂²×C₃₁	1	2	yes	complemented
155	#123	C₄×C₃₁	1	2	yes	—
156	C₂²⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
157	C₂²⋊Q₈	C₄×C₃₁	1	1	yes	characteristic
158	#121	C₄×C₃₁	1	1	yes	characteristic
159	#121	C₄×C₃₁	1	1	yes	characteristic
160	#120	C₄×C₃₁	1	2	yes	—
161	C₄×D₄	C₂²×C₃₁	1	2	yes	—
162	C₄×D₄	C₂²×C₃₁	1	2	yes	—
163	#117	C₂²×C₃₁	1	1	yes	characteristic
164	C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
165	C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
166	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
167	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
168	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
169	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
170	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	complemented
171	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	complemented
172	#97	C₂²×C₃₁	1	2	yes	complemented
173	C₄×C₈	C₂²×C₃₁	1	2	yes	complemented
174	C₃₁	#2976	1	1	yes	characteristic, direct product factor, Sylow
175	C₂⁴	C₂×C₄×C₃₁	1	1	yes	characteristic
176	#41	—	4	1	no	—
177	#41	—	4	1	no	—
178	C₂×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
179	C₂×D₄	C₂×C₄×C₃₁	1	2	yes	—
180	C₂×D₄	C₂×C₄×C₃₁	1	2	yes	—
181	C₂×D₄	C₂×C₄×C₃₁	1	2	yes	—
182	C₂×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
192	M₁₆	—	4	1	no	—
193	M₁₆	—	4	1	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	C₂³×C₃₁	1	2	yes	—
199	C₂×C₈	C₂³×C₃₁	1	2	yes	—
200	C₂×C₈	C₂³×C₃₁	1	2	yes	—
201	C₂×C₈	C₂³×C₃₁	1	2	yes	—
202	C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
203	C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
204	C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
205	C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
206	C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
207	C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
208	C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
209	C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
210	C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
211	C₄²	C₂³×C₃₁	1	2	yes	—
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂²×C₄×C₃₁	1	1	yes	characteristic
216	C₂³	C₂²×C₄×C₃₁	1	1	yes	characteristic
217	C₂³	C₂²×C₄×C₃₁	1	2	yes	—
218	C₂³	C₂²×C₄×C₃₁	1	1	yes	characteristic
219	Q₈	—	4	1	no	—
220	D₄	—	2	4	no	—
221	D₄	—	4	2	no	—
222	D₄	—	4	1	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	C₂²×C₄×C₃₁	1	2	yes	—
236	C₂×C₄	C₂²×C₄×C₃₁	1	2	yes	—
237	C₂×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₄×C₃₁	1	2	yes	—
240	C₂×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	4	4	no	—
258	C₂²	C₃₁×#131	1	2	yes	—
259	C₂²	C₂³×C₄×C₃₁	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
267	C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	4	1	no	—
270	C₂	—	4	1	no	—
271	C₂	—	4	2	no	—
272	C₂	C₃₁×#518	1	1	yes	characteristic
273	C₂	C₂×C₃₁×#131	1	1	yes	characteristic
274	C₂	C₃₁×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂³×C₄
prime radical	C₃₁×#525	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#436	C₂	semidirect
3. C₃₁×#435	C₂	semidirect
4. C₃₁×#435	C₂	semidirect
5. C₈×D₄×C₃₁	C₂	semidirect
6. C₃₁×#433	C₂	semidirect
7. C₃₁×#408	C₂	semidirect
8. C₃₁×#407	C₂	semidirect
9. C₂×C₃₁×C₂²⋊C₈	C₂	semidirect
14. C₂×M₁₆×C₃₁	C₂²	semidirect
17. C₂²×C₈×C₃₁	C₂²	semidirect
27. C₃₁×C₄⋊C₈	C₂²	semidirect
28. C₃₁×C₄⋊C₈	C₂²	semidirect
33. C₃₁×C₂²⋊C₈	C₂²	semidirect
34. C₃₁×C₂²⋊C₈	C₂²	semidirect
35. C₃₁×#97	C₂²	semidirect
36. C₄×C₈×C₃₁	C₂²	semidirect
108. #2976	C₃₁	direct (N × H)
132. #436	C₂×C₃₁	semidirect
133. #435	C₂×C₃₁	semidirect
134. #435	C₂×C₃₁	semidirect
135. C₈×D₄	C₂×C₃₁	semidirect
136. #433	C₂×C₃₁	semidirect
137. #408	C₂×C₃₁	semidirect
138. #407	C₂×C₃₁	semidirect
139. C₂×C₂²⋊C₈	C₂×C₃₁	semidirect
151. C₂×M₁₆	C₂²×C₃₁	semidirect
154. C₂²×C₈	C₂²×C₃₁	semidirect
164. C₄⋊C₈	C₂²×C₃₁	semidirect
165. C₄⋊C₈	C₂²×C₃₁	semidirect
170. C₂²⋊C₈	C₂²×C₃₁	semidirect
171. C₂²⋊C₈	C₂²×C₃₁	semidirect
172. #97	C₂²×C₃₁	semidirect
173. C₄×C₈	C₂²×C₃₁	semidirect
174. C₃₁	#2976	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×#525	2
C₃₁×#436	2
C₃₁×#435	2
C₈×D₄×C₃₁	2
C₃₁×#433	2
C₃₁×#408	2
C₃₁×#407	2
C₂×C₃₁×C₂²⋊C₈	2
#2976	31

Subgroups — C31×#2976 #16842909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2976 #16842909