Subgroups — C₃₁×#2896 #16842829

Order 3968.

772 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#419	C₂	1	2	yes	maximal, complemented
5	C₃₁×#418	C₂	1	2	yes	maximal, complemented
6	C₃₁×#417	C₂	1	4	yes	maximal, complemented
7	C₃₁×#416	C₂	1	4	yes	maximal, complemented
8	C₂×#41×C₃₁	—	2	4	no	—
9	C₂×#41×C₃₁	—	2	1	no	—
10	C₂×#41×C₃₁	—	2	1	no	—
11	C₂×#41×C₃₁	C₄	1	8	yes	complemented
12	C₂×#41×C₃₁	C₂²	1	2	yes	—
13	C₂×#41×C₃₁	C₂²	1	2	yes	—
14	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₃₁	—	2	1	no	—
16	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₃₁	—	2	1	no	—
18	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₃₁	—	2	4	no	—
20	C₂×M₁₆×C₃₁	C₂²	1	2	yes	—
21	C₂×M₁₆×C₃₁	C₂²	1	2	yes	complemented
22	C₂²×C₈×C₃₁	C₂²	1	2	yes	complemented
23	C₃₁×#117	—	2	1	no	—
24	C₃₁×#117	—	2	1	no	—
25	C₃₁×#117	C₂²	1	2	yes	—
26	C₃₁×#117	C₂²	1	2	yes	complemented
27	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
28	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
29	C₂×C₄²×C₃₁	—	2	1	no	—
30	C₃₁×#103	C₂²	1	8	yes	complemented
31	C₃₁×#102	C₂²	1	8	yes	complemented
32	C₂⁴×C₃₁	—	2	1	no	—
33	C₂⁴×C₃₁	D₄	1	1	yes	characteristic
34	#41×C₃₁	—	2	8	no	—
35	#41×C₃₁	—	2	8	no	—
36	#41×C₃₁	—	2	4	no	—
37	#41×C₃₁	—	2	4	no	—
38	#41×C₃₁	C₂×C₄	1	8	yes	—
39	#41×C₃₁	C₂×C₄	1	8	yes	complemented
40	C₂×Q₈×C₃₁	—	2	2	no	—
41	C₂×Q₈×C₃₁	—	2	1	no	—
42	C₂×Q₈×C₃₁	C₂×C₄	1	4	yes	complemented
43	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
44	C₂×D₄×C₃₁	—	2	8	no	—
45	C₂×D₄×C₃₁	—	2	2	no	—
46	C₂×D₄×C₃₁	—	2	2	no	—
47	C₂×D₄×C₃₁	—	2	1	no	—
48	C₂×D₄×C₃₁	—	2	2	no	—
49	C₂×D₄×C₃₁	C₂×C₄	1	4	yes	complemented
50	C₂×D₄×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₃₁	—	2	1	no	—
52	C₂²×C₄×C₃₁	—	2	8	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	C₂×C₄	1	4	yes	—
57	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	D₄	1	4	yes	—
61	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
64	M₁₆×C₃₁	—	2	4	no	—
65	M₁₆×C₃₁	—	2	4	no	—
66	C₂×C₈×C₃₁	—	2	4	no	—
67	C₂×C₈×C₃₁	C₂³	1	4	yes	—
68	C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
70	C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₄²×C₃₁	—	2	2	no	—
73	C₄²×C₃₁	—	2	2	no	—
74	C₂³×C₃₁	—	2	4	no	—
75	C₂³×C₃₁	—	2	2	no	—
76	C₂³×C₃₁	—	4	2	no	—
77	C₂³×C₃₁	—	4	4	no	—
78	C₂³×C₃₁	C₂²⋊C₄	1	4	yes	—
79	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₃₁	—	2	4	no	—
83	Q₈×C₃₁	—	2	2	no	—
84	Q₈×C₃₁	C₂²×C₄	1	4	yes	—
85	D₄×C₃₁	—	2	8	no	—
86	D₄×C₃₁	—	2	8	no	—
87	D₄×C₃₁	—	2	4	no	—
88	D₄×C₃₁	—	2	2	no	—
89	D₄×C₃₁	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₃₁	—	2	4	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	8	no	—
93	C₂×C₄×C₃₁	—	2	8	no	—
94	C₂×C₄×C₃₁	—	2	4	no	—
95	C₂×C₄×C₃₁	—	4	4	no	—
96	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
97	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
98	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
99	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
109	C₈×C₃₁	—	2	4	no	—
110	#2896	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₃₁	—	2	2	no	—
112	C₂²×C₃₁	—	2	2	no	—
113	C₂²×C₃₁	—	2	2	no	—
114	C₂²×C₃₁	—	2	4	no	—
115	C₂²×C₃₁	—	4	4	no	—
116	C₂²×C₃₁	—	4	8	no	—
117	C₂²×C₃₁	—	4	4	no	—
118	C₂²×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
119	C₂²×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂²×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
121	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
122	C₄×C₃₁	—	2	4	no	—
123	C₄×C₃₁	—	4	4	no	—
124	C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
125	C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
126	C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
127	C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
128	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
129	C₂²×#41	C₂×C₃₁	1	1	yes	characteristic
130	C₂²×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
131	C₂×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#419	C₂×C₃₁	1	2	yes	complemented
133	#418	C₂×C₃₁	1	2	yes	complemented
134	#417	C₂×C₃₁	1	4	yes	complemented
135	#416	C₂×C₃₁	1	4	yes	complemented
136	C₂×C₃₁	—	2	2	no	—
137	C₂×C₃₁	—	2	2	no	—
138	C₂×C₃₁	—	2	2	no	—
139	C₂×C₃₁	—	4	4	no	—
140	C₂×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₃₁	#575	1	2	yes	—
142	C₂×#41	—	2	4	no	—
143	C₂×#41	—	2	1	no	—
144	C₂×#41	—	2	1	no	—
145	C₂×#41	C₄×C₃₁	1	8	yes	complemented
146	C₂×#41	C₂²×C₃₁	1	2	yes	—
147	C₂×#41	C₂²×C₃₁	1	2	yes	—
148	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic
149	C₂²×D₄	—	2	1	no	—
150	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
151	C₂³×C₄	—	2	1	no	—
152	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×M₁₆	—	2	4	no	—
154	C₂×M₁₆	C₂²×C₃₁	1	2	yes	—
155	C₂×M₁₆	C₂²×C₃₁	1	2	yes	complemented
156	C₂²×C₈	C₂²×C₃₁	1	2	yes	complemented
157	#117	—	2	1	no	—
158	#117	—	2	1	no	—
159	#117	C₂²×C₃₁	1	2	yes	—
160	#117	C₂²×C₃₁	1	2	yes	complemented
161	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
162	C₂×C₂²⋊C₄	—	2	1	no	—
163	C₂×C₄²	—	2	1	no	—
164	#103	C₂²×C₃₁	1	8	yes	complemented
165	#102	C₂²×C₃₁	1	8	yes	complemented
166	C₃₁	#2896	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	—	2	1	no	—
168	C₂⁴	D₄×C₃₁	1	1	yes	characteristic
169	#41	—	2	8	no	—
170	#41	—	2	8	no	—
171	#41	—	2	4	no	—
172	#41	—	2	4	no	—
173	#41	C₂×C₄×C₃₁	1	8	yes	—
174	#41	C₂×C₄×C₃₁	1	8	yes	complemented
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
178	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
179	C₂×D₄	—	2	8	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	C₂×C₄×C₃₁	1	4	yes	complemented
185	C₂×D₄	C₂³×C₃₁	1	2	yes	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂×C₄×C₃₁	1	4	yes	—
192	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
193	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
194	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	D₄×C₃₁	1	4	yes	—
196	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
199	M₁₆	—	2	4	no	—
200	M₁₆	—	2	4	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	C₂³×C₃₁	1	4	yes	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	2	2	no	—
211	C₂³	—	4	2	no	—
212	C₂³	—	4	4	no	—
213	C₂³	C₂²⋊C₄×C₃₁	1	4	yes	—
214	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
215	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
217	Q₈	—	2	4	no	—
218	Q₈	—	2	2	no	—
219	Q₈	C₂²×C₄×C₃₁	1	4	yes	—
220	D₄	—	2	8	no	—
221	D₄	—	2	8	no	—
222	D₄	—	2	4	no	—
223	D₄	—	2	2	no	—
224	D₄	C₂²×C₄×C₃₁	1	4	yes	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	2	8	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
232	C₂×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
233	C₂×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
234	C₂×C₄	C₂²×C₄×C₃₁	1	2	yes	—
235	C₂×C₄	C₂²×C₄×C₃₁	1	2	yes	—
236	C₂×C₄	C₂²×C₄×C₃₁	1	2	yes	—
237	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
241	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
242	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
243	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
244	C₈	—	2	4	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	4	8	no	—
251	C₂²	—	4	4	no	—
252	C₂²	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
253	C₂²	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
254	C₂²	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
255	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
256	C₄	—	2	4	no	—
257	C₄	—	4	4	no	—
258	C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
259	C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
260	C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
261	C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
262	C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	2	2	no	—
265	C₂	—	2	2	no	—
266	C₂	—	4	4	no	—
267	C₂	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
268	C₂	C₃₁×#575	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₄
commutator	C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂²×#41×C₃₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₁₆×C₃₁	C₂	semidirect
3. C₂×C₃₁×#117	C₂	semidirect
4. C₃₁×#419	C₂	semidirect
5. C₃₁×#418	C₂	semidirect
6. C₃₁×#417	C₂	semidirect
7. C₃₁×#416	C₂	semidirect
11. C₂×#41×C₃₁	C₄	semidirect
21. C₂×M₁₆×C₃₁	C₂²	semidirect
22. C₂²×C₈×C₃₁	C₂²	semidirect
26. C₃₁×#117	C₂²	semidirect
27. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
30. C₃₁×#103	C₂²	semidirect
31. C₃₁×#102	C₂²	semidirect
39. #41×C₃₁	C₂×C₄	semidirect
42. C₂×Q₈×C₃₁	C₂×C₄	semidirect
49. C₂×D₄×C₃₁	C₂×C₄	semidirect
110. #2896	C₃₁	direct (N × H)
130. C₂²×M₁₆	C₂×C₃₁	semidirect
131. C₂×#117	C₂×C₃₁	semidirect
132. #419	C₂×C₃₁	semidirect
133. #418	C₂×C₃₁	semidirect
134. #417	C₂×C₃₁	semidirect
135. #416	C₂×C₃₁	semidirect
145. C₂×#41	C₄×C₃₁	semidirect
155. C₂×M₁₆	C₂²×C₃₁	semidirect
156. C₂²×C₈	C₂²×C₃₁	semidirect
160. #117	C₂²×C₃₁	semidirect
161. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
164. #103	C₂²×C₃₁	semidirect
165. #102	C₂²×C₃₁	semidirect
166. C₃₁	#2896	direct (N × H)
174. #41	C₂×C₄×C₃₁	semidirect
177. C₂×Q₈	C₂×C₄×C₃₁	semidirect
184. C₂×D₄	C₂×C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₃₁	2
C₂²×M₁₆×C₃₁	2
C₂×C₃₁×#117	2
C₃₁×#419	2
C₃₁×#418	2
C₃₁×#417	2
C₃₁×#416	2
#2896	31

Subgroups — C31×#2896 #16842829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2896 #16842829