Subgroups — C₂×C₃₁×#1721 #21476373

Order 7936.

1146 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#493	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#486	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1721	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×SD₁₆×C₃₁	—	2	2	no	—
8	C₂²×D₈×C₃₁	—	2	4	no	—
9	C₂²×D₈×C₃₁	—	2	2	no	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	2	yes	—
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₃₁×#493	—	2	4	no	—
14	C₃₁×#493	—	2	2	no	—
15	C₃₁×#493	C₂²	1	8	yes	complemented
16	C₃₁×#493	C₂²	1	4	yes	complemented
17	C₃₁×#492	—	2	2	no	—
18	C₃₁×#492	C₂²	1	4	yes	complemented
19	C₃₁×#486	—	2	4	no	—
20	C₃₁×#486	C₂²	1	8	yes	complemented
21	C₂×C₃₁×#102	—	2	4	no	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#322	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₃₁	—	2	1	no	—
26	C₂²×D₄×C₃₁	—	2	2	no	—
27	C₂²×D₄×C₃₁	—	2	2	no	—
28	C₂²×D₄×C₃₁	—	2	2	no	—
29	C₂²×D₄×C₃₁	—	2	2	no	—
30	C₂²×D₄×C₃₁	—	2	1	no	—
31	C₂×SD₁₆×C₃₁	—	2	8	no	—
32	C₂×SD₁₆×C₃₁	—	4	8	no	—
33	C₂×D₈×C₃₁	—	2	16	no	—
34	C₂×D₈×C₃₁	—	2	8	no	—
35	C₂×D₈×C₃₁	—	4	16	no	—
36	C₂×D₈×C₃₁	—	4	8	no	—
37	C₂²×C₈×C₃₁	D₄	1	4	yes	—
38	C₂²×C₈×C₃₁	D₄	1	2	yes	—
39	C₃₁×C₄⋊Q₈	—	2	2	no	—
40	C₃₁×C₄⋊Q₈	—	2	1	no	—
41	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₃₁×#127	—	2	2	no	—
43	C₃₁×#127	—	2	1	no	—
44	C₃₁×#127	—	2	2	no	—
45	C₃₁×#127	—	2	2	no	—
46	C₃₁×#127	—	2	2	no	—
47	C₃₁×#127	C₂³	1	4	yes	—
48	C₃₁×#127	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#102	—	2	16	no	—
52	C₄×C₈×C₃₁	D₄	1	4	yes	—
53	C₄×C₈×C₃₁	D₄	1	2	yes	—
54	C₄×C₈×C₃₁	C₂³	1	4	yes	—
55	C₄×C₈×C₃₁	C₂³	1	2	yes	—
56	C₂⁴×C₃₁	—	4	2	no	—
57	C₂⁴×C₃₁	—	4	1	no	—
58	C₂×Q₈×C₃₁	—	2	2	no	—
59	C₂×Q₈×C₃₁	—	4	1	no	—
60	C₂×Q₈×C₃₁	—	4	4	no	—
61	C₂×D₄×C₃₁	—	2	4	no	—
62	C₂×D₄×C₃₁	—	2	4	no	—
63	C₂×D₄×C₃₁	—	2	4	no	—
64	C₂×D₄×C₃₁	—	2	4	no	—
65	C₂×D₄×C₃₁	—	2	2	no	—
66	C₂×D₄×C₃₁	—	4	2	no	—
67	C₂×D₄×C₃₁	—	4	1	no	—
68	C₂×D₄×C₃₁	—	4	2	no	—
69	C₂×D₄×C₃₁	—	4	2	no	—
70	C₂×D₄×C₃₁	—	4	2	no	—
71	C₂×D₄×C₃₁	—	4	8	no	—
72	C₂×D₄×C₃₁	—	4	4	no	—
73	C₂×D₄×C₃₁	—	4	8	no	—
74	C₂×D₄×C₃₁	—	4	8	no	—
75	C₂×D₄×C₃₁	—	4	8	no	—
76	C₂²×C₄×C₃₁	—	4	1	no	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	SD₁₆×C₃₁	—	4	16	no	—
80	D₈×C₃₁	—	4	32	no	—
81	D₈×C₃₁	—	4	16	no	—
82	C₂×C₈×C₃₁	—	2	8	no	—
83	C₂×C₈×C₃₁	D₈	1	8	yes	complemented
84	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
85	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
86	C₄⋊C₄×C₃₁	—	2	4	no	—
87	C₄⋊C₄×C₃₁	—	4	2	no	—
88	C₄²×C₃₁	C₂×D₄	1	2	yes	—
89	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
91	C₂×#1721	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	4	4	no	—
93	C₂³×C₃₁	—	4	2	no	—
94	C₂³×C₃₁	—	8	4	no	—
95	C₂³×C₃₁	—	8	2	no	—
96	C₂³×C₃₁	—	8	4	no	—
97	C₂³×C₃₁	—	8	4	no	—
98	C₂³×C₃₁	—	8	4	no	—
99	C₂³×C₃₁	#127	1	1	yes	characteristic
100	Q₈×C₃₁	—	4	4	no	—
101	Q₈×C₃₁	—	8	2	no	—
102	D₄×C₃₁	—	4	8	no	—
103	D₄×C₃₁	—	4	4	no	—
104	D₄×C₃₁	—	4	8	no	—
105	D₄×C₃₁	—	4	8	no	—
106	D₄×C₃₁	—	4	8	no	—
107	D₄×C₃₁	—	8	4	no	—
108	D₄×C₃₁	—	8	2	no	—
109	D₄×C₃₁	—	8	4	no	—
110	D₄×C₃₁	—	8	4	no	—
111	D₄×C₃₁	—	8	4	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	8	2	no	—
114	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#127	1	2	yes	—
116	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
117	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	8	yes	—
118	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
120	C₈×C₃₁	—	2	8	no	—
121	C₈×C₃₁	C₂×D₈	1	8	yes	complemented
122	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
123	C₂×#493	C₂×C₃₁	1	2	yes	complemented
124	C₂×#492	C₂×C₃₁	1	1	yes	characteristic, complemented
125	C₂×#486	C₂×C₃₁	1	2	yes	complemented
126	#1721	C₂×C₃₁	1	8	yes	direct product factor
127	C₂×#322	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂²×C₃₁	—	8	4	no	—
129	C₂²×C₃₁	—	8	2	no	—
130	C₂²×C₃₁	—	8	4	no	—
131	C₂²×C₃₁	—	8	4	no	—
132	C₂²×C₃₁	—	8	4	no	—
133	C₂²×C₃₁	—	16	8	no	—
134	C₂²×C₃₁	—	16	4	no	—
135	C₂²×C₃₁	#493	1	2	yes	—
136	C₂²×C₃₁	#496	1	4	yes	—
137	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
138	C₄×C₃₁	—	8	2	no	—
139	C₄×C₃₁	#493	1	2	yes	—
140	C₄×C₃₁	#496	1	4	yes	—
141	C₄×C₃₁	C₂²×D₈	1	2	yes	—
142	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
143	C₂²×SD₁₆	—	2	2	no	—
144	C₂²×D₈	—	2	4	no	—
145	C₂²×D₈	—	2	2	no	—
146	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
147	C₂×#127	C₂²×C₃₁	1	2	yes	—
148	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
149	#493	—	2	4	no	—
150	#493	—	2	2	no	—
151	#493	C₂²×C₃₁	1	8	yes	complemented
152	#493	C₂²×C₃₁	1	4	yes	complemented
153	#492	—	2	2	no	—
154	#492	C₂²×C₃₁	1	4	yes	complemented
155	#486	—	2	4	no	—
156	#486	C₂²×C₃₁	1	8	yes	complemented
157	C₂×#102	—	2	4	no	—
158	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
160	#322	C₂²×C₃₁	1	4	yes	complemented
161	C₂×C₃₁	—	16	4	no	—
162	C₂×C₃₁	—	16	2	no	—
163	C₂×C₃₁	#1721	1	4	yes	direct product factor
164	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
165	C₂×C₃₁	C₂×#496	1	2	yes	—
166	C₂²×Q₈	—	2	1	no	—
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	—	2	2	no	—
169	C₂²×D₄	—	2	2	no	—
170	C₂²×D₄	—	2	1	no	—
171	C₂²×D₄	—	2	2	no	—
172	C₂×SD₁₆	—	2	8	no	—
173	C₂×SD₁₆	—	4	8	no	—
174	C₂×D₈	—	2	16	no	—
175	C₂×D₈	—	2	8	no	—
176	C₂×D₈	—	4	16	no	—
177	C₂×D₈	—	4	8	no	—
178	C₂²×C₈	D₄×C₃₁	1	4	yes	—
179	C₂²×C₈	D₄×C₃₁	1	2	yes	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	—	2	1	no	—
182	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
183	#127	—	2	2	no	—
184	#127	—	2	1	no	—
185	#127	—	2	2	no	—
186	#127	—	2	2	no	—
187	#127	—	2	2	no	—
188	#127	C₂³×C₃₁	1	4	yes	—
189	#127	C₂³×C₃₁	1	2	yes	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
192	#102	—	2	16	no	—
193	C₄×C₈	D₄×C₃₁	1	4	yes	—
194	C₄×C₈	D₄×C₃₁	1	2	yes	—
195	C₄×C₈	C₂³×C₃₁	1	4	yes	—
196	C₄×C₈	C₂³×C₃₁	1	2	yes	—
197	C₃₁	C₂×#1721	1	1	yes	characteristic, direct product factor, Sylow
198	C₂⁴	—	4	1	no	—
199	C₂⁴	—	4	2	no	—
200	C₂×Q₈	—	2	2	no	—
201	C₂×Q₈	—	4	1	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	4	2	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	8	no	—
214	C₂×D₄	—	4	4	no	—
215	C₂×D₄	—	4	8	no	—
216	C₂×D₄	—	4	8	no	—
217	C₂×D₄	—	4	8	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
221	SD₁₆	—	4	16	no	—
222	D₈	—	4	32	no	—
223	D₈	—	4	16	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	D₈×C₃₁	1	8	yes	complemented
226	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
227	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄²	C₂×D₄×C₃₁	1	2	yes	—
231	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
233	C₂³	—	4	4	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	8	4	no	—
236	C₂³	—	8	2	no	—
237	C₂³	—	8	4	no	—
238	C₂³	—	8	4	no	—
239	C₂³	—	8	4	no	—
240	C₂³	C₃₁×#127	1	1	yes	characteristic
241	Q₈	—	4	4	no	—
242	Q₈	—	8	2	no	—
243	D₄	—	4	8	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	8	no	—
246	D₄	—	4	8	no	—
247	D₄	—	4	8	no	—
248	D₄	—	8	4	no	—
249	D₄	—	8	2	no	—
250	D₄	—	8	4	no	—
251	D₄	—	8	4	no	—
252	D₄	—	8	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
256	C₂×C₄	C₃₁×#127	1	2	yes	—
257	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
258	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	8	yes	—
259	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
261	C₈	—	2	8	no	—
262	C₈	C₂×D₈×C₃₁	1	8	yes	complemented
263	C₂²	—	8	4	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	16	8	no	—
269	C₂²	—	16	4	no	—
270	C₂²	C₃₁×#493	1	2	yes	—
271	C₂²	C₃₁×#496	1	4	yes	—
272	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
273	C₄	—	8	2	no	—
274	C₄	C₃₁×#493	1	2	yes	—
275	C₄	C₃₁×#496	1	4	yes	—
276	C₄	C₂²×D₈×C₃₁	1	2	yes	—
277	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
278	C₂	—	16	4	no	—
279	C₂	—	16	2	no	—
280	C₂	C₃₁×#1721	1	4	yes	direct product factor
281	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
282	C₂	C₂×C₃₁×#496	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#493	C₂	semidirect
2. C₂×C₃₁×#493	C₂	semidirect
3. C₂×C₃₁×#492	C₂	semidirect
4. C₂×C₃₁×#486	C₂	semidirect
5. C₃₁×#1721	C₂	direct (N × H)
6. C₂×C₃₁×#322	C₂	semidirect
15. C₃₁×#493	C₂²	semidirect
16. C₃₁×#493	C₂²	semidirect
18. C₃₁×#492	C₂²	semidirect
20. C₃₁×#486	C₂²	semidirect
24. C₃₁×#322	C₂²	semidirect
83. C₂×C₈×C₃₁	D₈	semidirect
91. C₂×#1721	C₃₁	direct (N × H)
121. C₈×C₃₁	C₂×D₈	semidirect
122. C₂×#493	C₂×C₃₁	semidirect
123. C₂×#493	C₂×C₃₁	semidirect
124. C₂×#492	C₂×C₃₁	semidirect
125. C₂×#486	C₂×C₃₁	semidirect
126. #1721	C₂×C₃₁	direct (N × H)
127. C₂×#322	C₂×C₃₁	semidirect
151. #493	C₂²×C₃₁	semidirect
152. #493	C₂²×C₃₁	semidirect
154. #492	C₂²×C₃₁	semidirect
156. #486	C₂²×C₃₁	semidirect
160. #322	C₂²×C₃₁	semidirect
163. C₂×C₃₁	#1721	direct (N × H)
197. C₃₁	C₂×#1721	direct (N × H)
225. C₂×C₈	D₈×C₃₁	semidirect
262. C₈	C₂×D₈×C₃₁	semidirect
280. C₂	C₃₁×#1721	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×#493	2
C₂×C₃₁×#492	2
C₂×C₃₁×#486	2
C₃₁×#1721	2
C₂×C₃₁×#322	2
C₂×#1721	31

Subgroups — C2×C31×#1721 #21476373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1721 #21476373