Subgroups — C₂×C₃₁×#1722 #21476374

Order 7936.

914 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#486	C₂	1	2	yes	maximal
5	C₃₁×#1722	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₁₆×C₃₁	—	2	2	no	—
8	C₂²×SD₁₆×C₃₁	—	2	2	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₃₁×#500	—	2	4	no	—
13	C₃₁×#500	C₂²	1	8	yes	complemented
14	C₃₁×#494	—	2	2	no	—
15	C₃₁×#494	C₂²	1	4	yes	complemented
16	C₃₁×#492	—	2	2	no	—
17	C₃₁×#492	C₂²	1	4	yes	—
18	C₃₁×#486	—	2	4	no	—
19	C₃₁×#486	C₂²	1	8	yes	—
20	C₂×C₃₁×#107	—	2	4	no	—
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×Q₈×C₃₁	—	2	2	no	—
27	C₂²×D₄×C₃₁	—	2	2	no	—
28	C₂²×D₄×C₃₁	—	2	1	no	—
29	C₂×Q₁₆×C₃₁	—	2	8	no	—
30	C₂×Q₁₆×C₃₁	—	4	8	no	—
31	C₂×SD₁₆×C₃₁	—	2	8	no	—
32	C₂×SD₁₆×C₃₁	—	4	8	no	—
33	C₂²×C₈×C₃₁	D₄	1	4	yes	—
34	C₂²×C₈×C₃₁	D₄	1	2	yes	—
35	C₃₁×C₄⋊Q₈	—	2	2	no	—
36	C₃₁×C₄⋊Q₈	—	2	2	no	—
37	C₃₁×C₄⋊Q₈	—	2	2	no	—
38	C₃₁×C₄⋊Q₈	—	2	1	no	—
39	C₃₁×C₄⋊Q₈	—	2	2	no	—
40	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
41	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₃₁×#127	—	2	2	no	—
43	C₃₁×#127	—	2	1	no	—
44	C₃₁×#127	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×#107	—	2	16	no	—
50	C₃₁×#102	—	2	16	no	—
51	C₄×C₈×C₃₁	D₄	1	4	yes	—
52	C₄×C₈×C₃₁	D₄	1	2	yes	—
53	C₄×C₈×C₃₁	C₂³	1	4	yes	—
54	C₄×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂⁴×C₃₁	—	4	1	no	—
56	C₂×Q₈×C₃₁	—	2	2	no	—
57	C₂×Q₈×C₃₁	—	2	4	no	—
58	C₂×Q₈×C₃₁	—	4	2	no	—
59	C₂×Q₈×C₃₁	—	4	1	no	—
60	C₂×Q₈×C₃₁	—	4	4	no	—
61	C₂×Q₈×C₃₁	—	4	8	no	—
62	C₂×D₄×C₃₁	—	2	4	no	—
63	C₂×D₄×C₃₁	—	2	2	no	—
64	C₂×D₄×C₃₁	—	4	1	no	—
65	C₂×D₄×C₃₁	—	4	2	no	—
66	C₂×D₄×C₃₁	—	4	4	no	—
67	C₂×D₄×C₃₁	—	4	8	no	—
68	C₂²×C₄×C₃₁	—	4	2	no	—
69	C₂²×C₄×C₃₁	—	4	1	no	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	Q₁₆×C₃₁	—	4	16	no	—
73	SD₁₆×C₃₁	—	4	16	no	—
74	C₂×C₈×C₃₁	—	2	8	no	—
75	C₂×C₈×C₃₁	D₈	1	8	yes	—
76	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
77	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
78	C₄⋊C₄×C₃₁	—	2	4	no	—
79	C₄⋊C₄×C₃₁	—	2	4	no	—
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₄⋊C₄×C₃₁	—	4	2	no	—
82	C₄⋊C₄×C₃₁	—	4	2	no	—
83	C₄⋊C₄×C₃₁	—	4	2	no	—
84	C₄²×C₃₁	C₂×D₄	1	2	yes	—
85	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
87	C₂×#1722	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	4	2	no	—
89	C₂³×C₃₁	—	8	2	no	—
90	C₂³×C₃₁	—	8	4	no	—
91	C₂³×C₃₁	#127	1	1	yes	characteristic
92	Q₈×C₃₁	—	4	4	no	—
93	Q₈×C₃₁	—	4	8	no	—
94	Q₈×C₃₁	—	8	2	no	—
95	Q₈×C₃₁	—	8	4	no	—
96	D₄×C₃₁	—	4	4	no	—
97	D₄×C₃₁	—	4	8	no	—
98	D₄×C₃₁	—	8	2	no	—
99	D₄×C₃₁	—	8	4	no	—
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	8	2	no	—
103	C₂×C₄×C₃₁	—	8	4	no	—
104	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#127	1	2	yes	—
106	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
107	C₂×C₄×C₃₁	#137	1	8	yes	—
108	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
110	C₈×C₃₁	—	2	8	no	—
111	C₈×C₃₁	C₂×D₈	1	8	yes	—
112	C₂×#500	C₂×C₃₁	1	2	yes	complemented
113	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
114	C₂×#492	C₂×C₃₁	1	1	yes	characteristic
115	C₂×#486	C₂×C₃₁	1	2	yes	—
116	#1722	C₂×C₃₁	1	8	yes	direct product factor
117	C₂×#322	C₂×C₃₁	1	1	yes	characteristic, complemented
118	C₂²×C₃₁	—	8	2	no	—
119	C₂²×C₃₁	—	8	4	no	—
120	C₂²×C₃₁	—	16	4	no	—
121	C₂²×C₃₁	#493	1	2	yes	—
122	C₂²×C₃₁	#497	1	4	yes	—
123	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
124	C₄×C₃₁	—	8	2	no	—
125	C₄×C₃₁	—	8	4	no	—
126	C₄×C₃₁	#493	1	2	yes	—
127	C₄×C₃₁	#497	1	4	yes	—
128	C₄×C₃₁	C₂²×D₈	1	2	yes	—
129	C₄×C₃₁	C₂×#137	1	4	yes	—
130	C₂²×Q₁₆	—	2	2	no	—
131	C₂²×SD₁₆	—	2	2	no	—
132	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
133	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
134	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
135	#500	—	2	4	no	—
136	#500	C₂²×C₃₁	1	8	yes	complemented
137	#494	—	2	2	no	—
138	#494	C₂²×C₃₁	1	4	yes	complemented
139	#492	—	2	2	no	—
140	#492	C₂²×C₃₁	1	4	yes	—
141	#486	—	2	4	no	—
142	#486	C₂²×C₃₁	1	8	yes	—
143	C₂×#107	—	2	4	no	—
144	C₂×#102	—	2	4	no	—
145	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
146	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
147	#322	C₂²×C₃₁	1	4	yes	complemented
148	C₂×C₃₁	—	16	2	no	—
149	C₂×C₃₁	#1722	1	4	yes	direct product factor
150	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
151	C₂×C₃₁	C₂×#497	1	2	yes	—
152	C₂²×Q₈	—	2	1	no	—
153	C₂²×Q₈	—	2	2	no	—
154	C₂²×D₄	—	2	1	no	—
155	C₂²×D₄	—	2	2	no	—
156	C₂×Q₁₆	—	2	8	no	—
157	C₂×Q₁₆	—	4	8	no	—
158	C₂×SD₁₆	—	2	8	no	—
159	C₂×SD₁₆	—	4	8	no	—
160	C₂²×C₈	D₄×C₃₁	1	4	yes	—
161	C₂²×C₈	D₄×C₃₁	1	2	yes	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	—	2	1	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
168	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
169	#127	—	2	2	no	—
170	#127	—	2	1	no	—
171	#127	C₂³×C₃₁	1	2	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
176	#107	—	2	16	no	—
177	#102	—	2	16	no	—
178	C₄×C₈	D₄×C₃₁	1	4	yes	—
179	C₄×C₈	D₄×C₃₁	1	2	yes	—
180	C₄×C₈	C₂³×C₃₁	1	4	yes	—
181	C₄×C₈	C₂³×C₃₁	1	2	yes	—
182	C₃₁	C₂×#1722	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	—	4	1	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	1	no	—
188	C₂×Q₈	—	4	4	no	—
189	C₂×Q₈	—	4	8	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	—	4	8	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
200	Q₁₆	—	4	16	no	—
201	SD₁₆	—	4	16	no	—
202	C₂×C₈	—	2	8	no	—
203	C₂×C₈	D₈×C₃₁	1	8	yes	—
204	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
205	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄²	C₂×D₄×C₃₁	1	2	yes	—
213	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
214	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
215	C₂³	—	4	2	no	—
216	C₂³	—	8	2	no	—
217	C₂³	—	8	4	no	—
218	C₂³	C₃₁×#127	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	Q₈	—	4	8	no	—
221	Q₈	—	8	2	no	—
222	Q₈	—	8	4	no	—
223	D₄	—	4	4	no	—
224	D₄	—	4	8	no	—
225	D₄	—	8	2	no	—
226	D₄	—	8	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
232	C₂×C₄	C₃₁×#127	1	2	yes	—
233	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
234	C₂×C₄	C₃₁×#137	1	8	yes	—
235	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
237	C₈	—	2	8	no	—
238	C₈	C₂×D₈×C₃₁	1	8	yes	—
239	C₂²	—	8	2	no	—
240	C₂²	—	8	4	no	—
241	C₂²	—	16	4	no	—
242	C₂²	C₃₁×#493	1	2	yes	—
243	C₂²	C₃₁×#497	1	4	yes	—
244	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
245	C₄	—	8	2	no	—
246	C₄	—	8	4	no	—
247	C₄	C₃₁×#493	1	2	yes	—
248	C₄	C₃₁×#497	1	4	yes	—
249	C₄	C₂²×D₈×C₃₁	1	2	yes	—
250	C₄	C₂×C₃₁×#137	1	4	yes	—
251	C₂	—	16	2	no	—
252	C₂	C₃₁×#1722	1	4	yes	direct product factor
253	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
254	C₂	C₂×C₃₁×#497	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
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×#500	C₂	semidirect
2. C₂×C₃₁×#494	C₂	semidirect
5. C₃₁×#1722	C₂	direct (N × H)
6. C₂×C₃₁×#322	C₂	semidirect
13. C₃₁×#500	C₂²	semidirect
15. C₃₁×#494	C₂²	semidirect
24. C₃₁×#322	C₂²	semidirect
87. C₂×#1722	C₃₁	direct (N × H)
112. C₂×#500	C₂×C₃₁	semidirect
113. C₂×#494	C₂×C₃₁	semidirect
116. #1722	C₂×C₃₁	direct (N × H)
117. C₂×#322	C₂×C₃₁	semidirect
136. #500	C₂²×C₃₁	semidirect
138. #494	C₂²×C₃₁	semidirect
147. #322	C₂²×C₃₁	semidirect
149. C₂×C₃₁	#1722	direct (N × H)
182. C₃₁	C₂×#1722	direct (N × H)
252. C₂	C₃₁×#1722	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×#500	2
C₂×C₃₁×#494	2
C₂×C₃₁×#492	2
C₂×C₃₁×#486	2
C₃₁×#1722	2
C₂×C₃₁×#322	2
C₂×#1722	31

Subgroups — C2×C31×#1722 #21476374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1722 #21476374