Subgroups — C₂×C₃₁×#2249 #21490471

Order 7936.

854 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×SD₃₂	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#495	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2249	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂²×SD₁₆×C₃₁	—	2	2	no	—
9	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₃₁×#124	—	2	2	no	—
11	C₂×C₃₁×Q₃₂	—	2	4	no	—
12	C₂×C₃₁×Q₃₂	C₂²	1	4	yes	complemented
13	C₂×C₃₁×SD₃₂	—	2	8	no	—
14	C₂×C₃₁×SD₃₂	C₂²	1	8	yes	complemented
15	C₂×C₃₁×D₁₆	—	2	4	no	—
16	C₂×C₃₁×D₁₆	C₂²	1	4	yes	—
17	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#495	—	2	4	no	—
20	C₃₁×#495	C₂²	1	8	yes	complemented
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₃₁	C₂²	1	4	yes	complemented
23	C₂²×Q₈×C₃₁	—	2	2	no	—
24	C₂²×D₄×C₃₁	—	2	2	no	—
25	C₂×Q₁₆×C₃₁	—	2	4	no	—
26	C₂×Q₁₆×C₃₁	—	2	4	no	—
27	C₂×Q₁₆×C₃₁	—	2	2	no	—
28	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
29	C₂×SD₁₆×C₃₁	—	2	8	no	—
30	C₂×SD₁₆×C₃₁	—	4	8	no	—
31	C₂×D₈×C₃₁	—	2	4	no	—
32	C₂×D₈×C₃₁	—	2	4	no	—
33	C₂×D₈×C₃₁	—	2	2	no	—
34	C₂×D₈×C₃₁	C₂³	1	4	yes	—
35	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×#124	—	2	8	no	—
38	C₃₁×#124	—	4	4	no	—
39	C₂×C₂²⋊C₄×C₃₁	—	4	4	no	—
40	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
41	C₃₁×Q₃₂	—	2	8	no	—
42	C₃₁×SD₃₂	—	2	16	no	—
43	C₃₁×D₁₆	—	2	8	no	—
44	C₂×C₁₆×C₃₁	D₄	1	4	yes	—
45	C₂×C₁₆×C₃₁	D₄	1	4	yes	—
46	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
47	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
48	C₄×C₈×C₃₁	D₄	1	2	yes	—
49	C₄×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂⁴×C₃₁	—	4	2	no	—
51	C₂×Q₈×C₃₁	—	2	4	no	—
52	C₂×Q₈×C₃₁	—	4	2	no	—
53	C₂×Q₈×C₃₁	—	4	4	no	—
54	C₂×Q₈×C₃₁	—	4	4	no	—
55	C₂×D₄×C₃₁	—	2	4	no	—
56	C₂×D₄×C₃₁	—	4	4	no	—
57	C₂×D₄×C₃₁	—	4	4	no	—
58	C₂×D₄×C₃₁	—	4	2	no	—
59	C₂²×C₄×C₃₁	—	4	2	no	—
60	C₂²×C₄×C₃₁	D₈	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
62	Q₁₆×C₃₁	—	2	4	no	—
63	Q₁₆×C₃₁	—	2	4	no	—
64	Q₁₆×C₃₁	—	4	4	no	—
65	SD₁₆×C₃₁	—	4	16	no	—
66	D₈×C₃₁	—	2	4	no	—
67	D₈×C₃₁	—	2	4	no	—
68	D₈×C₃₁	—	4	4	no	—
69	C₂×C₈×C₃₁	D₈	1	4	yes	—
70	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
75	C₂²⋊C₄×C₃₁	—	4	16	no	—
76	C₄²×C₃₁	D₈	1	2	yes	—
77	C₄²×C₃₁	C₂×D₄	1	2	yes	—
78	C₁₆×C₃₁	C₂×D₄	1	4	yes	—
79	C₁₆×C₃₁	C₂×D₄	1	4	yes	—
80	C₂×#2249	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₃₁	—	4	4	no	—
82	C₂³×C₃₁	—	8	4	no	—
83	C₂³×C₃₁	—	8	8	no	—
84	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₃₁	—	4	4	no	—
86	Q₈×C₃₁	—	4	4	no	—
87	Q₈×C₃₁	—	8	4	no	—
88	D₄×C₃₁	—	4	4	no	—
89	D₄×C₃₁	—	4	4	no	—
90	D₄×C₃₁	—	8	4	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	4	4	no	—
93	C₂×C₄×C₃₁	—	8	4	no	—
94	C₂×C₄×C₃₁	#127	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
96	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
97	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₃₁	#127	1	2	yes	—
100	C₈×C₃₁	C₂×D₈	1	4	yes	—
101	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
102	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
103	C₂²×Q₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
104	C₂²×SD₃₂	C₂×C₃₁	1	2	yes	complemented
105	C₂²×D₁₆	C₂×C₃₁	1	1	yes	characteristic
106	C₂×#495	C₂×C₃₁	1	2	yes	complemented
107	#2249	C₂×C₃₁	1	8	yes	direct product factor
108	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
109	C₂²×C₃₁	—	8	4	no	—
110	C₂²×C₃₁	—	8	8	no	—
111	C₂²×C₃₁	—	16	8	no	—
112	C₂²×C₃₁	#493	1	2	yes	—
113	C₂²×C₃₁	#508	1	4	yes	—
114	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
115	C₄×C₃₁	—	2	4	no	—
116	C₄×C₃₁	—	8	4	no	—
117	C₄×C₃₁	#493	1	2	yes	—
118	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
119	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
120	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
121	C₂²×SD₁₆	—	2	2	no	—
122	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
123	C₂×#124	—	2	2	no	—
124	C₂×Q₃₂	—	2	4	no	—
125	C₂×Q₃₂	C₂²×C₃₁	1	4	yes	complemented
126	C₂×SD₃₂	—	2	8	no	—
127	C₂×SD₃₂	C₂²×C₃₁	1	8	yes	complemented
128	C₂×D₁₆	—	2	4	no	—
129	C₂×D₁₆	C₂²×C₃₁	1	4	yes	—
130	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
131	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
132	#495	—	2	4	no	—
133	#495	C₂²×C₃₁	1	8	yes	complemented
134	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
135	C₄×C₁₆	C₂²×C₃₁	1	4	yes	complemented
136	C₂×C₃₁	—	16	4	no	—
137	C₂×C₃₁	#2249	1	4	yes	direct product factor
138	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
139	C₂×C₃₁	C₂×#508	1	2	yes	—
140	C₂²×Q₈	—	2	2	no	—
141	C₂²×D₄	—	2	2	no	—
142	C₂×Q₁₆	—	2	4	no	—
143	C₂×Q₁₆	—	2	4	no	—
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
146	C₂×SD₁₆	—	2	8	no	—
147	C₂×SD₁₆	—	4	8	no	—
148	C₂×D₈	—	2	4	no	—
149	C₂×D₈	—	2	4	no	—
150	C₂×D₈	—	2	2	no	—
151	C₂×D₈	C₂³×C₃₁	1	4	yes	—
152	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
154	#124	—	2	8	no	—
155	#124	—	4	4	no	—
156	C₂×C₂²⋊C₄	—	4	4	no	—
157	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
158	Q₃₂	—	2	8	no	—
159	SD₃₂	—	2	16	no	—
160	D₁₆	—	2	8	no	—
161	C₂×C₁₆	D₄×C₃₁	1	4	yes	—
162	C₂×C₁₆	D₄×C₃₁	1	4	yes	—
163	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
164	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
165	C₄×C₈	D₄×C₃₁	1	2	yes	—
166	C₄×C₈	C₂³×C₃₁	1	2	yes	—
167	C₃₁	C₂×#2249	1	1	yes	characteristic, direct product factor, Sylow
168	C₂⁴	—	4	2	no	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	D₈×C₃₁	1	2	yes	—
179	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
180	Q₁₆	—	2	4	no	—
181	Q₁₆	—	2	4	no	—
182	Q₁₆	—	4	4	no	—
183	SD₁₆	—	4	16	no	—
184	D₈	—	2	4	no	—
185	D₈	—	2	4	no	—
186	D₈	—	4	4	no	—
187	C₂×C₈	D₈×C₃₁	1	4	yes	—
188	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
189	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
190	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
191	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
192	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
193	C₂²⋊C₄	—	4	16	no	—
194	C₄²	D₈×C₃₁	1	2	yes	—
195	C₄²	C₂×D₄×C₃₁	1	2	yes	—
196	C₁₆	C₂×D₄×C₃₁	1	4	yes	—
197	C₁₆	C₂×D₄×C₃₁	1	4	yes	—
198	C₂³	—	4	4	no	—
199	C₂³	—	8	4	no	—
200	C₂³	—	8	8	no	—
201	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
202	Q₈	—	4	4	no	—
203	Q₈	—	4	4	no	—
204	Q₈	—	8	4	no	—
205	D₄	—	4	4	no	—
206	D₄	—	4	4	no	—
207	D₄	—	8	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	4	no	—
211	C₂×C₄	C₃₁×#127	1	2	yes	—
212	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
213	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
215	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
216	C₈	C₃₁×#127	1	2	yes	—
217	C₈	C₂×D₈×C₃₁	1	4	yes	—
218	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
219	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
220	C₂²	—	8	4	no	—
221	C₂²	—	8	8	no	—
222	C₂²	—	16	8	no	—
223	C₂²	C₃₁×#493	1	2	yes	—
224	C₂²	C₃₁×#508	1	4	yes	—
225	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
226	C₄	—	2	4	no	—
227	C₄	—	8	4	no	—
228	C₄	C₃₁×#493	1	2	yes	—
229	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
230	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
231	C₂	—	16	4	no	—
232	C₂	C₃₁×#2249	1	4	yes	direct product factor
233	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
234	C₂	C₂×C₃₁×#508	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂²×C₃₁×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×Q₃₂	C₂	semidirect
2. C₂²×C₃₁×SD₃₂	C₂	semidirect
4. C₂×C₃₁×#495	C₂	semidirect
5. C₃₁×#2249	C₂	direct (N × H)
6. C₂×C₄×C₁₆×C₃₁	C₂	semidirect
12. C₂×C₃₁×Q₃₂	C₂²	semidirect
14. C₂×C₃₁×SD₃₂	C₂²	semidirect
20. C₃₁×#495	C₂²	semidirect
22. C₄×C₁₆×C₃₁	C₂²	semidirect
80. C₂×#2249	C₃₁	direct (N × H)
103. C₂²×Q₃₂	C₂×C₃₁	semidirect
104. C₂²×SD₃₂	C₂×C₃₁	semidirect
106. C₂×#495	C₂×C₃₁	semidirect
107. #2249	C₂×C₃₁	direct (N × H)
108. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
125. C₂×Q₃₂	C₂²×C₃₁	semidirect
127. C₂×SD₃₂	C₂²×C₃₁	semidirect
133. #495	C₂²×C₃₁	semidirect
135. C₄×C₁₆	C₂²×C₃₁	semidirect
137. C₂×C₃₁	#2249	direct (N × H)
167. C₃₁	C₂×#2249	direct (N × H)
232. C₂	C₃₁×#2249	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×Q₃₂	2
C₂²×C₃₁×SD₃₂	2
C₂²×C₃₁×D₁₆	2
C₂×C₃₁×#495	2
C₃₁×#2249	2
C₂×C₄×C₁₆×C₃₁	2
C₂×#2249	31

Subgroups — C2×C31×#2249 #21490471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2249 #21490471