Subgroups — C₂×C₃₁×#2251 #21490474

Order 7936.

826 conjugacy classes of subgroups.

The subgroup lattice has 272 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	2	yes	maximal, complemented
2	C₂²×C₃₁×SD₃₂	C₂	1	2	yes	maximal
3	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2251	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₁₆×C₃₁	—	2	1	no	—
8	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₃₁	—	2	1	no	—
11	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₃₁×#124	—	2	1	no	—
14	C₂×C₃₁×Q₃₂	—	2	8	no	—
15	C₂×C₃₁×Q₃₂	C₂²	1	8	yes	complemented
16	C₂×C₃₁×SD₃₂	—	2	8	no	—
17	C₂×C₃₁×SD₃₂	C₂²	1	8	yes	—
18	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#495	—	2	2	no	—
20	C₃₁×#495	C₂²	1	4	yes	—
21	C₃₁×#494	—	2	2	no	—
22	C₃₁×#494	C₂²	1	4	yes	complemented
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#346	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₃₁	—	2	1	no	—
26	C₂²×Q₈×C₃₁	—	2	2	no	—
27	C₂²×D₄×C₃₁	—	2	1	no	—
28	C₂×Q₁₆×C₃₁	—	2	4	no	—
29	C₂×Q₁₆×C₃₁	—	2	4	no	—
30	C₂×Q₁₆×C₃₁	—	2	4	no	—
31	C₂×Q₁₆×C₃₁	—	2	2	no	—
32	C₂×Q₁₆×C₃₁	—	2	4	no	—
33	C₂×Q₁₆×C₃₁	—	2	1	no	—
34	C₂×Q₁₆×C₃₁	—	4	4	no	—
35	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
36	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
37	C₂×SD₁₆×C₃₁	—	2	4	no	—
38	C₂×SD₁₆×C₃₁	—	4	4	no	—
39	C₂×D₈×C₃₁	—	2	4	no	—
40	C₂×D₈×C₃₁	—	2	1	no	—
41	C₂×D₈×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×C₄⋊Q₈	—	2	4	no	—
45	C₃₁×C₄⋊Q₈	—	4	2	no	—
46	C₃₁×#124	—	2	4	no	—
47	C₃₁×#124	—	4	2	no	—
48	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
49	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
50	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
51	C₃₁×Q₃₂	—	2	16	no	—
52	C₃₁×SD₃₂	—	2	16	no	—
53	C₂×C₁₆×C₃₁	D₄	1	8	yes	—
54	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
55	C₄×C₈×C₃₁	D₄	1	2	yes	—
56	C₄×C₈×C₃₁	C₂³	1	2	yes	—
57	C₂⁴×C₃₁	—	4	1	no	—
58	C₂×Q₈×C₃₁	—	2	2	no	—
59	C₂×Q₈×C₃₁	—	2	4	no	—
60	C₂×Q₈×C₃₁	—	4	1	no	—
61	C₂×Q₈×C₃₁	—	4	2	no	—
62	C₂×Q₈×C₃₁	—	4	4	no	—
63	C₂×Q₈×C₃₁	—	4	2	no	—
64	C₂×Q₈×C₃₁	—	4	2	no	—
65	C₂×Q₈×C₃₁	—	4	4	no	—
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	4	2	no	—
68	C₂×D₄×C₃₁	—	4	2	no	—
69	C₂×D₄×C₃₁	—	4	1	no	—
70	C₂²×C₄×C₃₁	—	4	1	no	—
71	C₂²×C₄×C₃₁	—	4	2	no	—
72	C₂²×C₄×C₃₁	D₈	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
74	Q₁₆×C₃₁	—	2	4	no	—
75	Q₁₆×C₃₁	—	2	4	no	—
76	Q₁₆×C₃₁	—	2	4	no	—
77	Q₁₆×C₃₁	—	4	4	no	—
78	Q₁₆×C₃₁	—	4	2	no	—
79	Q₁₆×C₃₁	—	4	8	no	—
80	SD₁₆×C₃₁	—	4	8	no	—
81	D₈×C₃₁	—	2	4	no	—
82	D₈×C₃₁	—	4	2	no	—
83	C₂×C₈×C₃₁	D₈	1	4	yes	—
84	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₃₁	—	4	8	no	—
89	C₂²⋊C₄×C₃₁	—	4	8	no	—
90	C₄²×C₃₁	D₈	1	2	yes	—
91	C₄²×C₃₁	C₂×D₄	1	2	yes	—
92	C₁₆×C₃₁	C₂×D₄	1	8	yes	—
93	C₂×#2251	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	4	2	no	—
95	C₂³×C₃₁	—	8	2	no	—
96	C₂³×C₃₁	—	8	4	no	—
97	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
98	Q₈×C₃₁	—	4	4	no	—
99	Q₈×C₃₁	—	4	2	no	—
100	Q₈×C₃₁	—	4	2	no	—
101	Q₈×C₃₁	—	4	4	no	—
102	Q₈×C₃₁	—	8	2	no	—
103	Q₈×C₃₁	—	8	4	no	—
104	D₄×C₃₁	—	4	2	no	—
105	D₄×C₃₁	—	4	2	no	—
106	D₄×C₃₁	—	8	2	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	4	no	—
110	C₂×C₄×C₃₁	—	8	2	no	—
111	C₂×C₄×C₃₁	—	8	4	no	—
112	C₂×C₄×C₃₁	#127	1	2	yes	—
113	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
114	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
117	C₈×C₃₁	#127	1	2	yes	—
118	C₈×C₃₁	C₂×D₈	1	4	yes	—
119	C₈×C₃₁	C₂²×D₄	1	2	yes	—
120	C₂²×Q₃₂	C₂×C₃₁	1	2	yes	complemented
121	C₂²×SD₃₂	C₂×C₃₁	1	2	yes	—
122	C₂×#495	C₂×C₃₁	1	1	yes	characteristic
123	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#2251	C₂×C₃₁	1	8	yes	direct product factor
125	C₂×#346	C₂×C₃₁	1	1	yes	characteristic, complemented
126	C₂²×C₃₁	—	8	2	no	—
127	C₂²×C₃₁	—	8	4	no	—
128	C₂²×C₃₁	—	16	4	no	—
129	C₂²×C₃₁	#493	1	2	yes	—
130	C₂²×C₃₁	#510	1	4	yes	—
131	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₃₁	—	2	4	no	—
133	C₄×C₃₁	—	8	2	no	—
134	C₄×C₃₁	—	8	4	no	—
135	C₄×C₃₁	#493	1	2	yes	—
136	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
137	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
138	C₂²×Q₁₆	—	2	1	no	—
139	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
140	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
141	C₂²×SD₁₆	—	2	1	no	—
142	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
143	C₂×C₄⋊Q₈	—	2	1	no	—
144	C₂×#124	—	2	1	no	—
145	C₂×Q₃₂	—	2	8	no	—
146	C₂×Q₃₂	C₂²×C₃₁	1	8	yes	complemented
147	C₂×SD₃₂	—	2	8	no	—
148	C₂×SD₃₂	C₂²×C₃₁	1	8	yes	—
149	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
150	#495	—	2	2	no	—
151	#495	C₂²×C₃₁	1	4	yes	—
152	#494	—	2	2	no	—
153	#494	C₂²×C₃₁	1	4	yes	complemented
154	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
155	#346	C₂²×C₃₁	1	4	yes	complemented
156	C₂×C₃₁	—	16	2	no	—
157	C₂×C₃₁	#2251	1	4	yes	direct product factor
158	C₂×C₃₁	C₂×#493	1	1	yes	characteristic
159	C₂×C₃₁	C₂×#510	1	2	yes	—
160	C₂²×Q₈	—	2	1	no	—
161	C₂²×Q₈	—	2	2	no	—
162	C₂²×D₄	—	2	1	no	—
163	C₂×Q₁₆	—	2	4	no	—
164	C₂×Q₁₆	—	2	4	no	—
165	C₂×Q₁₆	—	2	4	no	—
166	C₂×Q₁₆	—	2	4	no	—
167	C₂×Q₁₆	—	2	2	no	—
168	C₂×Q₁₆	—	2	1	no	—
169	C₂×Q₁₆	—	4	4	no	—
170	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
171	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
172	C₂×SD₁₆	—	2	4	no	—
173	C₂×SD₁₆	—	4	4	no	—
174	C₂×D₈	—	2	4	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	C₂³×C₃₁	1	2	yes	—
177	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	4	no	—
180	C₄⋊Q₈	—	4	2	no	—
181	#124	—	2	4	no	—
182	#124	—	4	2	no	—
183	C₂×C₄⋊C₄	—	4	2	no	—
184	C₂×C₂²⋊C₄	—	4	2	no	—
185	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
186	Q₃₂	—	2	16	no	—
187	SD₃₂	—	2	16	no	—
188	C₂×C₁₆	D₄×C₃₁	1	8	yes	—
189	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
190	C₄×C₈	D₄×C₃₁	1	2	yes	—
191	C₄×C₈	C₂³×C₃₁	1	2	yes	—
192	C₃₁	C₂×#2251	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	—	4	1	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	—	2	4	no	—
196	C₂×Q₈	—	4	1	no	—
197	C₂×Q₈	—	4	2	no	—
198	C₂×Q₈	—	4	4	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	D₈×C₃₁	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
210	Q₁₆	—	2	4	no	—
211	Q₁₆	—	2	4	no	—
212	Q₁₆	—	2	4	no	—
213	Q₁₆	—	4	4	no	—
214	Q₁₆	—	4	2	no	—
215	Q₁₆	—	4	8	no	—
216	SD₁₆	—	4	8	no	—
217	D₈	—	2	4	no	—
218	D₈	—	4	2	no	—
219	C₂×C₈	D₈×C₃₁	1	4	yes	—
220	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
222	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
223	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
224	C₄⋊C₄	—	4	8	no	—
225	C₂²⋊C₄	—	4	8	no	—
226	C₄²	D₈×C₃₁	1	2	yes	—
227	C₄²	C₂×D₄×C₃₁	1	2	yes	—
228	C₁₆	C₂×D₄×C₃₁	1	8	yes	—
229	C₂³	—	4	2	no	—
230	C₂³	—	8	2	no	—
231	C₂³	—	8	4	no	—
232	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
233	Q₈	—	4	4	no	—
234	Q₈	—	4	2	no	—
235	Q₈	—	4	2	no	—
236	Q₈	—	4	4	no	—
237	Q₈	—	8	2	no	—
238	Q₈	—	8	4	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	8	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	2	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	C₃₁×#127	1	2	yes	—
248	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
249	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
250	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
251	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
252	C₈	C₃₁×#127	1	2	yes	—
253	C₈	C₂×D₈×C₃₁	1	4	yes	—
254	C₈	C₂²×D₄×C₃₁	1	2	yes	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	16	4	no	—
258	C₂²	C₃₁×#493	1	2	yes	—
259	C₂²	C₃₁×#510	1	4	yes	—
260	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	8	2	no	—
263	C₄	—	8	4	no	—
264	C₄	C₃₁×#493	1	2	yes	—
265	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
266	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
267	C₂	—	16	2	no	—
268	C₂	C₃₁×#2251	1	4	yes	direct product factor
269	C₂	C₂×C₃₁×#493	1	1	yes	characteristic
270	C₂	C₂×C₃₁×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
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
4. C₂×C₃₁×#494	C₂	semidirect
5. C₃₁×#2251	C₂	direct (N × H)
6. C₂×C₃₁×#346	C₂	semidirect
15. C₂×C₃₁×Q₃₂	C₂²	semidirect
22. C₃₁×#494	C₂²	semidirect
24. C₃₁×#346	C₂²	semidirect
93. C₂×#2251	C₃₁	direct (N × H)
120. C₂²×Q₃₂	C₂×C₃₁	semidirect
123. C₂×#494	C₂×C₃₁	semidirect
124. #2251	C₂×C₃₁	direct (N × H)
125. C₂×#346	C₂×C₃₁	semidirect
146. C₂×Q₃₂	C₂²×C₃₁	semidirect
153. #494	C₂²×C₃₁	semidirect
155. #346	C₂²×C₃₁	semidirect
157. C₂×C₃₁	#2251	direct (N × H)
192. C₃₁	C₂×#2251	direct (N × H)
268. C₂	C₃₁×#2251	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
C₂×C₃₁×#495	2
C₂×C₃₁×#494	2
C₃₁×#2251	2
C₂×C₃₁×#346	2
C₂×#2251	31

Subgroups — C2×C31×#2251 #21490474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2251 #21490474