Subgroups — C₂×C₃₁×#2819 #21493439

Order 7936.

3342 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#124	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2819	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₃₁×#394	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×#393	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#392	C₂	1	4	yes	maximal, complemented
6	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
8	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
9	C₂³×D₄×C₃₁	C₂²	1	4	yes	—
10	C₂×C₃₁×#124	—	2	16	no	—
11	C₂×C₃₁×#124	C₂²	1	16	yes	complemented
12	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
13	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
14	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	—
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#394	C₂²	1	32	yes	complemented
18	C₃₁×#393	C₂²	1	32	yes	complemented
19	C₃₁×#392	C₂²	1	32	yes	complemented
20	C₃₁×#382	C₂²	1	8	yes	complemented
21	C₂×C₃₁×#95	C₂²	1	4	yes	—
22	C₂×C₃₁×#95	C₂²	1	4	yes	—
23	C₂⁵×C₃₁	C₂³	1	4	yes	—
24	C₂²×Q₈×C₃₁	—	2	8	no	—
25	C₂²×Q₈×C₃₁	—	4	8	no	—
26	C₂²×Q₈×C₃₁	C₂³	1	8	yes	—
27	C₂²×D₄×C₃₁	—	2	8	no	—
28	C₂²×D₄×C₃₁	—	4	8	no	—
29	C₂²×D₄×C₃₁	—	4	16	no	—
30	C₂²×D₄×C₃₁	—	4	8	no	—
31	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
32	C₂²×D₄×C₃₁	C₂³	1	16	yes	—
33	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
34	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
35	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
37	C₃₁×#124	—	2	64	no	—
38	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
40	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
41	C₂×C₂²⋊C₄×C₃₁	—	2	16	no	—
42	C₂×C₂²⋊C₄×C₃₁	—	4	16	no	—
43	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
44	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
45	C₂×C₂²⋊C₄×C₃₁	C₂³	1	32	yes	—
46	C₂×C₄²×C₃₁	D₄	1	8	yes	complemented
47	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
48	C₃₁×#95	C₂³	1	16	yes	—
49	C₃₁×#95	C₂³	1	16	yes	—
50	C₂⁴×C₃₁	—	2	8	no	—
51	C₂⁴×C₃₁	—	2	16	no	—
52	C₂⁴×C₃₁	—	2	16	no	—
53	C₂⁴×C₃₁	—	2	8	no	—
54	C₂⁴×C₃₁	—	2	8	no	—
55	C₂⁴×C₃₁	C₂⁴	1	8	yes	—
56	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
57	C₂×Q₈×C₃₁	—	2	16	no	—
58	C₂×Q₈×C₃₁	—	4	16	no	—
59	C₂×Q₈×C₃₁	—	4	32	no	—
60	C₂×D₄×C₃₁	—	2	16	no	—
61	C₂×D₄×C₃₁	—	4	16	no	—
62	C₂×D₄×C₃₁	—	4	32	no	—
63	C₂×D₄×C₃₁	—	4	16	no	—
64	C₂×D₄×C₃₁	—	4	32	no	—
65	C₂×D₄×C₃₁	—	4	64	no	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	C₂²×C₄×C₃₁	—	2	8	no	—
68	C₂²×C₄×C₃₁	—	2	2	no	—
69	C₂²×C₄×C₃₁	—	2	8	no	—
70	C₂²×C₄×C₃₁	—	2	16	no	—
71	C₂²×C₄×C₃₁	—	2	16	no	—
72	C₂²×C₄×C₃₁	—	2	8	no	—
73	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
75	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
76	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
77	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
78	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
79	C₄⋊C₄×C₃₁	—	4	8	no	—
80	C₂²⋊C₄×C₃₁	—	2	32	no	—
81	C₂²⋊C₄×C₃₁	—	2	32	no	—
82	C₂²⋊C₄×C₃₁	—	4	32	no	—
83	C₄²×C₃₁	C₂×D₄	1	16	yes	complemented
84	C₂×#2819	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₃₁	—	2	8	no	—
86	C₂³×C₃₁	—	2	16	no	—
87	C₂³×C₃₁	—	2	16	no	—
88	C₂³×C₃₁	—	2	8	no	—
89	C₂³×C₃₁	—	2	8	no	—
90	C₂³×C₃₁	—	4	16	no	—
91	C₂³×C₃₁	—	4	16	no	—
92	C₂³×C₃₁	—	4	16	no	—
93	C₂³×C₃₁	—	4	32	no	—
94	C₂³×C₃₁	—	4	32	no	—
95	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
96	C₂³×C₃₁	H(5,𝔽₂)	1	8	yes	—
97	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
98	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
99	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
100	Q₈×C₃₁	—	4	32	no	—
101	D₄×C₃₁	—	4	32	no	—
102	D₄×C₃₁	—	4	64	no	—
103	C₂×C₄×C₃₁	—	2	16	no	—
104	C₂×C₄×C₃₁	—	2	8	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	8	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	8	no	—
109	C₂×C₄×C₃₁	—	2	16	no	—
110	C₂×C₄×C₃₁	—	2	16	no	—
111	C₂×C₄×C₃₁	—	2	8	no	—
112	C₂×C₄×C₃₁	—	4	16	no	—
113	C₂×C₄×C₃₁	—	4	16	no	—
114	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
115	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
116	C₂²×#124	C₂×C₃₁	1	2	yes	complemented
117	#2819	C₂×C₃₁	1	16	yes	direct product factor
118	C₂×#394	C₂×C₃₁	1	4	yes	complemented
119	C₂×#393	C₂×C₃₁	1	4	yes	complemented
120	C₂×#392	C₂×C₃₁	1	4	yes	complemented
121	C₂×#382	C₂×C₃₁	1	1	yes	characteristic, complemented
122	C₂²×C₃₁	—	4	8	no	—
123	C₂²×C₃₁	—	4	8	no	—
124	C₂²×C₃₁	—	4	8	no	—
125	C₂²×C₃₁	—	4	16	no	—
126	C₂²×C₃₁	—	4	16	no	—
127	C₂²×C₃₁	—	8	32	no	—
128	C₂²×C₃₁	#535	1	8	yes	—
129	C₂²×C₃₁	#537	1	4	yes	—
130	C₂²×C₃₁	H(3,𝔽₄)	1	8	yes	—
131	C₂²×C₃₁	#562	1	8	yes	—
132	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
133	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	4	yes	—
134	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
135	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
136	C₄×C₃₁	—	2	8	no	—
137	C₄×C₃₁	—	2	4	no	—
138	C₄×C₃₁	—	4	8	no	—
139	C₄×C₃₁	—	4	8	no	—
140	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
141	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
142	C₂³×D₄	C₂²×C₃₁	1	4	yes	—
143	C₂×#124	—	2	16	no	—
144	C₂×#124	C₂²×C₃₁	1	16	yes	complemented
145	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
146	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
147	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
148	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	—
149	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
150	#394	C₂²×C₃₁	1	32	yes	complemented
151	#393	C₂²×C₃₁	1	32	yes	complemented
152	#392	C₂²×C₃₁	1	32	yes	complemented
153	#382	C₂²×C₃₁	1	8	yes	complemented
154	C₂×#95	C₂²×C₃₁	1	4	yes	—
155	C₂×#95	C₂²×C₃₁	1	4	yes	—
156	C₂×C₃₁	—	8	8	no	—
157	C₂×C₃₁	#2819	1	8	yes	direct product factor
158	C₂×C₃₁	C₂×#535	1	2	yes	—
159	C₂×C₃₁	C₂×#537	1	1	yes	characteristic
160	C₂×C₃₁	C₂×H(3,𝔽₄)	1	2	yes	—
161	C₂×C₃₁	C₂×#562	1	2	yes	—
162	C₂⁵	C₂³×C₃₁	1	4	yes	—
163	C₂²×Q₈	—	2	8	no	—
164	C₂²×Q₈	—	4	8	no	—
165	C₂²×Q₈	C₂³×C₃₁	1	8	yes	—
166	C₂²×D₄	—	2	8	no	—
167	C₂²×D₄	—	4	8	no	—
168	C₂²×D₄	—	4	16	no	—
169	C₂²×D₄	—	4	8	no	—
170	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
171	C₂²×D₄	C₂³×C₃₁	1	16	yes	—
172	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
173	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
174	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
175	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
176	#124	—	2	64	no	—
177	C₂×C₄⋊C₄	—	4	4	no	—
178	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
179	C₂×C₂²⋊C₄	—	2	16	no	—
180	C₂×C₂²⋊C₄	—	2	16	no	—
181	C₂×C₂²⋊C₄	—	4	16	no	—
182	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
183	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
184	C₂×C₂²⋊C₄	C₂³×C₃₁	1	32	yes	—
185	C₂×C₄²	D₄×C₃₁	1	8	yes	complemented
186	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
187	#95	C₂³×C₃₁	1	16	yes	—
188	#95	C₂³×C₃₁	1	16	yes	—
189	C₃₁	C₂×#2819	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴	—	2	8	no	—
191	C₂⁴	—	2	16	no	—
192	C₂⁴	—	2	16	no	—
193	C₂⁴	—	2	8	no	—
194	C₂⁴	—	2	8	no	—
195	C₂⁴	C₂⁴×C₃₁	1	8	yes	—
196	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
197	C₂×Q₈	—	2	16	no	—
198	C₂×Q₈	—	4	16	no	—
199	C₂×Q₈	—	4	32	no	—
200	C₂×D₄	—	2	16	no	—
201	C₂×D₄	—	4	16	no	—
202	C₂×D₄	—	4	32	no	—
203	C₂×D₄	—	4	16	no	—
204	C₂×D₄	—	4	32	no	—
205	C₂×D₄	—	4	64	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	8	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	16	no	—
211	C₂²×C₄	—	2	16	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
214	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
215	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
216	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
217	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
218	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
219	C₄⋊C₄	—	4	8	no	—
220	C₂²⋊C₄	—	2	32	no	—
221	C₂²⋊C₄	—	2	32	no	—
222	C₂²⋊C₄	—	4	32	no	—
223	C₄²	C₂×D₄×C₃₁	1	16	yes	complemented
224	C₂³	—	2	8	no	—
225	C₂³	—	2	16	no	—
226	C₂³	—	2	16	no	—
227	C₂³	—	2	8	no	—
228	C₂³	—	2	8	no	—
229	C₂³	—	4	16	no	—
230	C₂³	—	4	16	no	—
231	C₂³	—	4	16	no	—
232	C₂³	—	4	32	no	—
233	C₂³	—	4	32	no	—
234	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
235	C₂³	C₃₁×H(5,𝔽₂)	1	8	yes	—
236	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
237	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
238	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
239	Q₈	—	4	32	no	—
240	D₄	—	4	32	no	—
241	D₄	—	4	64	no	—
242	C₂×C₄	—	2	16	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	16	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	4	16	no	—
252	C₂×C₄	—	4	16	no	—
253	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
254	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
255	C₂²	—	4	8	no	—
256	C₂²	—	4	8	no	—
257	C₂²	—	4	8	no	—
258	C₂²	—	4	16	no	—
259	C₂²	—	4	16	no	—
260	C₂²	—	8	32	no	—
261	C₂²	C₃₁×#535	1	8	yes	—
262	C₂²	C₃₁×#537	1	4	yes	—
263	C₂²	C₃₁×H(3,𝔽₄)	1	8	yes	—
264	C₂²	C₃₁×#562	1	8	yes	—
265	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
266	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	4	yes	—
267	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
268	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
269	C₄	—	2	8	no	—
270	C₄	—	2	4	no	—
271	C₄	—	4	8	no	—
272	C₄	—	4	8	no	—
273	C₂	—	8	8	no	—
274	C₂	C₃₁×#2819	1	8	yes	direct product factor
275	C₂	C₂×C₃₁×#535	1	2	yes	—
276	C₂	C₂×C₃₁×#537	1	1	yes	characteristic
277	C₂	C₂×C₃₁×H(3,𝔽₄)	1	2	yes	—
278	C₂	C₂×C₃₁×#562	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×#124	C₂	semidirect
2. C₃₁×#2819	C₂	direct (N × H)
3. C₂×C₃₁×#394	C₂	semidirect
4. C₂×C₃₁×#393	C₂	semidirect
5. C₂×C₃₁×#392	C₂	semidirect
6. C₂×C₃₁×#382	C₂	semidirect
11. C₂×C₃₁×#124	C₂²	semidirect
17. C₃₁×#394	C₂²	semidirect
18. C₃₁×#393	C₂²	semidirect
19. C₃₁×#392	C₂²	semidirect
20. C₃₁×#382	C₂²	semidirect
46. C₂×C₄²×C₃₁	D₄	semidirect
83. C₄²×C₃₁	C₂×D₄	semidirect
84. C₂×#2819	C₃₁	direct (N × H)
116. C₂²×#124	C₂×C₃₁	semidirect
117. #2819	C₂×C₃₁	direct (N × H)
118. C₂×#394	C₂×C₃₁	semidirect
119. C₂×#393	C₂×C₃₁	semidirect
120. C₂×#392	C₂×C₃₁	semidirect
121. C₂×#382	C₂×C₃₁	semidirect
144. C₂×#124	C₂²×C₃₁	semidirect
150. #394	C₂²×C₃₁	semidirect
151. #393	C₂²×C₃₁	semidirect
152. #392	C₂²×C₃₁	semidirect
153. #382	C₂²×C₃₁	semidirect
157. C₂×C₃₁	#2819	direct (N × H)
185. C₂×C₄²	D₄×C₃₁	semidirect
189. C₃₁	C₂×#2819	direct (N × H)
223. C₄²	C₂×D₄×C₃₁	semidirect
274. C₂	C₃₁×#2819	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×#124	2
C₃₁×#2819	2
C₂×C₃₁×#394	2
C₂×C₃₁×#393	2
C₂×C₃₁×#392	2
C₂×C₃₁×#382	2
C₂×#2819	31

Subgroups — C2×C31×#2819 #21493439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2819 #21493439