Subgroups — D₃₁×#518 #16858894

Order 3968.

1920 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×D₃₁	C₂	1	1	yes	maximal, characteristic
2	#14521524	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521523	C₂	1	2	yes	maximal, complemented
4	#14521522	C₂	1	4	yes	maximal, complemented
5	#14521521	C₂	1	8	yes	maximal, complemented
6	C₄×D₄×D₃₁	C₂	1	8	yes	maximal, complemented
7	#117×D₃₁	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊C₄×D₃₁	C₂	1	4	yes	maximal, complemented
9	C₃₁×#518	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₃₁	C₂²	1	2	yes	—
11	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊D₃₁	C₂²	1	2	yes	—
13	C₂×D₄×D₃₁	C₄	1	16	yes	complemented
14	C₂×D₄×D₃₁	C₂²	1	4	yes	—
15	C₂×D₄×D₃₁	C₂²	1	4	yes	—
16	C₂²×D₁₂₄	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₃₁	C₂²	1	8	yes	complemented
19	C₄×D₄×C₃₁	C₂²	1	8	yes	complemented
20	C₃₁×#117	C₂²	1	2	yes	complemented
21	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
22	C₂×C₂²⋊Dic₃₁	C₂²	1	4	yes	complemented
23	D₄×Dic₃₁	C₂²	1	8	yes	complemented
24	C₄×C₂²⋊D₃₁	C₂²	1	16	yes	complemented
25	C₂×(C₂×D₃₁)⋊C₄	C₂²	1	8	yes	complemented
26	#11758503	C₂²	1	2	yes	complemented
27	#11758470	C₂²	1	8	yes	complemented
28	#11758469	C₂²	1	4	yes	complemented
29	C₄⋊C₄×D₃₁	C₂²	1	4	yes	complemented
30	(C₂²⋊D₃₁)⋊C₄	C₂²	1	16	yes	complemented
31	C₂²⋊C₄×D₃₁	C₂²	1	4	yes	—
32	C₂²⋊C₄×D₃₁	C₂²	1	8	yes	complemented
33	C₂²⋊C₄×D₃₁	C₂²	1	8	yes	complemented
34	C₂²⋊C₄×D₃₁	C₂²	1	8	yes	complemented
35	#11758454	C₂²	1	4	yes	complemented
36	C₄×D₁₂₄	C₂²	1	8	yes	complemented
37	#11758449	C₂²	1	4	yes	complemented
38	C₄²×D₃₁	C₂²	1	4	yes	complemented
39	C₂⁴×C₃₁	C₂³	1	2	yes	—
40	C₂³×D₃₁	—	4	8	no	—
41	C₂³×D₃₁	C₂×C₄	1	16	yes	complemented
42	C₂³×D₃₁	C₂³	1	1	yes	characteristic
43	C₂³×D₃₁	C₂³	1	4	yes	—
44	C₂³×D₃₁	C₂³	1	2	yes	—
45	C₂³×D₃₁	C₂³	1	4	yes	—
46	C₂×D₄×C₃₁	C₂×C₄	1	8	yes	complemented
47	C₂×D₄×C₃₁	C₂³	1	4	yes	—
48	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂³	1	8	yes	complemented
50	C₂×C₂²⋊D₃₁	C₂×C₄	1	16	yes	complemented
51	C₂×C₂²⋊D₃₁	C₂³	1	8	yes	—
52	C₂²×Dic₃₁	C₂³	1	1	yes	characteristic
53	C₂²×Dic₃₁	C₂³	1	8	yes	complemented
54	D₄×D₃₁	C₂×C₄	1	32	yes	complemented
55	D₄×D₃₁	C₂×C₄	1	32	yes	complemented
56	C₂×D₁₂₄	C₂×C₄	1	8	yes	complemented
57	C₂×D₁₂₄	C₂³	1	4	yes	—
58	C₂×C₄×D₃₁	—	4	16	no	—
59	C₂×C₄×D₃₁	C₂×C₄	1	8	yes	complemented
60	C₂×C₄×D₃₁	C₂³	1	8	yes	—
61	C₂×C₄×D₃₁	C₂³	1	2	yes	—
62	C₂×C₄×D₃₁	C₂³	1	2	yes	—
63	C₂×C₄×D₃₁	C₂³	1	16	yes	complemented
64	C₄⋊C₄×C₃₁	C₂³	1	4	yes	complemented
65	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
66	C₂²⋊C₄×C₃₁	C₂³	1	8	yes	complemented
67	C₄²×C₃₁	C₂³	1	4	yes	complemented
68	C₂²⋊Dic₃₁	C₂³	1	4	yes	—
69	C₂²⋊Dic₃₁	C₂³	1	8	yes	complemented
70	(C₂×D₃₁)⋊C₄	C₂³	1	8	yes	—
71	(C₂×D₃₁)⋊C₄	C₂³	1	16	yes	complemented
72	C₄⋊Dic₃₁	C₂³	1	4	yes	complemented
73	Dic₃₁⋊C₄	C₂³	1	8	yes	complemented
74	C₄×Dic₃₁	C₂³	1	4	yes	complemented
75	C₄×Dic₃₁	C₂³	1	8	yes	complemented
76	C₂³×C₃₁	—	4	4	no	—
77	C₂³×C₃₁	C₂²×C₄	1	8	yes	complemented
78	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
79	C₂³×C₃₁	C₂⁴	1	4	yes	—
80	C₂²×D₃₁	—	2	16	no	—
81	C₂²×D₃₁	—	2	4	no	—
82	C₂²×D₃₁	—	4	32	no	—
83	C₂²×D₃₁	—	4	4	no	—
84	C₂²×D₃₁	—	4	8	no	—
85	C₂²×D₃₁	C₂²×C₄	1	4	yes	—
86	C₂²×D₃₁	C₂²×C₄	1	16	yes	—
87	C₂²×D₃₁	C₂²×C₄	1	8	yes	complemented
88	C₂²×D₃₁	C₂²×C₄	1	16	yes	complemented
89	C₂²×D₃₁	C₂⁴	1	1	yes	characteristic
90	C₂²×D₃₁	C₂⁴	1	4	yes	—
91	C₂²×D₃₁	C₂⁴	1	1	yes	characteristic
92	D₄×C₃₁	C₂²×C₄	1	16	yes	complemented
93	C₂×C₄×C₃₁	—	4	8	no	—
94	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	complemented
95	C₂×C₄×C₃₁	C₂⁴	1	8	yes	—
96	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
97	C₂²⋊D₃₁	C₂²×C₄	1	32	yes	complemented
98	C₂×Dic₃₁	—	4	8	no	—
99	C₂×Dic₃₁	C₂²×C₄	1	4	yes	complemented
100	C₂×Dic₃₁	C₂⁴	1	8	yes	—
101	C₂×Dic₃₁	C₂⁴	1	2	yes	—
102	D₁₂₄	C₂²×C₄	1	16	yes	complemented
103	C₄×D₃₁	—	2	16	no	—
104	C₄×D₃₁	—	4	16	no	—
105	C₄×D₃₁	C₂²×C₄	1	8	yes	—
106	C₄×D₃₁	C₂²×C₄	1	8	yes	complemented
107	C₂×#518	—	31	1	no	maximal, Sylow
108	C₂²×C₃₁	—	2	8	no	—
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	—	4	8	no	—
111	C₂²×C₃₁	C₂³×C₄	1	2	yes	—
112	C₂²×C₃₁	C₂³×C₄	1	8	yes	—
113	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
114	C₂×D₃₁	—	2	8	no	—
115	C₂×D₃₁	—	2	2	no	—
116	C₂×D₃₁	—	2	8	no	—
117	C₂×D₃₁	—	2	32	no	—
118	C₂×D₃₁	—	4	16	no	—
119	C₂×D₃₁	—	4	8	no	—
120	C₂×D₃₁	C₂³×C₄	1	2	yes	—
121	C₂×D₃₁	C₂³×C₄	1	8	yes	—
122	C₂×D₃₁	C₂³×C₄	1	2	yes	—
123	C₂×D₃₁	H(5,𝔽₂)	1	4	yes	—
124	C₄×C₃₁	—	2	8	no	—
125	C₄×C₃₁	C₂³×C₄	1	4	yes	—
126	Dic₃₁	—	2	8	no	—
127	Dic₃₁	C₂³×C₄	1	4	yes	—
128	C₂³×D₄	—	31	1	no	—
129	#518	—	31	1	no	—
130	#518	—	31	8	no	—
131	#518	—	31	2	no	—
132	#518	—	31	4	no	—
133	#518	D₃₁	1	1	yes	characteristic, direct product factor
134	C₂×C₄×D₄	—	31	8	no	—
135	C₂×#117	—	31	2	no	—
136	C₂²×C₂²⋊C₄	—	31	4	no	—
137	C₂×C₃₁	—	2	2	no	—
138	C₂×C₃₁	—	2	8	no	—
139	C₂×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
140	C₂×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
141	D₃₁	—	2	2	no	—
142	D₃₁	—	2	8	no	—
143	D₃₁	#518	1	4	yes	direct product factor
144	C₂⁵	—	31	2	no	—
145	C₂²×D₄	—	31	2	no	—
146	C₂²×D₄	—	31	16	no	—
147	C₂²×D₄	—	31	4	no	—
148	C₂²×D₄	—	31	4	no	—
149	C₂²×D₄	—	31	1	no	—
150	C₂²×D₄	C₂×D₃₁	1	1	yes	characteristic
151	C₂³×C₄	—	31	1	no	—
152	C₂³×C₄	—	31	8	no	—
153	C₄×D₄	—	31	8	no	—
154	C₄×D₄	—	31	16	no	—
155	C₄×D₄	—	31	8	no	—
156	C₄×D₄	—	31	16	no	—
157	C₄×D₄	—	31	8	no	—
158	C₄×D₄	C₂×D₃₁	1	8	yes	complemented
159	#117	—	31	2	no	—
160	#117	—	31	4	no	—
161	#117	—	31	4	no	—
162	#117	—	31	4	no	—
163	#117	C₂×D₃₁	1	2	yes	complemented
164	C₂×C₄⋊C₄	—	31	4	no	—
165	C₂×C₂²⋊C₄	—	31	4	no	—
166	C₂×C₂²⋊C₄	—	31	8	no	—
167	C₂×C₂²⋊C₄	—	31	4	no	—
168	C₂×C₂²⋊C₄	—	31	8	no	—
169	C₂×C₂²⋊C₄	—	31	8	no	—
170	C₂×C₂²⋊C₄	—	31	8	no	—
171	C₂×C₂²⋊C₄	C₂×D₃₁	1	4	yes	complemented
172	C₂×C₄²	—	31	4	no	—
173	C₃₁	C₂×#518	1	1	yes	characteristic, complemented, Sylow
174	C₂⁴	—	31	1	no	—
175	C₂⁴	—	31	4	no	—
176	C₂⁴	—	31	16	no	—
177	C₂⁴	—	31	4	no	—
178	C₂⁴	—	31	2	no	—
179	C₂⁴	—	124	8	no	—
180	C₂⁴	C₂²×D₃₁	1	2	yes	—
181	C₂×D₄	—	31	16	no	—
182	C₂×D₄	—	31	8	no	—
183	C₂×D₄	—	31	32	no	—
184	C₂×D₄	—	31	32	no	—
185	C₂×D₄	—	31	8	no	—
186	C₂×D₄	—	31	4	no	—
187	C₂×D₄	C₄×D₃₁	1	8	yes	complemented
188	C₂×D₄	C₂²×D₃₁	1	4	yes	—
189	C₂²×C₄	—	31	1	no	—
190	C₂²×C₄	—	31	8	no	—
191	C₂²×C₄	—	31	2	no	—
192	C₂²×C₄	—	31	2	no	—
193	C₂²×C₄	—	31	8	no	—
194	C₂²×C₄	—	31	8	no	—
195	C₂²×C₄	—	31	16	no	—
196	C₂²×C₄	—	124	16	no	—
197	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₃₁	1	8	yes	complemented
199	C₄⋊C₄	—	31	8	no	—
200	C₄⋊C₄	—	31	4	no	—
201	C₄⋊C₄	C₂²×D₃₁	1	4	yes	complemented
202	C₂²⋊C₄	—	31	4	no	—
203	C₂²⋊C₄	—	31	8	no	—
204	C₂²⋊C₄	—	31	8	no	—
205	C₂²⋊C₄	—	31	16	no	—
206	C₂²⋊C₄	C₂²×D₃₁	1	4	yes	—
207	C₂²⋊C₄	C₂²×D₃₁	1	8	yes	complemented
208	C₄²	—	31	8	no	—
209	C₄²	—	31	4	no	—
210	C₄²	C₂²×D₃₁	1	4	yes	complemented
211	C₂³	—	4	4	no	—
212	C₂³	—	31	1	no	—
213	C₂³	—	31	4	no	—
214	C₂³	—	31	16	no	—
215	C₂³	—	31	1	no	—
216	C₂³	—	31	4	no	—
217	C₂³	—	31	16	no	—
218	C₂³	—	31	8	no	—
219	C₂³	—	62	16	no	—
220	C₂³	—	62	4	no	—
221	C₂³	—	124	8	no	—
222	C₂³	—	124	4	no	—
223	C₂³	—	124	32	no	—
224	C₂³	C₂×C₄×D₃₁	1	8	yes	complemented
225	C₂³	C₂³×D₃₁	1	1	yes	characteristic
226	C₂³	C₂³×D₃₁	1	4	yes	—
227	D₄	—	31	32	no	—
228	D₄	—	31	16	no	—
229	D₄	C₂×C₄×D₃₁	1	16	yes	complemented
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	—	31	4	no	—
232	C₂×C₄	—	31	2	no	—
233	C₂×C₄	—	31	8	no	—
234	C₂×C₄	—	31	8	no	—
235	C₂×C₄	—	31	8	no	—
236	C₂×C₄	—	62	16	no	—
237	C₂×C₄	—	124	16	no	—
238	C₂×C₄	—	124	8	no	—
239	C₂×C₄	C₂×C₄×D₃₁	1	4	yes	complemented
240	C₂×C₄	C₂³×D₃₁	1	8	yes	—
241	C₂×C₄	C₂³×D₃₁	1	2	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	8	no	—
245	C₂²	—	31	2	no	—
246	C₂²	—	31	4	no	—
247	C₂²	—	31	2	no	—
248	C₂²	—	31	8	no	—
249	C₂²	—	62	8	no	—
250	C₂²	—	62	32	no	—
251	C₂²	—	62	8	no	—
252	C₂²	—	62	2	no	—
253	C₂²	—	124	8	no	—
254	C₂²	—	124	16	no	—
255	C₂²	C₂²×C₄×D₃₁	1	2	yes	—
256	C₂²	C₂²×C₄×D₃₁	1	8	yes	—
257	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
258	C₄	—	2	8	no	—
259	C₄	—	31	4	no	—
260	C₄	—	62	8	no	—
261	C₄	C₂²×C₄×D₃₁	1	4	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	2	8	no	—
264	C₂	—	31	4	no	—
265	C₂	—	62	2	no	—
266	C₂	—	62	8	no	—
267	C₂	C₂³×C₄×D₃₁	1	1	yes	characteristic
268	C₂	H(5,𝔽₂)×D₃₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃₁
commutator	C₂×C₃₁	C₂⁴×C₄
Fitting	C₃₁×#518	C₂
socle, cosocle	C₂²×C₃₁	C₂⁵
prime radical	C₂²×D₄×D₃₁	C₂
hypercenter	#518	D₃₁
hypocenter	C₃₁	C₂×#518
LCS	G ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #518

Complements

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

N	H ≅ G/N	type
2. #14521524	C₂	semidirect
3. #14521523	C₂	semidirect
4. #14521522	C₂	semidirect
5. #14521521	C₂	semidirect
6. C₄×D₄×D₃₁	C₂	semidirect
7. #117×D₃₁	C₂	semidirect
8. C₂×C₂²⋊C₄×D₃₁	C₂	semidirect
9. C₃₁×#518	C₂	semidirect
13. C₂×D₄×D₃₁	C₄	semidirect
18. C₂²×C₄×D₃₁	C₂²	semidirect
19. C₄×D₄×C₃₁	C₂²	semidirect
20. C₃₁×#117	C₂²	semidirect
21. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
22. C₂×C₂²⋊Dic₃₁	C₂²	semidirect
23. D₄×Dic₃₁	C₂²	semidirect
24. C₄×C₂²⋊D₃₁	C₂²	semidirect
25. C₂×(C₂×D₃₁)⋊C₄	C₂²	semidirect
26. #11758503	C₂²	semidirect
27. #11758470	C₂²	semidirect
28. #11758469	C₂²	semidirect
29. C₄⋊C₄×D₃₁	C₂²	semidirect
30. (C₂²⋊D₃₁)⋊C₄	C₂²	semidirect
32. C₂²⋊C₄×D₃₁	C₂²	semidirect
33. C₂²⋊C₄×D₃₁	C₂²	semidirect
34. C₂²⋊C₄×D₃₁	C₂²	semidirect
35. #11758454	C₂²	semidirect
36. C₄×D₁₂₄	C₂²	semidirect
37. #11758449	C₂²	semidirect
38. C₄²×D₃₁	C₂²	semidirect
41. C₂³×D₃₁	C₂×C₄	semidirect
46. C₂×D₄×C₃₁	C₂×C₄	semidirect
49. C₂²×C₄×C₃₁	C₂³	semidirect
50. C₂×C₂²⋊D₃₁	C₂×C₄	semidirect
53. C₂²×Dic₃₁	C₂³	semidirect
54. D₄×D₃₁	C₂×C₄	semidirect
55. D₄×D₃₁	C₂×C₄	semidirect
56. C₂×D₁₂₄	C₂×C₄	semidirect
59. C₂×C₄×D₃₁	C₂×C₄	semidirect
63. C₂×C₄×D₃₁	C₂³	semidirect
64. C₄⋊C₄×C₃₁	C₂³	semidirect
66. C₂²⋊C₄×C₃₁	C₂³	semidirect
67. C₄²×C₃₁	C₂³	semidirect
69. C₂²⋊Dic₃₁	C₂³	semidirect
71. (C₂×D₃₁)⋊C₄	C₂³	semidirect
72. C₄⋊Dic₃₁	C₂³	semidirect
73. Dic₃₁⋊C₄	C₂³	semidirect
74. C₄×Dic₃₁	C₂³	semidirect
75. C₄×Dic₃₁	C₂³	semidirect
77. C₂³×C₃₁	C₂²×C₄	semidirect
87. C₂²×D₃₁	C₂²×C₄	semidirect
88. C₂²×D₃₁	C₂²×C₄	semidirect
92. D₄×C₃₁	C₂²×C₄	semidirect
94. C₂×C₄×C₃₁	C₂²×C₄	semidirect
97. C₂²⋊D₃₁	C₂²×C₄	semidirect
99. C₂×Dic₃₁	C₂²×C₄	semidirect
102. D₁₂₄	C₂²×C₄	semidirect
106. C₄×D₃₁	C₂²×C₄	semidirect
133. #518	D₃₁	direct (N × H)
143. D₃₁	#518	direct (N × H)
158. C₄×D₄	C₂×D₃₁	semidirect
163. #117	C₂×D₃₁	semidirect
171. C₂×C₂²⋊C₄	C₂×D₃₁	semidirect
173. C₃₁	C₂×#518	semidirect
187. C₂×D₄	C₄×D₃₁	semidirect
198. C₂²×C₄	C₂²×D₃₁	semidirect
201. C₄⋊C₄	C₂²×D₃₁	semidirect
207. C₂²⋊C₄	C₂²×D₃₁	semidirect
210. C₄²	C₂²×D₃₁	semidirect
224. C₂³	C₂×C₄×D₃₁	semidirect
229. D₄	C₂×C₄×D₃₁	semidirect
239. C₂×C₄	C₂×C₄×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄×D₃₁	2
#14521524	2
#14521523	2
#14521522	2
#14521521	2
C₄×D₄×D₃₁	2
#117×D₃₁	2
C₂×C₂²⋊C₄×D₃₁	2
C₃₁×#518	2
C₂×#518	31

Subgroups — D31×#518 #16858894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₁×#518 #16858894