Subgroups — D₃₁×#436 #16854394

Order 3968.

552 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₃₁	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₃₁	C₂	1	1	yes	maximal, characteristic
3	#14521128	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521127	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521126	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521125	C₂	1	2	yes	maximal, complemented
7	#14521124	C₂	1	2	yes	maximal, complemented
8	C₄⋊C₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊C₈×D₃₁	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₃₁	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₃₁	C₂²	1	2	yes	—
14	C₂×M₁₆×C₃₁	C₂²	1	2	yes	complemented
15	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	D₄×Dic₃₁	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₃₁	C₂²	1	2	yes	—
18	C₂×#92474	C₂²	1	2	yes	complemented
19	M₁₆×D₃₁	—	2	8	no	—
20	C₂×#92469	C₂²	1	2	yes	complemented
21	C₂×#92469	C₂²	1	2	yes	complemented
22	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₃₁	C₂²	1	2	yes	—
24	C₂×C₈×D₃₁	C₂²	1	1	yes	characteristic
25	#11758470	C₄	1	1	yes	characteristic
26	C₄⋊C₄×D₃₁	C₄	1	1	yes	characteristic
27	(C₂²⋊D₃₁)⋊C₄	C₄	1	2	yes	—
28	C₂²⋊C₄×D₃₁	C₄	1	2	yes	—
29	C₄×D₁₂₄	C₂²	1	1	yes	characteristic
30	C₄²×D₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
33	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
34	C₂²⋊(C₃₁⋊C₈)	C₂²	1	2	yes	complemented
35	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	complemented
36	(C₂×D₃₁)⋊C₈	C₂²	1	2	yes	complemented
37	Dic₃₁⋊C₈	C₂²	1	2	yes	complemented
38	C₈×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
39	C₈×Dic₃₁	C₂²	1	1	yes	characteristic, complemented
40	C₄⋊(C₃₁⋊C₈)	C₂²	1	1	yes	characteristic, complemented
41	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₃₁	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
45	C₂×C₂²⋊D₃₁	C₂×C₄	1	2	yes	—
46	C₂²×Dic₃₁	C₂×C₄	1	2	yes	—
47	D₄×D₃₁	—	2	4	no	—
48	C₂×D₁₂₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₃₁	—	2	4	no	—
50	C₂×C₄×D₃₁	—	2	4	no	—
51	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
54	C₂×C₄×D₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₃₁	C₂³	1	2	yes	—
56	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
57	M₁₆×C₃₁	—	2	4	no	—
58	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂³	1	2	yes	—
60	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
63	C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₃₁	C₂×C₄	1	2	yes	—
65	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
66	(C₂×D₃₁)⋊C₄	C₂×C₄	1	2	yes	—
67	C₄⋊Dic₃₁	C₂×C₄	1	1	yes	characteristic
68	Dic₃₁⋊C₄	C₂×C₄	1	2	yes	—
69	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₃₁	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
72	#92474	—	2	4	no	—
73	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₃₁⋊C₈	C₂³	1	2	yes	—
75	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
76	#92469	—	2	4	no	—
77	#92469	—	2	4	no	—
78	C₈×D₃₁	—	2	4	no	—
79	C₈×D₃₁	—	2	2	no	—
80	C₈×D₃₁	D₄	1	4	yes	complemented
81	C₂³×C₃₁	C₂²×C₄	1	2	yes	—
82	C₂²×D₃₁	—	2	4	no	—
83	C₂²×D₃₁	—	2	4	no	—
84	C₂²×D₃₁	—	2	4	no	—
85	C₂²×D₃₁	C₂²×C₄	1	2	yes	—
86	C₂²×D₃₁	C₂²×C₄	1	1	yes	characteristic
87	D₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₃₁	—	2	4	no	—
95	C₂×Dic₃₁	—	2	4	no	—
96	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₃₁	C₂²×C₄	1	2	yes	—
98	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₃₁	C₂²×C₄	1	1	yes	characteristic
100	D₁₂₄	—	2	2	no	—
101	C₄×D₃₁	—	2	4	no	—
102	C₄×D₃₁	—	2	2	no	—
103	C₄×D₃₁	—	2	2	no	—
104	C₄×D₃₁	—	2	2	no	—
105	C₄×D₃₁	—	2	4	no	—
106	C₄×D₃₁	M₁₆	1	4	yes	complemented
107	C₄×D₃₁	C₂×D₄	1	2	yes	—
108	C₄×D₃₁	#41	1	2	yes	—
109	C₈×C₃₁	—	2	2	no	—
110	C₈×C₃₁	—	2	1	no	—
111	C₈×C₃₁	C₂×D₄	1	2	yes	complemented
112	C₃₁⋊C₈	—	2	2	no	—
113	C₃₁⋊C₈	—	2	1	no	—
114	C₃₁⋊C₈	C₂×D₄	1	2	yes	complemented
115	C₂×#436	—	31	1	no	maximal, Sylow
116	C₂²×C₃₁	—	2	2	no	—
117	C₂²×C₃₁	—	2	2	no	—
118	C₂²×C₃₁	—	2	2	no	—
119	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
120	C₂×D₃₁	—	2	2	no	—
121	C₂×D₃₁	—	2	2	no	—
122	C₂×D₃₁	—	2	2	no	—
123	C₂×D₃₁	—	4	8	no	—
124	C₂×D₃₁	C₄×D₄	1	2	yes	—
125	C₂×D₃₁	C₂×M₁₆	1	2	yes	—
126	C₂×D₃₁	#131	1	2	yes	—
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	C₂×M₁₆	1	2	yes	complemented
130	C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
132	Dic₃₁	—	2	2	no	—
133	Dic₃₁	—	2	1	no	—
134	Dic₃₁	C₄×D₄	1	2	yes	—
135	Dic₃₁	C₂×M₁₆	1	2	yes	complemented
136	C₂²×M₁₆	—	31	2	no	—
137	C₂×C₄×D₄	—	31	1	no	—
138	#436	—	31	2	no	—
139	#436	—	31	1	no	—
140	#436	—	31	1	no	—
141	#436	—	31	2	no	—
142	#436	—	31	1	no	—
143	#436	D₃₁	1	1	yes	characteristic, direct product factor
144	C₂×C₄⋊C₈	—	31	1	no	—
145	C₂×C₂²⋊C₈	—	31	2	no	—
146	C₂×C₄×C₈	—	31	1	no	—
147	C₂×C₃₁	—	4	2	no	—
148	C₂×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
150	C₂×C₃₁	C₂×#131	1	1	yes	characteristic
151	D₃₁	—	4	2	no	—
152	D₃₁	#436	1	4	yes	direct product factor
153	C₂²×D₄	—	31	1	no	—
154	C₂³×C₄	—	31	2	no	—
155	C₂×M₁₆	—	31	2	no	—
156	C₂×M₁₆	—	31	2	no	—
157	C₂×M₁₆	—	31	2	no	—
158	C₂×M₁₆	—	62	8	no	—
159	C₂×M₁₆	C₂×D₃₁	1	2	yes	complemented
160	C₂²×C₈	—	31	1	no	—
161	C₂²×C₈	—	31	2	no	—
162	C₂²×C₈	—	31	1	no	—
163	C₄×D₄	—	31	1	no	—
164	C₄×D₄	—	31	2	no	—
165	C₄×D₄	—	31	1	no	—
166	C₄×D₄	—	31	2	no	—
167	C₄×D₄	—	31	1	no	—
168	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
169	C₂×C₄⋊C₄	—	31	1	no	—
170	C₂×C₂²⋊C₄	—	31	2	no	—
171	C₂×C₄²	—	31	1	no	—
172	C₄⋊C₈	—	31	2	no	—
173	C₄⋊C₈	—	31	1	no	—
174	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
175	C₂²⋊C₈	—	31	2	no	—
176	C₂²⋊C₈	—	31	2	no	—
177	C₂²⋊C₈	—	31	2	no	—
178	C₂²⋊C₈	C₂×D₃₁	1	2	yes	complemented
179	C₄×C₈	—	31	1	no	—
180	C₄×C₈	—	31	1	no	—
181	C₄×C₈	—	31	1	no	—
182	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic, complemented
183	C₃₁	C₂×#436	1	1	yes	characteristic, complemented, Sylow
184	C₂⁴	—	31	2	no	—
185	C₂×D₄	—	31	2	no	—
186	C₂×D₄	—	31	1	no	—
187	C₂×D₄	—	62	4	no	—
188	C₂×D₄	C₄×D₃₁	1	1	yes	characteristic
189	C₂²×C₄	—	31	1	no	—
190	C₂²×C₄	—	31	1	no	—
191	C₂²×C₄	—	31	2	no	—
192	C₂²×C₄	—	31	1	no	—
193	C₂²×C₄	—	31	2	no	—
194	C₂²×C₄	—	31	2	no	—
195	C₂²×C₄	—	31	2	no	—
196	C₂²×C₄	—	62	4	no	—
197	C₂²×C₄	—	62	4	no	—
198	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	62	4	no	—
201	M₁₆	—	62	4	no	—
202	M₁₆	—	62	4	no	—
203	C₂×C₈	—	31	1	no	—
204	C₂×C₈	—	31	4	no	—
205	C₂×C₈	—	31	1	no	—
206	C₂×C₈	—	31	2	no	—
207	C₂×C₈	—	62	2	no	—
208	C₂×C₈	—	62	4	no	—
209	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₃₁	1	2	yes	—
211	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
212	C₄⋊C₄	—	31	2	no	—
213	C₄⋊C₄	—	31	1	no	—
214	C₄⋊C₄	C₄×D₃₁	1	1	yes	characteristic
215	C₂²⋊C₄	—	31	2	no	—
216	C₂²⋊C₄	—	31	2	no	—
217	C₂²⋊C₄	—	31	2	no	—
218	C₂²⋊C₄	C₄×D₃₁	1	2	yes	—
219	C₄²	—	31	1	no	—
220	C₄²	—	31	1	no	—
221	C₄²	—	31	1	no	—
222	C₄²	C₂²×D₃₁	1	1	yes	characteristic
223	C₂³	—	31	1	no	—
224	C₂³	—	31	2	no	—
225	C₂³	—	62	4	no	—
226	C₂³	—	62	4	no	—
227	C₂³	—	62	4	no	—
228	C₂³	C₂×C₄×D₃₁	1	2	yes	—
229	D₄	—	2	2	no	—
230	D₄	—	62	4	no	—
231	D₄	—	62	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	31	1	no	—
235	C₂×C₄	—	31	1	no	—
236	C₂×C₄	—	31	4	no	—
237	C₂×C₄	—	31	2	no	—
238	C₂×C₄	—	31	2	no	—
239	C₂×C₄	—	31	1	no	—
240	C₂×C₄	—	31	2	no	—
241	C₂×C₄	—	62	4	no	—
242	C₂×C₄	—	62	2	no	—
243	C₂×C₄	—	62	4	no	—
244	C₂×C₄	—	62	4	no	—
245	C₂×C₄	—	62	2	no	—
246	C₂×C₄	—	62	2	no	—
247	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₃₁	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	31	2	no	—
254	C₈	—	62	2	no	—
255	C₈	—	62	1	no	—
256	C₈	D₄×D₃₁	1	2	yes	complemented
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	31	2	no	—
261	C₂²	—	31	2	no	—
262	C₂²	—	31	2	no	—
263	C₂²	—	62	2	no	—
264	C₂²	—	62	2	no	—
265	C₂²	—	62	2	no	—
266	C₂²	—	124	8	no	—
267	C₂²	C₂²×C₄×D₃₁	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	31	2	no	—
271	C₄	—	31	2	no	—
272	C₄	—	62	2	no	—
273	C₄	—	62	1	no	—
274	C₄	M₁₆×D₃₁	1	2	yes	complemented
275	C₄	C₂×D₄×D₃₁	1	1	yes	characteristic
276	C₄	#41×D₃₁	1	1	yes	characteristic
277	C₂	—	4	2	no	—
278	C₂	—	31	4	no	—
279	C₂	—	124	2	no	—
280	C₂	C₄×D₄×D₃₁	1	1	yes	characteristic
281	C₂	C₂×M₁₆×D₃₁	1	1	yes	characteristic
282	C₂	#131×D₃₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₁
commutator, socle	C₂²×C₃₁	C₂³×C₄
Fitting	C₃₁×#436	C₂
prime radical	C₂×D₄×D₃₁	C₄
cosocle	C₂×C₄×C₃₁	C₂⁴
hypercenter	#436	D₃₁
hypocenter	C₃₁	C₂×#436
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₃₁	C₂	semidirect
3. #14521128	C₂	semidirect
4. #14521127	C₂	semidirect
5. #14521126	C₂	semidirect
6. #14521125	C₂	semidirect
7. #14521124	C₂	semidirect
8. C₄⋊C₈×D₃₁	C₂	semidirect
9. C₂²⋊C₈×D₃₁	C₂	semidirect
10. C₄×C₈×D₃₁	C₂	semidirect
11. C₃₁×#436	C₂	semidirect
14. C₂×M₁₆×C₃₁	C₂²	semidirect
18. C₂×#92474	C₂²	semidirect
20. C₂×#92469	C₂²	semidirect
21. C₂×#92469	C₂²	semidirect
31. C₃₁×C₄⋊C₈	C₂²	semidirect
32. C₃₁×C₂²⋊C₈	C₂²	semidirect
33. C₄×C₈×C₃₁	C₂²	semidirect
34. C₂²⋊(C₃₁⋊C₈)	C₂²	semidirect
35. (C₂×D₃₁)⋊C₈	C₂²	semidirect
36. (C₂×D₃₁)⋊C₈	C₂²	semidirect
37. Dic₃₁⋊C₈	C₂²	semidirect
38. C₈×Dic₃₁	C₂²	semidirect
39. C₈×Dic₃₁	C₂²	semidirect
40. C₄⋊(C₃₁⋊C₈)	C₂²	semidirect
41. C₄×C₃₁⋊C₈	C₂²	semidirect
80. C₈×D₃₁	D₄	semidirect
106. C₄×D₃₁	M₁₆	semidirect
111. C₈×C₃₁	C₂×D₄	semidirect
114. C₃₁⋊C₈	C₂×D₄	semidirect
129. C₄×C₃₁	C₂×M₁₆	semidirect
135. Dic₃₁	C₂×M₁₆	semidirect
143. #436	D₃₁	direct (N × H)
152. D₃₁	#436	direct (N × H)
159. C₂×M₁₆	C₂×D₃₁	semidirect
174. C₄⋊C₈	C₂×D₃₁	semidirect
178. C₂²⋊C₈	C₂×D₃₁	semidirect
182. C₄×C₈	C₂×D₃₁	semidirect
183. C₃₁	C₂×#436	semidirect
256. C₈	D₄×D₃₁	semidirect
274. C₄	M₁₆×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₃₁	2
C₄×D₄×D₃₁	2
#14521128	2
#14521127	2
#14521126	2
#14521125	2
#14521124	2
C₄⋊C₈×D₃₁	2
C₂²⋊C₈×D₃₁	2
C₄×C₈×D₃₁	2
C₃₁×#436	2
C₂×#436	31

Subgroups — D31×#436 #16854394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₁×#436 #16854394