Subgroups — D₄×C₃₁×D₁₆ #21490604

Order 7936.

812 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	C₂²×C₃₁×D₁₆	C₂	1	2	yes	maximal, complemented
2	D₄×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2215	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2206	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2184	C₂	1	4	yes	maximal, complemented
7	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₃₁	C₂²	1	4	yes	complemented
10	D₄²×C₃₁	—	2	2	no	—
11	C₂×C₃₁×D₁₆	—	2	8	no	—
12	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×D₁₆	C₂²	1	4	yes	complemented
14	C₂×C₃₁×D₁₆	C₂²	1	2	yes	complemented
15	C₂×C₃₁×D₁₆	C₂²	1	2	yes	complemented
16	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	complemented
17	C₃₁×#493	C₂²	1	2	yes	complemented
18	C₃₁×#466	C₂²	1	4	yes	complemented
19	C₃₁×#459	—	2	2	no	—
20	C₃₁×#447	—	2	4	no	—
21	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
22	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#366	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
25	C₃₁×#357	C₂²	1	2	yes	complemented
26	C₃₁×#357	C₂²	1	4	yes	complemented
27	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
28	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
29	C₂²×D₄×C₃₁	—	2	4	no	—
30	C₂²×D₄×C₃₁	—	4	2	no	—
31	C₂×D₈×C₃₁	—	2	8	no	—
32	C₂×D₈×C₃₁	—	2	4	no	—
33	C₂×D₈×C₃₁	—	2	2	no	—
34	C₂×D₈×C₃₁	—	4	4	no	—
35	C₂×D₈×C₃₁	C₂³	1	4	yes	—
36	C₂×D₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₃₁×#127	—	2	2	no	—
39	C₃₁×#121	—	2	4	no	—
40	C₃₁×#121	—	4	2	no	—
41	C₃₁×#120	—	4	4	no	—
42	C₄×D₄×C₃₁	—	2	2	no	—
43	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
44	C₃₁×D₁₆	—	2	8	no	—
45	C₃₁×D₁₆	—	2	4	no	—
46	C₃₁×D₁₆	—	4	2	no	—
47	C₃₁×D₁₆	D₄	1	4	yes	direct product factor
48	C₂×C₁₆×C₃₁	—	2	4	no	—
49	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#107	C₂³	1	2	yes	—
53	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₃₁×#102	—	2	2	no	—
55	C₃₁×#102	—	2	4	no	—
56	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
57	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂⁴×C₃₁	—	4	4	no	—
59	C₂×D₄×C₃₁	—	2	2	no	—
60	C₂×D₄×C₃₁	—	2	4	no	—
61	C₂×D₄×C₃₁	—	4	8	no	—
62	C₂×D₄×C₃₁	—	4	4	no	—
63	C₂×D₄×C₃₁	—	4	4	no	—
64	C₂×D₄×C₃₁	—	4	2	no	—
65	C₂×D₄×C₃₁	—	4	2	no	—
66	C₂×D₄×C₃₁	—	8	4	no	—
67	C₂×D₄×C₃₁	—	8	8	no	—
68	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	—	4	2	no	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
71	D₈×C₃₁	—	2	4	no	—
72	D₈×C₃₁	—	2	2	no	—
73	D₈×C₃₁	—	2	4	no	—
74	D₈×C₃₁	—	4	8	no	—
75	D₈×C₃₁	—	8	2	no	—
76	D₈×C₃₁	C₂×D₄	1	4	yes	complemented
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
84	C₂²⋊C₄×C₃₁	—	4	2	no	—
85	C₂²⋊C₄×C₃₁	—	4	4	no	—
86	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
87	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
88	C₁₆×C₃₁	—	2	2	no	—
89	C₁₆×C₃₁	—	2	1	no	—
90	C₁₆×C₃₁	C₂×D₄	1	2	yes	complemented
91	D₄×D₁₆	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	4	2	no	—
93	C₂³×C₃₁	—	4	4	no	—
94	C₂³×C₃₁	—	8	8	no	—
95	C₂³×C₃₁	—	8	8	no	—
96	C₂³×C₃₁	—	16	4	no	—
97	C₂³×C₃₁	C₂×D₈	1	2	yes	—
98	D₄×C₃₁	—	2	4	no	—
99	D₄×C₃₁	—	4	4	no	—
100	D₄×C₃₁	—	4	2	no	—
101	D₄×C₃₁	—	4	4	no	—
102	D₄×C₃₁	—	8	8	no	—
103	D₄×C₃₁	—	8	8	no	—
104	D₄×C₃₁	—	8	4	no	—
105	D₄×C₃₁	—	16	2	no	—
106	D₄×C₃₁	D₁₆	1	4	yes	direct product factor
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	8	4	no	—
111	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₈×C₃₁	—	2	2	no	—
116	C₈×C₃₁	—	2	1	no	—
117	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
118	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²×D₁₆	C₂×C₃₁	1	2	yes	complemented
120	D₄×D₈	C₂×C₃₁	1	2	yes	complemented
121	#2246	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#2215	C₂×C₃₁	1	2	yes	complemented
123	#2206	C₂×C₃₁	1	2	yes	complemented
124	#2184	C₂×C₃₁	1	4	yes	complemented
125	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
126	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	4	4	no	—
130	C₂²×C₃₁	—	8	2	no	—
131	C₂²×C₃₁	—	8	2	no	—
132	C₂²×C₃₁	—	8	4	no	—
133	C₂²×C₃₁	—	8	4	no	—
134	C₂²×C₃₁	—	8	4	no	—
135	C₂²×C₃₁	—	16	16	no	—
136	C₂²×C₃₁	C₂×D₁₆	1	4	yes	complemented
137	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
138	C₄×C₃₁	—	2	1	no	—
139	C₄×C₃₁	—	2	2	no	—
140	C₄×C₃₁	—	8	2	no	—
141	C₄×C₃₁	C₂×D₁₆	1	2	yes	complemented
142	C₄×C₃₁	D₄²	1	1	yes	characteristic
143	C₄×C₃₁	#576	1	1	yes	characteristic
144	C₂²×D₈	C₂²×C₃₁	1	4	yes	complemented
145	D₄²	—	2	2	no	—
146	C₂×D₁₆	—	2	8	no	—
147	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×D₁₆	C₂²×C₃₁	1	4	yes	complemented
149	C₂×D₁₆	C₂²×C₃₁	1	2	yes	complemented
150	C₂×D₁₆	C₂²×C₃₁	1	2	yes	complemented
151	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	complemented
152	#493	C₂²×C₃₁	1	2	yes	complemented
153	#466	C₂²×C₃₁	1	4	yes	complemented
154	#459	—	2	2	no	—
155	#447	—	2	4	no	—
156	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
157	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
158	#366	C₂²×C₃₁	1	1	yes	characteristic, complemented
159	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
160	#357	C₂²×C₃₁	1	2	yes	complemented
161	#357	C₂²×C₃₁	1	4	yes	complemented
162	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
163	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
164	C₂×C₃₁	—	2	4	no	—
165	C₂×C₃₁	—	8	4	no	—
166	C₂×C₃₁	—	16	4	no	—
167	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
168	C₂×C₃₁	C₂²×D₁₆	1	1	yes	characteristic
169	C₂×C₃₁	#3415	1	1	yes	characteristic
170	C₂²×D₄	—	2	4	no	—
171	C₂²×D₄	—	4	2	no	—
172	C₂×D₈	—	2	8	no	—
173	C₂×D₈	—	2	4	no	—
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	—	4	4	no	—
176	C₂×D₈	C₂³×C₃₁	1	4	yes	—
177	C₂×D₈	C₂³×C₃₁	1	2	yes	—
178	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
179	#127	—	2	2	no	—
180	#121	—	2	4	no	—
181	#121	—	4	2	no	—
182	#120	—	4	4	no	—
183	C₄×D₄	—	2	2	no	—
184	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
185	D₁₆	—	2	8	no	—
186	D₁₆	—	2	4	no	—
187	D₁₆	—	4	2	no	—
188	D₁₆	D₄×C₃₁	1	4	yes	direct product factor
189	C₂×C₁₆	—	2	4	no	—
190	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
191	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
192	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
193	#107	C₂³×C₃₁	1	2	yes	—
194	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
195	#102	—	2	2	no	—
196	#102	—	2	4	no	—
197	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
198	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
199	C₃₁	D₄×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
200	C₂⁴	—	4	4	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	4	8	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	—	4	4	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂×D₄	—	8	4	no	—
209	C₂×D₄	—	8	8	no	—
210	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
213	D₈	—	2	4	no	—
214	D₈	—	2	2	no	—
215	D₈	—	2	4	no	—
216	D₈	—	4	8	no	—
217	D₈	—	8	2	no	—
218	D₈	C₂×D₄×C₃₁	1	4	yes	complemented
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
223	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
226	C₂²⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	4	no	—
228	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
229	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₁₆	—	2	2	no	—
231	C₁₆	—	2	1	no	—
232	C₁₆	C₂×D₄×C₃₁	1	2	yes	complemented
233	C₂³	—	4	2	no	—
234	C₂³	—	4	4	no	—
235	C₂³	—	8	8	no	—
236	C₂³	—	8	8	no	—
237	C₂³	—	16	4	no	—
238	C₂³	C₂×D₈×C₃₁	1	2	yes	—
239	D₄	—	2	4	no	—
240	D₄	—	4	4	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	8	8	no	—
244	D₄	—	8	8	no	—
245	D₄	—	8	4	no	—
246	D₄	—	16	2	no	—
247	D₄	C₃₁×D₁₆	1	4	yes	direct product factor
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
253	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
254	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
255	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	2	1	no	—
258	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
259	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	4	4	no	—
263	C₂²	—	8	2	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	4	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	16	16	no	—
269	C₂²	C₂×C₃₁×D₁₆	1	4	yes	complemented
270	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	2	no	—
273	C₄	—	8	2	no	—
274	C₄	C₂×C₃₁×D₁₆	1	2	yes	complemented
275	C₄	D₄²×C₃₁	1	1	yes	characteristic
276	C₄	C₃₁×#576	1	1	yes	characteristic
277	C₂	—	2	4	no	—
278	C₂	—	8	4	no	—
279	C₂	—	16	4	no	—
280	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
281	C₂	C₂²×C₃₁×D₁₆	1	1	yes	characteristic
282	C₂	C₃₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×D₁₆	C₂	semidirect
2. D₄×D₈×C₃₁	C₂	semidirect
3. C₃₁×#2246	C₂	semidirect
4. C₃₁×#2215	C₂	semidirect
5. C₃₁×#2206	C₂	semidirect
6. C₃₁×#2184	C₂	semidirect
7. C₄×C₃₁×D₁₆	C₂	semidirect
8. D₄×C₁₆×C₃₁	C₂	semidirect
9. C₂²×D₈×C₃₁	C₂²	semidirect
13. C₂×C₃₁×D₁₆	C₂²	semidirect
14. C₂×C₃₁×D₁₆	C₂²	semidirect
15. C₂×C₃₁×D₁₆	C₂²	semidirect
16. C₂²×C₁₆×C₃₁	C₂²	semidirect
17. C₃₁×#493	C₂²	semidirect
18. C₃₁×#466	C₂²	semidirect
21. C₄×D₈×C₃₁	C₂²	semidirect
23. C₃₁×#366	C₂²	semidirect
24. C₃₁×C₄⋊C₁₆	C₂²	semidirect
25. C₃₁×#357	C₂²	semidirect
26. C₃₁×#357	C₂²	semidirect
27. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
28. C₄×C₁₆×C₃₁	C₂²	semidirect
47. C₃₁×D₁₆	D₄	direct (N × H)
76. D₈×C₃₁	C₂×D₄	semidirect
90. C₁₆×C₃₁	C₂×D₄	semidirect
91. D₄×D₁₆	C₃₁	direct (N × H)
106. D₄×C₃₁	D₁₆	direct (N × H)
119. C₂²×D₁₆	C₂×C₃₁	semidirect
120. D₄×D₈	C₂×C₃₁	semidirect
121. #2246	C₂×C₃₁	semidirect
122. #2215	C₂×C₃₁	semidirect
123. #2206	C₂×C₃₁	semidirect
124. #2184	C₂×C₃₁	semidirect
125. C₄×D₁₆	C₂×C₃₁	semidirect
126. D₄×C₁₆	C₂×C₃₁	semidirect
136. C₂²×C₃₁	C₂×D₁₆	semidirect
141. C₄×C₃₁	C₂×D₁₆	semidirect
144. C₂²×D₈	C₂²×C₃₁	semidirect
148. C₂×D₁₆	C₂²×C₃₁	semidirect
149. C₂×D₁₆	C₂²×C₃₁	semidirect
150. C₂×D₁₆	C₂²×C₃₁	semidirect
151. C₂²×C₁₆	C₂²×C₃₁	semidirect
152. #493	C₂²×C₃₁	semidirect
153. #466	C₂²×C₃₁	semidirect
156. C₄×D₈	C₂²×C₃₁	semidirect
158. #366	C₂²×C₃₁	semidirect
159. C₄⋊C₁₆	C₂²×C₃₁	semidirect
160. #357	C₂²×C₃₁	semidirect
161. #357	C₂²×C₃₁	semidirect
162. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
163. C₄×C₁₆	C₂²×C₃₁	semidirect
188. D₁₆	D₄×C₃₁	direct (N × H)
199. C₃₁	D₄×D₁₆	direct (N × H)
218. D₈	C₂×D₄×C₃₁	semidirect
232. C₁₆	C₂×D₄×C₃₁	semidirect
247. D₄	C₃₁×D₁₆	direct (N × H)
269. C₂²	C₂×C₃₁×D₁₆	semidirect
274. C₄	C₂×C₃₁×D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×D₁₆	2
D₄×D₈×C₃₁	2
C₃₁×#2246	2
C₃₁×#2215	2
C₃₁×#2206	2
C₃₁×#2184	2
C₄×C₃₁×D₁₆	2
D₄×C₁₆×C₃₁	2
D₄×D₁₆	31

Subgroups — D4×C31×D16 #21490604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₃₁×D₁₆ #21490604