Subgroups — M₁₆×D₂₁₂ #20283453

Order 6784.

544 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₅₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₂₁₂	C₂	1	1	yes	maximal, characteristic
3	#16810702	C₂	1	2	yes	maximal, complemented
4	#16810696	C₂	1	4	yes	maximal, complemented
5	C₈×D₂₁₂	C₂	1	2	yes	maximal, complemented
6	#16810647	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810586	C₂	1	2	yes	maximal, complemented
8	C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₂₁₂	C₄	1	1	yes	characteristic
10	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
11	C₂×#11719917	C₂²	1	2	yes	—
12	C₂×#11719912	C₂²	1	4	yes	complemented
13	M₁₆×D₅₃	—	2	8	no	—
14	C₂×C₈×D₅₃	C₂²	1	4	yes	complemented
15	C₄×D₂₁₂	C₄	1	1	yes	characteristic
16	C₄×D₂₁₂	C₄	1	2	yes	—
17	C₄×D₂₁₂	C₄	1	1	yes	characteristic
18	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
19	C₄×D₂₁₂	C₂²	1	2	yes	—
20	C₄×D₂₁₂	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
22	C₂×#11719921	C₄	1	2	yes	—
23	C₄⋊#88322	C₂²	1	2	yes	complemented
24	C₄⋊#88322	C₂²	1	2	yes	complemented
25	#13133129	C₂²	1	4	yes	complemented
26	#13133129	C₂²	1	4	yes	complemented
27	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
30	#97×C₅₃	C₂²	1	2	yes	complemented
31	C₄×C₈×C₅₃	C₂²	1	2	yes	complemented
32	C₂³×D₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
37	C₂×D₂₁₂	—	2	4	no	—
38	C₂×D₂₁₂	C₂×C₄	1	2	yes	—
39	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
40	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₅₃	—	2	4	no	—
42	C₂×C₄×D₅₃	—	2	4	no	—
43	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₄×D₅₃	C₂³	1	2	yes	—
46	C₂×C₄×D₅₃	C₂³	1	2	yes	—
47	M₁₆×C₅₃	—	2	2	no	—
48	M₁₆×C₅₃	D₄	1	4	yes	complemented
49	C₂×C₈×C₅₃	C₂³	1	2	yes	—
50	C₂×C₈×C₅₃	C₂³	1	2	yes	—
51	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₅₃	C₂³	1	1	yes	characteristic
54	C₄²×C₅₃	C₂³	1	1	yes	characteristic
55	#11719921	C₂×C₄	1	2	yes	—
56	#11719921	C₂×C₄	1	2	yes	—
57	#11719921	C₂×C₄	1	2	yes	—
58	#11719921	C₂×C₄	1	2	yes	—
59	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
62	#11719917	—	2	4	no	—
63	C₂×#88322	C₂³	1	4	yes	—
64	#11719912	—	2	8	no	—
65	C₈×D₅₃	—	2	8	no	—
66	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×D₅₃	—	2	4	no	—
68	C₂²×D₅₃	—	2	4	no	—
69	C₂²×D₅₃	—	4	2	no	—
70	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
71	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	4	1	no	—
74	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₅₃	#41	1	2	yes	—
81	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₅₃	—	2	4	no	—
83	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
84	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
85	D₂₁₂	—	2	4	no	—
86	D₂₁₂	—	2	2	no	—
87	D₂₁₂	M₁₆	1	4	yes	direct product factor
88	C₄×D₅₃	—	2	4	no	—
89	C₄×D₅₃	—	2	4	no	—
90	C₄×D₅₃	—	2	2	no	—
91	C₄×D₅₃	—	2	2	no	—
92	C₄×D₅₃	—	2	2	no	—
93	C₄×D₅₃	—	2	2	no	—
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	C₂×D₄	1	4	yes	complemented
96	#88322	—	2	4	no	—
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	C₄×D₄	1	2	yes	—
100	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₅₃	—	2	2	no	—
102	C₂×D₅₃	—	2	2	no	—
103	C₂×D₅₃	—	2	2	no	—
104	C₂×D₅₃	—	2	2	no	—
105	C₂×D₅₃	—	2	2	no	—
106	C₂×D₅₃	—	4	8	no	—
107	C₂×D₅₃	C₂×M₁₆	1	4	yes	complemented
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	C₄×D₄	1	2	yes	—
112	C₄×C₅₃	C₂×M₁₆	1	2	yes	complemented
113	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
115	Dic₅₃	—	2	2	no	—
116	Dic₅₃	—	2	2	no	—
117	D₄×M₁₆	—	53	1	no	maximal, Sylow
118	C₂×C₅₃	—	2	2	no	—
119	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
121	C₂×C₅₃	#568	1	1	yes	characteristic
122	D₅₃	—	2	4	no	—
123	D₅₃	—	4	2	no	—
124	C₂²×M₁₆	—	53	2	no	—
125	C₂×C₄×D₄	—	53	1	no	—
126	#436	—	53	2	no	—
127	#435	—	53	4	no	—
128	C₈×D₄	—	53	2	no	—
129	#423	—	53	1	no	—
130	#407	—	53	2	no	—
131	C₄×M₁₆	D₅₃	1	1	yes	characteristic, complemented
132	C₅₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
133	C₂²×D₄	—	53	1	no	—
134	C₂³×C₄	—	53	2	no	—
135	C₂×M₁₆	—	53	2	no	—
136	C₂×M₁₆	—	53	4	no	—
137	C₂×M₁₆	—	106	8	no	—
138	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
139	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
140	C₂²×C₈	—	53	4	no	—
141	C₄×D₄	—	53	1	no	—
142	C₄×D₄	—	53	2	no	—
143	C₄×D₄	—	53	1	no	—
144	C₄×D₄	—	53	1	no	—
145	C₄×D₄	—	53	2	no	—
146	C₄×D₄	—	53	1	no	—
147	C₂×C₄⋊C₄	—	53	1	no	—
148	C₂×C₂²⋊C₄	—	53	2	no	—
149	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
150	C₄⋊C₈	—	53	2	no	—
151	C₄⋊C₈	—	53	2	no	—
152	C₂²⋊C₈	—	53	4	no	—
153	C₂²⋊C₈	—	53	4	no	—
154	#97	C₂×D₅₃	1	2	yes	complemented
155	C₄×C₈	C₂×D₅₃	1	2	yes	complemented
156	C₂⁴	—	53	2	no	—
157	C₂×D₄	—	53	1	no	—
158	C₂×D₄	—	53	2	no	—
159	C₂×D₄	—	53	1	no	—
160	C₂×D₄	—	106	4	no	—
161	C₂²×C₄	—	53	2	no	—
162	C₂²×C₄	—	53	2	no	—
163	C₂²×C₄	—	53	2	no	—
164	C₂²×C₄	—	53	2	no	—
165	C₂²×C₄	—	53	2	no	—
166	C₂²×C₄	—	106	4	no	—
167	C₂²×C₄	—	106	4	no	—
168	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
171	M₁₆	—	2	2	no	—
172	M₁₆	—	106	4	no	—
173	M₁₆	—	106	8	no	—
174	M₁₆	D₂₁₂	1	4	yes	direct product factor
175	C₂×C₈	—	53	4	no	—
176	C₂×C₈	—	106	8	no	—
177	C₂×C₈	C₂²×D₅₃	1	2	yes	—
178	C₂×C₈	C₂²×D₅₃	1	2	yes	—
179	C₄⋊C₄	—	53	2	no	—
180	C₄⋊C₄	—	53	1	no	—
181	C₄⋊C₄	—	53	1	no	—
182	C₂²⋊C₄	—	53	2	no	—
183	C₂²⋊C₄	—	53	2	no	—
184	C₂²⋊C₄	—	53	2	no	—
185	C₂²⋊C₄	—	53	2	no	—
186	C₄²	C₄×D₅₃	1	1	yes	characteristic
187	C₄²	C₄×D₅₃	1	1	yes	characteristic
188	C₄²	C₂²×D₅₃	1	1	yes	characteristic
189	C₄²	C₂²×D₅₃	1	1	yes	characteristic
190	C₂³	—	53	2	no	—
191	C₂³	—	53	2	no	—
192	C₂³	—	106	4	no	—
193	C₂³	—	106	4	no	—
194	C₂³	—	212	2	no	—
195	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
196	D₄	—	53	4	no	—
197	D₄	—	106	4	no	—
198	D₄	—	106	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	53	2	no	—
202	C₂×C₄	—	53	2	no	—
203	C₂×C₄	—	106	4	no	—
204	C₂×C₄	—	106	4	no	—
205	C₂×C₄	—	106	4	no	—
206	C₂×C₄	—	106	2	no	—
207	C₂×C₄	—	106	2	no	—
208	C₂×C₄	—	106	2	no	—
209	C₂×C₄	—	106	2	no	—
210	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
215	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
216	C₂×C₄	#11719945	1	2	yes	—
217	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	106	4	no	—
220	C₈	C₂×D₂₁₂	1	4	yes	complemented
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	53	4	no	—
224	C₂²	—	106	2	no	—
225	C₂²	—	106	2	no	—
226	C₂²	—	106	2	no	—
227	C₂²	—	106	2	no	—
228	C₂²	—	106	2	no	—
229	C₂²	—	212	8	no	—
230	C₂²	C₄×D₂₁₂	1	2	yes	—
231	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	106	2	no	—
236	C₄	—	106	2	no	—
237	C₄	C₄×D₂₁₂	1	2	yes	—
238	C₄	M₁₆×D₅₃	1	2	yes	complemented
239	C₄	C₂²×D₂₁₂	1	1	yes	characteristic
240	C₄	C₂×#11719945	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	—	106	4	no	—
243	C₂	—	212	2	no	—
244	C₂	C₂×C₄×D₂₁₂	1	1	yes	characteristic
245	C₂	C₂×M₁₆×D₅₃	1	1	yes	characteristic
246	C₂	#16811389	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₄×M₁₆×C₅₃	C₂
hypercenter	C₄×M₁₆	D₅₃
hypocenter	C₅₃	D₄×M₁₆
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₅₃	C₂	semidirect
3. #16810702	C₂	semidirect
4. #16810696	C₂	semidirect
5. C₈×D₂₁₂	C₂	semidirect
6. #16810647	C₂	semidirect
7. #16810586	C₂	semidirect
8. C₄×M₁₆×C₅₃	C₂	semidirect
12. C₂×#11719912	C₂²	semidirect
14. C₂×C₈×D₅₃	C₂²	semidirect
23. C₄⋊#88322	C₂²	semidirect
24. C₄⋊#88322	C₂²	semidirect
25. #13133129	C₂²	semidirect
26. #13133129	C₂²	semidirect
30. #97×C₅₃	C₂²	semidirect
31. C₄×C₈×C₅₃	C₂²	semidirect
48. M₁₆×C₅₃	D₄	semidirect
87. D₂₁₂	M₁₆	direct (N × H)
95. C₈×C₅₃	C₂×D₄	semidirect
107. C₂×D₅₃	C₂×M₁₆	semidirect
112. C₄×C₅₃	C₂×M₁₆	semidirect
131. C₄×M₁₆	D₅₃	semidirect
132. C₅₃	D₄×M₁₆	semidirect
154. #97	C₂×D₅₃	semidirect
155. C₄×C₈	C₂×D₅₃	semidirect
174. M₁₆	D₂₁₂	direct (N × H)
220. C₈	C₂×D₂₁₂	semidirect
238. C₄	M₁₆×D₅₃	semidirect

Subgroups — M16×D212 #20283453

Conjugacy classes of subgroups

Complements

Subgroups — M₁₆×D₂₁₂ #20283453