Subgroups — C₂×C₄²×D₂₁₂ #27542285

Order 13568.

5694 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	C₂²×C₄×D₂₁₂	C₂	1	3	yes	maximal
2	C₂²×C₄²×D₅₃	C₂	1	2	yes	maximal, complemented
3	C₄²×D₂₁₂	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄×C₄⋊Dic₅₃	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×#11719921	C₂	1	6	yes	maximal, complemented
6	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₂₁₂	C₂²	1	1	yes	characteristic
8	C₂³×C₄×D₅₃	C₂²	1	6	yes	—
9	C₂×C₄×D₂₁₂	C₄	1	48	yes	direct product factor
10	C₂×C₄×D₂₁₂	C₂²	1	24	yes	—
11	C₂²×C₄⋊Dic₅₃	C₂²	1	3	yes	—
12	C₂²×#11719921	C₂²	1	6	yes	—
13	C₂²×C₄×Dic₅₃	C₂²	1	6	yes	—
14	C₂×C₄²×D₅₃	—	2	16	no	—
15	C₂×C₄²×D₅₃	C₂²	1	16	yes	complemented
16	C₄×C₄⋊Dic₅₃	C₂²	1	24	yes	complemented
17	C₄×#11719921	C₂²	1	48	yes	complemented
18	C₂×#13133124	C₂²	1	6	yes	—
19	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
20	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
21	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₄³×C₅₃	C₂²	1	8	yes	complemented
23	C₂⁴×D₅₃	C₂³	1	2	yes	—
24	C₂²×D₂₁₂	C₂×C₄	1	24	yes	—
25	C₂²×D₂₁₂	C₂³	1	4	yes	—
26	C₂³×Dic₅₃	C₂³	1	6	yes	—
27	C₂²×C₄×D₅₃	—	2	24	no	—
28	C₂²×C₄×D₅₃	—	2	24	no	—
29	C₂²×C₄×D₅₃	C₂×C₄	1	48	yes	complemented
30	C₂²×C₄×D₅₃	C₂³	1	24	yes	—
31	C₄×D₂₁₂	C₂×C₄	1	192	yes	direct product factor
32	C₂×C₄⋊Dic₅₃	C₂×C₄	1	24	yes	complemented
33	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
34	C₂×#11719921	C₂×C₄	1	48	yes	complemented
35	C₂×#11719921	C₂³	1	24	yes	—
36	C₂×C₄×Dic₅₃	—	2	24	no	—
37	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
38	C₄²×D₅₃	—	2	64	no	—
39	#13133124	C₂³	1	24	yes	—
40	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
41	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
43	C₂×C₄²×C₅₃	—	2	12	no	—
44	C₂×C₄²×C₅₃	C₂×C₄	1	24	yes	complemented
45	C₂×C₄²×C₅₃	D₄	1	8	yes	complemented
46	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
47	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
48	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
49	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂³×D₅₃	—	2	4	no	—
51	C₂³×D₅₃	—	2	12	no	—
52	C₂³×D₅₃	C₂²×C₄	1	24	yes	—
53	C₂³×D₅₃	C₂⁴	1	4	yes	—
54	C₂²×C₄×C₅₃	—	2	12	no	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
56	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
59	C₂²×C₄×C₅₃	#41	1	12	yes	—
60	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
63	C₂²×Dic₅₃	—	2	12	no	—
64	C₂²×Dic₅₃	—	2	12	no	—
65	C₂²×Dic₅₃	C₂²×C₄	1	24	yes	—
66	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
67	C₂×D₂₁₂	C₄²	1	64	yes	direct product factor
68	C₂×D₂₁₂	C₂²×C₄	1	48	yes	—
69	C₂×C₄×D₅₃	—	2	192	no	—
70	C₂×C₄×D₅₃	—	2	48	no	—
71	C₂×C₄×D₅₃	—	2	48	no	—
72	C₂×C₄×D₅₃	C₂²×C₄	1	96	yes	complemented
73	C₄²×C₅₃	—	2	24	no	—
74	C₄²×C₅₃	C₂²×C₄	1	48	yes	complemented
75	C₄²×C₅₃	C₂×D₄	1	16	yes	complemented
76	#11719921	C₂²×C₄	1	96	yes	complemented
77	C₄⋊Dic₅₃	C₂²×C₄	1	48	yes	complemented
78	C₄×Dic₅₃	—	2	48	no	—
79	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
80	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
81	C₂³×C₅₃	C₂×#41	1	6	yes	—
82	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
83	C₂²×D₅₃	—	2	48	no	—
84	C₂²×D₅₃	—	2	48	no	—
85	C₂²×D₅₃	—	2	4	no	—
86	C₂²×D₅₃	—	2	12	no	—
87	C₂²×D₅₃	C₂×C₄²	1	32	yes	complemented
88	C₂²×D₅₃	C₂³×C₄	1	24	yes	—
89	C₂×C₄×C₅₃	—	2	24	no	—
90	C₂×C₄×C₅₃	—	2	12	no	—
91	C₂×C₄×C₅₃	C₂×C₄²	1	16	yes	complemented
92	C₂×C₄×C₅₃	C₄×D₄	1	48	yes	complemented
93	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
94	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
95	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
96	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
97	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
98	C₂×Dic₅₃	—	2	48	no	—
99	C₂×Dic₅₃	—	2	12	no	—
100	C₂×Dic₅₃	—	2	12	no	—
101	C₂×Dic₅₃	C₂³×C₄	1	24	yes	—
102	D₂₁₂	C₂×C₄²	1	64	yes	direct product factor
103	C₄×D₅₃	—	2	192	no	—
104	C₂×C₄²×D₄	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	C₂²×C₄²	1	4	yes	—
106	C₂²×C₅₃	C₂×C₄×D₄	1	12	yes	—
107	C₂²×C₅₃	C₄×#41	1	12	yes	—
108	C₂²×C₅₃	C₂⁴×C₄	1	3	yes	—
109	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
110	C₂²×C₅₃	C₂²×#41	1	3	yes	—
111	C₂×D₅₃	—	2	24	no	—
112	C₂×D₅₃	—	2	24	no	—
113	C₂×D₅₃	—	2	64	no	—
114	C₂×D₅₃	C₂²×C₄²	1	16	yes	complemented
115	C₄×C₅₃	—	2	12	no	—
116	C₄×C₅₃	C₂²×C₄²	1	8	yes	complemented
117	C₄×C₅₃	C₂×C₄×D₄	1	24	yes	complemented
118	Dic₅₃	—	2	24	no	—
119	C₂²×C₄×D₄	—	53	3	no	—
120	C₂³×C₄²	—	53	2	no	—
121	C₄²×D₄	—	53	16	no	—
122	C₂×C₄×C₄⋊C₄	—	53	3	no	—
123	C₂×C₄×C₂²⋊C₄	—	53	6	no	—
124	C₂×C₄³	D₅₃	1	1	yes	characteristic, complemented
125	C₂×C₅₃	C₄²×D₄	1	8	yes	complemented
126	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
127	C₂×C₅₃	C₂²×C₄×D₄	1	3	yes	—
128	C₂×C₅₃	C₂×C₄×#41	1	3	yes	—
129	D₅₃	—	2	16	no	—
130	C₂³×D₄	—	53	1	no	—
131	C₂⁴×C₄	—	53	6	no	—
132	C₂×C₄×D₄	—	53	48	no	—
133	C₂×C₄×D₄	—	53	24	no	—
134	C₂²×C₄⋊C₄	—	53	3	no	—
135	C₂²×C₂²⋊C₄	—	53	6	no	—
136	C₂²×C₄²	—	53	6	no	—
137	C₂²×C₄²	—	53	16	no	—
138	C₂²×C₄²	—	106	16	no	—
139	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
140	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
141	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
142	C₄×C₄⋊C₄	—	53	24	no	—
143	C₄×C₂²⋊C₄	—	53	48	no	—
144	C₂×#95	—	53	6	no	—
145	C₄³	C₂×D₅₃	1	8	yes	complemented
146	C₅₃	C₂×C₄²×D₄	1	1	yes	characteristic, complemented, Sylow
147	C₂⁵	—	53	2	no	—
148	C₂²×D₄	—	53	24	no	—
149	C₂²×D₄	—	53	4	no	—
150	C₂³×C₄	—	53	6	no	—
151	C₂³×C₄	—	53	48	no	—
152	C₂³×C₄	—	53	24	no	—
153	C₂³×C₄	—	106	24	no	—
154	C₂³×C₄	—	106	24	no	—
155	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
156	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
157	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
158	C₄×D₄	—	53	192	no	—
159	C₂×C₄⋊C₄	—	53	24	no	—
160	C₂×C₄⋊C₄	—	53	12	no	—
161	C₂×C₂²⋊C₄	—	53	48	no	—
162	C₂×C₂²⋊C₄	—	53	24	no	—
163	C₂×C₄²	—	2	12	no	—
164	C₂×C₄²	—	53	24	no	—
165	C₂×C₄²	—	106	24	no	—
166	C₂×C₄²	—	106	64	no	—
167	C₂×C₄²	C₄×D₅₃	1	24	yes	complemented
168	C₂×C₄²	D₂₁₂	1	8	yes	direct product factor
169	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
170	C₂×C₄²	C₂²×D₅₃	1	12	yes	—
171	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
172	#95	—	53	24	no	—
173	C₂⁴	—	53	24	no	—
174	C₂⁴	—	53	4	no	—
175	C₂⁴	—	106	4	no	—
176	C₂⁴	—	106	12	no	—
177	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
178	C₂×D₄	—	53	64	no	—
179	C₂×D₄	—	53	48	no	—
180	C₂²×C₄	—	2	12	no	—
181	C₂²×C₄	—	53	24	no	—
182	C₂²×C₄	—	53	12	no	—
183	C₂²×C₄	—	53	96	no	—
184	C₂²×C₄	—	106	12	no	—
185	C₂²×C₄	—	106	12	no	—
186	C₂²×C₄	—	106	192	no	—
187	C₂²×C₄	—	106	48	no	—
188	C₂²×C₄	—	106	48	no	—
189	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
190	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
191	C₂²×C₄	C₂×C₄×D₅₃	1	12	yes	—
192	C₂²×C₄	C₂×D₂₁₂	1	12	yes	—
193	C₂²×C₄	#11719945	1	12	yes	—
194	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
195	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
196	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
197	C₄⋊C₄	—	53	48	no	—
198	C₂²⋊C₄	—	53	96	no	—
199	C₄²	—	2	24	no	—
200	C₄²	—	106	48	no	—
201	C₄²	C₂×C₄×D₅₃	1	48	yes	complemented
202	C₄²	C₂×D₂₁₂	1	16	yes	direct product factor
203	C₂³	—	53	32	no	—
204	C₂³	—	53	24	no	—
205	C₂³	—	106	48	no	—
206	C₂³	—	106	48	no	—
207	C₂³	—	106	4	no	—
208	C₂³	—	106	12	no	—
209	C₂³	C₂²×C₄×D₅₃	1	6	yes	—
210	C₂³	C₂²×D₂₁₂	1	2	yes	—
211	C₂³	C₂×#11719945	1	6	yes	—
212	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
213	D₄	—	53	64	no	—
214	C₂×C₄	—	2	24	no	—
215	C₂×C₄	—	2	12	no	—
216	C₂×C₄	—	53	24	no	—
217	C₂×C₄	—	106	48	no	—
218	C₂×C₄	—	106	12	no	—
219	C₂×C₄	—	106	12	no	—
220	C₂×C₄	—	106	192	no	—
221	C₂×C₄	C₄²×D₅₃	1	16	yes	complemented
222	C₂×C₄	C₄×D₂₁₂	1	48	yes	direct product factor
223	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
224	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
225	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
226	C₂×C₄	C₂²×D₂₁₂	1	12	yes	—
227	C₂×C₄	C₂×#11719945	1	12	yes	—
228	C₂²	—	53	16	no	—
229	C₂²	—	106	64	no	—
230	C₂²	—	106	24	no	—
231	C₂²	—	106	24	no	—
232	C₂²	C₂×C₄²×D₅₃	1	4	yes	—
233	C₂²	C₂×C₄×D₂₁₂	1	12	yes	—
234	C₂²	C₄×#11719945	1	12	yes	—
235	C₂²	C₂³×C₄×D₅₃	1	3	yes	—
236	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
237	C₂²	C₂²×#11719945	1	3	yes	—
238	C₄	—	2	12	no	—
239	C₄	—	106	24	no	—
240	C₄	C₂×C₄²×D₅₃	1	8	yes	complemented
241	C₄	C₂×C₄×D₂₁₂	1	24	yes	direct product factor
242	C₂	—	106	16	no	—
243	C₂	C₄²×D₂₁₂	1	8	yes	direct product factor
244	C₂	C₂²×C₄²×D₅₃	1	1	yes	characteristic
245	C₂	C₂²×C₄×D₂₁₂	1	3	yes	—
246	C₂	C₂×C₄×#11719945	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₄³×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂³×D₂₁₂	C₂²
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	C₂×C₄²×D₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄²×D₅₃	C₂	semidirect
3. C₄²×D₂₁₂	C₂	direct (N × H)
4. C₂×C₄×C₄⋊Dic₅₃	C₂	semidirect
5. C₂×C₄×#11719921	C₂	semidirect
6. C₂×C₄³×C₅₃	C₂	semidirect
9. C₂×C₄×D₂₁₂	C₄	direct (N × H)
15. C₂×C₄²×D₅₃	C₂²	semidirect
16. C₄×C₄⋊Dic₅₃	C₂²	semidirect
17. C₄×#11719921	C₂²	semidirect
22. C₄³×C₅₃	C₂²	semidirect
29. C₂²×C₄×D₅₃	C₂×C₄	semidirect
31. C₄×D₂₁₂	C₂×C₄	direct (N × H)
32. C₂×C₄⋊Dic₅₃	C₂×C₄	semidirect
34. C₂×#11719921	C₂×C₄	semidirect
44. C₂×C₄²×C₅₃	C₂×C₄	semidirect
45. C₂×C₄²×C₅₃	D₄	semidirect
67. C₂×D₂₁₂	C₄²	direct (N × H)
72. C₂×C₄×D₅₃	C₂²×C₄	semidirect
74. C₄²×C₅₃	C₂²×C₄	semidirect
75. C₄²×C₅₃	C₂×D₄	semidirect
76. #11719921	C₂²×C₄	semidirect
77. C₄⋊Dic₅₃	C₂²×C₄	semidirect
87. C₂²×D₅₃	C₂×C₄²	semidirect
91. C₂×C₄×C₅₃	C₂×C₄²	semidirect
92. C₂×C₄×C₅₃	C₄×D₄	semidirect
102. D₂₁₂	C₂×C₄²	direct (N × H)
114. C₂×D₅₃	C₂²×C₄²	semidirect
116. C₄×C₅₃	C₂²×C₄²	semidirect
117. C₄×C₅₃	C₂×C₄×D₄	semidirect
124. C₂×C₄³	D₅₃	semidirect
125. C₂×C₅₃	C₄²×D₄	semidirect
145. C₄³	C₂×D₅₃	semidirect
146. C₅₃	C₂×C₄²×D₄	semidirect
167. C₂×C₄²	C₄×D₅₃	semidirect
168. C₂×C₄²	D₂₁₂	direct (N × H)
201. C₄²	C₂×C₄×D₅₃	semidirect
202. C₄²	C₂×D₂₁₂	direct (N × H)
221. C₂×C₄	C₄²×D₅₃	semidirect
222. C₂×C₄	C₄×D₂₁₂	direct (N × H)
240. C₄	C₂×C₄²×D₅₃	semidirect
241. C₄	C₂×C₄×D₂₁₂	direct (N × H)
243. C₂	C₄²×D₂₁₂	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×C₄×D₂₁₂	2
C₂²×C₄²×D₅₃	2
C₄²×D₂₁₂	2
C₂×C₄×C₄⋊Dic₅₃	2
C₂×C₄×#11719921	2
C₂×C₄³×C₅₃	2
C₂×C₄²×D₄	53

Subgroups — C2×C42×D212 #27542285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×D₂₁₂ #27542285