Subgroups — C₂×C₄×#13133176 #27542277

Order 13568.

4462 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133176	C₂	1	2	yes	maximal
2	C₄×#13133176	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×C₄⋊Dic₅₃	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₂²⋊Dic₅₃	C₂	1	4	yes	maximal
5	C₂×#16810469	C₂	1	2	yes	maximal, complemented
6	C₂×C₄²×Dic₅₃	C₂	1	2	yes	maximal, complemented
7	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13133176	C₄	1	32	yes	direct product factor
9	C₂×#13133176	C₂²	1	16	yes	—
10	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
11	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
12	C₂²×C₄×Dic₅₃	C₂²	1	8	yes	—
13	C₂²×C₄×Dic₅₃	C₂²	1	4	yes	—
14	C₄×C₄⋊Dic₅₃	C₂²	1	32	yes	complemented
15	C₄×C₂²⋊Dic₅₃	C₂²	1	32	yes	—
16	#16810469	C₂²	1	16	yes	complemented
17	C₂×#13133124	C₂²	1	8	yes	—
18	C₄²×Dic₅₃	C₂²	1	16	yes	complemented
19	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
20	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₅₃	C₄	1	16	yes	complemented
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
24	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄²×C₅₃	C₂²	1	8	yes	—
26	C₂³×Dic₅₃	C₂³	1	8	yes	—
27	#13133176	C₂×C₄	1	128	yes	direct product factor
28	C₂×C₄⋊Dic₅₃	C₂×C₄	1	32	yes	complemented
29	C₂×C₄⋊Dic₅₃	C₂³	1	16	yes	—
30	C₂×C₂²⋊Dic₅₃	C₂×C₄	1	32	yes	—
31	C₂×C₂²⋊Dic₅₃	C₂³	1	16	yes	—
32	C₂×C₄×Dic₅₃	—	2	32	no	—
33	C₂×C₄×Dic₅₃	C₂×C₄	1	32	yes	complemented
34	C₂×C₄×Dic₅₃	C₂³	1	32	yes	—
35	C₂×C₄×Dic₅₃	C₂³	1	16	yes	—
36	#13133124	C₂³	1	32	yes	—
37	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
38	C₂³×C₄×C₅₃	—	2	16	no	—
39	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	—
40	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	—
41	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
42	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
44	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
47	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
48	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	16	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	8	yes	—
51	C₂×C₄²×C₅₃	C₂×C₄	1	64	yes	complemented
52	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
53	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
54	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
55	C₂⁴×C₅₃	—	2	8	no	—
56	C₂⁴×C₅₃	C₂²×C₄	1	8	yes	—
57	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
58	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
59	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	—	2	8	no	—
61	C₂²×C₄×C₅₃	—	2	64	no	—
62	C₂²×C₄×C₅₃	—	2	32	no	—
63	C₂²×C₄×C₅₃	C₄²	1	64	yes	complemented
64	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	32	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	32	yes	—
71	C₂²×C₄×C₅₃	C₂²×C₄	1	16	yes	—
72	C₂²×C₄×C₅₃	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₅₃	#41	1	16	yes	—
74	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
75	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
77	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
78	C₂²×Dic₅₃	—	2	32	no	—
79	C₂²×Dic₅₃	C₂²×C₄	1	32	yes	—
80	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	16	yes	—
82	C₄²×C₅₃	C₂²×C₄	1	32	yes	—
83	C₄²×C₅₃	C₂²×C₄	1	16	yes	—
84	C₂²⋊Dic₅₃	C₂²×C₄	1	64	yes	—
85	C₄⋊Dic₅₃	C₂²×C₄	1	64	yes	complemented
86	C₄×Dic₅₃	—	2	64	no	—
87	C₄×Dic₅₃	C₂²×C₄	1	64	yes	complemented
88	C₂³×C₅₃	—	2	16	no	—
89	C₂³×C₅₃	—	2	32	no	—
90	C₂³×C₅₃	—	2	8	no	—
91	C₂³×C₅₃	C₂×C₄²	1	16	yes	—
92	C₂³×C₅₃	C₂³×C₄	1	8	yes	—
93	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
94	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
95	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
96	C₂³×C₅₃	C₂×#41	1	8	yes	—
97	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
98	C₂×C₄×C₅₃	—	2	16	no	—
99	C₂×C₄×C₅₃	—	2	8	no	—
100	C₂×C₄×C₅₃	—	2	64	no	—
101	C₂×C₄×C₅₃	C₂×C₄²	1	16	yes	—
102	C₂×C₄×C₅₃	C₂×C₄²	1	16	yes	—
103	C₂×C₄×C₅₃	C₂×C₄²	1	64	yes	complemented
104	C₂×C₄×C₅₃	#117	1	32	yes	complemented
105	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
106	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
107	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
108	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
109	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
110	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
111	C₂×C₄×C₅₃	C₂×#41	1	16	yes	—
112	C₂×Dic₅₃	—	2	64	no	—
113	C₂×Dic₅₃	—	2	32	no	—
114	C₂×Dic₅₃	C₂³×C₄	1	32	yes	—
115	C₂×C₄×#117	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	—	2	8	no	—
117	C₂²×C₅₃	—	2	16	no	—
118	C₂²×C₅₃	—	2	32	no	—
119	C₂²×C₅₃	C₂²×C₄²	1	8	yes	—
120	C₂²×C₅₃	C₂²×C₄²	1	4	yes	—
121	C₂²×C₅₃	C₂×#117	1	8	yes	—
122	C₂²×C₅₃	C₄×#41	1	16	yes	—
123	C₂²×C₅₃	C₂⁴×C₄	1	2	yes	—
124	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
125	C₂²×C₅₃	C₂²×#41	1	4	yes	—
126	C₄×C₅₃	—	2	8	no	—
127	C₄×C₅₃	C₂²×C₄²	1	8	yes	—
128	C₄×C₅₃	C₂²×C₄²	1	8	yes	—
129	C₄×C₅₃	C₂×#117	1	16	yes	complemented
130	Dic₅₃	—	2	32	no	—
131	C₂²×#117	—	53	2	no	—
132	C₂³×C₄²	D₅₃	1	1	yes	characteristic
133	C₄×#117	—	53	16	no	—
134	C₂×C₄×C₄⋊C₄	—	53	4	no	—
135	C₂×C₄×C₂²⋊C₄	—	53	4	no	—
136	C₂×#376	—	53	2	no	—
137	C₂×C₄³	—	53	2	no	—
138	C₂×C₅₃	—	2	8	no	—
139	C₂×C₅₃	C₄×#117	1	8	yes	complemented
140	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
141	C₂×C₅₃	C₂²×#117	1	2	yes	—
142	C₂×C₅₃	C₂×C₄×#41	1	4	yes	—
143	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
144	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
145	C₂×#117	—	53	32	no	—
146	C₂×#117	—	53	16	no	—
147	C₂²×C₄⋊C₄	—	53	4	no	—
148	C₂²×C₂²⋊C₄	—	53	4	no	—
149	C₂²×C₄²	—	53	4	no	—
150	C₂²×C₄²	—	53	8	no	—
151	C₂²×C₄²	Dic₅₃	1	16	yes	complemented
152	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
153	C₂²×C₄²	C₂×D₅₃	1	2	yes	—
154	C₂²×C₄²	C₂×D₅₃	1	8	yes	—
155	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
156	C₄×C₄⋊C₄	—	53	32	no	—
157	C₄×C₂²⋊C₄	—	53	32	no	—
158	#376	—	53	16	no	—
159	C₂×#95	—	53	8	no	—
160	C₄³	—	53	16	no	—
161	C₅₃	C₂×C₄×#117	1	1	yes	characteristic, complemented, Sylow
162	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
163	C₂³×C₄	—	2	16	no	—
164	C₂³×C₄	—	53	8	no	—
165	C₂³×C₄	C₄×D₅₃	1	16	yes	—
166	C₂³×C₄	C₂×Dic₅₃	1	16	yes	—
167	C₂³×C₄	C₂×Dic₅₃	1	8	yes	—
168	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
169	C₂³×C₄	C₂²×D₅₃	1	8	yes	—
170	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
171	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
172	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
173	C₂³×C₄	C₂²×D₅₃	1	2	yes	—
174	#117	—	53	128	no	—
175	C₂×C₄⋊C₄	—	53	32	no	—
176	C₂×C₄⋊C₄	—	53	16	no	—
177	C₂×C₂²⋊C₄	—	53	32	no	—
178	C₂×C₂²⋊C₄	—	53	16	no	—
179	C₂×C₄²	—	53	32	no	—
180	C₂×C₄²	—	53	16	no	—
181	C₂×C₄²	—	53	32	no	—
182	C₂×C₄²	—	106	32	no	—
183	C₂×C₄²	C₂×Dic₅₃	1	8	yes	—
184	C₂×C₄²	C₂×Dic₅₃	1	16	yes	—
185	C₂×C₄²	C₂×Dic₅₃	1	8	yes	—
186	C₂×C₄²	C₂×Dic₅₃	1	64	yes	complemented
187	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
188	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
189	C₂×C₄²	C₂²×D₅₃	1	8	yes	—
190	#95	—	53	32	no	—
191	C₂⁴	—	2	8	no	—
192	C₂⁴	C₂×C₄×D₅₃	1	8	yes	—
193	C₂⁴	C₂²×Dic₅₃	1	4	yes	—
194	C₂⁴	C₂³×D₅₃	1	2	yes	—
195	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	64	no	—
198	C₂²×C₄	—	2	32	no	—
199	C₂²×C₄	—	53	32	no	—
200	C₂²×C₄	—	53	16	no	—
201	C₂²×C₄	—	106	32	no	—
202	C₂²×C₄	C₄×Dic₅₃	1	64	yes	complemented
203	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
204	C₂²×C₄	C₂×C₄×D₅₃	1	8	yes	—
205	C₂²×C₄	C₂×C₄×D₅₃	1	32	yes	—
206	C₂²×C₄	#11719945	1	16	yes	—
207	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
208	C₂²×C₄	C₂²×Dic₅₃	1	32	yes	—
209	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
210	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
211	C₂²×C₄	C₂²×Dic₅₃	1	16	yes	—
212	C₂²×C₄	C₂²×Dic₅₃	1	8	yes	—
213	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
214	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
215	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
216	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
217	C₄⋊C₄	—	53	64	no	—
218	C₂²⋊C₄	—	53	64	no	—
219	C₄²	—	53	64	no	—
220	C₄²	—	106	64	no	—
221	C₄²	C₂²×Dic₅₃	1	16	yes	—
222	C₄²	C₂²×Dic₅₃	1	32	yes	—
223	C₄²	C₂²×Dic₅₃	1	16	yes	—
224	C₂³	—	2	16	no	—
225	C₂³	—	2	32	no	—
226	C₂³	—	2	8	no	—
227	C₂³	C₂×C₄×Dic₅₃	1	16	yes	—
228	C₂³	C₂²×C₄×D₅₃	1	8	yes	—
229	C₂³	C₂²×C₄×D₅₃	1	4	yes	—
230	C₂³	C₂³×Dic₅₃	1	4	yes	—
231	C₂³	C₂³×Dic₅₃	1	2	yes	—
232	C₂³	C₂×#11719945	1	8	yes	—
233	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
234	C₂×C₄	—	2	16	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	64	no	—
237	C₂×C₄	—	53	32	no	—
238	C₂×C₄	—	106	64	no	—
239	C₂×C₄	—	106	32	no	—
240	C₂×C₄	C₂×C₄×Dic₅₃	1	16	yes	—
241	C₂×C₄	C₂×C₄×Dic₅₃	1	16	yes	—
242	C₂×C₄	C₂×C₄×Dic₅₃	1	64	yes	complemented
243	C₂×C₄	#13133176	1	32	yes	direct product factor
244	C₂×C₄	C₂²×C₄×D₅₃	1	8	yes	—
245	C₂×C₄	C₂²×C₄×D₅₃	1	8	yes	—
246	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
247	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
248	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
249	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
250	C₂×C₄	C₂×#11719945	1	16	yes	—
251	C₂²	—	2	32	no	—
252	C₂²	—	2	8	no	—
253	C₂²	—	2	16	no	—
254	C₂²	C₂²×C₄×Dic₅₃	1	8	yes	—
255	C₂²	C₂²×C₄×Dic₅₃	1	4	yes	—
256	C₂²	C₂×#13133176	1	8	yes	—
257	C₂²	C₄×#11719945	1	16	yes	—
258	C₂²	C₂³×C₄×D₅₃	1	2	yes	—
259	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
260	C₂²	C₂²×#11719945	1	4	yes	—
261	C₄	—	2	8	no	—
262	C₄	—	106	32	no	—
263	C₄	C₂²×C₄×Dic₅₃	1	8	yes	—
264	C₄	C₂²×C₄×Dic₅₃	1	8	yes	—
265	C₄	C₂×#13133176	1	16	yes	direct product factor
266	C₂	—	2	8	no	—
267	C₂	C₄×#13133176	1	8	yes	direct product factor
268	C₂	C₂³×C₄×Dic₅₃	1	1	yes	characteristic
269	C₂	C₂²×#13133176	1	2	yes	—
270	C₂	C₂×C₄×#11719945	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×C₅₃	C₂³
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂³×C₄²	D₅₃
hypocenter	C₅₃	C₂×C₄×#117
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄² ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#13133176	C₂	direct (N × H)
3. C₂×C₄×C₄⋊Dic₅₃	C₂	semidirect
5. C₂×#16810469	C₂	semidirect
6. C₂×C₄²×Dic₅₃	C₂	semidirect
8. C₂×#13133176	C₄	direct (N × H)
14. C₄×C₄⋊Dic₅₃	C₂²	semidirect
16. #16810469	C₂²	semidirect
18. C₄²×Dic₅₃	C₂²	semidirect
21. C₂²×C₄²×C₅₃	C₄	semidirect
27. #13133176	C₂×C₄	direct (N × H)
28. C₂×C₄⋊Dic₅₃	C₂×C₄	semidirect
33. C₂×C₄×Dic₅₃	C₂×C₄	semidirect
51. C₂×C₄²×C₅₃	C₂×C₄	semidirect
63. C₂²×C₄×C₅₃	C₄²	semidirect
85. C₄⋊Dic₅₃	C₂²×C₄	semidirect
87. C₄×Dic₅₃	C₂²×C₄	semidirect
103. C₂×C₄×C₅₃	C₂×C₄²	semidirect
104. C₂×C₄×C₅₃	#117	semidirect
129. C₄×C₅₃	C₂×#117	semidirect
139. C₂×C₅₃	C₄×#117	semidirect
151. C₂²×C₄²	Dic₅₃	semidirect
161. C₅₃	C₂×C₄×#117	semidirect
186. C₂×C₄²	C₂×Dic₅₃	semidirect
202. C₂²×C₄	C₄×Dic₅₃	semidirect
242. C₂×C₄	C₂×C₄×Dic₅₃	semidirect
243. C₂×C₄	#13133176	direct (N × H)
265. C₄	C₂×#13133176	direct (N × H)
267. C₂	C₄×#13133176	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×#13133176	2
C₄×#13133176	2
C₂×C₄×C₄⋊Dic₅₃	2
C₂×C₄×C₂²⋊Dic₅₃	2
C₂×#16810469	2
C₂×C₄²×Dic₅₃	2
C₂³×C₄²×C₅₃	2
C₂×C₄×#117	53

Subgroups — C2×C4×#13133176 #27542277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#13133176 #27542277