Subgroups — C₄×#41×Dic₅₃ #27542294

Order 13568.

4086 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×Dic₅₃	C₂	1	2	yes	maximal
2	C₄×Q₈×Dic₅₃	C₂	1	4	yes	maximal, complemented
3	C₄×D₄×Dic₅₃	C₂	1	12	yes	maximal, complemented
4	#117×Dic₅₃	C₂	1	6	yes	maximal, complemented
5	C₄×#13133176	C₂	1	3	yes	maximal, complemented
6	C₂×C₄²×Dic₅₃	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#41×Dic₅₃	C₄	1	32	yes	direct product factor
9	C₂×Q₈×Dic₅₃	C₂²	1	4	yes	—
10	C₂×D₄×Dic₅₃	C₂²	1	12	yes	—
11	C₂×#13133176	C₂²	1	6	yes	—
12	C₂²×C₄×Dic₅₃	C₂²	1	12	yes	—
13	C₂²×C₄×Dic₅₃	C₂²	1	6	yes	—
14	C₄⋊C₄×Dic₅₃	C₂²	1	24	yes	—
15	C₄×C₄⋊Dic₅₃	C₂²	1	12	yes	—
16	C₂²⋊C₄×Dic₅₃	C₂²	1	24	yes	—
17	C₄×C₂²⋊Dic₅₃	C₂²	1	12	yes	—
18	#16810469	C₂²	1	6	yes	—
19	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
20	C₄²×Dic₅₃	C₂²	1	3	yes	—
21	C₄²×Dic₅₃	C₂²	1	6	yes	—
22	C₄²×Dic₅₃	C₂²	1	12	yes	—
23	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×#41×C₅₃	C₄	1	16	yes	complemented
25	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₅₃	C₂²	1	6	yes	—
27	C₂×#117×C₅₃	C₂²	1	3	yes	—
28	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
29	C₂³×Dic₅₃	C₂³	1	12	yes	—
30	Q₈×Dic₅₃	C₂×C₄	1	32	yes	complemented
31	D₄×Dic₅₃	C₂×C₄	1	96	yes	complemented
32	#13133176	C₂×C₄	1	48	yes	complemented
33	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
34	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	—
35	C₂×C₄×Dic₅₃	—	2	48	no	—
36	C₂×C₄×Dic₅₃	C₂×C₄	1	48	yes	complemented
37	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
38	C₂×C₄×Dic₅₃	C₂³	1	24	yes	—
39	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
40	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
41	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
42	C₂×C₄×Dic₅₃	C₂³	1	4	yes	—
43	#13133124	C₂³	1	24	yes	—
44	C₂×#41×C₅₃	C₂×C₄	1	16	yes	—
45	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
46	C₂²×Q₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₅₃	C₂³	1	3	yes	—
48	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
49	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
50	C₄×Q₈×C₅₃	C₂×C₄	1	16	yes	complemented
51	C₄×D₄×C₅₃	C₂×C₄	1	48	yes	complemented
52	#117×C₅₃	C₂×C₄	1	24	yes	complemented
53	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
54	C₂×C₂²⋊C₄×C₅₃	C₂³	1	6	yes	—
55	C₂×C₄²×C₅₃	C₂×C₄	1	24	yes	complemented
56	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
57	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
58	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
59	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
60	#41×C₅₃	C₄²	1	64	yes	complemented
61	C₂×Q₈×C₅₃	C₂²×C₄	1	8	yes	—
62	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
63	C₂×D₄×C₅₃	C₂²×C₄	1	24	yes	—
64	C₂×D₄×C₅₃	C₂²×C₄	1	12	yes	—
65	C₂²×C₄×C₅₃	—	2	24	no	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
68	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
69	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
70	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
72	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
73	C₂²×Dic₅₃	—	2	48	no	—
74	C₂²×Dic₅₃	C₂²×C₄	1	48	yes	—
75	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
76	C₂²×Dic₅₃	C₂⁴	1	4	yes	—
77	C₄⋊C₄×C₅₃	C₂²×C₄	1	24	yes	—
78	C₂²⋊C₄×C₅₃	C₂²×C₄	1	24	yes	—
79	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
80	C₄²×C₅₃	C₂²×C₄	1	12	yes	—
81	C₄²×C₅₃	C₂²×C₄	1	24	yes	—
82	C₂²⋊Dic₅₃	C₂²×C₄	1	48	yes	—
83	C₄⋊Dic₅₃	C₂²×C₄	1	48	yes	—
84	C₄×Dic₅₃	—	2	24	no	—
85	C₄×Dic₅₃	—	2	48	no	—
86	C₄×Dic₅₃	C₂²×C₄	1	8	yes	—
87	C₄×Dic₅₃	C₂²×C₄	1	24	yes	—
88	C₄×Dic₅₃	C₂²×C₄	1	48	yes	—
89	C₄×Dic₅₃	#41	1	16	yes	direct product factor
90	C₂³×C₅₃	—	2	12	no	—
91	C₂³×C₅₃	C₂³×C₄	1	12	yes	—
92	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
93	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
94	Q₈×C₅₃	C₂×C₄²	1	16	yes	complemented
95	D₄×C₅₃	C₂×C₄²	1	48	yes	complemented
96	C₂×C₄×C₅₃	—	2	12	no	—
97	C₂×C₄×C₅₃	—	2	48	no	—
98	C₂×C₄×C₅₃	C₂×C₄²	1	48	yes	complemented
99	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
100	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
101	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
102	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
104	C₂×C₄×C₅₃	C₂³×C₄	1	6	yes	—
105	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
106	C₂×Dic₅₃	—	2	24	no	—
107	C₂×Dic₅₃	—	2	96	no	—
108	C₂×Dic₅₃	C₂³×C₄	1	24	yes	—
109	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
110	C₂×Dic₅₃	C₂×#41	1	16	yes	—
111	C₄²×#41	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	24	no	—
113	C₂²×C₅₃	—	2	12	no	—
114	C₂²×C₅₃	C₂²×C₄²	1	12	yes	—
115	C₂²×C₅₃	C₂⁴×C₄	1	2	yes	—
116	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
117	C₂²×C₅₃	C₂²×#41	1	4	yes	—
118	C₄×C₅₃	—	2	12	no	—
119	C₄×C₅₃	C₂²×C₄²	1	4	yes	—
120	C₄×C₅₃	C₂²×C₄²	1	12	yes	—
121	C₄×C₅₃	C₄×#41	1	8	yes	complemented
122	Dic₅₃	—	2	24	no	—
123	Dic₅₃	C₄×#41	1	16	yes	direct product factor
124	C₂×C₄×#41	—	53	2	no	—
125	C₂×C₄×#41	D₅₃	1	1	yes	characteristic
126	C₄²×Q₈	—	53	4	no	—
127	C₄²×D₄	—	53	12	no	—
128	C₄×#117	—	53	3	no	—
129	C₄×#117	—	53	6	no	—
130	C₂×C₄³	—	53	3	no	—
131	C₂×C₅₃	—	2	12	no	—
132	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
133	C₂×C₅₃	C₂×C₄×#41	1	4	yes	—
134	C₂×C₅₃	C₂×C₄×#41	1	2	yes	—
135	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
136	C₄×#41	—	53	32	no	—
137	C₄×#41	Dic₅₃	1	16	yes	direct product factor
138	C₂×C₄×Q₈	—	53	4	no	—
139	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
140	C₂×C₄×D₄	—	53	12	no	—
141	C₂×C₄×D₄	C₂×D₅₃	1	6	yes	—
142	C₂×#117	—	53	6	no	—
143	C₂×#117	C₂×D₅₃	1	3	yes	—
144	C₂²×C₄²	—	53	6	no	—
145	C₂²×C₄²	—	53	12	no	—
146	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
147	C₄×C₄⋊C₄	—	53	24	no	—
148	C₄×C₄⋊C₄	—	53	12	no	—
149	C₄×C₂²⋊C₄	—	53	12	no	—
150	C₄×C₂²⋊C₄	—	53	24	no	—
151	#376	—	53	6	no	—
152	C₄³	—	53	6	no	—
153	C₄³	—	53	12	no	—
154	C₄³	—	53	1	no	—
155	C₄³	—	53	3	no	—
156	C₅₃	C₄²×#41	1	1	yes	characteristic, complemented, Sylow
157	C₂×#41	C₄×D₅₃	1	16	yes	—
158	C₂×#41	C₂×Dic₅₃	1	8	yes	—
159	C₂²×Q₈	C₂²×D₅₃	1	1	yes	characteristic
160	C₂²×D₄	C₂²×D₅₃	1	3	yes	—
161	C₂³×C₄	—	53	12	no	—
162	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
163	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
164	C₄×Q₈	—	53	32	no	—
165	C₄×Q₈	C₂×Dic₅₃	1	16	yes	complemented
166	C₄×D₄	—	53	96	no	—
167	C₄×D₄	C₂×Dic₅₃	1	48	yes	complemented
168	#117	—	53	48	no	—
169	#117	C₂×Dic₅₃	1	24	yes	complemented
170	C₂×C₄⋊C₄	—	53	12	no	—
171	C₂×C₄⋊C₄	C₂²×D₅₃	1	6	yes	—
172	C₂×C₂²⋊C₄	—	53	12	no	—
173	C₂×C₂²⋊C₄	C₂²×D₅₃	1	6	yes	—
174	C₂×C₄²	—	53	48	no	—
175	C₂×C₄²	—	53	6	no	—
176	C₂×C₄²	—	53	12	no	—
177	C₂×C₄²	—	53	2	no	—
178	C₂×C₄²	—	53	4	no	—
179	C₂×C₄²	—	53	24	no	—
180	C₂×C₄²	—	53	12	no	—
181	C₂×C₄²	—	106	48	no	—
182	C₂×C₄²	C₂×Dic₅₃	1	24	yes	complemented
183	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
184	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
185	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
186	#95	—	53	24	no	—
187	C₂⁴	C₂³×D₅₃	1	3	yes	—
188	#41	C₄×Dic₅₃	1	64	yes	direct product factor
189	C₂×Q₈	C₂×C₄×D₅₃	1	8	yes	—
190	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
191	C₂×D₄	C₂×C₄×D₅₃	1	24	yes	—
192	C₂×D₄	C₂²×Dic₅₃	1	12	yes	—
193	C₂²×C₄	—	2	24	no	—
194	C₂²×C₄	—	53	4	no	—
195	C₂²×C₄	—	53	48	no	—
196	C₂²×C₄	—	53	12	no	—
197	C₂²×C₄	—	106	48	no	—
198	C₂²×C₄	C₂×C₄×D₅₃	1	24	yes	—
199	C₂²×C₄	C₂²×Dic₅₃	1	24	yes	—
200	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
201	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
202	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
204	C₂²×C₄	C₂³×D₅₃	1	2	yes	—
205	C₄⋊C₄	—	53	48	no	—
206	C₄⋊C₄	C₂²×Dic₅₃	1	24	yes	—
207	C₂²⋊C₄	—	53	48	no	—
208	C₂²⋊C₄	C₂²×Dic₅₃	1	24	yes	—
209	C₄²	—	53	8	no	—
210	C₄²	—	53	24	no	—
211	C₄²	—	53	48	no	—
212	C₄²	—	53	16	no	—
213	C₄²	—	106	48	no	—
214	C₄²	—	106	24	no	—
215	C₄²	C₂²×Dic₅₃	1	4	yes	—
216	C₄²	C₂²×Dic₅₃	1	12	yes	—
217	C₄²	C₂²×Dic₅₃	1	24	yes	—
218	C₂³	—	2	12	no	—
219	C₂³	C₂²×C₄×D₅₃	1	12	yes	—
220	C₂³	C₂³×Dic₅₃	1	6	yes	—
221	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
222	Q₈	C₂×C₄×Dic₅₃	1	16	yes	complemented
223	D₄	C₂×C₄×Dic₅₃	1	48	yes	complemented
224	C₂×C₄	—	2	12	no	—
225	C₂×C₄	—	2	48	no	—
226	C₂×C₄	—	53	8	no	—
227	C₂×C₄	—	53	16	no	—
228	C₂×C₄	—	53	24	no	—
229	C₂×C₄	—	106	96	no	—
230	C₂×C₄	—	106	24	no	—
231	C₂×C₄	C₂×C₄×Dic₅₃	1	48	yes	complemented
232	C₂×C₄	C₂²×C₄×D₅₃	1	4	yes	—
233	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
234	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
235	C₂×C₄	C₂³×Dic₅₃	1	2	yes	—
236	C₂×C₄	C₂³×Dic₅₃	1	6	yes	—
237	C₂×C₄	C₂³×Dic₅₃	1	4	yes	—
238	C₂×C₄	#41×D₅₃	1	8	yes	—
239	C₂²	—	2	24	no	—
240	C₂²	—	2	12	no	—
241	C₂²	C₂²×C₄×Dic₅₃	1	12	yes	—
242	C₂²	C₂³×C₄×D₅₃	1	2	yes	—
243	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
244	C₂²	C₂×#41×D₅₃	1	4	yes	—
245	C₄	—	2	12	no	—
246	C₄	—	53	16	no	—
247	C₄	—	106	24	no	—
248	C₄	C₂²×C₄×Dic₅₃	1	4	yes	—
249	C₄	C₂²×C₄×Dic₅₃	1	12	yes	—
250	C₄	#41×Dic₅₃	1	8	yes	direct product factor
251	C₂	—	2	12	no	—
252	C₂	C₂³×C₄×Dic₅₃	1	1	yes	characteristic
253	C₂	C₄×#41×D₅₃	1	4	yes	—
254	C₂	C₂×#41×Dic₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator	C₂×C₅₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂×C₄×#41×C₅₃	C₂
socle, cosocle	C₂³×C₅₃	C₂⁵
prime radical	C₂²×#41×C₅₃	C₂²
hypercenter	C₂×C₄×#41	D₅₃
hypocenter	C₅₃	C₄²×#41
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×#41

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×Dic₅₃	C₂	semidirect
3. C₄×D₄×Dic₅₃	C₂	semidirect
4. #117×Dic₅₃	C₂	semidirect
5. C₄×#13133176	C₂	semidirect
6. C₂×C₄²×Dic₅₃	C₂	semidirect
8. #41×Dic₅₃	C₄	direct (N × H)
24. C₄×#41×C₅₃	C₄	semidirect
30. Q₈×Dic₅₃	C₂×C₄	semidirect
31. D₄×Dic₅₃	C₂×C₄	semidirect
32. #13133176	C₂×C₄	semidirect
36. C₂×C₄×Dic₅₃	C₂×C₄	semidirect
50. C₄×Q₈×C₅₃	C₂×C₄	semidirect
51. C₄×D₄×C₅₃	C₂×C₄	semidirect
52. #117×C₅₃	C₂×C₄	semidirect
55. C₂×C₄²×C₅₃	C₂×C₄	semidirect
60. #41×C₅₃	C₄²	semidirect
89. C₄×Dic₅₃	#41	direct (N × H)
94. Q₈×C₅₃	C₂×C₄²	semidirect
95. D₄×C₅₃	C₂×C₄²	semidirect
98. C₂×C₄×C₅₃	C₂×C₄²	semidirect
121. C₄×C₅₃	C₄×#41	semidirect
123. Dic₅₃	C₄×#41	direct (N × H)
137. C₄×#41	Dic₅₃	direct (N × H)
156. C₅₃	C₄²×#41	semidirect
165. C₄×Q₈	C₂×Dic₅₃	semidirect
167. C₄×D₄	C₂×Dic₅₃	semidirect
169. #117	C₂×Dic₅₃	semidirect
182. C₂×C₄²	C₂×Dic₅₃	semidirect
188. #41	C₄×Dic₅₃	direct (N × H)
222. Q₈	C₂×C₄×Dic₅₃	semidirect
223. D₄	C₂×C₄×Dic₅₃	semidirect
231. C₂×C₄	C₂×C₄×Dic₅₃	semidirect
250. C₄	#41×Dic₅₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#41×Dic₅₃	2
C₄×Q₈×Dic₅₃	2
C₄×D₄×Dic₅₃	2
#117×Dic₅₃	2
C₄×#13133176	2
C₂×C₄²×Dic₅₃	2
C₂×C₄×#41×C₅₃	2
C₄²×#41	53

Subgroups — C4×#41×Dic53 #27542294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#41×Dic₅₃ #27542294