Subgroups — C₂²×#16810492 #27542381

Order 13568.

5502 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11719919	C₂	1	4	yes	maximal
3	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810492	C₂	1	24	yes	maximal, direct product factor
5	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
7	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₅₃	—	2	8	no	—
9	C₂²×C₄⋊Dic₅₃	C₂²	1	12	yes	—
10	C₂²×#11719919	—	2	32	no	—
11	C₂²×#11719919	C₂²	1	48	yes	—
12	C₂²×C₄×Dic₅₃	C₄	1	16	yes	complemented
13	C₂²×C₄×Dic₅₃	C₂²	1	12	yes	—
14	#16810492	C₂²	1	64	yes	direct product factor
15	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₅₃	—	2	8	no	—
18	C₂²×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
19	C₂³×Dic₅₃	—	2	16	no	—
20	C₂³×Dic₅₃	—	2	8	no	—
21	C₂³×Dic₅₃	C₂×C₄	1	16	yes	—
22	C₂³×Dic₅₃	D₄	1	16	yes	—
23	C₂³×Dic₅₃	Q₈	1	16	yes	—
24	C₂³×Dic₅₃	C₂³	1	12	yes	—
25	C₂³×Dic₅₃	C₂³	1	12	yes	—
26	C₂×C₄⋊Dic₅₃	—	2	48	no	—
27	C₂×C₄⋊Dic₅₃	C₂³	1	16	yes	—
28	C₂×#11719919	—	2	192	no	—
29	C₂×#11719919	C₂³	1	64	yes	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	96	yes	complemented
31	C₂×C₄×Dic₅₃	C₂³	1	16	yes	—
32	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	—	2	8	no	—
34	C₂³×C₄×C₅₃	—	2	16	no	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
36	C₂³×C₄×C₅₃	D₄	1	8	yes	—
37	C₂³×C₄×C₅₃	Q₈	1	8	yes	—
38	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
39	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
40	C₂×C₄⋊C₄×C₅₃	—	2	48	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
42	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
43	C₂⁴×C₅₃	C₂×D₄	1	8	yes	—
44	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
45	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
46	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
47	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
48	C₂²×C₄×C₅₃	—	2	24	no	—
49	C₂²×C₄×C₅₃	—	2	48	no	—
50	C₂²×C₄×C₅₃	—	4	32	no	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	64	yes	complemented
52	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
54	C₂²×C₄×C₅₃	C₂×Q₈	1	24	yes	—
55	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
56	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
57	C₂²×Dic₅₃	—	2	48	no	—
58	C₂²×Dic₅₃	—	2	24	no	—
59	C₂²×Dic₅₃	—	4	32	no	—
60	C₂²×Dic₅₃	C₄⋊C₄	1	128	yes	complemented
61	C₂²×Dic₅₃	C₂²×C₄	1	48	yes	—
62	C₂²×Dic₅₃	C₂×D₄	1	48	yes	—
63	C₂²×Dic₅₃	C₂×Q₈	1	48	yes	—
64	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
65	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
66	C₄⋊C₄×C₅₃	—	2	32	no	—
67	C₄⋊Dic₅₃	—	2	32	no	—
68	#11719919	—	2	128	no	—
69	C₄×Dic₅₃	C₂²×C₄	1	64	yes	complemented
70	C₂³×C₅₃	C₂×C₄⋊C₄	1	32	yes	—
71	C₂³×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
72	C₂³×C₅₃	#127	1	16	yes	—
73	C₂³×C₅₃	C₄⋊Q₈	1	16	yes	—
74	C₂³×C₅₃	C₄⋊Q₈	1	32	yes	—
75	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
76	C₂³×C₅₃	C₂²×D₄	1	12	yes	—
77	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
78	C₂³×C₅₃	C₂²×Q₈	1	12	yes	—
79	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
80	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
81	C₂×C₄×C₅₃	—	2	16	no	—
82	C₂×C₄×C₅₃	—	2	8	no	—
83	C₂×C₄×C₅₃	—	4	48	no	—
84	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	96	yes	complemented
85	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
86	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
87	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
88	C₂×Dic₅₃	—	2	16	no	—
89	C₂×Dic₅₃	—	2	8	no	—
90	C₂×Dic₅₃	—	4	48	no	—
91	C₂×Dic₅₃	C₂×C₄⋊C₄	1	192	yes	complemented
92	C₂×Dic₅₃	C₂³×C₄	1	16	yes	—
93	C₂×Dic₅₃	C₂²×D₄	1	16	yes	—
94	C₂×Dic₅₃	C₂²×Q₈	1	16	yes	—
95	C₂²×#384	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	#384	1	64	yes	complemented
97	C₂²×C₅₃	C₂²×C₄⋊C₄	1	24	yes	—
98	C₂²×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
99	C₂²×C₅₃	C₂×#127	1	12	yes	—
100	C₂²×C₅₃	C₂×C₄⋊Q₈	1	12	yes	—
101	C₂²×C₅₃	C₂×C₄⋊Q₈	1	24	yes	—
102	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
103	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
104	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
105	C₂²×C₅₃	C₂³×Q₈	1	2	yes	—
106	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
107	C₄×C₅₃	—	4	8	no	—
108	C₄×C₅₃	C₂²×C₄⋊C₄	1	16	yes	complemented
109	Dic₅₃	—	4	8	no	—
110	Dic₅₃	C₂²×C₄⋊C₄	1	32	yes	complemented
111	C₂³×C₄⋊C₄	—	53	4	no	—
112	C₂³×C₄⋊C₄	—	53	1	no	—
113	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic
114	C₂³×C₄²	—	53	1	no	—
115	C₂×#384	—	53	24	no	—
116	C₂×C₅₃	C₂×#384	1	24	yes	complemented
117	C₂×C₅₃	C₂³×C₄⋊C₄	1	2	yes	—
118	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
120	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₅₃	C₂²×C₄⋊Q₈	1	2	yes	—
122	C₂⁴×C₄	—	53	2	no	—
123	C₂⁴×C₄	—	53	2	no	—
124	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
125	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
126	C₂²×C₄⋊C₄	—	2	8	no	—
127	C₂²×C₄⋊C₄	—	53	48	no	—
128	C₂²×C₄⋊C₄	—	53	12	no	—
129	C₂²×C₄⋊C₄	—	106	32	no	—
130	C₂²×C₄⋊C₄	—	106	8	no	—
131	C₂²×C₄⋊C₄	C₂×D₅₃	1	12	yes	—
132	C₂²×C₄²	—	53	16	no	—
133	C₂²×C₄²	—	53	12	no	—
134	#384	—	53	64	no	—
135	C₅₃	C₂²×#384	1	1	yes	characteristic, complemented, Sylow
136	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
137	C₂³×C₄	—	2	16	no	—
138	C₂³×C₄	—	2	8	no	—
139	C₂³×C₄	—	53	16	no	—
140	C₂³×C₄	—	53	16	no	—
141	C₂³×C₄	—	53	16	no	—
142	C₂³×C₄	—	53	12	no	—
143	C₂³×C₄	—	53	12	no	—
144	C₂³×C₄	—	106	16	no	—
145	C₂³×C₄	—	106	8	no	—
146	C₂³×C₄	Dic₁₀₆	1	8	yes	—
147	C₂³×C₄	C₄×D₅₃	1	8	yes	—
148	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
149	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
150	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
151	C₂×C₄⋊C₄	—	2	48	no	—
152	C₂×C₄⋊C₄	—	53	64	no	—
153	C₂×C₄⋊C₄	—	53	16	no	—
154	C₂×C₄⋊C₄	—	106	192	no	—
155	C₂×C₄⋊C₄	—	106	48	no	—
156	C₂×C₄⋊C₄	C₂²×D₅₃	1	16	yes	—
157	C₂×C₄²	—	53	96	no	—
158	C₂×C₄²	—	53	16	no	—
159	C₂⁴	C₂×Dic₁₀₆	1	4	yes	—
160	C₂⁴	C₂×C₄×D₅₃	1	4	yes	—
161	C₂⁴	D₄×D₅₃	1	8	yes	—
162	C₂⁴	Q₈×D₅₃	1	8	yes	—
163	C₂⁴	C₂×C₂²⋊D₅₃	1	4	yes	—
164	C₂⁴	C₂³×D₅₃	1	3	yes	—
165	C₂²×C₄	—	2	48	no	—
166	C₂²×C₄	—	2	24	no	—
167	C₂²×C₄	—	4	32	no	—
168	C₂²×C₄	—	53	128	no	—
169	C₂²×C₄	—	53	48	no	—
170	C₂²×C₄	—	53	48	no	—
171	C₂²×C₄	—	53	48	no	—
172	C₂²×C₄	—	53	8	no	—
173	C₂²×C₄	—	53	8	no	—
174	C₂²×C₄	—	106	48	no	—
175	C₂²×C₄	—	106	24	no	—
176	C₂²×C₄	—	212	32	no	—
177	C₂²×C₄	#11719919	1	64	yes	complemented
178	C₂²×C₄	C₂×Dic₁₀₆	1	24	yes	—
179	C₂²×C₄	C₂×C₄×D₅₃	1	24	yes	—
180	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	—
181	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
182	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
183	C₄⋊C₄	—	2	32	no	—
184	C₄⋊C₄	—	106	128	no	—
185	C₄⋊C₄	—	106	32	no	—
186	C₄²	—	53	64	no	—
187	C₂³	C₄⋊C₄×D₅₃	1	32	yes	—
188	C₂³	C₂×#11719919	1	16	yes	—
189	C₂³	#13133213	1	16	yes	—
190	C₂³	#13133215	1	16	yes	—
191	C₂³	#13133216	1	32	yes	—
192	C₂³	C₂²×C₄×D₅₃	1	6	yes	—
193	C₂³	C₂×D₄×D₅₃	1	12	yes	—
194	C₂³	C₂²×C₂²⋊D₅₃	1	6	yes	—
195	C₂³	C₂×Q₈×D₅₃	1	12	yes	—
196	C₂³	C₂²×Dic₁₀₆	1	6	yes	—
197	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	4	48	no	—
201	C₂×C₄	—	53	192	no	—
202	C₂×C₄	—	53	16	no	—
203	C₂×C₄	—	53	16	no	—
204	C₂×C₄	—	53	16	no	—
205	C₂×C₄	—	106	16	no	—
206	C₂×C₄	—	106	8	no	—
207	C₂×C₄	—	212	48	no	—
208	C₂×C₄	C₂×#11719919	1	96	yes	complemented
209	C₂×C₄	C₂²×C₄×D₅₃	1	8	yes	—
210	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
211	C₂×C₄	C₂²×Dic₁₀₆	1	8	yes	—
212	C₂²	#16810492	1	64	yes	direct product factor
213	C₂²	C₂×C₄⋊C₄×D₅₃	1	24	yes	—
214	C₂²	C₂²×#11719919	1	12	yes	—
215	C₂²	C₂×#13133213	1	12	yes	—
216	C₂²	C₂×#13133215	1	12	yes	—
217	C₂²	C₂×#13133216	1	24	yes	—
218	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
219	C₂²	C₂²×D₄×D₅₃	1	2	yes	—
220	C₂²	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
221	C₂²	C₂²×Q₈×D₅₃	1	2	yes	—
222	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
223	C₄	—	4	8	no	—
224	C₄	—	53	32	no	—
225	C₄	—	212	8	no	—
226	C₄	C₂²×#11719919	1	16	yes	complemented
227	C₂	C₂×#16810492	1	24	yes	direct product factor
228	C₂	C₂²×C₄⋊C₄×D₅₃	1	2	yes	—
229	C₂	C₂³×#11719919	1	1	yes	characteristic
230	C₂	C₂²×#13133213	1	1	yes	characteristic
231	C₂	C₂²×#13133215	1	1	yes	characteristic
232	C₂	C₂²×#13133216	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂³
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂²×#384
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₂×#16810492	C₂	direct (N × H)
12. C₂²×C₄×Dic₅₃	C₄	semidirect
14. #16810492	C₂²	direct (N × H)
30. C₂×C₄×Dic₅₃	C₂×C₄	semidirect
51. C₂²×C₄×C₅₃	C₄⋊C₄	semidirect
60. C₂²×Dic₅₃	C₄⋊C₄	semidirect
69. C₄×Dic₅₃	C₂²×C₄	semidirect
84. C₂×C₄×C₅₃	C₂×C₄⋊C₄	semidirect
91. C₂×Dic₅₃	C₂×C₄⋊C₄	semidirect
96. C₂²×C₅₃	#384	semidirect
108. C₄×C₅₃	C₂²×C₄⋊C₄	semidirect
110. Dic₅₃	C₂²×C₄⋊C₄	semidirect
116. C₂×C₅₃	C₂×#384	semidirect
135. C₅₃	C₂²×#384	semidirect
177. C₂²×C₄	#11719919	semidirect
208. C₂×C₄	C₂×#11719919	semidirect
212. C₂²	#16810492	direct (N × H)
226. C₄	C₂²×#11719919	semidirect
227. C₂	C₂×#16810492	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊Dic₅₃	2
C₂³×#11719919	2
C₂³×C₄×Dic₅₃	2
C₂×#16810492	2
C₂³×C₄⋊C₄×C₅₃	2
C₂²×#384	53

Subgroups — C22×#16810492 #27542381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810492 #27542381