Subgroups — C₂²×#16810488 #27542366

Order 13568.

4286 conjugacy classes of subgroups.

The subgroup lattice has 228 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×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810488	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#13133124	C₂	1	4	yes	maximal
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×C₄×Dic₅₃	C₄	1	16	yes	complemented
11	C₂²×C₄×Dic₅₃	C₂²	1	12	yes	—
12	#16810488	C₂²	1	64	yes	direct product factor
13	C₂×#13133124	C₂²	1	48	yes	—
14	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₅₃	—	2	8	no	—
17	C₂²×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
18	C₂³×Dic₅₃	—	2	8	no	—
19	C₂³×Dic₅₃	—	2	8	no	—
20	C₂³×Dic₅₃	—	2	8	no	—
21	C₂³×Dic₅₃	—	2	16	no	—
22	C₂³×Dic₅₃	C₂×C₄	1	16	yes	—
23	C₂³×Dic₅₃	C₂³	1	12	yes	—
24	C₂³×Dic₅₃	C₂³	1	12	yes	—
25	C₂×C₄⋊Dic₅₃	—	2	48	no	—
26	C₂×C₄⋊Dic₅₃	C₂³	1	16	yes	—
27	C₂×C₄×Dic₅₃	C₂×C₄	1	96	yes	complemented
28	C₂×C₄×Dic₅₃	C₂³	1	16	yes	—
29	#13133124	C₂³	1	64	yes	—
30	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₅₃	—	2	8	no	—
32	C₂³×C₄×C₅₃	—	2	16	no	—
33	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
34	C₂³×C₄×C₅₃	D₄	1	8	yes	—
35	C₂³×C₄×C₅₃	Q₈	1	8	yes	—
36	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
37	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
38	C₂×C₄⋊C₄×C₅₃	—	2	48	no	—
39	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
40	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
41	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
42	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
43	C₂⁴×C₅₃	#41	1	8	yes	—
44	C₂⁴×C₅₃	#41	1	8	yes	—
45	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
46	C₂²×C₄×C₅₃	—	2	24	no	—
47	C₂²×C₄×C₅₃	—	2	48	no	—
48	C₂²×C₄×C₅₃	—	4	32	no	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	64	yes	—
50	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
52	C₂²×C₄×C₅₃	C₂×Q₈	1	24	yes	—
53	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
54	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
55	C₂²×Dic₅₃	—	2	64	no	—
56	C₂²×Dic₅₃	—	2	24	no	—
57	C₂²×Dic₅₃	—	2	24	no	—
58	C₂²×Dic₅₃	—	2	24	no	—
59	C₂²×Dic₅₃	—	2	48	no	—
60	C₂²×Dic₅₃	—	4	32	no	—
61	C₂²×Dic₅₃	C₂²×C₄	1	48	yes	—
62	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
63	C₂²×Dic₅₃	C₂⁴	1	8	yes	—
64	C₄⋊C₄×C₅₃	—	2	32	no	—
65	C₄⋊Dic₅₃	—	2	32	no	—
66	C₄×Dic₅₃	C₂²×C₄	1	64	yes	complemented
67	C₂³×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
68	C₂³×C₅₃	#117	1	32	yes	—
69	C₂³×C₅₃	#124	1	16	yes	—
70	C₂³×C₅₃	#124	1	16	yes	—
71	C₂³×C₅₃	#125	1	32	yes	—
72	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
73	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
74	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
75	C₂³×C₅₃	C₂×#41	1	12	yes	—
76	C₂³×C₅₃	C₂×#41	1	12	yes	—
77	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	8	no	—
79	C₂×C₄×C₅₃	—	2	16	no	—
80	C₂×C₄×C₅₃	—	4	48	no	—
81	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	96	yes	—
82	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
83	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
84	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
85	C₂×Dic₅₃	—	2	96	no	—
86	C₂×Dic₅₃	—	2	8	no	—
87	C₂×Dic₅₃	—	2	8	no	—
88	C₂×Dic₅₃	—	2	8	no	—
89	C₂×Dic₅₃	—	2	16	no	—
90	C₂×Dic₅₃	—	4	48	no	—
91	C₂×Dic₅₃	C₂³×C₄	1	16	yes	—
92	C₂²×#382	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	#382	1	64	yes	complemented
94	C₂²×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
95	C₂²×C₅₃	C₂×#117	1	24	yes	—
96	C₂²×C₅₃	C₂×#124	1	12	yes	—
97	C₂²×C₅₃	C₂×#124	1	12	yes	—
98	C₂²×C₅₃	C₂×#125	1	24	yes	—
99	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
100	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
101	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
102	C₂²×C₅₃	C₂²×#41	1	2	yes	—
103	C₂²×C₅₃	C₂²×#41	1	2	yes	—
104	C₄×C₅₃	—	4	8	no	—
105	C₄×C₅₃	C₂²×C₄⋊C₄	1	16	yes	—
106	Dic₅₃	—	2	16	no	—
107	Dic₅₃	—	4	8	no	—
108	C₂³×C₄⋊C₄	—	53	1	no	—
109	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic
110	C₂³×C₄²	—	53	1	no	—
111	C₂×#382	—	53	24	no	—
112	C₂²×#95	—	53	4	no	—
113	C₂×C₅₃	C₂×#382	1	24	yes	complemented
114	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₅₃	C₂²×#117	1	2	yes	—
116	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
117	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
118	C₂×C₅₃	C₂²×#125	1	2	yes	—
119	C₂⁴×C₄	—	53	2	no	—
120	C₂⁴×C₄	—	53	2	no	—
121	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
122	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
123	C₂²×C₄⋊C₄	—	2	8	no	—
124	C₂²×C₄⋊C₄	—	53	12	no	—
125	C₂²×C₄⋊C₄	—	106	8	no	—
126	C₂²×C₄⋊C₄	C₂×D₅₃	1	12	yes	—
127	C₂²×C₄²	—	53	16	no	—
128	C₂²×C₄²	—	53	12	no	—
129	#382	—	53	64	no	—
130	C₂×#95	—	53	48	no	—
131	C₅₃	C₂²×#382	1	1	yes	characteristic, complemented, Sylow
132	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
133	C₂³×C₄	—	2	16	no	—
134	C₂³×C₄	—	2	8	no	—
135	C₂³×C₄	—	53	16	no	—
136	C₂³×C₄	—	53	12	no	—
137	C₂³×C₄	—	53	12	no	—
138	C₂³×C₄	—	106	8	no	—
139	C₂³×C₄	—	106	8	no	—
140	C₂³×C₄	—	106	16	no	—
141	C₂³×C₄	—	106	8	no	—
142	C₂³×C₄	Dic₁₀₆	1	8	yes	—
143	C₂³×C₄	C₄×D₅₃	1	8	yes	—
144	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
145	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
146	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
147	C₂×C₄⋊C₄	—	2	48	no	—
148	C₂×C₄⋊C₄	—	53	16	no	—
149	C₂×C₄⋊C₄	—	106	48	no	—
150	C₂×C₄⋊C₄	C₂²×D₅₃	1	16	yes	—
151	C₂×C₄²	—	53	96	no	—
152	C₂×C₄²	—	53	16	no	—
153	#95	—	53	64	no	—
154	C₂⁴	C₂×Dic₁₀₆	1	4	yes	—
155	C₂⁴	C₂×C₄×D₅₃	1	4	yes	—
156	C₂⁴	#11719947	1	8	yes	—
157	C₂⁴	#11719949	1	8	yes	—
158	C₂⁴	C₂×C₂²⋊D₅₃	1	4	yes	—
159	C₂⁴	C₂³×D₅₃	1	3	yes	—
160	C₂²×C₄	—	2	24	no	—
161	C₂²×C₄	—	2	48	no	—
162	C₂²×C₄	—	4	32	no	—
163	C₂²×C₄	—	53	48	no	—
164	C₂²×C₄	—	53	8	no	—
165	C₂²×C₄	—	53	8	no	—
166	C₂²×C₄	—	106	64	no	—
167	C₂²×C₄	—	106	24	no	—
168	C₂²×C₄	—	106	24	no	—
169	C₂²×C₄	—	106	24	no	—
170	C₂²×C₄	—	106	48	no	—
171	C₂²×C₄	—	212	32	no	—
172	C₂²×C₄	#11719919	1	64	yes	—
173	C₂²×C₄	C₂×Dic₁₀₆	1	24	yes	—
174	C₂²×C₄	C₂×C₄×D₅₃	1	24	yes	—
175	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	—
176	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
177	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
178	C₄⋊C₄	—	2	32	no	—
179	C₄⋊C₄	—	106	32	no	—
180	C₄²	—	53	64	no	—
181	C₂³	C₂×#11719919	1	16	yes	—
182	C₂³	#13133175	1	32	yes	—
183	C₂³	#13133204	1	16	yes	—
184	C₂³	#13133206	1	16	yes	—
185	C₂³	#13133208	1	32	yes	—
186	C₂³	C₂²×C₄×D₅₃	1	6	yes	—
187	C₂³	C₂²×C₂²⋊D₅₃	1	6	yes	—
188	C₂³	C₂²×Dic₁₀₆	1	6	yes	—
189	C₂³	C₂×#11719947	1	12	yes	—
190	C₂³	C₂×#11719949	1	12	yes	—
191	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	—	2	16	no	—
194	C₂×C₄	—	4	48	no	—
195	C₂×C₄	—	53	16	no	—
196	C₂×C₄	—	106	96	no	—
197	C₂×C₄	—	106	8	no	—
198	C₂×C₄	—	106	8	no	—
199	C₂×C₄	—	106	8	no	—
200	C₂×C₄	—	106	16	no	—
201	C₂×C₄	—	212	48	no	—
202	C₂×C₄	C₂×#11719919	1	96	yes	—
203	C₂×C₄	C₂²×C₄×D₅₃	1	8	yes	—
204	C₂×C₄	C₂²×C₂²⋊D₅₃	1	8	yes	—
205	C₂×C₄	C₂²×Dic₁₀₆	1	8	yes	—
206	C₂²	#16810488	1	64	yes	direct product factor
207	C₂²	C₂²×#11719919	1	12	yes	—
208	C₂²	C₂×#13133175	1	24	yes	—
209	C₂²	C₂×#13133204	1	12	yes	—
210	C₂²	C₂×#13133206	1	12	yes	—
211	C₂²	C₂×#13133208	1	24	yes	—
212	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
213	C₂²	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
214	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
215	C₂²	C₂²×#11719947	1	2	yes	—
216	C₂²	C₂²×#11719949	1	2	yes	—
217	C₄	—	4	8	no	—
218	C₄	—	106	16	no	—
219	C₄	—	212	8	no	—
220	C₄	C₂²×#11719919	1	16	yes	—
221	C₂	C₂×#16810488	1	24	yes	direct product factor
222	C₂	C₂³×#11719919	1	1	yes	characteristic
223	C₂	C₂²×#13133175	1	2	yes	—
224	C₂	C₂²×#13133204	1	1	yes	characteristic
225	C₂	C₂²×#13133206	1	1	yes	characteristic
226	C₂	C₂²×#13133208	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×#382
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
3. C₂×#16810488	C₂	direct (N × H)
10. C₂²×C₄×Dic₅₃	C₄	semidirect
12. #16810488	C₂²	direct (N × H)
27. C₂×C₄×Dic₅₃	C₂×C₄	semidirect
66. C₄×Dic₅₃	C₂²×C₄	semidirect
93. C₂²×C₅₃	#382	semidirect
113. C₂×C₅₃	C₂×#382	semidirect
131. C₅₃	C₂²×#382	semidirect
206. C₂²	#16810488	direct (N × H)
221. C₂	C₂×#16810488	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×C₄×Dic₅₃	2
C₂×#16810488	2
C₂²×#13133124	2
C₂³×C₄⋊C₄×C₅₃	2
C₂²×#382	53

Subgroups — C22×#16810488 #27542366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810488 #27542366