Subgroups — C₂²×C₅₃×#559 #27270222

Order 13568.

8710 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#559	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#124×C₅₃	C₂	1	4	yes	maximal, complemented
4	C₂²×#123×C₅₃	C₂	1	4	yes	maximal, complemented
5	C₂²×C₄×Q₈×C₅₃	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×D₄×C₅₃	C₂	1	4	yes	maximal, complemented
7	C₂³×Q₈×C₅₃	C₂²	1	2	yes	—
8	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
9	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₅₃	C₂²	1	4	yes	—
11	C₅₃×#559	C₂²	1	256	yes	direct product factor
12	C₂×#127×C₅₃	C₂²	1	24	yes	—
13	C₂×#124×C₅₃	C₂²	1	96	yes	complemented
14	C₂×#123×C₅₃	C₂²	1	96	yes	complemented
15	C₂×C₄×Q₈×C₅₃	C₂²	1	48	yes	complemented
16	C₂×C₄×D₄×C₅₃	C₂²	1	96	yes	complemented
17	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
19	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	8	yes	—
20	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
21	C₂²×C₄²×C₅₃	C₂²	1	4	yes	—
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂⁵×C₅₃	C₂³	1	4	yes	—
24	C₂²×Q₈×C₅₃	—	2	16	no	—
25	C₂²×Q₈×C₅₃	C₂³	1	24	yes	—
26	C₂²×D₄×C₅₃	—	2	32	no	—
27	C₂²×D₄×C₅₃	—	4	8	no	—
28	C₂²×D₄×C₅₃	C₂³	1	48	yes	—
29	C₂²×D₄×C₅₃	C₂³	1	24	yes	—
30	C₂³×C₄×C₅₃	—	4	16	no	—
31	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
33	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₅₃	C₂³	1	48	yes	—
35	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
36	#127×C₅₃	C₂³	1	64	yes	—
37	#124×C₅₃	C₂³	1	256	yes	complemented
38	#123×C₅₃	C₂³	1	256	yes	complemented
39	C₄×Q₈×C₅₃	C₂³	1	128	yes	complemented
40	C₄×D₄×C₅₃	C₂³	1	256	yes	complemented
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
42	C₂×C₄⋊C₄×C₅₃	C₂³	1	48	yes	—
43	C₂×C₂²⋊C₄×C₅₃	C₂³	1	96	yes	—
44	C₂×C₂²⋊C₄×C₅₃	C₂³	1	48	yes	—
45	C₂×C₄²×C₅₃	C₂³	1	48	yes	—
46	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
47	C₂⁴×C₅₃	—	2	32	no	—
48	C₂⁴×C₅₃	—	2	16	no	—
49	C₂⁴×C₅₃	C₂⁴	1	24	yes	—
50	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₂×Q₈×C₅₃	—	2	96	no	—
52	C₂×Q₈×C₅₃	C₂⁴	1	32	yes	—
53	C₂×D₄×C₅₃	—	2	192	no	—
54	C₂×D₄×C₅₃	—	4	48	no	—
55	C₂×D₄×C₅₃	C₂⁴	1	64	yes	—
56	C₂×D₄×C₅₃	C₂⁴	1	32	yes	—
57	C₂²×C₄×C₅₃	—	2	16	no	—
58	C₂²×C₄×C₅₃	—	2	16	no	—
59	C₂²×C₄×C₅₃	—	2	4	no	—
60	C₂²×C₄×C₅₃	—	4	96	no	—
61	C₂²×C₄×C₅₃	#41	1	16	yes	—
62	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
64	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	64	yes	—
66	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
67	C₄⋊C₄×C₅₃	C₂⁴	1	32	yes	—
68	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
69	C₂²⋊C₄×C₅₃	C₂⁴	1	128	yes	—
70	C₂²⋊C₄×C₅₃	C₂⁴	1	64	yes	—
71	C₄²×C₅₃	C₂⁴	1	64	yes	—
72	C₄²×C₅₃	C₂⁴	1	16	yes	—
73	C₂³×C₅₃	—	2	96	no	—
74	C₂³×C₅₃	—	2	48	no	—
75	C₂³×C₅₃	—	4	64	no	—
76	C₂³×C₅₃	C₂×#41	1	8	yes	—
77	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
78	C₂³×C₅₃	C₂⁵	1	16	yes	—
79	C₂³×C₅₃	C₂⁵	1	3	yes	—
80	Q₈×C₅₃	—	2	64	no	—
81	D₄×C₅₃	—	2	128	no	—
82	D₄×C₅₃	—	4	32	no	—
83	C₂×C₄×C₅₃	—	2	48	no	—
84	C₂×C₄×C₅₃	—	2	48	no	—
85	C₂×C₄×C₅₃	—	2	12	no	—
86	C₂×C₄×C₅₃	—	4	64	no	—
87	C₂×C₄×C₅₃	C₂×#41	1	48	yes	—
88	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
89	C₂×C₄×C₅₃	C₂⁵	1	4	yes	—
90	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
91	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
92	C₂²×#559	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂²×C₅₃	—	2	32	no	—
94	C₂²×C₅₃	—	2	16	no	—
95	C₂²×C₅₃	—	4	96	no	—
96	C₂²×C₅₃	#559	1	16	yes	direct product factor
97	C₂²×C₅₃	C₂²×#41	1	12	yes	—
98	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	6	yes	—
99	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
100	C₄×C₅₃	—	2	16	no	—
101	C₄×C₅₃	—	2	16	no	—
102	C₄×C₅₃	—	2	4	no	—
103	C₄×C₅₃	C₂²×#41	1	16	yes	—
104	C₂×#559	C₂×C₅₃	1	48	yes	direct product factor
105	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic
106	C₂²×#124	C₂×C₅₃	1	4	yes	complemented
107	C₂²×#123	C₂×C₅₃	1	4	yes	complemented
108	C₂²×C₄×Q₈	C₂×C₅₃	1	2	yes	complemented
109	C₂²×C₄×D₄	C₂×C₅₃	1	4	yes	complemented
110	C₂×C₅₃	—	4	16	no	—
111	C₂×C₅₃	C₂×#559	1	12	yes	direct product factor
112	C₂×C₅₃	C₂³×#41	1	2	yes	—
113	C₂×C₅₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
114	C₂³×Q₈	C₂²×C₅₃	1	2	yes	—
115	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
116	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
117	C₂⁴×C₄	C₂²×C₅₃	1	4	yes	—
118	#559	C₂²×C₅₃	1	256	yes	direct product factor
119	C₂×#127	C₂²×C₅₃	1	24	yes	—
120	C₂×#124	C₂²×C₅₃	1	96	yes	complemented
121	C₂×#123	C₂²×C₅₃	1	96	yes	complemented
122	C₂×C₄×Q₈	C₂²×C₅₃	1	48	yes	complemented
123	C₂×C₄×D₄	C₂²×C₅₃	1	96	yes	complemented
124	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
125	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
126	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	8	yes	—
127	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
128	C₂²×C₄²	C₂²×C₅₃	1	4	yes	—
129	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
130	C₅₃	C₂²×#559	1	1	yes	characteristic, direct product factor, Sylow
131	C₂⁵	C₂³×C₅₃	1	4	yes	—
132	C₂²×Q₈	—	2	16	no	—
133	C₂²×Q₈	C₂³×C₅₃	1	24	yes	—
134	C₂²×D₄	—	2	32	no	—
135	C₂²×D₄	—	4	8	no	—
136	C₂²×D₄	C₂³×C₅₃	1	48	yes	—
137	C₂²×D₄	C₂³×C₅₃	1	24	yes	—
138	C₂³×C₄	—	4	16	no	—
139	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
140	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
141	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
142	C₂³×C₄	C₂³×C₅₃	1	48	yes	—
143	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
144	#127	C₂³×C₅₃	1	64	yes	—
145	#124	C₂³×C₅₃	1	256	yes	complemented
146	#123	C₂³×C₅₃	1	256	yes	complemented
147	C₄×Q₈	C₂³×C₅₃	1	128	yes	complemented
148	C₄×D₄	C₂³×C₅₃	1	256	yes	complemented
149	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
150	C₂×C₄⋊C₄	C₂³×C₅₃	1	48	yes	—
151	C₂×C₂²⋊C₄	C₂³×C₅₃	1	96	yes	—
152	C₂×C₂²⋊C₄	C₂³×C₅₃	1	48	yes	—
153	C₂×C₄²	C₂³×C₅₃	1	48	yes	—
154	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
155	C₂⁴	—	2	32	no	—
156	C₂⁴	—	2	16	no	—
157	C₂⁴	C₂⁴×C₅₃	1	24	yes	—
158	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
159	C₂×Q₈	—	2	96	no	—
160	C₂×Q₈	C₂⁴×C₅₃	1	32	yes	—
161	C₂×D₄	—	2	192	no	—
162	C₂×D₄	—	4	48	no	—
163	C₂×D₄	C₂⁴×C₅₃	1	64	yes	—
164	C₂×D₄	C₂⁴×C₅₃	1	32	yes	—
165	C₂²×C₄	—	2	16	no	—
166	C₂²×C₄	—	2	16	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	4	96	no	—
169	C₂²×C₄	#41×C₅₃	1	16	yes	—
170	C₂²×C₄	C₂⁴×C₅₃	1	24	yes	—
171	C₂²×C₄	C₂⁴×C₅₃	1	24	yes	—
172	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
173	C₂²×C₄	C₂⁴×C₅₃	1	64	yes	—
174	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
175	C₄⋊C₄	C₂⁴×C₅₃	1	32	yes	—
176	C₄⋊C₄	C₂⁴×C₅₃	1	64	yes	—
177	C₂²⋊C₄	C₂⁴×C₅₃	1	128	yes	—
178	C₂²⋊C₄	C₂⁴×C₅₃	1	64	yes	—
179	C₄²	C₂⁴×C₅₃	1	64	yes	—
180	C₄²	C₂⁴×C₅₃	1	16	yes	—
181	C₂³	—	2	96	no	—
182	C₂³	—	2	48	no	—
183	C₂³	—	4	64	no	—
184	C₂³	C₂×#41×C₅₃	1	8	yes	—
185	C₂³	H(5,𝔽₂)×C₅₃	1	4	yes	—
186	C₂³	C₂⁵×C₅₃	1	16	yes	—
187	C₂³	C₂⁵×C₅₃	1	3	yes	—
188	Q₈	—	2	64	no	—
189	D₄	—	2	128	no	—
190	D₄	—	4	32	no	—
191	C₂×C₄	—	2	48	no	—
192	C₂×C₄	—	2	48	no	—
193	C₂×C₄	—	2	12	no	—
194	C₂×C₄	—	4	64	no	—
195	C₂×C₄	C₂×#41×C₅₃	1	48	yes	—
196	C₂×C₄	C₂⁵×C₅₃	1	16	yes	—
197	C₂×C₄	C₂⁵×C₅₃	1	16	yes	—
198	C₂×C₄	C₂⁵×C₅₃	1	4	yes	—
199	C₂×C₄	C₂⁵×C₅₃	1	8	yes	—
200	C₂²	—	2	32	no	—
201	C₂²	—	2	16	no	—
202	C₂²	—	4	96	no	—
203	C₂²	C₅₃×#559	1	16	yes	direct product factor
204	C₂²	C₂²×#41×C₅₃	1	12	yes	—
205	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	6	yes	—
206	C₂²	C₂⁶×C₅₃	1	1	yes	characteristic
207	C₄	—	2	16	no	—
208	C₄	—	2	16	no	—
209	C₄	—	2	4	no	—
210	C₄	C₂²×#41×C₅₃	1	16	yes	—
211	C₂	—	4	16	no	—
212	C₂	C₂×C₅₃×#559	1	12	yes	direct product factor
213	C₂	C₂³×#41×C₅₃	1	2	yes	—
214	C₂	C₂²×H(5,𝔽₂)×C₅₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅₃
prime radical	C₂²×#127×C₅₃	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅₃×#559	C₂	direct (N × H)
3. C₂²×#124×C₅₃	C₂	semidirect
4. C₂²×#123×C₅₃	C₂	semidirect
5. C₂²×C₄×Q₈×C₅₃	C₂	semidirect
6. C₂²×C₄×D₄×C₅₃	C₂	semidirect
11. C₅₃×#559	C₂²	direct (N × H)
13. C₂×#124×C₅₃	C₂²	semidirect
14. C₂×#123×C₅₃	C₂²	semidirect
15. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
16. C₂×C₄×D₄×C₅₃	C₂²	semidirect
37. #124×C₅₃	C₂³	semidirect
38. #123×C₅₃	C₂³	semidirect
39. C₄×Q₈×C₅₃	C₂³	semidirect
40. C₄×D₄×C₅₃	C₂³	semidirect
92. C₂²×#559	C₅₃	direct (N × H)
96. C₂²×C₅₃	#559	direct (N × H)
104. C₂×#559	C₂×C₅₃	direct (N × H)
106. C₂²×#124	C₂×C₅₃	semidirect
107. C₂²×#123	C₂×C₅₃	semidirect
108. C₂²×C₄×Q₈	C₂×C₅₃	semidirect
109. C₂²×C₄×D₄	C₂×C₅₃	semidirect
111. C₂×C₅₃	C₂×#559	direct (N × H)
118. #559	C₂²×C₅₃	direct (N × H)
120. C₂×#124	C₂²×C₅₃	semidirect
121. C₂×#123	C₂²×C₅₃	semidirect
122. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
123. C₂×C₄×D₄	C₂²×C₅₃	semidirect
130. C₅₃	C₂²×#559	direct (N × H)
145. #124	C₂³×C₅₃	semidirect
146. #123	C₂³×C₅₃	semidirect
147. C₄×Q₈	C₂³×C₅₃	semidirect
148. C₄×D₄	C₂³×C₅₃	semidirect
203. C₂²	C₅₃×#559	direct (N × H)
212. C₂	C₂×C₅₃×#559	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×C₅₃×#559	2
C₂²×#127×C₅₃	2
C₂²×#124×C₅₃	2
C₂²×#123×C₅₃	2
C₂²×C₄×Q₈×C₅₃	2
C₂²×C₄×D₄×C₅₃	2
C₂²×#559	53

Subgroups — C22×C53×#559 #27270222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#559 #27270222