Subgroups — C₂×C₅₃×#2860 #27244141

Order 13568.

2838 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#126×C₅₃	C₂	1	2	yes	maximal, complemented
2	C₅₃×#2860	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₅₃×#390	C₂	1	3	yes	maximal
4	C₂×C₅₃×#389	C₂	1	3	yes	maximal, complemented
5	C₂×C₅₃×#388	C₂	1	6	yes	maximal, complemented
6	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#126×C₅₃	—	2	16	no	—
9	C₂×#126×C₅₃	C₂²	1	16	yes	complemented
10	C₂²×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
11	C₂²×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
12	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	—
13	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	—
14	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#390	C₂²	1	24	yes	—
18	C₅₃×#389	C₂²	1	24	yes	complemented
19	C₅₃×#388	C₂²	1	48	yes	complemented
20	C₂×#95×C₅₃	C₂²	1	6	yes	—
21	C₄³×C₅₃	C₂²	1	8	yes	complemented
22	C₂⁵×C₅₃	C₂³	1	2	yes	—
23	C₂²×D₄×C₅₃	—	2	12	no	—
24	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
25	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
27	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	#126×C₅₃	—	2	64	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	24	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	12	no	—
32	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
34	C₂×C₂²⋊C₄×C₅₃	—	2	24	no	—
35	C₂×C₂²⋊C₄×C₅₃	—	2	24	no	—
36	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
37	C₂×C₂²⋊C₄×C₅₃	C₂³	1	24	yes	—
38	C₂×C₄²×C₅₃	—	2	12	no	—
39	C₂×C₄²×C₅₃	—	2	12	no	—
40	C₂×C₄²×C₅₃	D₄	1	8	yes	complemented
41	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
44	#95×C₅₃	C₂³	1	24	yes	—
45	C₂⁴×C₅₃	—	2	4	no	—
46	C₂⁴×C₅₃	—	2	12	no	—
47	C₂⁴×C₅₃	—	2	12	no	—
48	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
49	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×D₄×C₅₃	—	2	24	no	—
51	C₂×D₄×C₅₃	—	4	16	no	—
52	C₂²×C₄×C₅₃	—	2	6	no	—
53	C₂²×C₄×C₅₃	—	2	6	no	—
54	C₂²×C₄×C₅₃	—	2	6	no	—
55	C₂²×C₄×C₅₃	—	2	6	no	—
56	C₂²×C₄×C₅₃	—	2	12	no	—
57	C₂²×C₄×C₅₃	—	2	12	no	—
58	C₂²×C₄×C₅₃	—	2	12	no	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
60	C₂²×C₄×C₅₃	#41	1	12	yes	—
61	C₂²×C₄×C₅₃	#41	1	12	yes	—
62	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
64	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
66	C₄⋊C₄×C₅₃	—	2	48	no	—
67	C₄⋊C₄×C₅₃	—	2	24	no	—
68	C₂²⋊C₄×C₅₃	—	2	48	no	—
69	C₂²⋊C₄×C₅₃	—	2	48	no	—
70	C₄²×C₅₃	—	2	24	no	—
71	C₄²×C₅₃	—	2	24	no	—
72	C₄²×C₅₃	C₂×D₄	1	16	yes	complemented
73	C₂³×C₅₃	—	2	4	no	—
74	C₂³×C₅₃	—	2	12	no	—
75	C₂³×C₅₃	—	2	12	no	—
76	C₂³×C₅₃	—	4	8	no	—
77	C₂³×C₅₃	—	4	24	no	—
78	C₂³×C₅₃	—	4	24	no	—
79	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
80	C₂³×C₅₃	C₂×#41	1	6	yes	—
81	C₂³×C₅₃	C₂×#41	1	6	yes	—
82	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
83	D₄×C₅₃	—	4	16	no	—
84	C₂×C₄×C₅₃	—	2	24	no	—
85	C₂×C₄×C₅₃	—	2	24	no	—
86	C₂×C₄×C₅₃	—	2	6	no	—
87	C₂×C₄×C₅₃	—	2	6	no	—
88	C₂×C₄×C₅₃	—	2	6	no	—
89	C₂×C₄×C₅₃	—	2	6	no	—
90	C₂×C₄×C₅₃	—	2	12	no	—
91	C₂×C₄×C₅₃	—	2	12	no	—
92	C₂×C₄×C₅₃	—	2	12	no	—
93	C₂×C₄×C₅₃	—	4	24	no	—
94	C₂×C₄×C₅₃	#126	1	16	yes	complemented
95	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
96	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
97	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
98	C₂×#2860	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₃	—	4	4	no	—
100	C₂²×C₅₃	—	4	12	no	—
101	C₂²×C₅₃	—	4	12	no	—
102	C₂²×C₅₃	—	8	16	no	—
103	C₂²×C₅₃	C₂×#126	1	4	yes	—
104	C₂²×C₅₃	#529	1	12	yes	—
105	C₂²×C₅₃	#532	1	12	yes	—
106	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
107	C₂²×C₅₃	C₂²×#41	1	3	yes	—
108	C₂²×C₅₃	C₂²×#41	1	3	yes	—
109	C₄×C₅₃	—	2	12	no	—
110	C₄×C₅₃	—	2	12	no	—
111	C₄×C₅₃	—	4	12	no	—
112	C₄×C₅₃	C₂×#126	1	8	yes	complemented
113	C₂²×#126	C₂×C₅₃	1	2	yes	complemented
114	#2860	C₂×C₅₃	1	16	yes	direct product factor
115	C₂×#390	C₂×C₅₃	1	3	yes	—
116	C₂×#389	C₂×C₅₃	1	3	yes	complemented
117	C₂×#388	C₂×C₅₃	1	6	yes	complemented
118	C₂×C₄³	C₂×C₅₃	1	1	yes	characteristic, complemented
119	C₂×C₅₃	—	8	4	no	—
120	C₂×C₅₃	#2860	1	8	yes	direct product factor
121	C₂×C₅₃	C₂²×#126	1	1	yes	characteristic
122	C₂×C₅₃	C₂×#529	1	3	yes	—
123	C₂×C₅₃	C₂×#532	1	3	yes	—
124	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
125	C₂×#126	—	2	16	no	—
126	C₂×#126	C₂²×C₅₃	1	16	yes	complemented
127	C₂²×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
128	C₂²×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
129	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	6	yes	—
130	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	6	yes	—
131	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
132	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
133	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
134	#390	C₂²×C₅₃	1	24	yes	—
135	#389	C₂²×C₅₃	1	24	yes	complemented
136	#388	C₂²×C₅₃	1	48	yes	complemented
137	C₂×#95	C₂²×C₅₃	1	6	yes	—
138	C₄³	C₂²×C₅₃	1	8	yes	complemented
139	C₅₃	C₂×#2860	1	1	yes	characteristic, direct product factor, Sylow
140	C₂⁵	C₂³×C₅₃	1	2	yes	—
141	C₂²×D₄	—	2	12	no	—
142	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
143	C₂³×C₄	C₂³×C₅₃	1	6	yes	—
144	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
145	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
146	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
147	#126	—	2	64	no	—
148	C₂×C₄⋊C₄	—	2	24	no	—
149	C₂×C₄⋊C₄	—	2	12	no	—
150	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
151	C₂×C₄⋊C₄	C₂³×C₅₃	1	12	yes	—
152	C₂×C₂²⋊C₄	—	2	24	no	—
153	C₂×C₂²⋊C₄	—	2	24	no	—
154	C₂×C₂²⋊C₄	C₂³×C₅₃	1	24	yes	—
155	C₂×C₂²⋊C₄	C₂³×C₅₃	1	24	yes	—
156	C₂×C₄²	—	2	12	no	—
157	C₂×C₄²	—	2	12	no	—
158	C₂×C₄²	D₄×C₅₃	1	8	yes	complemented
159	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
160	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
161	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
162	#95	C₂³×C₅₃	1	24	yes	—
163	C₂⁴	—	2	4	no	—
164	C₂⁴	—	2	12	no	—
165	C₂⁴	—	2	12	no	—
166	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
167	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
168	C₂×D₄	—	2	24	no	—
169	C₂×D₄	—	4	16	no	—
170	C₂²×C₄	—	2	6	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	2	6	no	—
173	C₂²×C₄	—	2	6	no	—
174	C₂²×C₄	—	2	12	no	—
175	C₂²×C₄	—	2	12	no	—
176	C₂²×C₄	—	2	12	no	—
177	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
178	C₂²×C₄	#41×C₅₃	1	12	yes	—
179	C₂²×C₄	#41×C₅₃	1	12	yes	—
180	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
181	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
182	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
183	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
184	C₄⋊C₄	—	2	48	no	—
185	C₄⋊C₄	—	2	24	no	—
186	C₂²⋊C₄	—	2	48	no	—
187	C₂²⋊C₄	—	2	48	no	—
188	C₄²	—	2	24	no	—
189	C₄²	—	2	24	no	—
190	C₄²	C₂×D₄×C₅₃	1	16	yes	complemented
191	C₂³	—	2	4	no	—
192	C₂³	—	2	12	no	—
193	C₂³	—	2	12	no	—
194	C₂³	—	4	8	no	—
195	C₂³	—	4	24	no	—
196	C₂³	—	4	24	no	—
197	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
198	C₂³	C₂×#41×C₅₃	1	6	yes	—
199	C₂³	C₂×#41×C₅₃	1	6	yes	—
200	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
201	D₄	—	4	16	no	—
202	C₂×C₄	—	2	24	no	—
203	C₂×C₄	—	2	24	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	2	6	no	—
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	2	6	no	—
208	C₂×C₄	—	2	12	no	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	2	12	no	—
211	C₂×C₄	—	4	24	no	—
212	C₂×C₄	#126×C₅₃	1	16	yes	complemented
213	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
214	C₂×C₄	C₂×#41×C₅₃	1	12	yes	—
215	C₂×C₄	C₂×#41×C₅₃	1	12	yes	—
216	C₂²	—	4	4	no	—
217	C₂²	—	4	12	no	—
218	C₂²	—	4	12	no	—
219	C₂²	—	8	16	no	—
220	C₂²	C₂×#126×C₅₃	1	4	yes	—
221	C₂²	C₅₃×#529	1	12	yes	—
222	C₂²	C₅₃×#532	1	12	yes	—
223	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
224	C₂²	C₂²×#41×C₅₃	1	3	yes	—
225	C₂²	C₂²×#41×C₅₃	1	3	yes	—
226	C₄	—	2	12	no	—
227	C₄	—	2	12	no	—
228	C₄	—	4	12	no	—
229	C₄	C₂×#126×C₅₃	1	8	yes	complemented
230	C₂	—	8	4	no	—
231	C₂	C₅₃×#2860	1	8	yes	direct product factor
232	C₂	C₂²×#126×C₅₃	1	1	yes	characteristic
233	C₂	C₂×C₅₃×#529	1	3	yes	—
234	C₂	C₂×C₅₃×#532	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#126×C₅₃	C₂	semidirect
2. C₅₃×#2860	C₂	direct (N × H)
4. C₂×C₅₃×#389	C₂	semidirect
5. C₂×C₅₃×#388	C₂	semidirect
6. C₂×C₄³×C₅₃	C₂	semidirect
9. C₂×#126×C₅₃	C₂²	semidirect
18. C₅₃×#389	C₂²	semidirect
19. C₅₃×#388	C₂²	semidirect
21. C₄³×C₅₃	C₂²	semidirect
40. C₂×C₄²×C₅₃	D₄	semidirect
72. C₄²×C₅₃	C₂×D₄	semidirect
94. C₂×C₄×C₅₃	#126	semidirect
98. C₂×#2860	C₅₃	direct (N × H)
112. C₄×C₅₃	C₂×#126	semidirect
113. C₂²×#126	C₂×C₅₃	semidirect
114. #2860	C₂×C₅₃	direct (N × H)
116. C₂×#389	C₂×C₅₃	semidirect
117. C₂×#388	C₂×C₅₃	semidirect
118. C₂×C₄³	C₂×C₅₃	semidirect
120. C₂×C₅₃	#2860	direct (N × H)
126. C₂×#126	C₂²×C₅₃	semidirect
135. #389	C₂²×C₅₃	semidirect
136. #388	C₂²×C₅₃	semidirect
138. C₄³	C₂²×C₅₃	semidirect
139. C₅₃	C₂×#2860	direct (N × H)
158. C₂×C₄²	D₄×C₅₃	semidirect
190. C₄²	C₂×D₄×C₅₃	semidirect
212. C₂×C₄	#126×C₅₃	semidirect
229. C₄	C₂×#126×C₅₃	semidirect
231. C₂	C₅₃×#2860	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×#126×C₅₃	2
C₅₃×#2860	2
C₂×C₅₃×#390	2
C₂×C₅₃×#389	2
C₂×C₅₃×#388	2
C₂×C₄³×C₅₃	2
C₂×#2860	53

Subgroups — C2×C53×#2860 #27244141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2860 #27244141