Subgroups — C₂×C₅₃×#2641 #27242132

Order 13568.

3326 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×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#2641	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#394	C₂	1	8	yes	maximal, complemented
5	C₂×C₅₃×#390	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₅₃	C₂²	1	2	yes	—
8	C₂³×D₄×C₅₃	C₂²	1	4	yes	—
9	C₂×#127×C₅₃	—	2	8	no	—
10	C₂×#127×C₅₃	C₂²	1	8	yes	—
11	C₂×#125×C₅₃	—	2	8	no	—
12	C₂×#125×C₅₃	C₂²	1	8	yes	complemented
13	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	16	yes	—
16	C₂²×C₄²×C₅₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#394	C₂²	1	64	yes	complemented
19	C₅₃×#390	C₂²	1	32	yes	complemented
20	C₅₃×#376	C₂²	1	8	yes	complemented
21	C₂×#95×C₅₃	C₂²	1	4	yes	—
22	C₂⁵×C₅₃	C₂³	1	4	yes	—
23	C₂²×D₄×C₅₃	—	2	16	no	—
24	C₂²×D₄×C₅₃	—	2	8	no	—
25	C₂²×D₄×C₅₃	—	2	16	no	—
26	C₂²×D₄×C₅₃	—	4	16	no	—
27	C₂²×D₄×C₅₃	C₂³	1	8	yes	—
28	C₂²×D₄×C₅₃	C₂³	1	16	yes	—
29	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
30	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
31	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
32	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
33	#127×C₅₃	—	2	32	no	—
34	#125×C₅₃	—	2	32	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	16	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	8	no	—
37	C₂×C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
38	C₂×C₄⋊C₄×C₅₃	C₂³	1	16	yes	—
39	C₂×C₂²⋊C₄×C₅₃	—	4	32	no	—
40	C₂×C₂²⋊C₄×C₅₃	C₂³	1	64	yes	—
41	C₂×C₄²×C₅₃	—	2	8	no	—
42	C₂×C₄²×C₅₃	D₄	1	8	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	8	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
45	#95×C₅₃	C₂³	1	16	yes	—
46	C₂⁴×C₅₃	—	2	8	no	—
47	C₂⁴×C₅₃	—	2	16	no	—
48	C₂⁴×C₅₃	—	2	32	no	—
49	C₂⁴×C₅₃	C₂⁴	1	8	yes	—
50	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₂×D₄×C₅₃	—	2	32	no	—
52	C₂×D₄×C₅₃	—	2	16	no	—
53	C₂×D₄×C₅₃	—	2	32	no	—
54	C₂×D₄×C₅₃	—	4	32	no	—
55	C₂×D₄×C₅₃	—	4	64	no	—
56	C₂×D₄×C₅₃	—	4	32	no	—
57	C₂²×C₄×C₅₃	—	2	8	no	—
58	C₂²×C₄×C₅₃	—	2	8	no	—
59	C₂²×C₄×C₅₃	—	2	8	no	—
60	C₂²×C₄×C₅₃	—	2	16	no	—
61	C₂²×C₄×C₅₃	—	2	16	no	—
62	C₂²×C₄×C₅₃	C₂×D₄	1	8	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	4	yes	—
64	C₂²×C₄×C₅₃	#41	1	8	yes	—
65	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
66	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
67	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
68	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
69	C₄⋊C₄×C₅₃	—	2	32	no	—
70	C₄⋊C₄×C₅₃	—	2	16	no	—
71	C₂²⋊C₄×C₅₃	—	4	64	no	—
72	C₄²×C₅₃	—	2	16	no	—
73	C₄²×C₅₃	C₂×D₄	1	16	yes	—
74	C₂³×C₅₃	—	2	8	no	—
75	C₂³×C₅₃	—	2	16	no	—
76	C₂³×C₅₃	—	2	32	no	—
77	C₂³×C₅₃	—	4	16	no	—
78	C₂³×C₅₃	—	4	32	no	—
79	C₂³×C₅₃	—	4	64	no	—
80	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
81	C₂³×C₅₃	C₂×#41	1	4	yes	—
82	C₂³×C₅₃	H(5,𝔽₂)	1	8	yes	—
83	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
84	D₄×C₅₃	—	4	64	no	—
85	D₄×C₅₃	—	4	32	no	—
86	C₂×C₄×C₅₃	—	2	16	no	—
87	C₂×C₄×C₅₃	—	2	8	no	—
88	C₂×C₄×C₅₃	—	2	8	no	—
89	C₂×C₄×C₅₃	—	2	8	no	—
90	C₂×C₄×C₅₃	—	2	16	no	—
91	C₂×C₄×C₅₃	—	2	16	no	—
92	C₂×C₄×C₅₃	—	4	16	no	—
93	C₂×C₄×C₅₃	—	4	16	no	—
94	C₂×C₄×C₅₃	#124	1	16	yes	—
95	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
96	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
97	C₂×C₄×C₅₃	C₂×#41	1	8	yes	—
98	C₂×#2641	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₅₃	—	4	8	no	—
100	C₂²×C₅₃	—	4	16	no	—
101	C₂²×C₅₃	—	4	32	no	—
102	C₂²×C₅₃	—	8	32	no	—
103	C₂²×C₅₃	C₂×#124	1	4	yes	—
104	C₂²×C₅₃	#535	1	8	yes	—
105	C₂²×C₅₃	#540	1	16	yes	—
106	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
107	C₂²×C₅₃	C₂²×#41	1	2	yes	—
108	C₂²×C₅₃	C₂×H(5,𝔽₂)	1	4	yes	—
109	C₄×C₅₃	—	2	8	no	—
110	C₄×C₅₃	—	4	8	no	—
111	C₄×C₅₃	—	4	8	no	—
112	C₄×C₅₃	C₂×#124	1	8	yes	—
113	C₂²×#127	C₂×C₅₃	1	1	yes	characteristic
114	C₂²×#125	C₂×C₅₃	1	1	yes	characteristic, complemented
115	#2641	C₂×C₅₃	1	16	yes	direct product factor
116	C₂×#394	C₂×C₅₃	1	8	yes	complemented
117	C₂×#390	C₂×C₅₃	1	4	yes	complemented
118	C₂×#376	C₂×C₅₃	1	1	yes	characteristic, complemented
119	C₂×C₅₃	—	8	8	no	—
120	C₂×C₅₃	#2641	1	8	yes	direct product factor
121	C₂×C₅₃	C₂²×#124	1	1	yes	characteristic
122	C₂×C₅₃	C₂×#535	1	2	yes	—
123	C₂×C₅₃	C₂×#540	1	4	yes	—
124	C₂³×D₄	C₂²×C₅₃	1	2	yes	—
125	C₂³×D₄	C₂²×C₅₃	1	4	yes	—
126	C₂×#127	—	2	8	no	—
127	C₂×#127	C₂²×C₅₃	1	8	yes	—
128	C₂×#125	—	2	8	no	—
129	C₂×#125	C₂²×C₅₃	1	8	yes	complemented
130	C₂²×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
131	C₂²×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
132	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	16	yes	—
133	C₂²×C₄²	C₂²×C₅₃	1	2	yes	—
134	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
135	#394	C₂²×C₅₃	1	64	yes	complemented
136	#390	C₂²×C₅₃	1	32	yes	complemented
137	#376	C₂²×C₅₃	1	8	yes	complemented
138	C₂×#95	C₂²×C₅₃	1	4	yes	—
139	C₅₃	C₂×#2641	1	1	yes	characteristic, direct product factor, Sylow
140	C₂⁵	C₂³×C₅₃	1	4	yes	—
141	C₂²×D₄	—	2	16	no	—
142	C₂²×D₄	—	2	8	no	—
143	C₂²×D₄	—	2	16	no	—
144	C₂²×D₄	—	4	16	no	—
145	C₂²×D₄	C₂³×C₅₃	1	8	yes	—
146	C₂²×D₄	C₂³×C₅₃	1	16	yes	—
147	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
148	C₂³×C₄	C₂³×C₅₃	1	4	yes	—
149	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
150	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
151	#127	—	2	32	no	—
152	#125	—	2	32	no	—
153	C₂×C₄⋊C₄	—	2	16	no	—
154	C₂×C₄⋊C₄	—	2	8	no	—
155	C₂×C₄⋊C₄	C₂³×C₅₃	1	8	yes	—
156	C₂×C₄⋊C₄	C₂³×C₅₃	1	16	yes	—
157	C₂×C₂²⋊C₄	—	4	32	no	—
158	C₂×C₂²⋊C₄	C₂³×C₅₃	1	64	yes	—
159	C₂×C₄²	—	2	8	no	—
160	C₂×C₄²	D₄×C₅₃	1	8	yes	—
161	C₂×C₄²	C₂³×C₅₃	1	8	yes	—
162	C₂×C₄²	C₂³×C₅₃	1	4	yes	—
163	#95	C₂³×C₅₃	1	16	yes	—
164	C₂⁴	—	2	8	no	—
165	C₂⁴	—	2	16	no	—
166	C₂⁴	—	2	32	no	—
167	C₂⁴	C₂⁴×C₅₃	1	8	yes	—
168	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
169	C₂×D₄	—	2	32	no	—
170	C₂×D₄	—	2	16	no	—
171	C₂×D₄	—	2	32	no	—
172	C₂×D₄	—	4	32	no	—
173	C₂×D₄	—	4	64	no	—
174	C₂×D₄	—	4	32	no	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	8	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	—	2	16	no	—
179	C₂²×C₄	—	2	16	no	—
180	C₂²×C₄	C₂×D₄×C₅₃	1	8	yes	—
181	C₂²×C₄	C₂×D₄×C₅₃	1	4	yes	—
182	C₂²×C₄	#41×C₅₃	1	8	yes	—
183	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
184	C₂²×C₄	C₂⁴×C₅₃	1	8	yes	—
185	C₂²×C₄	C₂⁴×C₅₃	1	4	yes	—
186	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
187	C₄⋊C₄	—	2	32	no	—
188	C₄⋊C₄	—	2	16	no	—
189	C₂²⋊C₄	—	4	64	no	—
190	C₄²	—	2	16	no	—
191	C₄²	C₂×D₄×C₅₃	1	16	yes	—
192	C₂³	—	2	8	no	—
193	C₂³	—	2	16	no	—
194	C₂³	—	2	32	no	—
195	C₂³	—	4	16	no	—
196	C₂³	—	4	32	no	—
197	C₂³	—	4	64	no	—
198	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
199	C₂³	C₂×#41×C₅₃	1	4	yes	—
200	C₂³	H(5,𝔽₂)×C₅₃	1	8	yes	—
201	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
202	D₄	—	4	64	no	—
203	D₄	—	4	32	no	—
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	8	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	2	16	no	—
209	C₂×C₄	—	2	16	no	—
210	C₂×C₄	—	4	16	no	—
211	C₂×C₄	—	4	16	no	—
212	C₂×C₄	#124×C₅₃	1	16	yes	—
213	C₂×C₄	C₂²×D₄×C₅₃	1	8	yes	—
214	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
215	C₂×C₄	C₂×#41×C₅₃	1	8	yes	—
216	C₂²	—	4	32	no	—
217	C₂²	—	4	8	no	—
218	C₂²	—	4	16	no	—
219	C₂²	—	8	32	no	—
220	C₂²	C₂×#124×C₅₃	1	4	yes	—
221	C₂²	C₅₃×#535	1	8	yes	—
222	C₂²	C₅₃×#540	1	16	yes	—
223	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
224	C₂²	C₂²×#41×C₅₃	1	2	yes	—
225	C₂²	C₂×H(5,𝔽₂)×C₅₃	1	4	yes	—
226	C₄	—	2	8	no	—
227	C₄	—	4	8	no	—
228	C₄	—	4	8	no	—
229	C₄	C₂×#124×C₅₃	1	8	yes	—
230	C₂	—	8	8	no	—
231	C₂	C₅₃×#2641	1	8	yes	direct product factor
232	C₂	C₂²×#124×C₅₃	1	1	yes	characteristic
233	C₂	C₂×C₅₃×#535	1	2	yes	—
234	C₂	C₂×C₅₃×#540	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂²×#127×C₅₃	C₂
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
2. C₂²×#125×C₅₃	C₂	semidirect
3. C₅₃×#2641	C₂	direct (N × H)
4. C₂×C₅₃×#394	C₂	semidirect
5. C₂×C₅₃×#390	C₂	semidirect
6. C₂×C₅₃×#376	C₂	semidirect
12. C₂×#125×C₅₃	C₂²	semidirect
18. C₅₃×#394	C₂²	semidirect
19. C₅₃×#390	C₂²	semidirect
20. C₅₃×#376	C₂²	semidirect
98. C₂×#2641	C₅₃	direct (N × H)
114. C₂²×#125	C₂×C₅₃	semidirect
115. #2641	C₂×C₅₃	direct (N × H)
116. C₂×#394	C₂×C₅₃	semidirect
117. C₂×#390	C₂×C₅₃	semidirect
118. C₂×#376	C₂×C₅₃	semidirect
120. C₂×C₅₃	#2641	direct (N × H)
129. C₂×#125	C₂²×C₅₃	semidirect
135. #394	C₂²×C₅₃	semidirect
136. #390	C₂²×C₅₃	semidirect
137. #376	C₂²×C₅₃	semidirect
139. C₅₃	C₂×#2641	direct (N × H)
231. C₂	C₅₃×#2641	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×#127×C₅₃	2
C₂²×#125×C₅₃	2
C₅₃×#2641	2
C₂×C₅₃×#394	2
C₂×C₅₃×#390	2
C₂×C₅₃×#376	2
C₂×#2641	53

Subgroups — C2×C53×#2641 #27242132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2641 #27242132