Subgroups — C₅₃×#2406 #20271343

Order 6784.

864 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×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#123×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#395	C₂	1	2	yes	maximal, complemented
5	C₅₃×#394	C₂	1	2	yes	maximal, complemented
6	C₅₃×#386	C₂	1	2	yes	maximal, complemented
7	C₅₃×#385	C₂	1	4	yes	maximal, complemented
8	C₅₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₅₃	—	2	8	no	—
10	C₂×#41×C₅₃	—	2	4	no	—
11	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₅₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
18	#123×C₅₃	—	2	8	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
20	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
23	C₂×C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
25	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
26	#95×C₅₃	C₂²	1	2	yes	—
27	#95×C₅₃	C₂²	1	2	yes	—
28	#95×C₅₃	C₂²	1	4	yes	—
29	C₂⁴×C₅₃	C₂³	1	2	yes	—
30	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
31	#41×C₅₃	—	2	32	no	—
32	C₂×Q₈×C₅₃	—	2	4	no	—
33	C₂×Q₈×C₅₃	—	2	2	no	—
34	C₂×D₄×C₅₃	—	2	4	no	—
35	C₂×D₄×C₅₃	—	2	4	no	—
36	C₂×D₄×C₅₃	—	2	4	no	—
37	C₂×D₄×C₅₃	—	2	2	no	—
38	C₂×D₄×C₅₃	D₄	1	8	yes	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	4	no	—
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	—	2	4	no	—
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	2	4	no	—
47	C₂²×C₄×C₅₃	D₄	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
49	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
54	C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄⋊C₄×C₅₃	—	2	8	no	—
57	C₂²⋊C₄×C₅₃	—	2	8	no	—
58	C₂²⋊C₄×C₅₃	—	2	4	no	—
59	C₂²⋊C₄×C₅₃	—	2	8	no	—
60	C₂³×C₅₃	—	2	2	no	—
61	C₂³×C₅₃	—	2	2	no	—
62	C₂³×C₅₃	—	2	2	no	—
63	C₂³×C₅₃	—	2	2	no	—
64	C₂³×C₅₃	—	2	4	no	—
65	C₂³×C₅₃	—	2	4	no	—
66	C₂³×C₅₃	C₂×D₄	1	4	yes	—
67	C₂³×C₅₃	C₂×D₄	1	2	yes	—
68	C₂³×C₅₃	C₂×D₄	1	4	yes	—
69	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
70	Q₈×C₅₃	—	2	8	no	—
71	D₄×C₅₃	—	2	8	no	—
72	D₄×C₅₃	—	2	16	no	—
73	C₂×C₄×C₅₃	—	2	8	no	—
74	C₂×C₄×C₅₃	—	2	16	no	—
75	C₂×C₄×C₅₃	—	2	8	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	4	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
90	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₅₃	#41	1	4	yes	complemented
92	C₂²×C₅₃	—	2	2	no	—
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	—	2	8	no	—
95	C₂²×C₅₃	—	2	4	no	—
96	C₂²×C₅₃	—	2	8	no	—
97	C₂²×C₅₃	—	4	2	no	—
98	C₂²×C₅₃	—	4	2	no	—
99	C₂²×C₅₃	—	4	4	no	—
100	C₂²×C₅₃	#120	1	4	yes	—
101	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
102	C₂²×C₅₃	C₂²×D₄	1	2	yes	—
103	C₂²×C₅₃	C₂×#41	1	2	yes	—
104	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	4	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	2	4	no	—
109	C₄×C₅₃	—	4	2	no	—
110	C₄×C₅₃	—	4	4	no	—
111	#2406	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₅₃	—	2	4	no	—
113	C₂×C₅₃	—	4	4	no	—
114	C₂×C₅₃	C₂×#120	1	1	yes	characteristic
115	C₂×C₅₃	#525	1	1	yes	characteristic
116	C₂×C₅₃	#536	1	1	yes	characteristic
117	C₂×C₅₃	#547	1	4	yes	—
118	C₂²×#41	C₂×C₅₃	1	1	yes	characteristic
119	C₂×#123	C₂×C₅₃	1	2	yes	complemented
120	C₂²×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic, complemented
121	#395	C₂×C₅₃	1	2	yes	complemented
122	#394	C₂×C₅₃	1	2	yes	complemented
123	#386	C₂×C₅₃	1	2	yes	complemented
124	#385	C₂×C₅₃	1	4	yes	complemented
125	#381	C₂×C₅₃	1	1	yes	characteristic, complemented
126	C₅₃	#2406	1	1	yes	characteristic, direct product factor, Sylow
127	C₂×#41	—	2	8	no	—
128	C₂×#41	—	2	4	no	—
129	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
130	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
131	C₂²×D₄	C₂²×C₅₃	1	2	yes	—
132	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
133	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
134	C₂³×C₄	C₂²×C₅₃	1	2	yes	—
135	C₂³×C₄	C₂²×C₅₃	1	1	yes	characteristic
136	#123	—	2	8	no	—
137	C₂×C₄⋊C₄	—	2	4	no	—
138	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
139	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
140	C₂×C₄⋊C₄	C₂²×C₅₃	1	4	yes	—
141	C₂×C₂²⋊C₄	C₂²×C₅₃	1	2	yes	—
142	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
143	C₂×C₂²⋊C₄	C₂²×C₅₃	1	4	yes	—
144	#95	C₂²×C₅₃	1	2	yes	—
145	#95	C₂²×C₅₃	1	2	yes	—
146	#95	C₂²×C₅₃	1	4	yes	—
147	C₂⁴	C₂³×C₅₃	1	2	yes	—
148	C₂⁴	C₂³×C₅₃	1	1	yes	characteristic
149	#41	—	2	32	no	—
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	2	2	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	2	4	no	—
154	C₂×D₄	—	2	4	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	D₄×C₅₃	1	8	yes	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	D₄×C₅₃	1	4	yes	—
166	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
167	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
168	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
170	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	2	8	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₂²⋊C₄	—	2	8	no	—
177	C₂²⋊C₄	—	2	8	no	—
178	C₂³	—	2	2	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	2	4	no	—
184	C₂³	C₂×D₄×C₅₃	1	4	yes	—
185	C₂³	C₂×D₄×C₅₃	1	4	yes	—
186	C₂³	C₂×D₄×C₅₃	1	2	yes	—
187	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
188	Q₈	—	2	8	no	—
189	D₄	—	2	8	no	—
190	D₄	—	2	16	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	16	no	—
193	C₂×C₄	—	2	8	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₅₃	1	4	yes	—
208	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
209	C₂×C₄	#41×C₅₃	1	4	yes	complemented
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	8	no	—
213	C₂²	—	2	4	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	#120×C₅₃	1	4	yes	—
219	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
220	C₂²	C₂²×D₄×C₅₃	1	2	yes	—
221	C₂²	C₂×#41×C₅₃	1	2	yes	—
222	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	4	no	—
227	C₄	—	4	2	no	—
228	C₄	—	4	4	no	—
229	C₂	—	2	4	no	—
230	C₂	—	4	4	no	—
231	C₂	C₂×#120×C₅₃	1	1	yes	characteristic
232	C₂	C₅₃×#525	1	1	yes	characteristic
233	C₂	C₅₃×#536	1	1	yes	characteristic
234	C₂	C₅₃×#547	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅₃
prime radical	C₂²×#41×C₅₃	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#123×C₅₃	C₂	semidirect
3. C₂²×C₄⋊C₄×C₅₃	C₂	semidirect
4. C₅₃×#395	C₂	semidirect
5. C₅₃×#394	C₂	semidirect
6. C₅₃×#386	C₂	semidirect
7. C₅₃×#385	C₂	semidirect
8. C₅₃×#381	C₂	semidirect
91. C₂×C₄×C₅₃	#41	semidirect
111. #2406	C₅₃	direct (N × H)
119. C₂×#123	C₂×C₅₃	semidirect
120. C₂²×C₄⋊C₄	C₂×C₅₃	semidirect
121. #395	C₂×C₅₃	semidirect
122. #394	C₂×C₅₃	semidirect
123. #386	C₂×C₅₃	semidirect
124. #385	C₂×C₅₃	semidirect
125. #381	C₂×C₅₃	semidirect
126. C₅₃	#2406	direct (N × H)
209. C₂×C₄	#41×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41×C₅₃	2
C₂×#123×C₅₃	2
C₂²×C₄⋊C₄×C₅₃	2
C₅₃×#395	2
C₅₃×#394	2
C₅₃×#386	2
C₅₃×#385	2
C₅₃×#381	2
#2406	53

Subgroups — C53×#2406 #20271343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2406 #20271343