Subgroups — C₅₃×#2925 #20271862

Order 6784.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#433	C₂	1	4	yes	maximal, complemented
3	C₅₃×#432	C₂	1	2	yes	maximal, complemented
4	C₅₃×#431	C₂	1	2	yes	maximal, complemented
5	C₅₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#408	C₂	1	2	yes	maximal
8	C₅₃×#405	C₂	1	2	yes	maximal, complemented
9	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
12	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
14	C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
15	C₄×D₄×C₅₃	C₄	1	2	yes	—
16	C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
17	C₄×D₄×C₅₃	C₄	1	2	yes	—
18	#117×C₅₃	C₂²	1	1	yes	characteristic
19	#117×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
22	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
25	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
26	C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
27	C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
28	C₂²⋊C₈×C₅₃	C₂²	1	4	yes	—
29	#97×C₅₃	C₂²	1	4	yes	—
30	C₄×C₈×C₅₃	C₂²	1	4	yes	—
31	#41×C₅₃	—	2	2	no	—
32	#41×C₅₃	—	2	2	no	—
33	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	M₁₆×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	—	2	4	no	—
44	C₂×C₈×C₅₃	C₂³	1	2	yes	—
45	C₂×C₈×C₅₃	C₂³	1	4	yes	—
46	C₂×C₈×C₅₃	C₂³	1	2	yes	—
47	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
52	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₄²×C₅₃	C₂×C₄	1	2	yes	—
55	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂×C₄	1	2	yes	—
57	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂³	1	2	yes	—
59	C₄²×C₅₃	C₂³	1	1	yes	characteristic
60	C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	Q₈×C₅₃	—	2	2	no	—
64	D₄×C₅₃	—	2	4	no	—
65	D₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	4	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	4	1	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#41	1	2	yes	—
81	C₂×C₄×C₅₃	#41	1	2	yes	—
82	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
83	C₈×C₅₃	—	2	2	no	—
84	C₈×C₅₃	—	2	2	no	—
85	C₈×C₅₃	—	2	4	no	—
86	C₂²×C₅₃	—	2	1	no	—
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	—	2	2	no	—
90	C₂²×C₅₃	—	2	2	no	—
91	C₂²×C₅₃	#117	1	2	yes	—
92	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
93	C₄×C₅₃	—	2	2	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	2	2	no	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	#117	1	2	yes	—
100	C₄×C₅₃	#131	1	2	yes	—
101	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
102	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
103	#2925	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂×C₅₃	—	2	2	no	—
105	C₂×C₅₃	—	4	2	no	—
106	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
107	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
108	C₂×C₅₃	#568	1	1	yes	characteristic
109	C₄×#41	C₂×C₅₃	1	1	yes	characteristic
110	#433	C₂×C₅₃	1	4	yes	complemented
111	#432	C₂×C₅₃	1	2	yes	complemented
112	#431	C₂×C₅₃	1	2	yes	complemented
113	#423	C₂×C₅₃	1	1	yes	characteristic, complemented
114	C₂×C₄⋊C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
115	#408	C₂×C₅₃	1	2	yes	—
116	#405	C₂×C₅₃	1	2	yes	complemented
117	C₅₃	#2925	1	1	yes	characteristic, direct product factor, Sylow
118	C₂×#41	C₂²×C₅₃	1	1	yes	characteristic
119	C₂×M₁₆	C₂²×C₅₃	1	2	yes	—
120	C₂²×C₈	C₂²×C₅₃	1	2	yes	—
121	C₄×Q₈	C₄×C₅₃	1	1	yes	characteristic
122	C₄×Q₈	C₄×C₅₃	1	1	yes	characteristic
123	C₄×D₄	C₄×C₅₃	1	2	yes	—
124	C₄×D₄	C₄×C₅₃	1	1	yes	characteristic
125	C₄×D₄	C₄×C₅₃	1	2	yes	—
126	C₄×D₄	C₄×C₅₃	1	1	yes	characteristic
127	#117	C₂²×C₅₃	1	1	yes	characteristic
128	#117	C₂²×C₅₃	1	2	yes	—
129	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
130	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
131	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
132	C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
133	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
134	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
135	C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
136	C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
137	C₂²⋊C₈	C₂²×C₅₃	1	4	yes	—
138	#97	C₂²×C₅₃	1	4	yes	—
139	C₄×C₈	C₂²×C₅₃	1	4	yes	—
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
143	C₂×D₄	C₂×C₄×C₅₃	1	2	yes	—
144	C₂×D₄	C₂×C₄×C₅₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₄×C₅₃	1	2	yes	—
146	C₂²×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₄×C₅₃	1	2	yes	—
148	C₂²×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
150	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
151	M₁₆	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₂³×C₅₃	1	2	yes	—
154	C₂×C₈	C₂³×C₅₃	1	4	yes	—
155	C₂×C₈	C₂³×C₅₃	1	2	yes	—
156	C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
157	C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
158	C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
159	C₄⋊C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
160	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
161	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
162	C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
163	C₄²	C₂×C₄×C₅₃	1	2	yes	—
164	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
165	C₄²	C₂×C₄×C₅₃	1	2	yes	—
166	C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
167	C₄²	C₂³×C₅₃	1	2	yes	—
168	C₄²	C₂³×C₅₃	1	1	yes	characteristic
169	C₄²	C₂³×C₅₃	1	1	yes	characteristic
170	C₂³	C₂²×C₄×C₅₃	1	2	yes	—
171	C₂³	C₂²×C₄×C₅₃	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	D₄	—	2	4	no	—
174	D₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
183	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
184	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
185	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
186	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
189	C₂×C₄	#41×C₅₃	1	2	yes	—
190	C₂×C₄	#41×C₅₃	1	2	yes	—
191	C₂×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	2	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	#117×C₅₃	1	2	yes	—
201	C₂²	C₂³×C₄×C₅₃	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	#117×C₅₃	1	2	yes	—
209	C₄	#131×C₅₃	1	2	yes	—
210	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
211	C₄	C₂×#41×C₅₃	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	4	2	no	—
214	C₂	C₂×#117×C₅₃	1	1	yes	characteristic
215	C₂	C₂×#131×C₅₃	1	1	yes	characteristic
216	C₂	C₅₃×#568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₅₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₅₃
socle	C₂²×C₅₃	C₂³×C₄
prime radical	C₂×#41×C₅₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#433	C₂	semidirect
3. C₅₃×#432	C₂	semidirect
4. C₅₃×#431	C₂	semidirect
5. C₅₃×#423	C₂	semidirect
6. C₂×C₄⋊C₈×C₅₃	C₂	semidirect
8. C₅₃×#405	C₂	semidirect
103. #2925	C₅₃	direct (N × H)
110. #433	C₂×C₅₃	semidirect
111. #432	C₂×C₅₃	semidirect
112. #431	C₂×C₅₃	semidirect
113. #423	C₂×C₅₃	semidirect
114. C₂×C₄⋊C₈	C₂×C₅₃	semidirect
116. #405	C₂×C₅₃	semidirect
117. C₅₃	#2925	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#41×C₅₃	2
C₅₃×#433	2
C₅₃×#432	2
C₅₃×#431	2
C₅₃×#423	2
C₂×C₄⋊C₈×C₅₃	2
C₅₃×#408	2
C₅₃×#405	2
#2925	53

Subgroups — C53×#2925 #20271862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2925 #20271862