Subgroups — C₅₃×#7554 #27215066

Order 13568.

3158 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#95×C₅₃	C₂	1	6	yes	maximal
3	C₂⁴×C₄×C₅₃	C₂²	1	4	yes	—
4	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
5	C₂²×C₄²×C₅₃	—	2	12	no	—
6	C₂²×C₄²×C₅₃	C₄	1	4	yes	complemented
7	C₂×#95×C₅₃	—	2	12	no	—
8	C₂×#95×C₅₃	—	2	12	no	—
9	C₂×#95×C₅₃	C₄	1	24	yes	complemented
10	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
11	C₂³×C₄×C₅₃	—	2	12	no	—
12	C₂³×C₄×C₅₃	—	2	12	no	—
13	C₂³×C₄×C₅₃	—	2	12	no	—
14	C₂³×C₄×C₅₃	—	2	12	no	—
15	C₂³×C₄×C₅₃	—	2	24	no	—
16	C₂³×C₄×C₅₃	C₂×C₄	1	24	yes	—
17	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
18	C₂³×C₄×C₅₃	C₂×C₄	1	12	yes	—
19	C₂³×C₄×C₅₃	D₄	1	12	yes	—
20	C₂³×C₄×C₅₃	D₄	1	6	yes	—
21	C₂³×C₄×C₅₃	Q₈	1	6	yes	—
22	C₂×C₄²×C₅₃	—	2	32	no	—
23	C₂×C₄²×C₅₃	—	2	24	no	—
24	#95×C₅₃	—	2	48	no	—
25	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂⁴×C₅₃	C₂²×C₄	1	6	yes	—
27	C₂⁴×C₅₃	C₂×D₄	1	6	yes	—
28	C₂⁴×C₅₃	C₂×D₄	1	3	yes	—
29	C₂⁴×C₅₃	C₂×Q₈	1	3	yes	—
30	C₂⁴×C₅₃	#41	1	12	yes	—
31	C₂²×C₄×C₅₃	—	2	12	no	—
32	C₂²×C₄×C₅₃	—	2	12	no	—
33	C₂²×C₄×C₅₃	—	2	48	no	—
34	C₂²×C₄×C₅₃	—	2	24	no	—
35	C₂²×C₄×C₅₃	—	2	24	no	—
36	C₂²×C₄×C₅₃	—	2	24	no	—
37	C₂²×C₄×C₅₃	—	2	24	no	—
38	C₂²×C₄×C₅₃	—	2	24	no	—
39	C₂²×C₄×C₅₃	—	2	24	no	—
40	C₂²×C₄×C₅₃	—	2	48	no	—
41	C₂²×C₄×C₅₃	—	2	48	no	—
42	C₂²×C₄×C₅₃	—	4	4	no	—
43	C₂²×C₄×C₅₃	—	4	24	no	—
44	C₂²×C₄×C₅₃	—	4	12	no	—
45	C₂²×C₄×C₅₃	—	4	12	no	—
46	C₂²×C₄×C₅₃	—	4	8	no	—
47	C₂²×C₄×C₅₃	—	4	4	no	—
48	C₂²×C₄×C₅₃	C₄²	1	12	yes	—
49	C₂²×C₄×C₅₃	C₄²	1	16	yes	complemented
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	24	yes	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	12	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	12	yes	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	24	yes	—
54	C₄²×C₅₃	—	2	32	no	—
55	C₂³×C₅₃	C₂×C₄²	1	3	yes	—
56	C₂³×C₅₃	C₂×C₄²	1	4	yes	—
57	C₂³×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
58	C₂³×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
59	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
60	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
61	C₂³×C₅₃	#117	1	6	yes	—
62	C₂³×C₅₃	#117	1	12	yes	—
63	C₂³×C₅₃	C₄×D₄	1	24	yes	—
64	C₂³×C₅₃	C₄×D₄	1	12	yes	—
65	C₂³×C₅₃	C₄×Q₈	1	12	yes	—
66	C₂³×C₅₃	#120	1	4	yes	—
67	C₂³×C₅₃	#121	1	12	yes	—
68	C₂³×C₅₃	C₂²⋊Q₈	1	12	yes	—
69	C₂³×C₅₃	#123	1	12	yes	—
70	C₂³×C₅₃	#124	1	6	yes	—
71	C₂³×C₅₃	#125	1	6	yes	—
72	C₂³×C₅₃	#126	1	8	yes	—
73	C₂³×C₅₃	#127	1	1	yes	characteristic
74	C₂³×C₅₃	C₄⋊Q₈	1	3	yes	—
75	C₂×C₄×C₅₃	—	2	24	no	—
76	C₂×C₄×C₅₃	—	2	12	no	—
77	C₂×C₄×C₅₃	—	2	24	no	—
78	C₂×C₄×C₅₃	—	2	12	no	—
79	C₂×C₄×C₅₃	—	2	48	no	—
80	C₂×C₄×C₅₃	—	2	48	no	—
81	C₂×C₄×C₅₃	—	2	48	no	—
82	C₂×C₄×C₅₃	—	4	24	no	—
83	C₂×C₄×C₅₃	—	4	48	no	—
84	C₂×C₄×C₅₃	—	4	24	no	—
85	C₂×C₄×C₅₃	#95	1	24	yes	complemented
86	#7554	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₅₃	C₄³	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×#95	1	3	yes	—
89	C₂²×C₅₃	#376	1	3	yes	—
90	C₂²×C₅₃	C₄×C₂²⋊C₄	1	6	yes	—
91	C₂²×C₅₃	C₄×C₂²⋊C₄	1	12	yes	—
92	C₂²×C₅₃	C₄×C₄⋊C₄	1	6	yes	—
93	C₂²×C₅₃	C₄×C₄⋊C₄	1	12	yes	—
94	C₂²×C₅₃	#379	1	1	yes	characteristic
95	C₂²×C₅₃	#380	1	6	yes	—
96	C₂²×C₅₃	#381	1	3	yes	—
97	C₂²×C₅₃	#382	1	3	yes	—
98	C₂²×C₅₃	#383	1	2	yes	—
99	C₂²×C₅₃	#384	1	1	yes	characteristic
100	C₂²×C₅₃	#385	1	12	yes	—
101	C₂²×C₅₃	#386	1	12	yes	—
102	C₂²×C₅₃	#387	1	24	yes	—
103	C₂²×C₅₃	#388	1	24	yes	—
104	C₂²×C₅₃	#389	1	12	yes	—
105	C₂²×C₅₃	#390	1	6	yes	—
106	C₂²×C₅₃	#391	1	6	yes	—
107	C₄×C₅₃	—	2	24	no	—
108	C₄×C₅₃	—	4	16	no	—
109	C₂³×C₄²	C₂×C₅₃	1	1	yes	characteristic
110	C₂²×#95	C₂×C₅₃	1	6	yes	—
111	C₂×C₅₃	C₄×#95	1	3	yes	—
112	C₂×C₅₃	C₂²⋊#95	1	3	yes	—
113	C₂×C₅₃	#1434	1	6	yes	—
114	C₂×C₅₃	C₄⋊#95	1	3	yes	—
115	C₂×C₅₃	#1436	1	12	yes	—
116	C₂×C₅₃	#1437	1	4	yes	—
117	C₂⁴×C₄	C₂²×C₅₃	1	4	yes	—
118	C₂⁴×C₄	C₂²×C₅₃	1	3	yes	—
119	C₂²×C₄²	—	2	12	no	—
120	C₂²×C₄²	C₄×C₅₃	1	4	yes	complemented
121	C₂×#95	—	2	12	no	—
122	C₂×#95	—	2	12	no	—
123	C₂×#95	C₄×C₅₃	1	24	yes	complemented
124	C₅₃	#7554	1	1	yes	characteristic, direct product factor, Sylow
125	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
126	C₂³×C₄	—	2	12	no	—
127	C₂³×C₄	—	2	12	no	—
128	C₂³×C₄	—	2	12	no	—
129	C₂³×C₄	—	2	12	no	—
130	C₂³×C₄	—	2	24	no	—
131	C₂³×C₄	C₂×C₄×C₅₃	1	24	yes	—
132	C₂³×C₄	C₂×C₄×C₅₃	1	12	yes	—
133	C₂³×C₄	C₂×C₄×C₅₃	1	6	yes	—
134	C₂³×C₄	D₄×C₅₃	1	6	yes	—
135	C₂³×C₄	D₄×C₅₃	1	12	yes	—
136	C₂³×C₄	Q₈×C₅₃	1	6	yes	—
137	C₂×C₄²	—	2	32	no	—
138	C₂×C₄²	—	2	24	no	—
139	#95	—	2	48	no	—
140	C₂⁴	C₂²×C₄×C₅₃	1	1	yes	characteristic
141	C₂⁴	C₂²×C₄×C₅₃	1	6	yes	—
142	C₂⁴	C₂×D₄×C₅₃	1	6	yes	—
143	C₂⁴	C₂×D₄×C₅₃	1	3	yes	—
144	C₂⁴	C₂×Q₈×C₅₃	1	3	yes	—
145	C₂⁴	#41×C₅₃	1	12	yes	—
146	C₂²×C₄	—	2	12	no	—
147	C₂²×C₄	—	2	12	no	—
148	C₂²×C₄	—	2	48	no	—
149	C₂²×C₄	—	2	24	no	—
150	C₂²×C₄	—	2	24	no	—
151	C₂²×C₄	—	2	24	no	—
152	C₂²×C₄	—	2	24	no	—
153	C₂²×C₄	—	2	24	no	—
154	C₂²×C₄	—	2	24	no	—
155	C₂²×C₄	—	2	48	no	—
156	C₂²×C₄	—	2	48	no	—
157	C₂²×C₄	—	4	4	no	—
158	C₂²×C₄	—	4	24	no	—
159	C₂²×C₄	—	4	12	no	—
160	C₂²×C₄	—	4	12	no	—
161	C₂²×C₄	—	4	8	no	—
162	C₂²×C₄	—	4	4	no	—
163	C₂²×C₄	C₄²×C₅₃	1	12	yes	—
164	C₂²×C₄	C₄²×C₅₃	1	16	yes	complemented
165	C₂²×C₄	C₂²⋊C₄×C₅₃	1	24	yes	—
166	C₂²×C₄	C₂²⋊C₄×C₅₃	1	12	yes	—
167	C₂²×C₄	C₄⋊C₄×C₅₃	1	12	yes	—
168	C₂²×C₄	C₄⋊C₄×C₅₃	1	24	yes	—
169	C₄²	—	2	32	no	—
170	C₂³	C₂×C₄²×C₅₃	1	3	yes	—
171	C₂³	C₂×C₄²×C₅₃	1	4	yes	—
172	C₂³	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
173	C₂³	C₂×C₂²⋊C₄×C₅₃	1	6	yes	—
174	C₂³	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
175	C₂³	C₂×C₄⋊C₄×C₅₃	1	6	yes	—
176	C₂³	#117×C₅₃	1	6	yes	—
177	C₂³	#117×C₅₃	1	12	yes	—
178	C₂³	C₄×D₄×C₅₃	1	24	yes	—
179	C₂³	C₄×D₄×C₅₃	1	12	yes	—
180	C₂³	C₄×Q₈×C₅₃	1	12	yes	—
181	C₂³	#120×C₅₃	1	4	yes	—
182	C₂³	#121×C₅₃	1	12	yes	—
183	C₂³	C₂²⋊Q₈×C₅₃	1	12	yes	—
184	C₂³	#123×C₅₃	1	12	yes	—
185	C₂³	#124×C₅₃	1	6	yes	—
186	C₂³	#125×C₅₃	1	6	yes	—
187	C₂³	#126×C₅₃	1	8	yes	—
188	C₂³	#127×C₅₃	1	1	yes	characteristic
189	C₂³	C₄⋊Q₈×C₅₃	1	3	yes	—
190	C₂×C₄	—	2	24	no	—
191	C₂×C₄	—	2	12	no	—
192	C₂×C₄	—	2	24	no	—
193	C₂×C₄	—	2	12	no	—
194	C₂×C₄	—	2	48	no	—
195	C₂×C₄	—	2	48	no	—
196	C₂×C₄	—	2	48	no	—
197	C₂×C₄	—	4	24	no	—
198	C₂×C₄	—	4	48	no	—
199	C₂×C₄	—	4	24	no	—
200	C₂×C₄	#95×C₅₃	1	24	yes	complemented
201	C₂²	C₄³×C₅₃	1	1	yes	characteristic
202	C₂²	C₂×#95×C₅₃	1	3	yes	—
203	C₂²	C₅₃×#376	1	3	yes	—
204	C₂²	C₄×C₂²⋊C₄×C₅₃	1	6	yes	—
205	C₂²	C₄×C₂²⋊C₄×C₅₃	1	12	yes	—
206	C₂²	C₄×C₄⋊C₄×C₅₃	1	6	yes	—
207	C₂²	C₄×C₄⋊C₄×C₅₃	1	12	yes	—
208	C₂²	C₅₃×#379	1	1	yes	characteristic
209	C₂²	C₅₃×#380	1	6	yes	—
210	C₂²	C₅₃×#381	1	3	yes	—
211	C₂²	C₅₃×#382	1	3	yes	—
212	C₂²	C₅₃×#383	1	2	yes	—
213	C₂²	C₅₃×#384	1	1	yes	characteristic
214	C₂²	C₅₃×#385	1	12	yes	—
215	C₂²	C₅₃×#386	1	12	yes	—
216	C₂²	C₅₃×#387	1	24	yes	—
217	C₂²	C₅₃×#388	1	24	yes	—
218	C₂²	C₅₃×#389	1	12	yes	—
219	C₂²	C₅₃×#390	1	6	yes	—
220	C₂²	C₅₃×#391	1	6	yes	—
221	C₄	—	2	24	no	—
222	C₄	—	4	16	no	—
223	C₂	C₄×#95×C₅₃	1	3	yes	—
224	C₂	C₅₃×C₂²⋊#95	1	3	yes	—
225	C₂	C₅₃×#1434	1	6	yes	—
226	C₂	C₅₃×C₄⋊#95	1	3	yes	—
227	C₂	C₅₃×#1436	1	12	yes	—
228	C₂	C₅₃×#1437	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₅₃	C₂³
commutator	C₂²	C₄³×C₅₃
Frattini, cosocle	C₂⁵	C₂³×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₂²×C₄²×C₅₃	C₄	semidirect
9. C₂×#95×C₅₃	C₄	semidirect
49. C₂²×C₄×C₅₃	C₄²	semidirect
85. C₂×C₄×C₅₃	#95	semidirect
86. #7554	C₅₃	direct (N × H)
120. C₂²×C₄²	C₄×C₅₃	semidirect
123. C₂×#95	C₄×C₅₃	semidirect
124. C₅₃	#7554	direct (N × H)
164. C₂²×C₄	C₄²×C₅₃	semidirect
200. C₂×C₄	#95×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₄²×C₅₃	2
C₂²×#95×C₅₃	2
#7554	53

Subgroups — C53×#7554 #27215066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#7554 #27215066