Subgroups — C₅₃×#17353 #27224865

Order 13568.

1238 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#520	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2982	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1460	C₂	1	6	yes	maximal, complemented
4	C₅₃×#1458	C₂	1	6	yes	maximal, complemented
5	C₂²×#41×C₅₃	C₄	1	1	yes	characteristic
6	C₅₃×#520	—	2	8	no	—
7	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₅₃	C₄	1	3	yes	—
10	C₂×C₄×D₄×C₅₃	C₂²	1	3	yes	—
11	C₂×#117×C₅₃	C₂²	1	3	yes	—
12	C₂²×C₄⋊C₄×C₅₃	C₄	1	3	yes	—
13	C₈×Q₈×C₅₃	—	2	8	no	—
14	C₅₃×#431	C₂²	1	12	yes	—
15	C₅₃×#324	C₂²	1	4	yes	—
16	C₅₃×#323	C₂²	1	12	yes	—
17	C₂×#41×C₅₃	—	2	4	no	—
18	C₂×#41×C₅₃	—	2	4	no	—
19	C₂×#41×C₅₃	C₈	1	8	yes	complemented
20	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×D₄×C₅₃	C₂×C₄	1	3	yes	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
24	C₄×Q₈×C₅₃	—	2	4	no	—
25	C₄×Q₈×C₅₃	D₄	1	8	yes	—
26	C₄×D₄×C₅₃	—	2	12	no	—
27	C₄×D₄×C₅₃	—	2	12	no	—
28	#117×C₅₃	—	2	12	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	12	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
33	C₂×C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
35	C₂×C₂²⋊C₄×C₅₃	C₂³	1	3	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
38	C₄⋊C₈×C₅₃	—	2	12	no	—
39	C₂²⋊C₈×C₅₃	C₂³	1	8	yes	—
40	C₄×C₈×C₅₃	—	2	12	no	—
41	C₂⁴×C₅₃	C₂²×C₄	1	3	yes	—
42	#41×C₅₃	—	2	32	no	—
43	C₂×Q₈×C₅₃	C₂²⋊C₄	1	4	yes	—
44	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
45	C₂×Q₈×C₅₃	M₁₆	1	4	yes	—
46	C₂×D₄×C₅₃	—	2	6	no	—
47	C₂×D₄×C₅₃	—	2	6	no	—
48	C₂×D₄×C₅₃	C₂×C₈	1	12	yes	—
49	C₂²×C₄×C₅₃	—	2	6	no	—
50	C₂²×C₄×C₅₃	—	2	6	no	—
51	C₂²×C₄×C₅₃	—	2	12	no	—
52	C₂²×C₄×C₅₃	—	8	6	no	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	12	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
59	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₅₃	—	2	8	no	—
61	C₄⋊C₄×C₅₃	—	2	6	no	—
62	C₄⋊C₄×C₅₃	—	2	6	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	12	yes	—
65	C₂²⋊C₄×C₅₃	—	2	6	no	—
66	C₂²⋊C₄×C₅₃	—	2	6	no	—
67	C₄²×C₅₃	—	2	6	no	—
68	C₄²×C₅₃	C₂×D₄	1	12	yes	—
69	C₂³×C₅₃	—	2	3	no	—
70	C₂³×C₅₃	—	2	3	no	—
71	C₂³×C₅₃	—	4	3	no	—
72	C₂³×C₅₃	—	4	3	no	—
73	C₂³×C₅₃	C₂²×C₈	1	6	yes	—
74	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
75	Q₈×C₅₃	C₂²⋊C₈	1	16	yes	—
76	D₄×C₅₃	—	2	24	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	24	no	—
80	C₂×C₄×C₅₃	—	2	3	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	4	3	no	—
83	C₂×C₄×C₅₃	—	4	1	no	—
84	C₂×C₄×C₅₃	—	4	3	no	—
85	C₂×C₄×C₅₃	—	4	1	no	—
86	C₂×C₄×C₅₃	—	8	12	no	—
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
88	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
89	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
91	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
92	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
93	C₈×C₅₃	—	4	8	no	—
94	#17353	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂²×C₅₃	—	2	1	no	—
96	C₂²×C₅₃	—	2	1	no	—
97	C₂²×C₅₃	—	2	6	no	—
98	C₂²×C₅₃	—	4	6	no	—
99	C₂²×C₅₃	—	4	6	no	—
100	C₂²×C₅₃	—	4	3	no	—
101	C₂²×C₅₃	—	4	3	no	—
102	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
104	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	2	6	no	—
108	C₄×C₅₃	—	4	1	no	—
109	C₄×C₅₃	—	4	3	no	—
110	C₄×C₅₃	C₂×C₂²⋊C₈	1	12	yes	—
111	C₂×#520	C₂×C₅₃	1	1	yes	characteristic
112	#2982	C₂×C₅₃	1	2	yes	complemented
113	#1460	C₂×C₅₃	1	6	yes	complemented
114	#1458	C₂×C₅₃	1	6	yes	complemented
115	C₂×C₅₃	—	2	1	no	—
116	C₂×C₅₃	—	2	1	no	—
117	C₂×C₅₃	—	4	6	no	—
118	C₂×C₅₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
119	C₂×C₅₃	#2884	1	1	yes	characteristic
120	C₂×C₅₃	#2888	1	1	yes	characteristic
121	C₂²×#41	C₄×C₅₃	1	1	yes	characteristic
122	#520	—	2	8	no	—
123	C₂×C₄×Q₈	C₄×C₅₃	1	1	yes	characteristic
124	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
125	C₂×C₄×D₄	C₄×C₅₃	1	3	yes	—
126	C₂×C₄×D₄	C₂²×C₅₃	1	3	yes	—
127	C₂×#117	C₂²×C₅₃	1	3	yes	—
128	C₂²×C₄⋊C₄	C₄×C₅₃	1	3	yes	—
129	C₈×Q₈	—	2	8	no	—
130	#431	C₂²×C₅₃	1	12	yes	—
131	#324	C₂²×C₅₃	1	4	yes	—
132	#323	C₂²×C₅₃	1	12	yes	—
133	C₅₃	#17353	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×#41	—	2	4	no	—
135	C₂×#41	—	2	4	no	—
136	C₂×#41	C₈×C₅₃	1	8	yes	complemented
137	C₂²×Q₈	C₂×C₄×C₅₃	1	1	yes	characteristic
138	C₂²×D₄	C₂×C₄×C₅₃	1	3	yes	—
139	C₂³×C₄	C₂×C₄×C₅₃	1	3	yes	—
140	C₂³×C₄	C₂×C₄×C₅₃	1	6	yes	—
141	C₄×Q₈	—	2	4	no	—
142	C₄×Q₈	D₄×C₅₃	1	8	yes	—
143	C₄×D₄	—	2	12	no	—
144	C₄×D₄	—	2	12	no	—
145	#117	—	2	12	no	—
146	C₂×C₄⋊C₄	—	2	12	no	—
147	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	3	yes	—
148	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	3	yes	—
149	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	3	yes	—
150	C₂×C₄⋊C₄	C₂³×C₅₃	1	1	yes	characteristic
151	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	3	yes	—
152	C₂×C₂²⋊C₄	C₂³×C₅₃	1	3	yes	—
153	C₂×C₄²	C₂×C₄×C₅₃	1	3	yes	—
154	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
155	C₄⋊C₈	—	2	12	no	—
156	C₂²⋊C₈	C₂³×C₅₃	1	8	yes	—
157	C₄×C₈	—	2	12	no	—
158	C₂⁴	C₂²×C₄×C₅₃	1	3	yes	—
159	#41	—	2	32	no	—
160	C₂×Q₈	C₂²⋊C₄×C₅₃	1	4	yes	—
161	C₂×Q₈	C₂×C₈×C₅₃	1	4	yes	—
162	C₂×Q₈	M₁₆×C₅₃	1	4	yes	—
163	C₂×D₄	—	2	6	no	—
164	C₂×D₄	—	2	6	no	—
165	C₂×D₄	C₂×C₈×C₅₃	1	12	yes	—
166	C₂²×C₄	—	2	6	no	—
167	C₂²×C₄	—	2	6	no	—
168	C₂²×C₄	—	2	12	no	—
169	C₂²×C₄	—	8	6	no	—
170	C₂²×C₄	C₂×C₈×C₅₃	1	12	yes	—
171	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
174	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
175	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
176	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
177	C₂×C₈	—	2	8	no	—
178	C₄⋊C₄	—	2	6	no	—
179	C₄⋊C₄	—	2	6	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	12	yes	—
182	C₂²⋊C₄	—	2	6	no	—
183	C₂²⋊C₄	—	2	6	no	—
184	C₄²	—	2	6	no	—
185	C₄²	C₂×D₄×C₅₃	1	12	yes	—
186	C₂³	—	2	3	no	—
187	C₂³	—	2	3	no	—
188	C₂³	—	4	3	no	—
189	C₂³	—	4	3	no	—
190	C₂³	C₂²×C₈×C₅₃	1	6	yes	—
191	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
192	Q₈	C₂²⋊C₈×C₅₃	1	16	yes	—
193	D₄	—	2	24	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	24	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	3	no	—
199	C₂×C₄	—	4	3	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	3	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	8	12	no	—
204	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	6	yes	—
205	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	6	yes	—
206	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
207	C₂×C₄	C₂²×C₈×C₅₃	1	6	yes	—
208	C₂×C₄	C₂×M₁₆×C₅₃	1	6	yes	—
209	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
210	C₈	—	4	8	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	2	6	no	—
214	C₂²	—	4	6	no	—
215	C₂²	—	4	6	no	—
216	C₂²	—	4	3	no	—
217	C₂²	—	4	3	no	—
218	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
219	C₂²	C₂³×C₈×C₅₃	1	1	yes	characteristic
220	C₂²	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	2	2	no	—
223	C₄	—	2	6	no	—
224	C₄	—	4	1	no	—
225	C₄	—	4	3	no	—
226	C₄	C₂×C₂²⋊C₈×C₅₃	1	12	yes	—
227	C₂	—	2	1	no	—
228	C₂	—	2	1	no	—
229	C₂	—	4	6	no	—
230	C₂	C₂²×C₂²⋊C₈×C₅₃	1	1	yes	characteristic
231	C₂	C₅₃×#2884	1	1	yes	characteristic
232	C₂	C₅₃×#2888	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₈×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
prime radical	C₂²×#41×C₅₃	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄×Q₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2982	C₂	semidirect
3. C₅₃×#1460	C₂	semidirect
4. C₅₃×#1458	C₂	semidirect
19. C₂×#41×C₅₃	C₈	semidirect
94. #17353	C₅₃	direct (N × H)
112. #2982	C₂×C₅₃	semidirect
113. #1460	C₂×C₅₃	semidirect
114. #1458	C₂×C₅₃	semidirect
133. C₅₃	#17353	direct (N × H)
136. C₂×#41	C₈×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#520	2
C₅₃×#2982	2
C₅₃×#1460	2
C₅₃×#1458	2
#17353	53

Subgroups — C53×#17353 #27224865

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17353 #27224865