Subgroups — C₅₃×#17782 #27225294

Order 13568.

1244 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	C₅₃×#3498	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2991	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1560	C₂	1	3	yes	maximal, complemented
4	C₅₃×#1545	C₂	1	6	yes	maximal, complemented
5	C₅₃×#1543	C₂	1	3	yes	maximal, complemented
6	C₂×H(5,𝔽₂)×C₅₃	C₄	1	2	yes	—
7	C₅₃×#550	—	2	2	no	—
8	D₄²×C₅₃	—	2	6	no	—
9	C₅₃×#535	C₄	1	6	yes	—
10	C₅₃×#532	C₂²	1	3	yes	—
11	C₂×#127×C₅₃	C₂²	1	3	yes	—
12	C₄×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₈×Q₈×C₅₃	—	2	2	no	—
14	C₅₃×#436	—	2	6	no	—
15	C₅₃×#431	C₂²	1	6	yes	complemented
16	C₅₃×#423	C₂²	1	6	yes	complemented
17	C₅₃×#333	C₂²	1	1	yes	characteristic, complemented
18	C₅₃×#333	C₂²	1	3	yes	complemented
19	C₅₃×#331	C₂²	1	6	yes	complemented
20	C₅₃×#331	C₂²	1	6	yes	complemented
21	H(5,𝔽₂)×C₅₃	—	4	8	no	—
22	C₂×#41×C₅₃	—	2	2	no	—
23	C₂×#41×C₅₃	C₂×C₄	1	6	yes	—
24	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×D₄×C₅₃	—	2	6	no	—
26	C₂²×D₄×C₅₃	—	2	6	no	—
27	C₂²×D₄×C₅₃	C₂×C₄	1	6	yes	—
28	C₂×M₁₆×C₅₃	—	2	6	no	—
29	C₄⋊Q₈×C₅₃	C₂³	1	1	yes	characteristic
30	#127×C₅₃	—	2	6	no	—
31	#127×C₅₃	C₂×C₄	1	6	yes	—
32	#127×C₅₃	C₂³	1	3	yes	—
33	#124×C₅₃	C₂×C₄	1	6	yes	—
34	#121×C₅₃	—	2	12	no	—
35	#121×C₅₃	—	2	6	no	—
36	#120×C₅₃	—	2	12	no	—
37	C₄×Q₈×C₅₃	D₄	1	2	yes	—
38	C₄×D₄×C₅₃	—	2	6	no	—
39	C₄×D₄×C₅₃	D₄	1	6	yes	—
40	#117×C₅₃	C₂×C₄	1	3	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
42	C₄⋊C₈×C₅₃	—	2	6	no	—
43	C₄⋊C₈×C₅₃	C₂³	1	6	yes	—
44	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
45	C₂²⋊C₈×C₅₃	—	2	6	no	—
46	C₄×C₈×C₅₃	—	2	6	no	—
47	C₂⁴×C₅₃	—	2	6	no	—
48	#41×C₅₃	—	2	4	no	—
49	#41×C₅₃	—	4	12	no	—
50	#41×C₅₃	—	4	8	no	—
51	C₂×Q₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
53	C₂×D₄×C₅₃	—	2	6	no	—
54	C₂×D₄×C₅₃	—	2	6	no	—
55	C₂×D₄×C₅₃	—	2	12	no	—
56	C₂×D₄×C₅₃	—	2	6	no	—
57	C₂×D₄×C₅₃	—	4	12	no	—
58	C₂×D₄×C₅₃	—	4	24	no	—
59	C₂×D₄×C₅₃	—	8	6	no	—
60	C₂×D₄×C₅₃	—	8	6	no	—
61	C₂×D₄×C₅₃	C₂²⋊C₄	1	3	yes	—
62	C₂×D₄×C₅₃	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₅₃	—	2	6	no	—
64	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
66	M₁₆×C₅₃	—	8	6	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	—	2	6	no	—
69	C₄⋊C₄×C₅₃	—	2	2	no	—
70	C₄⋊C₄×C₅₃	C₂²⋊C₄	1	6	yes	—
71	C₂²⋊C₄×C₅₃	—	2	12	no	—
72	C₂²⋊C₄×C₅₃	C₂²⋊C₄	1	6	yes	—
73	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
74	C₄²×C₅₃	C₂×D₄	1	6	yes	—
75	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
76	C₂³×C₅₃	—	2	6	no	—
77	C₂³×C₅₃	—	4	12	no	—
78	C₂³×C₅₃	—	8	12	no	—
79	C₂³×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
80	Q₈×C₅₃	—	4	2	no	—
81	Q₈×C₅₃	#100	1	4	yes	—
82	D₄×C₅₃	—	2	6	no	—
83	D₄×C₅₃	—	4	6	no	—
84	D₄×C₅₃	—	4	12	no	—
85	D₄×C₅₃	—	4	6	no	—
86	D₄×C₅₃	—	4	6	no	—
87	D₄×C₅₃	—	4	12	no	—
88	D₄×C₅₃	—	8	12	no	—
89	C₂×C₄×C₅₃	—	2	6	no	—
90	C₂×C₄×C₅₃	—	2	6	no	—
91	C₂×C₄×C₅₃	—	2	6	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	4	12	no	—
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
96	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
98	C₈×C₅₃	—	4	6	no	—
99	C₈×C₅₃	—	4	2	no	—
100	#17782	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₅₃	—	2	3	no	—
102	C₂²×C₅₃	—	2	6	no	—
103	C₂²×C₅₃	—	4	6	no	—
104	C₂²×C₅₃	—	4	12	no	—
105	C₂²×C₅₃	—	16	12	no	—
106	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	2	6	no	—
109	C₄×C₅₃	—	4	2	no	—
110	C₄×C₅₃	C₂×#100	1	6	yes	—
111	C₄×C₅₃	#576	1	2	yes	—
112	C₄×C₅₃	#577	1	2	yes	—
113	#3498	C₂×C₅₃	1	1	yes	characteristic
114	#2991	C₂×C₅₃	1	2	yes	complemented
115	#1560	C₂×C₅₃	1	3	yes	complemented
116	#1545	C₂×C₅₃	1	6	yes	complemented
117	#1543	C₂×C₅₃	1	3	yes	complemented
118	C₂×C₅₃	—	4	3	no	—
119	C₂×C₅₃	—	8	6	no	—
120	C₂×C₅₃	C₂²×#100	1	1	yes	characteristic
121	C₂×C₅₃	#2897	1	2	yes	—
122	C₂×H(5,𝔽₂)	C₄×C₅₃	1	2	yes	—
123	#550	—	2	2	no	—
124	D₄²	—	2	6	no	—
125	#535	C₄×C₅₃	1	6	yes	—
126	#532	C₂²×C₅₃	1	3	yes	—
127	C₂×#127	C₂²×C₅₃	1	3	yes	—
128	C₄×#41	C₂²×C₅₃	1	1	yes	characteristic
129	C₈×Q₈	—	2	2	no	—
130	#436	—	2	6	no	—
131	#431	C₂²×C₅₃	1	6	yes	complemented
132	#423	C₂²×C₅₃	1	6	yes	complemented
133	#333	C₂²×C₅₃	1	1	yes	characteristic, complemented
134	#333	C₂²×C₅₃	1	3	yes	complemented
135	#331	C₂²×C₅₃	1	6	yes	complemented
136	#331	C₂²×C₅₃	1	6	yes	complemented
137	C₅₃	#17782	1	1	yes	characteristic, direct product factor, Sylow
138	H(5,𝔽₂)	—	4	8	no	—
139	C₂×#41	—	2	2	no	—
140	C₂×#41	C₂×C₄×C₅₃	1	6	yes	—
141	C₂×#41	C₂×C₄×C₅₃	1	1	yes	characteristic
142	C₂²×D₄	—	2	6	no	—
143	C₂²×D₄	—	2	6	no	—
144	C₂²×D₄	C₂×C₄×C₅₃	1	6	yes	—
145	C₂×M₁₆	—	2	6	no	—
146	C₄⋊Q₈	C₂³×C₅₃	1	1	yes	characteristic
147	#127	—	2	6	no	—
148	#127	C₂×C₄×C₅₃	1	6	yes	—
149	#127	C₂³×C₅₃	1	3	yes	—
150	#124	C₂×C₄×C₅₃	1	6	yes	—
151	#121	—	2	6	no	—
152	#121	—	2	12	no	—
153	#120	—	2	12	no	—
154	C₄×Q₈	D₄×C₅₃	1	2	yes	—
155	C₄×D₄	—	2	6	no	—
156	C₄×D₄	D₄×C₅₃	1	6	yes	—
157	#117	C₂×C₄×C₅₃	1	3	yes	—
158	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
159	C₄⋊C₈	—	2	6	no	—
160	C₄⋊C₈	C₂³×C₅₃	1	6	yes	—
161	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
162	C₂²⋊C₈	—	2	6	no	—
163	C₄×C₈	—	2	6	no	—
164	C₂⁴	—	2	6	no	—
165	#41	—	2	4	no	—
166	#41	—	4	12	no	—
167	#41	—	4	8	no	—
168	C₂×Q₈	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
169	C₂×Q₈	C₂²×C₄×C₅₃	1	2	yes	—
170	C₂×D₄	—	2	6	no	—
171	C₂×D₄	—	2	6	no	—
172	C₂×D₄	—	2	12	no	—
173	C₂×D₄	—	2	6	no	—
174	C₂×D₄	—	4	12	no	—
175	C₂×D₄	—	4	24	no	—
176	C₂×D₄	—	8	6	no	—
177	C₂×D₄	—	8	6	no	—
178	C₂×D₄	C₂²⋊C₄×C₅₃	1	3	yes	—
179	C₂×D₄	C₂²×C₄×C₅₃	1	6	yes	—
180	C₂²×C₄	—	2	6	no	—
181	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
182	C₂²×C₄	C₂×D₄×C₅₃	1	6	yes	—
183	M₁₆	—	8	6	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	6	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	C₂²⋊C₄×C₅₃	1	6	yes	—
188	C₂²⋊C₄	—	2	12	no	—
189	C₂²⋊C₄	C₂²⋊C₄×C₅₃	1	6	yes	—
190	C₄²	C₂²×C₄×C₅₃	1	3	yes	—
191	C₄²	C₂×D₄×C₅₃	1	6	yes	—
192	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
193	C₂³	—	2	6	no	—
194	C₂³	—	4	12	no	—
195	C₂³	—	8	12	no	—
196	C₂³	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
197	Q₈	—	4	2	no	—
198	Q₈	#100×C₅₃	1	4	yes	—
199	D₄	—	2	6	no	—
200	D₄	—	4	6	no	—
201	D₄	—	4	12	no	—
202	D₄	—	4	6	no	—
203	D₄	—	4	6	no	—
204	D₄	—	4	12	no	—
205	D₄	—	8	12	no	—
206	C₂×C₄	—	2	6	no	—
207	C₂×C₄	—	2	6	no	—
208	C₂×C₄	—	2	6	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	12	no	—
211	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	6	yes	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	3	yes	—
213	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
215	C₈	—	4	6	no	—
216	C₈	—	4	2	no	—
217	C₂²	—	2	3	no	—
218	C₂²	—	2	6	no	—
219	C₂²	—	4	6	no	—
220	C₂²	—	4	12	no	—
221	C₂²	—	16	12	no	—
222	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	6	no	—
225	C₄	—	4	2	no	—
226	C₄	C₂×#100×C₅₃	1	6	yes	—
227	C₄	C₅₃×#576	1	2	yes	—
228	C₄	C₅₃×#577	1	2	yes	—
229	C₂	—	4	3	no	—
230	C₂	—	8	6	no	—
231	C₂	C₂²×#100×C₅₃	1	1	yes	characteristic
232	C₂	C₅₃×#2897	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
prime radical	C₅₃×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄×#41×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅₃×#2991	C₂	semidirect
3. C₅₃×#1560	C₂	semidirect
4. C₅₃×#1545	C₂	semidirect
5. C₅₃×#1543	C₂	semidirect
15. C₅₃×#431	C₂²	semidirect
16. C₅₃×#423	C₂²	semidirect
17. C₅₃×#333	C₂²	semidirect
18. C₅₃×#333	C₂²	semidirect
19. C₅₃×#331	C₂²	semidirect
20. C₅₃×#331	C₂²	semidirect
100. #17782	C₅₃	direct (N × H)
114. #2991	C₂×C₅₃	semidirect
115. #1560	C₂×C₅₃	semidirect
116. #1545	C₂×C₅₃	semidirect
117. #1543	C₂×C₅₃	semidirect
131. #431	C₂²×C₅₃	semidirect
132. #423	C₂²×C₅₃	semidirect
133. #333	C₂²×C₅₃	semidirect
134. #333	C₂²×C₅₃	semidirect
135. #331	C₂²×C₅₃	semidirect
136. #331	C₂²×C₅₃	semidirect
137. C₅₃	#17782	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3498	2
C₅₃×#2991	2
C₅₃×#1560	2
C₅₃×#1545	2
C₅₃×#1543	2
#17782	53

Subgroups — C53×#17782 #27225294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#17782 #27225294