Subgroups — C₅₃×#34173 #27241685

Order 13568.

5126 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#519	C₂	1	2	yes	maximal, complemented
3	Q₈×C₂²⋊C₄×C₅₃	C₂	1	16	yes	maximal, complemented
4	C₅₃×#2310	C₂	1	6	yes	maximal, complemented
5	C₅₃×#2280	C₂	1	6	yes	maximal
6	C₂³×Q₈×C₅₃	—	2	16	no	—
7	C₂³×Q₈×C₅₃	C₄	1	16	yes	complemented
8	C₂³×Q₈×C₅₃	C₂²	1	3	yes	—
9	C₂³×Q₈×C₅₃	C₂²	1	8	yes	—
10	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
12	C₅₃×#519	—	2	16	no	—
13	C₂×C₄×Q₈×C₅₃	C₂²	1	16	yes	complemented
14	C₂×#117×C₅₃	C₂²	1	12	yes	—
15	C₅₃×#391	C₂²	1	48	yes	complemented
16	C₅₃×#381	C₂²	1	12	yes	—
17	C₅₃×#380	C₂²	1	24	yes	—
18	C₅₃×#379	C₂²	1	4	yes	—
19	C₄×C₂²⋊C₄×C₅₃	C₂²	1	24	yes	—
20	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×Q₈×C₅₃	—	2	128	no	—
22	C₂²×Q₈×C₅₃	—	2	64	no	—
23	C₂²×Q₈×C₅₃	—	2	24	no	—
24	C₂²×Q₈×C₅₃	C₂×C₄	1	64	yes	—
25	C₂²×Q₈×C₅₃	C₂×C₄	1	24	yes	—
26	C₂²×Q₈×C₅₃	C₂×C₄	1	8	yes	—
27	C₂²×Q₈×C₅₃	D₄	1	16	yes	—
28	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
29	C₂²×Q₈×C₅₃	C₂³	1	4	yes	—
30	C₂³×C₄×C₅₃	—	2	24	no	—
31	C₂³×C₄×C₅₃	C₂×C₄	1	24	yes	—
32	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
33	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
34	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
35	C₄×Q₈×C₅₃	—	2	64	no	—
36	#117×C₅₃	—	2	48	no	—
37	C₂×C₄⋊C₄×C₅₃	C₂³	1	24	yes	—
38	C₂×C₂²⋊C₄×C₅₃	C₂³	1	8	yes	—
39	C₂×C₂²⋊C₄×C₅₃	C₂³	1	16	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	24	yes	—
41	#95×C₅₃	C₂³	1	48	yes	—
42	C₂⁴×C₅₃	—	2	4	no	—
43	C₂⁴×C₅₃	—	2	8	no	—
44	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
45	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂⁴×C₅₃	C₂⁴	1	2	yes	—
47	C₂×Q₈×C₅₃	—	2	256	no	—
48	C₂×Q₈×C₅₃	—	2	96	no	—
49	C₂×Q₈×C₅₃	—	2	48	no	—
50	C₂×Q₈×C₅₃	C₂²⋊C₄	1	64	yes	complemented
51	C₂×Q₈×C₅₃	C₂²×C₄	1	48	yes	—
52	C₂×Q₈×C₅₃	C₂²×C₄	1	16	yes	—
53	C₂×Q₈×C₅₃	C₂×D₄	1	32	yes	—
54	C₂²×C₄×C₅₃	—	2	12	no	—
55	C₂²×C₄×C₅₃	—	2	96	no	—
56	C₂²×C₄×C₅₃	—	2	48	no	—
57	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	48	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
61	C₂²×C₄×C₅₃	C₂⁴	1	16	yes	—
62	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
63	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
64	C₄⋊C₄×C₅₃	—	2	48	no	—
65	C₂²⋊C₄×C₅₃	—	2	16	no	—
66	C₂²⋊C₄×C₅₃	—	2	32	no	—
67	C₄²×C₅₃	—	2	48	no	—
68	C₂³×C₅₃	—	2	32	no	—
69	C₂³×C₅₃	—	2	8	no	—
70	C₂³×C₅₃	—	2	16	no	—
71	C₂³×C₅₃	—	2	4	no	—
72	C₂³×C₅₃	—	2	4	no	—
73	C₂³×C₅₃	—	2	4	no	—
74	C₂³×C₅₃	C₂³×C₄	1	2	yes	—
75	C₂³×C₅₃	C₂³×C₄	1	4	yes	—
76	C₂³×C₅₃	C₂²×D₄	1	4	yes	—
77	C₂³×C₅₃	2₋¹⁺⁴	1	8	yes	—
78	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
79	Q₈×C₅₃	—	2	96	no	—
80	Q₈×C₅₃	C₂×C₂²⋊C₄	1	64	yes	—
81	C₂×C₄×C₅₃	—	2	24	no	—
82	C₂×C₄×C₅₃	—	2	12	no	—
83	C₂×C₄×C₅₃	—	2	96	no	—
84	C₂×C₄×C₅₃	—	2	16	no	—
85	C₂×C₄×C₅₃	—	2	32	no	—
86	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	48	yes	—
87	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
88	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
89	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
90	#34173	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂²×C₅₃	—	2	32	no	—
92	C₂²×C₅₃	—	2	4	no	—
93	C₂²×C₅₃	—	2	4	no	—
94	C₂²×C₅₃	—	2	4	no	—
95	C₂²×C₅₃	—	2	8	no	—
96	C₂²×C₅₃	—	2	16	no	—
97	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	4	yes	—
98	C₂²×C₅₃	#519	1	8	yes	complemented
99	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
100	C₂²×C₅₃	C₂³×D₄	1	2	yes	—
101	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	4	yes	—
102	C₄×C₅₃	—	2	12	no	—
103	C₄×C₅₃	—	4	16	no	—
104	C₄×C₅₃	C₂²×C₂²⋊C₄	1	24	yes	—
105	C₂⁴×Q₈	C₂×C₅₃	1	1	yes	characteristic
106	C₂×#519	C₂×C₅₃	1	2	yes	complemented
107	Q₈×C₂²⋊C₄	C₂×C₅₃	1	16	yes	complemented
108	#2310	C₂×C₅₃	1	6	yes	complemented
109	#2280	C₂×C₅₃	1	6	yes	—
110	C₂×C₅₃	—	2	4	no	—
111	C₂×C₅₃	—	2	8	no	—
112	C₂×C₅₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₅₃	C₂×#519	1	2	yes	—
114	C₂×C₅₃	#3486	1	4	yes	—
115	C₂³×Q₈	—	2	16	no	—
116	C₂³×Q₈	C₄×C₅₃	1	16	yes	complemented
117	C₂³×Q₈	C₂²×C₅₃	1	3	yes	—
118	C₂³×Q₈	C₂²×C₅₃	1	8	yes	—
119	C₂³×Q₈	C₂²×C₅₃	1	1	yes	characteristic
120	C₂⁴×C₄	C₂²×C₅₃	1	3	yes	—
121	#519	—	2	16	no	—
122	C₂×C₄×Q₈	C₂²×C₅₃	1	16	yes	complemented
123	C₂×#117	C₂²×C₅₃	1	12	yes	—
124	#391	C₂²×C₅₃	1	48	yes	complemented
125	#381	C₂²×C₅₃	1	12	yes	—
126	#380	C₂²×C₅₃	1	24	yes	—
127	#379	C₂²×C₅₃	1	4	yes	—
128	C₄×C₂²⋊C₄	C₂²×C₅₃	1	24	yes	—
129	C₅₃	#34173	1	1	yes	characteristic, direct product factor, Sylow
130	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
131	C₂²×Q₈	—	2	64	no	—
132	C₂²×Q₈	—	2	128	no	—
133	C₂²×Q₈	—	2	24	no	—
134	C₂²×Q₈	C₂×C₄×C₅₃	1	64	yes	—
135	C₂²×Q₈	C₂×C₄×C₅₃	1	24	yes	—
136	C₂²×Q₈	C₂×C₄×C₅₃	1	8	yes	—
137	C₂²×Q₈	D₄×C₅₃	1	16	yes	—
138	C₂²×Q₈	C₂³×C₅₃	1	12	yes	—
139	C₂²×Q₈	C₂³×C₅₃	1	4	yes	—
140	C₂³×C₄	—	2	24	no	—
141	C₂³×C₄	C₂×C₄×C₅₃	1	24	yes	—
142	C₂³×C₄	C₂³×C₅₃	1	12	yes	—
143	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
144	C₂³×C₄	C₂³×C₅₃	1	3	yes	—
145	C₄×Q₈	—	2	64	no	—
146	#117	—	2	48	no	—
147	C₂×C₄⋊C₄	C₂³×C₅₃	1	24	yes	—
148	C₂×C₂²⋊C₄	C₂³×C₅₃	1	8	yes	—
149	C₂×C₂²⋊C₄	C₂³×C₅₃	1	16	yes	—
150	C₂×C₄²	C₂³×C₅₃	1	24	yes	—
151	#95	C₂³×C₅₃	1	48	yes	—
152	C₂⁴	—	2	4	no	—
153	C₂⁴	—	2	8	no	—
154	C₂⁴	C₂²×C₄×C₅₃	1	4	yes	—
155	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
156	C₂⁴	C₂⁴×C₅₃	1	2	yes	—
157	C₂×Q₈	—	2	256	no	—
158	C₂×Q₈	—	2	96	no	—
159	C₂×Q₈	—	2	48	no	—
160	C₂×Q₈	C₂²⋊C₄×C₅₃	1	64	yes	complemented
161	C₂×Q₈	C₂²×C₄×C₅₃	1	48	yes	—
162	C₂×Q₈	C₂²×C₄×C₅₃	1	16	yes	—
163	C₂×Q₈	C₂×D₄×C₅₃	1	32	yes	—
164	C₂²×C₄	—	2	12	no	—
165	C₂²×C₄	—	2	96	no	—
166	C₂²×C₄	—	2	48	no	—
167	C₂²×C₄	C₂²×C₄×C₅₃	1	12	yes	—
168	C₂²×C₄	C₂²×C₄×C₅₃	1	48	yes	—
169	C₂²×C₄	C₂²×C₄×C₅₃	1	12	yes	—
170	C₂²×C₄	C₂×D₄×C₅₃	1	24	yes	—
171	C₂²×C₄	C₂⁴×C₅₃	1	16	yes	—
172	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
173	C₂²×C₄	C₂⁴×C₅₃	1	6	yes	—
174	C₄⋊C₄	—	2	48	no	—
175	C₂²⋊C₄	—	2	16	no	—
176	C₂²⋊C₄	—	2	32	no	—
177	C₄²	—	2	48	no	—
178	C₂³	—	2	32	no	—
179	C₂³	—	2	8	no	—
180	C₂³	—	2	16	no	—
181	C₂³	—	2	4	no	—
182	C₂³	—	2	4	no	—
183	C₂³	—	2	4	no	—
184	C₂³	C₂³×C₄×C₅₃	1	2	yes	—
185	C₂³	C₂³×C₄×C₅₃	1	4	yes	—
186	C₂³	C₂²×D₄×C₅₃	1	4	yes	—
187	C₂³	2₋¹⁺⁴×C₅₃	1	8	yes	—
188	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
189	Q₈	—	2	96	no	—
190	Q₈	C₂×C₂²⋊C₄×C₅₃	1	64	yes	—
191	C₂×C₄	—	2	24	no	—
192	C₂×C₄	—	2	12	no	—
193	C₂×C₄	—	2	96	no	—
194	C₂×C₄	—	2	16	no	—
195	C₂×C₄	—	2	32	no	—
196	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	48	yes	—
197	C₂×C₄	C₂³×C₄×C₅₃	1	12	yes	—
198	C₂×C₄	C₂³×C₄×C₅₃	1	12	yes	—
199	C₂×C₄	C₂²×D₄×C₅₃	1	24	yes	—
200	C₂²	—	2	32	no	—
201	C₂²	—	2	8	no	—
202	C₂²	—	2	16	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	4	yes	—
207	C₂²	C₅₃×#519	1	8	yes	complemented
208	C₂²	C₂⁴×C₄×C₅₃	1	1	yes	characteristic
209	C₂²	C₂³×D₄×C₅₃	1	2	yes	—
210	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	4	yes	—
211	C₄	—	2	12	no	—
212	C₄	—	4	16	no	—
213	C₄	C₂²×C₂²⋊C₄×C₅₃	1	24	yes	—
214	C₂	—	2	4	no	—
215	C₂	—	2	8	no	—
216	C₂	C₂³×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
217	C₂	C₂×C₅₃×#519	1	2	yes	—
218	C₂	C₅₃×#3486	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂⁵×C₅₃	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₃×#519	C₂	semidirect
3. Q₈×C₂²⋊C₄×C₅₃	C₂	semidirect
4. C₅₃×#2310	C₂	semidirect
7. C₂³×Q₈×C₅₃	C₄	semidirect
13. C₂×C₄×Q₈×C₅₃	C₂²	semidirect
15. C₅₃×#391	C₂²	semidirect
50. C₂×Q₈×C₅₃	C₂²⋊C₄	semidirect
90. #34173	C₅₃	direct (N × H)
98. C₂²×C₅₃	#519	semidirect
106. C₂×#519	C₂×C₅₃	semidirect
107. Q₈×C₂²⋊C₄	C₂×C₅₃	semidirect
108. #2310	C₂×C₅₃	semidirect
116. C₂³×Q₈	C₄×C₅₃	semidirect
122. C₂×C₄×Q₈	C₂²×C₅₃	semidirect
124. #391	C₂²×C₅₃	semidirect
129. C₅₃	#34173	direct (N × H)
160. C₂×Q₈	C₂²⋊C₄×C₅₃	semidirect
207. C₂²	C₅₃×#519	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₅₃	2
C₂×C₅₃×#519	2
Q₈×C₂²⋊C₄×C₅₃	2
C₅₃×#2310	2
C₅₃×#2280	2
#34173	53

Subgroups — C53×#34173 #27241685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#34173 #27241685