Subgroups — C₅₃×#8232 #27215744

Order 13568.

516 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1767	C₂	1	2	yes	maximal
3	C₂×C₅₃×#333	C₂	1	2	yes	maximal
4	C₅₃×#1296	C₂	1	2	yes	maximal
5	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
8	C₂×#97×C₅₃	—	2	2	no	—
9	C₅₃×#391	C₄	1	2	yes	—
10	C₅₃×#389	—	2	2	no	—
11	C₅₃×#384	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
13	C₄³×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#336	—	2	2	no	—
15	C₅₃×#333	C₄	1	8	yes	complemented
16	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂²×C₈×C₅₃	—	2	4	no	—
18	C₄⋊Q₈×C₅₃	—	2	2	no	—
19	C₄⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
20	C₄×Q₈×C₅₃	—	4	4	no	—
21	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
22	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅₃	—	2	2	no	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	D₄	1	2	yes	—
29	C₂×C₄²×C₅₃	D₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	2	4	no	—
32	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
33	#97×C₅₃	—	4	4	no	—
34	#95×C₅₃	—	2	4	no	—
35	C₂×Q₈×C₅₃	—	2	2	no	—
36	C₂×Q₈×C₅₃	—	4	2	no	—
37	C₂×Q₈×C₅₃	—	4	2	no	—
38	C₂×Q₈×C₅₃	C₄²	1	4	yes	—
39	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₅₃	—	2	8	no	—
47	C₂×C₈×C₅₃	—	4	4	no	—
48	C₂×C₈×C₅₃	—	4	4	no	—
49	C₄⋊C₄×C₅₃	—	2	8	no	—
50	C₄⋊C₄×C₅₃	—	4	4	no	—
51	C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₄⋊C₄×C₅₃	—	4	2	no	—
53	C₄⋊C₄×C₅₃	—	4	2	no	—
54	C₄²×C₅₃	—	2	2	no	—
55	C₄²×C₅₃	—	2	2	no	—
56	C₄²×C₅₃	—	2	4	no	—
57	C₄²×C₅₃	—	2	2	no	—
58	C₄²×C₅₃	—	4	1	no	—
59	C₄²×C₅₃	—	4	4	no	—
60	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
61	C₄²×C₅₃	#41	1	2	yes	—
62	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	Q₈×C₅₃	—	4	4	no	—
64	Q₈×C₅₃	—	4	4	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	4	2	no	—
75	C₂×C₄×C₅₃	—	4	2	no	—
76	C₂×C₄×C₅₃	—	4	2	no	—
77	C₂×C₄×C₅₃	—	4	4	no	—
78	C₂×C₄×C₅₃	#100	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
82	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₅₃	#137	1	4	yes	—
84	C₂×C₄×C₅₃	#137	1	2	yes	—
85	C₈×C₅₃	—	4	8	no	—
86	#8232	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₅₃	C₂×#100	1	1	yes	characteristic
89	C₂²×C₅₃	#413	1	1	yes	characteristic
90	C₂²×C₅₃	#417	1	2	yes	—
91	C₂²×C₅₃	#419	1	2	yes	—
92	C₄×C₅₃	—	2	2	no	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	4	2	no	—
96	C₄×C₅₃	—	4	1	no	—
97	C₄×C₅₃	—	4	4	no	—
98	C₄×C₅₃	—	8	2	no	—
99	C₄×C₅₃	#440	1	4	yes	—
100	C₄×C₅₃	#441	1	4	yes	—
101	C₄×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
102	#1767	C₂×C₅₃	1	2	yes	—
103	C₂×#333	C₂×C₅₃	1	2	yes	—
104	#1296	C₂×C₅₃	1	2	yes	—
105	C₂×C₅₃	#1558	1	1	yes	characteristic
106	C₂×C₅₃	#1559	1	2	yes	—
107	C₂×C₅₃	#1560	1	1	yes	characteristic
108	C₂×C₅₃	C₄×#100	1	1	yes	characteristic
109	C₂×C₅₃	#1763	1	2	yes	—
110	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
111	C₂×C₄×Q₈	C₄×C₅₃	1	2	yes	—
112	C₂×C₄⋊C₈	C₂²×C₅₃	1	4	yes	—
113	C₂×#97	—	2	2	no	—
114	#391	C₄×C₅₃	1	2	yes	—
115	#389	—	2	2	no	—
116	#384	C₂²×C₅₃	1	1	yes	characteristic
117	C₄×C₄⋊C₄	—	2	2	no	—
118	C₄³	C₂²×C₅₃	1	1	yes	characteristic
119	#336	—	2	2	no	—
120	#333	C₄×C₅₃	1	8	yes	complemented
121	C₅₃	#8232	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
123	C₂²×C₈	—	2	4	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	C₂×C₄×C₅₃	1	4	yes	—
126	C₄×Q₈	—	4	4	no	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
131	C₂×C₄²	—	2	2	no	—
132	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
133	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
134	C₂×C₄²	D₄×C₅₃	1	2	yes	—
135	C₂×C₄²	D₄×C₅₃	1	2	yes	—
136	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
137	C₄⋊C₈	—	2	4	no	—
138	C₄⋊C₈	C₂×C₄×C₅₃	1	8	yes	—
139	#97	—	4	4	no	—
140	#95	—	2	4	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	C₄²×C₅₃	1	4	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
148	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²⋊C₄×C₅₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	—	4	4	no	—
155	C₄⋊C₄	—	2	8	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄²	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	4	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	4	no	—
166	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
167	C₄²	#41×C₅₃	1	2	yes	—
168	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
169	Q₈	—	4	4	no	—
170	Q₈	—	4	4	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	#100×C₅₃	1	2	yes	—
185	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
186	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
187	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
188	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
189	C₂×C₄	#137×C₅₃	1	4	yes	—
190	C₂×C₄	#137×C₅₃	1	2	yes	—
191	C₈	—	4	8	no	—
192	C₂²	C₄×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
193	C₂²	C₂×#100×C₅₃	1	1	yes	characteristic
194	C₂²	C₅₃×#413	1	1	yes	characteristic
195	C₂²	C₅₃×#417	1	2	yes	—
196	C₂²	C₅₃×#419	1	2	yes	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	4	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	1	no	—
203	C₄	—	8	2	no	—
204	C₄	C₅₃×#440	1	4	yes	—
205	C₄	C₅₃×#441	1	4	yes	—
206	C₂	C₅₃×#1558	1	1	yes	characteristic
207	C₂	C₅₃×#1559	1	2	yes	—
208	C₂	C₅₃×#1560	1	1	yes	characteristic
209	C₂	C₄×#100×C₅₃	1	1	yes	characteristic
210	C₂	C₅₃×#1763	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₅₃×#333	C₄	semidirect
86. #8232	C₅₃	direct (N × H)
120. #333	C₄×C₅₃	semidirect
121. C₅₃	#8232	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×C₄⋊Q₈×C₅₃	2
C₅₃×#1767	2
C₂×C₅₃×#333	2
C₅₃×#1296	2
#8232	53

Subgroups — C53×#8232 #27215744

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8232 #27215744