Subgroups — C₂×C₅₃×#1593 #27225400

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#446	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1593	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×Q₈⋊C₈	C₂	1	2	yes	maximal
6	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#446	—	2	4	no	—
10	C₅₃×#446	C₂²	1	8	yes	—
11	C₄×Q₁₆×C₅₃	C₄	1	4	yes	—
12	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	—
13	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₅₃	—	2	2	no	—
15	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×#103×C₅₃	C₄	1	2	yes	—
17	C₂×#97×C₅₃	—	2	2	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₅₃×#334	C₂²	1	4	yes	—
22	C₅₃×Q₈⋊C₈	C₂²	1	8	yes	—
23	C₅₃×#322	C₂²	1	4	yes	—
24	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆×C₅₃	—	2	4	no	—
26	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	2	yes	—
32	C₄×Q₈×C₅₃	—	2	2	no	—
33	C₄×Q₈×C₅₃	—	2	2	no	—
34	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
35	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
36	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	#107×C₅₃	C₂×C₄	1	4	yes	—
41	C₄⋊C₈×C₅₃	—	2	8	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
44	#103×C₅₃	C₂×C₄	1	8	yes	—
45	#97×C₅₃	—	2	8	no	—
46	C₄×C₈×C₅₃	—	2	1	no	—
47	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₅₃	D₄	1	2	yes	—
49	C₄×C₈×C₅₃	C₂³	1	2	yes	—
50	C₄×C₈×C₅₃	C₂³	1	2	yes	—
51	C₄×C₈×C₅₃	C₂³	1	2	yes	—
52	C₂×Q₈×C₅₃	—	2	2	no	—
53	C₂×Q₈×C₅₃	—	2	4	no	—
54	C₂×Q₈×C₅₃	—	2	4	no	—
55	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	Q₁₆×C₅₃	—	2	8	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	2	2	no	—
66	C₂×C₈×C₅₃	—	4	2	no	—
67	C₂×C₈×C₅₃	—	4	4	no	—
68	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
72	C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₄⋊C₄×C₅₃	—	2	2	no	—
74	C₄⋊C₄×C₅₃	—	2	4	no	—
75	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
76	C₄²×C₅₃	—	2	4	no	—
77	C₄²×C₅₃	—	2	4	no	—
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₃	#41	1	1	yes	characteristic
81	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
82	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
83	Q₈×C₅₃	—	2	4	no	—
84	Q₈×C₅₃	—	2	4	no	—
85	Q₈×C₅₃	—	4	4	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	—	4	4	no	—
91	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
95	C₂×C₄×C₅₃	#131	1	2	yes	—
96	C₂×C₄×C₅₃	#137	1	4	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	2	4	no	—
102	C₈×C₅₃	—	4	2	no	—
103	C₈×C₅₃	—	4	4	no	—
104	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
105	C₂×#1593	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₅₃	#436	1	2	yes	—
107	C₂²×C₅₃	#441	1	2	yes	—
108	C₂²×C₅₃	#444	1	2	yes	—
109	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	4	4	no	—
113	C₄×C₅₃	—	4	4	no	—
114	C₄×C₅₃	#436	1	2	yes	—
115	C₄×C₅₃	#441	1	2	yes	—
116	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
117	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
118	C₄×C₅₃	C₂×#137	1	2	yes	—
119	C₂×#446	C₂×C₅₃	1	2	yes	—
120	C₂×C₄×Q₁₆	C₂×C₅₃	1	1	yes	characteristic
121	#1593	C₂×C₅₃	1	8	yes	direct product factor
122	C₂×#334	C₂×C₅₃	1	1	yes	characteristic
123	C₂×Q₈⋊C₈	C₂×C₅₃	1	2	yes	—
124	C₂×#322	C₂×C₅₃	1	1	yes	characteristic
125	C₂×C₅₃	#1593	1	4	yes	direct product factor
126	C₂×C₅₃	C₂×#436	1	1	yes	characteristic
127	C₂×C₅₃	C₂×#441	1	1	yes	characteristic
128	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
129	C₂²×Q₁₆	C₄×C₅₃	1	1	yes	characteristic
130	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
131	#446	—	2	4	no	—
132	#446	C₂²×C₅₃	1	8	yes	—
133	C₄×Q₁₆	C₄×C₅₃	1	4	yes	—
134	C₄×Q₁₆	C₂²×C₅₃	1	4	yes	—
135	C₂×#107	C₄×C₅₃	1	1	yes	characteristic
136	C₂×C₄⋊C₈	—	2	2	no	—
137	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
138	C₂×#103	C₄×C₅₃	1	2	yes	—
139	C₂×#97	—	2	2	no	—
140	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
143	#334	C₂²×C₅₃	1	4	yes	—
144	Q₈⋊C₈	C₂²×C₅₃	1	8	yes	—
145	#322	C₂²×C₅₃	1	4	yes	—
146	C₅₃	C₂×#1593	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
148	C₂×Q₁₆	—	2	4	no	—
149	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
153	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
154	C₂²×C₈	D₄×C₅₃	1	2	yes	—
155	C₄×Q₈	—	2	2	no	—
156	C₄×Q₈	—	2	2	no	—
157	C₄×Q₈	C₂×C₄×C₅₃	1	4	yes	—
158	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
163	#107	C₂×C₄×C₅₃	1	4	yes	—
164	C₄⋊C₈	—	2	8	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
167	#103	C₂×C₄×C₅₃	1	8	yes	—
168	#97	—	2	8	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
171	C₄×C₈	D₄×C₅₃	1	2	yes	—
172	C₄×C₈	C₂³×C₅₃	1	2	yes	—
173	C₄×C₈	C₂³×C₅₃	1	2	yes	—
174	C₄×C₈	C₂³×C₅₃	1	2	yes	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	C₂²×C₄×C₅₃	1	4	yes	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
183	C₂²×C₄	#41×C₅₃	1	1	yes	characteristic
184	Q₁₆	—	2	8	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	4	no	—
191	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
192	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
193	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
194	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	C₂²×C₄×C₅₃	1	4	yes	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
203	C₄²	#41×C₅₃	1	1	yes	characteristic
204	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
205	C₂³	C₄×D₄×C₅₃	1	1	yes	characteristic
206	Q₈	—	2	4	no	—
207	Q₈	—	2	4	no	—
208	Q₈	—	4	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
215	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
216	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
217	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
218	C₂×C₄	#131×C₅₃	1	2	yes	—
219	C₂×C₄	#137×C₅₃	1	4	yes	—
220	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂×#41×C₅₃	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	4	no	—
225	C₈	—	4	2	no	—
226	C₈	—	4	4	no	—
227	C₈	C₂×M₁₆×C₅₃	1	4	yes	—
228	C₂²	C₅₃×#436	1	2	yes	—
229	C₂²	C₅₃×#441	1	2	yes	—
230	C₂²	C₅₃×#444	1	2	yes	—
231	C₂²	C₂×C₄×D₄×C₅₃	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	4	no	—
235	C₄	—	4	4	no	—
236	C₄	C₅₃×#436	1	2	yes	—
237	C₄	C₅₃×#441	1	2	yes	—
238	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
239	C₄	C₂×#131×C₅₃	1	1	yes	characteristic
240	C₄	C₂×#137×C₅₃	1	2	yes	—
241	C₂	C₅₃×#1593	1	4	yes	direct product factor
242	C₂	C₂×C₅₃×#436	1	1	yes	characteristic
243	C₂	C₂×C₅₃×#441	1	1	yes	characteristic
244	C₂	C₂×C₅₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₄²	C₂⁴×C₅₃
socle, prime radical	C₂³×C₅₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1593	C₂	direct (N × H)
105. C₂×#1593	C₅₃	direct (N × H)
121. #1593	C₂×C₅₃	direct (N × H)
125. C₂×C₅₃	#1593	direct (N × H)
146. C₅₃	C₂×#1593	direct (N × H)
241. C₂	C₅₃×#1593	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#446	2
C₂×C₄×Q₁₆×C₅₃	2
C₅₃×#1593	2
C₂×C₅₃×#334	2
C₂×C₅₃×Q₈⋊C₈	2
C₂×C₅₃×#322	2
C₂×#1593	53

Subgroups — C2×C53×#1593 #27225400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1593 #27225400