Subgroups — C₂×C₅₃×#1837 #27228278

Order 13568.

870 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	C₂×C₅₃×#423	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1837	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#341	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₅₃	C₂²	1	4	yes	—
6	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
7	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×#423	—	2	4	no	—
9	C₅₃×#423	C₂²	1	8	yes	—
10	C₂×C₄⋊C₈×C₅₃	—	2	4	no	—
11	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
15	C₅₃×#341	C₂²	1	16	yes	—
16	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
17	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₅₃	—	2	4	no	—
19	C₂×M₁₆×C₅₃	—	2	16	no	—
20	C₂×M₁₆×C₅₃	C₂³	1	8	yes	—
21	C₂²×C₈×C₅₃	—	2	4	no	—
22	C₂²×C₈×C₅₃	—	2	8	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	2	yes	—
30	C₂×C₄²×C₅₃	D₄	1	2	yes	—
31	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
33	C₄⋊C₈×C₅₃	—	2	16	no	—
34	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
35	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
36	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	2	8	no	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
46	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	#41	1	8	yes	—
48	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
49	M₁₆×C₅₃	—	2	16	no	—
50	M₁₆×C₅₃	—	4	8	no	—
51	C₂×C₈×C₅₃	—	2	8	no	—
52	C₂×C₈×C₅₃	—	2	8	no	—
53	C₂×C₈×C₅₃	—	2	8	no	—
54	C₂×C₈×C₅₃	—	2	8	no	—
55	C₂×C₈×C₅₃	—	4	8	no	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
57	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
58	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
59	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
60	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
61	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
62	C₄²×C₅₃	C₂×D₄	1	4	yes	—
63	C₂³×C₅₃	—	2	4	no	—
64	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂³×C₅₃	#125	1	4	yes	—
67	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	2	8	no	—
70	C₂×C₄×C₅₃	—	2	8	no	—
71	C₂×C₄×C₅₃	#108	1	8	yes	—
72	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
76	C₂×C₄×C₅₃	#117	1	4	yes	—
77	C₂×C₄×C₅₃	#117	1	4	yes	—
78	C₂×C₄×C₅₃	#124	1	4	yes	—
79	C₂×C₄×C₅₃	#124	1	4	yes	—
80	C₂×C₄×C₅₃	#125	1	4	yes	—
81	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
84	C₈×C₅₃	—	2	8	no	—
85	C₈×C₅₃	—	4	8	no	—
86	C₂×#1837	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂²×C₅₃	—	2	8	no	—
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	—	2	2	no	—
90	C₂²×C₅₃	#382	1	2	yes	—
91	C₂²×C₅₃	#382	1	2	yes	—
92	C₂²×C₅₃	C₂×#108	1	4	yes	—
93	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₅₃	C₂×#125	1	2	yes	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	#382	1	2	yes	—
97	C₄×C₅₃	#382	1	2	yes	—
98	C₄×C₅₃	C₂×#108	1	8	yes	—
99	C₄×C₅₃	C₂×#117	1	2	yes	—
100	C₄×C₅₃	C₂×#124	1	2	yes	—
101	C₂×#423	C₂×C₅₃	1	2	yes	—
102	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
103	#1837	C₂×C₅₃	1	8	yes	direct product factor
104	C₂×#341	C₂×C₅₃	1	4	yes	—
105	C₂×C₅₃	—	2	2	no	—
106	C₂×C₅₃	—	2	2	no	—
107	C₂×C₅₃	#1837	1	4	yes	direct product factor
108	C₂×C₅₃	C₂×#382	1	1	yes	characteristic
109	C₂×C₅₃	C₂²×#108	1	2	yes	—
110	C₂²×M₁₆	C₂²×C₅₃	1	4	yes	—
111	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
112	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
113	#423	—	2	4	no	—
114	#423	C₂²×C₅₃	1	8	yes	—
115	C₂×C₄⋊C₈	—	2	4	no	—
116	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
117	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	—
118	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
120	#341	C₂²×C₅₃	1	16	yes	—
121	C₅₃	C₂×#1837	1	1	yes	characteristic, direct product factor, Sylow
122	C₂³×C₄	Q₈×C₅₃	1	2	yes	—
123	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
124	C₂×M₁₆	—	2	4	no	—
125	C₂×M₁₆	—	2	16	no	—
126	C₂×M₁₆	C₂³×C₅₃	1	8	yes	—
127	C₂²×C₈	—	2	8	no	—
128	C₂²×C₈	—	2	4	no	—
129	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
130	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
131	C₂²×C₈	C₂³×C₅₃	1	4	yes	—
132	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
133	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
134	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
135	C₂×C₄²	D₄×C₅₃	1	2	yes	—
136	C₂×C₄²	D₄×C₅₃	1	2	yes	—
137	C₂×C₄²	Q₈×C₅₃	1	2	yes	—
138	C₂×C₄²	C₂³×C₅₃	1	2	yes	—
139	C₄⋊C₈	—	2	16	no	—
140	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
141	C₄×C₈	C₂×C₄×C₅₃	1	8	yes	—
142	C₂⁴	C₂×Q₈×C₅₃	1	1	yes	characteristic
143	C₂²×C₄	—	2	8	no	—
144	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
145	C₂²×C₄	C₄⋊C₄×C₅₃	1	4	yes	—
146	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
147	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
148	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×Q₈×C₅₃	1	4	yes	—
152	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
153	C₂²×C₄	#41×C₅₃	1	8	yes	—
154	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
155	M₁₆	—	2	16	no	—
156	M₁₆	—	4	8	no	—
157	C₂×C₈	—	2	8	no	—
158	C₂×C₈	—	2	8	no	—
159	C₂×C₈	—	2	8	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	—	4	8	no	—
162	C₂×C₈	C₂²×C₄×C₅₃	1	8	yes	—
163	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
164	C₄²	C₄⋊C₄×C₅₃	1	4	yes	—
165	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
166	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
167	C₄²	C₂²×C₄×C₅₃	1	2	yes	—
168	C₄²	C₂×D₄×C₅₃	1	4	yes	—
169	C₂³	—	2	4	no	—
170	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
171	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
172	C₂³	#125×C₅₃	1	4	yes	—
173	C₂³	C₂²×Q₈×C₅₃	1	1	yes	characteristic
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	#108×C₅₃	1	8	yes	—
178	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
179	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	4	yes	—
182	C₂×C₄	#117×C₅₃	1	4	yes	—
183	C₂×C₄	#117×C₅₃	1	4	yes	—
184	C₂×C₄	#124×C₅₃	1	4	yes	—
185	C₂×C₄	#124×C₅₃	1	4	yes	—
186	C₂×C₄	#125×C₅₃	1	4	yes	—
187	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
188	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
189	C₂×C₄	C₂×#41×C₅₃	1	4	yes	—
190	C₈	—	2	8	no	—
191	C₈	—	4	8	no	—
192	C₂²	—	2	8	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	C₅₃×#382	1	2	yes	—
196	C₂²	C₅₃×#382	1	2	yes	—
197	C₂²	C₂×#108×C₅₃	1	4	yes	—
198	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
199	C₂²	C₂×#125×C₅₃	1	2	yes	—
200	C₄	—	2	4	no	—
201	C₄	C₅₃×#382	1	2	yes	—
202	C₄	C₅₃×#382	1	2	yes	—
203	C₄	C₂×#108×C₅₃	1	8	yes	—
204	C₄	C₂×#117×C₅₃	1	2	yes	—
205	C₄	C₂×#124×C₅₃	1	2	yes	—
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	C₅₃×#1837	1	4	yes	direct product factor
209	C₂	C₂×C₅₃×#382	1	1	yes	characteristic
210	C₂	C₂²×#108×C₅₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1837	C₂	direct (N × H)
86. C₂×#1837	C₅₃	direct (N × H)
103. #1837	C₂×C₅₃	direct (N × H)
107. C₂×C₅₃	#1837	direct (N × H)
121. C₅₃	C₂×#1837	direct (N × H)
208. C₂	C₅₃×#1837	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×#423	2
C₂²×C₄×C₈×C₅₃	2
C₅₃×#1837	2
C₂×C₅₃×#341	2
C₂×#1837	53

Subgroups — C2×C53×#1837 #27228278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1837 #27228278