Subgroups — C₂×C₅₃×#2245 #27241212

Order 13568.

582 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#2245	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅₃×#358	C₂	1	4	yes	maximal
5	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₅₃	—	2	1	no	—
7	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
9	C₂×#125×C₅₃	—	2	1	no	—
10	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
11	C₅₃×#499	—	2	2	no	—
12	C₅₃×#499	C₂²	1	4	yes	—
13	C₅₃×#494	—	2	2	no	—
14	C₅₃×#494	C₂²	1	4	yes	—
15	C₂×#107×C₅₃	C₂²	1	2	yes	—
16	C₂×#106×C₅₃	—	2	1	no	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#358	C₂²	1	16	yes	—
19	C₅₃×#346	C₂²	1	4	yes	—
20	C₂²×Q₈×C₅₃	—	2	2	no	—
21	C₂×Q₁₆×C₅₃	—	2	2	no	—
22	C₂×Q₁₆×C₅₃	—	2	4	no	—
23	C₂×Q₁₆×C₅₃	—	2	8	no	—
24	C₂×Q₁₆×C₅₃	—	4	4	no	—
25	C₂×Q₁₆×C₅₃	C₂³	1	4	yes	—
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊Q₈×C₅₃	—	2	4	no	—
29	C₄⋊Q₈×C₅₃	—	4	2	no	—
30	#125×C₅₃	—	2	4	no	—
31	#125×C₅₃	—	4	2	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₂×C₁₆×C₅₃	—	2	4	no	—
37	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
38	#107×C₅₃	—	2	2	no	—
39	#107×C₅₃	C₂³	1	4	yes	—
40	#106×C₅₃	—	2	4	no	—
41	C₄×C₈×C₅₃	D₄	1	2	yes	—
42	C₄×C₈×C₅₃	C₂³	1	2	yes	—
43	C₂×Q₈×C₅₃	—	2	4	no	—
44	C₂×Q₈×C₅₃	—	4	2	no	—
45	C₂×Q₈×C₅₃	—	4	4	no	—
46	C₂×Q₈×C₅₃	—	4	4	no	—
47	C₂²×C₄×C₅₃	—	4	2	no	—
48	C₂²×C₄×C₅₃	—	4	2	no	—
49	C₂²×C₄×C₅₃	D₈	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	Q₁₆×C₅₃	—	2	8	no	—
52	Q₁₆×C₅₃	—	4	4	no	—
53	Q₁₆×C₅₃	—	4	8	no	—
54	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₅₃	#41	1	4	yes	—
58	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₅₃	—	2	4	no	—
60	C₄⋊C₄×C₅₃	—	4	2	no	—
61	C₄⋊C₄×C₅₃	—	4	8	no	—
62	C₄⋊C₄×C₅₃	—	4	8	no	—
63	C₄²×C₅₃	D₈	1	2	yes	—
64	C₄²×C₅₃	C₂×D₄	1	2	yes	—
65	C₁₆×C₅₃	—	2	4	no	—
66	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
67	Q₈×C₅₃	—	4	4	no	—
68	Q₈×C₅₃	—	4	4	no	—
69	Q₈×C₅₃	—	8	4	no	—
70	C₂×C₄×C₅₃	—	2	4	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	—	8	4	no	—
74	C₂×C₄×C₅₃	—	8	4	no	—
75	C₂×C₄×C₅₃	#124	1	2	yes	—
76	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
77	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
80	C₈×C₅₃	#124	1	2	yes	—
81	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
82	C₈×C₅₃	C₂×#41	1	2	yes	—
83	C₂×#2245	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₅₃	#486	1	2	yes	—
85	C₂²×C₅₃	#510	1	4	yes	—
86	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
87	C₄×C₅₃	—	2	4	no	—
88	C₄×C₅₃	—	8	4	no	—
89	C₄×C₅₃	—	8	4	no	—
90	C₄×C₅₃	#486	1	2	yes	—
91	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
92	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
93	C₂×#499	C₂×C₅₃	1	1	yes	characteristic
94	C₂×#494	C₂×C₅₃	1	1	yes	characteristic
95	#2245	C₂×C₅₃	1	8	yes	direct product factor
96	C₂×#358	C₂×C₅₃	1	4	yes	—
97	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
98	C₂×C₅₃	#2245	1	4	yes	direct product factor
99	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
100	C₂×C₅₃	C₂×#510	1	2	yes	—
101	C₂²×Q₁₆	—	2	1	no	—
102	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
103	C₂×C₄⋊Q₈	—	2	1	no	—
104	C₂×#125	—	2	1	no	—
105	C₂²×C₁₆	C₂²×C₅₃	1	2	yes	—
106	#499	—	2	2	no	—
107	#499	C₂²×C₅₃	1	4	yes	—
108	#494	—	2	2	no	—
109	#494	C₂²×C₅₃	1	4	yes	—
110	C₂×#107	C₂²×C₅₃	1	2	yes	—
111	C₂×#106	—	2	1	no	—
112	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
113	#358	C₂²×C₅₃	1	16	yes	—
114	#346	C₂²×C₅₃	1	4	yes	—
115	C₅₃	C₂×#2245	1	1	yes	characteristic, direct product factor, Sylow
116	C₂²×Q₈	—	2	2	no	—
117	C₂×Q₁₆	—	2	4	no	—
118	C₂×Q₁₆	—	2	2	no	—
119	C₂×Q₁₆	—	2	8	no	—
120	C₂×Q₁₆	—	4	4	no	—
121	C₂×Q₁₆	C₂³×C₅₃	1	4	yes	—
122	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
123	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	4	2	no	—
126	#125	—	2	4	no	—
127	#125	—	4	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	4	2	no	—
130	C₂×C₄⋊C₄	—	4	2	no	—
131	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
132	C₂×C₁₆	—	2	4	no	—
133	C₂×C₁₆	C₂³×C₅₃	1	4	yes	—
134	#107	—	2	2	no	—
135	#107	C₂³×C₅₃	1	4	yes	—
136	#106	—	2	4	no	—
137	C₄×C₈	D₄×C₅₃	1	2	yes	—
138	C₄×C₈	C₂³×C₅₃	1	2	yes	—
139	C₂×Q₈	—	2	4	no	—
140	C₂×Q₈	—	4	2	no	—
141	C₂×Q₈	—	4	4	no	—
142	C₂×Q₈	—	4	4	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	—	4	2	no	—
145	C₂²×C₄	D₈×C₅₃	1	2	yes	—
146	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
147	Q₁₆	—	2	8	no	—
148	Q₁₆	—	4	4	no	—
149	Q₁₆	—	4	8	no	—
150	C₂×C₈	SD₁₆×C₅₃	1	4	yes	—
151	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
152	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
153	C₂×C₈	#41×C₅₃	1	4	yes	—
154	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	8	no	—
158	C₄⋊C₄	—	4	8	no	—
159	C₄²	D₈×C₅₃	1	2	yes	—
160	C₄²	C₂×D₄×C₅₃	1	2	yes	—
161	C₁₆	—	2	4	no	—
162	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
163	Q₈	—	4	4	no	—
164	Q₈	—	4	4	no	—
165	Q₈	—	8	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	—	8	4	no	—
170	C₂×C₄	—	8	4	no	—
171	C₂×C₄	#124×C₅₃	1	2	yes	—
172	C₂×C₄	C₂×D₈×C₅₃	1	4	yes	—
173	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
174	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
175	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
176	C₈	#124×C₅₃	1	2	yes	—
177	C₈	C₂×SD₁₆×C₅₃	1	4	yes	—
178	C₈	C₂×#41×C₅₃	1	2	yes	—
179	C₂²	C₅₃×#486	1	2	yes	—
180	C₂²	C₅₃×#510	1	4	yes	—
181	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
182	C₄	—	2	4	no	—
183	C₄	—	8	4	no	—
184	C₄	—	8	4	no	—
185	C₄	C₅₃×#486	1	2	yes	—
186	C₄	C₂×#124×C₅₃	1	1	yes	characteristic
187	C₄	C₂²×SD₁₆×C₅₃	1	1	yes	characteristic
188	C₂	C₅₃×#2245	1	4	yes	direct product factor
189	C₂	C₂×C₅₃×#486	1	1	yes	characteristic
190	C₂	C₂×C₅₃×#510	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#2245	C₂	direct (N × H)
83. C₂×#2245	C₅₃	direct (N × H)
95. #2245	C₂×C₅₃	direct (N × H)
98. C₂×C₅₃	#2245	direct (N × H)
115. C₅₃	C₂×#2245	direct (N × H)
188. C₂	C₅₃×#2245	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×#499	2
C₂×C₅₃×#494	2
C₅₃×#2245	2
C₂×C₅₃×#358	2
C₂×C₅₃×#346	2
C₂×#2245	53

Subgroups — C2×C53×#2245 #27241212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2245 #27241212