Subgroups — C₄×C₅₃×#341 #27218216

Order 13568.

910 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₅₃	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#341	C₂	1	4	yes	maximal
4	C₂²×M₁₆×C₅₃	C₂²	1	4	yes	—
5	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
7	C₄×M₁₆×C₅₃	—	2	4	no	—
8	C₄×M₁₆×C₅₃	C₄	1	8	yes	—
9	C₂×#97×C₅₃	—	2	2	no	—
10	C₂×C₄×C₈×C₅₃	—	2	4	no	—
11	C₂×C₄×C₈×C₅₃	—	2	2	no	—
12	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
13	C₅₃×#341	C₄	1	16	yes	direct product factor
14	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
15	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×M₁₆×C₅₃	—	2	4	no	—
17	C₂×M₁₆×C₅₃	—	2	8	no	—
18	C₂×M₁₆×C₅₃	C₂×C₄	1	16	yes	—
19	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
20	C₂²×C₈×C₅₃	—	2	4	no	—
21	C₂²×C₈×C₅₃	—	2	4	no	—
22	C₂²×C₈×C₅₃	—	2	4	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅₃	D₄	1	4	yes	—
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	#97×C₅₃	—	2	8	no	—
31	C₄×C₈×C₅₃	—	2	8	no	—
32	C₄×C₈×C₅₃	—	2	8	no	—
33	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	8	no	—
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	#41	1	4	yes	—
44	M₁₆×C₅₃	—	2	8	no	—
45	M₁₆×C₅₃	—	2	16	no	—
46	M₁₆×C₅₃	C₄²	1	16	yes	—
47	C₂×C₈×C₅₃	—	2	4	no	—
48	C₂×C₈×C₅₃	—	2	8	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	—	2	16	no	—
51	C₂×C₈×C₅₃	—	2	16	no	—
52	C₂×C₈×C₅₃	C₄²	1	8	yes	—
53	C₄²×C₅₃	C₄²	1	4	yes	—
54	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
57	C₂³×C₅₃	—	2	4	no	—
58	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
59	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂³×C₅₃	C₄×Q₈	1	4	yes	—
61	C₂×C₄×C₅₃	—	2	4	no	—
62	C₂×C₄×C₅₃	—	2	16	no	—
63	C₂×C₄×C₅₃	#95	1	4	yes	—
64	C₂×C₄×C₅₃	#95	1	4	yes	—
65	C₂×C₄×C₅₃	#108	1	8	yes	—
66	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
69	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
71	C₂×C₄×C₅₃	#117	1	2	yes	—
72	C₂×C₄×C₅₃	#117	1	4	yes	—
73	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
74	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
75	C₈×C₅₃	—	2	8	no	—
76	C₈×C₅₃	—	2	16	no	—
77	C₄×#341	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂²×C₅₃	—	2	4	no	—
79	C₂²×C₅₃	—	2	8	no	—
80	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
81	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
82	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
83	C₂²×C₅₃	C₂×#108	1	4	yes	—
84	C₄×C₅₃	—	2	4	no	—
85	C₄×C₅₃	#341	1	8	yes	direct product factor
86	C₄×C₅₃	C₄³	1	1	yes	characteristic
87	C₄×C₅₃	#376	1	1	yes	characteristic
88	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
89	C₄×C₅₃	C₄×C₂²⋊C₄	1	4	yes	—
90	C₂×C₄×M₁₆	C₂×C₅₃	1	2	yes	—
91	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
92	C₂×#341	C₂×C₅₃	1	4	yes	—
93	C₂×C₅₃	—	2	4	no	—
94	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
95	C₂×C₅₃	C₂×#341	1	2	yes	—
96	C₂×C₅₃	C₄×#108	1	4	yes	—
97	C₂²×M₁₆	C₂²×C₅₃	1	4	yes	—
98	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
99	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
100	C₄×M₁₆	—	2	4	no	—
101	C₄×M₁₆	C₄×C₅₃	1	8	yes	—
102	C₂×#97	—	2	2	no	—
103	C₂×C₄×C₈	—	2	4	no	—
104	C₂×C₄×C₈	—	2	2	no	—
105	C₂×C₄×C₈	C₄×C₅₃	1	4	yes	—
106	#341	C₄×C₅₃	1	16	yes	direct product factor
107	C₅₃	C₄×#341	1	1	yes	characteristic, direct product factor, Sylow
108	C₂³×C₄	Q₈×C₅₃	1	2	yes	—
109	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
110	C₂×M₁₆	—	2	4	no	—
111	C₂×M₁₆	—	2	8	no	—
112	C₂×M₁₆	C₂×C₄×C₅₃	1	16	yes	—
113	C₂×M₁₆	C₂×C₄×C₅₃	1	8	yes	—
114	C₂²×C₈	—	2	4	no	—
115	C₂²×C₈	—	2	4	no	—
116	C₂²×C₈	—	2	4	no	—
117	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
118	C₂²×C₈	C₂×C₄×C₅₃	1	4	yes	—
119	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
120	C₂×C₄²	C₂×C₄×C₅₃	1	4	yes	—
121	C₂×C₄²	D₄×C₅₃	1	4	yes	—
122	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
123	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
124	#97	—	2	8	no	—
125	C₄×C₈	—	2	8	no	—
126	C₄×C₈	—	2	8	no	—
127	C₂⁴	C₂×Q₈×C₅₃	1	1	yes	characteristic
128	C₂²×C₄	—	2	8	no	—
129	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
130	C₂²×C₄	C₄⋊C₄×C₅₃	1	8	yes	—
131	C₂²×C₄	C₄⋊C₄×C₅₃	1	2	yes	—
132	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
133	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
134	C₂²×C₄	C₂²×C₄×C₅₃	1	4	yes	—
135	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
137	C₂²×C₄	#41×C₅₃	1	4	yes	—
138	M₁₆	—	2	8	no	—
139	M₁₆	—	2	16	no	—
140	M₁₆	C₄²×C₅₃	1	16	yes	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	8	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	16	no	—
145	C₂×C₈	—	2	16	no	—
146	C₂×C₈	C₄²×C₅₃	1	8	yes	—
147	C₄²	C₄²×C₅₃	1	4	yes	—
148	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
149	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
150	C₄²	C₂²⋊C₄×C₅₃	1	4	yes	—
151	C₂³	—	2	4	no	—
152	C₂³	C₂×C₄⋊C₄×C₅₃	1	2	yes	—
153	C₂³	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
154	C₂³	C₄×Q₈×C₅₃	1	4	yes	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	16	no	—
157	C₂×C₄	#95×C₅₃	1	4	yes	—
158	C₂×C₄	#95×C₅₃	1	4	yes	—
159	C₂×C₄	#108×C₅₃	1	8	yes	—
160	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄²×C₅₃	1	2	yes	—
162	C₂×C₄	C₂×C₄²×C₅₃	1	4	yes	—
163	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
164	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
165	C₂×C₄	#117×C₅₃	1	2	yes	—
166	C₂×C₄	#117×C₅₃	1	4	yes	—
167	C₂×C₄	C₄×D₄×C₅₃	1	4	yes	—
168	C₂×C₄	C₄×D₄×C₅₃	1	8	yes	—
169	C₈	—	2	8	no	—
170	C₈	—	2	16	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	2	8	no	—
173	C₂²	C₂×#95×C₅₃	1	1	yes	characteristic
174	C₂²	C₄×C₄⋊C₄×C₅₃	1	4	yes	—
175	C₂²	C₄×C₄⋊C₄×C₅₃	1	2	yes	—
176	C₂²	C₂×#108×C₅₃	1	4	yes	—
177	C₄	—	2	4	no	—
178	C₄	C₅₃×#341	1	8	yes	direct product factor
179	C₄	C₄³×C₅₃	1	1	yes	characteristic
180	C₄	C₅₃×#376	1	1	yes	characteristic
181	C₄	C₄×C₂²⋊C₄×C₅₃	1	2	yes	—
182	C₄	C₄×C₂²⋊C₄×C₅₃	1	4	yes	—
183	C₂	—	2	4	no	—
184	C₂	C₄×#95×C₅₃	1	1	yes	characteristic
185	C₂	C₂×C₅₃×#341	1	2	yes	—
186	C₂	C₄×#108×C₅₃	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₅₃	D₄
commutator	C₄	C₄³×C₅₃
Frattini, cosocle	C₂³×C₄	C₂³×C₅₃
socle	C₂³×C₅₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₂×Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅₃ ◁ C₂²×C₄²×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₅₃×#341	C₄	direct (N × H)
77. C₄×#341	C₅₃	direct (N × H)
85. C₄×C₅₃	#341	direct (N × H)
106. #341	C₄×C₅₃	direct (N × H)
107. C₅₃	C₄×#341	direct (N × H)
178. C₄	C₅₃×#341	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×C₄×M₁₆×C₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₂×C₅₃×#341	2
C₄×#341	53

Subgroups — C4×C53×#341 #27218216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅₃×#341 #27218216