Subgroups — C₂²×C₅₃×#335 #27225296

Order 13568.

942 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₈×C₅₃	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#335	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₅₃	—	2	2	no	—
5	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₅₃	—	2	6	no	—
9	C₂×C₄⋊C₈×C₅₃	C₂²	1	12	yes	—
10	C₂×C₄×C₈×C₅₃	C₄	1	6	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	6	yes	—
12	C₅₃×#335	C₂²	1	16	yes	direct product factor
13	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
15	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	—	2	4	no	—
17	C₂²×C₈×C₅₃	—	2	12	no	—
18	C₂²×C₈×C₅₃	—	4	8	no	—
19	C₂²×C₈×C₅₃	C₈	1	8	yes	complemented
20	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	6	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
23	C₂×C₄²×C₅₃	D₄	1	3	yes	—
24	C₂×C₄²×C₅₃	Q₈	1	3	yes	—
25	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
26	C₄⋊C₈×C₅₃	—	2	12	no	—
27	C₄⋊C₈×C₅₃	C₂³	1	8	yes	—
28	C₄×C₈×C₅₃	C₂×C₄	1	12	yes	—
29	C₄×C₈×C₅₃	C₂³	1	4	yes	—
30	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	—	2	4	no	—
32	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
33	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
34	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
35	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
37	C₂²×C₄×C₅₃	D₈	1	4	yes	—
38	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
42	C₂×C₈×C₅₃	—	2	12	no	—
43	C₂×C₈×C₅₃	—	2	8	no	—
44	C₂×C₈×C₅₃	—	4	24	no	—
45	C₂×C₈×C₅₃	C₂×C₈	1	24	yes	complemented
46	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
47	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
48	C₄²×C₅₃	C₄⋊C₄	1	6	yes	—
49	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
50	C₄²×C₅₃	C₂×D₄	1	3	yes	—
51	C₄²×C₅₃	C₂×Q₈	1	3	yes	—
52	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
54	C₂³×C₅₃	#107	1	4	yes	—
55	C₂³×C₅₃	#108	1	4	yes	—
56	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
57	C₂×C₄×C₅₃	—	2	6	no	—
58	C₂×C₄×C₅₃	—	2	6	no	—
59	C₂×C₄×C₅₃	C₄⋊C₈	1	12	yes	—
60	C₂×C₄×C₅₃	#107	1	12	yes	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
64	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
65	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
66	C₂×C₄×C₅₃	C₂×D₈	1	6	yes	—
67	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
68	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
71	C₈×C₅₃	—	2	4	no	—
72	C₈×C₅₃	—	4	8	no	—
73	C₈×C₅₃	C₂²×C₈	1	8	yes	complemented
74	C₂²×#335	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂²×C₅₃	#335	1	16	yes	direct product factor
76	C₂²×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
77	C₂²×C₅₃	C₂×#107	1	6	yes	—
78	C₂²×C₅₃	C₂×#108	1	6	yes	—
79	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₅₃	—	2	3	no	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
83	C₄×C₅₃	C₂×#107	1	6	yes	—
84	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
85	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
86	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
87	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
88	C₂²×C₄⋊C₈	C₂×C₅₃	1	2	yes	—
89	C₂²×C₄×C₈	C₂×C₅₃	1	1	yes	characteristic
90	C₂×#335	C₂×C₅₃	1	12	yes	direct product factor
91	C₂×C₅₃	C₂×#335	1	12	yes	direct product factor
92	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
93	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
94	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
95	C₂³×C₈	—	2	2	no	—
96	C₂³×C₈	C₄×C₅₃	1	1	yes	characteristic
97	C₂³×C₈	C₄×C₅₃	1	1	yes	characteristic
98	C₂²×C₄²	C₂²×C₅₃	1	1	yes	characteristic
99	C₂×C₄⋊C₈	—	2	6	no	—
100	C₂×C₄⋊C₈	C₂²×C₅₃	1	12	yes	—
101	C₂×C₄×C₈	C₄×C₅₃	1	6	yes	—
102	C₂×C₄×C₈	C₂²×C₅₃	1	6	yes	—
103	#335	C₂²×C₅₃	1	16	yes	direct product factor
104	C₅₃	C₂²×#335	1	1	yes	characteristic, direct product factor, Sylow
105	C₂³×C₄	C₂×C₄×C₅₃	1	1	yes	characteristic
106	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
107	C₂³×C₄	Q₈×C₅₃	1	1	yes	characteristic
108	C₂²×C₈	—	2	4	no	—
109	C₂²×C₈	—	2	12	no	—
110	C₂²×C₈	—	4	8	no	—
111	C₂²×C₈	C₈×C₅₃	1	8	yes	complemented
112	C₂²×C₈	C₂×C₄×C₅₃	1	6	yes	—
113	C₂²×C₈	C₂×C₄×C₅₃	1	6	yes	—
114	C₂×C₄²	C₂×C₄×C₅₃	1	3	yes	—
115	C₂×C₄²	D₄×C₅₃	1	3	yes	—
116	C₂×C₄²	Q₈×C₅₃	1	3	yes	—
117	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
118	C₄⋊C₈	—	2	12	no	—
119	C₄⋊C₈	C₂³×C₅₃	1	8	yes	—
120	C₄×C₈	C₂×C₄×C₅₃	1	12	yes	—
121	C₄×C₈	C₂³×C₅₃	1	4	yes	—
122	C₂⁴	C₄⋊C₄×C₅₃	1	1	yes	characteristic
123	C₂²×C₄	—	2	4	no	—
124	C₂²×C₄	C₄⋊C₄×C₅₃	1	3	yes	—
125	C₂²×C₄	C₄⋊C₄×C₅₃	1	3	yes	—
126	C₂²×C₄	C₄⋊C₄×C₅₃	1	3	yes	—
127	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	—
128	C₂²×C₄	M₁₆×C₅₃	1	4	yes	—
129	C₂²×C₄	D₈×C₅₃	1	4	yes	—
130	C₂²×C₄	Q₁₆×C₅₃	1	4	yes	—
131	C₂²×C₄	C₂²×C₄×C₅₃	1	3	yes	—
132	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
133	C₂²×C₄	C₂×Q₈×C₅₃	1	3	yes	—
134	C₂×C₈	—	2	12	no	—
135	C₂×C₈	—	2	8	no	—
136	C₂×C₈	—	4	24	no	—
137	C₂×C₈	C₂×C₈×C₅₃	1	24	yes	complemented
138	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
139	C₂×C₈	C₂²×C₄×C₅₃	1	4	yes	—
140	C₄²	C₄⋊C₄×C₅₃	1	6	yes	—
141	C₄²	C₂²×C₄×C₅₃	1	3	yes	—
142	C₄²	C₂×D₄×C₅₃	1	3	yes	—
143	C₄²	C₂×Q₈×C₅₃	1	3	yes	—
144	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
145	C₂³	C₄⋊C₈×C₅₃	1	4	yes	—
146	C₂³	#107×C₅₃	1	4	yes	—
147	C₂³	#108×C₅₃	1	4	yes	—
148	C₂³	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
149	C₂×C₄	—	2	6	no	—
150	C₂×C₄	—	2	6	no	—
151	C₂×C₄	C₄⋊C₈×C₅₃	1	12	yes	—
152	C₂×C₄	#107×C₅₃	1	12	yes	—
153	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
154	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
155	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	3	yes	—
156	C₂×C₄	C₂²×C₈×C₅₃	1	6	yes	—
157	C₂×C₄	C₂×M₁₆×C₅₃	1	6	yes	—
158	C₂×C₄	C₂×D₈×C₅₃	1	6	yes	—
159	C₂×C₄	C₂×Q₁₆×C₅₃	1	6	yes	—
160	C₂×C₄	C₂³×C₄×C₅₃	1	1	yes	characteristic
161	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
162	C₂×C₄	C₂²×Q₈×C₅₃	1	1	yes	characteristic
163	C₈	—	2	4	no	—
164	C₈	—	4	8	no	—
165	C₈	C₂²×C₈×C₅₃	1	8	yes	complemented
166	C₂²	C₅₃×#335	1	16	yes	direct product factor
167	C₂²	C₂×C₄⋊C₈×C₅₃	1	6	yes	—
168	C₂²	C₂×#107×C₅₃	1	6	yes	—
169	C₂²	C₂×#108×C₅₃	1	6	yes	—
170	C₂²	C₂²×C₄⋊C₄×C₅₃	1	1	yes	characteristic
171	C₄	—	2	3	no	—
172	C₄	—	2	1	no	—
173	C₄	C₂×C₄⋊C₈×C₅₃	1	6	yes	—
174	C₄	C₂×#107×C₅₃	1	6	yes	—
175	C₄	C₂³×C₈×C₅₃	1	1	yes	characteristic
176	C₄	C₂²×M₁₆×C₅₃	1	1	yes	characteristic
177	C₄	C₂²×D₈×C₅₃	1	1	yes	characteristic
178	C₄	C₂²×Q₁₆×C₅₃	1	1	yes	characteristic
179	C₂	C₂×C₅₃×#335	1	12	yes	direct product factor
180	C₂	C₂²×C₄⋊C₈×C₅₃	1	1	yes	characteristic
181	C₂	C₂²×#107×C₅₃	1	1	yes	characteristic
182	C₂	C₂²×#108×C₅₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₅₃×#335	C₂	direct (N × H)
12. C₅₃×#335	C₂²	direct (N × H)
19. C₂²×C₈×C₅₃	C₈	semidirect
45. C₂×C₈×C₅₃	C₂×C₈	semidirect
73. C₈×C₅₃	C₂²×C₈	semidirect
74. C₂²×#335	C₅₃	direct (N × H)
75. C₂²×C₅₃	#335	direct (N × H)
90. C₂×#335	C₂×C₅₃	direct (N × H)
91. C₂×C₅₃	C₂×#335	direct (N × H)
103. #335	C₂²×C₅₃	direct (N × H)
104. C₅₃	C₂²×#335	direct (N × H)
111. C₂²×C₈	C₈×C₅₃	semidirect
137. C₂×C₈	C₂×C₈×C₅₃	semidirect
165. C₈	C₂²×C₈×C₅₃	semidirect
166. C₂²	C₅₃×#335	direct (N × H)
179. C₂	C₂×C₅₃×#335	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₈×C₅₃	2
C₂²×C₄×C₈×C₅₃	2
C₂×C₅₃×#335	2
C₂²×#335	53

Subgroups — C22×C53×#335 #27225296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅₃×#335 #27225296