Subgroups — C₄²×D₈×C₅₃ #27228009

Order 13568.

1350 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×D₈×C₅₃	C₂	1	3	yes	maximal
2	C₄²×D₄×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₄×#107×C₅₃	C₂	1	3	yes	maximal, complemented
4	C₄×#102×C₅₃	C₂	1	6	yes	maximal, complemented
5	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₅₃	C₂²	1	6	yes	—
8	C₂²×C₄²×C₅₃	—	2	2	no	—
9	C₄×D₈×C₅₃	C₄	1	24	yes	direct product factor
10	C₂×#107×C₅₃	C₂²	1	3	yes	—
11	C₂×#102×C₅₃	C₂²	1	6	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
14	C₄×C₄⋊C₄×C₅₃	C₂²	1	6	yes	—
15	C₄×C₂²⋊C₄×C₅₃	—	2	6	no	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#340	C₂²	1	6	yes	—
18	C₂²×D₄×C₅₃	C₂³	1	2	yes	—
19	C₂³×C₄×C₅₃	—	2	6	no	—
20	C₂×D₈×C₅₃	C₂×C₄	1	12	yes	—
21	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
22	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₄×D₄×C₅₃	—	2	12	no	—
24	C₄×D₄×C₅₃	C₂×C₄	1	24	yes	complemented
25	C₂×C₄⋊C₄×C₅₃	C₂³	1	6	yes	—
26	C₂×C₂²⋊C₄×C₅₃	—	2	6	no	—
27	C₂×C₄²×C₅₃	—	2	6	no	—
28	C₂×C₄²×C₅₃	—	4	8	no	—
29	C₂×C₄²×C₅₃	D₄	1	3	yes	—
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
32	#107×C₅₃	C₂×C₄	1	12	yes	complemented
33	#102×C₅₃	C₂×C₄	1	24	yes	complemented
34	C₄×C₈×C₅₃	—	2	6	no	—
35	C₄×C₈×C₅₃	C₂×C₄	1	12	yes	complemented
36	#95×C₅₃	—	2	6	no	—
37	C₂⁴×C₅₃	—	2	2	no	—
38	C₂×D₄×C₅₃	—	2	6	no	—
39	C₂×D₄×C₅₃	C₂²×C₄	1	12	yes	—
40	C₂²×C₄×C₅₃	—	2	6	no	—
41	C₂²×C₄×C₅₃	—	2	24	no	—
42	C₂²×C₄×C₅₃	—	4	12	no	—
43	C₂²×C₄×C₅₃	—	4	12	no	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
46	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
47	D₈×C₅₃	C₄²	1	16	yes	direct product factor
48	C₂×C₈×C₅₃	—	2	6	no	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	6	yes	—
51	C₄⋊C₄×C₅₃	—	2	6	no	—
52	C₄⋊C₄×C₅₃	C₂²×C₄	1	12	yes	—
53	C₂²⋊C₄×C₅₃	—	2	24	no	—
54	C₄²×C₅₃	—	2	6	no	—
55	C₄²×C₅₃	—	4	12	no	—
56	C₄²×C₅₃	D₈	1	4	yes	direct product factor
57	C₄²×C₅₃	C₂²×C₄	1	6	yes	—
58	C₄²×C₅₃	#41	1	6	yes	—
59	C₂³×C₅₃	—	2	12	no	—
60	C₂³×C₅₃	—	4	2	no	—
61	C₂³×C₅₃	—	4	6	no	—
62	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
63	D₄×C₅₃	—	2	12	no	—
64	D₄×C₅₃	C₂×C₄²	1	8	yes	complemented
65	C₂×C₄×C₅₃	—	2	6	no	—
66	C₂×C₄×C₅₃	—	2	12	no	—
67	C₂×C₄×C₅₃	—	4	6	no	—
68	C₂×C₄×C₅₃	—	4	6	no	—
69	C₂×C₄×C₅₃	—	4	48	no	—
70	C₂×C₄×C₅₃	C₄×D₄	1	6	yes	—
71	C₂×C₄×C₅₃	C₄×D₄	1	6	yes	—
72	C₂×C₄×C₅₃	C₂×D₈	1	6	yes	—
73	C₂×C₄×C₅₃	#135	1	6	yes	—
74	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
75	C₂×C₄×C₅₃	C₂×#41	1	3	yes	—
76	C₈×C₅₃	—	2	6	no	—
77	C₈×C₅₃	C₂×C₄²	1	4	yes	complemented
78	C₄²×D₈	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₅₃	—	2	8	no	—
80	C₂²×C₅₃	—	4	12	no	—
81	C₂²×C₅₃	—	4	12	no	—
82	C₂²×C₅₃	C₂×C₄×D₄	1	3	yes	—
83	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×#135	1	3	yes	—
85	C₄×C₅₃	—	2	6	no	—
86	C₄×C₅₃	—	4	12	no	—
87	C₄×C₅₃	C₄×D₈	1	12	yes	direct product factor
88	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
89	C₄×C₅₃	C₄×#41	1	3	yes	—
90	C₂×C₄×D₈	C₂×C₅₃	1	3	yes	—
91	C₄²×D₄	C₂×C₅₃	1	2	yes	complemented
92	C₄×#107	C₂×C₅₃	1	3	yes	complemented
93	C₄×#102	C₂×C₅₃	1	6	yes	complemented
94	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic, complemented
95	C₂×C₅₃	—	4	8	no	—
96	C₂×C₅₃	C₄²×D₄	1	1	yes	characteristic
97	C₂×C₅₃	C₂×C₄×D₈	1	3	yes	—
98	C₂×C₅₃	C₄×#135	1	3	yes	—
99	C₂²×D₈	C₂²×C₅₃	1	1	yes	characteristic
100	C₂×C₄×D₄	C₂²×C₅₃	1	6	yes	—
101	C₂²×C₄²	—	2	2	no	—
102	C₄×D₈	C₄×C₅₃	1	24	yes	direct product factor
103	C₂×#107	C₂²×C₅₃	1	3	yes	—
104	C₂×#102	C₂²×C₅₃	1	6	yes	—
105	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	—
106	C₂×C₄×C₈	C₂²×C₅₃	1	3	yes	—
107	C₄×C₄⋊C₄	C₂²×C₅₃	1	6	yes	—
108	C₄×C₂²⋊C₄	—	2	6	no	—
109	C₄³	C₂²×C₅₃	1	1	yes	characteristic
110	#340	C₂²×C₅₃	1	6	yes	—
111	C₅₃	C₄²×D₈	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×D₄	C₂³×C₅₃	1	2	yes	—
113	C₂³×C₄	—	2	6	no	—
114	C₂×D₈	C₂×C₄×C₅₃	1	12	yes	—
115	C₂²×C₈	C₂³×C₅₃	1	3	yes	—
116	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
117	C₄×D₄	—	2	12	no	—
118	C₄×D₄	C₂×C₄×C₅₃	1	24	yes	complemented
119	C₂×C₄⋊C₄	C₂³×C₅₃	1	6	yes	—
120	C₂×C₂²⋊C₄	—	2	6	no	—
121	C₂×C₄²	—	2	6	no	—
122	C₂×C₄²	—	4	8	no	—
123	C₂×C₄²	D₄×C₅₃	1	3	yes	—
124	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
125	C₂×C₄²	C₂³×C₅₃	1	3	yes	—
126	#107	C₂×C₄×C₅₃	1	12	yes	complemented
127	#102	C₂×C₄×C₅₃	1	24	yes	complemented
128	C₄×C₈	—	2	6	no	—
129	C₄×C₈	C₂×C₄×C₅₃	1	12	yes	complemented
130	#95	—	2	6	no	—
131	C₂⁴	—	2	2	no	—
132	C₂×D₄	—	2	6	no	—
133	C₂×D₄	C₂²×C₄×C₅₃	1	12	yes	—
134	C₂²×C₄	—	2	6	no	—
135	C₂²×C₄	—	2	24	no	—
136	C₂²×C₄	—	4	12	no	—
137	C₂²×C₄	—	4	12	no	—
138	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
139	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
140	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
141	D₈	C₄²×C₅₃	1	16	yes	direct product factor
142	C₂×C₈	—	2	6	no	—
143	C₂×C₈	C₂²×C₄×C₅₃	1	6	yes	—
144	C₂×C₈	C₂²×C₄×C₅₃	1	6	yes	—
145	C₄⋊C₄	—	2	6	no	—
146	C₄⋊C₄	C₂²×C₄×C₅₃	1	12	yes	—
147	C₂²⋊C₄	—	2	24	no	—
148	C₄²	—	2	6	no	—
149	C₄²	—	4	12	no	—
150	C₄²	D₈×C₅₃	1	4	yes	direct product factor
151	C₄²	C₂²×C₄×C₅₃	1	6	yes	—
152	C₄²	#41×C₅₃	1	6	yes	—
153	C₂³	—	2	12	no	—
154	C₂³	—	4	2	no	—
155	C₂³	—	4	6	no	—
156	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
157	D₄	—	2	12	no	—
158	D₄	C₂×C₄²×C₅₃	1	8	yes	complemented
159	C₂×C₄	—	2	6	no	—
160	C₂×C₄	—	2	12	no	—
161	C₂×C₄	—	4	6	no	—
162	C₂×C₄	—	4	6	no	—
163	C₂×C₄	—	4	48	no	—
164	C₂×C₄	C₄×D₄×C₅₃	1	6	yes	—
165	C₂×C₄	C₄×D₄×C₅₃	1	6	yes	—
166	C₂×C₄	C₂×D₈×C₅₃	1	6	yes	—
167	C₂×C₄	#135×C₅₃	1	6	yes	—
168	C₂×C₄	C₂³×C₄×C₅₃	1	3	yes	—
169	C₂×C₄	C₂×#41×C₅₃	1	3	yes	—
170	C₈	—	2	6	no	—
171	C₈	C₂×C₄²×C₅₃	1	4	yes	complemented
172	C₂²	—	2	8	no	—
173	C₂²	—	4	12	no	—
174	C₂²	—	4	12	no	—
175	C₂²	C₂×C₄×D₄×C₅₃	1	3	yes	—
176	C₂²	C₂²×D₈×C₅₃	1	1	yes	characteristic
177	C₂²	C₂×#135×C₅₃	1	3	yes	—
178	C₄	—	2	6	no	—
179	C₄	—	4	12	no	—
180	C₄	C₄×D₈×C₅₃	1	12	yes	direct product factor
181	C₄	C₂²×C₄²×C₅₃	1	1	yes	characteristic
182	C₄	C₄×#41×C₅₃	1	3	yes	—
183	C₂	—	4	8	no	—
184	C₂	C₄²×D₄×C₅₃	1	1	yes	characteristic
185	C₂	C₂×C₄×D₈×C₅₃	1	3	yes	—
186	C₂	C₄×#135×C₅₃	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²×C₅₃	D₄
commutator	C₄	C₂²×C₄²×C₅₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂²×D₄
prime radical	C₂²×D₈×C₅₃	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅₃ ◁ C₄³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄²×D₄×C₅₃	C₂	semidirect
3. C₄×#107×C₅₃	C₂	semidirect
4. C₄×#102×C₅₃	C₂	semidirect
5. C₄²×C₈×C₅₃	C₂	semidirect
9. C₄×D₈×C₅₃	C₄	direct (N × H)
24. C₄×D₄×C₅₃	C₂×C₄	semidirect
32. #107×C₅₃	C₂×C₄	semidirect
33. #102×C₅₃	C₂×C₄	semidirect
35. C₄×C₈×C₅₃	C₂×C₄	semidirect
47. D₈×C₅₃	C₄²	direct (N × H)
56. C₄²×C₅₃	D₈	direct (N × H)
64. D₄×C₅₃	C₂×C₄²	semidirect
77. C₈×C₅₃	C₂×C₄²	semidirect
78. C₄²×D₈	C₅₃	direct (N × H)
87. C₄×C₅₃	C₄×D₈	direct (N × H)
91. C₄²×D₄	C₂×C₅₃	semidirect
92. C₄×#107	C₂×C₅₃	semidirect
93. C₄×#102	C₂×C₅₃	semidirect
94. C₄²×C₈	C₂×C₅₃	semidirect
102. C₄×D₈	C₄×C₅₃	direct (N × H)
111. C₅₃	C₄²×D₈	direct (N × H)
118. C₄×D₄	C₂×C₄×C₅₃	semidirect
126. #107	C₂×C₄×C₅₃	semidirect
127. #102	C₂×C₄×C₅₃	semidirect
129. C₄×C₈	C₂×C₄×C₅₃	semidirect
141. D₈	C₄²×C₅₃	direct (N × H)
150. C₄²	D₈×C₅₃	direct (N × H)
158. D₄	C₂×C₄²×C₅₃	semidirect
171. C₈	C₂×C₄²×C₅₃	semidirect
180. C₄	C₄×D₈×C₅₃	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×D₈×C₅₃	2
C₄²×D₄×C₅₃	2
C₄×#107×C₅₃	2
C₄×#102×C₅₃	2
C₄²×C₈×C₅₃	2
C₄²×D₈	53

Subgroups — C42×D8×C53 #27228009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×D₈×C₅₃ #27228009