Subgroups — C₄²×C₄⋊C₄×C₅₃ #27221260

Order 13568.

2478 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
10. C₄×C₄⋊C₄×C₅₃	C₄	direct (N × H)
13. C₄³×C₅₃	C₄	semidirect
44. C₄⋊C₄×C₅₃	C₄²	direct (N × H)
47. C₄²×C₅₃	C₄²	semidirect
48. C₄²×C₅₃	C₄⋊C₄	direct (N × H)
67. C₄²×C₄⋊C₄	C₅₃	direct (N × H)
77. C₄×C₅₃	C₄³	semidirect
78. C₄×C₅₃	C₄×C₄⋊C₄	direct (N × H)
93. C₄×C₄⋊C₄	C₄×C₅₃	direct (N × H)
96. C₄³	C₄×C₅₃	semidirect
97. C₅₃	C₄²×C₄⋊C₄	direct (N × H)
128. C₄⋊C₄	C₄²×C₅₃	direct (N × H)
131. C₄²	C₄²×C₅₃	semidirect
132. C₄²	C₄⋊C₄×C₅₃	direct (N × H)
160. C₄	C₄³×C₅₃	semidirect
161. C₄	C₄×C₄⋊C₄×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×C₄⋊C₄×C₅₃	2
C₂×C₄³×C₅₃	2
C₄×#95×C₅₃	2
C₄²×C₄⋊C₄	53

Subgroups — C42×C4⋊C4×C53 #27221260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₄⋊C₄×C₅₃ #27221260