Subgroups — C₂³×#13133146 #28247699

Order 13568.

3870 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13133146	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₅₃	C₄	1	1	yes	characteristic
6	C₂⁴×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂³×#88322	—	2	8	no	—
8	C₂³×#88322	C₂²	1	14	yes	—
9	C₂²×C₄×Dic₅₃	C₄	1	14	yes	—
10	C₂²×C₄×Dic₅₃	C₂²	1	14	yes	—
11	C₂×#13133146	C₂²	1	112	yes	direct product factor
12	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₅₃	—	2	8	no	—
14	C₂³×C₈×C₅₃	C₂²	1	14	yes	—
15	C₂³×Dic₅₃	—	2	8	no	—
16	C₂³×Dic₅₃	C₈	1	16	yes	complemented
17	C₂³×Dic₅₃	C₂×C₄	1	14	yes	—
18	C₂³×Dic₅₃	C₂×C₄	1	14	yes	—
19	C₂²×#88322	—	2	56	no	—
20	C₂²×#88322	C₂³	1	28	yes	—
21	C₂×C₄×Dic₅₃	C₂×C₄	1	84	yes	—
22	C₂×C₄×Dic₅₃	C₂³	1	28	yes	—
23	#13133146	C₂³	1	64	yes	direct product factor
24	C₂⁵×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	C₂×C₄	1	7	yes	—
26	C₂³×C₄×C₅₃	D₄	1	8	yes	—
27	C₂³×C₄×C₅₃	Q₈	1	8	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	7	yes	—
29	C₂²×C₈×C₅₃	—	2	56	no	—
30	C₂²×C₈×C₅₃	C₂³	1	28	yes	—
31	C₂⁴×C₅₃	C₄⋊C₄	1	8	yes	—
32	C₂⁴×C₅₃	C₂×C₈	1	8	yes	—
33	C₂⁴×C₅₃	M₁₆	1	8	yes	—
34	C₂⁴×C₅₃	C₂²×C₄	1	7	yes	—
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	56	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	21	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	28	yes	—
38	C₂²×C₄×C₅₃	C₂×Q₈	1	28	yes	—
39	C₂²×C₄×C₅₃	C₂⁴	1	7	yes	—
40	C₂²×Dic₅₃	—	2	56	no	—
41	C₂²×Dic₅₃	C₂×C₈	1	112	yes	complemented
42	C₂²×Dic₅₃	C₂²×C₄	1	28	yes	—
43	C₂²×Dic₅₃	C₂²×C₄	1	28	yes	—
44	C₂×C₈×C₅₃	—	2	56	no	—
45	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
46	C₄×Dic₅₃	C₂²×C₄	1	56	yes	—
47	C₄×Dic₅₃	C₂⁴	1	8	yes	—
48	C₂×#88322	—	2	56	no	—
49	C₂×#88322	C₂⁴	1	8	yes	—
50	C₂³×C₅₃	C₄⋊C₈	1	64	yes	complemented
51	C₂³×C₅₃	C₂×C₄⋊C₄	1	28	yes	—
52	C₂³×C₅₃	C₂²×C₈	1	28	yes	—
53	C₂³×C₅₃	C₂×M₁₆	1	28	yes	—
54	C₂³×C₅₃	C₂³×C₄	1	7	yes	—
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	84	yes	—
56	C₂×C₄×C₅₃	C₂³×C₄	1	7	yes	—
57	C₂×C₄×C₅₃	C₂²×D₄	1	14	yes	—
58	C₂×C₄×C₅₃	C₂²×Q₈	1	14	yes	—
59	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
60	C₂×Dic₅₃	—	2	56	no	—
61	C₂×Dic₅₃	C₂²×C₈	1	112	yes	complemented
62	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
63	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
64	C₈×C₅₃	—	2	8	no	—
65	#88322	—	2	8	no	—
66	C₂³×C₄⋊C₈	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	C₂×C₄⋊C₈	1	112	yes	complemented
68	C₂²×C₅₃	C₂²×C₄⋊C₄	1	14	yes	—
69	C₂²×C₅₃	C₂³×C₈	1	14	yes	—
70	C₂²×C₅₃	C₂²×M₁₆	1	14	yes	—
71	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
72	C₄×C₅₃	C₂²×C₄⋊C₄	1	14	yes	—
73	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
74	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
75	Dic₅₃	—	2	8	no	—
76	Dic₅₃	C₂³×C₈	1	16	yes	complemented
77	C₂⁴×C₈	—	53	1	no	—
78	C₂⁴×C₈	D₅₃	1	1	yes	characteristic
79	C₂³×C₄²	—	53	1	no	—
80	C₂²×C₄⋊C₈	—	53	28	no	—
81	C₂×C₅₃	C₂²×C₄⋊C₈	1	28	yes	complemented
82	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
84	C₂×C₅₃	C₂³×M₁₆	1	1	yes	characteristic
85	C₂⁴×C₄	—	53	1	no	—
86	C₂⁴×C₄	—	53	1	no	—
87	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
88	C₂³×C₈	—	2	8	no	—
89	C₂³×C₈	—	53	14	no	—
90	C₂³×C₈	—	106	8	no	—
91	C₂³×C₈	C₂×D₅₃	1	14	yes	—
92	C₂²×C₄²	—	53	14	no	—
93	C₂²×C₄²	—	53	14	no	—
94	C₂×C₄⋊C₈	—	53	112	no	—
95	C₅₃	C₂³×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
96	C₂⁵	C₄×D₅₃	1	1	yes	characteristic
97	C₂³×C₄	—	53	14	no	—
98	C₂³×C₄	—	53	16	no	—
99	C₂³×C₄	—	53	14	no	—
100	C₂³×C₄	—	106	8	no	—
101	C₂³×C₄	Dic₁₀₆	1	8	yes	—
102	C₂³×C₄	C₄×D₅₃	1	7	yes	—
103	C₂³×C₄	C₂²⋊D₅₃	1	8	yes	—
104	C₂³×C₄	C₂²×D₅₃	1	7	yes	—
105	C₂²×C₈	—	2	56	no	—
106	C₂²×C₈	—	53	28	no	—
107	C₂²×C₈	—	106	56	no	—
108	C₂²×C₈	C₂²×D₅₃	1	28	yes	—
109	C₂×C₄²	—	53	84	no	—
110	C₂×C₄²	—	53	28	no	—
111	C₄⋊C₈	—	53	64	no	—
112	C₂⁴	C₈×D₅₃	1	8	yes	—
113	C₂⁴	#11719912	1	8	yes	—
114	C₂⁴	#11719919	1	8	yes	—
115	C₂⁴	C₂×C₄×D₅₃	1	7	yes	—
116	C₂²×C₄	—	53	112	no	—
117	C₂²×C₄	—	53	28	no	—
118	C₂²×C₄	—	53	28	no	—
119	C₂²×C₄	—	106	56	no	—
120	C₂²×C₄	#11719919	1	56	yes	—
121	C₂²×C₄	C₂×Dic₁₀₆	1	28	yes	—
122	C₂²×C₄	C₂×C₄×D₅₃	1	21	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₅₃	1	28	yes	—
124	C₂²×C₄	C₂³×D₅₃	1	7	yes	—
125	C₂×C₈	—	2	56	no	—
126	C₂×C₈	—	53	8	no	—
127	C₂×C₈	—	106	56	no	—
128	C₂×C₈	C₂³×D₅₃	1	8	yes	—
129	C₄²	—	53	56	no	—
130	C₄²	—	53	8	no	—
131	C₂³	#13133146	1	64	yes	direct product factor
132	C₂³	C₂×#11719919	1	28	yes	—
133	C₂³	C₂×C₈×D₅₃	1	28	yes	—
134	C₂³	C₂×#11719912	1	28	yes	—
135	C₂³	C₂²×C₄×D₅₃	1	7	yes	—
136	C₂×C₄	—	53	112	no	—
137	C₂×C₄	—	53	8	no	—
138	C₂×C₄	—	53	8	no	—
139	C₂×C₄	—	106	56	no	—
140	C₂×C₄	C₂×#11719919	1	84	yes	—
141	C₂×C₄	C₂²×C₄×D₅₃	1	7	yes	—
142	C₂×C₄	C₂²×C₂²⋊D₅₃	1	14	yes	—
143	C₂×C₄	C₂²×Dic₁₀₆	1	14	yes	—
144	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
145	C₈	—	2	8	no	—
146	C₈	—	106	8	no	—
147	C₂²	C₂×#13133146	1	112	yes	direct product factor
148	C₂²	C₂²×#11719919	1	14	yes	—
149	C₂²	C₂²×C₈×D₅₃	1	14	yes	—
150	C₂²	C₂²×#11719912	1	14	yes	—
151	C₂²	C₂³×C₄×D₅₃	1	1	yes	characteristic
152	C₄	—	53	16	no	—
153	C₄	—	106	8	no	—
154	C₄	C₂²×#11719919	1	14	yes	—
155	C₄	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
156	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
157	C₂	C₂²×#13133146	1	28	yes	direct product factor
158	C₂	C₂³×#11719919	1	1	yes	characteristic
159	C₂	C₂³×C₈×D₅₃	1	1	yes	characteristic
160	C₂	C₂³×#11719912	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₅₃
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂⁴×C₈×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂×C₄
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂⁴×C₈	D₅₃
hypocenter	C₅₃	C₂³×C₄⋊C₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
3. C₂²×#13133146	C₂	direct (N × H)
11. C₂×#13133146	C₂²	direct (N × H)
16. C₂³×Dic₅₃	C₈	semidirect
23. #13133146	C₂³	direct (N × H)
41. C₂²×Dic₅₃	C₂×C₈	semidirect
50. C₂³×C₅₃	C₄⋊C₈	semidirect
61. C₂×Dic₅₃	C₂²×C₈	semidirect
67. C₂²×C₅₃	C₂×C₄⋊C₈	semidirect
76. Dic₅₃	C₂³×C₈	semidirect
81. C₂×C₅₃	C₂²×C₄⋊C₈	semidirect
95. C₅₃	C₂³×C₄⋊C₈	semidirect
131. C₂³	#13133146	direct (N × H)
147. C₂²	C₂×#13133146	direct (N × H)
157. C₂	C₂²×#13133146	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#88322	2
C₂³×C₄×Dic₅₃	2
C₂²×#13133146	2
C₂⁴×C₈×C₅₃	2
C₂³×C₄⋊C₈	53

Subgroups — C23×#13133146 #28247699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133146 #28247699