Subgroups — C₂³×#13133208 #28290759

Order 13568.

8414 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	C₂²×#13133208	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
3	C₂³×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×#11719919	C₂	1	2	yes	maximal
5	C₂³×C₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
8	C₂⁴×Dic₅₃	C₂²	1	1	yes	characteristic
9	C₂⁴×Dic₅₃	C₂²	1	2	yes	—
10	C₂×#13133208	C₂²	1	448	yes	direct product factor
11	C₂²×C₄⋊Dic₅₃	C₂²	1	56	yes	—
12	C₂²×C₄⋊Dic₅₃	C₂²	1	28	yes	—
13	C₂²×#11719919	C₂²	1	56	yes	—
14	C₂²×C₄×Dic₅₃	C₂²	1	28	yes	—
15	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₅₃	C₂²	1	28	yes	—
18	C₂³×Dic₅₃	—	2	8	no	—
19	C₂³×Dic₅₃	—	2	8	no	—
20	C₂³×Dic₅₃	—	2	16	no	—
21	C₂³×Dic₅₃	C₂³	1	14	yes	—
22	C₂³×Dic₅₃	C₂³	1	14	yes	—
23	C₂³×Dic₅₃	C₂³	1	28	yes	—
24	#13133208	C₂³	1	512	yes	direct product factor
25	C₂×C₄⋊Dic₅₃	C₂³	1	224	yes	—
26	C₂×C₄⋊Dic₅₃	C₂³	1	112	yes	—
27	C₂×#11719919	C₂³	1	224	yes	—
28	C₂×C₄×Dic₅₃	C₂³	1	112	yes	—
29	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	—	2	16	no	—
31	C₂³×C₄×C₅₃	Q₈	1	16	yes	—
32	C₂³×C₄×C₅₃	C₂³	1	28	yes	—
33	C₂³×C₄×C₅₃	C₂³	1	14	yes	—
34	C₂×C₄⋊C₄×C₅₃	C₂³	1	112	yes	—
35	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
36	C₂⁴×C₅₃	#41	1	8	yes	—
37	C₂⁴×C₅₃	#41	1	8	yes	—
38	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
39	C₂²×C₄×C₅₃	—	2	112	no	—
40	C₂²×C₄×C₅₃	C₂×Q₈	1	112	yes	—
41	C₂²×C₄×C₅₃	C₂⁴	1	56	yes	—
42	C₂²×C₄×C₅₃	C₂⁴	1	28	yes	—
43	C₂²×Dic₅₃	—	2	56	no	—
44	C₂²×Dic₅₃	—	2	56	no	—
45	C₂²×Dic₅₃	—	2	112	no	—
46	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
47	C₂²×Dic₅₃	C₂⁴	1	28	yes	—
48	C₂²×Dic₅₃	C₂⁴	1	56	yes	—
49	C₄⋊C₄×C₅₃	C₂⁴	1	64	yes	—
50	C₄⋊Dic₅₃	C₂⁴	1	128	yes	—
51	C₄⋊Dic₅₃	C₂⁴	1	64	yes	—
52	#11719919	C₂⁴	1	128	yes	—
53	C₄×Dic₅₃	C₂⁴	1	64	yes	—
54	C₂³×C₅₃	#125	1	64	yes	complemented
55	C₂³×C₅₃	C₂²×Q₈	1	28	yes	—
56	C₂³×C₅₃	C₂×#41	1	28	yes	—
57	C₂³×C₅₃	C₂×#41	1	28	yes	—
58	C₂³×C₅₃	C₂⁵	1	7	yes	—
59	C₂×C₄×C₅₃	—	2	112	no	—
60	C₂×C₄×C₅₃	C₂²×Q₈	1	112	yes	—
61	C₂×C₄×C₅₃	C₂⁵	1	16	yes	—
62	C₂×C₄×C₅₃	C₂⁵	1	8	yes	—
63	C₂×Dic₅₃	—	2	56	no	—
64	C₂×Dic₅₃	—	2	56	no	—
65	C₂×Dic₅₃	—	2	112	no	—
66	C₂×Dic₅₃	C₂⁵	1	8	yes	—
67	C₂×Dic₅₃	C₂⁵	1	8	yes	—
68	C₂×Dic₅₃	C₂⁵	1	16	yes	—
69	C₂³×#125	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	C₂×#125	1	112	yes	complemented
71	C₂²×C₅₃	C₂³×Q₈	1	14	yes	—
72	C₂²×C₅₃	C₂²×#41	1	14	yes	—
73	C₂²×C₅₃	C₂²×#41	1	14	yes	—
74	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
75	C₄×C₅₃	—	2	16	no	—
76	C₄×C₅₃	C₂³×Q₈	1	16	yes	—
77	Dic₅₃	—	2	8	no	—
78	Dic₅₃	—	2	8	no	—
79	Dic₅₃	—	2	16	no	—
80	C₂²×#125	—	53	56	no	—
81	C₂³×C₄⋊C₄	—	53	2	no	—
82	C₂³×C₄⋊C₄	—	53	2	no	—
83	C₂³×C₄⋊C₄	—	53	1	no	—
84	C₂³×C₄⋊C₄	D₅₃	1	1	yes	characteristic
85	C₂³×C₄²	—	53	1	no	—
86	C₂×C₅₃	C₂²×#125	1	28	yes	complemented
87	C₂×C₅₃	C₂⁴×Q₈	1	1	yes	characteristic
88	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
89	C₂×C₅₃	C₂³×#41	1	1	yes	characteristic
90	C₂⁴×C₄	—	53	1	no	—
91	C₂⁴×C₄	—	53	1	no	—
92	C₂⁴×C₄	—	53	2	no	—
93	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
94	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
95	C₂×#125	—	53	448	no	—
96	C₂²×C₄⋊C₄	—	53	56	no	—
97	C₂²×C₄⋊C₄	—	53	28	no	—
98	C₂²×C₄⋊C₄	—	53	56	no	—
99	C₂²×C₄⋊C₄	C₂×D₅₃	1	28	yes	—
100	C₂²×C₄²	—	53	28	no	—
101	C₅₃	C₂³×#125	1	1	yes	characteristic, complemented, Sylow
102	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
103	C₂³×C₄	—	2	16	no	—
104	C₂³×C₄	—	53	28	no	—
105	C₂³×C₄	—	53	14	no	—
106	C₂³×C₄	—	53	14	no	—
107	C₂³×C₄	—	106	8	no	—
108	C₂³×C₄	—	106	8	no	—
109	C₂³×C₄	—	106	16	no	—
110	C₂³×C₄	Dic₁₀₆	1	16	yes	—
111	C₂³×C₄	C₂²×D₅₃	1	28	yes	—
112	C₂³×C₄	C₂²×D₅₃	1	14	yes	—
113	#125	—	53	512	no	—
114	C₂×C₄⋊C₄	—	53	224	no	—
115	C₂×C₄⋊C₄	—	53	224	no	—
116	C₂×C₄⋊C₄	—	53	112	no	—
117	C₂×C₄⋊C₄	C₂²×D₅₃	1	112	yes	—
118	C₂×C₄²	—	53	112	no	—
119	C₂⁴	C₂×Dic₁₀₆	1	8	yes	—
120	C₂⁴	#11719947	1	8	yes	—
121	C₂⁴	#11719949	1	8	yes	—
122	C₂⁴	C₂³×D₅₃	1	7	yes	—
123	C₂²×C₄	—	2	112	no	—
124	C₂²×C₄	—	53	28	no	—
125	C₂²×C₄	—	53	28	no	—
126	C₂²×C₄	—	53	56	no	—
127	C₂²×C₄	—	106	56	no	—
128	C₂²×C₄	—	106	112	no	—
129	C₂²×C₄	—	106	56	no	—
130	C₂²×C₄	C₂×Dic₁₀₆	1	112	yes	—
131	C₂²×C₄	C₂³×D₅₃	1	56	yes	—
132	C₂²×C₄	C₂³×D₅₃	1	28	yes	—
133	C₄⋊C₄	—	53	128	no	—
134	C₄⋊C₄	—	53	128	no	—
135	C₄⋊C₄	—	53	64	no	—
136	C₄⋊C₄	C₂³×D₅₃	1	64	yes	—
137	C₄²	—	53	64	no	—
138	C₂³	#13133208	1	64	yes	direct product factor
139	C₂³	C₂²×Dic₁₀₆	1	28	yes	—
140	C₂³	C₂×#11719947	1	28	yes	—
141	C₂³	C₂×#11719949	1	28	yes	—
142	C₂³	C₂⁴×D₅₃	1	7	yes	—
143	C₂×C₄	—	2	112	no	—
144	C₂×C₄	—	53	8	no	—
145	C₂×C₄	—	53	8	no	—
146	C₂×C₄	—	53	16	no	—
147	C₂×C₄	—	106	56	no	—
148	C₂×C₄	—	106	56	no	—
149	C₂×C₄	—	106	112	no	—
150	C₂×C₄	C₂²×Dic₁₀₆	1	112	yes	—
151	C₂×C₄	C₂⁴×D₅₃	1	16	yes	—
152	C₂×C₄	C₂⁴×D₅₃	1	8	yes	—
153	C₂²	C₂×#13133208	1	112	yes	direct product factor
154	C₂²	C₂³×Dic₁₀₆	1	14	yes	—
155	C₂²	C₂²×#11719947	1	14	yes	—
156	C₂²	C₂²×#11719949	1	14	yes	—
157	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
158	C₄	—	2	16	no	—
159	C₄	—	106	8	no	—
160	C₄	—	106	8	no	—
161	C₄	—	106	16	no	—
162	C₄	C₂³×Dic₁₀₆	1	16	yes	—
163	C₂	C₂²×#13133208	1	28	yes	direct product factor
164	C₂	C₂⁴×Dic₁₀₆	1	1	yes	characteristic
165	C₂	C₂³×#11719947	1	1	yes	characteristic
166	C₂	C₂³×#11719949	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂³×C₄⋊C₄	D₅₃
hypocenter	C₅₃	C₂³×#125
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#13133208	C₂	direct (N × H)
10. C₂×#13133208	C₂²	direct (N × H)
24. #13133208	C₂³	direct (N × H)
54. C₂³×C₅₃	#125	semidirect
70. C₂²×C₅₃	C₂×#125	semidirect
86. C₂×C₅₃	C₂²×#125	semidirect
101. C₅₃	C₂³×#125	semidirect
138. C₂³	#13133208	direct (N × H)
153. C₂²	C₂×#13133208	direct (N × H)
163. C₂	C₂²×#13133208	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×#13133208	2
C₂³×C₄⋊Dic₅₃	2
C₂³×#11719919	2
C₂³×C₄×Dic₅₃	2
C₂³×C₄⋊C₄×C₅₃	2
C₂³×#125	53

Subgroups — C23×#13133208 #28290759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13133208 #28290759