Subgroups — C₂²×#16810489 #27542367

Order 13568.

4286 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊Dic₅₃	C₂	1	2	yes	maximal
2	C₂×#16810489	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133124	C₂	1	4	yes	maximal
4	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₅₃	C₂²	1	4	yes	—
6	C₂²×C₄⋊Dic₅₃	—	2	16	no	—
7	C₂²×C₄⋊Dic₅₃	C₂²	1	24	yes	—
8	#16810489	C₂²	1	64	yes	direct product factor
9	C₂×#13133124	C₂²	1	48	yes	—
10	C₂⁴×C₄×C₅₃	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₅₃	C₄	1	16	yes	complemented
13	C₂²×C₄²×C₅₃	C₂²	1	12	yes	—
14	C₂³×Dic₅₃	—	2	16	no	—
15	C₂³×Dic₅₃	—	2	32	no	—
16	C₂³×Dic₅₃	C₂³	1	24	yes	—
17	C₂×C₄⋊Dic₅₃	—	2	96	no	—
18	C₂×C₄⋊Dic₅₃	C₂³	1	32	yes	—
19	#13133124	C₂³	1	64	yes	—
20	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₅₃	—	2	16	no	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	16	yes	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₅₃	D₄	1	8	yes	—
25	C₂³×C₄×C₅₃	Q₈	1	8	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	12	yes	—
27	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	96	yes	complemented
29	C₂×C₄²×C₅₃	C₂³	1	16	yes	—
30	C₂⁴×C₅₃	C₂²×C₄	1	4	yes	—
31	C₂⁴×C₅₃	C₂×D₄	1	4	yes	—
32	C₂⁴×C₅₃	C₂×Q₈	1	4	yes	—
33	C₂⁴×C₅₃	#41	1	16	yes	—
34	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
35	C₂²×C₄×C₅₃	—	2	64	no	—
36	C₂²×C₄×C₅₃	—	2	48	no	—
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	64	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	48	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	24	yes	—
40	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
41	C₂²×C₄×C₅₃	C₂×Q₈	1	24	yes	—
42	C₂²×C₄×C₅₃	C₂⁴	1	8	yes	—
43	C₂²×C₄×C₅₃	C₂⁴	1	4	yes	—
44	C₂²×Dic₅₃	—	2	48	no	—
45	C₂²×Dic₅₃	—	2	96	no	—
46	C₂²×Dic₅₃	—	4	64	no	—
47	C₂²×Dic₅₃	C₂⁴	1	16	yes	—
48	C₄²×C₅₃	C₂²×C₄	1	64	yes	complemented
49	C₄⋊Dic₅₃	—	2	64	no	—
50	C₂³×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
51	C₂³×C₅₃	#117	1	32	yes	—
52	C₂³×C₅₃	#124	1	32	yes	—
53	C₂³×C₅₃	#125	1	32	yes	—
54	C₂³×C₅₃	C₂³×C₄	1	6	yes	—
55	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
56	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
57	C₂³×C₅₃	C₂×#41	1	24	yes	—
58	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
59	C₂×C₄×C₅₃	—	2	96	no	—
60	C₂×C₄×C₅₃	—	2	16	no	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	96	yes	—
62	C₂×C₄×C₅₃	C₂³×C₄	1	16	yes	—
63	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
64	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
65	C₂×C₄×C₅₃	C₂²×Q₈	1	8	yes	—
66	C₂×Dic₅₃	—	2	16	no	—
67	C₂×Dic₅₃	—	2	32	no	—
68	C₂×Dic₅₃	—	4	96	no	—
69	C₂²×#382	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	#382	1	64	yes	complemented
71	C₂²×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
72	C₂²×C₅₃	C₂×#117	1	24	yes	—
73	C₂²×C₅₃	C₂×#124	1	24	yes	—
74	C₂²×C₅₃	C₂×#125	1	24	yes	—
75	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
76	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
77	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
78	C₂²×C₅₃	C₂²×#41	1	4	yes	—
79	C₄×C₅₃	—	2	16	no	—
80	C₄×C₅₃	C₂²×C₄⋊C₄	1	16	yes	—
81	Dic₅₃	—	4	16	no	—
82	C₂³×C₄⋊C₄	—	53	2	no	—
83	C₂³×C₄²	D₅₃	1	1	yes	characteristic
84	C₂×#382	—	53	24	no	—
85	C₂²×#95	—	53	4	no	—
86	C₂×C₅₃	C₂×#382	1	24	yes	complemented
87	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₅₃	C₂²×#117	1	2	yes	—
89	C₂×C₅₃	C₂²×#124	1	2	yes	—
90	C₂×C₅₃	C₂²×#125	1	2	yes	—
91	C₂⁴×C₄	—	53	4	no	—
92	C₂⁴×C₄	C₂×D₅₃	1	2	yes	—
93	C₂⁴×C₄	C₂×D₅₃	1	1	yes	characteristic
94	C₂²×C₄⋊C₄	—	53	24	no	—
95	C₂²×C₄⋊C₄	—	106	16	no	—
96	C₂²×C₄²	Dic₅₃	1	16	yes	complemented
97	C₂²×C₄²	C₂×D₅₃	1	12	yes	—
98	#382	—	53	64	no	—
99	C₂×#95	—	53	48	no	—
100	C₅₃	C₂²×#382	1	1	yes	characteristic, complemented, Sylow
101	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
102	C₂³×C₄	—	2	16	no	—
103	C₂³×C₄	—	53	24	no	—
104	C₂³×C₄	—	106	16	no	—
105	C₂³×C₄	—	106	32	no	—
106	C₂³×C₄	Dic₁₀₆	1	8	yes	—
107	C₂³×C₄	D₂₁₂	1	8	yes	—
108	C₂³×C₄	C₂×Dic₅₃	1	16	yes	—
109	C₂³×C₄	C₂×Dic₅₃	1	8	yes	—
110	C₂³×C₄	C₂²×D₅₃	1	12	yes	—
111	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
112	C₂×C₄⋊C₄	—	53	32	no	—
113	C₂×C₄⋊C₄	—	106	96	no	—
114	C₂×C₄²	C₂×Dic₅₃	1	96	yes	complemented
115	C₂×C₄²	C₂²×D₅₃	1	16	yes	—
116	#95	—	53	64	no	—
117	C₂⁴	C₂×Dic₁₀₆	1	4	yes	—
118	C₂⁴	C₂×D₂₁₂	1	4	yes	—
119	C₂⁴	#11719945	1	16	yes	—
120	C₂⁴	C₂²×Dic₅₃	1	4	yes	—
121	C₂⁴	C₂³×D₅₃	1	3	yes	—
122	C₂²×C₄	—	2	64	no	—
123	C₂²×C₄	—	2	48	no	—
124	C₂²×C₄	—	53	16	no	—
125	C₂²×C₄	—	106	48	no	—
126	C₂²×C₄	—	106	96	no	—
127	C₂²×C₄	—	212	64	no	—
128	C₂²×C₄	C₄⋊Dic₅₃	1	64	yes	—
129	C₂²×C₄	C₂×Dic₁₀₆	1	24	yes	—
130	C₂²×C₄	C₂×D₂₁₂	1	24	yes	—
131	C₂²×C₄	C₂²×Dic₅₃	1	48	yes	—
132	C₂²×C₄	C₂²×Dic₅₃	1	24	yes	—
133	C₂²×C₄	C₂³×D₅₃	1	8	yes	—
134	C₂²×C₄	C₂³×D₅₃	1	4	yes	—
135	C₄⋊C₄	—	106	64	no	—
136	C₄²	C₂²×Dic₅₃	1	64	yes	complemented
137	C₂³	C₂×C₄⋊Dic₅₃	1	16	yes	—
138	C₂³	#13133176	1	32	yes	—
139	C₂³	#13133205	1	32	yes	—
140	C₂³	#13133209	1	32	yes	—
141	C₂³	C₂³×Dic₅₃	1	6	yes	—
142	C₂³	C₂²×D₂₁₂	1	6	yes	—
143	C₂³	C₂²×Dic₁₀₆	1	6	yes	—
144	C₂³	C₂×#11719945	1	24	yes	—
145	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
146	C₂×C₄	—	2	96	no	—
147	C₂×C₄	—	2	16	no	—
148	C₂×C₄	—	106	16	no	—
149	C₂×C₄	—	106	32	no	—
150	C₂×C₄	—	212	96	no	—
151	C₂×C₄	C₂×C₄⋊Dic₅₃	1	96	yes	—
152	C₂×C₄	C₂³×Dic₅₃	1	16	yes	—
153	C₂×C₄	C₂³×Dic₅₃	1	8	yes	—
154	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
155	C₂×C₄	C₂²×Dic₁₀₆	1	8	yes	—
156	C₂²	#16810489	1	64	yes	direct product factor
157	C₂²	C₂²×C₄⋊Dic₅₃	1	12	yes	—
158	C₂²	C₂×#13133176	1	24	yes	—
159	C₂²	C₂×#13133205	1	24	yes	—
160	C₂²	C₂×#13133209	1	24	yes	—
161	C₂²	C₂⁴×Dic₅₃	1	1	yes	characteristic
162	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
163	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
164	C₂²	C₂²×#11719945	1	4	yes	—
165	C₄	—	2	16	no	—
166	C₄	—	212	16	no	—
167	C₄	C₂²×C₄⋊Dic₅₃	1	16	yes	—
168	C₂	C₂×#16810489	1	24	yes	direct product factor
169	C₂	C₂³×C₄⋊Dic₅₃	1	1	yes	characteristic
170	C₂	C₂²×#13133176	1	2	yes	—
171	C₂	C₂²×#13133205	1	2	yes	—
172	C₂	C₂²×#13133209	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle, prime radical	C₂⁵×C₅₃	C₂³
cosocle	C₂³×C₅₃	C₂⁵
hypercenter	C₂³×C₄²	D₅₃
hypocenter	C₅₃	C₂²×#382
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#16810489	C₂	direct (N × H)
8. #16810489	C₂²	direct (N × H)
12. C₂²×C₄²×C₅₃	C₄	semidirect
28. C₂×C₄²×C₅₃	C₂×C₄	semidirect
48. C₄²×C₅₃	C₂²×C₄	semidirect
70. C₂²×C₅₃	#382	semidirect
86. C₂×C₅₃	C₂×#382	semidirect
96. C₂²×C₄²	Dic₅₃	semidirect
100. C₅₃	C₂²×#382	semidirect
114. C₂×C₄²	C₂×Dic₅₃	semidirect
136. C₄²	C₂²×Dic₅₃	semidirect
156. C₂²	#16810489	direct (N × H)
168. C₂	C₂×#16810489	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊Dic₅₃	2
C₂×#16810489	2
C₂²×#13133124	2
C₂³×C₄²×C₅₃	2
C₂²×#382	53

Subgroups — C22×#16810489 #27542367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810489 #27542367