Subgroups — C₂²×#16810441 #27537066

Order 13568.

862 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	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810441	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	C₄	1	1	yes	characteristic
6	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
7	C₂²×#11719908	—	2	4	no	—
8	C₂²×#11719908	C₂²	1	6	yes	—
9	C₂×C₈×Dic₅₃	C₄	1	6	yes	—
10	C₂×C₈×Dic₅₃	C₂²	1	6	yes	—
11	#16810441	C₂²	1	16	yes	direct product factor
12	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₃	—	2	4	no	—
14	C₂²×C₁₆×C₅₃	C₂²	1	6	yes	—
15	C₂³×Dic₅₃	C₈	1	1	yes	characteristic
16	C₂³×Dic₅₃	C₈	1	1	yes	characteristic
17	C₂²×#88322	—	2	4	no	—
18	C₂²×#88322	C₂×C₄	1	6	yes	—
19	C₂×C₄×Dic₅₃	C₈	1	6	yes	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
21	C₂×#11719908	—	2	12	no	—
22	C₂×#11719908	C₂³	1	4	yes	—
23	C₈×Dic₅₃	C₂×C₄	1	12	yes	—
24	C₈×Dic₅₃	C₂³	1	4	yes	—
25	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
27	C₂²×C₈×C₅₃	D₄	1	4	yes	—
28	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
30	C₂×C₁₆×C₅₃	—	2	12	no	—
31	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
32	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
34	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
35	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
37	C₂²×Dic₅₃	—	2	4	no	—
38	C₂²×Dic₅₃	C₁₆	1	8	yes	complemented
39	C₂²×Dic₅₃	C₂×C₈	1	6	yes	—
40	C₂²×Dic₅₃	C₂×C₈	1	6	yes	—
41	C₂×C₈×C₅₃	C₄⋊C₄	1	12	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
43	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
44	C₂×C₈×C₅₃	C₂×Q₈	1	6	yes	—
45	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₄×Dic₅₃	C₂×C₈	1	12	yes	—
47	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
48	C₂×#88322	—	2	12	no	—
49	C₂×#88322	C₂²×C₄	1	4	yes	—
50	C₁₆×C₅₃	—	2	4	no	—
51	#11719908	—	2	4	no	—
52	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
53	C₂³×C₅₃	C₂×C₁₆	1	4	yes	—
54	C₂³×C₅₃	M₃₂	1	4	yes	—
55	C₂³×C₅₃	C₂²×C₈	1	3	yes	—
56	C₂×C₄×C₅₃	C₄⋊C₈	1	12	yes	—
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
58	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
59	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
60	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
61	C₂×Dic₅₃	—	2	12	no	—
62	C₂×Dic₅₃	C₂×C₁₆	1	24	yes	complemented
63	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
64	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
65	C₈×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
66	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
67	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
68	#88322	—	2	4	no	—
69	C₂²×C₄⋊C₁₆	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	C₄⋊C₁₆	1	16	yes	complemented
71	C₂²×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
72	C₂²×C₅₃	C₂²×C₁₆	1	6	yes	—
73	C₂²×C₅₃	C₂×M₃₂	1	6	yes	—
74	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
75	C₄×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
76	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
78	Dic₅₃	—	2	4	no	—
79	Dic₅₃	C₂²×C₁₆	1	8	yes	complemented
80	C₂³×C₁₆	—	53	1	no	—
81	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
82	C₂²×C₄×C₈	—	53	1	no	—
83	C₂×C₄⋊C₁₆	—	53	12	no	—
84	C₂×C₅₃	C₂×C₄⋊C₁₆	1	12	yes	complemented
85	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
86	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
87	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
88	C₂³×C₈	—	53	1	no	—
89	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
90	C₂²×C₄²	—	53	1	no	—
91	C₂²×C₁₆	—	2	4	no	—
92	C₂²×C₁₆	—	53	6	no	—
93	C₂²×C₁₆	—	106	4	no	—
94	C₂²×C₁₆	C₂×D₅₃	1	6	yes	—
95	C₂×C₄×C₈	—	53	6	no	—
96	C₂×C₄×C₈	—	53	6	no	—
97	C₄⋊C₁₆	—	53	16	no	—
98	C₅₃	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	—	53	1	no	—
100	C₂³×C₄	—	53	1	no	—
101	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
102	C₂²×C₈	—	53	6	no	—
103	C₂²×C₈	—	106	4	no	—
104	C₂²×C₈	Dic₁₀₆	1	4	yes	—
105	C₂²×C₈	C₄×D₅₃	1	3	yes	—
106	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
107	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
108	C₂×C₄²	—	53	6	no	—
109	C₂×C₄²	—	53	6	no	—
110	C₂×C₁₆	—	2	12	no	—
111	C₂×C₁₆	—	53	4	no	—
112	C₂×C₁₆	—	106	12	no	—
113	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
114	C₄×C₈	—	53	12	no	—
115	C₄×C₈	—	53	4	no	—
116	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
117	C₂²×C₄	—	53	8	no	—
118	C₂²×C₄	—	53	6	no	—
119	C₂²×C₄	—	53	6	no	—
120	C₂²×C₄	—	106	4	no	—
121	C₂²×C₄	C₈×D₅₃	1	3	yes	—
122	C₂²×C₄	#11719912	1	4	yes	—
123	C₂²×C₄	#11719919	1	4	yes	—
124	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
125	C₂×C₈	—	53	4	no	—
126	C₂×C₈	—	106	12	no	—
127	C₂×C₈	#11719919	1	12	yes	—
128	C₂×C₈	C₂×Dic₁₀₆	1	6	yes	—
129	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
130	C₂×C₈	C₂×C₂²⋊D₅₃	1	6	yes	—
131	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
132	C₄²	—	53	12	no	—
133	C₄²	—	53	4	no	—
134	C₁₆	—	2	4	no	—
135	C₁₆	—	106	4	no	—
136	C₂³	#13133146	1	4	yes	—
137	C₂³	C₁₆×D₅₃	1	4	yes	—
138	C₂³	#13133156	1	4	yes	—
139	C₂³	C₂×C₈×D₅₃	1	3	yes	—
140	C₂×C₄	—	53	24	no	—
141	C₂×C₄	—	53	4	no	—
142	C₂×C₄	—	53	4	no	—
143	C₂×C₄	—	106	12	no	—
144	C₂×C₄	#13133146	1	12	yes	—
145	C₂×C₄	C₂×#11719919	1	6	yes	—
146	C₂×C₄	C₂×C₈×D₅₃	1	3	yes	—
147	C₂×C₄	C₂×#11719912	1	6	yes	—
148	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
149	C₈	—	106	4	no	—
150	C₈	C₂×#11719919	1	6	yes	—
151	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
152	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
153	C₂²	#16810441	1	16	yes	direct product factor
154	C₂²	C₂×#13133146	1	6	yes	—
155	C₂²	C₂×C₁₆×D₅₃	1	6	yes	—
156	C₂²	C₂×#13133156	1	6	yes	—
157	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
158	C₄	—	53	8	no	—
159	C₄	—	106	4	no	—
160	C₄	C₂×#13133146	1	6	yes	—
161	C₄	C₂²×#11719919	1	1	yes	characteristic
162	C₄	C₂²×#11719912	1	1	yes	characteristic
163	C₂	C₂×#16810441	1	12	yes	direct product factor
164	C₂	C₂²×#13133146	1	1	yes	characteristic
165	C₂	C₂²×C₁₆×D₅₃	1	1	yes	characteristic
166	C₂	C₂²×#13133156	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	C₂²×C₄⋊C₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#16810441	C₂	direct (N × H)
11. #16810441	C₂²	direct (N × H)
38. C₂²×Dic₅₃	C₁₆	semidirect
62. C₂×Dic₅₃	C₂×C₁₆	semidirect
70. C₂²×C₅₃	C₄⋊C₁₆	semidirect
79. Dic₅₃	C₂²×C₁₆	semidirect
84. C₂×C₅₃	C₂×C₄⋊C₁₆	semidirect
98. C₅₃	C₂²×C₄⋊C₁₆	semidirect
153. C₂²	#16810441	direct (N × H)
163. C₂	C₂×#16810441	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×#11719908	2
C₂²×C₈×Dic₅₃	2
C₂×#16810441	2
C₂³×C₁₆×C₅₃	2
C₂²×C₄⋊C₁₆	53

Subgroups — C22×#16810441 #27537066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810441 #27537066