Subgroups — #28247794

Order 13568.

2564 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811390	C₂	1	2	yes	maximal
2	#20283326	C₂	1	16	yes	maximal, complemented
3	C₂×#16810672	C₂	1	12	yes	maximal, complemented
4	C₂×C₅₃×#568	C₂	1	1	yes	maximal, characteristic
5	#16811390	—	2	8	no	—
6	#16811390	C₂²	1	16	yes	—
7	C₂×#13133224	—	2	8	no	—
8	C₂²×#11719917	C₂²	1	12	yes	—
9	#16810672	C₂²	1	16	yes	—
10	#16810672	C₂²	1	48	yes	complemented
11	C₂×#13133153	C₂²	1	48	yes	complemented
12	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#568	C₄	1	16	yes	—
14	C₂×#131×C₅₃	C₂²	1	8	yes	—
15	C₂²×M₁₆×C₅₃	C₂²	1	6	yes	—
16	#13133224	—	2	64	no	—
17	C₂×#11719917	—	2	12	no	—
18	C₂×#11719917	—	2	24	no	—
19	C₂×#11719917	C₂³	1	8	yes	—
20	C₂×#11719917	C₂³	1	48	yes	—
21	C₂²×#88322	—	2	12	no	—
22	#13133153	C₂³	1	64	yes	—
23	C₂×#41×C₅₃	C₂×C₄	1	8	yes	—
24	C₂×#41×C₅₃	D₄	1	8	yes	—
25	C₂×#41×C₅₃	Q₈	1	4	yes	—
26	C₂×#41×C₅₃	C₂³	1	4	yes	—
27	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
28	C₂²×D₄×C₅₃	Q₈	1	3	yes	—
29	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
30	#131×C₅₃	C₂×C₄	1	64	yes	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	48	yes	—
32	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
33	C₂×M₁₆×C₅₃	C₂³	1	12	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	12	yes	—
35	C₂⁴×C₅₃	C₂×Q₈	1	3	yes	—
36	#41×C₅₃	C₄⋊C₄	1	32	yes	—
37	#41×C₅₃	C₂²×C₄	1	16	yes	—
38	#41×C₅₃	C₂×D₄	1	16	yes	—
39	C₂×Q₈×C₅₃	C₄⋊C₄	1	8	yes	—
40	C₂×Q₈×C₅₃	C₂×Q₈	1	4	yes	—
41	C₂×D₄×C₅₃	C₄⋊C₄	1	24	yes	—
42	C₂×D₄×C₅₃	C₂×Q₈	1	12	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
45	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
46	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
47	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
48	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
49	M₁₆×C₅₃	C₂²×C₄	1	16	yes	—
50	M₁₆×C₅₃	C₂²×C₄	1	48	yes	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	48	yes	—
52	C₂×C₈×C₅₃	C₂⁴	1	8	yes	—
53	#11719917	—	2	8	no	—
54	#11719917	—	2	48	no	—
55	#11719917	C₂⁴	1	16	yes	—
56	C₂×#88322	—	2	8	no	—
57	C₂×#88322	—	2	48	no	—
58	C₂³×C₅₃	—	8	3	no	—
59	C₂³×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
60	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
61	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
62	Q₈×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
63	D₄×C₅₃	C₂×C₄⋊C₄	1	48	yes	—
64	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
65	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	24	yes	—
66	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
67	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
68	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
69	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
70	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
72	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
73	C₈×C₅₃	C₂³×C₄	1	16	yes	—
74	#88322	—	2	16	no	—
75	#60113	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	—	4	1	no	—
77	C₂²×C₅₃	—	4	12	no	—
78	C₂²×C₅₃	—	4	6	no	—
79	C₂²×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
80	C₂²×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
81	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
82	C₄×C₅₃	C₂²×C₄⋊C₄	1	2	yes	—
83	C₄×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
84	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
85	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
86	C₂×#568	—	53	2	no	—
87	C₂×#568	D₅₃	1	1	yes	characteristic
88	#2916	—	53	16	no	—
89	C₂×#430	—	53	12	no	—
90	C₂×C₅₃	—	2	2	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	—	2	12	no	—
93	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
95	#568	—	53	16	no	—
96	#568	—	106	8	no	—
97	#568	Dic₅₃	1	16	yes	—
98	C₂×#131	—	106	8	no	—
99	C₂×#131	C₂×D₅₃	1	8	yes	—
100	C₂²×M₁₆	—	53	12	no	—
101	C₂²×M₁₆	C₂×D₅₃	1	6	yes	—
102	#430	—	53	16	no	—
103	#430	—	53	48	no	—
104	C₂×#108	—	53	48	no	—
105	C₅₃	#60113	1	1	yes	characteristic, complemented, Sylow
106	C₂×#41	Dic₁₀₆	1	4	yes	—
107	C₂×#41	D₂₁₂	1	8	yes	—
108	C₂×#41	C₂×Dic₅₃	1	8	yes	—
109	C₂×#41	C₂²×D₅₃	1	4	yes	—
110	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
111	C₂²×D₄	Dic₁₀₆	1	3	yes	—
112	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
113	#131	—	106	64	no	—
114	#131	C₂×Dic₅₃	1	64	yes	—
115	C₂×M₁₆	—	53	8	no	—
116	C₂×M₁₆	—	53	48	no	—
117	C₂×M₁₆	—	106	12	no	—
118	C₂×M₁₆	—	106	24	no	—
119	C₂×M₁₆	C₂×Dic₅₃	1	48	yes	—
120	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
121	C₂×M₁₆	C₂²×D₅₃	1	12	yes	—
122	C₂²×C₈	—	106	12	no	—
123	C₂²×C₈	C₂²×D₅₃	1	12	yes	—
124	#108	—	53	64	no	—
125	C₂⁴	C₂×Dic₁₀₆	1	3	yes	—
126	#41	C₄⋊Dic₅₃	1	32	yes	—
127	#41	C₂×D₂₁₂	1	16	yes	—
128	#41	C₂²×Dic₅₃	1	16	yes	—
129	C₂×Q₈	C₄⋊Dic₅₃	1	8	yes	—
130	C₂×Q₈	C₂×Dic₁₀₆	1	4	yes	—
131	C₂×D₄	C₄⋊Dic₅₃	1	24	yes	—
132	C₂×D₄	C₂×Dic₁₀₆	1	12	yes	—
133	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
134	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
135	C₂²×C₄	C₂×D₂₁₂	1	12	yes	—
136	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
137	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
138	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
139	M₁₆	—	53	16	no	—
140	M₁₆	—	106	8	no	—
141	M₁₆	—	106	48	no	—
142	M₁₆	C₂²×Dic₅₃	1	16	yes	—
143	M₁₆	C₂²×Dic₅₃	1	48	yes	—
144	C₂×C₈	—	106	8	no	—
145	C₂×C₈	—	106	48	no	—
146	C₂×C₈	C₂²×Dic₅₃	1	48	yes	—
147	C₂×C₈	C₂³×D₅₃	1	8	yes	—
148	C₂³	—	8	3	no	—
149	C₂³	C₂×C₄⋊Dic₅₃	1	12	yes	—
150	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
151	C₂³	C₂²×Dic₁₀₆	1	6	yes	—
152	Q₈	C₂×C₄⋊Dic₅₃	1	16	yes	—
153	D₄	C₂×C₄⋊Dic₅₃	1	48	yes	—
154	C₂×C₄	C₂×C₄⋊Dic₅₃	1	12	yes	—
155	C₂×C₄	C₂×C₄⋊Dic₅₃	1	24	yes	—
156	C₂×C₄	C₂³×Dic₅₃	1	12	yes	—
157	C₂×C₄	C₂³×Dic₅₃	1	2	yes	—
158	C₂×C₄	C₂²×D₂₁₂	1	2	yes	—
159	C₂×C₄	C₂²×D₂₁₂	1	12	yes	—
160	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
161	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
162	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
163	C₈	—	106	16	no	—
164	C₈	C₂³×Dic₅₃	1	16	yes	—
165	C₂²	—	4	1	no	—
166	C₂²	—	4	12	no	—
167	C₂²	—	4	6	no	—
168	C₂²	C₂²×C₄⋊Dic₅₃	1	2	yes	—
169	C₂²	C₂²×C₄⋊Dic₅₃	1	12	yes	—
170	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
171	C₄	C₂²×C₄⋊Dic₅₃	1	2	yes	—
172	C₄	C₂²×C₄⋊Dic₅₃	1	12	yes	—
173	C₄	C₂⁴×Dic₅₃	1	1	yes	characteristic
174	C₄	C₂³×D₂₁₂	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	2	12	no	—
178	C₂	C₂³×C₄⋊Dic₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂³×D₂₁₂
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#568	C₂
socle	C₂×C₅₃	C₂³×C₄⋊C₄
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂×#568	D₅₃
hypocenter	C₅₃	#60113
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
2. #20283326	C₂	semidirect
3. C₂×#16810672	C₂	semidirect
10. #16810672	C₂²	semidirect
11. C₂×#13133153	C₂²	semidirect
105. C₅₃	#60113	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811390	2
#20283326	2
C₂×#16810672	2
C₂×C₅₃×#568	2
#60113	53