Subgroups — #27299488

Order 13568.

310 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×#16810453	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810398	C₂	1	1	yes	maximal, characteristic
3	#20274596	C₂	1	2	yes	maximal
4	#20272810	C₂	1	2	yes	maximal
5	C₅₃×#2158	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂×#13133153	C₂²	1	2	yes	—
8	#16810579	—	2	2	no	—
9	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
10	#16810453	C₄	1	4	yes	complemented
11	#16810398	C₄	1	4	yes	complemented
12	#16810382	—	2	2	no	—
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#427	C₂²	1	2	yes	—
15	C₅₃×#367	C₄	1	2	yes	complemented
16	C₅₃×#366	C₄	1	2	yes	complemented
17	C₂×#11719917	—	2	2	no	—
18	C₂²×#88322	D₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
20	C₂×#11719908	—	2	1	no	—
21	C₂×#11719908	—	2	1	no	—
22	C₂×#11719908	C₂×C₄	1	2	yes	—
23	#13133153	—	2	2	no	—
24	#13133153	C₂×C₄	1	4	yes	—
25	#13133128	—	4	2	no	—
26	C₄×#88322	—	4	2	no	—
27	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
28	C₈×Dic₅₃	Q₈	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	#117×C₅₃	—	2	2	no	—
31	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#107×C₅₃	C₂×C₄	1	4	yes	—
36	#106×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	—	2	1	no	—
39	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
44	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₂²⋊C₄×C₅₃	—	4	2	no	—
47	C₄²×C₅₃	—	4	2	no	—
48	C₄×Dic₅₃	—	2	1	no	—
49	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
50	#11719917	—	2	4	no	—
51	#11719917	—	4	2	no	—
52	C₂×#88322	—	4	2	no	—
53	C₂×#88322	C₄⋊C₄	1	2	yes	—
54	C₂×#88322	D₈	1	2	yes	—
55	C₂×#88322	Q₁₆	1	2	yes	—
56	C₁₆×C₅₃	C₄²	1	2	yes	—
57	C₁₆×C₅₃	C₄²	1	2	yes	—
58	#11719908	—	2	2	no	—
59	C₂³×C₅₃	#135	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	4	4	no	—
61	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	—	4	2	no	—
69	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₈×C₅₃	#117	1	1	yes	characteristic
71	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
72	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
73	#88322	—	4	4	no	—
74	#88322	#107	1	4	yes	—
75	#12671	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	—	2	2	no	—
77	C₂²×C₅₃	#427	1	1	yes	characteristic
78	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
79	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
80	C₄×C₅₃	—	8	4	no	—
81	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
83	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
84	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
85	Dic₅₃	—	4	2	no	—
86	C₂×#368	—	53	1	no	—
87	#2158	D₅₃	1	1	yes	characteristic
88	C₂×#346	—	53	1	no	—
89	#1772	—	53	2	no	—
90	#1382	—	53	2	no	—
91	C₂×C₅₃	—	2	1	no	—
92	C₂×C₅₃	—	2	1	no	—
93	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
94	C₂×C₅₃	#2179	1	2	yes	—
95	C₂²×C₁₆	—	53	1	no	—
96	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
97	C₂×#108	—	53	2	no	—
98	#427	C₂×D₅₃	1	2	yes	—
99	#405	—	106	2	no	—
100	C₂×C₄×C₈	—	53	1	no	—
101	#368	—	53	4	no	—
102	#367	Dic₅₃	1	2	yes	complemented
103	#366	Dic₅₃	1	2	yes	complemented
104	#346	—	53	4	no	—
105	#339	—	106	2	no	—
106	C₅₃	#12671	1	1	yes	characteristic, complemented, Sylow
107	C₂×M₁₆	—	106	2	no	—
108	C₂²×C₈	—	53	1	no	—
109	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
110	#117	—	2	2	no	—
111	C₂×C₄²	—	53	1	no	—
112	C₂×C₁₆	—	53	2	no	—
113	C₂×C₁₆	—	106	1	no	—
114	C₂×C₁₆	—	106	1	no	—
115	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
116	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
117	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
118	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
119	#108	—	53	4	no	—
120	#108	—	106	2	no	—
121	#107	C₂×Dic₅₃	1	4	yes	—
122	#106	—	2	2	no	—
123	#97	—	212	2	no	—
124	C₄×C₈	—	53	2	no	—
125	C₄×C₈	—	53	2	no	—
126	C₄×C₈	—	212	2	no	—
127	C₂²×C₄	—	106	1	no	—
128	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
129	M₁₆	—	106	4	no	—
130	M₁₆	—	212	2	no	—
131	C₂×C₈	—	53	2	no	—
132	C₂×C₈	—	53	2	no	—
133	C₂×C₈	—	53	2	no	—
134	C₂×C₈	—	212	2	no	—
135	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
137	C₂×C₈	#11719947	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
139	C₂×C₈	#11719949	1	1	yes	characteristic
140	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
141	C₄⋊C₄	—	2	4	no	—
142	C₂²⋊C₄	—	4	2	no	—
143	C₄²	—	4	2	no	—
144	C₄²	—	53	2	no	—
145	C₄²	—	106	1	no	—
146	C₁₆	—	106	2	no	—
147	C₁₆	C₄×Dic₅₃	1	2	yes	—
148	C₁₆	C₄×Dic₅₃	1	2	yes	—
149	C₂³	#13133236	1	1	yes	characteristic
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	—	106	2	no	—
152	C₂×C₄	—	212	2	no	—
153	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
154	C₂×C₄	#13133179	1	1	yes	characteristic
155	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
156	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
157	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
158	C₂×C₄	#13133240	1	1	yes	characteristic
159	C₈	—	53	4	no	—
160	C₈	—	212	4	no	—
161	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
162	C₈	#13133175	1	1	yes	characteristic
163	C₈	#13133182	1	1	yes	characteristic
164	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
165	C₂²	—	2	2	no	—
166	C₂²	#16810660	1	1	yes	characteristic
167	C₂²	D₈×Dic₅₃	1	1	yes	characteristic
168	C₂²	#16810717	1	1	yes	characteristic
169	C₄	—	8	4	no	—
170	C₄	—	212	2	no	—
171	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
172	C₄	#107×D₅₃	1	1	yes	characteristic
173	C₄	#16810705	1	1	yes	characteristic
174	C₄	Q₁₆×Dic₅₃	1	1	yes	characteristic
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	#107×Dic₅₃	1	1	yes	characteristic
178	C₂	#20276601	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2158	C₂
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂³×C₅₃	#135
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2158	D₅₃
hypocenter	C₅₃	#12671
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
10. #16810453	C₄	semidirect
11. #16810398	C₄	semidirect
15. C₅₃×#367	C₄	semidirect
16. C₅₃×#366	C₄	semidirect
102. #367	Dic₅₃	semidirect
103. #366	Dic₅₃	semidirect
106. C₅₃	#12671	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810453	2
C₂×#16810398	2
#20274596	2
#20272810	2
C₅₃×#2158	2
#12671	53