Subgroups — #27300503

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810441	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
9	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
10	C₄⋊#11719908	—	2	2	no	—
11	#16810441	—	2	8	no	—
12	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	—	2	2	no	—
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₅₃	C₈	1	2	yes	—
18	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×#11719908	—	2	4	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₄×#88322	—	2	2	no	—
23	C₈×Dic₅₃	—	2	4	no	—
24	C₈×Dic₅₃	D₄	1	4	yes	—
25	C₈×Dic₅₃	Q₈	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	—	2	2	no	—
30	C₂×C₁₆×C₅₃	—	2	4	no	—
31	C₄×C₈×C₅₃	D₄	1	2	yes	—
32	C₄×C₈×C₅₃	Q₈	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
37	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
41	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
44	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₄²×C₅₃	M₁₆	1	2	yes	—
46	C₄×Dic₅₃	—	2	2	no	—
47	C₄×Dic₅₃	—	2	4	no	—
48	C₄×Dic₅₃	C₁₆	1	4	yes	complemented
49	C₄×Dic₅₃	C₄⋊C₄	1	4	yes	—
50	C₄×Dic₅₃	M₁₆	1	4	yes	—
51	C₂×#88322	—	2	2	no	—
52	C₂×#88322	—	2	2	no	—
53	C₂×#88322	C₄⋊C₄	1	4	yes	—
54	C₁₆×C₅₃	—	4	2	no	—
55	#11719908	—	4	2	no	—
56	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
61	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
66	C₂×Dic₅₃	—	2	2	no	—
67	C₂×Dic₅₃	—	2	2	no	—
68	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
69	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
70	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
71	C₂×Dic₅₃	M₃₂	1	4	yes	—
72	C₈×C₅₃	—	2	2	no	—
73	C₈×C₅₃	#127	1	1	yes	characteristic
74	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
76	#88322	—	2	4	no	—
77	#12898	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
79	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
80	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
81	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
82	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	C₄⋊C₁₆	1	4	yes	complemented
85	C₄×C₅₃	#384	1	1	yes	characteristic
86	C₄×C₅₃	#423	1	2	yes	—
87	C₄×C₅₃	#423	1	1	yes	characteristic
88	Dic₅₃	—	2	4	no	—
89	Dic₅₃	C₄⋊C₁₆	1	8	yes	complemented
90	C₂×C₄⋊C₁₆	—	53	4	no	—
91	C₂×C₄⋊C₁₆	—	53	1	no	—
92	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
93	C₄²×C₈	—	53	1	no	—
94	C₂×C₅₃	#1842	1	1	yes	characteristic
95	C₂×C₅₃	C₂×C₄⋊C₁₆	1	2	yes	—
96	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
97	C₂×C₅₃	#2150	1	2	yes	—
98	C₂×C₅₃	#2150	1	1	yes	characteristic
99	C₂²×C₁₆	—	53	2	no	—
100	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
101	C₂×C₄×C₈	—	53	2	no	—
102	C₂×C₄×C₈	—	53	2	no	—
103	C₂×C₄×C₈	—	53	1	no	—
104	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
105	C₄³	—	53	1	no	—
106	C₄⋊C₁₆	—	2	2	no	—
107	C₄⋊C₁₆	—	106	8	no	—
108	C₄⋊C₁₆	—	106	2	no	—
109	C₅₃	#12898	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₈	—	53	2	no	—
111	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
112	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
113	C₂×C₄²	—	53	2	no	—
114	C₂×C₄²	—	53	1	no	—
115	C₂×C₄²	—	53	2	no	—
116	C₂×C₄²	—	53	1	no	—
117	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
118	C₂×C₁₆	—	2	4	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	106	4	no	—
121	C₂×C₁₆	—	106	2	no	—
122	C₄×C₈	—	53	4	no	—
123	C₄×C₈	—	53	4	no	—
124	C₄×C₈	—	106	4	no	—
125	C₄×C₈	—	106	2	no	—
126	C₄×C₈	Dic₁₀₆	1	2	yes	—
127	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
128	C₂²×C₄	—	53	2	no	—
129	C₂²×C₄	—	53	2	no	—
130	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
131	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	53	4	no	—
135	C₂×C₈	—	106	2	no	—
136	C₂×C₈	—	106	2	no	—
137	C₂×C₈	#11719919	1	2	yes	—
138	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
139	C₂×C₈	D₄×D₅₃	1	2	yes	—
140	C₂×C₈	Q₈×D₅₃	1	2	yes	—
141	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
142	C₄²	—	53	4	no	—
143	C₄²	—	53	4	no	—
144	C₄²	—	53	4	no	—
145	C₄²	—	106	4	no	—
146	C₄²	—	106	2	no	—
147	C₄²	#11719912	1	2	yes	—
148	C₄²	#11719919	1	2	yes	—
149	C₁₆	—	4	2	no	—
150	C₁₆	—	212	2	no	—
151	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	53	4	no	—
154	C₂×C₄	—	53	4	no	—
155	C₂×C₄	—	53	4	no	—
156	C₂×C₄	—	53	4	no	—
157	C₂×C₄	—	106	2	no	—
158	C₂×C₄	—	106	2	no	—
159	C₂×C₄	#13133146	1	2	yes	—
160	C₂×C₄	#13133146	1	2	yes	—
161	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
162	C₂×C₄	#13133156	1	2	yes	—
163	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
164	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
165	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
166	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	106	4	no	—
169	C₈	#13133213	1	1	yes	characteristic
170	C₈	#13133215	1	1	yes	characteristic
171	C₈	#13133216	1	2	yes	—
172	C₂²	C₄⋊C₈×D₅₃	1	2	yes	—
173	C₂²	C₂×#13133146	1	1	yes	characteristic
174	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
175	C₂²	M₃₂×D₅₃	1	2	yes	—
176	C₂²	C₂×#13133156	1	1	yes	characteristic
177	C₄	—	2	2	no	—
178	C₄	—	53	8	no	—
179	C₄	—	106	4	no	—
180	C₄	#16810441	1	4	yes	complemented
181	C₄	#16810492	1	1	yes	characteristic
182	C₄	#16810645	1	2	yes	—
183	C₄	#16810646	1	1	yes	characteristic
184	C₂	#20274866	1	1	yes	characteristic
185	C₂	C₄⋊C₁₆×D₅₃	1	2	yes	—
186	C₂	C₂×#16810441	1	1	yes	characteristic
187	C₂	#20276469	1	2	yes	—
188	C₂	#20276470	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#12898
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
48. C₄×Dic₅₃	C₁₆	semidirect
84. C₄×C₅₃	C₄⋊C₁₆	semidirect
89. Dic₅₃	C₄⋊C₁₆	semidirect
109. C₅₃	#12898	semidirect
180. C₄	#16810441	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₂×#16810441	2
C₄×C₈×Dic₅₃	2
C₂×C₅₃×C₄⋊C₁₆	2
#12898	53