Subgroups — #27271027

Order 13568.

274 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810399	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133151	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	—	2	1	no	—
5	C₂×#13133151	C₄	1	2	yes	—
6	C₄×C₄⋊Dic₅₃	—	2	1	no	—
7	#16810399	—	2	1	no	—
8	#16810399	C₄	1	2	yes	complemented
9	#16810386	—	2	1	no	—
10	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#334	—	2	1	no	—
13	C₅₃×#334	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₅₃	—	2	2	no	—
15	C₂×C₄×Dic₅₃	—	4	1	no	—
16	C₂×#11719908	—	2	4	no	—
17	C₂×#11719908	—	2	2	no	—
18	#13133151	—	2	2	no	—
19	#13133151	C₈	1	4	yes	complemented
20	#13133124	—	4	1	no	—
21	C₂²×C₈×C₅₃	—	4	1	no	—
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	C₄⋊C₈×C₅₃	—	2	2	no	—
26	C₄⋊C₈×C₅₃	—	4	1	no	—
27	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
34	C₂²×Dic₅₃	—	4	2	no	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	—	4	2	no	—
37	C₂×C₈×C₅₃	—	8	2	no	—
38	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
44	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
47	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
48	C₄⋊Dic₅₃	—	2	4	no	—
49	C₄⋊Dic₅₃	—	4	2	no	—
50	C₄×Dic₅₃	—	8	2	no	—
51	#11719908	—	2	4	no	—
52	C₂³×C₅₃	#102	1	1	yes	characteristic
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
58	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#104	1	2	yes	—
62	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
64	C₂×Dic₅₃	—	4	4	no	—
65	C₂×Dic₅₃	—	8	2	no	—
66	C₂×Dic₅₃	—	8	2	no	—
67	C₈×C₅₃	—	2	1	no	—
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	8	2	no	—
70	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
71	C₈×C₅₃	#97	1	1	yes	characteristic
72	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
73	#7178	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	D₄⋊C₈	1	2	yes	—
75	C₂²×C₅₃	#340	1	1	yes	characteristic
76	C₂²×C₅₃	#360	1	2	yes	—
77	C₂²×C₅₃	#361	1	1	yes	characteristic
78	C₂²×C₅₃	#362	1	1	yes	characteristic
79	C₄×C₅₃	—	2	1	no	—
80	C₄×C₅₃	—	2	1	no	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	—	2	1	no	—
83	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
84	C₄×C₅₃	#336	1	1	yes	characteristic
85	C₄×C₅₃	#339	1	1	yes	characteristic
86	Dic₅₃	—	8	4	no	—
87	C₂×#346	—	53	1	no	—
88	C₄×#107	—	53	1	no	—
89	C₂×#334	D₅₃	1	1	yes	characteristic
90	C₂×C₅₃	#1279	1	1	yes	characteristic
91	C₂×C₅₃	#1333	1	2	yes	—
92	C₂×C₅₃	#1334	1	2	yes	—
93	C₂×C₅₃	#1386	1	1	yes	characteristic
94	C₂×C₅₃	#1387	1	1	yes	characteristic
95	C₂²×C₁₆	—	106	1	no	—
96	C₂×#107	—	53	2	no	—
97	C₂×C₄⋊C₈	—	2	1	no	—
98	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
99	C₄×C₄⋊C₄	—	106	1	no	—
100	#346	—	53	2	no	—
101	#346	—	106	1	no	—
102	#340	—	106	1	no	—
103	#334	—	2	1	no	—
104	#334	Dic₅₃	1	2	yes	complemented
105	C₅₃	#7178	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	4	1	no	—
107	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
108	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	106	2	no	—
110	C₂×C₄²	—	212	1	no	—
111	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
112	C₂×C₁₆	—	106	4	no	—
113	C₂×C₁₆	—	106	2	no	—
114	#107	—	53	4	no	—
115	#107	—	106	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	4	1	no	—
118	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
119	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
122	#95	—	212	1	no	—
123	C₂²×C₄	—	212	2	no	—
124	C₂²×C₄	#11719921	1	1	yes	characteristic
125	C₂²×C₄	#11719922	1	1	yes	characteristic
126	C₂²×C₄	#11719924	1	1	yes	characteristic
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	8	2	no	—
130	C₂×C₈	C₈×D₅₃	1	2	yes	—
131	C₂×C₈	#11719912	1	2	yes	—
132	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
133	C₂×C₈	#11719919	1	1	yes	characteristic
134	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
135	C₂×C₈	#11719921	1	1	yes	characteristic
136	C₄⋊C₄	—	106	4	no	—
137	C₄⋊C₄	—	212	2	no	—
138	C₄²	—	424	2	no	—
139	C₄²	#11719919	1	1	yes	characteristic
140	C₄²	#11719923	1	1	yes	characteristic
141	C₄²	#11719925	1	1	yes	characteristic
142	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
143	C₁₆	—	106	4	no	—
144	C₂³	#13133137	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	424	2	no	—
149	C₂×C₄	—	424	2	no	—
150	C₂×C₄	—	212	4	no	—
151	C₂×C₄	#13133124	1	1	yes	characteristic
152	C₂×C₄	#13133129	1	2	yes	—
153	C₂×C₄	#13133138	1	1	yes	characteristic
154	C₂×C₄	#13133140	1	1	yes	characteristic
155	C₂×C₄	#13133141	1	1	yes	characteristic
156	C₂×C₄	#13133143	1	2	yes	—
157	C₂×C₄	#13133148	1	1	yes	characteristic
158	C₂×C₄	#13133150	1	1	yes	characteristic
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	8	2	no	—
162	C₈	C₈×Dic₅₃	1	1	yes	characteristic
163	C₈	#13133127	1	1	yes	characteristic
164	C₈	#13133146	1	2	yes	—
165	C₂²	#16810350	1	2	yes	—
166	C₂²	#16810385	1	1	yes	characteristic
167	C₂²	#16810432	1	2	yes	—
168	C₂²	#16810435	1	1	yes	characteristic
169	C₂²	#16810438	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	424	4	no	—
175	C₄	#16810353	1	2	yes	—
176	C₄	#16810377	1	1	yes	characteristic
177	C₄	#16810384	1	1	yes	characteristic
178	C₂	#20272554	1	1	yes	characteristic
179	C₂	#20272681	1	2	yes	—
180	C₂	#20272684	1	2	yes	—
181	C₂	#20272822	1	1	yes	characteristic
182	C₂	#20272825	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133137
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle, prime radical	C₂³×C₅₃	#102
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	#7178
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
8. #16810399	C₄	semidirect
13. C₅₃×#334	C₄	semidirect
19. #13133151	C₈	semidirect
104. #334	Dic₅₃	semidirect
105. C₅₃	#7178	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810399	2
C₄×#13133151	2
C₂×C₅₃×#334	2
#7178	53