Subgroups — #27302710

Order 13568.

342 conjugacy classes of subgroups.

The subgroup lattice has 196 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	#20274887	C₂	1	1	yes	maximal, characteristic
3	#20272810	C₂	1	4	yes	maximal
4	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133153	—	2	1	no	—
7	C₂×#13133153	C₂²	1	2	yes	—
8	#16810647	—	2	1	no	—
9	#16810399	—	2	1	no	—
10	#16810399	—	2	1	no	—
11	C₅₃×#533	—	2	1	no	—
12	C₅₃×#500	C₄	1	2	yes	—
13	C₅₃×#499	—	2	1	no	—
14	C₅₃×#499	C₄	1	2	yes	—
15	C₅₃×#498	—	2	1	no	—
16	C₅₃×#427	—	2	1	no	—
17	C₅₃×#427	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#11719917	—	2	2	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₂×#11719908	Q₈	1	4	yes	complemented
23	#13133153	—	2	4	no	—
24	#13133153	—	4	2	no	—
25	C₄⋊#88322	—	4	2	no	—
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊Q₈×C₅₃	—	2	2	no	—
29	#125×C₅₃	—	2	2	no	—
30	C₂²⋊Q₈×C₅₃	—	4	2	no	—
31	C₄×Q₈×C₅₃	—	4	2	no	—
32	#117×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
34	#107×C₅₃	—	2	2	no	—
35	#107×C₅₃	C₂×C₄	1	4	yes	—
36	#106×C₅₃	—	2	2	no	—
37	#106×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₅₃	—	4	2	no	—
43	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
48	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₅₃	#41	1	2	yes	—
53	C₄⋊C₄×C₅₃	—	2	4	no	—
54	C₄⋊C₄×C₅₃	—	4	4	no	—
55	C₄⋊C₄×C₅₃	—	4	2	no	—
56	C₂²⋊C₄×C₅₃	—	4	2	no	—
57	C₄²×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₃	D₈	1	2	yes	—
61	#11719917	—	4	2	no	—
62	#11719917	—	4	2	no	—
63	C₂×#88322	—	4	2	no	—
64	#11719908	—	2	4	no	—
65	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	Q₈×C₅₃	—	8	4	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	—	4	4	no	—
69	C₂×C₄×C₅₃	#102	1	2	yes	—
70	C₂×C₄×C₅₃	#102	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₈×C₅₃	#103	1	2	yes	—
78	C₈×C₅₃	#103	1	2	yes	—
79	C₈×C₅₃	C₄×Q₈	1	2	yes	—
80	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
81	#88322	—	8	2	no	—
82	#13286	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
85	C₂²×C₅₃	#487	1	1	yes	characteristic
86	C₂²×C₅₃	#498	1	1	yes	characteristic
87	C₄×C₅₃	—	2	2	no	—
88	C₄×C₅₃	—	2	2	no	—
89	C₄×C₅₃	—	8	4	no	—
90	C₄×C₅₃	#391	1	1	yes	characteristic
91	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
92	C₄×C₅₃	#486	1	1	yes	characteristic
93	C₄×C₅₃	#500	1	1	yes	characteristic
94	#3160	D₅₃	1	1	yes	characteristic
95	C₂×#346	—	53	1	no	—
96	#1848	—	53	1	no	—
97	#1382	—	53	4	no	—
98	C₂×C₅₃	—	2	1	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	#1990	1	1	yes	characteristic
101	C₂×C₅₃	#2143	1	2	yes	—
102	#533	—	2	1	no	—
103	C₂²×C₁₆	—	53	2	no	—
104	#500	Dic₅₃	1	2	yes	—
105	#499	—	2	1	no	—
106	#499	Dic₅₃	1	2	yes	—
107	#498	—	2	1	no	—
108	C₂×#108	—	53	2	no	—
109	C₂×#108	—	106	1	no	—
110	#427	—	2	1	no	—
111	#427	C₂×D₅₃	1	2	yes	—
112	#423	—	106	1	no	—
113	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
114	#346	—	106	1	no	—
115	#346	—	106	1	no	—
116	C₅₃	#13286	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	106	2	no	—
118	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
120	C₄⋊Q₈	—	2	2	no	—
121	#125	—	2	2	no	—
122	C₂²⋊Q₈	—	4	2	no	—
123	C₄×Q₈	—	4	2	no	—
124	#117	—	2	2	no	—
125	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₂×C₁₆	—	53	4	no	—
127	C₂×C₁₆	—	106	2	no	—
128	C₂×C₁₆	—	106	2	no	—
129	#108	—	106	4	no	—
130	#108	—	212	2	no	—
131	#107	—	2	2	no	—
132	#107	C₂×Dic₅₃	1	4	yes	—
133	#106	—	2	2	no	—
134	#106	—	2	2	no	—
135	C₄⋊C₈	—	212	2	no	—
136	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
137	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂×Q₈	—	4	2	no	—
141	C₂²×C₄	#11719923	1	2	yes	—
142	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
143	M₁₆	—	212	2	no	—
144	M₁₆	—	212	2	no	—
145	C₂×C₈	—	212	2	no	—
146	C₂×C₈	#11719924	1	2	yes	—
147	C₂×C₈	#11719925	1	2	yes	—
148	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
149	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₅₃	1	2	yes	—
151	C₂×C₈	#11719949	1	2	yes	—
152	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
153	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₂²⋊C₄	—	4	2	no	—
158	C₄²	—	4	2	no	—
159	C₄²	#11719922	1	2	yes	—
160	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
161	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
162	C₁₆	—	106	4	no	—
163	C₂³	C₂×#11719923	1	1	yes	characteristic
164	Q₈	—	8	4	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	#13133138	1	2	yes	—
168	C₂×C₄	#13133138	1	2	yes	—
169	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
170	C₂×C₄	#13133206	1	1	yes	characteristic
171	C₂×C₄	#13133215	1	1	yes	characteristic
172	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
173	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
174	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
175	C₈	—	424	2	no	—
176	C₈	#13133141	1	2	yes	—
177	C₈	#13133141	1	2	yes	—
178	C₈	Q₈×Dic₅₃	1	2	yes	—
179	C₈	#13133193	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#13133138	1	1	yes	characteristic
182	C₂²	#16810975	1	1	yes	characteristic
183	C₂²	#16811019	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	2	2	no	—
186	C₄	—	8	4	no	—
187	C₄	#16810530	1	1	yes	characteristic
188	C₄	C₂×#13133141	1	1	yes	characteristic
189	C₄	#16810971	1	1	yes	characteristic
190	C₄	#16811026	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#20275677	1	1	yes	characteristic
194	C₂	#20276443	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#3160	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#3160	D₅₃
hypocenter	C₅₃	#13286
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
22. C₂×#11719908	Q₈	semidirect
116. C₅₃	#13286	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
#20274887	2
#20272810	2
C₅₃×#3160	2
#13286	53