Subgroups — #27302347

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811003	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810438	C₂	1	4	yes	maximal
3	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₂	—	2	1	no	—
6	C₂²×Dic₂₁₂	C₂²	1	2	yes	—
7	C₂×#13133217	—	2	1	no	—
8	C₂×#13133157	C₂²	1	2	yes	—
9	#16811003	—	2	1	no	—
10	#16811003	—	2	1	no	—
11	#16811003	C₄	1	2	yes	—
12	#16811003	C₄	1	2	yes	—
13	#16810438	—	2	8	no	—
14	#16810399	—	2	1	no	—
15	C₄×#11719908	—	2	1	no	—
16	C₂×#108×C₅₃	—	2	1	no	—
17	C₂×#108×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#423	—	2	1	no	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₀₆	—	2	2	no	—
21	C₂×Dic₂₁₂	—	2	2	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	—	2	2	no	—
24	C₂×Dic₂₁₂	—	2	4	no	—
25	C₂×Dic₂₁₂	—	2	4	no	—
26	C₂×Dic₂₁₂	—	4	2	no	—
27	C₂×Dic₂₁₂	—	4	2	no	—
28	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
29	#13133217	—	2	2	no	—
30	#13133217	—	2	2	no	—
31	#13133217	—	4	2	no	—
32	C₂×C₄⋊Dic₅₃	—	4	2	no	—
33	#13133157	—	2	2	no	—
34	#13133157	D₄	1	4	yes	—
35	C₂×#11719908	—	2	2	no	—
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
40	#108×C₅₃	—	2	4	no	—
41	#108×C₅₃	—	4	2	no	—
42	C₄⋊C₈×C₅₃	—	4	2	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	4	2	no	—
50	C₂×Dic₁₀₆	—	2	4	no	—
51	C₂×Dic₁₀₆	—	4	2	no	—
52	C₂×Dic₁₀₆	—	4	2	no	—
53	C₂×Dic₁₀₆	—	4	2	no	—
54	C₂×Dic₁₀₆	—	4	2	no	—
55	C₂×Dic₁₀₆	—	4	2	no	—
56	M₁₆×C₅₃	—	4	2	no	—
57	M₁₆×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	D₈	1	2	yes	—
62	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	#41	1	2	yes	—
67	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₅₃	D₈	1	2	yes	—
70	C₄⋊Dic₅₃	—	4	4	no	—
71	C₄⋊Dic₅₃	—	4	4	no	—
72	Dic₂₁₂	—	2	8	no	—
73	Dic₂₁₂	—	4	4	no	—
74	Dic₂₁₂	—	4	4	no	—
75	Dic₂₁₂	—	4	4	no	—
76	#11719908	—	2	4	no	—
77	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	#102	1	2	yes	—
80	C₂×C₄×C₅₃	#102	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×Dic₅₃	—	4	4	no	—
88	C₂×Dic₅₃	—	8	2	no	—
89	C₂×Dic₅₃	—	8	2	no	—
90	Dic₁₀₆	—	4	4	no	—
91	Dic₁₀₆	—	4	4	no	—
92	Dic₁₀₆	—	8	2	no	—
93	Dic₁₀₆	—	8	2	no	—
94	C₈×C₅₃	—	8	2	no	—
95	C₈×C₅₃	#102	1	2	yes	—
96	C₈×C₅₃	#102	1	2	yes	—
97	C₈×C₅₃	C₄×D₄	1	2	yes	—
98	C₈×C₅₃	#121	1	2	yes	—
99	#13212	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	—	2	2	no	—
101	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
102	C₂²×C₅₃	#486	1	1	yes	characteristic
103	C₂²×C₅₃	#492	1	1	yes	characteristic
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	#362	1	4	yes	complemented
106	C₄×C₅₃	#390	1	1	yes	characteristic
107	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
108	C₄×C₅₃	#486	1	1	yes	characteristic
109	C₄×C₅₃	#493	1	1	yes	characteristic
110	Dic₅₃	—	8	4	no	—
111	C₂×#494	—	53	1	no	—
112	C₂×#362	—	53	4	no	—
113	C₄×M₃₂	—	53	1	no	—
114	#1848	D₅₃	1	1	yes	characteristic
115	C₂×C₅₃	—	2	1	no	—
116	C₂×C₅₃	—	2	1	no	—
117	C₂×C₅₃	#1957	1	1	yes	characteristic
118	C₂×C₅₃	C₂×#362	1	2	yes	—
119	C₂²×Q₁₆	—	53	2	no	—
120	C₂²×Q₁₆	—	106	1	no	—
121	C₂×C₄⋊Q₈	—	106	1	no	—
122	C₂×M₃₂	—	53	2	no	—
123	#494	—	53	2	no	—
124	#494	—	53	2	no	—
125	#494	—	106	1	no	—
126	#494	—	106	1	no	—
127	C₂×#108	—	2	1	no	—
128	C₂×#108	C₂×D₅₃	1	2	yes	—
129	#423	—	2	1	no	—
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	#362	—	106	8	no	—
132	#346	—	106	1	no	—
133	C₄×C₁₆	—	106	1	no	—
134	C₅₃	#13212	1	1	yes	characteristic, complemented, Sylow
135	C₂²×Q₈	—	106	2	no	—
136	C₂×Q₁₆	—	53	4	no	—
137	C₂×Q₁₆	—	106	2	no	—
138	C₂×Q₁₆	—	106	4	no	—
139	C₂×Q₁₆	—	106	4	no	—
140	C₂×Q₁₆	—	106	2	no	—
141	C₂×Q₁₆	—	106	2	no	—
142	C₂×Q₁₆	—	212	2	no	—
143	C₂×Q₁₆	—	212	2	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
147	C₄⋊Q₈	—	106	2	no	—
148	C₄⋊Q₈	—	106	2	no	—
149	C₄⋊Q₈	—	212	2	no	—
150	C₂×C₄⋊C₄	—	212	2	no	—
151	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
152	M₃₂	—	53	4	no	—
153	M₃₂	—	106	2	no	—
154	C₂×C₁₆	—	106	2	no	—
155	#108	—	2	4	no	—
156	#108	—	4	2	no	—
157	C₄⋊C₈	—	4	2	no	—
158	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
159	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
160	C₄×C₈	D₂₁₂	1	1	yes	characteristic
161	C₄×C₈	D₂₁₂	1	1	yes	characteristic
162	C₂×Q₈	—	106	4	no	—
163	C₂×Q₈	—	212	2	no	—
164	C₂×Q₈	—	212	2	no	—
165	C₂×Q₈	—	212	2	no	—
166	C₂×Q₈	—	212	2	no	—
167	C₂×Q₈	—	212	2	no	—
168	C₂²×C₄	—	212	2	no	—
169	C₂²×C₄	#11719924	1	2	yes	—
170	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
171	Q₁₆	—	106	8	no	—
172	Q₁₆	—	212	4	no	—
173	Q₁₆	—	212	4	no	—
174	Q₁₆	—	212	4	no	—
175	M₁₆	—	4	2	no	—
176	M₁₆	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	#11719921	1	1	yes	characteristic
179	C₂×C₈	#11719921	1	1	yes	characteristic
180	C₂×C₈	#11719922	1	2	yes	—
181	C₂×C₈	#11719924	1	2	yes	—
182	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
183	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
184	C₂×C₈	D₄×D₅₃	1	2	yes	—
185	C₂×C₈	#11719949	1	2	yes	—
186	C₄⋊C₄	—	212	4	no	—
187	C₄⋊C₄	—	212	4	no	—
188	C₄²	#11719921	1	1	yes	characteristic
189	C₄²	#11719921	1	1	yes	characteristic
190	C₄²	#11719922	1	2	yes	—
191	C₁₆	—	106	4	no	—
192	C₂³	C₂×#11719924	1	1	yes	characteristic
193	Q₈	—	424	2	no	—
194	Q₈	—	424	2	no	—
195	Q₈	—	212	4	no	—
196	Q₈	—	212	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	424	2	no	—
199	C₂×C₄	—	424	2	no	—
200	C₂×C₄	—	212	4	no	—
201	C₂×C₄	#13133137	1	2	yes	—
202	C₂×C₄	#13133137	1	2	yes	—
203	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
204	C₂×C₄	#13133206	1	1	yes	characteristic
205	C₂×C₄	#13133213	1	1	yes	characteristic
206	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
207	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
208	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
209	C₈	—	8	2	no	—
210	C₈	#13133137	1	2	yes	—
211	C₈	#13133137	1	2	yes	—
212	C₈	#13133179	1	2	yes	—
213	C₈	#13133192	1	2	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	C₂×#13133137	1	1	yes	characteristic
216	C₂²	#16810971	1	1	yes	characteristic
217	C₂²	#16810997	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	424	4	no	—
220	C₄	#16810438	1	4	yes	complemented
221	C₄	#16810524	1	1	yes	characteristic
222	C₄	C₂×#13133137	1	1	yes	characteristic
223	C₄	#16810971	1	1	yes	characteristic
224	C₄	#16810999	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	#20275520	1	1	yes	characteristic
228	C₂	C₂×#16810438	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#13133137
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1848	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1848	D₅₃
hypocenter	C₅₃	#13212
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
105. C₄×C₅₃	#362	semidirect
134. C₅₃	#13212	semidirect
220. C₄	#16810438	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811003	2
C₂×#16810438	2
C₄×#13133157	2
C₅₃×#1848	2
#13212	53