Subgroups — #27536800

Order 13568.

1114 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	#20289426	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811040	C₂	1	1	yes	maximal, characteristic, complemented
3	#20276344	C₂	1	6	yes	maximal, complemented
4	C₂×#16810408	C₂	1	6	yes	maximal, complemented
5	C₅₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811459	C₄	1	1	yes	characteristic
7	#16811390	C₄	1	1	yes	characteristic
8	#16811389	C₄	1	6	yes	—
9	#16811388	—	2	4	no	—
10	C₂×#13133223	C₂²	1	6	yes	—
11	#131×D₅₃	—	2	4	no	—
12	#16811040	—	2	4	no	—
13	C₂×#13133157	C₂²	1	3	yes	complemented
14	C₂²×#11719908	C₂²	1	3	yes	complemented
15	#16810408	C₂²	1	4	yes	complemented
16	#16810408	C₂²	1	12	yes	complemented
17	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#504	—	2	4	no	—
19	C₂×M₃₂×C₅₃	C₂²	1	6	yes	complemented
20	#13133265	—	2	1	no	—
21	#13133264	C₈	1	4	yes	—
22	#13133263	C₈	1	4	yes	—
23	#13133262	—	2	3	no	—
24	C₂×#11719945	C₂×C₄	1	6	yes	—
25	#41×D₅₃	—	2	4	no	—
26	#13133225	—	2	12	no	—
27	#13133224	—	2	4	no	—
28	#13133223	—	2	6	no	—
29	#13133223	—	2	6	no	—
30	#13133223	C₂×C₄	1	12	yes	—
31	#13133223	C₂³	1	4	yes	—
32	C₂×#11719917	C₂×C₄	1	6	yes	—
33	C₂×#11719912	—	2	6	no	—
34	M₁₆×D₅₃	—	2	12	no	—
35	C₂×C₈×D₅₃	—	2	6	no	—
36	#13133157	—	2	6	no	—
37	C₂×#11719908	—	2	6	no	—
38	C₂×#11719908	C₂³	1	1	yes	characteristic
39	C₂×#11719908	C₂³	1	3	yes	—
40	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
41	#131×C₅₃	D₄	1	4	yes	—
42	#131×C₅₃	D₄	1	4	yes	—
43	C₂×M₁₆×C₅₃	C₂×C₄	1	3	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
45	M₃₂×C₅₃	—	2	6	no	—
46	M₃₂×C₅₃	C₂³	1	4	yes	—
47	C₂×C₁₆×C₅₃	—	2	6	no	—
48	#41×C₅₃	C₂²⋊C₄	1	4	yes	—
49	#41×C₅₃	C₂×C₈	1	4	yes	—
50	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
51	C₂×D₄×C₅₃	M₁₆	1	3	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
53	C₂×C₂²⋊D₅₃	—	2	6	no	—
54	#11719949	—	2	4	no	—
55	Q₈×D₅₃	—	2	4	no	—
56	#11719947	—	2	12	no	—
57	D₄×D₅₃	—	2	12	no	—
58	#11719945	—	2	6	no	—
59	#11719945	—	2	6	no	—
60	#11719945	C₂×C₈	1	12	yes	—
61	#11719945	C₂²×C₄	1	4	yes	—
62	C₂×D₂₁₂	C₂×C₈	1	6	yes	—
63	C₂×C₄×D₅₃	—	2	6	no	—
64	C₂×Dic₁₀₆	C₂×C₈	1	6	yes	—
65	M₁₆×C₅₃	C₂²⋊C₄	1	12	yes	—
66	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
67	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
68	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
69	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
70	#11719917	—	2	6	no	—
71	#11719917	C₂²×C₄	1	4	yes	—
72	C₂×#88322	—	2	6	no	—
73	#11719912	—	2	12	no	—
74	#11719912	—	2	4	no	—
75	C₈×D₅₃	—	2	12	no	—
76	C₈×D₅₃	—	2	4	no	—
77	C₁₆×C₅₃	—	2	4	no	—
78	#11719908	—	2	1	no	—
79	#11719908	—	2	3	no	—
80	C₂³×C₅₃	C₂×M₁₆	1	3	yes	—
81	C₂²×D₅₃	—	2	6	no	—
82	Q₈×C₅₃	C₂²⋊C₈	1	4	yes	—
83	D₄×C₅₃	C₂²⋊C₈	1	12	yes	—
84	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
85	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
86	C₂×C₄×C₅₃	C₂×M₁₆	1	3	yes	—
87	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
88	C₂²⋊D₅₃	—	2	4	no	—
89	C₂²⋊D₅₃	—	2	12	no	—
90	C₂×Dic₅₃	—	2	6	no	—
91	D₂₁₂	—	2	6	no	—
92	D₂₁₂	C₂²×C₈	1	4	yes	—
93	C₄×D₅₃	—	2	4	no	—
94	C₄×D₅₃	—	2	12	no	—
95	Dic₁₀₆	—	2	6	no	—
96	Dic₁₀₆	C₂²×C₈	1	4	yes	—
97	C₈×C₅₃	C₂×C₂²⋊C₄	1	6	yes	—
98	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₅₃	C₂²×D₄	1	1	yes	characteristic
100	#88322	—	2	4	no	—
101	#33340	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	4	3	no	—
103	C₂²×C₅₃	C₂×C₂²⋊C₈	1	6	yes	—
104	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
105	C₂×D₅₃	—	2	4	no	—
106	C₂×D₅₃	—	8	6	no	—
107	C₄×C₅₃	C₂×C₂²⋊C₈	1	6	yes	—
108	C₄×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
110	Dic₅₃	—	2	4	no	—
111	#3573	—	53	1	no	—
112	#3407	D₅₃	1	1	yes	characteristic, complemented
113	C₂×#504	—	53	1	no	—
114	#2117	—	53	6	no	—
115	C₂×#350	—	53	6	no	—
116	C₂×C₅₃	—	2	6	no	—
117	C₂×C₅₃	—	2	1	no	—
118	C₂×C₅₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
119	D₅₃	—	4	4	no	—
120	#585	—	53	1	no	—
121	#568	—	53	1	no	—
122	#568	—	53	6	no	—
123	#568	—	106	4	no	—
124	C₂×#131	—	53	6	no	—
125	C₂×#131	—	106	4	no	—
126	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
127	#504	—	2	4	no	—
128	#504	—	106	4	no	—
129	C₂×M₃₂	—	53	3	no	—
130	C₂×M₃₂	C₂×D₅₃	1	6	yes	complemented
131	C₂²×C₁₆	—	53	3	no	—
132	#350	—	53	4	no	—
133	#350	—	53	12	no	—
134	C₅₃	#33340	1	1	yes	characteristic, complemented, Sylow
135	2₋¹⁺⁴	—	53	4	no	—
136	2₋¹⁺⁴	—	106	1	no	—
137	H(5,𝔽₂)	—	53	4	no	—
138	H(5,𝔽₂)	—	106	3	no	—
139	C₂×#41	—	53	6	no	—
140	C₂×#41	—	106	4	no	—
141	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
142	#131	—	53	12	no	—
143	#131	—	53	4	no	—
144	#131	—	106	4	no	—
145	#131	—	106	12	no	—
146	#131	—	106	6	no	—
147	#131	—	106	6	no	—
148	#131	D₂₁₂	1	4	yes	—
149	#131	C₂²⋊D₅₃	1	4	yes	—
150	C₂×M₁₆	—	53	6	no	—
151	C₂×M₁₆	—	106	12	no	—
152	C₂×M₁₆	—	106	6	no	—
153	C₂×M₁₆	C₄×D₅₃	1	3	yes	—
154	C₂²×C₈	—	106	6	no	—
155	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
156	M₃₂	—	2	6	no	—
157	M₃₂	—	106	6	no	—
158	M₃₂	C₂²×D₅₃	1	4	yes	—
159	C₂×C₁₆	—	2	6	no	—
160	C₂×C₁₆	—	53	1	no	—
161	C₂×C₁₆	—	53	3	no	—
162	C₂×C₁₆	—	106	6	no	—
163	#41	—	53	12	no	—
164	#41	—	53	4	no	—
165	#41	—	106	4	no	—
166	#41	—	106	12	no	—
167	#41	—	106	6	no	—
168	#41	—	106	6	no	—
169	#41	C₈×D₅₃	1	4	yes	—
170	#41	#11719921	1	4	yes	—
171	C₂×Q₈	—	53	6	no	—
172	C₂×Q₈	—	106	4	no	—
173	C₂×Q₈	#11719912	1	1	yes	characteristic
174	C₂×D₄	—	53	6	no	—
175	C₂×D₄	—	106	6	no	—
176	C₂×D₄	—	106	12	no	—
177	C₂×D₄	#11719912	1	3	yes	—
178	C₂²×C₄	—	106	6	no	—
179	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
180	M₁₆	—	53	4	no	—
181	M₁₆	—	106	6	no	—
182	M₁₆	—	106	12	no	—
183	M₁₆	—	106	4	no	—
184	M₁₆	#11719921	1	12	yes	—
185	C₂×C₈	—	106	6	no	—
186	C₂×C₈	—	106	12	no	—
187	C₂×C₈	—	106	4	no	—
188	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
189	C₂×C₈	C₂×D₂₁₂	1	6	yes	—
190	C₂×C₈	C₂×C₂²⋊D₅₃	1	6	yes	—
191	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
192	C₁₆	—	2	4	no	—
193	C₁₆	—	106	1	no	—
194	C₁₆	—	106	3	no	—
195	C₂³	—	106	6	no	—
196	C₂³	C₂×#11719912	1	3	yes	—
197	Q₈	—	53	4	no	—
198	Q₈	—	106	6	no	—
199	Q₈	#13133129	1	4	yes	—
200	D₄	—	53	4	no	—
201	D₄	—	106	12	no	—
202	D₄	—	106	4	no	—
203	D₄	—	106	6	no	—
204	D₄	#13133129	1	12	yes	—
205	C₂×C₄	—	106	6	no	—
206	C₂×C₄	—	106	4	no	—
207	C₂×C₄	—	106	12	no	—
208	C₂×C₄	C₂×#11719921	1	6	yes	—
209	C₂×C₄	C₂×C₈×D₅₃	1	6	yes	—
210	C₂×C₄	C₂×#11719912	1	3	yes	—
211	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
212	C₈	—	106	4	no	—
213	C₈	C₂×#11719921	1	6	yes	—
214	C₈	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
215	C₈	C₂²×D₂₁₂	1	1	yes	characteristic
216	C₂²	—	4	3	no	—
217	C₂²	—	106	4	no	—
218	C₂²	—	424	6	no	—
219	C₂²	C₂×#13133129	1	6	yes	—
220	C₂²	C₂²×#11719912	1	1	yes	characteristic
221	C₄	—	106	4	no	—
222	C₄	C₂×#13133129	1	6	yes	—
223	C₄	C₂²×#11719921	1	1	yes	characteristic
224	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	6	no	—
227	C₂	—	212	4	no	—
228	C₂	C₂²×#13133129	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#3407	C₂
socle	C₂×C₅₃	C₂²×C₂²⋊C₈
prime radical	#16811459	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#3407	D₅₃
hypocenter	C₅₃	#33340
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ #3407

Complements

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

N	H ≅ G/N	type
2. C₂×#16811040	C₂	semidirect
3. #20276344	C₂	semidirect
4. C₂×#16810408	C₂	semidirect
5. C₅₃×#3407	C₂	semidirect
13. C₂×#13133157	C₂²	semidirect
14. C₂²×#11719908	C₂²	semidirect
15. #16810408	C₂²	semidirect
16. #16810408	C₂²	semidirect
19. C₂×M₃₂×C₅₃	C₂²	semidirect
112. #3407	D₅₃	semidirect
130. C₂×M₃₂	C₂×D₅₃	semidirect
134. C₅₃	#33340	semidirect

Maximal subgroups

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

subgroup	index
#20289426	2
C₂×#16811040	2
#20276344	2
C₂×#16810408	2
C₅₃×#3407	2
#33340	53