Subgroups — #27282811

Order 13568.

322 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274887	C₂	1	1	yes	maximal, characteristic
2	#20273489	C₂	1	2	yes	maximal
3	#20273463	C₂	1	1	yes	maximal, characteristic
4	#20272618	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133153	—	2	1	no	—
8	C₂×#13133153	—	2	1	no	—
9	#16810647	C₂²	1	2	yes	—
10	#16810376	C₄	1	2	yes	complemented
11	#16810375	—	2	4	no	—
12	#16810374	C₄	1	2	yes	complemented
13	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	—	2	1	no	—
18	C₂×#11719917	—	2	2	no	—
19	#13133153	—	4	2	no	—
20	#13133153	—	4	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	C₂×C₄	1	4	yes	—
24	C₂²⋊#88322	—	2	2	no	—
25	C₄×#88322	—	2	2	no	—
26	C₂²×C₈×C₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄²×C₅₃	—	2	1	no	—
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₅₃	—	2	1	no	—
33	C₄⋊C₈×C₅₃	D₄	1	2	yes	—
34	C₄×C₈×C₅₃	—	4	2	no	—
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
38	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
39	#95×C₅₃	Q₈	1	2	yes	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	4	1	no	—
48	C₂×C₈×C₅₃	—	4	1	no	—
49	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	D₈	1	2	yes	—
54	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
55	C₄⋊C₄×C₅₃	—	2	4	no	—
56	C₄²×C₅₃	—	4	1	no	—
57	C₄²×C₅₃	—	4	1	no	—
58	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₅₃	#41	1	1	yes	characteristic
62	#11719917	—	4	2	no	—
63	#11719917	—	4	2	no	—
64	C₂×#88322	—	2	4	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	4	no	—
69	C₂×C₄×C₅₃	—	4	1	no	—
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
82	C₈×C₅₃	—	2	1	no	—
83	C₈×C₅₃	—	2	1	no	—
84	C₈×C₅₃	—	4	2	no	—
85	C₈×C₅₃	#107	1	2	yes	—
86	C₈×C₅₃	#107	1	2	yes	—
87	#88322	—	4	4	no	—
88	#88322	—	8	2	no	—
89	#9627	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	—	2	1	no	—
91	C₂²×C₅₃	—	2	1	no	—
92	C₂²×C₅₃	#389	1	1	yes	characteristic
93	C₂²×C₅₃	#474	1	1	yes	characteristic
94	C₂²×C₅₃	#477	1	1	yes	characteristic
95	C₄×C₅₃	—	2	1	no	—
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	4	4	no	—
99	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
100	C₄×C₅₃	#428	1	1	yes	characteristic
101	C₄×C₅₃	#443	1	2	yes	—
102	C₄×C₅₃	#459	1	1	yes	characteristic
103	C₄×C₅₃	#462	1	1	yes	characteristic
104	#1848	—	53	1	no	—
105	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic
106	#1571	—	53	2	no	—
107	#1564	—	53	1	no	—
108	#1303	—	53	2	no	—
109	C₂×C₅₃	—	2	1	no	—
110	C₂×C₅₃	—	2	1	no	—
111	C₂×C₅₃	#1945	1	1	yes	characteristic
112	C₂×C₅₃	#1947	1	1	yes	characteristic
113	C₂×C₅₃	#1953	1	1	yes	characteristic
114	#431	—	53	2	no	—
115	C₂×#108	—	106	1	no	—
116	C₂×#108	—	106	1	no	—
117	#423	—	53	2	no	—
118	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
119	C₂×C₄×C₈	—	2	1	no	—
120	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
121	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
122	#336	—	2	1	no	—
123	#335	—	53	2	no	—
124	#335	—	106	4	no	—
125	#334	—	53	2	no	—
126	C₅₃	#9627	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₁₆	—	106	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
134	#108	—	212	2	no	—
135	#108	—	212	2	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	53	4	no	—
138	C₄⋊C₈	—	106	2	no	—
139	C₄⋊C₈	—	106	2	no	—
140	C₄⋊C₈	D₂₁₂	1	2	yes	—
141	C₂²⋊C₈	—	106	2	no	—
142	C₄×C₈	—	4	2	no	—
143	C₄×C₈	—	106	2	no	—
144	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
145	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
146	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
147	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
148	#95	Dic₁₀₆	1	2	yes	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
151	C₂²×C₄	#11719945	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
153	M₁₆	—	212	2	no	—
154	M₁₆	—	212	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	106	4	no	—
161	C₂×C₈	—	212	2	no	—
162	C₂×C₈	#11719919	1	1	yes	characteristic
163	C₂×C₈	#11719919	1	1	yes	characteristic
164	C₂×C₈	#11719919	1	1	yes	characteristic
165	C₂×C₈	#11719919	1	1	yes	characteristic
166	C₂×C₈	#11719922	1	2	yes	—
167	C₂×C₈	#11719925	1	2	yes	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	1	no	—
171	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
172	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
173	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
174	C₄²	#11719945	1	1	yes	characteristic
175	C₂³	#13133196	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
182	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
183	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
184	C₂×C₄	#13133191	1	1	yes	characteristic
185	C₂×C₄	#13133209	1	1	yes	characteristic
186	C₂×C₄	#13133217	1	1	yes	characteristic
187	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
188	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
189	C₂×C₄	#13133242	1	1	yes	characteristic
190	C₂×C₄	#13133247	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	2	no	—
194	C₈	—	424	2	no	—
195	C₈	—	212	4	no	—
196	C₈	#13133150	1	2	yes	—
197	C₈	#13133150	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#16810521	1	1	yes	characteristic
201	C₂²	#16810903	1	1	yes	characteristic
202	C₂²	#16810918	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	4	4	no	—
207	C₄	C₂×#13133150	1	1	yes	characteristic
208	C₄	#16810666	1	1	yes	characteristic
209	C₄	#16810741	1	2	yes	—
210	C₄	#16810818	1	1	yes	characteristic
211	C₄	#16810833	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#20275463	1	1	yes	characteristic
215	C₂	#20275473	1	1	yes	characteristic
216	C₂	#20275505	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133191
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂³×C₅₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#9627
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
10. #16810376	C₄	semidirect
12. #16810374	C₄	semidirect
126. C₅₃	#9627	semidirect

Maximal subgroups

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

subgroup	index
#20274887	2
#20273489	2
#20273463	2
#20272618	2
C₈×C₄⋊C₄×C₅₃	2
#9627	53