Subgroups — #27537455

Order 13568.

718 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	C₂²×D₈₄₈	C₂	1	1	yes	maximal, characteristic
2	#20283568	C₂	1	2	yes	maximal, complemented
3	#20276592	C₂	1	4	yes	maximal, complemented
4	C₄×D₈₄₈	C₂	1	4	yes	maximal, complemented
5	#20276499	C₂	1	1	yes	maximal, characteristic, complemented
6	#20276432	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₂₄	C₂²	1	2	yes	—
9	#16811097	—	2	2	no	—
10	C₂×D₈₄₈	C₄	1	8	yes	complemented
11	C₂×D₈₄₈	C₂²	1	2	yes	—
12	C₂×D₈₄₈	C₂²	1	2	yes	—
13	#16810736	—	2	4	no	—
14	C₄×D₄₂₄	C₂²	1	8	yes	complemented
15	#16810661	C₂²	1	2	yes	—
16	#16810629	—	2	2	no	—
17	#16810451	C₂²	1	2	yes	complemented
18	#16810449	C₂²	1	2	yes	complemented
19	#16810425	C₂²	1	8	yes	complemented
20	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#405	C₂²	1	1	yes	characteristic
23	C₅₃×#346	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₅₃	C₂²	1	2	yes	complemented
25	C₂²×D₂₁₂	—	2	2	no	—
26	C₂×D₄₂₄	—	2	2	no	—
27	C₂×D₄₂₄	—	2	4	no	—
28	C₂×D₄₂₄	C₂×C₄	1	8	yes	complemented
29	C₂×D₄₂₄	C₂³	1	4	yes	—
30	C₄×D₂₁₂	—	2	8	no	—
31	#13133179	—	4	4	no	—
32	#13133176	—	2	2	no	—
33	C₂²⋊C₄×D₅₃	—	4	4	no	—
34	D₈₄₈	C₂×C₄	1	8	yes	complemented
35	D₈₄₈	C₂×C₄	1	8	yes	complemented
36	#13133151	C₂³	1	4	yes	—
37	#13133149	—	2	2	no	—
38	#13133139	—	2	8	no	—
39	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	#117×C₅₃	D₄	1	1	yes	characteristic
42	M₃₂×C₅₃	—	2	2	no	—
43	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
44	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
45	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
47	#97×C₅₃	D₄	1	2	yes	—
48	C₄×C₈×C₅₃	C₂³	1	2	yes	—
49	C₂³×D₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×D₂₁₂	—	2	8	no	—
52	C₂×D₂₁₂	—	2	4	no	—
53	C₂×D₂₁₂	—	4	2	no	—
54	C₂×D₂₁₂	—	4	4	no	—
55	C₂×C₄×D₅₃	—	4	8	no	—
56	M₁₆×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	#41	1	2	yes	—
61	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	D₈	1	2	yes	—
63	C₂²⋊C₄×C₅₃	D₈	1	2	yes	—
64	C₄²×C₅₃	C₂×D₄	1	2	yes	—
65	C₂²⋊Dic₅₃	—	4	2	no	—
66	#11719921	—	4	8	no	—
67	C₄⋊Dic₅₃	—	2	4	no	—
68	C₄×Dic₅₃	—	4	2	no	—
69	D₄₂₄	—	2	8	no	—
70	D₄₂₄	—	2	4	no	—
71	D₄₂₄	C₂²×C₄	1	8	yes	—
72	C₁₆×C₅₃	—	2	2	no	—
73	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
74	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
75	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
76	C₂²×D₅₃	—	4	4	no	—
77	C₂²×D₅₃	—	4	8	no	—
78	C₂²×D₅₃	—	16	4	no	—
79	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
81	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#135	1	2	yes	—
83	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₅₃	—	4	4	no	—
85	D₂₁₂	—	2	8	no	—
86	D₂₁₂	—	4	4	no	—
87	D₂₁₂	—	4	8	no	—
88	C₄×D₅₃	—	16	8	no	—
89	C₈×C₅₃	—	2	2	no	—
90	C₈×C₅₃	C₄×D₄	1	2	yes	—
91	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
93	#33456	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	2	no	—
95	C₂²×C₅₃	C₄×D₈	1	2	yes	—
96	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
97	C₂×D₅₃	—	4	8	no	—
98	C₂×D₅₃	—	8	8	no	—
99	C₂×D₅₃	—	16	8	no	—
100	C₄×C₅₃	—	2	4	no	—
101	C₄×C₅₃	C₄×D₈	1	2	yes	—
102	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
103	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
104	Dic₅₃	—	8	4	no	—
105	C₂²×D₁₆	—	53	1	no	—
106	#2948	—	53	2	no	—
107	#2177	—	53	4	no	—
108	C₄×D₁₆	—	53	4	no	—
109	#2158	—	53	1	no	—
110	#2140	—	53	2	no	—
111	#2108	D₅₃	1	1	yes	characteristic, complemented
112	C₂×C₅₃	—	2	2	no	—
113	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
114	C₂×C₅₃	#3415	1	2	yes	—
115	D₅₃	—	8	8	no	—
116	C₂²×D₈	—	53	2	no	—
117	#518	—	106	2	no	—
118	C₂×D₁₆	—	53	8	no	—
119	C₂×D₁₆	—	53	2	no	—
120	C₂×D₁₆	—	53	2	no	—
121	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
122	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
123	#442	—	106	4	no	—
124	C₄×D₈	—	53	8	no	—
125	#427	—	53	2	no	—
126	#418	—	106	2	no	—
127	#405	C₂×D₅₃	1	1	yes	characteristic
128	#367	—	53	2	no	—
129	#366	—	53	2	no	—
130	#357	—	53	8	no	—
131	#346	C₂×D₅₃	1	2	yes	complemented
132	C₄×C₁₆	C₂×D₅₃	1	2	yes	complemented
133	C₅₃	#33456	1	1	yes	characteristic, complemented, Sylow
134	C₂²×D₄	—	106	2	no	—
135	C₂×D₈	—	53	8	no	—
136	C₂×D₈	—	53	4	no	—
137	C₂×D₈	—	106	2	no	—
138	C₂×D₈	—	106	4	no	—
139	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
141	C₄×D₄	—	106	8	no	—
142	C₄×D₄	—	212	4	no	—
143	#117	—	106	2	no	—
144	#117	D₂₁₂	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	—	212	4	no	—
146	D₁₆	—	53	8	no	—
147	D₁₆	—	53	8	no	—
148	M₃₂	—	2	2	no	—
149	C₂×C₁₆	C₄×D₅₃	1	4	yes	complemented
150	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
151	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
152	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
153	#107	—	53	4	no	—
154	#106	—	106	2	no	—
155	#102	—	106	8	no	—
156	#97	D₂₁₂	1	2	yes	—
157	C₄×C₈	C₂²×D₅₃	1	2	yes	—
158	C₂⁴	—	212	2	no	—
159	C₂×D₄	—	106	8	no	—
160	C₂×D₄	—	106	4	no	—
161	C₂×D₄	—	212	2	no	—
162	C₂×D₄	—	212	4	no	—
163	C₂²×C₄	—	212	8	no	—
164	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
165	D₈	—	53	8	no	—
166	D₈	—	106	8	no	—
167	D₈	—	106	4	no	—
168	M₁₆	—	2	2	no	—
169	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
170	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
171	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
172	C₂×C₈	#11719945	1	2	yes	—
173	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
174	C₄⋊C₄	—	106	4	no	—
175	C₄⋊C₄	D₄₂₄	1	2	yes	—
176	C₂²⋊C₄	—	212	2	no	—
177	C₂²⋊C₄	—	212	8	no	—
178	C₂²⋊C₄	D₄₂₄	1	2	yes	—
179	C₄²	—	212	2	no	—
180	C₄²	C₂×D₂₁₂	1	2	yes	—
181	C₁₆	—	2	2	no	—
182	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
183	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
184	C₂³	—	848	4	no	—
185	C₂³	—	212	8	no	—
186	C₂³	—	212	4	no	—
187	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
188	D₄	—	106	8	no	—
189	D₄	—	212	4	no	—
190	D₄	—	212	8	no	—
191	C₂×C₄	—	848	8	no	—
192	C₂×C₄	—	212	4	no	—
193	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
194	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
195	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
196	C₂×C₄	#13133239	1	2	yes	—
197	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₄×D₂₁₂	1	2	yes	—
200	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
201	C₈	C₂×#11719945	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	848	8	no	—
204	C₂²	—	424	8	no	—
205	C₂²	—	212	8	no	—
206	C₂²	C₄×D₄₂₄	1	2	yes	—
207	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	424	4	no	—
210	C₄	C₄×D₄₂₄	1	2	yes	—
211	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
212	C₄	C₂×#13133239	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	424	8	no	—
215	C₂	C₂×C₄×D₄₂₄	1	1	yes	characteristic
216	C₂	#20287856	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2108	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂²×D₈₄₈	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#33456
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #20283568	C₂	semidirect
3. #20276592	C₂	semidirect
4. C₄×D₈₄₈	C₂	semidirect
5. #20276499	C₂	semidirect
6. #20276432	C₂	semidirect
7. C₅₃×#2108	C₂	semidirect
10. C₂×D₈₄₈	C₄	semidirect
14. C₄×D₄₂₄	C₂²	semidirect
17. #16810451	C₂²	semidirect
18. #16810449	C₂²	semidirect
19. #16810425	C₂²	semidirect
23. C₅₃×#346	C₂²	semidirect
24. C₄×C₁₆×C₅₃	C₂²	semidirect
28. C₂×D₄₂₄	C₂×C₄	semidirect
34. D₈₄₈	C₂×C₄	semidirect
35. D₈₄₈	C₂×C₄	semidirect
43. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
111. #2108	D₅₃	semidirect
131. #346	C₂×D₅₃	semidirect
132. C₄×C₁₆	C₂×D₅₃	semidirect
133. C₅₃	#33456	semidirect
149. C₂×C₁₆	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₈₄₈	2
#20283568	2
#20276592	2
C₄×D₈₄₈	2
#20276499	2
#20276432	2
C₅₃×#2108	2
#33456	53