Subgroups — C₂×#20276498 #27537120

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810660	C₂	1	2	yes	maximal
3	#20276498	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810451	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810448	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
8	C₂×#13133175	—	2	2	no	—
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₅₃	C₄	1	8	yes	complemented
11	C₂×C₁₆×D₅₃	C₂²	1	4	yes	—
12	#16810660	—	2	4	no	—
13	#16810660	C₂²	1	8	yes	—
14	C₂×#13133151	C₂²	1	2	yes	—
15	C₂×#13133148	—	2	2	no	—
16	#16810451	C₂²	1	4	yes	complemented
17	#16810448	C₂²	1	8	yes	complemented
18	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#107×C₅₃	C₂²	1	2	yes	—
20	C₅₃×#367	C₂²	1	4	yes	complemented
21	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
22	C₂²×#88322	D₄	1	1	yes	characteristic
23	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
24	C₂×C₈×D₅₃	D₄	1	2	yes	—
25	C₂×C₈×D₅₃	Q₈	1	4	yes	—
26	C₂×C₈×D₅₃	C₂³	1	2	yes	—
27	#13133175	—	2	8	no	—
28	#13133175	—	4	4	no	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×#11719921	—	4	2	no	—
31	C₂×C₄×Dic₅₃	—	4	2	no	—
32	C₂×#11719908	C₂×C₄	1	4	yes	—
33	C₂×#11719908	C₂³	1	2	yes	—
34	C₁₆×D₅₃	C₂×C₄	1	16	yes	complemented
35	#13133151	—	2	2	no	—
36	#13133151	C₂³	1	4	yes	—
37	#13133148	—	2	8	no	—
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
41	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
42	#107×C₅₃	—	2	2	no	—
43	#107×C₅₃	C₂³	1	4	yes	—
44	C₂³×D₅₃	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₅₃	—	4	2	no	—
48	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
49	C₂×C₄×D₅₃	C₄⋊C₄	1	4	yes	—
50	C₂×C₄×D₅₃	D₈	1	2	yes	—
51	C₂×C₄×D₅₃	Q₁₆	1	4	yes	—
52	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₅₃	—	2	4	no	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	#11719921	—	4	8	no	—
60	C₄⋊Dic₅₃	—	2	4	no	—
61	C₄⋊Dic₅₃	—	4	2	no	—
62	C₄×Dic₅₃	—	4	8	no	—
63	C₂×#88322	C₄⋊C₄	1	4	yes	—
64	C₂×#88322	C₂×D₄	1	2	yes	—
65	C₈×D₅₃	C₄⋊C₄	1	8	yes	—
66	C₈×D₅₃	C₂²×C₄	1	4	yes	—
67	C₈×D₅₃	C₂×Q₈	1	4	yes	—
68	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
69	#11719908	C₂²×C₄	1	4	yes	—
70	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
71	C₂²×D₅₃	—	2	4	no	—
72	C₂²×D₅₃	#107	1	4	yes	—
73	C₂²×D₅₃	C₂×D₈	1	2	yes	—
74	C₂×C₄×C₅₃	—	4	4	no	—
75	C₂×C₄×C₅₃	—	8	4	no	—
76	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
79	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×Dic₅₃	—	4	4	no	—
81	C₂×Dic₅₃	—	8	4	no	—
82	C₂×Dic₅₃	#107	1	4	yes	—
83	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
84	C₄×D₅₃	#107	1	8	yes	—
85	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
86	C₄×D₅₃	C₂×Q₁₆	1	4	yes	—
87	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
88	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
89	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
90	#88322	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×#2158	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	C₂×#107	1	2	yes	—
93	C₂²×C₅₃	#508	1	4	yes	—
94	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
95	C₂×D₅₃	—	2	4	no	—
96	C₂×D₅₃	—	2	8	no	—
97	C₂×D₅₃	C₂×#107	1	4	yes	—
98	C₄×C₅₃	—	8	4	no	—
99	C₄×C₅₃	C₂×#107	1	2	yes	—
100	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
102	Dic₅₃	—	8	4	no	—
103	Dic₅₃	C₂×#107	1	4	yes	—
104	C₂³×C₁₆	—	53	1	no	—
105	C₂×#427	—	53	2	no	—
106	#2158	—	53	8	no	—
107	C₂×#367	—	53	1	no	—
108	C₂×#367	D₅₃	1	1	yes	characteristic, complemented
109	C₂×#366	—	53	2	no	—
110	C₂×C₅₃	#2158	1	4	yes	complemented
111	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
112	C₂×C₅₃	C₂×#508	1	2	yes	—
113	D₅₃	—	2	4	no	—
114	C₂³×C₈	—	53	1	no	—
115	C₂×#117	—	106	2	no	—
116	C₂²×C₁₆	—	53	1	no	—
117	C₂²×C₁₆	—	53	8	no	—
118	C₂²×C₁₆	—	53	4	no	—
119	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
120	#427	—	53	8	no	—
121	#427	—	106	4	no	—
122	C₂×#107	—	53	2	no	—
123	C₂×#107	C₂×D₅₃	1	2	yes	—
124	C₂×#106	—	106	2	no	—
125	#367	—	53	4	no	—
126	#367	C₂×D₅₃	1	4	yes	complemented
127	#366	—	53	8	no	—
128	C₅₃	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
129	C₂³×C₄	—	53	1	no	—
130	C₂²×C₈	—	53	1	no	—
131	C₂²×C₈	—	53	4	no	—
132	C₂²×C₈	—	53	2	no	—
133	C₂²×C₈	—	53	4	no	—
134	C₂²×C₈	—	53	2	no	—
135	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
136	#117	—	106	8	no	—
137	#117	—	212	4	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	106	2	no	—
140	C₂×C₂²⋊C₄	—	212	2	no	—
141	C₂×C₄²	—	212	2	no	—
142	C₂×C₁₆	—	53	4	no	—
143	C₂×C₁₆	—	53	2	no	—
144	C₂×C₁₆	—	53	16	no	—
145	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
146	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
147	#107	—	2	2	no	—
148	#107	—	53	4	no	—
149	#107	—	106	2	no	—
150	#107	C₂²×D₅₃	1	4	yes	—
151	#106	—	106	8	no	—
152	C₂⁴	—	53	1	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	—	53	1	no	—
155	C₂²×C₄	—	53	4	no	—
156	C₂²×C₄	—	53	2	no	—
157	C₂²×C₄	—	53	4	no	—
158	C₂²×C₄	—	53	2	no	—
159	C₂²×C₄	—	212	2	no	—
160	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
161	C₂×C₈	—	53	4	no	—
162	C₂×C₈	—	53	2	no	—
163	C₂×C₈	—	53	8	no	—
164	C₂×C₈	—	53	4	no	—
165	C₂×C₈	—	53	4	no	—
166	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
167	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
168	C₂×C₈	Q₈×D₅₃	1	2	yes	—
169	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	106	4	no	—
173	C₄⋊C₄	—	212	2	no	—
174	C₂²⋊C₄	—	212	8	no	—
175	C₄²	—	212	8	no	—
176	C₁₆	—	53	4	no	—
177	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
178	C₂³	—	53	2	no	—
179	C₂³	—	53	4	no	—
180	C₂³	—	106	4	no	—
181	C₂³	D₈×D₅₃	1	1	yes	characteristic
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	8	4	no	—
184	C₂×C₄	—	53	2	no	—
185	C₂×C₄	—	53	4	no	—
186	C₂×C₄	—	53	4	no	—
187	C₂×C₄	—	53	4	no	—
188	C₂×C₄	—	53	8	no	—
189	C₂×C₄	—	424	4	no	—
190	C₂×C₄	—	212	4	no	—
191	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
192	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
193	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
194	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
195	C₈	—	53	4	no	—
196	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
197	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
198	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
199	C₂²	—	53	4	no	—
200	C₂²	—	106	8	no	—
201	C₂²	—	106	4	no	—
202	C₂²	#107×D₅₃	1	2	yes	—
203	C₂²	#16811054	1	4	yes	—
204	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
205	C₄	—	8	4	no	—
206	C₄	—	53	4	no	—
207	C₄	—	424	4	no	—
208	C₄	#107×D₅₃	1	2	yes	—
209	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
210	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
211	C₂	—	106	4	no	—
212	C₂	#20276498	1	4	yes	direct product factor
213	C₂	C₂×#107×D₅₃	1	1	yes	characteristic
214	C₂	C₂×#16811054	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#367	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂³×D₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#367	D₅₃
hypocenter	C₅₃	C₂×#2158
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
3. #20276498	C₂	direct (N × H)
4. C₂×#16810451	C₂	semidirect
5. C₂×#16810448	C₂	semidirect
6. C₂×C₅₃×#367	C₂	semidirect
10. C₂×C₁₆×D₅₃	C₄	semidirect
16. #16810451	C₂²	semidirect
17. #16810448	C₂²	semidirect
20. C₅₃×#367	C₂²	semidirect
34. C₁₆×D₅₃	C₂×C₄	semidirect
108. C₂×#367	D₅₃	semidirect
110. C₂×C₅₃	#2158	semidirect
126. #367	C₂×D₅₃	semidirect
128. C₅₃	C₂×#2158	semidirect
212. C₂	#20276498	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₅₃	2
C₂×#16810660	2
#20276498	2
C₂×#16810451	2
C₂×#16810448	2
C₂×C₅₃×#367	2
C₂×#2158	53

Subgroups — C2×#20276498 #27537120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276498 #27537120