Subgroups — C₂×#20276496 #27537118

Order 13568.

694 conjugacy classes of subgroups.

The subgroup lattice has 224 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	#20276496	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810450	C₂	1	2	yes	maximal, complemented
5	C₂×#16810449	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅₃×#366	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	#16810450	C₂²	1	8	yes	complemented
17	#16810449	C₂²	1	4	yes	complemented
18	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#107×C₅₃	C₂²	1	2	yes	—
20	C₅₃×#366	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	2	no	—
72	C₂²×D₅₃	—	2	2	no	—
73	C₂²×D₅₃	#107	1	4	yes	—
74	C₂²×D₅₃	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₅₃	—	4	4	no	—
76	C₂×C₄×C₅₃	—	8	4	no	—
77	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
80	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₅₃	—	4	4	no	—
82	C₂×Dic₅₃	—	8	4	no	—
83	C₂×Dic₅₃	#107	1	4	yes	—
84	C₂×Dic₅₃	C₂×D₈	1	2	yes	—
85	C₄×D₅₃	#107	1	8	yes	—
86	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
87	C₄×D₅₃	C₂×Q₁₆	1	4	yes	—
88	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
89	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
91	#88322	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×#2158	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	C₂×#107	1	2	yes	—
94	C₂²×C₅₃	#508	1	2	yes	—
95	C₂²×C₅₃	#508	1	2	yes	—
96	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
97	C₂×D₅₃	—	2	2	no	—
98	C₂×D₅₃	—	2	2	no	—
99	C₂×D₅₃	—	2	8	no	—
100	C₂×D₅₃	C₂×#107	1	4	yes	—
101	C₄×C₅₃	—	8	4	no	—
102	C₄×C₅₃	C₂×#107	1	2	yes	—
103	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
104	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
105	Dic₅₃	—	8	4	no	—
106	Dic₅₃	C₂×#107	1	4	yes	—
107	C₂³×C₁₆	—	53	1	no	—
108	C₂×#427	—	53	2	no	—
109	#2158	—	53	8	no	—
110	C₂×#367	—	53	2	no	—
111	C₂×#366	—	53	1	no	—
112	C₂×#366	D₅₃	1	1	yes	characteristic, complemented
113	C₂×C₅₃	#2158	1	4	yes	complemented
114	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
115	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#508	1	1	yes	characteristic
117	D₅₃	—	2	4	no	—
118	C₂³×C₈	—	53	1	no	—
119	C₂×#117	—	106	2	no	—
120	C₂²×C₁₆	—	53	1	no	—
121	C₂²×C₁₆	—	53	8	no	—
122	C₂²×C₁₆	—	53	4	no	—
123	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
124	#427	—	53	8	no	—
125	#427	—	106	4	no	—
126	C₂×#107	—	53	2	no	—
127	C₂×#107	C₂×D₅₃	1	2	yes	—
128	C₂×#106	—	106	2	no	—
129	#367	—	53	8	no	—
130	#366	—	53	4	no	—
131	#366	C₂×D₅₃	1	4	yes	complemented
132	C₅₃	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₄	—	53	1	no	—
134	C₂²×C₈	—	53	1	no	—
135	C₂²×C₈	—	53	4	no	—
136	C₂²×C₈	—	53	2	no	—
137	C₂²×C₈	—	53	4	no	—
138	C₂²×C₈	—	53	2	no	—
139	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
140	#117	—	106	8	no	—
141	#117	—	212	4	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	106	2	no	—
144	C₂×C₂²⋊C₄	—	212	2	no	—
145	C₂×C₄²	—	212	2	no	—
146	C₂×C₁₆	—	53	4	no	—
147	C₂×C₁₆	—	53	2	no	—
148	C₂×C₁₆	—	53	16	no	—
149	C₂×C₁₆	C₄×D₅₃	1	4	yes	—
150	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
151	#107	—	2	2	no	—
152	#107	—	53	4	no	—
153	#107	—	106	2	no	—
154	#107	C₂²×D₅₃	1	4	yes	—
155	#106	—	106	8	no	—
156	C₂⁴	—	53	1	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	53	4	no	—
160	C₂²×C₄	—	53	2	no	—
161	C₂²×C₄	—	53	4	no	—
162	C₂²×C₄	—	53	2	no	—
163	C₂²×C₄	—	212	2	no	—
164	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
165	C₂×C₈	—	53	4	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	53	8	no	—
168	C₂×C₈	—	53	4	no	—
169	C₂×C₈	—	53	4	no	—
170	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
171	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
172	C₂×C₈	Q₈×D₅₃	1	2	yes	—
173	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	106	4	no	—
177	C₄⋊C₄	—	212	2	no	—
178	C₂²⋊C₄	—	212	8	no	—
179	C₄²	—	212	8	no	—
180	C₁₆	—	53	4	no	—
181	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
182	C₂³	—	53	2	no	—
183	C₂³	—	53	4	no	—
184	C₂³	—	106	2	no	—
185	C₂³	—	106	2	no	—
186	C₂³	D₈×D₅₃	1	1	yes	characteristic
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	8	4	no	—
189	C₂×C₄	—	53	2	no	—
190	C₂×C₄	—	53	4	no	—
191	C₂×C₄	—	53	4	no	—
192	C₂×C₄	—	53	4	no	—
193	C₂×C₄	—	53	8	no	—
194	C₂×C₄	—	424	4	no	—
195	C₂×C₄	—	212	4	no	—
196	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
197	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
198	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
199	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
200	C₈	—	53	4	no	—
201	C₈	C₄⋊C₄×D₅₃	1	2	yes	—
202	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
203	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
204	C₂²	—	53	4	no	—
205	C₂²	—	106	8	no	—
206	C₂²	—	106	2	no	—
207	C₂²	—	106	2	no	—
208	C₂²	#107×D₅₃	1	2	yes	—
209	C₂²	#16811052	1	2	yes	—
210	C₂²	#16811056	1	2	yes	—
211	C₂²	C₂×D₈×D₅₃	1	1	yes	characteristic
212	C₄	—	8	4	no	—
213	C₄	—	53	4	no	—
214	C₄	—	424	4	no	—
215	C₄	#107×D₅₃	1	2	yes	—
216	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
217	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
218	C₂	—	106	4	no	—
219	C₂	#20276496	1	4	yes	direct product factor
220	C₂	C₂×#107×D₅₃	1	1	yes	characteristic
221	C₂	C₂×#16811052	1	1	yes	characteristic
222	C₂	C₂×#16811056	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#366	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂³×D₅₃	D₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#366	D₅₃
hypocenter	C₅₃	C₂×#2158
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
3. #20276496	C₂	direct (N × H)
4. C₂×#16810450	C₂	semidirect
5. C₂×#16810449	C₂	semidirect
6. C₂×C₅₃×#366	C₂	semidirect
10. C₂×C₁₆×D₅₃	C₄	semidirect
16. #16810450	C₂²	semidirect
17. #16810449	C₂²	semidirect
20. C₅₃×#366	C₂²	semidirect
34. C₁₆×D₅₃	C₂×C₄	semidirect
112. C₂×#366	D₅₃	semidirect
113. C₂×C₅₃	#2158	semidirect
131. #366	C₂×D₅₃	semidirect
132. C₅₃	C₂×#2158	semidirect
219. C₂	#20276496	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
#20276496	2
C₂×#16810450	2
C₂×#16810449	2
C₂×C₅₃×#366	2
C₂×#2158	53

Subgroups — C2×#20276496 #27537118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276496 #27537118