Subgroups — C₂×#20289909 #28313737

Order 13568.

1182 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₃⋊C₈	C₂	1	2	yes	maximal
2	#20289909	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16811464	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×D₅₃⋊C₈	C₄	1	4	yes	—
6	C₄×D₅₃⋊C₈	C₄	1	8	yes	—
7	C₄×D₅₃⋊C₈	C₂²	1	8	yes	—
8	C₂×C₄×#88324	C₄	1	4	yes	—
9	C₂×C₄×#88324	C₂²	1	4	yes	—
10	#16811464	C₂²	1	16	yes	complemented
11	C₂²×C₈×D₅₃	C₄	1	2	yes	—
12	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
15	C₄×C₈×D₅₃	C₄	1	4	yes	—
16	C₄×C₈×D₅₃	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
18	C₂×D₅₃⋊C₈	—	2	16	no	—
19	C₂×D₅₃⋊C₈	C₂×C₄	1	16	yes	—
20	C₂²×#88324	C₂×C₄	1	8	yes	—
21	C₄×#88324	C₂×C₄	1	16	yes	—
22	C₄×#88324	C₂×C₄	1	8	yes	—
23	C₄×#88324	C₂³	1	8	yes	—
24	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
26	C₂²×#88322	C₂×C₄	1	2	yes	—
27	C₂×C₈×D₅₃	C₈	1	16	yes	complemented
28	C₂×C₈×D₅₃	C₂×C₄	1	8	yes	—
29	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
31	C₄²×D₅₃	C₂×C₄	1	2	yes	—
32	C₄²×D₅₃	C₂×C₄	1	4	yes	—
33	C₄²×D₅₃	C₂³	1	2	yes	—
34	C₄×#88322	C₂×C₄	1	4	yes	—
35	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
36	C₈×Dic₅₃	C₂³	1	4	yes	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
40	C₂³×D₅₃	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
44	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×D₅₃	C₄²	1	2	yes	—
46	C₂×C₄×D₅₃	C₄²	1	4	yes	—
47	C₂×C₄×D₅₃	C₂×C₈	1	8	yes	—
48	C₂×C₄×D₅₃	M₁₆	1	16	yes	—
49	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
50	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
51	C₂×#88324	—	2	16	no	—
52	C₂×#88324	C₂²×C₄	1	16	yes	—
53	D₅₃⋊C₈	—	2	32	no	—
54	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	complemented
55	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
56	C₄²×C₅₃	C₄²	1	2	yes	—
57	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
58	C₄×Dic₅₃	C₄²	1	4	yes	—
59	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
61	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
62	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
63	C₂×#88322	C₂×C₈	1	8	yes	complemented
64	C₂×#88322	C₂²×C₄	1	4	yes	—
65	C₈×D₅₃	C₂×C₈	1	32	yes	complemented
66	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
67	C₂²×D₅₃	C₄×C₈	1	4	yes	—
68	C₂²×D₅₃	#97	1	8	yes	—
69	C₂²×D₅₃	C₂×C₄²	1	2	yes	—
70	C₂×C₄×C₅₃	#97	1	8	yes	—
71	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
73	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
74	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
76	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
77	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
78	C₂×Dic₅₃	C₂×M₁₆	1	8	yes	—
79	C₂×Dic₅₃	C₂³×C₄	1	2	yes	—
80	C₄×D₅₃	C₄×C₈	1	8	yes	—
81	C₄×D₅₃	#97	1	16	yes	—
82	C₄×D₅₃	C₂²×C₈	1	8	yes	—
83	C₄×D₅₃	C₂×M₁₆	1	16	yes	—
84	#88324	—	2	16	no	—
85	C₈×C₅₃	C₂²×C₈	1	8	yes	complemented
86	#88322	C₂²×C₈	1	8	yes	complemented
87	C₂²×#322	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
89	C₂²×C₅₃	C₂×#97	1	4	yes	—
90	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
91	C₂×D₅₃	#322	1	16	yes	complemented
92	C₂×D₅₃	C₂×C₄×C₈	1	4	yes	—
93	C₂×D₅₃	C₂×#97	1	8	yes	—
94	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
95	C₄×C₅₃	C₂×#97	1	8	yes	—
96	C₄×C₅₃	C₂³×C₈	1	2	yes	—
97	Dic₅₃	C₂×C₄×C₈	1	4	yes	—
98	Dic₅₃	C₂×#97	1	4	yes	—
99	Dic₅₃	C₂²×M₁₆	1	4	yes	—
100	C₂²×C₄×C₈	—	53	2	no	—
101	C₂²×C₄×C₈	—	53	1	no	—
102	C₂×#322	—	53	4	no	—
103	C₂×#322	—	53	8	no	—
104	C₂×C₅₃	C₂×#322	1	4	yes	complemented
105	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
106	C₂×C₅₃	C₂²×#97	1	2	yes	—
107	D₅₃	C₂×#322	1	8	yes	complemented
108	C₂³×C₈	—	53	2	no	—
109	C₂³×C₈	—	53	4	no	—
110	C₂²×C₄²	—	53	1	no	—
111	C₂×C₄×C₈	—	53	4	no	—
112	C₂×C₄×C₈	—	53	4	no	—
113	C₂×C₄×C₈	—	53	8	no	—
114	C₂×C₄×C₈	—	53	8	no	—
115	C₂×C₄×C₈	—	53	2	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	—	53	4	no	—
118	C₂×C₄×C₈	—	53	4	no	—
119	C₂×C₄×C₈	#6157	1	1	yes	characteristic
120	#322	—	53	16	no	—
121	C₅₃	C₂²×#322	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₄	—	53	2	no	—
123	C₂³×C₄	—	53	1	no	—
124	C₂²×C₈	—	53	2	no	—
125	C₂²×C₈	—	53	8	no	—
126	C₂²×C₈	—	53	16	no	—
127	C₂²×C₈	—	53	8	no	—
128	C₂²×C₈	—	53	16	no	—
129	C₂²×C₈	—	106	16	no	—
130	C₂²×C₈	C₂×#6157	1	2	yes	—
131	C₂×C₄²	—	53	2	no	—
132	C₂×C₄²	—	53	1	no	—
133	C₂×C₄²	—	53	4	no	—
134	C₂×C₄²	—	53	2	no	—
135	C₂×C₄²	—	53	2	no	—
136	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
137	C₄×C₈	—	53	8	no	—
138	C₄×C₈	—	53	16	no	—
139	C₄×C₈	—	53	8	no	—
140	C₄×C₈	—	53	4	no	—
141	C₄×C₈	—	53	4	no	—
142	C₄×C₈	—	53	4	no	—
143	C₄×C₈	C₂×#6157	1	4	yes	—
144	C₂⁴	—	53	1	no	—
145	C₂²×C₄	—	53	1	no	—
146	C₂²×C₄	—	53	2	no	—
147	C₂²×C₄	—	53	16	no	—
148	C₂²×C₄	—	53	2	no	—
149	C₂²×C₄	—	53	4	no	—
150	C₂²×C₄	—	53	4	no	—
151	C₂²×C₄	—	53	8	no	—
152	C₂²×C₄	—	53	2	no	—
153	C₂²×C₄	C₄×#6157	1	2	yes	—
154	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
155	C₂×C₈	—	53	8	no	—
156	C₂×C₈	—	53	4	no	—
157	C₂×C₈	—	53	32	no	—
158	C₂×C₈	—	53	16	no	—
159	C₂×C₈	—	106	16	no	—
160	C₂×C₈	—	106	32	no	—
161	C₂×C₈	D₅₃⋊C₈	1	8	yes	complemented
162	C₂×C₈	C₂²×#6157	1	4	yes	—
163	C₄²	—	53	2	no	—
164	C₄²	—	53	4	no	—
165	C₄²	—	53	4	no	—
166	C₄²	—	53	1	no	—
167	C₄²	—	53	1	no	—
168	C₄²	C₄×#6157	1	2	yes	—
169	C₄²	C₂²×#6157	1	2	yes	—
170	C₂³	—	53	4	no	—
171	C₂³	—	53	8	no	—
172	C₂³	—	53	2	no	—
173	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
174	C₂×C₄	—	53	4	no	—
175	C₂×C₄	—	53	8	no	—
176	C₂×C₄	—	53	2	no	—
177	C₂×C₄	—	53	2	no	—
178	C₂×C₄	—	53	2	no	—
179	C₂×C₄	—	53	8	no	—
180	C₂×C₄	—	53	16	no	—
181	C₂×C₄	—	53	16	no	—
182	C₂×C₄	—	53	8	no	—
183	C₂×C₄	#13133271	1	8	yes	—
184	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×#6157	1	4	yes	—
186	C₂×C₄	C₂×D₅₃⋊C₈	1	4	yes	—
187	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
188	C₈	—	53	8	no	—
189	C₈	—	106	16	no	—
190	C₈	C₂×D₅₃⋊C₈	1	8	yes	complemented
191	C₂²	—	53	16	no	—
192	C₂²	—	53	4	no	—
193	C₂²	—	53	8	no	—
194	C₂²	C₄×D₅₃⋊C₈	1	2	yes	—
195	C₂²	C₂×#13133271	1	4	yes	—
196	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
197	C₄	—	53	4	no	—
198	C₄	—	53	4	no	—
199	C₄	—	53	4	no	—
200	C₄	C₄×D₅₃⋊C₈	1	2	yes	—
201	C₄	C₂×#13133271	1	8	yes	—
202	C₄	C₂²×D₅₃⋊C₈	1	2	yes	—
203	C₂	—	53	8	no	—
204	C₂	#20289909	1	4	yes	direct product factor
205	C₂	C₂×C₄×D₅₃⋊C₈	1	1	yes	characteristic
206	C₂	C₂²×#13133271	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂³×D₅₃	C₄²
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₂²×#322
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20289909	C₂	direct (N × H)
3. C₂×#16811464	C₂	semidirect
10. #16811464	C₂²	semidirect
27. C₂×C₈×D₅₃	C₈	semidirect
54. C₂×C₈×C₅₃	C₂×C₈	semidirect
63. C₂×#88322	C₂×C₈	semidirect
65. C₈×D₅₃	C₂×C₈	semidirect
85. C₈×C₅₃	C₂²×C₈	semidirect
86. #88322	C₂²×C₈	semidirect
91. C₂×D₅₃	#322	semidirect
104. C₂×C₅₃	C₂×#322	semidirect
107. D₅₃	C₂×#322	semidirect
121. C₅₃	C₂²×#322	semidirect
161. C₂×C₈	D₅₃⋊C₈	semidirect
190. C₈	C₂×D₅₃⋊C₈	semidirect
204. C₂	#20289909	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅₃⋊C₈	2
#20289909	2
C₂×#16811464	2
C₂×C₄×C₈×D₅₃	2
C₂²×#322	53

Subgroups — C2×#20289909 #28313737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289909 #28313737