Subgroups — C₂²×C₄×D₅₃⋊C₈ #28329168

Order 13568.

4638 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₅₃⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×D₅₃⋊C₈	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄×#88324	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄²×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×D₅₃⋊C₈	C₄	1	32	yes	direct product factor
6	C₂²×D₅₃⋊C₈	C₂²	1	24	yes	—
7	C₂³×#88324	C₂²	1	4	yes	—
8	C₄×D₅₃⋊C₈	C₂²	1	64	yes	direct product factor
9	C₂×C₄×#88324	C₂²	1	48	yes	complemented
10	C₂³×C₄×D₅₃	C₄	1	2	yes	—
11	C₂³×C₄×D₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
14	C₂×C₄²×D₅₃	C₄	1	12	yes	—
15	C₂×C₄²×D₅₃	C₂²	1	12	yes	—
16	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
17	C₂×D₅₃⋊C₈	C₂×C₄	1	192	yes	direct product factor
18	C₂×D₅₃⋊C₈	C₂³	1	32	yes	—
19	C₂²×#88324	C₂×C₄	1	32	yes	complemented
20	C₂²×#88324	C₂³	1	24	yes	—
21	C₄×#88324	C₂³	1	64	yes	complemented
22	C₂⁴×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×Dic₅₃	C₂×C₄	1	2	yes	—
24	C₂³×Dic₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×D₅₃	C₈	1	32	yes	complemented
26	C₂²×C₄×D₅₃	C₂×C₄	1	6	yes	—
27	C₂²×C₄×D₅₃	C₂×C₄	1	16	yes	—
28	C₂²×C₄×D₅₃	C₂×C₄	1	24	yes	—
29	C₂²×C₄×D₅₃	C₂³	1	6	yes	—
30	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
31	C₂×C₄×Dic₅₃	C₂×C₄	1	12	yes	—
32	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
33	C₄²×D₅₃	C₂×C₄	1	48	yes	—
34	C₄²×D₅₃	C₂³	1	16	yes	—
35	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
38	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂³×D₅₃	C₄²	1	8	yes	—
40	C₂³×D₅₃	C₂×C₈	1	16	yes	—
41	C₂³×D₅₃	C₂²×C₄	1	6	yes	—
42	C₂²×C₄×C₅₃	C₄²	1	8	yes	—
43	C₂²×C₄×C₅₃	C₂×C₈	1	16	yes	complemented
44	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
46	C₂²×Dic₅₃	C₂×C₈	1	16	yes	complemented
47	C₂²×Dic₅₃	C₂²×C₄	1	3	yes	—
48	C₂²×Dic₅₃	C₂²×C₄	1	8	yes	—
49	C₂²×Dic₅₃	C₂²×C₄	1	12	yes	—
50	C₂²×Dic₅₃	C₂⁴	1	3	yes	—
51	C₂×C₄×D₅₃	C₄²	1	48	yes	—
52	C₂×C₄×D₅₃	C₂×C₈	1	192	yes	complemented
53	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
54	C₂×C₄×D₅₃	C₂²×C₄	1	48	yes	—
55	C₂×C₄×D₅₃	C₂²×C₄	1	32	yes	—
56	C₂×C₄×D₅₃	C₂⁴	1	4	yes	—
57	C₂×#88324	C₂²×C₄	1	96	yes	complemented
58	C₂×#88324	C₂⁴	1	16	yes	—
59	D₅₃⋊C₈	C₂²×C₄	1	128	yes	direct product factor
60	C₄²×C₅₃	C₂²×C₄	1	16	yes	—
61	C₄×Dic₅₃	C₂²×C₄	1	24	yes	—
62	C₄×Dic₅₃	C₂²×C₄	1	16	yes	—
63	C₄×Dic₅₃	C₂⁴	1	8	yes	—
64	C₂³×C₅₃	C₂×C₄²	1	4	yes	—
65	C₂³×C₅₃	C₂²×C₈	1	8	yes	—
66	C₂³×C₅₃	C₂³×C₄	1	3	yes	—
67	C₂²×D₅₃	C₄×C₈	1	64	yes	complemented
68	C₂²×D₅₃	C₂×C₄²	1	24	yes	—
69	C₂²×D₅₃	C₂²×C₈	1	48	yes	—
70	C₂²×D₅₃	C₂³×C₄	1	4	yes	—
71	C₂×C₄×C₅₃	C₂×C₄²	1	24	yes	—
72	C₂×C₄×C₅₃	C₂²×C₈	1	48	yes	complemented
73	C₂×C₄×C₅₃	C₂³×C₄	1	8	yes	—
74	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
75	C₂×Dic₅₃	C₂×C₄²	1	12	yes	—
76	C₂×Dic₅₃	C₂²×C₈	1	48	yes	complemented
77	C₂×Dic₅₃	C₂³×C₄	1	8	yes	—
78	C₂×Dic₅₃	C₂³×C₄	1	12	yes	—
79	C₂×Dic₅₃	C₂³×C₄	1	3	yes	—
80	C₂×Dic₅₃	C₂⁵	1	1	yes	characteristic
81	C₄×D₅₃	C₂×C₄²	1	48	yes	—
82	C₄×D₅₃	C₂²×C₈	1	128	yes	complemented
83	C₄×D₅₃	C₂³×C₄	1	16	yes	—
84	#88324	C₂³×C₄	1	32	yes	complemented
85	C₂³×C₄×C₈	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	C₂×C₄×C₈	1	16	yes	complemented
87	C₂²×C₅₃	C₂²×C₄²	1	6	yes	—
88	C₂²×C₅₃	C₂³×C₈	1	12	yes	—
89	C₂²×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
90	C₂×D₅₃	C₂×C₄×C₈	1	96	yes	complemented
91	C₂×D₅₃	C₂²×C₄²	1	8	yes	—
92	C₂×D₅₃	C₂³×C₈	1	16	yes	—
93	C₄×C₅₃	C₂²×C₄²	1	8	yes	—
94	C₄×C₅₃	C₂³×C₈	1	16	yes	complemented
95	Dic₅₃	C₂²×C₄²	1	6	yes	—
96	Dic₅₃	C₂³×C₈	1	16	yes	complemented
97	Dic₅₃	C₂⁴×C₄	1	2	yes	—
98	C₂⁴×C₈	—	53	2	no	—
99	C₂³×C₄²	—	53	1	no	—
100	C₂²×C₄×C₈	—	53	4	no	—
101	C₂²×C₄×C₈	—	53	24	no	—
102	C₂×C₅₃	C₂²×C₄×C₈	1	12	yes	complemented
103	C₂×C₅₃	C₂³×C₄²	1	1	yes	characteristic
104	C₂×C₅₃	C₂⁴×C₈	1	2	yes	—
105	D₅₃	C₂²×C₄×C₈	1	16	yes	complemented
106	C₂⁴×C₄	—	53	2	no	—
107	C₂⁴×C₄	—	53	1	no	—
108	C₂³×C₈	—	53	4	no	—
109	C₂³×C₈	—	53	24	no	—
110	C₂³×C₈	—	53	32	no	—
111	C₂²×C₄²	—	53	2	no	—
112	C₂²×C₄²	—	53	1	no	—
113	C₂²×C₄²	—	53	12	no	—
114	C₂²×C₄²	—	53	12	no	—
115	C₂²×C₄²	#6157	1	1	yes	characteristic
116	C₂×C₄×C₈	—	53	48	no	—
117	C₂×C₄×C₈	—	53	64	no	—
118	C₅₃	C₂³×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
119	C₂⁵	—	53	1	no	—
120	C₂³×C₄	—	53	1	no	—
121	C₂³×C₄	—	53	2	no	—
122	C₂³×C₄	—	53	16	no	—
123	C₂³×C₄	—	53	6	no	—
124	C₂³×C₄	—	53	6	no	—
125	C₂³×C₄	—	53	24	no	—
126	C₂³×C₄	—	53	32	no	—
127	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
128	C₂³×C₄	C₂×#6157	1	2	yes	—
129	C₂²×C₈	—	53	24	no	—
130	C₂²×C₈	—	53	32	no	—
131	C₂²×C₈	—	53	32	no	—
132	C₂²×C₈	—	53	192	no	—
133	C₂×C₄²	—	53	12	no	—
134	C₂×C₄²	—	53	12	no	—
135	C₂×C₄²	—	53	12	no	—
136	C₂×C₄²	—	53	48	no	—
137	C₂×C₄²	—	53	16	no	—
138	C₂×C₄²	C₂×#6157	1	12	yes	—
139	C₄×C₈	—	53	64	no	—
140	C₂⁴	—	53	8	no	—
141	C₂⁴	—	53	16	no	—
142	C₂⁴	—	53	6	no	—
143	C₂⁴	C₂²×#6157	1	1	yes	characteristic
144	C₂²×C₄	—	53	8	no	—
145	C₂²×C₄	—	53	16	no	—
146	C₂²×C₄	—	53	3	no	—
147	C₂²×C₄	—	53	12	no	—
148	C₂²×C₄	—	53	3	no	—
149	C₂²×C₄	—	53	48	no	—
150	C₂²×C₄	—	53	48	no	—
151	C₂²×C₄	—	53	192	no	—
152	C₂²×C₄	—	53	12	no	—
153	C₂²×C₄	—	53	32	no	—
154	C₂²×C₄	—	53	4	no	—
155	C₂²×C₄	D₅₃⋊C₈	1	16	yes	direct product factor
156	C₂²×C₄	C₄×#6157	1	8	yes	—
157	C₂²×C₄	C₂²×#6157	1	6	yes	—
158	C₂²×C₄	C₂²×#6157	1	12	yes	—
159	C₂×C₈	—	53	96	no	—
160	C₂×C₈	—	53	16	no	—
161	C₂×C₈	—	53	128	no	—
162	C₄²	—	53	8	no	—
163	C₄²	—	53	16	no	—
164	C₄²	—	53	24	no	—
165	C₄²	C₂²×#6157	1	16	yes	—
166	C₂³	—	53	64	no	—
167	C₂³	—	53	24	no	—
168	C₂³	—	53	48	no	—
169	C₂³	—	53	4	no	—
170	C₂³	C₂×C₄×#6157	1	4	yes	—
171	C₂³	C₂×D₅₃⋊C₈	1	8	yes	—
172	C₂³	C₂³×#6157	1	3	yes	—
173	C₂×C₄	—	53	12	no	—
174	C₂×C₄	—	53	12	no	—
175	C₂×C₄	—	53	48	no	—
176	C₂×C₄	—	53	3	no	—
177	C₂×C₄	—	53	8	no	—
178	C₂×C₄	—	53	1	no	—
179	C₂×C₄	—	53	48	no	—
180	C₂×C₄	—	53	16	no	—
181	C₂×C₄	—	53	128	no	—
182	C₂×C₄	C₂×C₄×#6157	1	24	yes	—
183	C₂×C₄	C₂×D₅₃⋊C₈	1	48	yes	direct product factor
184	C₂×C₄	C₂³×#6157	1	4	yes	—
185	C₂×C₄	C₂³×#6157	1	8	yes	—
186	C₈	—	53	32	no	—
187	C₂²	—	53	8	no	—
188	C₂²	—	53	16	no	—
189	C₂²	—	53	96	no	—
190	C₂²	C₄×D₅₃⋊C₈	1	16	yes	direct product factor
191	C₂²	C₂²×C₄×#6157	1	6	yes	—
192	C₂²	C₂²×D₅₃⋊C₈	1	12	yes	—
193	C₂²	C₂⁴×#6157	1	1	yes	characteristic
194	C₄	—	53	2	no	—
195	C₄	—	53	6	no	—
196	C₄	—	53	16	no	—
197	C₄	C₂²×C₄×#6157	1	8	yes	—
198	C₄	C₂²×D₅₃⋊C₈	1	16	yes	direct product factor
199	C₂	—	53	16	no	—
200	C₂	C₂×C₄×D₅₃⋊C₈	1	12	yes	direct product factor
201	C₂	C₂³×C₄×#6157	1	1	yes	characteristic
202	C₂	C₂³×D₅₃⋊C₈	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	#6157
commutator, hypocenter	C₅₃	C₂³×C₄×C₈
Frattini	C₂²	C₂⁴×#6157
Fitting	C₂²×C₄²×C₅₃	C₄
socle	C₂⁴×C₅₃	C₂²×C₄
prime radical	C₂⁴×D₅₃	C₂×C₄
cosocle	C₂×Dic₅₃	C₂⁵
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₅₃⋊C₈	C₂	direct (N × H)
3. C₂²×C₄×#88324	C₂	semidirect
5. C₂²×D₅₃⋊C₈	C₄	direct (N × H)
8. C₄×D₅₃⋊C₈	C₂²	direct (N × H)
9. C₂×C₄×#88324	C₂²	semidirect
17. C₂×D₅₃⋊C₈	C₂×C₄	direct (N × H)
19. C₂²×#88324	C₂×C₄	semidirect
21. C₄×#88324	C₂³	semidirect
25. C₂²×C₄×D₅₃	C₈	semidirect
43. C₂²×C₄×C₅₃	C₂×C₈	semidirect
46. C₂²×Dic₅₃	C₂×C₈	semidirect
52. C₂×C₄×D₅₃	C₂×C₈	semidirect
57. C₂×#88324	C₂²×C₄	semidirect
59. D₅₃⋊C₈	C₂²×C₄	direct (N × H)
67. C₂²×D₅₃	C₄×C₈	semidirect
72. C₂×C₄×C₅₃	C₂²×C₈	semidirect
76. C₂×Dic₅₃	C₂²×C₈	semidirect
82. C₄×D₅₃	C₂²×C₈	semidirect
84. #88324	C₂³×C₄	semidirect
86. C₂²×C₅₃	C₂×C₄×C₈	semidirect
90. C₂×D₅₃	C₂×C₄×C₈	semidirect
94. C₄×C₅₃	C₂³×C₈	semidirect
96. Dic₅₃	C₂³×C₈	semidirect
102. C₂×C₅₃	C₂²×C₄×C₈	semidirect
105. D₅₃	C₂²×C₄×C₈	semidirect
118. C₅₃	C₂³×C₄×C₈	semidirect
155. C₂²×C₄	D₅₃⋊C₈	direct (N × H)
183. C₂×C₄	C₂×D₅₃⋊C₈	direct (N × H)
190. C₂²	C₄×D₅₃⋊C₈	direct (N × H)
198. C₄	C₂²×D₅₃⋊C₈	direct (N × H)
200. C₂	C₂×C₄×D₅₃⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₅₃⋊C₈	2
C₂×C₄×D₅₃⋊C₈	2
C₂²×C₄×#88324	2
C₂²×C₄²×D₅₃	2
C₂³×C₄×C₈	53

Subgroups — C22×C4×D53⋊C8 #28329168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×D₅₃⋊C₈ #28329168