Subgroups — C₈×C₄⋊C₈×C₅₃ #27215771

Order 13568.

550 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₈×C₅₃	C₂	1	2	yes	maximal
2	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈²×C₅₃	C₂	1	2	yes	maximal
4	C₅₃×#1273	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₅₃	C₄	1	4	yes	—
6	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	C₄	1	4	yes	—
13	C₈²×C₅₃	—	2	4	no	—
14	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₈	1	16	yes	direct product factor
23	C₄×C₈×C₅₃	—	2	4	no	—
24	C₄×C₈×C₅₃	—	2	4	no	—
25	C₄×C₈×C₅₃	—	2	2	no	—
26	C₄×C₈×C₅₃	—	2	4	no	—
27	C₄×C₈×C₅₃	C₈	1	8	yes	complemented
28	C₄×C₈×C₅₃	D₄	1	2	yes	—
29	C₄×C₈×C₅₃	Q₈	1	2	yes	—
30	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
31	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	—	2	8	no	—
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	8	no	—
39	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
41	C₂×C₈×C₅₃	C₂×C₈	1	16	yes	—
42	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	—	2	4	no	—
46	C₄²×C₅₃	C₂×C₈	1	4	yes	—
47	C₄²×C₅₃	C₂×C₈	1	8	yes	—
48	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
50	C₄²×C₅₃	#41	1	2	yes	—
51	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	2	no	—
55	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
56	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
57	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
58	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
65	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
66	C₂×C₄×C₅₃	#131	1	4	yes	—
67	C₈×C₅₃	—	2	4	no	—
68	C₈×C₅₃	—	2	16	no	—
69	C₈×C₅₃	C₄⋊C₈	1	8	yes	direct product factor
70	C₈×C₄⋊C₈	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
73	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
74	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
75	C₂²×C₅₃	#405	1	2	yes	—
76	C₄×C₅₃	—	2	4	no	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	C₈²	1	4	yes	complemented
79	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
80	C₄×C₅₃	#431	1	2	yes	—
81	C₄×C₅₃	C₈×D₄	1	4	yes	—
82	C₄×C₅₃	C₈×Q₈	1	4	yes	—
83	C₄×C₄⋊C₈	C₂×C₅₃	1	2	yes	—
84	C₄²×C₈	C₂×C₅₃	1	1	yes	characteristic
85	C₂×C₈²	C₂×C₅₃	1	2	yes	—
86	#1273	C₂×C₅₃	1	2	yes	—
87	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
88	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
89	C₂×C₅₃	#1450	1	1	yes	characteristic
90	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
91	C₂×C₅₃	C₈×C₄⋊C₄	1	2	yes	—
92	C₂×C₄⋊C₈	C₄×C₅₃	1	4	yes	—
93	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
94	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
95	C₂×C₄×C₈	C₂²×C₅₃	1	4	yes	—
96	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
98	C₄³	C₂²×C₅₃	1	1	yes	characteristic
99	#336	C₄×C₅₃	1	4	yes	—
100	C₈²	—	2	4	no	—
101	C₅₃	C₈×C₄⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
102	C₂²×C₈	C₂×C₄×C₅₃	1	8	yes	—
103	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
104	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
105	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
106	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
107	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
108	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
109	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
110	C₄⋊C₈	C₈×C₅₃	1	16	yes	direct product factor
111	C₄×C₈	—	2	4	no	—
112	C₄×C₈	—	2	4	no	—
113	C₄×C₈	—	2	2	no	—
114	C₄×C₈	—	2	4	no	—
115	C₄×C₈	C₈×C₅₃	1	8	yes	complemented
116	C₄×C₈	D₄×C₅₃	1	2	yes	—
117	C₄×C₈	Q₈×C₅₃	1	2	yes	—
118	C₂²×C₄	C₄²×C₅₃	1	2	yes	—
119	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
120	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
121	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
122	C₂²×C₄	C₂²×C₄×C₅₃	1	2	yes	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	8	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	C₄⋊C₄×C₅₃	1	4	yes	—
128	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
129	C₂×C₈	C₂×C₈×C₅₃	1	16	yes	—
130	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
131	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	4	no	—
134	C₄²	C₂×C₈×C₅₃	1	4	yes	—
135	C₄²	C₂×C₈×C₅₃	1	8	yes	—
136	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
137	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
138	C₄²	#41×C₅₃	1	2	yes	—
139	C₂³	C₂×C₄²×C₅₃	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	C₄×C₈×C₅₃	1	4	yes	—
144	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
145	C₂×C₄	C₄×C₈×C₅₃	1	4	yes	—
146	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
147	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
148	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
149	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
150	C₂×C₄	C₄×Q₈×C₅₃	1	2	yes	—
151	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
152	C₂×C₄	C₂²×C₈×C₅₃	1	4	yes	—
153	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
154	C₂×C₄	#131×C₅₃	1	4	yes	—
155	C₈	—	2	4	no	—
156	C₈	—	2	16	no	—
157	C₈	C₄⋊C₈×C₅₃	1	8	yes	direct product factor
158	C₂²	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
159	C₂²	C₂×C₄×C₈×C₅₃	1	2	yes	—
160	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
161	C₂²	C₄×M₁₆×C₅₃	1	1	yes	characteristic
162	C₂²	C₅₃×#405	1	2	yes	—
163	C₄	—	2	4	no	—
164	C₄	—	2	2	no	—
165	C₄	C₈²×C₅₃	1	4	yes	complemented
166	C₄	C₂×C₄⋊C₈×C₅₃	1	2	yes	—
167	C₄	C₅₃×#431	1	2	yes	—
168	C₄	C₈×D₄×C₅₃	1	4	yes	—
169	C₄	C₈×Q₈×C₅₃	1	4	yes	—
170	C₂	C₂×C₈²×C₅₃	1	1	yes	characteristic
171	C₂	C₈×M₁₆×C₅₃	1	2	yes	—
172	C₂	C₅₃×#1450	1	1	yes	characteristic
173	C₂	C₄×C₄⋊C₈×C₅₃	1	1	yes	characteristic
174	C₂	C₈×C₄⋊C₄×C₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈×C₅₃	C₂²
commutator	C₂	C₂×C₈²×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂×C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
22. C₄⋊C₈×C₅₃	C₈	direct (N × H)
27. C₄×C₈×C₅₃	C₈	semidirect
69. C₈×C₅₃	C₄⋊C₈	direct (N × H)
70. C₈×C₄⋊C₈	C₅₃	direct (N × H)
78. C₄×C₅₃	C₈²	semidirect
101. C₅₃	C₈×C₄⋊C₈	direct (N × H)
110. C₄⋊C₈	C₈×C₅₃	direct (N × H)
115. C₄×C₈	C₈×C₅₃	semidirect
157. C₈	C₄⋊C₈×C₅₃	direct (N × H)
165. C₄	C₈²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₈×C₅₃	2
C₄²×C₈×C₅₃	2
C₂×C₈²×C₅₃	2
C₅₃×#1273	2
C₈×C₄⋊C₈	53

Subgroups — C8×C4⋊C8×C53 #27215771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₄⋊C₈×C₅₃ #27215771