Subgroups — C₅₃×#12674 #27220186

Order 13568.

358 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅₃×#367	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₄×#107×C₅₃	C₂	1	2	yes	maximal
5	C₅₃×#1380	C₂	1	2	yes	maximal
6	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#107×C₅₃	C₂²	1	2	yes	—
9	C₂×#107×C₅₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊C₄×C₅₃	—	2	2	no	—
12	C₅₃×#367	C₄	1	4	yes	complemented
13	C₅₃×#366	C₄	1	4	yes	complemented
14	C₅₃×#346	C₄	1	4	yes	complemented
15	C₅₃×#340	—	2	2	no	—
16	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₂×C₄²×C₅₃	—	4	2	no	—
21	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₅₃	—	2	1	no	—
23	C₂×C₁₆×C₅₃	—	2	1	no	—
24	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
28	#107×C₅₃	—	2	2	no	—
29	#107×C₅₃	—	2	2	no	—
30	#107×C₅₃	C₂×C₄	1	4	yes	—
31	#107×C₅₃	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₅₃	Q₈	1	2	yes	—
34	#95×C₅₃	—	4	2	no	—
35	C₂²×C₄×C₅₃	—	4	2	no	—
36	C₂²×C₄×C₅₃	—	4	2	no	—
37	C₂²×C₄×C₅₃	D₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	2	1	no	—
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
47	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
48	C₄⋊C₄×C₅₃	—	2	4	no	—
49	C₄⋊C₄×C₅₃	—	2	4	no	—
50	C₄⋊C₄×C₅₃	—	4	2	no	—
51	C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₄²×C₅₃	—	8	4	no	—
53	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₄²×C₅₃	Q₁₆	1	2	yes	—
55	C₁₆×C₅₃	—	2	2	no	—
56	C₁₆×C₅₃	C₄²	1	4	yes	complemented
57	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	4	4	no	—
61	C₂×C₄×C₅₃	—	4	4	no	—
62	C₂×C₄×C₅₃	—	8	2	no	—
63	C₂×C₄×C₅₃	—	8	2	no	—
64	C₂×C₄×C₅₃	—	8	2	no	—
65	C₂×C₄×C₅₃	—	8	2	no	—
66	C₂×C₄×C₅₃	#107	1	4	yes	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
73	C₈×C₅₃	—	2	2	no	—
74	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
75	C₈×C₅₃	#117	1	1	yes	characteristic
76	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
77	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
78	#12674	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
80	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
81	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
82	C₂²×C₅₃	#509	1	2	yes	—
83	C₂²×C₅₃	#510	1	2	yes	—
84	C₄×C₅₃	—	2	4	no	—
85	C₄×C₅₃	—	8	4	no	—
86	C₄×C₅₃	—	8	4	no	—
87	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₅₃	#427	1	1	yes	characteristic
89	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
90	C₄×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
91	C₂×#367	C₂×C₅₃	1	1	yes	characteristic
92	C₂×#366	C₂×C₅₃	1	1	yes	characteristic
93	C₂×#346	C₂×C₅₃	1	1	yes	characteristic
94	C₄×#107	C₂×C₅₃	1	2	yes	—
95	#1380	C₂×C₅₃	1	2	yes	—
96	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
97	C₂×C₅₃	#2159	1	2	yes	—
98	C₂×C₅₃	#2175	1	2	yes	—
99	C₂×C₅₃	#2176	1	1	yes	characteristic
100	C₂×C₅₃	#2177	1	1	yes	characteristic
101	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
102	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
103	C₂×#107	C₂²×C₅₃	1	2	yes	—
104	C₂×#107	C₂²×C₅₃	1	2	yes	—
105	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	2	2	no	—
107	#367	C₄×C₅₃	1	4	yes	complemented
108	#366	C₄×C₅₃	1	4	yes	complemented
109	#346	C₄×C₅₃	1	4	yes	complemented
110	#340	—	2	2	no	—
111	C₅₃	#12674	1	1	yes	characteristic, direct product factor, Sylow
112	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
113	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄²	—	4	2	no	—
117	C₂×C₄²	D₄×C₅₃	1	1	yes	characteristic
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
121	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
122	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
123	C₂×C₁₆	C₂×C₄×C₅₃	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	C₂×C₄×C₅₃	1	4	yes	—
127	#107	C₂×C₄×C₅₃	1	4	yes	—
128	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
129	C₄×C₈	Q₈×C₅₃	1	2	yes	—
130	#95	—	4	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	D₈×C₅₃	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	C₄⋊C₄×C₅₃	1	2	yes	—
138	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
140	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
141	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
142	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
143	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄²	—	8	4	no	—
149	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
150	C₄²	Q₁₆×C₅₃	1	2	yes	—
151	C₁₆	—	2	2	no	—
152	C₁₆	C₄²×C₅₃	1	4	yes	complemented
153	C₂³	C₂×D₈×C₅₃	1	1	yes	characteristic
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	4	4	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	8	2	no	—
160	C₂×C₄	—	8	2	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	#107×C₅₃	1	4	yes	—
163	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
164	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
165	C₂×C₄	C₄×D₄×C₅₃	1	1	yes	characteristic
166	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
167	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
168	C₂×C₄	#135×C₅₃	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₂×C₄²×C₅₃	1	1	yes	characteristic
171	C₈	#117×C₅₃	1	1	yes	characteristic
172	C₈	C₄×Q₈×C₅₃	1	1	yes	characteristic
173	C₈	C₄×Q₈×C₅₃	1	1	yes	characteristic
174	C₂²	C₂×#107×C₅₃	1	1	yes	characteristic
175	C₂²	C₄×D₈×C₅₃	1	1	yes	characteristic
176	C₂²	C₄×D₈×C₅₃	1	1	yes	characteristic
177	C₂²	C₅₃×#509	1	2	yes	—
178	C₂²	C₅₃×#510	1	2	yes	—
179	C₄	—	2	4	no	—
180	C₄	—	8	4	no	—
181	C₄	—	8	4	no	—
182	C₄	C₄×C₄⋊C₄×C₅₃	1	1	yes	characteristic
183	C₄	C₅₃×#427	1	1	yes	characteristic
184	C₄	C₄×Q₁₆×C₅₃	1	1	yes	characteristic
185	C₄	C₄×Q₁₆×C₅₃	1	1	yes	characteristic
186	C₂	C₄×#107×C₅₃	1	1	yes	characteristic
187	C₂	C₅₃×#2159	1	2	yes	—
188	C₂	C₅₃×#2175	1	2	yes	—
189	C₂	C₅₃×#2176	1	1	yes	characteristic
190	C₂	C₅₃×#2177	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂×D₈
commutator	C₈	C₂×C₄²×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ C₂×C₄×C₈×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₅₃×#367	C₄	semidirect
13. C₅₃×#366	C₄	semidirect
14. C₅₃×#346	C₄	semidirect
56. C₁₆×C₅₃	C₄²	semidirect
78. #12674	C₅₃	direct (N × H)
107. #367	C₄×C₅₃	semidirect
108. #366	C₄×C₅₃	semidirect
109. #346	C₄×C₅₃	semidirect
111. C₅₃	#12674	direct (N × H)
152. C₁₆	C₄²×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅₃×#367	2
C₂×C₅₃×#366	2
C₂×C₅₃×#346	2
C₄×#107×C₅₃	2
C₅₃×#1380	2
#12674	53

Subgroups — C53×#12674 #27220186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12674 #27220186