Subgroups — C₅₃×#2356 #20271293

Order 6784.

464 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#391	C₂	1	2	yes	maximal
3	C₅₃×#387	C₂	1	4	yes	maximal
4	C₅₃×#383	C₂	1	2	yes	maximal
5	C₅₃×#382	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊C₄×C₅₃	C₂	1	2	yes	maximal
8	C₅₃×#376	C₂	1	2	yes	maximal
9	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₅₃	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	4	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	#95×C₅₃	C₂²	1	1	yes	characteristic
20	#95×C₅₃	C₂²	1	2	yes	—
21	#95×C₅₃	C₂²	1	4	yes	—
22	#95×C₅₃	C₂²	1	1	yes	characteristic
23	#95×C₅₃	C₂²	1	4	yes	—
24	#95×C₅₃	C₂²	1	2	yes	—
25	#95×C₅₃	C₂²	1	4	yes	—
26	C₂×Q₈×C₅₃	—	2	2	no	—
27	C₂×Q₈×C₅₃	—	2	2	no	—
28	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₅₃	—	2	4	no	—
38	C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
40	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
41	C₄²×C₅₃	—	2	8	no	—
42	C₄²×C₅₃	—	2	4	no	—
43	C₄²×C₅₃	C₂×C₄	1	8	yes	—
44	C₄²×C₅₃	C₂×C₄	1	4	yes	—
45	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
46	Q₈×C₅₃	—	2	8	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	—	2	2	no	—
49	C₂×C₄×C₅₃	—	2	4	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	2	2	no	—
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	—	2	4	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	4	no	—
60	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₅₃	#41	1	4	yes	—
66	C₂×C₄×C₅₃	#41	1	2	yes	—
67	C₂×C₄×C₅₃	#41	1	2	yes	—
68	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
69	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
70	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
71	C₂²×C₅₃	C₂×#41	1	2	yes	—
72	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
73	C₄×C₅₃	—	2	4	no	—
74	C₄×C₅₃	—	2	4	no	—
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	2	4	no	—
78	C₄×C₅₃	—	4	2	no	—
79	C₄×C₅₃	—	4	4	no	—
80	C₄×C₅₃	#117	1	4	yes	—
81	#2356	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
83	C₂×C₅₃	C₄×#41	1	1	yes	characteristic
84	C₂×C₅₃	#519	1	1	yes	characteristic
85	C₂×C₅₃	#552	1	2	yes	—
86	C₂×C₅₃	#556	1	2	yes	—
87	C₂×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic
88	#391	C₂×C₅₃	1	2	yes	—
89	#387	C₂×C₅₃	1	4	yes	—
90	#383	C₂×C₅₃	1	2	yes	—
91	#382	C₂×C₅₃	1	1	yes	characteristic
92	C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
93	C₄×C₄⋊C₄	C₂×C₅₃	1	2	yes	—
94	#376	C₂×C₅₃	1	2	yes	—
95	C₅₃	#2356	1	1	yes	characteristic, direct product factor, Sylow
96	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
97	C₄×Q₈	C₄×C₅₃	1	8	yes	complemented
98	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
99	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
100	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
101	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
102	C₂×C₄²	C₂²×C₅₃	1	4	yes	—
103	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
104	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
105	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
106	#95	C₂²×C₅₃	1	1	yes	characteristic
107	#95	C₂²×C₅₃	1	2	yes	—
108	#95	C₂²×C₅₃	1	4	yes	—
109	#95	C₂²×C₅₃	1	1	yes	characteristic
110	#95	C₂²×C₅₃	1	4	yes	—
111	#95	C₂²×C₅₃	1	2	yes	—
112	#95	C₂²×C₅₃	1	4	yes	—
113	C₂×Q₈	—	2	2	no	—
114	C₂×Q₈	—	2	2	no	—
115	C₂×Q₈	C₂×C₄×C₅₃	1	4	yes	—
116	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
117	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
118	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
119	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
120	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
121	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
122	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
123	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
124	C₄⋊C₄	—	2	4	no	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	C₂×C₄×C₅₃	1	8	yes	—
127	C₄⋊C₄	C₂×C₄×C₅₃	1	4	yes	—
128	C₄²	—	2	8	no	—
129	C₄²	—	2	4	no	—
130	C₄²	C₂×C₄×C₅₃	1	8	yes	—
131	C₄²	C₂×C₄×C₅₃	1	4	yes	—
132	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
133	Q₈	—	2	8	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
148	C₂×C₄	C₂²×C₄×C₅₃	1	4	yes	—
149	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
150	C₂×C₄	C₂²×C₄×C₅₃	1	4	yes	—
151	C₂×C₄	C₂²×C₄×C₅₃	1	2	yes	—
152	C₂×C₄	#41×C₅₃	1	4	yes	—
153	C₂×C₄	#41×C₅₃	1	2	yes	—
154	C₂×C₄	#41×C₅₃	1	2	yes	—
155	C₂²	C₂³×C₄×C₅₃	1	1	yes	characteristic
156	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
157	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
158	C₂²	C₂×#41×C₅₃	1	2	yes	—
159	C₂²	2₋¹⁺⁴×C₅₃	1	2	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	2	4	no	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	4	2	no	—
166	C₄	—	4	4	no	—
167	C₄	#117×C₅₃	1	4	yes	—
168	C₂	C₂×#117×C₅₃	1	1	yes	characteristic
169	C₂	C₄×#41×C₅₃	1	1	yes	characteristic
170	C₂	C₅₃×#519	1	1	yes	characteristic
171	C₂	C₅₃×#552	1	2	yes	—
172	C₂	C₅₃×#556	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	C₂⁴
commutator	C₂²	C₂³×C₄×C₅₃
Frattini, cosocle	C₂³	C₂⁴×C₅₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₄×Q₈×C₅₃	C₄	semidirect
81. #2356	C₅₃	direct (N × H)
95. C₅₃	#2356	direct (N × H)
97. C₄×Q₈	C₄×C₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₅₃	2
C₅₃×#391	2
C₅₃×#387	2
C₅₃×#383	2
C₅₃×#382	2
C₄×C₄⋊C₄×C₅₃	2
C₅₃×#376	2
#2356	53

Subgroups — C53×#2356 #20271293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2356 #20271293