Subgroups — C₅₃×#2567 #20271504

Order 6784.

500 conjugacy classes of subgroups.

The subgroup lattice has 164 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	C₂×C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#395	C₂	1	4	yes	maximal
4	C₅₃×#391	C₂	1	2	yes	maximal
5	C₅₃×#389	C₂	1	4	yes	maximal
6	C₅₃×#384	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
10	C₄⋊Q₈×C₅₃	—	2	4	no	—
11	C₄×Q₈×C₅₃	—	2	4	no	—
12	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	8	yes	—
17	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	#95×C₅₃	C₂²	1	2	yes	—
22	#95×C₅₃	C₂²	1	8	yes	—
23	C₂×Q₈×C₅₃	—	2	2	no	—
24	C₂×Q₈×C₅₃	—	2	4	no	—
25	C₂×Q₈×C₅₃	—	2	2	no	—
26	C₂×Q₈×C₅₃	—	4	4	no	—
27	C₂×Q₈×C₅₃	Q₈	1	4	yes	complemented
28	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
29	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
33	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	—	2	4	no	—
36	C₄⋊C₄×C₅₃	—	2	2	no	—
37	C₄⋊C₄×C₅₃	—	2	4	no	—
38	C₄⋊C₄×C₅₃	—	2	4	no	—
39	C₄⋊C₄×C₅₃	—	4	8	no	—
40	C₄⋊C₄×C₅₃	Q₈	1	8	yes	complemented
41	C₄²×C₅₃	—	2	4	no	—
42	C₄²×C₅₃	—	2	4	no	—
43	C₄²×C₅₃	D₄	1	4	yes	—
44	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
45	Q₈×C₅₃	—	2	8	no	—
46	Q₈×C₅₃	—	4	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	4	no	—
52	C₂×C₄×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	—	2	2	no	—
54	C₂×C₄×C₅₃	—	2	8	no	—
55	C₂×C₄×C₅₃	—	2	4	no	—
56	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
60	C₂×C₄×C₅₃	C₂×Q₈	1	8	yes	—
61	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
63	C₂×C₄×C₅₃	#41	1	2	yes	—
64	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
65	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
66	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
67	C₂²×C₅₃	C₂×#41	1	1	yes	characteristic
68	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
69	C₄×C₅₃	—	2	4	no	—
70	C₄×C₅₃	—	2	2	no	—
71	C₄×C₅₃	—	2	8	no	—
72	C₄×C₅₃	—	4	2	no	—
73	C₄×C₅₃	—	4	4	no	—
74	C₄×C₅₃	C₂²⋊Q₈	1	4	yes	—
75	C₄×C₅₃	C₄⋊Q₈	1	4	yes	—
76	#2567	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₅₃	#535	1	1	yes	characteristic
80	C₂×C₅₃	#554	1	2	yes	—
81	C₂×C₅₃	Q₈²	1	2	yes	—
82	C₂×C₄⋊Q₈	C₂×C₅₃	1	1	yes	characteristic
83	C₂×C₄×Q₈	C₂×C₅₃	1	1	yes	characteristic
84	#395	C₂×C₅₃	1	4	yes	—
85	#391	C₂×C₅₃	1	2	yes	—
86	#389	C₂×C₅₃	1	4	yes	—
87	#384	C₂×C₅₃	1	2	yes	—
88	C₄×C₄⋊C₄	C₂×C₅₃	1	1	yes	characteristic
89	C₅₃	#2567	1	1	yes	characteristic, direct product factor, Sylow
90	C₂²×Q₈	C₂²×C₅₃	1	1	yes	characteristic
91	C₂²×Q₈	C₂²×C₅₃	1	2	yes	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₄×Q₈	—	2	4	no	—
94	C₂×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
95	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
96	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
97	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
98	C₂×C₄⋊C₄	C₂²×C₅₃	1	8	yes	—
99	C₂×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
100	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
101	C₂×C₄²	C₂²×C₅₃	1	2	yes	—
102	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
103	#95	C₂²×C₅₃	1	2	yes	—
104	#95	C₂²×C₅₃	1	8	yes	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	2	4	no	—
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	—	4	4	no	—
109	C₂×Q₈	Q₈×C₅₃	1	4	yes	complemented
110	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
111	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
112	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
113	C₂²×C₄	C₂³×C₅₃	1	2	yes	—
114	C₂²×C₄	C₂³×C₅₃	1	4	yes	—
115	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
116	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	4	8	no	—
122	C₄⋊C₄	Q₈×C₅₃	1	8	yes	complemented
123	C₄²	—	2	4	no	—
124	C₄²	—	2	4	no	—
125	C₄²	D₄×C₅₃	1	4	yes	—
126	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
127	Q₈	—	2	8	no	—
128	Q₈	—	4	8	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
139	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
140	C₂×C₄	C₂×D₄×C₅₃	1	2	yes	—
141	C₂×C₄	C₂×Q₈×C₅₃	1	8	yes	—
142	C₂×C₄	C₂×Q₈×C₅₃	1	4	yes	—
143	C₂×C₄	C₂×Q₈×C₅₃	1	2	yes	—
144	C₂×C₄	C₂×Q₈×C₅₃	1	4	yes	—
145	C₂×C₄	#41×C₅₃	1	2	yes	—
146	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
147	C₂²	C₂²×Q₈×C₅₃	1	1	yes	characteristic
148	C₂²	C₂²×Q₈×C₅₃	1	2	yes	—
149	C₂²	C₂×#41×C₅₃	1	1	yes	characteristic
150	C₂²	H(5,𝔽₂)×C₅₃	1	2	yes	—
151	C₄	—	2	4	no	—
152	C₄	—	2	2	no	—
153	C₄	—	2	8	no	—
154	C₄	—	4	2	no	—
155	C₄	—	4	4	no	—
156	C₄	C₂²⋊Q₈×C₅₃	1	4	yes	—
157	C₄	C₄⋊Q₈×C₅₃	1	4	yes	—
158	C₂	C₂×C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
159	C₂	C₂×C₄⋊Q₈×C₅₃	1	1	yes	characteristic
160	C₂	C₅₃×#535	1	1	yes	characteristic
161	C₂	C₅₃×#554	1	2	yes	—
162	C₂	Q₈²×C₅₃	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
27. C₂×Q₈×C₅₃	Q₈	semidirect
40. C₄⋊C₄×C₅₃	Q₈	semidirect
76. #2567	C₅₃	direct (N × H)
89. C₅₃	#2567	direct (N × H)
109. C₂×Q₈	Q₈×C₅₃	semidirect
122. C₄⋊C₄	Q₈×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
C₂×C₄×Q₈×C₅₃	2
C₅₃×#395	2
C₅₃×#391	2
C₅₃×#389	2
C₅₃×#384	2
C₄×C₄⋊C₄×C₅₃	2
#2567	53

Subgroups — C53×#2567 #20271504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#2567 #20271504