Subgroups — C₂×C₅₃×#1446 #27221361

Order 13568.

4830 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	C₂⁴×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#95×C₅₃	C₂	1	6	yes	maximal
3	C₅₃×#1446	C₂	1	8	yes	maximal, direct product factor
4	C₂³×Q₈×C₅₃	—	2	24	no	—
5	C₂³×Q₈×C₅₃	C₄	1	8	yes	complemented
6	C₂³×Q₈×C₅₃	C₂²	1	4	yes	—
7	C₂⁴×C₄×C₅₃	C₂²	1	4	yes	—
8	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
9	C₂×#95×C₅₃	—	2	24	no	—
10	C₂×#95×C₅₃	C₂²	1	24	yes	—
11	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
12	C₂²×Q₈×C₅₃	—	2	128	no	—
13	C₂²×Q₈×C₅₃	—	2	96	no	—
14	C₂²×Q₈×C₅₃	—	2	48	no	—
15	C₂²×Q₈×C₅₃	C₂×C₄	1	16	yes	complemented
16	C₂³×C₄×C₅₃	—	2	24	no	—
17	C₂³×C₄×C₅₃	—	2	24	no	—
18	C₂³×C₄×C₅₃	—	2	24	no	—
19	C₂³×C₄×C₅₃	C₂×C₄	1	12	yes	—
20	C₂³×C₄×C₅₃	D₄	1	24	yes	—
21	C₂³×C₄×C₅₃	Q₈	1	16	yes	complemented
22	C₂³×C₄×C₅₃	C₂³	1	8	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
24	#95×C₅₃	—	2	48	no	—
25	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
26	C₂⁴×C₅₃	C₂×D₄	1	12	yes	—
27	C₂⁴×C₅₃	C₂×Q₈	1	8	yes	—
28	C₂⁴×C₅₃	#41	1	8	yes	—
29	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
30	C₂×Q₈×C₅₃	—	2	128	no	—
31	C₂×Q₈×C₅₃	—	2	256	no	—
32	C₂×Q₈×C₅₃	—	2	96	no	—
33	C₂²×C₄×C₅₃	—	2	48	no	—
34	C₂²×C₄×C₅₃	—	2	96	no	—
35	C₂²×C₄×C₅₃	—	2	24	no	—
36	C₂²×C₄×C₅₃	—	2	96	no	—
37	C₂²×C₄×C₅₃	—	2	24	no	—
38	C₂²×C₄×C₅₃	—	2	24	no	—
39	C₂²×C₄×C₅₃	—	4	16	no	—
40	C₂²×C₄×C₅₃	—	4	48	no	—
41	C₂²×C₄×C₅₃	C₂²⋊C₄	1	48	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
43	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
44	C₂²×C₄×C₅₃	C₂×Q₈	1	16	yes	complemented
45	C₂³×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
46	C₂³×C₅₃	C₄×Q₈	1	16	yes	—
47	C₂³×C₅₃	#120	1	16	yes	—
48	C₂³×C₅₃	C₂²⋊Q₈	1	48	yes	—
49	C₂³×C₅₃	#124	1	24	yes	—
50	C₂³×C₅₃	C₄⋊Q₈	1	24	yes	—
51	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
52	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
53	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
54	C₂³×C₅₃	C₂×#41	1	4	yes	—
55	Q₈×C₅₃	—	2	128	no	—
56	C₂×C₄×C₅₃	—	2	24	no	—
57	C₂×C₄×C₅₃	—	2	48	no	—
58	C₂×C₄×C₅₃	—	2	96	no	—
59	C₂×C₄×C₅₃	—	2	48	no	—
60	C₂×C₄×C₅₃	—	4	8	no	—
61	C₂×C₄×C₅₃	—	4	24	no	—
62	C₂×C₄×C₅₃	—	4	64	no	—
63	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	24	yes	—
64	C₂×#1446	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂²×C₅₃	#379	1	8	yes	—
66	C₂²×C₅₃	#391	1	48	yes	—
67	C₂²×C₅₃	#393	1	32	yes	—
68	C₂²×C₅₃	#395	1	32	yes	—
69	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	3	yes	—
70	C₂²×C₅₃	C₂×C₄×Q₈	1	4	yes	—
71	C₂²×C₅₃	C₂×#120	1	4	yes	—
72	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	12	yes	—
73	C₂²×C₅₃	C₂×#124	1	6	yes	—
74	C₂²×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
75	C₄×C₅₃	—	2	24	no	—
76	C₄×C₅₃	—	4	16	no	—
77	C₂⁴×Q₈	C₂×C₅₃	1	1	yes	characteristic
78	C₂²×#95	C₂×C₅₃	1	6	yes	—
79	#1446	C₂×C₅₃	1	8	yes	direct product factor
80	C₂×C₅₃	#1446	1	16	yes	direct product factor
81	C₂×C₅₃	C₂×#379	1	1	yes	characteristic
82	C₂×C₅₃	C₂×#391	1	6	yes	—
83	C₂×C₅₃	C₂×#393	1	4	yes	—
84	C₂×C₅₃	C₂×#395	1	4	yes	—
85	C₂³×Q₈	—	2	24	no	—
86	C₂³×Q₈	C₄×C₅₃	1	8	yes	complemented
87	C₂³×Q₈	C₂²×C₅₃	1	4	yes	—
88	C₂⁴×C₄	C₂²×C₅₃	1	4	yes	—
89	C₂⁴×C₄	C₂²×C₅₃	1	3	yes	—
90	C₂×#95	—	2	24	no	—
91	C₂×#95	C₂²×C₅₃	1	24	yes	—
92	C₅₃	C₂×#1446	1	1	yes	characteristic, direct product factor, Sylow
93	C₂⁵	C₂³×C₅₃	1	1	yes	characteristic
94	C₂²×Q₈	—	2	128	no	—
95	C₂²×Q₈	—	2	48	no	—
96	C₂²×Q₈	—	2	96	no	—
97	C₂²×Q₈	C₂×C₄×C₅₃	1	16	yes	complemented
98	C₂³×C₄	—	2	24	no	—
99	C₂³×C₄	—	2	24	no	—
100	C₂³×C₄	—	2	24	no	—
101	C₂³×C₄	C₂×C₄×C₅₃	1	12	yes	—
102	C₂³×C₄	D₄×C₅₃	1	24	yes	—
103	C₂³×C₄	Q₈×C₅₃	1	16	yes	complemented
104	C₂³×C₄	C₂³×C₅₃	1	8	yes	—
105	C₂³×C₄	C₂³×C₅₃	1	6	yes	—
106	#95	—	2	48	no	—
107	C₂⁴	C₂²×C₄×C₅₃	1	2	yes	—
108	C₂⁴	C₂×D₄×C₅₃	1	12	yes	—
109	C₂⁴	C₂×Q₈×C₅₃	1	8	yes	—
110	C₂⁴	#41×C₅₃	1	8	yes	—
111	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
112	C₂×Q₈	—	2	128	no	—
113	C₂×Q₈	—	2	256	no	—
114	C₂×Q₈	—	2	96	no	—
115	C₂²×C₄	—	2	48	no	—
116	C₂²×C₄	—	2	96	no	—
117	C₂²×C₄	—	2	24	no	—
118	C₂²×C₄	—	2	96	no	—
119	C₂²×C₄	—	2	24	no	—
120	C₂²×C₄	—	2	24	no	—
121	C₂²×C₄	—	4	16	no	—
122	C₂²×C₄	—	4	48	no	—
123	C₂²×C₄	C₂²⋊C₄×C₅₃	1	48	yes	—
124	C₂²×C₄	C₂²×C₄×C₅₃	1	12	yes	—
125	C₂²×C₄	C₂×D₄×C₅₃	1	24	yes	—
126	C₂²×C₄	C₂×Q₈×C₅₃	1	16	yes	complemented
127	C₂³	C₂×C₂²⋊C₄×C₅₃	1	12	yes	—
128	C₂³	C₄×Q₈×C₅₃	1	16	yes	—
129	C₂³	#120×C₅₃	1	16	yes	—
130	C₂³	C₂²⋊Q₈×C₅₃	1	48	yes	—
131	C₂³	#124×C₅₃	1	24	yes	—
132	C₂³	C₄⋊Q₈×C₅₃	1	24	yes	—
133	C₂³	C₂³×C₄×C₅₃	1	1	yes	characteristic
134	C₂³	C₂²×D₄×C₅₃	1	6	yes	—
135	C₂³	C₂²×Q₈×C₅₃	1	4	yes	—
136	C₂³	C₂×#41×C₅₃	1	4	yes	—
137	Q₈	—	2	128	no	—
138	C₂×C₄	—	2	24	no	—
139	C₂×C₄	—	2	48	no	—
140	C₂×C₄	—	2	96	no	—
141	C₂×C₄	—	2	48	no	—
142	C₂×C₄	—	4	8	no	—
143	C₂×C₄	—	4	24	no	—
144	C₂×C₄	—	4	64	no	—
145	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	24	yes	—
146	C₂²	C₅₃×#379	1	8	yes	—
147	C₂²	C₅₃×#391	1	48	yes	—
148	C₂²	C₅₃×#393	1	32	yes	—
149	C₂²	C₅₃×#395	1	32	yes	—
150	C₂²	C₂²×C₂²⋊C₄×C₅₃	1	3	yes	—
151	C₂²	C₂×C₄×Q₈×C₅₃	1	4	yes	—
152	C₂²	C₂×#120×C₅₃	1	4	yes	—
153	C₂²	C₂×C₂²⋊Q₈×C₅₃	1	12	yes	—
154	C₂²	C₂×#124×C₅₃	1	6	yes	—
155	C₂²	C₂×C₄⋊Q₈×C₅₃	1	6	yes	—
156	C₄	—	2	24	no	—
157	C₄	—	4	16	no	—
158	C₂	C₅₃×#1446	1	16	yes	direct product factor
159	C₂	C₂×C₅₃×#379	1	1	yes	characteristic
160	C₂	C₂×C₅₃×#391	1	6	yes	—
161	C₂	C₂×C₅₃×#393	1	4	yes	—
162	C₂	C₂×C₅₃×#395	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₅₃	C₂³
commutator	C₂³	C₂³×C₄×C₅₃
Frattini, cosocle	C₂⁴	C₂⁴×C₅₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅₃×#1446	C₂	direct (N × H)
5. C₂³×Q₈×C₅₃	C₄	semidirect
15. C₂²×Q₈×C₅₃	C₂×C₄	semidirect
21. C₂³×C₄×C₅₃	Q₈	semidirect
44. C₂²×C₄×C₅₃	C₂×Q₈	semidirect
64. C₂×#1446	C₅₃	direct (N × H)
79. #1446	C₂×C₅₃	direct (N × H)
80. C₂×C₅₃	#1446	direct (N × H)
86. C₂³×Q₈	C₄×C₅₃	semidirect
92. C₅₃	C₂×#1446	direct (N × H)
97. C₂²×Q₈	C₂×C₄×C₅₃	semidirect
103. C₂³×C₄	Q₈×C₅₃	semidirect
126. C₂²×C₄	C₂×Q₈×C₅₃	semidirect
158. C₂	C₅₃×#1446	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₅₃	2
C₂²×#95×C₅₃	2
C₅₃×#1446	2
C₂×#1446	53

Subgroups — C2×C53×#1446 #27221361

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#1446 #27221361