Subgroups — C₂×C₅₃×#2658 #27242142

Order 13568.

3590 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	C₂²×C₄⋊Q₈×C₅₃	C₂	1	3	yes	maximal, complemented
2	C₂²×#124×C₅₃	C₂	1	3	yes	maximal
3	C₅₃×#2658	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅₃×#399	C₂	1	8	yes	maximal, complemented
5	C₂×C₅₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Q₈×C₅₃	C₂²	1	3	yes	—
7	C₂³×D₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Q₈×C₅₃	—	2	24	no	—
9	C₂×C₄⋊Q₈×C₅₃	C₂²	1	24	yes	complemented
10	C₂×#124×C₅₃	—	2	24	no	—
11	C₂×#124×C₅₃	C₂²	1	24	yes	—
12	C₂²×C₄⋊C₄×C₅₃	C₂²	1	12	yes	—
13	C₂²×C₂²⋊C₄×C₅₃	C₂²	1	12	yes	—
14	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
15	C₅₃×#399	C₂²	1	64	yes	complemented
16	C₅₃×#376	C₂²	1	8	yes	complemented
17	C₂×#95×C₅₃	C₂²	1	4	yes	—
18	C₂⁵×C₅₃	C₂³	1	2	yes	—
19	C₂²×Q₈×C₅₃	—	2	12	no	—
20	C₂²×Q₈×C₅₃	—	2	24	no	—
21	C₂²×Q₈×C₅₃	C₂³	1	12	yes	—
22	C₂²×D₄×C₅₃	—	2	12	no	—
23	C₂²×D₄×C₅₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
25	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊Q₈×C₅₃	—	2	96	no	—
28	#124×C₅₃	—	2	96	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	48	no	—
30	C₂×C₄⋊C₄×C₅₃	C₂³	1	48	yes	—
31	C₂×C₂²⋊C₄×C₅₃	—	2	48	no	—
32	C₂×C₂²⋊C₄×C₅₃	C₂³	1	48	yes	—
33	C₂×C₄²×C₅₃	D₄	1	24	yes	—
34	C₂×C₄²×C₅₃	C₂³	1	12	yes	—
35	#95×C₅₃	C₂³	1	16	yes	—
36	C₂⁴×C₅₃	—	2	12	no	—
37	C₂⁴×C₅₃	—	2	16	no	—
38	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
39	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
40	C₂×Q₈×C₅₃	—	2	24	no	—
41	C₂×Q₈×C₅₃	—	2	48	no	—
42	C₂×Q₈×C₅₃	—	4	48	no	—
43	C₂×D₄×C₅₃	—	2	24	no	—
44	C₂×D₄×C₅₃	—	4	16	no	—
45	C₂²×C₄×C₅₃	—	2	24	no	—
46	C₂²×C₄×C₅₃	—	2	12	no	—
47	C₂²×C₄×C₅₃	—	2	24	no	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
49	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
51	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
52	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
53	C₄⋊C₄×C₅₃	—	2	96	no	—
54	C₂²⋊C₄×C₅₃	—	2	96	no	—
55	C₄²×C₅₃	C₂×D₄	1	48	yes	—
56	C₂³×C₅₃	—	2	12	no	—
57	C₂³×C₅₃	—	2	16	no	—
58	C₂³×C₅₃	—	4	8	no	—
59	C₂³×C₅₃	—	4	48	no	—
60	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
61	C₂³×C₅₃	2₋¹⁺⁴	1	8	yes	—
62	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
63	Q₈×C₅₃	—	4	48	no	—
64	D₄×C₅₃	—	4	16	no	—
65	C₂×C₄×C₅₃	—	2	48	no	—
66	C₂×C₄×C₅₃	—	2	24	no	—
67	C₂×C₄×C₅₃	—	2	12	no	—
68	C₂×C₄×C₅₃	—	2	24	no	—
69	C₂×C₄×C₅₃	—	4	24	no	—
70	C₂×C₄×C₅₃	#127	1	16	yes	—
71	C₂×C₄×C₅₃	C₂²×D₄	1	24	yes	—
72	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
73	C₂×#2658	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂²×C₅₃	—	4	4	no	—
75	C₂²×C₅₃	—	4	24	no	—
76	C₂²×C₅₃	—	8	16	no	—
77	C₂²×C₅₃	C₂×#127	1	4	yes	—
78	C₂²×C₅₃	#536	1	24	yes	—
79	C₂²×C₅₃	C₂³×D₄	1	3	yes	—
80	C₂²×C₅₃	C₂×2₋¹⁺⁴	1	4	yes	—
81	C₄×C₅₃	—	2	24	no	—
82	C₄×C₅₃	—	4	12	no	—
83	C₄×C₅₃	C₂×#127	1	8	yes	—
84	C₂²×C₄⋊Q₈	C₂×C₅₃	1	3	yes	complemented
85	C₂²×#124	C₂×C₅₃	1	3	yes	—
86	#2658	C₂×C₅₃	1	16	yes	direct product factor
87	C₂×#399	C₂×C₅₃	1	8	yes	complemented
88	C₂×#376	C₂×C₅₃	1	1	yes	characteristic, complemented
89	C₂×C₅₃	—	8	4	no	—
90	C₂×C₅₃	#2658	1	8	yes	direct product factor
91	C₂×C₅₃	C₂²×#127	1	1	yes	characteristic
92	C₂×C₅₃	C₂×#536	1	6	yes	—
93	C₂³×Q₈	C₂²×C₅₃	1	3	yes	—
94	C₂³×D₄	C₂²×C₅₃	1	1	yes	characteristic
95	C₂×C₄⋊Q₈	—	2	24	no	—
96	C₂×C₄⋊Q₈	C₂²×C₅₃	1	24	yes	complemented
97	C₂×#124	—	2	24	no	—
98	C₂×#124	C₂²×C₅₃	1	24	yes	—
99	C₂²×C₄⋊C₄	C₂²×C₅₃	1	12	yes	—
100	C₂²×C₂²⋊C₄	C₂²×C₅₃	1	12	yes	—
101	C₂²×C₄²	C₂²×C₅₃	1	3	yes	—
102	#399	C₂²×C₅₃	1	64	yes	complemented
103	#376	C₂²×C₅₃	1	8	yes	complemented
104	C₂×#95	C₂²×C₅₃	1	4	yes	—
105	C₅₃	C₂×#2658	1	1	yes	characteristic, direct product factor, Sylow
106	C₂⁵	C₂³×C₅₃	1	2	yes	—
107	C₂²×Q₈	—	2	12	no	—
108	C₂²×Q₈	—	2	24	no	—
109	C₂²×Q₈	C₂³×C₅₃	1	12	yes	—
110	C₂²×D₄	—	2	12	no	—
111	C₂²×D₄	C₂³×C₅₃	1	4	yes	—
112	C₂³×C₄	C₂³×C₅₃	1	6	yes	—
113	C₂³×C₄	C₂³×C₅₃	1	6	yes	—
114	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
115	C₄⋊Q₈	—	2	96	no	—
116	#124	—	2	96	no	—
117	C₂×C₄⋊C₄	—	2	48	no	—
118	C₂×C₄⋊C₄	C₂³×C₅₃	1	48	yes	—
119	C₂×C₂²⋊C₄	—	2	48	no	—
120	C₂×C₂²⋊C₄	C₂³×C₅₃	1	48	yes	—
121	C₂×C₄²	D₄×C₅₃	1	24	yes	—
122	C₂×C₄²	C₂³×C₅₃	1	12	yes	—
123	#95	C₂³×C₅₃	1	16	yes	—
124	C₂⁴	—	2	12	no	—
125	C₂⁴	—	2	16	no	—
126	C₂⁴	C₂⁴×C₅₃	1	4	yes	—
127	C₂⁴	C₂⁴×C₅₃	1	1	yes	characteristic
128	C₂×Q₈	—	2	24	no	—
129	C₂×Q₈	—	2	48	no	—
130	C₂×Q₈	—	4	48	no	—
131	C₂×D₄	—	2	24	no	—
132	C₂×D₄	—	4	16	no	—
133	C₂²×C₄	—	2	24	no	—
134	C₂²×C₄	—	2	12	no	—
135	C₂²×C₄	—	2	24	no	—
136	C₂²×C₄	C₂×D₄×C₅₃	1	24	yes	—
137	C₂²×C₄	C₂×D₄×C₅₃	1	12	yes	—
138	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
139	C₂²×C₄	C₂⁴×C₅₃	1	12	yes	—
140	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
141	C₄⋊C₄	—	2	96	no	—
142	C₂²⋊C₄	—	2	96	no	—
143	C₄²	C₂×D₄×C₅₃	1	48	yes	—
144	C₂³	—	2	12	no	—
145	C₂³	—	2	16	no	—
146	C₂³	—	4	8	no	—
147	C₂³	—	4	48	no	—
148	C₂³	C₂²×D₄×C₅₃	1	6	yes	—
149	C₂³	2₋¹⁺⁴×C₅₃	1	8	yes	—
150	C₂³	C₂⁵×C₅₃	1	1	yes	characteristic
151	Q₈	—	4	48	no	—
152	D₄	—	4	16	no	—
153	C₂×C₄	—	2	48	no	—
154	C₂×C₄	—	2	24	no	—
155	C₂×C₄	—	2	12	no	—
156	C₂×C₄	—	2	24	no	—
157	C₂×C₄	—	4	24	no	—
158	C₂×C₄	#127×C₅₃	1	16	yes	—
159	C₂×C₄	C₂²×D₄×C₅₃	1	24	yes	—
160	C₂×C₄	C₂²×D₄×C₅₃	1	12	yes	—
161	C₂²	—	4	24	no	—
162	C₂²	—	4	4	no	—
163	C₂²	—	8	16	no	—
164	C₂²	C₂×#127×C₅₃	1	4	yes	—
165	C₂²	C₅₃×#536	1	24	yes	—
166	C₂²	C₂³×D₄×C₅₃	1	3	yes	—
167	C₂²	C₂×2₋¹⁺⁴×C₅₃	1	4	yes	—
168	C₄	—	2	24	no	—
169	C₄	—	4	12	no	—
170	C₄	C₂×#127×C₅₃	1	8	yes	—
171	C₂	—	8	4	no	—
172	C₂	C₅₃×#2658	1	8	yes	direct product factor
173	C₂	C₂²×#127×C₅₃	1	1	yes	characteristic
174	C₂	C₂×C₅₃×#536	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₅₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅₃
prime radical	C₂³×D₄×C₅₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Q₈×C₅₃	C₂	semidirect
3. C₅₃×#2658	C₂	direct (N × H)
4. C₂×C₅₃×#399	C₂	semidirect
5. C₂×C₅₃×#376	C₂	semidirect
9. C₂×C₄⋊Q₈×C₅₃	C₂²	semidirect
15. C₅₃×#399	C₂²	semidirect
16. C₅₃×#376	C₂²	semidirect
73. C₂×#2658	C₅₃	direct (N × H)
84. C₂²×C₄⋊Q₈	C₂×C₅₃	semidirect
86. #2658	C₂×C₅₃	direct (N × H)
87. C₂×#399	C₂×C₅₃	semidirect
88. C₂×#376	C₂×C₅₃	semidirect
90. C₂×C₅₃	#2658	direct (N × H)
96. C₂×C₄⋊Q₈	C₂²×C₅₃	semidirect
102. #399	C₂²×C₅₃	semidirect
103. #376	C₂²×C₅₃	semidirect
105. C₅₃	C₂×#2658	direct (N × H)
172. C₂	C₅₃×#2658	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₅₃	2
C₂²×#124×C₅₃	2
C₅₃×#2658	2
C₂×C₅₃×#399	2
C₂×C₅₃×#376	2
C₂×#2658	53

Subgroups — C2×C53×#2658 #27242142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅₃×#2658 #27242142