Subgroups — #28301453

Order 13568.

574 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	C₄²×#88324	C₂	1	2	yes	maximal
2	#20289537	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88324	C₄	1	4	yes	—
5	C₂×C₄×#88324	C₄	1	4	yes	—
6	C₂×C₄×#88324	C₂²	1	4	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
9	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
10	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂²×#88324	C₂×C₄	1	8	yes	—
13	C₄×#88324	—	2	8	no	—
14	C₄×#88324	—	2	8	no	—
15	C₄×#88324	C₈	1	16	yes	complemented
16	C₂²×#88322	C₂×C₄	1	2	yes	—
17	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
21	C₄×#88322	C₈	1	4	yes	complemented
22	C₈×Dic₅₃	—	2	4	no	—
23	C₈×Dic₅₃	—	2	4	no	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
27	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×Dic₅₃	C₄²	1	2	yes	—
31	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
32	C₂×#88324	—	2	8	no	—
33	C₂×#88324	—	2	8	no	—
34	C₂×#88324	C₂×C₈	1	16	yes	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
38	C₄²×C₅₃	C₂×C₈	1	2	yes	—
39	C₄²×C₅₃	C₂×C₈	1	2	yes	—
40	C₄×Dic₅₃	C₂×C₈	1	8	yes	—
41	C₄×Dic₅₃	M₁₆	1	4	yes	—
42	C₄×Dic₅₃	M₁₆	1	8	yes	—
43	C₄×Dic₅₃	#41	1	4	yes	—
44	C₂×#88322	—	2	2	no	—
45	C₂×#88322	—	2	2	no	—
46	C₂×#88322	C₂×C₈	1	4	yes	—
47	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
49	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
50	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
51	C₂×C₄×C₅₃	#97	1	4	yes	—
52	C₂×C₄×C₅₃	#117	1	2	yes	—
53	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
56	C₂×Dic₅₃	C₄×C₈	1	4	yes	—
57	C₂×Dic₅₃	#97	1	4	yes	—
58	C₂×Dic₅₃	#97	1	4	yes	—
59	C₂×Dic₅₃	#117	1	4	yes	—
60	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
61	C₂×Dic₅₃	C₂×M₁₆	1	4	yes	—
62	#88324	—	2	16	no	—
63	C₈×C₅₃	—	2	4	no	—
64	#88322	—	2	4	no	—
65	#8131	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	#376	1	1	yes	characteristic
67	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
68	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₂²×C₅₃	C₂×#97	1	2	yes	—
70	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
71	C₄×C₅₃	C₈²	1	4	yes	—
72	C₄×C₅₃	#322	1	4	yes	—
73	C₄×C₅₃	#431	1	2	yes	—
74	C₄×C₅₃	#431	1	2	yes	—
75	Dic₅₃	#322	1	8	yes	—
76	Dic₅₃	#431	1	8	yes	—
77	C₄²×C₈	—	53	2	no	—
78	C₄²×C₈	—	53	1	no	—
79	#1273	—	53	4	no	—
80	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
81	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
82	C₂×C₅₃	C₂×#322	1	2	yes	—
83	C₂×C₅₃	#1744	1	2	yes	—
84	C₂×C₅₃	#1744	1	1	yes	characteristic
85	C₂×C₄×C₈	—	53	4	no	—
86	C₂×C₄×C₈	—	53	1	no	—
87	C₂×C₄×C₈	—	53	4	no	—
88	C₂×C₄×C₈	—	53	4	no	—
89	C₂×C₄×C₈	—	53	2	no	—
90	C₂×C₄×C₈	—	53	2	no	—
91	C₂×C₄×C₈	#6157	1	1	yes	characteristic
92	C₄³	—	53	1	no	—
93	C₅₃	#8131	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	53	8	no	—
95	C₂²×C₈	—	53	2	no	—
96	C₂²×C₈	C₂×#6157	1	2	yes	—
97	C₂×C₄²	—	53	2	no	—
98	C₂×C₄²	—	53	2	no	—
99	C₂×C₄²	—	53	1	no	—
100	C₂×C₄²	—	53	1	no	—
101	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
102	C₄×C₈	—	53	16	no	—
103	C₄×C₈	—	53	4	no	—
104	C₄×C₈	—	106	4	no	—
105	C₄×C₈	—	106	4	no	—
106	C₄×C₈	—	106	8	no	—
107	C₄×C₈	—	106	8	no	—
108	C₄×C₈	#88324	1	4	yes	complemented
109	C₂²×C₄	—	53	2	no	—
110	C₂²×C₄	—	53	2	no	—
111	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
112	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
113	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	53	16	no	—
117	C₂×C₈	—	53	4	no	—
118	C₂×C₈	—	106	8	no	—
119	C₂×C₈	—	106	8	no	—
120	C₂×C₈	—	106	2	no	—
121	C₂×C₈	—	106	2	no	—
122	C₂×C₈	C₂×#88324	1	4	yes	—
123	C₄²	—	53	8	no	—
124	C₄²	—	53	8	no	—
125	C₄²	—	53	4	no	—
126	C₄²	—	53	4	no	—
127	C₄²	D₅₃⋊C₈	1	2	yes	—
128	C₄²	C₂×#88324	1	2	yes	—
129	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
130	C₂×C₄	—	53	4	no	—
131	C₂×C₄	—	53	4	no	—
132	C₂×C₄	—	53	4	no	—
133	C₂×C₄	—	53	4	no	—
134	C₂×C₄	—	53	4	no	—
135	C₂×C₄	—	53	4	no	—
136	C₂×C₄	C₈×#6157	1	4	yes	—
137	C₂×C₄	C₄×#88324	1	2	yes	—
138	C₂×C₄	C₄×#88324	1	2	yes	—
139	C₂×C₄	#13133271	1	4	yes	—
140	C₂×C₄	#13133296	1	2	yes	—
141	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
142	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
143	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	106	16	no	—
146	C₈	—	106	4	no	—
147	C₂²	#16811559	1	1	yes	characteristic
148	C₂²	C₂×C₈×#6157	1	2	yes	—
149	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
150	C₂²	C₂×#13133271	1	2	yes	—
151	C₂²	#16811582	1	1	yes	characteristic
152	C₄	—	53	8	no	—
153	C₄	—	53	8	no	—
154	C₄	C₈×#88324	1	4	yes	—
155	C₄	#16811464	1	4	yes	—
156	C₄	#16811639	1	2	yes	—
157	C₄	#16811641	1	2	yes	—
158	C₂	C₂×C₈×#88324	1	1	yes	characteristic
159	C₂	C₂×#16811464	1	1	yes	characteristic
160	C₂	#20289911	1	2	yes	—
161	C₂	#20290428	1	2	yes	—
162	C₂	#20290429	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×#6157
commutator	C₂×C₅₃	C₂×C₈²
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#8131
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
15. C₄×#88324	C₈	semidirect
21. C₄×#88322	C₈	semidirect
26. C₄×C₈×C₅₃	C₈	semidirect
93. C₅₃	#8131	semidirect
108. C₄×C₈	#88324	semidirect

Maximal subgroups

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

subgroup	index
C₄²×#88324	2
#20289537	2
C₄×C₈×Dic₅₃	2
#8131	53