Subgroups — #27537737

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276624	C₂	1	8	yes	maximal, complemented
2	#20276518	C₂	1	4	yes	maximal
3	#20276514	C₂	1	2	yes	maximal
4	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11719908	C₂²	1	4	yes	—
6	C₄⋊#11719908	C₂²	1	8	yes	—
7	C₂²⋊#11719908	C₂²	1	4	yes	—
8	#16810399	C₂²	1	8	yes	—
9	C₄×#11719908	C₂²	1	4	yes	—
10	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
11	C₈×Q₈×C₅₃	C₄	1	4	yes	—
12	C₈×Q₈×C₅₃	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
14	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
17	C₂×#11719908	C₂³	1	8	yes	—
18	C₂²×Q₈×C₅₃	C₈	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₅₃	C₈	1	4	yes	—
23	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
25	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	C₂×C₄	1	8	yes	—
29	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₅₃	C₂³	1	2	yes	—
33	C₄×C₈×C₅₃	C₂³	1	4	yes	—
34	C₂×Q₈×C₅₃	—	2	4	no	—
35	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
36	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	—	4	1	no	—
43	C₂×C₈×C₅₃	—	4	2	no	—
44	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
47	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
49	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
50	C₄²×C₅₃	C₂×C₈	1	4	yes	—
51	C₄²×C₅₃	C₂×C₈	1	2	yes	—
52	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
53	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
54	#11719908	—	2	8	no	—
55	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	Q₈×C₅₃	—	2	8	no	—
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	4	2	no	—
61	C₂×C₄×C₅₃	—	4	1	no	—
62	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
63	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
64	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
65	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
66	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
68	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
69	C₈×C₅₃	—	2	1	no	—
70	C₈×C₅₃	—	2	2	no	—
71	C₈×C₅₃	—	2	4	no	—
72	C₈×C₅₃	2₋¹⁺⁴	1	2	yes	—
73	#33497	—	53	1	no	maximal, Sylow
74	C₂²×C₅₃	—	2	1	no	—
75	C₂²×C₅₃	—	2	1	no	—
76	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
77	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
78	C₄×C₅₃	—	2	4	no	—
79	C₄×C₅₃	—	2	4	no	—
80	C₄×C₅₃	—	2	2	no	—
81	C₄×C₅₃	—	2	1	no	—
82	C₄×C₅₃	C₂×M₃₂	1	4	yes	—
83	C₄×C₅₃	#519	1	1	yes	characteristic
84	C₄×C₅₃	#568	1	1	yes	characteristic
85	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic
86	#2183	—	53	8	no	—
87	#2163	—	53	4	no	—
88	#2162	—	53	2	no	—
89	C₂×C₅₃	—	2	2	no	—
90	C₂×C₅₃	#2982	1	1	yes	characteristic
91	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
92	C₂×C₅₃	#3407	1	1	yes	characteristic
93	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
94	C₈×Q₈	Dic₅₃	1	4	yes	—
95	C₈×Q₈	C₂×D₅₃	1	4	yes	—
96	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
97	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
98	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
99	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
100	C₄⋊C₁₆	—	53	4	no	—
101	C₄⋊C₁₆	—	53	8	no	—
102	C₂²⋊C₁₆	—	53	4	no	—
103	#346	—	53	8	no	—
104	C₄×C₁₆	—	53	4	no	—
105	C₅₃	#33497	1	1	yes	characteristic, complemented, Sylow
106	C₂²×Q₈	#88322	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
108	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
109	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
110	C₄×Q₈	#88322	1	4	yes	—
111	C₄×Q₈	C₂×Dic₅₃	1	4	yes	—
112	C₂×C₄⋊C₄	#88322	1	2	yes	—
113	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
114	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
115	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
116	C₂×C₁₆	—	53	8	no	—
117	C₄⋊C₈	C₂×Dic₅₃	1	8	yes	—
118	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
119	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
120	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
121	C₄×C₈	C₂²×D₅₃	1	2	yes	—
122	C₄×C₈	C₂²×D₅₃	1	4	yes	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	C₂×#88322	1	4	yes	—
125	C₂²×C₄	C₂×#88322	1	2	yes	—
126	C₂²×C₄	C₂×#88322	1	2	yes	—
127	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
128	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
129	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
134	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
135	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
136	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
137	C₄⋊C₄	C₂×#88322	1	8	yes	—
138	C₄⋊C₄	C₂×#88322	1	4	yes	—
139	C₄²	C₂×#88322	1	4	yes	—
140	C₄²	C₂×#88322	1	2	yes	—
141	C₄²	C₂²×Dic₅₃	1	4	yes	—
142	C₄²	C₂²×Dic₅₃	1	2	yes	—
143	C₁₆	—	106	8	no	—
144	C₂³	C₂²×#88322	1	1	yes	characteristic
145	Q₈	—	2	8	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	#13133157	1	4	yes	—
152	C₂×C₄	C₂²×#88322	1	4	yes	—
153	C₂×C₄	C₂²×#88322	1	2	yes	—
154	C₂×C₄	C₂²×#88322	1	4	yes	—
155	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
156	C₂×C₄	C₂²×#88322	1	2	yes	—
157	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
158	C₈	—	2	1	no	—
159	C₈	—	2	2	no	—
160	C₈	—	2	4	no	—
161	C₈	#13133265	1	2	yes	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	C₂×#13133157	1	2	yes	—
165	C₂²	C₂³×#88322	1	1	yes	characteristic
166	C₄	—	2	4	no	—
167	C₄	—	2	4	no	—
168	C₄	—	2	2	no	—
169	C₄	—	2	1	no	—
170	C₄	C₂×#13133157	1	4	yes	—
171	C₄	#16811101	1	1	yes	characteristic
172	C₄	#16811390	1	1	yes	characteristic
173	C₂	—	2	2	no	—
174	C₂	#20283796	1	1	yes	characteristic
175	C₂	C₂²×#13133157	1	1	yes	characteristic
176	C₂	#20287818	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₅₃	C₂
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	#33497
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
1. #20276624	C₂	semidirect
105. C₅₃	#33497	semidirect

Maximal subgroups

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

subgroup	index
#20276624	2
#20276518	2
#20276514	2
C₂×C₈×Q₈×C₅₃	2
#33497	53