Subgroups — #27410682

Order 13568.

664 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275652	C₂	1	4	yes	maximal
2	#20275274	C₂	1	8	yes	maximal
3	#20274858	C₂	1	2	yes	maximal
4	C₄×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	4	yes	—
6	C₂×C₄×#88322	C₂²	1	4	yes	—
7	#16810378	C₂²	1	8	yes	—
8	#16810378	C₂²	1	8	yes	—
9	#16810378	C₂²	1	4	yes	—
10	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#391	C₄	1	2	yes	—
13	C₅₃×#389	C₄	1	4	yes	—
14	C₅₃×#384	C₄	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×#88322	C₂³	1	8	yes	—
18	C₄⋊#88322	—	4	4	no	—
19	C₄×#88322	—	4	4	no	—
20	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₄⋊Q₈×C₅₃	C₈	1	8	yes	complemented
22	C₄×Q₈×C₅₃	—	4	2	no	—
23	C₄×Q₈×C₅₃	—	4	2	no	—
24	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂³	1	4	yes	—
33	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
34	#95×C₅₃	C₂×C₄	1	8	yes	—
35	C₂×Q₈×C₅₃	—	4	2	no	—
36	C₂×Q₈×C₅₃	—	4	2	no	—
37	C₂×Q₈×C₅₃	C₂×C₈	1	8	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₄⋊C₄×C₅₃	—	4	4	no	—
46	C₄⋊C₄×C₅₃	—	4	2	no	—
47	C₄⋊C₄×C₅₃	—	4	4	no	—
48	C₄⋊C₄×C₅₃	C₂×C₈	1	16	yes	—
49	C₄²×C₅₃	—	2	2	no	—
50	C₄²×C₅₃	—	2	4	no	—
51	C₄²×C₅₃	—	2	4	no	—
52	C₄²×C₅₃	—	2	4	no	—
53	C₄²×C₅₃	—	4	1	no	—
54	C₄²×C₅₃	—	4	2	no	—
55	C₄²×C₅₃	C₂×C₈	1	4	yes	—
56	C₂×#88322	—	2	8	no	—
57	C₂×#88322	—	2	16	no	—
58	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
59	Q₈×C₅₃	—	4	8	no	—
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	4	no	—
62	C₂×C₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	4	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	—	2	2	no	—
72	C₂×C₄×C₅₃	—	2	8	no	—
73	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
75	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
76	C₂×C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
77	C₂×C₄×C₅₃	2₋¹⁺⁴	1	4	yes	—
78	#88322	—	4	8	no	—
79	#23363	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	#518	1	1	yes	characteristic
81	C₂²×C₅₃	#519	1	2	yes	—
82	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
83	C₂²×C₅₃	#568	1	1	yes	characteristic
84	C₂²×C₅₃	#568	1	2	yes	—
85	C₄×C₅₃	—	2	8	no	—
86	C₄×C₅₃	—	2	2	no	—
87	C₄×C₅₃	—	2	4	no	—
88	C₄×C₅₃	—	4	4	no	—
89	C₄×C₅₃	—	4	1	no	—
90	C₄×C₅₃	—	4	2	no	—
91	C₄×C₅₃	#563	1	4	yes	—
92	C₄×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
93	#1984	—	53	4	no	—
94	#1916	—	53	8	no	—
95	#1839	—	53	2	no	—
96	C₂×C₅₃	#2381	1	1	yes	characteristic
97	C₂×C₅₃	#2972	1	1	yes	characteristic
98	C₂×C₅₃	#2980	1	1	yes	characteristic
99	C₂×C₅₃	#2982	1	2	yes	—
100	C₂×C₅₃	#2987	1	2	yes	—
101	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
102	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
103	C₂×C₄⋊C₈	—	53	4	no	—
104	C₂×C₄×C₈	—	53	4	no	—
105	#391	Dic₅₃	1	2	yes	—
106	#389	Dic₅₃	1	4	yes	—
107	#384	Dic₅₃	1	1	yes	characteristic
108	C₄×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
109	C₄³	C₂×D₅₃	1	1	yes	characteristic
110	#336	—	53	8	no	—
111	#336	—	53	4	no	—
112	#336	—	53	8	no	—
113	C₅₃	#23363	1	1	yes	characteristic, complemented, Sylow
114	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
115	C₂²×C₈	—	53	8	no	—
116	C₄⋊Q₈	#88322	1	8	yes	complemented
117	C₄×Q₈	—	4	2	no	—
118	C₄×Q₈	—	4	2	no	—
119	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
120	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
121	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
122	C₂×C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
123	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
124	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
125	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
126	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
127	C₂×C₄²	C₂²×D₅₃	1	4	yes	—
128	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
129	C₄⋊C₈	—	212	4	no	—
130	C₄×C₈	—	212	4	no	—
131	#95	C₂×Dic₅₃	1	8	yes	—
132	C₂×Q₈	—	4	2	no	—
133	C₂×Q₈	—	4	2	no	—
134	C₂×Q₈	C₂×#88322	1	8	yes	—
135	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
136	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
137	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
138	C₂²×C₄	C₂²×Dic₅₃	1	4	yes	—
139	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
140	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
141	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
142	C₂×C₈	—	106	8	no	—
143	C₂×C₈	—	106	16	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	C₂×#88322	1	16	yes	—
148	C₄²	—	2	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	2	no	—
151	C₄²	—	2	4	no	—
152	C₄²	—	4	2	no	—
153	C₄²	—	4	1	no	—
154	C₄²	C₂×#88322	1	4	yes	—
155	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
156	Q₈	—	4	8	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	8	no	—
170	C₂×C₄	C₂²×#88322	1	4	yes	—
171	C₂×C₄	C₂²×#88322	1	2	yes	—
172	C₂×C₄	C₂²×#88322	1	8	yes	—
173	C₂×C₄	#13133262	1	2	yes	—
174	C₂×C₄	#13133265	1	4	yes	—
175	C₈	—	212	8	no	—
176	C₂²	#16811098	1	1	yes	characteristic
177	C₂²	#16811101	1	2	yes	—
178	C₂²	C₂³×#88322	1	1	yes	characteristic
179	C₂²	#16811390	1	1	yes	characteristic
180	C₂²	#16811390	1	2	yes	—
181	C₄	—	2	8	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	4	no	—
184	C₄	—	4	1	no	—
185	C₄	—	4	4	no	—
186	C₄	—	4	2	no	—
187	C₄	#16811372	1	4	yes	—
188	C₂	#20277896	1	1	yes	characteristic
189	C₂	#20283739	1	1	yes	characteristic
190	C₂	#20283786	1	1	yes	characteristic
191	C₂	#20283796	1	2	yes	—
192	C₂	#20283821	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₈
Fitting	C₄×C₄⋊Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂³×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#23363
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
21. C₄⋊Q₈×C₅₃	C₈	semidirect
113. C₅₃	#23363	semidirect
116. C₄⋊Q₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20275652	2
#20275274	2
#20274858	2
C₄×C₄⋊Q₈×C₅₃	2
#23363	53