Subgroups — #28311427

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811546	C₂	1	2	yes	maximal
2	C₂×#16811545	C₂	1	2	yes	maximal
3	#20291049	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133291	C₂²	1	4	yes	—
6	#16811546	—	2	4	no	—
7	#16811545	—	2	4	no	—
8	#16811544	—	2	4	no	—
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
16	#13133291	—	2	4	no	—
17	#13133291	—	2	4	no	—
18	C₂×#11719910	—	2	4	no	—
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂³	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	—	2	2	no	—
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×#88322	D₄	1	2	yes	—
25	C₄×#88322	Q₈	1	2	yes	—
26	C₈×Dic₅₃	—	2	2	no	—
27	C₈×Dic₅₃	C₈	1	2	yes	—
28	C₈×Dic₅₃	C₈	1	2	yes	—
29	C₈×Dic₅₃	D₄	1	2	yes	—
30	C₈×Dic₅₃	D₄	1	2	yes	—
31	C₈×Dic₅₃	Q₈	1	2	yes	—
32	C₈×Dic₅₃	Q₈	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	—	2	1	no	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₅₃	—	2	1	no	—
40	C₂²×Dic₅₃	M₁₆	1	2	yes	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	—	2	1	no	—
43	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
47	C₄²×C₅₃	—	2	1	no	—
48	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₄×Dic₅₃	—	2	8	no	—
50	C₄×Dic₅₃	—	2	1	no	—
51	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
53	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
54	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
55	C₂×#88322	C₄⋊C₄	1	2	yes	—
56	C₂×#88322	C₂×C₈	1	2	yes	—
57	C₂×#88322	M₁₆	1	2	yes	—
58	C₂×#88322	C₂×D₄	1	1	yes	characteristic
59	C₂×#88322	C₂×D₄	1	1	yes	characteristic
60	C₂×#88322	C₂×D₄	1	1	yes	characteristic
61	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
62	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
63	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
64	#11719910	—	4	4	no	—
65	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	4	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×Dic₅₃	—	2	4	no	—
75	C₂×Dic₅₃	—	2	2	no	—
76	C₂×Dic₅₃	—	2	4	no	—
77	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
78	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
79	C₈×C₅₃	—	2	2	no	—
80	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
81	#88322	C₄⋊C₈	1	4	yes	—
82	#88322	#127	1	1	yes	characteristic
83	#88322	C₄⋊Q₈	1	1	yes	characteristic
84	#88322	C₄⋊Q₈	1	1	yes	characteristic
85	#88322	C₄⋊Q₈	1	1	yes	characteristic
86	#12900	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	—	2	2	no	—
88	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₅₃	#423	1	1	yes	characteristic
90	C₂²×C₅₃	#423	1	1	yes	characteristic
91	C₄×C₅₃	—	2	4	no	—
92	C₄×C₅₃	#384	1	1	yes	characteristic
93	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄×C₅₃	#423	1	1	yes	characteristic
96	Dic₅₃	—	2	4	no	—
97	Dic₅₃	—	2	2	no	—
98	Dic₅₃	#364	1	4	yes	complemented
99	C₂×#364	—	53	2	no	—
100	C₂×#364	—	53	2	no	—
101	#2150	—	53	2	no	—
102	C₄²×C₈	—	53	1	no	—
103	C₂×C₅₃	—	2	1	no	—
104	C₂×C₅₃	—	2	1	no	—
105	C₂×C₅₃	#1842	1	1	yes	characteristic
106	C₂×C₅₃	C₂×#364	1	2	yes	—
107	C₂×M₃₂	—	53	4	no	—
108	C₂×C₄×C₈	—	53	1	no	—
109	C₂×C₄×C₈	—	53	1	no	—
110	C₂×C₄×C₈	—	53	1	no	—
111	C₂×C₄×C₈	—	53	1	no	—
112	C₂×C₄×C₈	—	53	1	no	—
113	C₂×C₄×C₈	#6157	1	1	yes	characteristic
114	C₄³	—	53	1	no	—
115	#364	—	106	4	no	—
116	#364	—	106	4	no	—
117	C₄⋊C₁₆	—	106	4	no	—
118	C₅₃	#12900	1	1	yes	characteristic, complemented, Sylow
119	C₂²×C₈	—	53	1	no	—
120	C₂²×C₈	—	53	1	no	—
121	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
122	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
123	C₂×C₄²	—	53	1	no	—
124	C₂×C₄²	—	53	1	no	—
125	C₂×C₄²	—	106	2	no	—
126	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
127	M₃₂	—	106	4	no	—
128	M₃₂	—	106	4	no	—
129	C₂×C₁₆	—	106	4	no	—
130	C₄×C₈	—	2	2	no	—
131	C₄×C₈	—	53	2	no	—
132	C₄×C₈	—	53	2	no	—
133	C₄×C₈	—	53	2	no	—
134	C₄×C₈	—	53	2	no	—
135	C₄×C₈	—	53	2	no	—
136	C₄×C₈	—	53	2	no	—
137	C₄×C₈	—	53	2	no	—
138	C₄×C₈	—	53	2	no	—
139	C₄×C₈	—	106	2	no	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	53	2	no	—
142	C₂²×C₄	—	106	1	no	—
143	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	53	1	no	—
147	C₂×C₈	—	53	1	no	—
148	C₂×C₈	—	53	2	no	—
149	C₂×C₈	—	53	1	no	—
150	C₂×C₈	—	53	1	no	—
151	C₂×C₈	—	53	1	no	—
152	C₂×C₈	—	53	1	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	2	no	—
155	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
156	C₂×C₈	C₄⋊#6157	1	2	yes	—
157	C₂×C₈	C₄⋊#6157	1	2	yes	—
158	C₂×C₈	#11719940	1	2	yes	—
159	C₄²	—	2	1	no	—
160	C₄²	—	53	2	no	—
161	C₄²	—	53	1	no	—
162	C₄²	—	53	1	no	—
163	C₄²	—	53	2	no	—
164	C₄²	—	106	8	no	—
165	C₄²	—	106	1	no	—
166	C₄²	C₄⋊#6157	1	2	yes	—
167	C₁₆	—	212	4	no	—
168	C₂³	D₅₃⋊M₁₆	1	1	yes	characteristic
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	53	2	no	—
172	C₂×C₄	—	53	2	no	—
173	C₂×C₄	—	106	4	no	—
174	C₂×C₄	—	106	4	no	—
175	C₂×C₄	—	106	2	no	—
176	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
179	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
181	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	53	1	no	—
184	C₈	—	53	1	no	—
185	C₈	—	53	1	no	—
186	C₈	—	53	1	no	—
187	C₈	—	53	4	no	—
188	C₈	#13133284	1	4	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
191	C₂²	#16811626	1	1	yes	characteristic
192	C₂²	#16811628	1	1	yes	characteristic
193	C₄	—	2	4	no	—
194	C₄	—	53	4	no	—
195	C₄	—	106	4	no	—
196	C₄	—	106	2	no	—
197	C₄	#16811569	1	1	yes	characteristic
198	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
199	C₄	C₂×#13133284	1	1	yes	characteristic
200	C₄	#16811626	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#20290648	1	1	yes	characteristic
204	C₂	D₅₃⋊#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#12900
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
98. Dic₅₃	#364	semidirect
118. C₅₃	#12900	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811546	2
C₂×#16811545	2
#20291049	2
C₄×C₈×Dic₅₃	2
#12900	53