Subgroups — #27299413

Order 13568.

358 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810443	C₂	1	2	yes	maximal
2	M₃₂×Dic₅₃	C₂	1	2	yes	maximal
3	#20274401	C₂	1	1	yes	maximal, characteristic
4	#20272797	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133128	C₄	1	1	yes	characteristic
8	C₂×#13133127	C₄	1	1	yes	characteristic
9	C₂×#13133127	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	#16810469	C₄	1	1	yes	characteristic
13	#16810443	C₄	1	8	yes	complemented
14	#16810398	—	2	2	no	—
15	C₁₆×Dic₅₃	—	2	2	no	—
16	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	#13133157	—	2	2	no	—
23	#13133157	C₂×C₄	1	4	yes	—
24	C₂×#11719908	—	2	2	no	—
25	#13133128	C₈	1	4	yes	—
26	#13133127	—	2	2	no	—
27	#13133127	C₈	1	4	yes	—
28	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
29	C₈×Dic₅₃	D₄	1	2	yes	—
30	C₈×Dic₅₃	Q₈	1	2	yes	—
31	#13133124	—	2	2	no	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	M₃₂×C₅₃	—	2	2	no	—
36	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₂²×C₄×C₅₃	—	2	1	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	—	2	1	no	—
42	C₂²×Dic₅₃	M₁₆	1	2	yes	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	2	yes	—
50	C₄²×C₅₃	—	2	1	no	—
51	C₄²×C₅₃	C₂×C₈	1	2	yes	—
52	C₄×Dic₅₃	—	2	1	no	—
53	C₄×Dic₅₃	C₄²	1	2	yes	—
54	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
55	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
56	C₂×#88322	C₄²	1	2	yes	—
57	C₂×#88322	C₄⋊C₄	1	2	yes	—
58	C₂×#88322	C₂×C₈	1	2	yes	—
59	C₂×#88322	C₂×C₈	1	2	yes	—
60	C₂×#88322	C₂×C₈	1	2	yes	—
61	C₂×#88322	M₁₆	1	2	yes	—
62	C₁₆×C₅₃	—	2	4	no	—
63	#11719908	—	2	4	no	—
64	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	4	2	no	—
67	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂²×C₈	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×Dic₅₃	—	2	2	no	—
74	C₂×Dic₅₃	—	4	2	no	—
75	C₂×Dic₅₃	—	4	2	no	—
76	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
77	C₈×C₅₃	—	2	2	no	—
78	C₈×C₅₃	C₄×D₄	1	2	yes	—
79	C₈×C₅₃	C₄×Q₈	1	2	yes	—
80	#88322	C₄×C₈	1	4	yes	—
81	#88322	C₄⋊C₈	1	4	yes	—
82	#12656	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
85	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₅₃	#431	1	1	yes	characteristic
87	C₄×C₅₃	—	4	2	no	—
88	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
90	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₄×C₅₃	#431	1	1	yes	characteristic
92	Dic₅₃	—	2	4	no	—
93	Dic₅₃	—	4	2	no	—
94	C₂×#364	—	53	2	no	—
95	C₄×M₃₂	—	53	2	no	—
96	#1726	—	53	1	no	—
97	#1376	—	53	1	no	—
98	#1376	D₅₃	1	1	yes	characteristic
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	—	2	1	no	—
101	C₂×C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic
102	C₂×C₅₃	#2153	1	2	yes	—
103	C₂×M₃₂	—	53	2	no	—
104	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
105	C₂×#97	—	53	1	no	—
106	C₂×#97	—	53	1	no	—
107	C₂×#97	—	53	1	no	—
108	C₂×C₄×C₈	—	53	1	no	—
109	C₂×C₄×C₈	—	53	1	no	—
110	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
111	#376	—	53	1	no	—
112	#364	—	53	8	no	—
113	#346	—	106	2	no	—
114	C₄×C₁₆	—	106	2	no	—
115	C₅₃	#12656	1	1	yes	characteristic, complemented, Sylow
116	C₂²×C₈	—	53	1	no	—
117	C₂²×C₈	—	53	1	no	—
118	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
120	C₂×C₄²	—	53	1	no	—
121	C₂×C₄²	—	53	1	no	—
122	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
123	M₃₂	—	2	2	no	—
124	M₃₂	—	53	4	no	—
125	M₃₂	—	106	2	no	—
126	M₃₂	C₄×D₅₃	1	4	yes	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	106	2	no	—
129	#97	—	53	4	no	—
130	#97	—	53	4	no	—
131	#97	—	106	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	—	53	4	no	—
134	C₄×C₈	—	53	2	no	—
135	C₄×C₈	—	53	2	no	—
136	#95	—	106	2	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	53	2	no	—
139	C₂²×C₄	—	106	1	no	—
140	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	53	2	no	—
144	C₂×C₈	—	53	2	no	—
145	C₂×C₈	—	53	2	no	—
146	C₂×C₈	—	53	2	no	—
147	C₂×C₈	—	53	2	no	—
148	C₂×C₈	—	53	2	no	—
149	C₂×C₈	C₈×D₅₃	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
151	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
152	C₂×C₈	#11719947	1	2	yes	—
153	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
154	C₄²	—	2	1	no	—
155	C₄²	—	53	2	no	—
156	C₄²	—	53	2	no	—
157	C₄²	—	53	2	no	—
158	C₄²	—	106	1	no	—
159	C₄²	C₈×D₅₃	1	2	yes	—
160	C₁₆	—	2	4	no	—
161	C₁₆	—	106	4	no	—
162	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	53	4	no	—
166	C₂×C₄	—	106	2	no	—
167	C₂×C₄	—	212	2	no	—
168	C₂×C₄	—	212	2	no	—
169	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
170	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
171	C₂×C₄	#13133173	1	1	yes	characteristic
172	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
174	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	53	4	no	—
177	C₈	—	53	4	no	—
178	C₈	#13133177	1	2	yes	—
179	C₈	#13133182	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	#16810575	1	1	yes	characteristic
182	C₂²	C₄⋊C₈×D₅₃	1	1	yes	characteristic
183	C₂²	#16810673	1	1	yes	characteristic
184	C₄	—	4	2	no	—
185	C₄	—	106	4	no	—
186	C₄	—	212	2	no	—
187	C₄	#16810474	1	1	yes	characteristic
188	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
189	C₄	C₄⋊C₈×D₅₃	1	1	yes	characteristic
190	C₄	#16810673	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#20274563	1	1	yes	characteristic
194	C₂	#20276479	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1376	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1376	D₅₃
hypocenter	C₅₃	#12656
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
13. #16810443	C₄	semidirect
115. C₅₃	#12656	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810443	2
M₃₂×Dic₅₃	2
#20274401	2
#20272797	2
C₅₃×#1376	2
#12656	53