Subgroups — #27300445

Order 13568.

298 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×#16810444	C₂	1	1	yes	maximal, characteristic
2	#20274401	C₂	1	1	yes	maximal, characteristic
3	#20272798	C₂	1	2	yes	maximal
4	#20272794	C₂	1	2	yes	maximal
5	C₂×C₅₃×#364	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133128	C₄	1	1	yes	characteristic
8	C₂×#13133127	C₂²	1	1	yes	characteristic
9	C₂×#13133127	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	#16810469	C₄	1	1	yes	characteristic
13	#16810444	—	2	2	no	—
14	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#364	—	2	2	no	—
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	#13133157	—	2	2	no	—
22	#13133157	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	#13133128	—	2	2	no	—
25	#13133127	—	2	1	no	—
26	#13133127	—	2	1	no	—
27	#13133127	—	2	1	no	—
28	#13133127	—	2	1	no	—
29	C₈×Dic₅₃	—	2	2	no	—
30	C₈×Dic₅₃	C₈	1	4	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	—	2	2	no	—
37	C₂×C₁₆×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₄×C₈×C₅₃	Q₈	1	2	yes	—
40	C₂²×C₄×C₅₃	—	2	1	no	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	2	1	no	—
43	C₂²×Dic₅₃	M₁₆	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	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	1	yes	characteristic
50	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
51	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
53	C₄²×C₅₃	—	2	1	no	—
54	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₄×Dic₅₃	—	2	1	no	—
56	C₄×Dic₅₃	—	2	1	no	—
57	C₄×Dic₅₃	—	2	1	no	—
58	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
59	C₂×#88322	—	2	1	no	—
60	C₂×#88322	—	2	1	no	—
61	C₂×#88322	—	2	1	no	—
62	C₂×#88322	—	2	1	no	—
63	C₂×#88322	C₂×C₈	1	2	yes	—
64	C₂×#88322	M₁₆	1	2	yes	—
65	C₁₆×C₅₃	—	4	2	no	—
66	#11719908	—	4	2	no	—
67	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	4	2	no	—
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	2	no	—
77	C₂×Dic₅₃	—	2	2	no	—
78	C₂×Dic₅₃	—	4	2	no	—
79	C₂×Dic₅₃	—	4	2	no	—
80	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
81	C₈×C₅₃	#124	1	1	yes	characteristic
82	C₈×C₅₃	#124	1	1	yes	characteristic
83	C₈×C₅₃	#125	1	1	yes	characteristic
84	C₈×C₅₃	#125	1	1	yes	characteristic
85	#88322	—	2	2	no	—
86	#88322	—	2	2	no	—
87	#12881	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	#423	1	1	yes	characteristic
90	C₂²×C₅₃	#431	1	1	yes	characteristic
91	C₂²×C₅₃	#432	1	1	yes	characteristic
92	C₄×C₅₃	—	4	2	no	—
93	C₄×C₅₃	#382	1	1	yes	characteristic
94	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄×C₅₃	#431	1	1	yes	characteristic
96	C₄×C₅₃	#432	1	1	yes	characteristic
97	Dic₅₃	—	4	2	no	—
98	Dic₅₃	—	4	2	no	—
99	C₂×#364	—	53	1	no	—
100	C₂×#364	D₅₃	1	1	yes	characteristic
101	#1726	—	53	1	no	—
102	#1377	—	53	2	no	—
103	#1375	—	53	2	no	—
104	C₂×C₅₃	—	2	1	no	—
105	C₂×C₅₃	—	2	1	no	—
106	C₂×C₅₃	#1831	1	1	yes	characteristic
107	C₂×C₅₃	#2166	1	2	yes	—
108	C₂×M₃₂	—	53	2	no	—
109	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
110	C₂×#97	—	53	1	no	—
111	C₂×#97	—	53	1	no	—
112	C₂×#97	—	53	1	no	—
113	C₂×C₄×C₈	—	53	1	no	—
114	C₂×C₄×C₈	—	53	1	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	#376	—	53	1	no	—
117	#364	—	2	2	no	—
118	#364	—	106	2	no	—
119	C₅₃	#12881	1	1	yes	characteristic, complemented, Sylow
120	C₂²×C₈	—	53	1	no	—
121	C₂²×C₈	—	53	1	no	—
122	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
123	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
124	C₂×C₄²	—	53	1	no	—
125	C₂×C₄²	—	53	1	no	—
126	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
127	M₃₂	—	2	2	no	—
128	M₃₂	—	2	2	no	—
129	M₃₂	—	106	2	no	—
130	M₃₂	—	106	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	106	2	no	—
133	#97	—	106	2	no	—
134	#97	—	106	1	no	—
135	#97	—	106	1	no	—
136	#97	—	106	1	no	—
137	#97	—	106	1	no	—
138	C₄×C₈	—	53	4	no	—
139	C₄×C₈	—	106	2	no	—
140	C₄×C₈	Dic₁₀₆	1	2	yes	—
141	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
142	#95	—	106	2	no	—
143	C₂²×C₄	—	2	1	no	—
144	C₂²×C₄	—	53	2	no	—
145	C₂²×C₄	—	106	1	no	—
146	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
147	C₂×C₈	—	53	2	no	—
148	C₂×C₈	—	53	2	no	—
149	C₂×C₈	—	106	1	no	—
150	C₂×C₈	—	106	1	no	—
151	C₂×C₈	—	106	1	no	—
152	C₂×C₈	—	106	1	no	—
153	C₂×C₈	C₈×D₅₃	1	2	yes	—
154	C₂×C₈	#11719912	1	2	yes	—
155	C₂×C₈	#11719919	1	2	yes	—
156	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
157	C₂×C₈	#11719947	1	1	yes	characteristic
158	C₂×C₈	#11719947	1	1	yes	characteristic
159	C₂×C₈	#11719949	1	1	yes	characteristic
160	C₂×C₈	#11719949	1	1	yes	characteristic
161	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
162	C₄²	—	2	1	no	—
163	C₄²	—	53	2	no	—
164	C₄²	—	106	1	no	—
165	C₄²	—	106	1	no	—
166	C₄²	—	106	1	no	—
167	C₄²	#11719919	1	2	yes	—
168	C₁₆	—	4	2	no	—
169	C₁₆	—	212	2	no	—
170	C₂³	M₁₆×D₅₃	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	106	2	no	—
174	C₂×C₄	—	106	2	no	—
175	C₂×C₄	—	212	2	no	—
176	C₂×C₄	—	212	2	no	—
177	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
178	C₂×C₄	#13133175	1	1	yes	characteristic
179	C₂×C₄	#13133175	1	1	yes	characteristic
180	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
181	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
183	C₈	—	106	2	no	—
184	C₈	—	106	2	no	—
185	C₈	#13133146	1	4	yes	—
186	C₈	#13133204	1	1	yes	characteristic
187	C₈	#13133206	1	1	yes	characteristic
188	C₈	#13133208	1	1	yes	characteristic
189	C₈	#13133208	1	1	yes	characteristic
190	C₂²	—	2	2	no	—
191	C₂²	#16810646	1	1	yes	characteristic
192	C₂²	#16810675	1	1	yes	characteristic
193	C₂²	#16810679	1	1	yes	characteristic
194	C₄	—	4	2	no	—
195	C₄	—	212	2	no	—
196	C₄	—	212	2	no	—
197	C₄	#16810488	1	1	yes	characteristic
198	C₄	C₂×#13133146	1	1	yes	characteristic
199	C₄	#16810675	1	1	yes	characteristic
200	C₄	#16810679	1	1	yes	characteristic
201	C₂	—	2	1	no	—
202	C₂	—	2	1	no	—
203	C₂	#20274831	1	1	yes	characteristic
204	C₂	#20276530	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133175
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#364	C₂
socle	C₂²×C₅₃	#432
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#364	D₅₃
hypocenter	C₅₃	#12881
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
119. C₅₃	#12881	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810444	2
#20274401	2
#20272798	2
#20272794	2
C₂×C₅₃×#364	2
#12881	53