Subgroups — #27299424

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810400	C₂	1	2	yes	maximal
3	C₄×#13133127	C₂	1	1	yes	maximal, characteristic
4	#20272794	C₂	1	2	yes	maximal
5	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	C₂×#13133127	C₄	1	1	yes	characteristic
8	C₂×#13133127	C₄	1	1	yes	characteristic
9	C₂×#13133127	C₂²	1	1	yes	characteristic
10	C₂×#13133127	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88322	C₄	1	1	yes	characteristic
12	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
13	C₄⋊#11719908	C₄	1	4	yes	complemented
14	#16810400	—	2	4	no	—
15	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×C₄⋊C₁₆	C₄	1	4	yes	complemented
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	—	2	2	no	—
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	#13133157	—	2	2	no	—
24	#13133157	—	2	2	no	—
25	C₂×#11719908	C₂×C₄	1	4	yes	—
26	#13133127	—	2	1	no	—
27	#13133127	—	2	1	no	—
28	#13133127	—	2	2	no	—
29	#13133127	—	2	2	no	—
30	#13133127	D₄	1	2	yes	—
31	#13133127	Q₈	1	2	yes	—
32	C₄×#88322	—	2	1	no	—
33	C₄×#88322	—	2	1	no	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	M₃₂×C₅₃	—	2	2	no	—
38	M₃₂×C₅₃	—	2	2	no	—
39	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	—	2	1	no	—
45	C₂²×Dic₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	C₄²	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
54	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
55	C₄²×C₅₃	C₄²	1	2	yes	—
56	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
57	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
58	C₄×Dic₅₃	—	2	1	no	—
59	C₄×Dic₅₃	—	2	1	no	—
60	C₄×Dic₅₃	—	4	4	no	—
61	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
62	C₄×Dic₅₃	M₁₆	1	2	yes	—
63	C₂×#88322	—	2	1	no	—
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	—	2	1	no	—
66	C₂×#88322	—	2	1	no	—
67	C₂×#88322	C₄⋊C₄	1	2	yes	—
68	C₂×#88322	M₁₆	1	2	yes	—
69	C₁₆×C₅₃	—	2	4	no	—
70	#11719908	—	2	4	no	—
71	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	—	2	4	no	—
73	C₂×C₄×C₅₃	#97	1	2	yes	—
74	C₂×C₄×C₅₃	#97	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×Dic₅₃	—	2	2	no	—
82	C₂×Dic₅₃	—	2	2	no	—
83	C₂×Dic₅₃	—	4	2	no	—
84	C₂×Dic₅₃	—	4	2	no	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
87	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
88	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
89	C₈×C₅₃	C₄×Q₈	1	1	yes	characteristic
90	#88322	—	2	2	no	—
91	#88322	—	2	2	no	—
92	#12658	—	53	1	no	maximal, Sylow
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
95	C₂²×C₅₃	#423	1	1	yes	characteristic
96	C₂²×C₅₃	#432	1	1	yes	characteristic
97	C₄×C₅₃	—	2	2	no	—
98	C₄×C₅₃	#347	1	4	yes	complemented
99	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
101	C₄×C₅₃	#423	1	1	yes	characteristic
102	C₄×C₅₃	#432	1	1	yes	characteristic
103	Dic₅₃	—	4	2	no	—
104	Dic₅₃	—	4	2	no	—
105	#2150	—	53	1	no	—
106	#2150	D₅₃	1	1	yes	characteristic
107	C₂×#347	—	53	2	no	—
108	C₄×#97	—	53	1	no	—
109	#1375	—	53	2	no	—
110	C₂×C₅₃	—	2	1	no	—
111	C₂×C₅₃	—	2	1	no	—
112	C₂×C₅₃	#1767	1	1	yes	characteristic
113	C₂×C₅₃	C₂×#347	1	2	yes	—
114	C₂×M₃₂	—	53	2	no	—
115	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
116	C₂×#97	—	53	1	no	—
117	C₂×#97	—	53	1	no	—
118	C₂×#97	—	53	1	no	—
119	C₂×#97	—	53	1	no	—
120	C₂×C₄×C₈	—	53	1	no	—
121	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
122	C₄³	—	53	1	no	—
123	C₄⋊C₁₆	—	53	4	no	—
124	C₄⋊C₁₆	Dic₅₃	1	4	yes	complemented
125	#347	—	106	4	no	—
126	C₅₃	#12658	1	1	yes	characteristic, complemented, Sylow
127	C₂²×C₈	—	53	1	no	—
128	C₂²×C₈	—	53	1	no	—
129	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
131	C₂×C₄²	—	53	1	no	—
132	C₂×C₄²	—	53	1	no	—
133	C₂×C₄²	—	106	2	no	—
134	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
135	M₃₂	—	2	2	no	—
136	M₃₂	—	2	2	no	—
137	M₃₂	—	106	2	no	—
138	M₃₂	—	106	2	no	—
139	C₂×C₁₆	—	53	4	no	—
140	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
141	#97	—	53	2	no	—
142	#97	—	53	2	no	—
143	#97	—	106	2	no	—
144	#97	—	106	2	no	—
145	#97	—	106	1	no	—
146	#97	—	106	1	no	—
147	C₄×C₈	—	106	1	no	—
148	C₄×C₈	—	106	1	no	—
149	C₄×C₈	C₄×D₅₃	1	2	yes	—
150	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
151	C₂²×C₄	—	106	1	no	—
152	C₂²×C₄	—	106	1	no	—
153	C₂²×C₄	#11719912	1	2	yes	—
154	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	53	2	no	—
158	C₂×C₈	—	53	2	no	—
159	C₂×C₈	—	106	1	no	—
160	C₂×C₈	—	106	1	no	—
161	C₂×C₈	—	106	1	no	—
162	C₂×C₈	—	106	1	no	—
163	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
164	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
165	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
166	C₂×C₈	#11719947	1	1	yes	characteristic
167	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
168	C₂×C₈	#11719949	1	1	yes	characteristic
169	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
170	C₄²	—	53	2	no	—
171	C₄²	—	53	2	no	—
172	C₄²	—	106	1	no	—
173	C₄²	—	106	1	no	—
174	C₄²	—	212	4	no	—
175	C₄²	#11719912	1	1	yes	characteristic
176	C₄²	#11719912	1	1	yes	characteristic
177	C₄²	C₄×Dic₅₃	1	2	yes	—
178	C₁₆	—	2	4	no	—
179	C₁₆	—	106	4	no	—
180	C₂³	C₂×#11719912	1	1	yes	characteristic
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	106	2	no	—
183	C₂×C₄	—	106	2	no	—
184	C₂×C₄	—	212	2	no	—
185	C₂×C₄	—	212	2	no	—
186	C₂×C₄	#13133127	1	2	yes	—
187	C₂×C₄	#13133127	1	2	yes	—
188	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
189	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
190	C₂×C₄	#13133175	1	1	yes	characteristic
191	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
192	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
193	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	106	2	no	—
196	C₈	—	106	2	no	—
197	C₈	#13133179	1	1	yes	characteristic
198	C₈	D₄×Dic₅₃	1	1	yes	characteristic
199	C₈	#13133182	1	1	yes	characteristic
200	C₈	Q₈×Dic₅₃	1	1	yes	characteristic
201	C₂²	—	2	2	no	—
202	C₂²	C₂×#13133127	1	1	yes	characteristic
203	C₂²	#16810645	1	1	yes	characteristic
204	C₂²	#16810679	1	1	yes	characteristic
205	C₄	—	2	2	no	—
206	C₄	—	212	2	no	—
207	C₄	—	212	2	no	—
208	C₄	#16810400	1	4	yes	complemented
209	C₄	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
210	C₄	M₁₆×Dic₅₃	1	1	yes	characteristic
211	C₄	#16810645	1	1	yes	characteristic
212	C₄	#16810679	1	1	yes	characteristic
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	#20274570	1	1	yes	characteristic
216	C₂	C₂×#16810400	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#12658
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #2150

Complements

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

N	H ≅ G/N	type
13. C₄⋊#11719908	C₄	semidirect
17. C₅₃×C₄⋊C₁₆	C₄	semidirect
98. C₄×C₅₃	#347	semidirect
124. C₄⋊C₁₆	Dic₅₃	semidirect
126. C₅₃	#12658	semidirect
208. C₄	#16810400	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
C₂×#16810400	2
C₄×#13133127	2
#20272794	2
C₅₃×#2150	2
#12658	53