Subgroups — #27270648

Order 13568.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274568	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	1	no	—
5	C₂×C₄⋊#88322	C₄	1	2	yes	—
6	C₂×#13133128	—	2	1	no	—
7	C₂×C₄×#88322	—	2	1	no	—
8	#16810378	—	2	1	no	—
9	#16810378	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₅₃	—	2	1	no	—
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	—	2	1	no	—
13	C₅₃×#336	C₄	1	2	yes	—
14	C₂²×#88322	—	2	2	no	—
15	C₂²×#88322	—	2	2	no	—
16	C₄⋊#88322	—	2	4	no	—
17	C₄⋊#88322	—	2	2	no	—
18	C₄⋊#88322	C₈	1	4	yes	complemented
19	#13133128	—	4	1	no	—
20	#13133128	—	4	1	no	—
21	C₄×#88322	—	4	1	no	—
22	C₄×#88322	—	4	1	no	—
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
31	C₄×C₈×C₅₃	—	2	4	no	—
32	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	2	4	no	—
40	C₂×C₈×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₄²×C₅₃	—	2	2	no	—
43	C₄²×C₅₃	—	2	2	no	—
44	C₄²×C₅₃	—	2	2	no	—
45	C₄²×C₅₃	—	4	1	no	—
46	C₄²×C₅₃	—	4	1	no	—
47	C₄²×C₅₃	C₂×C₈	1	2	yes	—
48	C₄²×C₅₃	M₁₆	1	2	yes	—
49	C₄²×C₅₃	D₈	1	1	yes	characteristic
50	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
51	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
52	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
53	C₂×#88322	—	2	4	no	—
54	C₂×#88322	—	2	4	no	—
55	C₂×#88322	—	4	2	no	—
56	C₂×#88322	—	4	2	no	—
57	C₂×#88322	—	4	2	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂³×C₅₃	#95	1	1	yes	characteristic
60	C₂×C₄×C₅₃	—	2	1	no	—
61	C₂×C₄×C₅₃	—	2	1	no	—
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
76	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
77	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#104	1	2	yes	—
80	C₂×C₄×C₅₃	#104	1	2	yes	—
81	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
82	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#108	1	2	yes	—
85	C₈×C₅₃	—	4	4	no	—
86	#88322	—	4	4	no	—
87	#88322	—	4	4	no	—
88	#7027	—	53	1	no	maximal, Sylow
89	C₂²×C₅₃	#336	1	1	yes	characteristic
90	C₂²×C₅₃	#339	1	1	yes	characteristic
91	C₂²×C₅₃	#339	1	1	yes	characteristic
92	C₂²×C₅₃	#340	1	1	yes	characteristic
93	C₂²×C₅₃	#341	1	1	yes	characteristic
94	C₂²×C₅₃	#344	1	1	yes	characteristic
95	C₂²×C₅₃	#344	1	1	yes	characteristic
96	C₄×C₅₃	—	2	1	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	2	no	—
99	C₄×C₅₃	—	2	1	no	—
100	C₄×C₅₃	—	2	1	no	—
101	C₄×C₅₃	—	2	2	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
106	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
107	#1767	—	53	1	no	—
108	C₄×C₄⋊C₈	—	53	1	no	—
109	#1744	D₅₃	1	1	yes	characteristic
110	C₂×C₅₃	#1278	1	1	yes	characteristic
111	C₂×C₅₃	#1279	1	1	yes	characteristic
112	C₂×C₅₃	#1281	1	1	yes	characteristic
113	C₂×C₅₃	#1300	1	1	yes	characteristic
114	C₂×C₅₃	#1301	1	1	yes	characteristic
115	C₂×C₅₃	#1302	1	1	yes	characteristic
116	C₂×C₅₃	#1303	1	1	yes	characteristic
117	C₂×C₄⋊C₈	—	53	2	no	—
118	C₂×C₄⋊C₈	—	106	1	no	—
119	C₂×#97	—	106	1	no	—
120	C₂×C₄×C₈	—	2	1	no	—
121	C₂×C₄×C₈	—	106	1	no	—
122	C₄³	C₂×D₅₃	1	1	yes	characteristic
123	#336	—	2	1	no	—
124	#336	—	53	2	no	—
125	#336	—	106	1	no	—
126	#336	Dic₅₃	1	2	yes	—
127	C₅₃	#7027	1	1	yes	characteristic, complemented, Sylow
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	106	2	no	—
130	C₂²×C₈	—	106	2	no	—
131	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
132	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
133	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
134	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
135	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
136	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
137	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
138	C₄⋊C₈	—	53	4	no	—
139	C₄⋊C₈	—	106	4	no	—
140	C₄⋊C₈	—	106	2	no	—
141	#97	—	212	1	no	—
142	#97	—	212	1	no	—
143	C₄×C₈	—	2	4	no	—
144	C₄×C₈	—	212	1	no	—
145	C₄×C₈	—	212	1	no	—
146	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
147	C₂²×C₄	#11719919	1	1	yes	characteristic
148	C₂²×C₄	#11719919	1	1	yes	characteristic
149	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
150	C₂²×C₄	#11719921	1	1	yes	characteristic
151	C₂²×C₄	#11719921	1	1	yes	characteristic
152	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	106	4	no	—
157	C₂×C₈	—	106	4	no	—
158	C₂×C₈	—	212	2	no	—
159	C₂×C₈	—	212	2	no	—
160	C₂×C₈	—	212	2	no	—
161	C₂×C₈	—	212	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	—	4	1	no	—
166	C₄²	—	4	1	no	—
167	C₄²	C₈×D₅₃	1	2	yes	—
168	C₄²	#11719912	1	2	yes	—
169	C₄²	#11719913	1	1	yes	characteristic
170	C₄²	#11719913	1	1	yes	characteristic
171	C₄²	D₄₂₄	1	1	yes	characteristic
172	C₄²	Dic₂₁₂	1	1	yes	characteristic
173	C₂³	#13133124	1	1	yes	characteristic
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	C₈×Dic₅₃	1	1	yes	characteristic
187	C₂×C₄	#13133127	1	1	yes	characteristic
188	C₂×C₄	#13133129	1	2	yes	—
189	C₂×C₄	#13133139	1	1	yes	characteristic
190	C₂×C₄	#13133139	1	1	yes	characteristic
191	C₂×C₄	#13133142	1	1	yes	characteristic
192	C₂×C₄	#13133142	1	1	yes	characteristic
193	C₂×C₄	#13133144	1	2	yes	—
194	C₂×C₄	#13133145	1	2	yes	—
195	C₂×C₄	#13133146	1	2	yes	—
196	C₂×C₄	#13133149	1	1	yes	characteristic
197	C₂×C₄	#13133151	1	1	yes	characteristic
198	C₂×C₄	#13133152	1	2	yes	—
199	C₈	—	4	4	no	—
200	C₈	—	212	4	no	—
201	C₈	—	212	4	no	—
202	C₂²	#16810377	1	1	yes	characteristic
203	C₂²	#16810382	1	1	yes	characteristic
204	C₂²	#16810383	1	1	yes	characteristic
205	C₂²	#16810386	1	1	yes	characteristic
206	C₂²	#16810387	1	1	yes	characteristic
207	C₂²	#16810393	1	1	yes	characteristic
208	C₂²	#16810395	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	#16810352	1	2	yes	—
219	C₄	#16810355	1	2	yes	—
220	C₂	#20272551	1	1	yes	characteristic
221	C₂	#20272555	1	1	yes	characteristic
222	C₂	#20272559	1	1	yes	characteristic
223	C₂	#20272608	1	1	yes	characteristic
224	C₂	#20272611	1	1	yes	characteristic
225	C₂	#20272614	1	1	yes	characteristic
226	C₂	#20272617	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719921
commutator	C₂×C₄×C₅₃	C₄×C₈
Frattini	C₄³	C₂×D₅₃
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#7027
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

Complements

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

N	H ≅ G/N	type
18. C₄⋊#88322	C₈	semidirect
127. C₅₃	#7027	semidirect

Maximal subgroups

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

subgroup	index
#20274568	2
C₄×C₄⋊#88322	2
C₅₃×#1744	2
#7027	53