Subgroups — #27273517

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	2	yes	maximal
2	#20272546	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810680	—	2	2	no	—
5	#16810680	—	2	2	no	—
6	#16810676	C₄	1	4	yes	—
7	#16810676	C₂²	1	4	yes	—
8	#16810647	—	2	2	no	—
9	#16810587	—	2	2	no	—
10	C₄×#11719917	C₄	1	4	yes	—
11	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×M₁₆×C₅₃	—	2	2	no	—
14	C₄×M₁₆×C₅₃	—	2	2	no	—
15	C₂×#97×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
17	C₂×#11719917	—	2	4	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₄⋊#88322	—	2	4	no	—
20	C₂²⋊#88322	—	2	4	no	—
21	C₂²⋊#88322	—	2	4	no	—
22	#13133128	—	2	4	no	—
23	C₄×#88322	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
27	C₂×M₁₆×C₅₃	—	2	8	no	—
28	C₂×M₁₆×C₅₃	—	2	4	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	D₄	1	2	yes	—
34	C₂×C₄²×C₅₃	D₄	1	2	yes	—
35	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#97×C₅₃	—	2	4	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	M₁₆×C₅₃	—	2	16	no	—
58	C₂×C₈×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
61	C₄²×C₅₃	C₄²	1	1	yes	characteristic
62	C₄²×C₅₃	C₄²	1	1	yes	characteristic
63	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₅₃	C₂×D₄	1	2	yes	—
71	#11719917	—	4	8	no	—
72	C₂×#88322	—	2	8	no	—
73	C₂³×C₅₃	—	2	2	no	—
74	C₂³×C₅₃	—	4	1	no	—
75	C₂³×C₅₃	—	4	1	no	—
76	C₂³×C₅₃	#95	1	2	yes	—
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	4	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	—	2	4	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	1	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	#95	1	2	yes	—
90	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
99	C₈×C₅₃	—	2	8	no	—
100	#88322	—	4	8	no	—
101	#7862	—	53	1	no	maximal, Sylow
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	—	2	1	no	—
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	4	4	no	—
108	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
109	C₂²×C₅₃	#423	1	2	yes	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	2	1	no	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	H(3,ℤ/4)	1	2	yes	—
116	C₄×C₅₃	#338	1	2	yes	—
117	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
118	C₄×C₅₃	#407	1	2	yes	—
119	#2920	—	53	2	no	—
120	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
121	#1275	—	53	4	no	—
122	C₂×C₅₃	—	2	2	no	—
123	C₂×C₅₃	—	4	2	no	—
124	C₂×C₅₃	#1728	1	1	yes	characteristic
125	C₂×C₅₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#338	1	1	yes	characteristic
127	C₂²×M₁₆	Dic₅₃	1	2	yes	—
128	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
129	#432	—	106	2	no	—
130	#432	—	106	2	no	—
131	#431	—	53	4	no	—
132	#431	—	53	4	no	—
133	#423	—	106	2	no	—
134	#407	—	106	2	no	—
135	C₄×M₁₆	—	2	2	no	—
136	C₄×M₁₆	—	2	2	no	—
137	C₄×M₁₆	—	53	4	no	—
138	C₂×#97	C₂×D₅₃	1	2	yes	—
139	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
140	C₅₃	#7862	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
142	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
143	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
144	C₂×M₁₆	—	2	4	no	—
145	C₂×M₁₆	—	2	8	no	—
146	C₂×M₁₆	—	106	4	no	—
147	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
148	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
149	C₂×C₄²	C₄×D₅₃	1	2	yes	—
150	C₂×C₄²	C₄×D₅₃	1	2	yes	—
151	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
153	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
154	C₂×C₄²	C₂²⋊D₅₃	1	2	yes	—
155	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
156	C₄⋊C₈	—	106	4	no	—
157	C₄⋊C₈	—	106	4	no	—
158	C₂²⋊C₈	—	106	4	no	—
159	C₂²⋊C₈	—	106	4	no	—
160	#97	—	2	4	no	—
161	#97	—	106	4	no	—
162	C₄×C₈	—	2	2	no	—
163	C₄×C₈	—	2	2	no	—
164	C₄×C₈	—	53	8	no	—
165	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
170	C₂²×C₄	#11719919	1	2	yes	—
171	C₂²×C₄	#11719919	1	2	yes	—
172	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
175	C₂²×C₄	#11719921	1	2	yes	—
176	C₂²×C₄	#11719921	1	2	yes	—
177	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
178	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
181	M₁₆	—	2	16	no	—
182	M₁₆	—	212	8	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	106	8	no	—
186	C₂×C₈	#11719917	1	8	yes	complemented
187	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
188	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
189	C₄²	#11719919	1	2	yes	—
190	C₄²	#11719919	1	2	yes	—
191	C₄²	#11719921	1	2	yes	—
192	C₄²	#11719921	1	2	yes	—
193	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
194	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
195	C₄²	C₂×C₄×D₅₃	1	2	yes	—
196	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	#13133124	1	2	yes	—
201	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	#13133124	1	1	yes	characteristic
214	C₂×C₄	#13133124	1	1	yes	characteristic
215	C₂×C₄	#13133124	1	2	yes	—
216	C₂×C₄	#13133124	1	1	yes	characteristic
217	C₂×C₄	#13133124	1	1	yes	characteristic
218	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
219	C₂×C₄	C₂×#11719921	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂×#11719919	1	2	yes	—
222	C₂×C₄	C₂×#11719917	1	4	yes	—
223	C₈	—	2	8	no	—
224	C₈	—	212	8	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	2	no	—
230	C₂²	—	4	4	no	—
231	C₂²	C₂×#13133124	1	1	yes	characteristic
232	C₂²	#16810647	1	2	yes	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	#16810380	1	2	yes	—
239	C₄	#16810381	1	2	yes	—
240	C₄	C₄×#11719917	1	2	yes	—
241	C₄	#16810587	1	2	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	#20274409	1	1	yes	characteristic
245	C₂	C₂×#16810380	1	1	yes	characteristic
246	C₂	C₂×#16810381	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₅₃
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#7862
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20272546	C₂	semidirect
60. C₂×C₈×C₅₃	M₁₆	semidirect
140. C₅₃	#7862	semidirect
186. C₂×C₈	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
#20272546	2
C₂×C₄×M₁₆×C₅₃	2
#7862	53