Subgroups — #27300021

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	2	yes	maximal, complemented
2	C₂×#16810405	C₂	1	2	yes	maximal
3	#20272799	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	4	yes	—
6	C₄⋊#11719908	—	2	4	no	—
7	#16810405	—	2	4	no	—
8	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
11	C₄×M₁₆×C₅₃	—	2	4	no	—
12	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
13	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	#13133157	—	2	4	no	—
17	#13133157	—	2	4	no	—
18	C₂×#11719908	—	2	4	no	—
19	C₂³×C₄×C₅₃	C₈	1	2	yes	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₅₃	—	2	2	no	—
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	C₂×M₁₆×C₅₃	—	2	2	no	—
26	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
27	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	—	2	1	no	—
34	C₂×C₄²×C₅₃	—	2	1	no	—
35	C₂×C₄²×C₅₃	C₈	1	2	yes	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	#97×C₅₃	—	2	2	no	—
39	#97×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	D₄	1	4	yes	—
42	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	—	4	4	no	—
50	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₅₃	—	2	8	no	—
57	M₁₆×C₅₃	—	2	4	no	—
58	M₁₆×C₅₃	—	2	4	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	—	2	1	no	—
63	C₂×C₈×C₅₃	—	2	1	no	—
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
68	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₅₃	#41	1	2	yes	—
72	C₄²×C₅₃	—	2	2	no	—
73	C₄²×C₅₃	—	2	2	no	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
78	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
79	#11719908	—	4	4	no	—
80	C₂³×C₅₃	—	2	1	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂³×C₅₃	—	4	2	no	—
83	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
84	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	4	4	no	—
94	C₂×C₄×C₅₃	—	4	4	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
98	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
105	C₈×C₅₃	—	2	2	no	—
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	—	2	2	no	—
108	C₈×C₅₃	#121	1	2	yes	—
109	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
110	#12785	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	4	2	no	—
114	C₂²×C₅₃	—	4	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₂²×C₅₃	#431	1	1	yes	characteristic
119	C₄×C₅₃	—	2	2	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	4	2	no	—
122	C₄×C₅₃	#349	1	4	yes	complemented
123	C₄×C₅₃	#380	1	1	yes	characteristic
124	C₄×C₅₃	#407	1	1	yes	characteristic
125	C₄×C₅₃	#423	1	1	yes	characteristic
126	C₄×C₅₃	#432	1	1	yes	characteristic
127	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
128	#2150	—	53	2	no	—
129	C₂×#349	—	53	2	no	—
130	#1377	—	53	2	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	2	1	no	—
133	C₂×C₅₃	—	4	2	no	—
134	C₂×C₅₃	#1797	1	1	yes	characteristic
135	C₂×C₅₃	C₂×#349	1	2	yes	—
136	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
137	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
138	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
139	C₂×M₃₂	—	53	4	no	—
140	C₄×M₁₆	—	2	4	no	—
141	C₂×#97	Dic₅₃	1	1	yes	characteristic
142	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
143	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
144	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
145	C₄⋊C₁₆	—	106	4	no	—
146	#349	—	106	4	no	—
147	C₅₃	#12785	1	1	yes	characteristic, complemented, Sylow
148	C₂³×C₄	#88322	1	2	yes	—
149	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
156	C₂×M₁₆	D₂₁₂	1	2	yes	—
157	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
159	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	C₂×C₄²	—	2	1	no	—
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	#88322	1	2	yes	—
165	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
167	M₃₂	—	106	4	no	—
168	M₃₂	—	106	4	no	—
169	C₂×C₁₆	—	106	4	no	—
170	#97	—	2	2	no	—
171	#97	—	2	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	C₂²⋊D₅₃	1	4	yes	—
174	C₂⁴	C₂×#88322	1	1	yes	characteristic
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	C₂×#88322	1	2	yes	—
183	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
184	C₂²×C₄	C₂×#88322	1	2	yes	—
185	C₂²×C₄	#11719917	1	2	yes	—
186	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
187	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
188	M₁₆	—	2	8	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	4	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	#11719917	1	2	yes	—
198	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
199	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
200	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
201	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
202	C₂×C₈	#11719945	1	2	yes	—
203	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	#11719917	1	1	yes	characteristic
209	C₄²	#11719917	1	1	yes	characteristic
210	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
211	C₁₆	—	212	4	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₄⋊#88322	1	2	yes	—
216	C₂³	C₂²×#88322	1	1	yes	characteristic
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂²⋊#88322	1	2	yes	—
229	C₂×C₄	C₂²⋊#88322	1	2	yes	—
230	C₂×C₄	C₄⋊#88322	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	#13133176	1	1	yes	characteristic
234	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
235	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
236	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	#13133191	1	2	yes	—
241	C₈	#13133196	1	2	yes	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
248	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
249	C₂²	#16810676	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	#16810405	1	4	yes	complemented
254	C₄	#16810483	1	1	yes	characteristic
255	C₄	#16810587	1	1	yes	characteristic
256	C₄	#16810647	1	1	yes	characteristic
257	C₄	#16810680	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	#20274695	1	1	yes	characteristic
262	C₂	C₂×#16810405	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#12785
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. #20276471	C₂	semidirect
3. #20272799	C₂	semidirect
122. C₄×C₅₃	#349	semidirect
147. C₅₃	#12785	semidirect
253. C₄	#16810405	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×#16810405	2
#20272799	2
C₂×C₄×M₁₆×C₅₃	2
#12785	53