Subgroups — #27300023

Order 13568.

454 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810581	C₂	1	1	yes	maximal, characteristic
2	#20276470	C₂	1	2	yes	maximal, complemented
3	C₂×#16810405	C₂	1	1	yes	maximal, characteristic
4	#20272798	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#349	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	C₄	1	1	yes	characteristic
7	C₂×#13133176	C₄	1	1	yes	characteristic
8	C₂×#13133157	C₂²	1	2	yes	—
9	#16810581	—	2	4	no	—
10	C₂×#13133127	C₄	1	1	yes	characteristic
11	C₂×#13133127	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16810441	—	2	4	no	—
15	#16810405	—	2	2	no	—
16	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#349	—	2	2	no	—
19	C₂²×#88322	—	2	2	no	—
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	2	no	—
22	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	#13133176	—	2	2	no	—
25	#13133176	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	C₈	1	2	yes	—
27	C₂×C₂²⋊Dic₅₃	C₈	1	2	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	#13133157	—	2	2	no	—
31	#13133157	—	2	2	no	—
32	C₂×#11719908	—	2	2	no	—
33	#13133127	—	2	2	no	—
34	#13133127	—	2	2	no	—
35	C₈×Dic₅₃	—	2	2	no	—
36	C₈×Dic₅₃	D₄	1	4	yes	—
37	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
41	C₂×M₁₆×C₅₃	Q₈	1	2	yes	—
42	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	M₃₂×C₅₃	—	2	2	no	—
45	M₃₂×C₅₃	—	2	2	no	—
46	C₂×C₁₆×C₅₃	—	2	2	no	—
47	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
57	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
58	M₁₆×C₅₃	—	2	4	no	—
59	M₁₆×C₅₃	—	2	4	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₅₃	#41	1	2	yes	—
67	C₂²⋊Dic₅₃	—	2	4	no	—
68	C₄⋊Dic₅₃	—	2	4	no	—
69	C₄×Dic₅₃	—	2	1	no	—
70	C₄×Dic₅₃	—	2	1	no	—
71	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
72	C₄×Dic₅₃	M₁₆	1	2	yes	—
73	C₂×#88322	—	2	8	no	—
74	C₂×#88322	—	2	1	no	—
75	C₂×#88322	—	2	1	no	—
76	C₂×#88322	—	2	1	no	—
77	C₂×#88322	—	2	1	no	—
78	C₂×#88322	C₂²⋊C₄	1	2	yes	—
79	C₂×#88322	M₁₆	1	2	yes	—
80	C₁₆×C₅₃	—	4	2	no	—
81	#11719908	—	4	2	no	—
82	C₂³×C₅₃	—	2	1	no	—
83	C₂³×C₅₃	—	2	1	no	—
84	C₂³×C₅₃	—	4	2	no	—
85	C₂³×C₅₃	C₄⋊C₈	1	2	yes	—
86	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	—	2	1	no	—
88	C₂×C₄×C₅₃	—	2	1	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	C₄⋊C₈	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×Dic₅₃	—	2	2	no	—
101	C₂×Dic₅₃	—	4	2	no	—
102	C₂×Dic₅₃	—	4	2	no	—
103	C₂×Dic₅₃	C₂²⋊C₈	1	4	yes	—
104	C₈×C₅₃	—	2	2	no	—
105	C₈×C₅₃	#121	1	2	yes	—
106	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
107	#88322	—	2	2	no	—
108	#88322	—	2	2	no	—
109	#12786	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	—	4	2	no	—
113	C₂²×C₅₃	—	4	2	no	—
114	C₂²×C₅₃	—	4	2	no	—
115	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
116	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₂²×C₅₃	#431	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	#380	1	1	yes	characteristic
120	C₄×C₅₃	#407	1	1	yes	characteristic
121	C₄×C₅₃	#423	1	1	yes	characteristic
122	C₄×C₅₃	#432	1	1	yes	characteristic
123	Dic₅₃	—	2	4	no	—
124	Dic₅₃	—	4	2	no	—
125	C₂×#405	—	53	1	no	—
126	#2150	—	53	2	no	—
127	C₂×#349	—	53	1	no	—
128	C₂×#349	D₅₃	1	1	yes	characteristic
129	#1377	—	53	2	no	—
130	C₂×C₅₃	—	2	1	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	4	2	no	—
133	C₂×C₅₃	#1797	1	1	yes	characteristic
134	C₂×C₅₃	#2115	1	2	yes	—
135	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
136	C₂³×C₈	—	53	1	no	—
137	C₂×#117	—	53	1	no	—
138	C₂×M₃₂	—	53	2	no	—
139	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
140	#405	—	106	4	no	—
141	C₂×#97	—	53	1	no	—
142	C₂×#97	—	53	1	no	—
143	C₂×C₄×C₈	—	53	1	no	—
144	C₂×C₄×C₈	—	53	1	no	—
145	C₄⋊C₁₆	—	106	4	no	—
146	#349	—	2	2	no	—
147	#349	—	106	2	no	—
148	C₅₃	#12786	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
150	C₂×M₁₆	—	2	2	no	—
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	Dic₁₀₆	1	2	yes	—
153	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
154	C₂²×C₈	—	53	1	no	—
155	C₂²×C₈	—	53	1	no	—
156	C₂²×C₈	—	106	2	no	—
157	C₂²×C₈	—	106	2	no	—
158	C₂²×C₈	—	106	2	no	—
159	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
160	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
161	#117	—	106	2	no	—
162	#117	—	106	2	no	—
163	C₂×C₄⋊C₄	—	53	2	no	—
164	C₂×C₂²⋊C₄	—	53	2	no	—
165	C₂×C₄²	—	53	1	no	—
166	C₂×C₄²	—	53	1	no	—
167	M₃₂	—	2	2	no	—
168	M₃₂	—	2	2	no	—
169	M₃₂	—	106	2	no	—
170	M₃₂	—	106	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	106	2	no	—
173	#97	—	106	2	no	—
174	#97	—	106	2	no	—
175	C₄×C₈	—	53	4	no	—
176	C₄×C₈	—	106	2	no	—
177	C₂⁴	C₈×D₅₃	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	53	2	no	—
183	C₂²×C₄	—	53	2	no	—
184	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
185	C₂²×C₄	#11719912	1	2	yes	—
186	C₂²×C₄	#11719919	1	2	yes	—
187	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
188	M₁₆	—	2	4	no	—
189	M₁₆	—	2	4	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	53	2	no	—
193	C₂×C₈	—	53	2	no	—
194	C₂×C₈	—	106	8	no	—
195	C₂×C₈	—	106	1	no	—
196	C₂×C₈	—	106	1	no	—
197	C₂×C₈	—	106	1	no	—
198	C₂×C₈	—	106	1	no	—
199	C₂×C₈	#11719919	1	2	yes	—
200	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
201	C₂×C₈	D₄×D₅₃	1	2	yes	—
202	C₂×C₈	#11719947	1	2	yes	—
203	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
204	C₄⋊C₄	—	106	4	no	—
205	C₂²⋊C₄	—	106	4	no	—
206	C₄²	—	53	2	no	—
207	C₄²	—	53	2	no	—
208	C₄²	—	106	1	no	—
209	C₄²	—	106	1	no	—
210	C₁₆	—	4	2	no	—
211	C₁₆	—	212	2	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	4	2	no	—
215	C₂³	#13133146	1	2	yes	—
216	C₂³	C₂×C₈×D₅₃	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	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	53	4	no	—
224	C₂×C₄	—	106	2	no	—
225	C₂×C₄	—	212	2	no	—
226	C₂×C₄	—	212	2	no	—
227	C₂×C₄	#13133146	1	2	yes	—
228	C₂×C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
229	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
230	C₂×C₄	#13133173	1	1	yes	characteristic
231	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
232	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	106	2	no	—
236	C₈	—	106	2	no	—
237	C₈	#13133189	1	2	yes	—
238	C₈	#13133194	1	2	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	C₂²⋊C₈×D₅₃	1	1	yes	characteristic
245	C₂²	C₂×#13133146	1	1	yes	characteristic
246	C₂²	#16810673	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	106	4	no	—
249	C₄	—	212	2	no	—
250	C₄	#16810481	1	1	yes	characteristic
251	C₄	#16810585	1	1	yes	characteristic
252	C₄	#16810646	1	1	yes	characteristic
253	C₄	#16810677	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	#20274693	1	1	yes	characteristic
258	C₂	#20276334	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₅₃
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#349	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#349	D₅₃
hypocenter	C₅₃	#12786
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
2. #20276470	C₂	semidirect
4. #20272798	C₂	semidirect
148. C₅₃	#12786	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810581	2
#20276470	2
C₂×#16810405	2
#20272798	2
C₂×C₅₃×#349	2
#12786	53