Subgroups — #27349814

Order 13568.

814 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283676	C₂	1	2	yes	maximal
2	C₄×#13133223	C₂	1	1	yes	maximal, characteristic
3	#20273019	C₂	1	2	yes	maximal, complemented
4	#20273013	C₂	1	2	yes	maximal, complemented
5	#20273005	C₂	1	2	yes	maximal, complemented
6	C₈×#11719912	C₂	1	4	yes	maximal, complemented
7	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133223	C₄	1	2	yes	—
10	C₄×#11719945	C₂²	1	1	yes	characteristic
11	#16810750	C₄	1	4	yes	—
12	#16810700	C₄	1	4	yes	—
13	#16810691	C₄	1	8	yes	—
14	#16810674	C₂²	1	4	yes	—
15	#16810578	C₄	1	2	yes	—
16	#16810577	C₄	1	4	yes	—
17	C₄×#11719917	C₂²	1	1	yes	characteristic
18	C₄×#11719912	C₂²	1	4	yes	—
19	C₄×#11719912	C₂²	1	2	yes	—
20	C₂×C₄×#88322	C₂²	1	2	yes	—
21	C₄×C₈×D₅₃	C₂²	1	2	yes	—
22	#16810345	C₂²	1	4	yes	—
23	#16810344	C₂²	1	4	yes	—
24	C₈×#88322	C₂²	1	2	yes	—
25	C₈×#88322	C₂²	1	1	yes	characteristic
26	C₈×#88322	C₂²	1	1	yes	characteristic
27	C₅₃×#431	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#322	C₂²	1	2	yes	—
30	C₈²×C₅₃	C₂²	1	2	yes	—
31	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
32	#13133223	—	2	8	no	—
33	C₂×#11719917	C₂×C₄	1	2	yes	—
34	C₂×#11719912	C₂×C₄	1	4	yes	—
35	C₂×#11719912	C₂×C₄	1	8	yes	—
36	C₂²×#88322	C₂×C₄	1	4	yes	—
37	C₂×C₈×D₅₃	C₂×C₄	1	4	yes	—
38	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
39	C₄×D₂₁₂	C₂×C₄	1	2	yes	—
40	C₄×C₂²⋊D₅₃	C₂×C₄	1	4	yes	—
41	#13133176	C₂×C₄	1	1	yes	characteristic
42	#13133174	C₂×C₄	1	2	yes	—
43	C₄²×D₅₃	C₂³	1	2	yes	—
44	#13133146	C₂×C₄	1	8	yes	—
45	#13133129	C₂×C₄	1	8	yes	—
46	#13133128	C₂×C₄	1	2	yes	—
47	#13133127	C₂×C₄	1	4	yes	—
48	#13133127	C₂×C₄	1	8	yes	—
49	C₄×#88322	C₂×C₄	1	4	yes	—
50	C₄×#88322	C₂³	1	2	yes	—
51	C₄×#88322	C₂³	1	2	yes	—
52	C₄×#88322	C₂³	1	2	yes	—
53	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
57	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
58	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₅₃	C₂³	1	4	yes	—
61	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₂²⋊D₅₃	C₄²	1	2	yes	—
65	#11719945	—	2	4	no	—
66	C₂×D₂₁₂	C₄²	1	1	yes	characteristic
67	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
68	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
69	C₂×Dic₁₀₆	C₄²	1	1	yes	characteristic
70	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
71	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
74	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
77	C₂²⋊Dic₅₃	C₄²	1	2	yes	—
78	#11719921	C₄²	1	4	yes	—
79	C₄⋊Dic₅₃	C₄²	1	2	yes	—
80	#11719919	C₄²	1	4	yes	—
81	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
82	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
83	#11719917	—	2	4	no	—
84	C₂×#88322	—	2	8	no	—
85	C₂×#88322	C₂²×C₄	1	4	yes	—
86	C₂×#88322	C₂²×C₄	1	4	yes	—
87	C₂×#88322	C₂²×C₄	1	4	yes	—
88	#11719912	—	2	8	no	—
89	#11719912	—	2	16	no	—
90	C₈×D₅₃	—	2	8	no	—
91	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
92	C₂²×D₅₃	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
99	C₂²⋊D₅₃	—	2	4	no	—
100	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
101	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
102	D₂₁₂	—	2	2	no	—
103	C₄×D₅₃	—	2	4	no	—
104	C₄×D₅₃	—	2	8	no	—
105	Dic₁₀₆	—	2	2	no	—
106	C₈×C₅₃	—	2	8	no	—
107	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
108	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
109	#88322	—	2	4	no	—
110	#88322	—	2	4	no	—
111	#88322	#131	1	8	yes	complemented
112	#17335	—	53	1	no	maximal, Sylow
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
115	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
116	C₂×D₅₃	—	2	2	no	—
117	C₂×D₅₃	—	2	4	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
120	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
121	C₄×C₅₃	C₂×#131	1	4	yes	—
122	Dic₅₃	—	2	2	no	—
123	Dic₅₃	—	2	4	no	—
124	#2965	—	53	2	no	—
125	C₄×#131	—	53	1	no	—
126	#1455	—	53	2	no	—
127	#1453	—	53	2	no	—
128	#1451	—	53	2	no	—
129	#1450	D₅₃	1	1	yes	characteristic, complemented
130	C₈×M₁₆	—	53	4	no	—
131	C₂×C₈²	—	53	1	no	—
132	C₂×C₅₃	—	2	2	no	—
133	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
134	C₂×C₅₃	C₄×#131	1	2	yes	—
135	D₅₃	—	4	2	no	—
136	C₂×#131	—	53	2	no	—
137	C₄×#41	—	53	1	no	—
138	#446	—	53	4	no	—
139	#436	—	53	4	no	—
140	#435	—	53	8	no	—
141	#431	—	53	4	no	—
142	#431	C₂×D₅₃	1	2	yes	—
143	#405	—	53	2	no	—
144	#405	—	53	4	no	—
145	C₄×M₁₆	—	53	1	no	—
146	C₄×M₁₆	—	53	2	no	—
147	C₄×M₁₆	—	53	4	no	—
148	C₂×C₄×C₈	—	53	2	no	—
149	C₂×C₄×C₈	—	53	2	no	—
150	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
151	#322	—	53	4	no	—
152	#322	—	53	4	no	—
153	#322	C₂×D₅₃	1	2	yes	—
154	C₈²	—	53	1	no	—
155	C₈²	—	53	1	no	—
156	C₈²	—	53	2	no	—
157	C₈²	C₂×D₅₃	1	2	yes	—
158	C₅₃	#17335	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	—	53	1	no	—
160	#131	—	106	8	no	—
161	C₂×M₁₆	—	53	2	no	—
162	C₂×M₁₆	—	53	4	no	—
163	C₂×M₁₆	—	53	8	no	—
164	C₂²×C₈	—	53	4	no	—
165	C₂²×C₈	—	53	4	no	—
166	C₂²×C₈	C₄×D₅₃	1	2	yes	—
167	C₄×Q₈	—	53	2	no	—
168	C₄×D₄	—	53	4	no	—
169	C₄×D₄	—	53	2	no	—
170	#117	—	53	1	no	—
171	#117	—	53	2	no	—
172	C₂×C₄²	—	53	2	no	—
173	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
174	C₄⋊C₈	—	53	8	no	—
175	C₄⋊C₈	C₄×D₅₃	1	4	yes	—
176	C₂²⋊C₈	—	53	8	no	—
177	C₂²⋊C₈	C₄×D₅₃	1	4	yes	—
178	#97	—	53	4	no	—
179	#97	—	53	8	no	—
180	#97	—	53	2	no	—
181	C₄×C₈	—	53	4	no	—
182	C₄×C₈	—	53	4	no	—
183	C₄×C₈	—	53	2	no	—
184	C₄×C₈	—	53	2	no	—
185	C₄×C₈	—	53	2	no	—
186	C₄×C₈	C₄×D₅₃	1	2	yes	—
187	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
188	C₄×C₈	C₂²×D₅₃	1	4	yes	—
189	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
190	#41	—	106	4	no	—
191	C₂×Q₈	—	53	1	no	—
192	C₂×D₄	—	53	2	no	—
193	C₂×D₄	—	53	1	no	—
194	C₂²×C₄	—	53	4	no	—
195	C₂²×C₄	—	53	2	no	—
196	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
198	M₁₆	—	106	16	no	—
199	M₁₆	—	106	4	no	—
200	M₁₆	—	106	8	no	—
201	C₂×C₈	—	53	4	no	—
202	C₂×C₈	—	53	4	no	—
203	C₂×C₈	—	53	4	no	—
204	C₂×C₈	—	106	8	no	—
205	C₂×C₈	—	106	8	no	—
206	C₂×C₈	#11719912	1	8	yes	complemented
207	C₂×C₈	C₂×C₄×D₅₃	1	8	yes	—
208	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
209	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
210	C₄⋊C₄	—	53	4	no	—
211	C₄⋊C₄	—	53	2	no	—
212	C₂²⋊C₄	—	53	2	no	—
213	C₂²⋊C₄	—	53	4	no	—
214	C₄²	—	53	2	no	—
215	C₄²	—	53	4	no	—
216	C₄²	C₂×C₄×D₅₃	1	2	yes	—
217	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
218	C₄²	C₂³×D₅₃	1	1	yes	characteristic
219	C₂³	—	53	2	no	—
220	C₂³	C₄²×D₅₃	1	1	yes	characteristic
221	Q₈	—	106	2	no	—
222	D₄	—	106	4	no	—
223	D₄	—	106	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	53	4	no	—
226	C₂×C₄	—	53	2	no	—
227	C₂×C₄	—	106	4	no	—
228	C₂×C₄	—	106	8	no	—
229	C₂×C₄	C₄²×D₅₃	1	2	yes	—
230	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
231	C₂×C₄	C₂×#11719912	1	4	yes	—
232	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
234	C₈	—	2	8	no	—
235	C₈	—	53	8	no	—
236	C₈	—	106	4	no	—
237	C₈	—	106	4	no	—
238	C₈	C₂×#11719912	1	4	yes	—
239	C₈	C₂×#11719912	1	4	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	106	4	no	—
242	C₂²	—	106	2	no	—
243	C₂²	C₄×#11719912	1	2	yes	—
244	C₂²	C₂×C₄²×D₅₃	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	106	4	no	—
247	C₄	—	106	2	no	—
248	C₄	C₄×#11719912	1	2	yes	—
249	C₄	C₂²×#11719912	1	2	yes	—
250	C₄	#131×D₅₃	1	4	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	212	2	no	—
253	C₂	C₂×C₄×#11719912	1	1	yes	characteristic
254	C₂	#20283137	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#1450	C₂
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1450	D₅₃
hypocenter	C₅₃	#17335
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #20273019	C₂	semidirect
4. #20273013	C₂	semidirect
5. #20273005	C₂	semidirect
6. C₈×#11719912	C₂	semidirect
7. C₂×C₈×#88322	C₂	semidirect
8. C₅₃×#1450	C₂	semidirect
70. C₂×C₈×C₅₃	M₁₆	semidirect
111. #88322	#131	semidirect
129. #1450	D₅₃	semidirect
158. C₅₃	#17335	semidirect
206. C₂×C₈	#11719912	semidirect

Maximal subgroups

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

subgroup	index
#20283676	2
C₄×#13133223	2
#20273019	2
#20273013	2
#20273005	2
C₈×#11719912	2
C₂×C₈×#88322	2
C₅₃×#1450	2
#17335	53