Subgroups — #27282948

Order 13568.

502 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	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#20275650	C₂	1	2	yes	maximal
3	C₂×#16810353	C₂	1	2	yes	maximal
4	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₄	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	C₂²	1	2	yes	—
8	C₄×Dic₂₁₂	—	2	2	no	—
9	C₄×Dic₂₁₂	—	2	2	no	—
10	C₂×#13133151	C₄	1	1	yes	characteristic
11	C₂×#13133142	C₄	1	2	yes	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
14	#16810377	—	2	4	no	—
15	#16810353	—	2	4	no	—
16	C₈×#88322	—	2	2	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#335	—	2	2	no	—
20	C₂²×Dic₁₀₆	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₁₂	—	2	2	no	—
22	C₂×Dic₂₁₂	—	2	2	no	—
23	C₂×Dic₂₁₂	C₈	1	4	yes	complemented
24	C₂²×#88322	—	2	1	no	—
25	C₂²×#88322	D₄	1	2	yes	—
26	C₄×Dic₁₀₆	—	2	2	no	—
27	C₄×Dic₁₀₆	—	2	2	no	—
28	C₄×Dic₁₀₆	—	2	2	no	—
29	C₄×Dic₁₀₆	—	2	2	no	—
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂×#11719919	—	2	2	no	—
32	C₂×C₄×Dic₅₃	—	2	2	no	—
33	#13133151	—	2	2	no	—
34	#13133142	—	2	4	no	—
35	C₄×#88322	—	2	1	no	—
36	C₄×#88322	—	2	1	no	—
37	C₄×#88322	—	2	1	no	—
38	C₄×#88322	D₄	1	2	yes	—
39	C₂²×C₈×C₅₃	—	2	2	no	—
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄⋊C₈×C₅₃	—	2	2	no	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
50	C₂²×Dic₅₃	—	2	2	no	—
51	C₂²×Dic₅₃	—	2	2	no	—
52	C₂×Dic₁₀₆	—	2	2	no	—
53	C₂×Dic₁₀₆	—	2	2	no	—
54	C₂×Dic₁₀₆	—	2	2	no	—
55	C₂×Dic₁₀₆	—	2	2	no	—
56	C₂×Dic₁₀₆	—	2	2	no	—
57	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	—	4	4	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	#41	1	1	yes	characteristic
70	C₄⋊Dic₅₃	—	2	2	no	—
71	C₄⋊Dic₅₃	—	2	2	no	—
72	#11719919	—	4	2	no	—
73	#11719919	—	4	2	no	—
74	C₄×Dic₅₃	—	4	2	no	—
75	C₄×Dic₅₃	—	4	2	no	—
76	C₂×#88322	—	2	4	no	—
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	—	2	2	no	—
79	C₂×#88322	—	2	2	no	—
80	C₂×#88322	Q₁₆	1	4	yes	complemented
81	Dic₂₁₂	—	2	8	no	—
82	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
95	C₂×C₄×C₅₃	#135	1	2	yes	—
96	C₂×Dic₅₃	—	2	4	no	—
97	C₂×Dic₅₃	—	4	2	no	—
98	C₂×Dic₅₃	—	4	2	no	—
99	C₂×Dic₅₃	—	4	2	no	—
100	C₂×Dic₅₃	—	4	4	no	—
101	Dic₁₀₆	—	2	4	no	—
102	Dic₁₀₆	—	2	4	no	—
103	Dic₁₀₆	—	4	2	no	—
104	Dic₁₀₆	—	4	2	no	—
105	C₈×C₅₃	—	2	2	no	—
106	C₈×C₅₃	—	8	2	no	—
107	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
108	#88322	—	2	4	no	—
109	#88322	—	2	4	no	—
110	#9656	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	#390	1	1	yes	characteristic
112	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
113	C₂²×C₅₃	#436	1	1	yes	characteristic
114	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
115	C₂²×C₅₃	#443	1	2	yes	—
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₄×C₅₃	#408	1	1	yes	characteristic
120	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
121	C₄×C₅₃	#436	1	1	yes	characteristic
122	C₄×C₅₃	#467	1	2	yes	—
123	C₄×C₅₃	#494	1	1	yes	characteristic
124	C₄×C₅₃	#495	1	1	yes	characteristic
125	Dic₅₃	—	4	4	no	—
126	Dic₅₃	—	8	2	no	—
127	C₂×C₄×Q₁₆	—	53	1	no	—
128	#1984	—	53	2	no	—
129	C₂×#335	D₅₃	1	1	yes	characteristic
130	C₂×Q₈⋊C₈	—	53	2	no	—
131	C₂×C₈²	—	53	1	no	—
132	C₂×C₅₃	C₈×Q₁₆	1	2	yes	—
133	C₂×C₅₃	#1591	1	2	yes	—
134	C₂×C₅₃	#1954	1	1	yes	characteristic
135	C₂×C₅₃	#1969	1	1	yes	characteristic
136	C₂×C₅₃	#1974	1	1	yes	characteristic
137	C₂²×Q₁₆	—	53	1	no	—
138	C₂×C₄×Q₈	—	53	2	no	—
139	C₄×Q₁₆	—	106	2	no	—
140	C₄×Q₁₆	—	106	2	no	—
141	C₂×#107	—	53	1	no	—
142	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
143	C₂×#103	—	53	2	no	—
144	C₂×C₄×C₈	—	53	1	no	—
145	C₂×C₄×C₈	—	53	1	no	—
146	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
147	#336	—	106	4	no	—
148	#335	—	2	2	no	—
149	Q₈⋊C₈	—	106	4	no	—
150	C₈²	—	106	2	no	—
151	C₅₃	#9656	1	1	yes	characteristic, complemented, Sylow
152	C₂²×Q₈	—	53	2	no	—
153	C₂×Q₁₆	—	53	4	no	—
154	C₂×Q₁₆	—	106	2	no	—
155	C₂×Q₁₆	—	106	2	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	53	2	no	—
158	C₂²×C₈	—	106	1	no	—
159	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
160	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
161	C₄×Q₈	—	106	2	no	—
162	C₄×Q₈	—	106	2	no	—
163	C₄×Q₈	—	106	2	no	—
164	C₄×Q₈	—	106	2	no	—
165	C₂×C₄⋊C₄	—	53	2	no	—
166	C₂×C₄⋊C₄	—	106	2	no	—
167	C₂×C₄²	—	106	2	no	—
168	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
169	#107	—	106	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	#103	—	106	4	no	—
173	C₄×C₈	—	53	2	no	—
174	C₄×C₈	—	106	1	no	—
175	C₄×C₈	—	106	1	no	—
176	C₄×C₈	—	106	1	no	—
177	C₄×C₈	D₂₁₂	1	2	yes	—
178	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
179	C₂×Q₈	—	53	4	no	—
180	C₂×Q₈	—	106	2	no	—
181	C₂×Q₈	—	106	2	no	—
182	C₂×Q₈	—	106	2	no	—
183	C₂×Q₈	—	106	2	no	—
184	C₂×Q₈	—	106	2	no	—
185	C₂²×C₄	—	106	2	no	—
186	C₂²×C₄	—	106	2	no	—
187	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
188	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
189	C₂²×C₄	#11719949	1	1	yes	characteristic
190	Q₁₆	—	106	8	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	53	4	no	—
196	C₂×C₈	—	106	4	no	—
197	C₂×C₈	—	106	2	no	—
198	C₂×C₈	—	106	2	no	—
199	C₂×C₈	—	106	2	no	—
200	C₂×C₈	C₈×D₅₃	1	2	yes	—
201	C₂×C₈	#11719912	1	2	yes	—
202	C₂×C₈	#11719921	1	2	yes	—
203	C₂×C₈	#11719921	1	2	yes	—
204	C₄⋊C₄	—	106	2	no	—
205	C₄⋊C₄	—	106	2	no	—
206	C₄⋊C₄	—	212	2	no	—
207	C₄⋊C₄	—	212	2	no	—
208	C₄²	—	212	2	no	—
209	C₄²	—	212	2	no	—
210	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
211	C₄²	D₄×D₅₃	1	1	yes	characteristic
212	C₄²	#11719949	1	1	yes	characteristic
213	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
214	C₂³	#13133179	1	1	yes	characteristic
215	Q₈	—	106	4	no	—
216	Q₈	—	106	4	no	—
217	Q₈	—	212	2	no	—
218	Q₈	—	212	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	106	4	no	—
221	C₂×C₄	—	212	2	no	—
222	C₂×C₄	—	212	4	no	—
223	C₂×C₄	—	212	2	no	—
224	C₂×C₄	—	212	2	no	—
225	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
226	C₂×C₄	#13133179	1	1	yes	characteristic
227	C₂×C₄	#13133192	1	1	yes	characteristic
228	C₂×C₄	#13133192	1	1	yes	characteristic
229	C₂×C₄	#13133206	1	1	yes	characteristic
230	C₂×C₄	#13133213	1	1	yes	characteristic
231	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
233	C₂×C₄	#13133225	1	1	yes	characteristic
234	C₂×C₄	#13133225	1	1	yes	characteristic
235	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
236	C₂×C₄	#13133236	1	2	yes	—
237	C₈	—	2	2	no	—
238	C₈	—	8	2	no	—
239	C₈	—	106	4	no	—
240	C₈	—	106	4	no	—
241	C₈	#13133129	1	4	yes	—
242	C₂²	#16810524	1	1	yes	characteristic
243	C₂²	#16810688	1	1	yes	characteristic
244	C₂²	#16810700	1	1	yes	characteristic
245	C₂²	#16810717	1	2	yes	—
246	C₂²	#16810739	1	2	yes	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	424	2	no	—
250	C₄	—	212	4	no	—
251	C₄	C₂×#13133129	1	1	yes	characteristic
252	C₄	#16810591	1	1	yes	characteristic
253	C₄	#16810688	1	1	yes	characteristic
254	C₄	#16810700	1	1	yes	characteristic
255	C₄	#16810865	1	2	yes	—
256	C₄	#16811001	1	1	yes	characteristic
257	C₄	#16811006	1	1	yes	characteristic
258	C₂	#20273521	1	2	yes	—
259	C₂	#20273611	1	2	yes	—
260	C₂	#20275508	1	1	yes	characteristic
261	C₂	#20275574	1	1	yes	characteristic
262	C₂	#20275601	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	#9656
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
23. C₂×Dic₂₁₂	C₈	semidirect
80. C₂×#88322	Q₁₆	semidirect
151. C₅₃	#9656	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₂	2
#20275650	2
C₂×#16810353	2
C₂×C₈×#88322	2
C₂×C₅₃×#335	2
#9656	53