Subgroups — #27282960

Order 13568.

486 conjugacy classes of subgroups.

The subgroup lattice has 258 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	#20275651	C₂	1	2	yes	maximal
3	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810355	C₂	1	2	yes	maximal
5	C₂×C₅₃×#322	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×C₄⋊#88322	C₂²	1	2	yes	—
12	C₂×#13133142	C₄	1	2	yes	—
13	#16810377	—	2	4	no	—
14	#16810374	—	2	2	no	—
15	#16810355	—	2	4	no	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#322	—	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	2	no	—
25	C₄×Dic₁₀₆	—	2	2	no	—
26	C₄×Dic₁₀₆	—	2	2	no	—
27	C₄×Dic₁₀₆	—	2	2	no	—
28	C₄×Dic₁₀₆	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
30	C₂×#11719919	—	2	2	no	—
31	C₂×C₄×Dic₅₃	—	2	2	no	—
32	#13133151	—	2	2	no	—
33	C₄⋊#88322	—	2	2	no	—
34	C₄⋊#88322	—	2	2	no	—
35	#13133142	—	2	4	no	—
36	C₂²×C₈×C₅₃	—	2	1	no	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	D₄	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	—	2	1	no	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
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	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₅₃	#41	1	1	yes	characteristic
72	C₄⋊Dic₅₃	—	2	2	no	—
73	C₄⋊Dic₅₃	—	2	2	no	—
74	#11719919	—	4	2	no	—
75	#11719919	—	4	2	no	—
76	C₄×Dic₅₃	—	4	2	no	—
77	C₄×Dic₅₃	—	4	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	—	4	4	no	—
80	Dic₂₁₂	—	2	8	no	—
81	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#137	1	4	yes	—
94	C₂×Dic₅₃	—	2	4	no	—
95	C₂×Dic₅₃	—	4	2	no	—
96	C₂×Dic₅₃	—	4	2	no	—
97	C₂×Dic₅₃	—	4	2	no	—
98	C₂×Dic₅₃	—	4	4	no	—
99	Dic₁₀₆	—	2	4	no	—
100	Dic₁₀₆	—	2	4	no	—
101	Dic₁₀₆	—	4	2	no	—
102	Dic₁₀₆	—	4	2	no	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	—	4	2	no	—
105	C₈×C₅₃	—	4	2	no	—
106	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
107	#88322	—	8	2	no	—
108	#9658	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	#390	1	1	yes	characteristic
110	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
111	C₂²×C₅₃	#436	1	1	yes	characteristic
112	C₂²×C₅₃	#441	1	2	yes	—
113	C₂²×C₅₃	#444	1	2	yes	—
114	C₄×C₅₃	—	2	1	no	—
115	C₄×C₅₃	—	2	1	no	—
116	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
117	C₄×C₅₃	#408	1	1	yes	characteristic
118	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
119	C₄×C₅₃	#436	1	1	yes	characteristic
120	C₄×C₅₃	#470	1	2	yes	—
121	C₄×C₅₃	#497	1	2	yes	—
122	Dic₅₃	—	4	4	no	—
123	Dic₅₃	—	8	2	no	—
124	C₂×C₄×Q₁₆	—	53	1	no	—
125	#1984	—	53	2	no	—
126	C₂×#334	—	53	1	no	—
127	C₂×Q₈⋊C₈	—	53	2	no	—
128	C₂×#322	D₅₃	1	1	yes	characteristic
129	C₂×C₅₃	#1579	1	2	yes	—
130	C₂×C₅₃	#1593	1	2	yes	—
131	C₂×C₅₃	#1954	1	1	yes	characteristic
132	C₂×C₅₃	#1971	1	1	yes	characteristic
133	C₂×C₅₃	#1975	1	1	yes	characteristic
134	C₂²×Q₁₆	—	53	1	no	—
135	C₂×C₄×Q₈	—	53	2	no	—
136	C₄×Q₁₆	—	106	2	no	—
137	C₄×Q₁₆	—	106	2	no	—
138	C₂×#107	—	53	1	no	—
139	C₂×C₄⋊C₈	—	53	2	no	—
140	C₂×#103	—	53	2	no	—
141	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
144	#336	—	106	4	no	—
145	#334	—	106	2	no	—
146	Q₈⋊C₈	—	106	4	no	—
147	#322	—	2	2	no	—
148	C₅₃	#9658	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	—	53	2	no	—
150	C₂×Q₁₆	—	53	4	no	—
151	C₂×Q₁₆	—	106	2	no	—
152	C₂×Q₁₆	—	106	2	no	—
153	C₂²×C₈	—	2	1	no	—
154	C₂²×C₈	—	106	2	no	—
155	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
156	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
157	C₂²×C₈	D₂₁₂	1	2	yes	—
158	C₄×Q₈	—	106	2	no	—
159	C₄×Q₈	—	106	2	no	—
160	C₄×Q₈	—	106	2	no	—
161	C₄×Q₈	—	106	2	no	—
162	C₂×C₄⋊C₄	—	53	2	no	—
163	C₂×C₄⋊C₄	—	106	2	no	—
164	C₂×C₄²	—	106	2	no	—
165	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
166	#107	—	106	2	no	—
167	C₄⋊C₈	—	106	2	no	—
168	C₄⋊C₈	—	106	2	no	—
169	#103	—	106	4	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	C₄×C₈	D₂₁₂	1	2	yes	—
174	C₄×C₈	D₂₁₂	1	2	yes	—
175	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
176	C₂×Q₈	—	53	4	no	—
177	C₂×Q₈	—	106	2	no	—
178	C₂×Q₈	—	106	2	no	—
179	C₂×Q₈	—	106	2	no	—
180	C₂×Q₈	—	106	2	no	—
181	C₂×Q₈	—	106	2	no	—
182	C₂²×C₄	—	106	2	no	—
183	C₂²×C₄	—	106	2	no	—
184	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
186	C₂²×C₄	#11719945	1	1	yes	characteristic
187	Q₁₆	—	106	8	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	212	2	no	—
195	C₂×C₈	—	212	4	no	—
196	C₂×C₈	C₈×D₅₃	1	2	yes	—
197	C₂×C₈	#11719912	1	2	yes	—
198	C₂×C₈	#11719921	1	2	yes	—
199	C₂×C₈	#11719921	1	2	yes	—
200	C₄⋊C₄	—	106	2	no	—
201	C₄⋊C₄	—	106	2	no	—
202	C₄⋊C₄	—	212	2	no	—
203	C₄⋊C₄	—	212	2	no	—
204	C₄²	—	212	2	no	—
205	C₄²	—	212	2	no	—
206	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
207	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
208	C₄²	#11719945	1	1	yes	characteristic
209	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
211	Q₈	—	106	4	no	—
212	Q₈	—	106	4	no	—
213	Q₈	—	212	2	no	—
214	Q₈	—	212	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	106	4	no	—
217	C₂×C₄	—	212	2	no	—
218	C₂×C₄	—	212	4	no	—
219	C₂×C₄	—	212	2	no	—
220	C₂×C₄	—	212	2	no	—
221	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
222	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
223	C₂×C₄	#13133191	1	1	yes	characteristic
224	C₂×C₄	#13133191	1	1	yes	characteristic
225	C₂×C₄	#13133205	1	1	yes	characteristic
226	C₂×C₄	#13133214	1	1	yes	characteristic
227	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
228	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
229	C₂×C₄	#13133223	1	1	yes	characteristic
230	C₂×C₄	#13133223	1	1	yes	characteristic
231	C₂×C₄	#13133250	1	4	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	4	2	no	—
234	C₈	—	4	2	no	—
235	C₈	—	424	2	no	—
236	C₈	#13133129	1	4	yes	—
237	C₂²	#16810525	1	1	yes	characteristic
238	C₂²	C₈×D₂₁₂	1	1	yes	characteristic
239	C₂²	#16810702	1	1	yes	characteristic
240	C₂²	#16810731	1	2	yes	—
241	C₂²	#16810746	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	424	2	no	—
245	C₄	—	212	4	no	—
246	C₄	C₂×#13133129	1	1	yes	characteristic
247	C₄	#16810589	1	1	yes	characteristic
248	C₄	C₈×D₂₁₂	1	1	yes	characteristic
249	C₄	#16810702	1	1	yes	characteristic
250	C₄	#16810881	1	2	yes	—
251	C₄	#16811017	1	2	yes	—
252	C₂	#20273535	1	2	yes	—
253	C₂	#20273625	1	2	yes	—
254	C₂	#20275509	1	1	yes	characteristic
255	C₂	#20275586	1	1	yes	characteristic
256	C₂	#20275608	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#9658
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
23. C₂×Dic₂₁₂	C₈	semidirect
148. C₅₃	#9658	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
#20275651	2
C₂×#16810374	2
C₂×#16810355	2
C₂×C₅₃×#322	2
#9658	53