Subgroups — #27282994

Order 13568.

482 conjugacy classes of subgroups.

The subgroup lattice has 242 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	4	no	—
23	C₂²×#88322	—	2	2	no	—
24	C₄×Dic₁₀₆	—	2	2	no	—
25	C₄×Dic₁₀₆	—	2	2	no	—
26	C₄×Dic₁₀₆	—	2	2	no	—
27	C₄×Dic₁₀₆	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
29	C₂×#11719919	—	2	2	no	—
30	C₂×C₄×Dic₅₃	—	2	2	no	—
31	#13133151	—	2	2	no	—
32	C₄⋊#88322	—	2	2	no	—
33	C₄⋊#88322	—	2	2	no	—
34	#13133142	—	2	4	no	—
35	C₂²×C₈×C₅₃	—	2	1	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	2	yes	—
39	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₅₃	—	2	1	no	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	2	2	no	—
50	C₂²×Dic₅₃	—	2	2	no	—
51	C₂×Dic₁₀₆	—	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	M₁₆	1	4	yes	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	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	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₅₃	#41	1	1	yes	characteristic
69	C₄⋊Dic₅₃	—	2	2	no	—
70	C₄⋊Dic₅₃	—	2	2	no	—
71	#11719919	—	4	2	no	—
72	#11719919	—	4	2	no	—
73	C₄×Dic₅₃	—	4	2	no	—
74	C₄×Dic₅₃	—	4	2	no	—
75	C₂×#88322	—	4	2	no	—
76	C₂×#88322	—	4	4	no	—
77	Dic₂₁₂	—	2	8	no	—
78	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
87	C₂×C₄×C₅₃	#131	1	2	yes	—
88	C₂×C₄×C₅₃	#137	1	4	yes	—
89	C₂×Dic₅₃	—	2	4	no	—
90	C₂×Dic₅₃	—	4	2	no	—
91	C₂×Dic₅₃	—	4	2	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	C₂×Dic₅₃	—	4	4	no	—
94	Dic₁₀₆	—	2	4	no	—
95	Dic₁₀₆	—	2	4	no	—
96	Dic₁₀₆	—	4	2	no	—
97	Dic₁₀₆	—	4	2	no	—
98	C₈×C₅₃	—	2	4	no	—
99	C₈×C₅₃	—	2	2	no	—
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	4	2	no	—
102	#88322	—	8	2	no	—
103	#9664	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	#390	1	1	yes	characteristic
105	C₂²×C₅₃	#435	1	2	yes	—
106	C₂²×C₅₃	#441	1	2	yes	—
107	C₂²×C₅₃	#443	1	2	yes	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	#407	1	1	yes	characteristic
111	C₄×C₅₃	#408	1	1	yes	characteristic
112	C₄×C₅₃	#435	1	2	yes	—
113	C₄×C₅₃	#470	1	2	yes	—
114	C₄×C₅₃	#497	1	2	yes	—
115	Dic₅₃	—	4	4	no	—
116	Dic₅₃	—	8	2	no	—
117	C₂×C₄×Q₁₆	—	53	1	no	—
118	#1984	—	53	2	no	—
119	C₂×#334	—	53	1	no	—
120	C₂×Q₈⋊C₈	—	53	2	no	—
121	C₂×#322	D₅₃	1	1	yes	characteristic
122	C₂×C₅₃	#1587	1	4	yes	—
123	C₂×C₅₃	#1955	1	1	yes	characteristic
124	C₂×C₅₃	#1971	1	1	yes	characteristic
125	C₂×C₅₃	#1974	1	1	yes	characteristic
126	C₂²×Q₁₆	—	53	1	no	—
127	C₂×C₄×Q₈	—	53	2	no	—
128	C₄×Q₁₆	—	106	2	no	—
129	C₄×Q₁₆	—	106	2	no	—
130	C₂×#107	—	53	1	no	—
131	C₂×C₄⋊C₈	—	53	2	no	—
132	C₂×#103	—	53	2	no	—
133	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
134	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
135	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
136	#336	—	106	4	no	—
137	#334	—	106	2	no	—
138	Q₈⋊C₈	—	106	4	no	—
139	#322	—	2	2	no	—
140	C₅₃	#9664	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₈	—	53	2	no	—
142	C₂×Q₁₆	—	106	4	no	—
143	C₂×Q₁₆	—	106	2	no	—
144	C₂²×C₈	—	2	1	no	—
145	C₂²×C₈	—	106	2	no	—
146	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
147	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
148	C₂²×C₈	D₂₁₂	1	2	yes	—
149	C₄×Q₈	—	106	2	no	—
150	C₄×Q₈	—	106	2	no	—
151	C₄×Q₈	—	106	2	no	—
152	C₄×Q₈	—	106	2	no	—
153	C₂×C₄⋊C₄	—	53	2	no	—
154	C₂×C₄⋊C₄	—	106	2	no	—
155	C₂×C₄²	—	106	2	no	—
156	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
157	#107	—	106	2	no	—
158	C₄⋊C₈	—	106	2	no	—
159	C₄⋊C₈	—	106	2	no	—
160	#103	—	106	4	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	D₂₁₂	1	2	yes	—
165	C₄×C₈	D₂₁₂	1	2	yes	—
166	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
167	C₂×Q₈	—	53	4	no	—
168	C₂×Q₈	—	106	2	no	—
169	C₂×Q₈	—	106	2	no	—
170	C₂×Q₈	—	106	2	no	—
171	C₂×Q₈	—	106	2	no	—
172	C₂×Q₈	—	106	2	no	—
173	C₂²×C₄	—	106	2	no	—
174	C₂²×C₄	—	106	2	no	—
175	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
177	C₂²×C₄	#11719945	1	1	yes	characteristic
178	Q₁₆	—	106	8	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	212	2	no	—
186	C₂×C₈	—	212	4	no	—
187	C₂×C₈	#11719921	1	2	yes	—
188	C₂×C₈	#11719921	1	2	yes	—
189	C₄⋊C₄	—	106	2	no	—
190	C₄⋊C₄	—	106	2	no	—
191	C₄⋊C₄	—	212	2	no	—
192	C₄⋊C₄	—	212	2	no	—
193	C₄²	—	212	2	no	—
194	C₄²	—	212	2	no	—
195	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
196	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
197	C₄²	#11719945	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
199	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
200	Q₈	—	106	4	no	—
201	Q₈	—	106	4	no	—
202	Q₈	—	212	2	no	—
203	Q₈	—	212	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	106	4	no	—
206	C₂×C₄	—	212	2	no	—
207	C₂×C₄	—	212	4	no	—
208	C₂×C₄	—	212	2	no	—
209	C₂×C₄	—	212	2	no	—
210	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
211	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
212	C₂×C₄	#13133191	1	1	yes	characteristic
213	C₂×C₄	#13133191	1	1	yes	characteristic
214	C₂×C₄	#13133205	1	1	yes	characteristic
215	C₂×C₄	#13133214	1	1	yes	characteristic
216	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
217	C₂×C₄	#13133225	1	2	yes	—
218	C₂×C₄	#13133250	1	4	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	2	2	no	—
221	C₈	—	2	2	no	—
222	C₈	—	4	2	no	—
223	C₈	—	424	2	no	—
224	C₂²	#16810525	1	1	yes	characteristic
225	C₂²	#16810696	1	2	yes	—
226	C₂²	#16810731	1	2	yes	—
227	C₂²	#16810741	1	2	yes	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	424	2	no	—
231	C₄	—	212	4	no	—
232	C₄	#16810586	1	1	yes	characteristic
233	C₄	#16810591	1	1	yes	characteristic
234	C₄	#16810696	1	2	yes	—
235	C₄	#16810881	1	2	yes	—
236	C₄	#16811017	1	2	yes	—
237	C₂	#20273588	1	4	yes	—
238	C₂	#20275513	1	1	yes	characteristic
239	C₂	#20275586	1	1	yes	characteristic
240	C₂	#20275603	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
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	#9664
LCS	G ▷ C₂²×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
140. C₅₃	#9664	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
#9664	53