Subgroups — #27282961

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	#20275652	C₂	1	2	yes	maximal
2	C₂×#16810374	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	2	yes	maximal
4	C₂×#16810344	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₂²	1	2	yes	—
7	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	#16810378	—	2	4	no	—
10	#16810374	—	2	2	no	—
11	Q₈⋊#88322	—	2	4	no	—
12	#16810344	—	2	2	no	—
13	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₅₃	—	2	2	no	—
16	C₄×Q₁₆×C₅₃	—	2	2	no	—
17	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
18	C₂×#103×C₅₃	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×#88322	—	2	2	no	—
21	C₂²×#88322	—	2	1	no	—
22	C₂²×#88322	D₄	1	2	yes	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	D₄	1	2	yes	—
29	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×Q₁₆×C₅₃	—	2	2	no	—
31	C₂×Q₁₆×C₅₃	—	2	2	no	—
32	C₂×Q₁₆×C₅₃	C₈	1	4	yes	complemented
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	C₄×Q₈×C₅₃	—	2	2	no	—
37	C₄×Q₈×C₅₃	—	2	2	no	—
38	C₄×Q₈×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₅₃	—	2	2	no	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#107×C₅₃	—	2	2	no	—
44	#103×C₅₃	—	2	4	no	—
45	C₄×C₈×C₅₃	D₄	1	2	yes	—
46	C₄×C₈×C₅₃	D₄	1	2	yes	—
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	—	2	2	no	—
52	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
58	Q₁₆×C₅₃	—	2	8	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	—	4	2	no	—
70	C₄²×C₅₃	—	4	2	no	—
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	#41	1	1	yes	characteristic
75	C₂×#88322	—	2	2	no	—
76	C₂×#88322	—	2	2	no	—
77	C₂×#88322	—	2	4	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂×#88322	—	4	4	no	—
81	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
82	Q₈×C₅₃	—	2	4	no	—
83	Q₈×C₅₃	—	2	4	no	—
84	Q₈×C₅₃	—	4	2	no	—
85	Q₈×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	4	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#137	1	4	yes	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
105	#88322	—	4	2	no	—
106	#88322	—	4	2	no	—
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	—	4	4	no	—
117	C₄×C₅₃	—	8	2	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	#470	1	2	yes	—
123	C₄×C₅₃	#497	1	2	yes	—
124	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
125	#1984	—	53	2	no	—
126	C₂×#334	—	53	1	no	—
127	C₂×Q₈⋊C₈	—	53	2	no	—
128	C₂×#322	—	53	1	no	—
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	Dic₅₃	1	1	yes	characteristic
135	C₂×C₄×Q₈	C₂×D₅₃	1	2	yes	—
136	C₄×Q₁₆	—	2	2	no	—
137	C₄×Q₁₆	—	2	2	no	—
138	C₂×#107	Dic₅₃	1	1	yes	characteristic
139	C₂×C₄⋊C₈	—	53	2	no	—
140	C₂×#103	Dic₅₃	1	2	yes	—
141	C₂×C₄×C₈	—	53	1	no	—
142	C₂×C₄×C₈	—	53	1	no	—
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	—	106	2	no	—
148	C₅₃	#9658	1	1	yes	characteristic, complemented, Sylow
149	C₂²×Q₈	C₂×Dic₅₃	1	2	yes	—
150	C₂×Q₁₆	—	2	2	no	—
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×Q₁₆	#88322	1	4	yes	complemented
153	C₂²×C₈	—	53	2	no	—
154	C₂²×C₈	—	106	1	no	—
155	C₂²×C₈	—	106	2	no	—
156	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
157	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
158	C₄×Q₈	—	2	2	no	—
159	C₄×Q₈	—	2	2	no	—
160	C₄×Q₈	—	2	2	no	—
161	C₄×Q₈	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
166	#107	—	2	2	no	—
167	C₄⋊C₈	—	106	2	no	—
168	C₄⋊C₈	—	106	2	no	—
169	#103	—	2	4	no	—
170	C₄×C₈	—	53	2	no	—
171	C₄×C₈	—	106	1	no	—
172	C₄×C₈	—	106	1	no	—
173	C₄×C₈	—	106	1	no	—
174	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
175	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
176	C₂×Q₈	—	2	2	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₂×#88322	1	4	yes	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
185	C₂²×C₄	#11719947	1	1	yes	characteristic
186	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
187	Q₁₆	—	2	8	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	106	2	no	—
191	C₂×C₈	—	106	2	no	—
192	C₂×C₈	—	106	4	no	—
193	C₂×C₈	—	212	2	no	—
194	C₂×C₈	—	212	4	no	—
195	C₂×C₈	—	212	2	no	—
196	C₂×C₈	C₂×#88322	1	2	yes	—
197	C₂×C₈	#11719917	1	2	yes	—
198	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
199	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄²	—	4	2	no	—
205	C₄²	—	4	2	no	—
206	C₄²	D₄×D₅₃	1	1	yes	characteristic
207	C₄²	#11719947	1	1	yes	characteristic
208	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
209	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
210	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
211	Q₈	—	2	4	no	—
212	Q₈	—	2	4	no	—
213	Q₈	—	4	2	no	—
214	Q₈	—	4	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
222	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
223	C₂×C₄	#13133188	1	1	yes	characteristic
224	C₂×C₄	#13133188	1	1	yes	characteristic
225	C₂×C₄	#13133204	1	1	yes	characteristic
226	C₂×C₄	#13133213	1	1	yes	characteristic
227	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
228	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
229	C₂×C₄	#13133224	1	1	yes	characteristic
230	C₂×C₄	#13133224	1	1	yes	characteristic
231	C₂×C₄	#13133249	1	4	yes	—
232	C₈	—	2	2	no	—
233	C₈	—	424	2	no	—
234	C₈	—	212	2	no	—
235	C₈	—	212	2	no	—
236	C₈	C₂²⋊#88322	1	4	yes	—
237	C₂²	#16810526	1	1	yes	characteristic
238	C₂²	D₄×#88322	1	1	yes	characteristic
239	C₂²	#16810701	1	1	yes	characteristic
240	C₂²	#16810730	1	2	yes	—
241	C₂²	#16810745	1	2	yes	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	4	no	—
245	C₄	—	8	2	no	—
246	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
247	C₄	#16810590	1	1	yes	characteristic
248	C₄	D₄×#88322	1	1	yes	characteristic
249	C₄	#16810701	1	1	yes	characteristic
250	C₄	#16810878	1	2	yes	—
251	C₄	#16811016	1	2	yes	—
252	C₂	#20273534	1	2	yes	—
253	C₂	#20273624	1	2	yes	—
254	C₂	#20275510	1	1	yes	characteristic
255	C₂	#20275587	1	1	yes	characteristic
256	C₂	#20275609	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×C₄×Q₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#9658
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
32. C₂×Q₁₆×C₅₃	C₈	semidirect
148. C₅₃	#9658	semidirect
152. C₂×Q₁₆	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20275652	2
C₂×#16810374	2
C₂×Q₈⋊#88322	2
C₂×#16810344	2
C₂×C₄×Q₁₆×C₅₃	2
#9658	53