Subgroups — #27298975

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133151	C₂	1	1	yes	maximal, characteristic
3	#20272807	C₂	1	4	yes	maximal, complemented
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133176	—	2	1	no	—
6	C₂²×C₄⋊Dic₅₃	—	2	1	no	—
7	#16810661	—	2	2	no	—
8	#16810661	C₄	1	4	yes	complemented
9	C₂×#13133151	—	2	2	no	—
10	C₂×#13133151	—	2	1	no	—
11	C₂×#13133151	C₄	1	4	yes	complemented
12	C₂×#13133151	C₂²	1	2	yes	—
13	C₂×#13133151	C₂²	1	2	yes	—
14	C₂×#13133149	—	2	1	no	—
15	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₅₃	—	2	2	no	—
17	C₂×M₃₂×C₅₃	—	2	2	no	—
18	C₂×M₃₂×C₅₃	C₄	1	4	yes	complemented
19	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
20	C₂³×Dic₅₃	—	4	1	no	—
21	#13133176	—	2	4	no	—
22	#13133176	—	4	2	no	—
23	C₂×C₄⋊Dic₅₃	—	2	4	no	—
24	C₂×C₄⋊Dic₅₃	—	2	2	no	—
25	C₂×C₄⋊Dic₅₃	—	2	2	no	—
26	C₂×C₄⋊Dic₅₃	—	4	1	no	—
27	C₂×C₄⋊Dic₅₃	—	4	2	no	—
28	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
29	C₂×C₄×Dic₅₃	—	4	1	no	—
30	#13133151	—	2	4	no	—
31	#13133151	—	2	2	no	—
32	#13133151	—	2	2	no	—
33	#13133151	C₂×C₄	1	4	yes	—
34	#13133151	C₂×C₄	1	4	yes	complemented
35	#13133149	—	2	4	no	—
36	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₅₃	D₄	1	2	yes	—
41	C₂²×C₈×C₅₃	D₄	1	2	yes	—
42	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	M₃₂×C₅₃	—	2	8	no	—
46	C₂×C₁₆×C₅₃	—	2	2	no	—
47	C₂×C₁₆×C₅₃	—	2	2	no	—
48	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
49	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
55	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
56	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×Dic₅₃	—	4	4	no	—
59	C₂²×Dic₅₃	—	4	2	no	—
60	C₂²×Dic₅₃	—	4	2	no	—
61	C₂²×Dic₅₃	—	8	4	no	—
62	C₂×C₈×C₅₃	C₄²	1	4	yes	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
66	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
68	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
75	C₂²⋊Dic₅₃	—	4	4	no	—
76	C₄⋊Dic₅₃	—	2	4	no	—
77	C₄⋊Dic₅₃	—	2	4	no	—
78	C₄⋊Dic₅₃	—	4	4	no	—
79	C₄⋊Dic₅₃	—	4	2	no	—
80	C₄⋊Dic₅₃	—	4	2	no	—
81	C₄×Dic₅₃	—	4	4	no	—
82	C₁₆×C₅₃	—	2	4	no	—
83	C₂³×C₅₃	—	2	2	no	—
84	C₂³×C₅₃	—	2	2	no	—
85	C₂³×C₅₃	#102	1	2	yes	—
86	C₂³×C₅₃	#102	1	2	yes	—
87	C₂³×C₅₃	#107	1	2	yes	—
88	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#95	1	4	yes	—
90	C₂×C₄×C₅₃	#102	1	2	yes	—
91	C₂×C₄×C₅₃	#102	1	2	yes	—
92	C₂×C₄×C₅₃	#103	1	4	yes	—
93	C₂×C₄×C₅₃	#103	1	4	yes	—
94	C₂×C₄×C₅₃	#106	1	4	yes	—
95	C₂×C₄×C₅₃	#107	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×Dic₅₃	—	4	4	no	—
103	C₂×Dic₅₃	—	4	4	no	—
104	C₂×Dic₅₃	—	8	8	no	—
105	C₂×Dic₅₃	—	8	2	no	—
106	C₂×Dic₅₃	—	8	4	no	—
107	C₂×Dic₅₃	—	8	2	no	—
108	C₈×C₅₃	#95	1	4	yes	—
109	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
110	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	#12557	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	—	2	4	no	—
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	—	2	4	no	—
118	C₂²×C₅₃	#340	1	4	yes	—
119	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
121	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
122	C₂²×C₅₃	#509	1	2	yes	—
123	C₂²×C₅₃	#510	1	2	yes	—
124	C₄×C₅₃	#340	1	4	yes	—
125	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
126	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
127	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
128	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
129	Dic₅₃	—	8	4	no	—
130	Dic₅₃	—	8	4	no	—
131	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
132	C₂×#427	—	53	1	no	—
133	C₂²×#107	—	53	1	no	—
134	#1380	—	53	4	no	—
135	C₂×C₅₃	—	2	4	no	—
136	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
137	C₂×C₅₃	#2139	1	2	yes	—
138	C₂×C₅₃	#2140	1	1	yes	characteristic
139	C₂×C₅₃	#2141	1	1	yes	characteristic
140	C₂×C₅₃	#2159	1	2	yes	—
141	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
142	C₂×#117	—	106	1	no	—
143	C₂²×C₄⋊C₄	—	106	1	no	—
144	C₂×M₃₂	—	2	2	no	—
145	C₂×M₃₂	—	2	2	no	—
146	C₂×M₃₂	Dic₅₃	1	4	yes	complemented
147	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
148	#427	—	53	4	no	—
149	#427	—	106	2	no	—
150	C₂×#107	—	53	4	no	—
151	C₂×#107	—	53	2	no	—
152	C₂×#107	—	53	2	no	—
153	C₂×#107	—	106	1	no	—
154	C₂×#107	—	106	2	no	—
155	C₂×#106	—	106	1	no	—
156	C₅₃	#12557	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₄	—	212	1	no	—
158	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
159	C₂²×C₈	Dic₁₀₆	1	2	yes	—
160	C₂²×C₈	C₄×D₅₃	1	2	yes	—
161	C₂²×C₈	C₄×D₅₃	1	2	yes	—
162	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
163	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
164	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
165	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
166	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
167	#117	—	106	4	no	—
168	#117	—	212	2	no	—
169	C₂×C₄⋊C₄	—	106	2	no	—
170	C₂×C₄⋊C₄	—	106	2	no	—
171	C₂×C₄⋊C₄	—	106	4	no	—
172	C₂×C₄⋊C₄	—	212	2	no	—
173	C₂×C₄⋊C₄	—	212	1	no	—
174	C₂×C₂²⋊C₄	—	212	1	no	—
175	C₂×C₄²	—	212	1	no	—
176	M₃₂	—	2	8	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	2	2	no	—
179	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	complemented
180	#107	—	53	4	no	—
181	#107	—	53	4	no	—
182	#107	—	106	2	no	—
183	#107	—	106	2	no	—
184	#107	—	106	4	no	—
185	#106	—	106	4	no	—
186	C₂⁴	D₄₂₄	1	1	yes	characteristic
187	C₂²×C₄	—	424	4	no	—
188	C₂²×C₄	—	212	4	no	—
189	C₂²×C₄	—	212	2	no	—
190	C₂²×C₄	—	212	2	no	—
191	C₂²×C₄	#11719913	1	2	yes	—
192	C₂²×C₄	#11719913	1	2	yes	—
193	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
194	C₂²×C₄	Dic₂₁₂	1	2	yes	—
195	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
196	C₂²×C₄	#11719921	1	2	yes	—
197	C₂²×C₄	#11719921	1	2	yes	—
198	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
199	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
200	C₂×C₈	#11719919	1	4	yes	—
201	C₂×C₈	#11719919	1	4	yes	—
202	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
203	C₂×C₈	#11719921	1	2	yes	—
204	C₂×C₈	#11719921	1	2	yes	—
205	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
206	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
209	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
210	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
211	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
212	C₄⋊C₄	—	106	4	no	—
213	C₄⋊C₄	—	106	4	no	—
214	C₄⋊C₄	—	212	4	no	—
215	C₄⋊C₄	—	212	2	no	—
216	C₄⋊C₄	—	212	2	no	—
217	C₂²⋊C₄	—	212	4	no	—
218	C₄²	—	212	4	no	—
219	C₁₆	—	2	4	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	#13133139	1	2	yes	—
223	C₂³	#13133139	1	2	yes	—
224	C₂³	#13133151	1	2	yes	—
225	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
226	C₂×C₄	—	424	8	no	—
227	C₂×C₄	—	424	4	no	—
228	C₂×C₄	—	424	2	no	—
229	C₂×C₄	—	424	2	no	—
230	C₂×C₄	—	212	4	no	—
231	C₂×C₄	—	212	4	no	—
232	C₂×C₄	#13133124	1	4	yes	—
233	C₂×C₄	#13133139	1	2	yes	—
234	C₂×C₄	#13133139	1	2	yes	—
235	C₂×C₄	#13133142	1	4	yes	—
236	C₂×C₄	#13133142	1	4	yes	—
237	C₂×C₄	#13133149	1	4	yes	—
238	C₂×C₄	#13133151	1	2	yes	—
239	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
240	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
243	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
244	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
245	C₈	#13133124	1	4	yes	—
246	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
247	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
248	C₈	C₂×#11719919	1	1	yes	characteristic
249	C₈	C₂×#11719919	1	1	yes	characteristic
250	C₂²	—	2	4	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	#16810386	1	4	yes	—
255	C₂²	C₂×#13133139	1	1	yes	characteristic
256	C₂²	C₂×#13133139	1	1	yes	characteristic
257	C₂²	C₂×#13133151	1	1	yes	characteristic
258	C₂²	#16811061	1	2	yes	—
259	C₂²	#16811066	1	2	yes	—
260	C₄	—	424	4	no	—
261	C₄	—	424	4	no	—
262	C₄	#16810386	1	4	yes	—
263	C₄	C₂×#13133124	1	1	yes	characteristic
264	C₄	C₂×#13133142	1	1	yes	characteristic
265	C₄	C₂×#13133142	1	1	yes	characteristic
266	C₄	C₂×#13133149	1	1	yes	characteristic
267	C₂	—	2	4	no	—
268	C₂	C₂×#16810386	1	1	yes	characteristic
269	C₂	#20276428	1	2	yes	—
270	C₂	#20276432	1	1	yes	characteristic
271	C₂	#20276436	1	1	yes	characteristic
272	C₂	#20276503	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	#12557
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272807	C₂	semidirect
8. #16810661	C₄	semidirect
11. C₂×#13133151	C₄	semidirect
18. C₂×M₃₂×C₅₃	C₄	semidirect
34. #13133151	C₂×C₄	semidirect
48. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
146. C₂×M₃₂	Dic₅₃	semidirect
156. C₅₃	#12557	semidirect
179. C₂×C₁₆	C₂×Dic₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810661	2
C₂²×#13133151	2
#20272807	2
C₂²×M₃₂×C₅₃	2
#12557	53