Subgroups — #27539616

Order 13568.

514 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	#20285229	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811028	C₂	1	1	yes	maximal, characteristic
3	#20277001	C₂	1	4	yes	maximal, complemented
4	#20276997	C₂	1	2	yes	maximal, complemented
5	#20276990	C₂	1	2	yes	maximal, complemented
6	#20276503	C₂	1	4	yes	maximal
7	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#16811180	—	2	1	no	—
9	C₂×#13133217	—	2	1	no	—
10	#16811028	—	2	1	no	—
11	#16811028	—	2	1	no	—
12	#16811028	C₂²	1	1	yes	characteristic
13	#16811028	C₂²	1	2	yes	—
14	#16811028	C₂²	1	2	yes	—
15	#16811028	C₂²	1	1	yes	characteristic
16	#16811025	—	2	1	no	—
17	#16811025	C₂²	1	2	yes	—
18	#16811021	—	2	1	no	—
19	#16810661	—	2	1	no	—
20	#16810661	C₂²	1	2	yes	—
21	C₂×#13133151	—	2	1	no	—
22	C₂×#13133151	C₂²	1	2	yes	—
23	#16810451	C₂²	1	8	yes	—
24	#16810449	C₂²	1	8	yes	—
25	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#346	C₂²	1	2	yes	—
28	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
29	C₂²×Dic₁₀₆	—	4	1	no	—
30	#13133217	—	2	1	no	—
31	#13133217	—	2	1	no	—
32	#13133217	—	2	2	no	—
33	#13133217	—	2	2	no	—
34	#13133217	—	4	1	no	—
35	#13133217	—	4	1	no	—
36	#13133209	—	2	2	no	—
37	#13133196	—	4	1	no	—
38	#13133196	—	4	1	no	—
39	C₄×Dic₁₀₆	—	4	1	no	—
40	C₄×Dic₁₀₆	—	4	1	no	—
41	#13133176	—	2	2	no	—
42	C₂×C₄⋊Dic₅₃	—	2	2	no	—
43	C₂×C₄⋊Dic₅₃	—	4	1	no	—
44	#13133151	—	2	1	no	—
45	#13133151	—	2	1	no	—
46	#13133151	—	2	2	no	—
47	#13133151	—	2	2	no	—
48	#13133151	—	2	2	no	—
49	#13133151	C₂³	1	2	yes	—
50	#13133151	C₂³	1	2	yes	—
51	#13133151	C₂³	1	4	yes	—
52	#13133149	—	2	2	no	—
53	#13133149	—	2	2	no	—
54	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
57	M₃₂×C₅₃	Q₈	1	8	yes	complemented
58	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
59	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₅₃	—	4	2	no	—
67	C₂×Dic₁₀₆	—	4	1	no	—
68	C₂×Dic₁₀₆	—	4	1	no	—
69	C₂×Dic₁₀₆	—	4	2	no	—
70	C₂×Dic₁₀₆	—	4	2	no	—
71	C₂×Dic₁₀₆	—	8	4	no	—
72	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
78	C₄²×C₅₃	D₈	1	1	yes	characteristic
79	C₄²×C₅₃	D₈	1	1	yes	characteristic
80	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂²⋊Dic₅₃	—	4	2	no	—
83	C₄⋊Dic₅₃	—	2	2	no	—
84	C₄⋊Dic₅₃	—	2	2	no	—
85	C₄⋊Dic₅₃	—	2	4	no	—
86	C₄⋊Dic₅₃	—	4	2	no	—
87	C₄⋊Dic₅₃	—	4	2	no	—
88	C₄⋊Dic₅₃	—	4	2	no	—
89	C₄⋊Dic₅₃	—	4	1	no	—
90	C₄⋊Dic₅₃	—	4	1	no	—
91	#11719919	—	4	2	no	—
92	#11719919	—	4	2	no	—
93	C₄×Dic₅₃	—	4	2	no	—
94	C₁₆×C₅₃	C₂×Q₈	1	8	yes	complemented
95	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₂×Dic₅₃	—	4	4	no	—
107	C₂×Dic₅₃	—	4	2	no	—
108	C₂×Dic₅₃	—	4	2	no	—
109	C₂×Dic₅₃	—	8	4	no	—
110	Dic₁₀₆	—	8	4	no	—
111	Dic₁₀₆	—	8	2	no	—
112	Dic₁₀₆	—	8	4	no	—
113	Dic₁₀₆	—	8	2	no	—
114	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
115	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
116	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
117	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
118	#33731	—	53	1	no	maximal, Sylow
119	C₂²×C₅₃	—	2	1	no	—
120	C₂²×C₅₃	—	2	1	no	—
121	C₂²×C₅₃	#500	1	2	yes	—
122	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	#500	1	2	yes	—
126	C₄×C₅₃	#509	1	2	yes	—
127	C₄×C₅₃	#510	1	2	yes	—
128	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
130	Dic₅₃	—	8	4	no	—
131	Dic₅₃	—	8	2	no	—
132	Dic₅₃	—	8	2	no	—
133	#3160	—	53	1	no	—
134	C₂×#500	—	53	1	no	—
135	#2255	—	53	4	no	—
136	#2254	—	53	2	no	—
137	#2252	—	53	2	no	—
138	#2159	—	53	4	no	—
139	C₄×M₃₂	D₅₃	1	1	yes	characteristic
140	C₂×C₅₃	—	2	2	no	—
141	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
142	C₂×C₅₃	C₂×#509	1	1	yes	characteristic
143	C₂×C₅₃	C₂×#510	1	1	yes	characteristic
144	#533	—	106	1	no	—
145	C₂×C₄⋊Q₈	—	106	1	no	—
146	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
147	#500	—	53	2	no	—
148	#500	—	53	2	no	—
149	#500	—	53	1	no	—
150	#500	—	53	1	no	—
151	#500	—	106	1	no	—
152	#500	—	106	1	no	—
153	#499	—	53	2	no	—
154	#499	—	106	1	no	—
155	#498	—	106	1	no	—
156	#427	—	53	2	no	—
157	#427	—	106	1	no	—
158	C₂×#107	—	53	2	no	—
159	C₂×#107	—	106	1	no	—
160	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
161	#367	—	53	8	no	—
162	#366	—	53	8	no	—
163	#346	C₂×D₅₃	1	2	yes	—
164	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
165	C₅₃	#33731	1	1	yes	characteristic, complemented, Sylow
166	C₂²×Q₈	—	212	1	no	—
167	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
169	C₄⋊Q₈	—	106	2	no	—
170	C₄⋊Q₈	—	106	2	no	—
171	C₄⋊Q₈	—	106	1	no	—
172	C₄⋊Q₈	—	106	1	no	—
173	C₄⋊Q₈	—	212	1	no	—
174	C₄⋊Q₈	—	212	1	no	—
175	#125	—	106	2	no	—
176	C₂²⋊Q₈	—	212	1	no	—
177	C₂²⋊Q₈	—	212	1	no	—
178	C₄×Q₈	—	212	1	no	—
179	C₄×Q₈	—	212	1	no	—
180	#117	—	106	2	no	—
181	C₂×C₄⋊C₄	—	106	2	no	—
182	C₂×C₄⋊C₄	—	212	1	no	—
183	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
184	M₃₂	Dic₁₀₆	1	8	yes	complemented
185	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
186	#107	—	53	2	no	—
187	#107	—	53	4	no	—
188	#107	—	53	2	no	—
189	#107	—	106	2	no	—
190	#107	—	106	2	no	—
191	#107	—	106	2	no	—
192	#107	—	106	1	no	—
193	#107	—	106	1	no	—
194	#106	—	106	2	no	—
195	#106	—	106	2	no	—
196	C₄×C₈	D₂₁₂	1	1	yes	characteristic
197	C₄×C₈	D₂₁₂	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
200	C₂×Q₈	—	424	4	no	—
201	C₂×Q₈	—	212	1	no	—
202	C₂×Q₈	—	212	1	no	—
203	C₂×Q₈	—	212	2	no	—
204	C₂×Q₈	—	212	2	no	—
205	C₂²×C₄	—	212	2	no	—
206	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
207	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
209	C₂×C₈	Dic₂₁₂	1	4	yes	—
210	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
211	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
212	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
213	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
214	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
215	C₄⋊C₄	—	106	2	no	—
216	C₄⋊C₄	—	106	2	no	—
217	C₄⋊C₄	—	106	4	no	—
218	C₄⋊C₄	—	212	2	no	—
219	C₄⋊C₄	—	212	2	no	—
220	C₄⋊C₄	—	212	2	no	—
221	C₄⋊C₄	—	212	2	no	—
222	C₄⋊C₄	—	212	2	no	—
223	C₄⋊C₄	—	212	1	no	—
224	C₄⋊C₄	—	212	1	no	—
225	C₂²⋊C₄	—	212	2	no	—
226	C₄²	—	212	2	no	—
227	C₄²	D₄₂₄	1	1	yes	characteristic
228	C₄²	D₄₂₄	1	1	yes	characteristic
229	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
230	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
231	C₁₆	C₂×Dic₁₀₆	1	8	yes	complemented
232	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
233	Q₈	—	424	4	no	—
234	Q₈	—	424	4	no	—
235	Q₈	—	424	2	no	—
236	Q₈	—	424	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	424	4	no	—
240	C₂×C₄	—	212	2	no	—
241	C₂×C₄	—	212	2	no	—
242	C₂×C₄	—	212	4	no	—
243	C₂×C₄	#13133217	1	2	yes	—
244	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
246	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
248	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
249	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
250	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
251	C₈	#13133217	1	2	yes	—
252	C₈	C₂×Dic₂₁₂	1	2	yes	—
253	C₈	C₂×Dic₂₁₂	1	2	yes	—
254	C₈	C₂²×Dic₁₀₆	1	2	yes	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	#16811028	1	2	yes	—
258	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	424	4	no	—
262	C₄	—	424	2	no	—
263	C₄	—	424	2	no	—
264	C₄	#16811028	1	2	yes	—
265	C₄	#16811061	1	2	yes	—
266	C₄	#16811066	1	2	yes	—
267	C₄	C₂×#13133217	1	1	yes	characteristic
268	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	C₂×#16811028	1	1	yes	characteristic
271	C₂	C₂×#16811061	1	1	yes	characteristic
272	C₂	C₂×#16811066	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	#33731
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20277001	C₂	semidirect
4. #20276997	C₂	semidirect
5. #20276990	C₂	semidirect
57. M₃₂×C₅₃	Q₈	semidirect
94. C₁₆×C₅₃	C₂×Q₈	semidirect
165. C₅₃	#33731	semidirect
184. M₃₂	Dic₁₀₆	semidirect
231. C₁₆	C₂×Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
C₂×#16811028	2
#20277001	2
#20276997	2
#20276990	2
#20276503	2
C₄×M₃₂×C₅₃	2
#33731	53