Subgroups — #27537404

Order 13568.

682 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811055	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133239	C₂	1	2	yes	maximal, complemented
3	#20276592	C₂	1	2	yes	maximal, complemented
4	#20276587	C₂	1	2	yes	maximal, complemented
5	#20276581	C₂	1	4	yes	maximal, complemented
6	#20276499	C₂	1	1	yes	maximal, characteristic, complemented
7	#20276423	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13133239	C₂²	1	2	yes	—
10	C₄×#11719945	—	2	2	no	—
11	#16811055	C₄	1	8	yes	complemented
12	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
13	C₂×#13133162	C₂²	1	2	yes	—
14	C₂×D₈₄₈	C₂²	1	1	yes	characteristic
15	C₄×Dic₂₁₂	C₂²	1	4	yes	—
16	C₄×#11719913	—	2	4	no	—
17	C₄×D₄₂₄	C₂²	1	4	yes	—
18	#16810661	C₂²	1	2	yes	—
19	#16810620	—	2	2	no	—
20	#16810451	C₂²	1	2	yes	—
21	#16810449	C₂²	1	2	yes	—
22	#16810428	C₂²	1	4	yes	—
23	#16810425	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#346	C₂²	1	2	yes	—
28	C₅₃×#346	C₂²	1	2	yes	—
29	C₂×#11719945	—	2	2	no	—
30	#13133239	—	2	4	no	—
31	#13133239	C₂×C₄	1	8	yes	complemented
32	C₂×Dic₂₁₂	C₂³	1	2	yes	—
33	C₂×#11719913	—	2	2	no	—
34	C₂×D₄₂₄	C₂³	1	2	yes	—
35	C₄×Dic₁₀₆	—	2	4	no	—
36	C₄×D₂₁₂	—	2	4	no	—
37	C₄×C₂²⋊D₅₃	—	4	4	no	—
38	#13133176	—	2	2	no	—
39	#13133174	—	4	2	no	—
40	C₄²×D₅₃	—	4	2	no	—
41	Dic₄₂₄	C₂×C₄	1	4	yes	complemented
42	#13133162	C₂×C₄	1	8	yes	complemented
43	D₈₄₈	C₂×C₄	1	4	yes	complemented
44	#13133151	C₂³	1	4	yes	—
45	#13133149	—	2	2	no	—
46	#13133142	—	2	4	no	—
47	#13133139	—	2	4	no	—
48	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₅₃	—	2	2	no	—
52	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	complemented
53	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
54	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₅₃	D₄	1	2	yes	—
57	C₄×C₈×C₅₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₅₃	D₈	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₃	—	4	2	no	—
61	#11719945	—	2	8	no	—
62	#11719945	—	4	4	no	—
63	C₂×D₂₁₂	—	2	2	no	—
64	C₂×C₄×D₅₃	—	4	2	no	—
65	C₂×C₄×D₅₃	—	4	4	no	—
66	C₂×Dic₁₀₆	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	#41	1	2	yes	—
72	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
73	C₄²×C₅₃	D₈	1	2	yes	—
74	C₄²×C₅₃	C₂×D₄	1	2	yes	—
75	C₂²⋊Dic₅₃	—	4	2	no	—
76	#11719921	—	4	4	no	—
77	C₄⋊Dic₅₃	—	2	4	no	—
78	#11719919	—	4	4	no	—
79	C₄×Dic₅₃	—	4	2	no	—
80	C₄×Dic₅₃	—	4	4	no	—
81	Dic₂₁₂	—	2	2	no	—
82	Dic₂₁₂	C₂²×C₄	1	4	yes	—
83	D₄₂₄	—	2	2	no	—
84	D₄₂₄	C₂²×C₄	1	4	yes	—
85	#11719913	—	2	8	no	—
86	C₁₆×C₅₃	—	2	2	no	—
87	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
88	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
90	C₂²×D₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
94	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#135	1	2	yes	—
96	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
97	C₂²⋊D₅₃	—	4	8	no	—
98	C₂×Dic₅₃	—	4	2	no	—
99	C₂×Dic₅₃	—	4	4	no	—
100	D₂₁₂	—	2	4	no	—
101	D₂₁₂	—	4	2	no	—
102	C₄×D₅₃	—	4	8	no	—
103	C₄×D₅₃	—	8	8	no	—
104	Dic₁₀₆	—	2	4	no	—
105	Dic₁₀₆	—	4	2	no	—
106	C₈×C₅₃	—	2	2	no	—
107	C₈×C₅₃	C₄×D₄	1	2	yes	—
108	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
110	#33451	—	53	1	no	maximal, Sylow
111	C₂²×C₅₃	—	2	2	no	—
112	C₂²×C₅₃	C₄×D₈	1	2	yes	—
113	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
114	C₂×D₅₃	—	4	4	no	—
115	C₂×D₅₃	—	8	4	no	—
116	C₄×C₅₃	—	2	4	no	—
117	C₄×C₅₃	C₄×D₈	1	2	yes	—
118	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
119	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
120	Dic₅₃	—	4	4	no	—
121	Dic₅₃	—	8	4	no	—
122	C₂×#508	—	53	1	no	—
123	C₄×#135	—	53	2	no	—
124	#2177	—	53	2	no	—
125	#2176	—	53	2	no	—
126	#2175	—	53	4	no	—
127	#2158	—	53	1	no	—
128	#2138	—	53	2	no	—
129	C₂×#346	D₅₃	1	1	yes	characteristic, complemented
130	C₂×C₅₃	—	2	2	no	—
131	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
132	C₂×C₅₃	#3414	1	2	yes	—
133	D₅₃	—	8	4	no	—
134	C₂×#135	—	53	2	no	—
135	C₄×#41	—	106	2	no	—
136	#508	—	53	8	no	—
137	C₂×Q₃₂	—	53	1	no	—
138	C₂×SD₃₂	—	53	2	no	—
139	C₂×D₁₆	—	53	1	no	—
140	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
141	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
142	C₄×Q₁₆	—	53	4	no	—
143	C₄×SD₁₆	—	106	4	no	—
144	C₄×D₈	—	53	4	no	—
145	#427	—	53	2	no	—
146	#416	—	106	2	no	—
147	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
148	#367	—	53	2	no	—
149	#366	—	53	2	no	—
150	#358	—	53	4	no	—
151	#357	—	53	4	no	—
152	#346	C₂×D₅₃	1	2	yes	—
153	#346	C₂×D₅₃	1	2	yes	—
154	C₅₃	#33451	1	1	yes	characteristic, complemented, Sylow
155	C₂×#41	—	106	2	no	—
156	#135	—	53	8	no	—
157	#135	—	106	4	no	—
158	C₂×Q₁₆	—	53	2	no	—
159	C₂×SD₁₆	—	106	2	no	—
160	C₂×D₈	—	53	2	no	—
161	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
162	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	C₄×Q₈	—	106	4	no	—
164	C₄×D₄	—	106	4	no	—
165	C₄×D₄	—	212	4	no	—
166	#117	—	106	2	no	—
167	#117	—	212	2	no	—
168	C₂×C₄²	—	212	2	no	—
169	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
170	Q₃₂	—	53	4	no	—
171	SD₃₂	—	53	8	no	—
172	D₁₆	—	53	4	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₄×D₅₃	1	4	yes	complemented
175	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
176	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
178	#107	—	53	4	no	—
179	#106	—	106	2	no	—
180	#103	—	106	4	no	—
181	#102	—	106	4	no	—
182	C₄×C₈	D₂₁₂	1	2	yes	—
183	C₄×C₈	C₂²×D₅₃	1	2	yes	—
184	#41	—	106	8	no	—
185	#41	—	212	4	no	—
186	C₂×Q₈	—	106	2	no	—
187	C₂×D₄	—	106	2	no	—
188	C₂×D₄	—	212	2	no	—
189	C₂²×C₄	—	212	4	no	—
190	C₂²×C₄	—	212	2	no	—
191	C₂²×C₄	D₄₂₄	1	2	yes	—
192	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
193	Q₁₆	—	53	4	no	—
194	Q₁₆	—	106	2	no	—
195	SD₁₆	—	106	8	no	—
196	D₈	—	53	4	no	—
197	D₈	—	106	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
200	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
201	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
202	C₂×C₈	#11719945	1	2	yes	—
203	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
204	C₄⋊C₄	—	106	4	no	—
205	C₄⋊C₄	—	212	4	no	—
206	C₂²⋊C₄	—	212	2	no	—
207	C₂²⋊C₄	—	212	4	no	—
208	C₄²	—	212	4	no	—
209	C₄²	—	212	2	no	—
210	C₄²	D₄₂₄	1	2	yes	—
211	C₄²	C₂×D₂₁₂	1	2	yes	—
212	C₁₆	—	2	2	no	—
213	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
214	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
215	C₂³	—	212	2	no	—
216	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
217	Q₈	—	106	4	no	—
218	Q₈	—	212	2	no	—
219	D₄	—	106	4	no	—
220	D₄	—	212	8	no	—
221	D₄	—	212	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	424	8	no	—
224	C₂×C₄	—	212	4	no	—
225	C₂×C₄	—	212	2	no	—
226	C₂×C₄	—	212	8	no	—
227	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
228	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
229	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
230	C₂×C₄	#13133239	1	2	yes	—
231	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	C₄×D₂₁₂	1	2	yes	—
234	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
235	C₈	C₂×#11719945	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	424	4	no	—
238	C₂²	—	212	4	no	—
239	C₂²	C₄×D₄₂₄	1	2	yes	—
240	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	424	4	no	—
243	C₄	—	212	4	no	—
244	C₄	C₄×D₄₂₄	1	2	yes	—
245	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
246	C₄	C₂×#13133239	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	424	4	no	—
249	C₂	C₂×C₄×D₄₂₄	1	1	yes	characteristic
250	C₂	#20287851	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂×#16811055	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#33451
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₄×#13133239	C₂	semidirect
3. #20276592	C₂	semidirect
4. #20276587	C₂	semidirect
5. #20276581	C₂	semidirect
6. #20276499	C₂	semidirect
7. #20276423	C₂	semidirect
8. C₂×C₅₃×#346	C₂	semidirect
11. #16811055	C₄	semidirect
31. #13133239	C₂×C₄	semidirect
41. Dic₄₂₄	C₂×C₄	semidirect
42. #13133162	C₂×C₄	semidirect
43. D₈₄₈	C₂×C₄	semidirect
52. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
129. C₂×#346	D₅₃	semidirect
154. C₅₃	#33451	semidirect
174. C₂×C₁₆	C₄×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811055	2
C₄×#13133239	2
#20276592	2
#20276587	2
#20276581	2
#20276499	2
#20276423	2
C₂×C₅₃×#346	2
#33451	53