Subgroups — #27300289

Order 13568.

358 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810453	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276499	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274769	C₂	1	2	yes	maximal
4	#20272830	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	C₂²	1	2	yes	—
7	#16810661	C₂²	1	2	yes	—
8	#16810640	—	2	2	no	—
9	#16810453	—	2	1	no	—
10	#16810453	—	2	1	no	—
11	#16810451	—	2	1	no	—
12	#16810449	—	2	1	no	—
13	#16810395	—	2	2	no	—
14	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#504	—	2	1	no	—
16	C₅₃×#504	—	2	1	no	—
17	C₅₃×#504	C₄	1	2	yes	complemented
18	C₅₃×#504	C₄	1	2	yes	complemented
19	C₂×M₃₂×C₅₃	—	2	1	no	—
20	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₃	—	2	1	no	—
22	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#11719917	—	2	2	no	—
24	#13133176	—	2	2	no	—
25	#13133153	—	2	2	no	—
26	#13133153	—	2	2	no	—
27	#13133151	—	2	2	no	—
28	#13133149	—	2	2	no	—
29	#13133144	—	4	4	no	—
30	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
31	#131×C₅₃	—	2	1	no	—
32	#131×C₅₃	—	2	1	no	—
33	#131×C₅₃	C₂×C₄	1	2	yes	—
34	#131×C₅₃	Q₈	1	2	yes	—
35	C₂×M₁₆×C₅₃	—	2	1	no	—
36	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	—	2	1	no	—
38	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
39	M₃₂×C₅₃	—	2	1	no	—
40	M₃₂×C₅₃	—	2	2	no	—
41	M₃₂×C₅₃	—	2	2	no	—
42	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×C₁₆×C₅₃	—	2	1	no	—
44	C₂×C₁₆×C₅₃	—	2	2	no	—
45	C₂×C₁₆×C₅₃	—	2	2	no	—
46	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
49	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
50	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
51	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
52	#41×C₅₃	—	2	1	no	—
53	#41×C₅₃	—	2	1	no	—
54	#41×C₅₃	C₄⋊C₄	1	2	yes	—
55	#41×C₅₃	D₈	1	2	yes	—
56	C₂×Q₈×C₅₃	Q₁₆	1	1	yes	characteristic
57	C₂×D₄×C₅₃	—	2	1	no	—
58	C₂×D₄×C₅₃	Q₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	—	2	1	no	—
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	M₁₆×C₅₃	—	2	1	no	—
62	M₁₆×C₅₃	—	2	2	no	—
63	M₁₆×C₅₃	—	2	2	no	—
64	M₁₆×C₅₃	C₄⋊C₄	1	2	yes	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	—	2	2	no	—
67	C₂×C₈×C₅₃	—	2	2	no	—
68	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
74	C₂²⋊Dic₅₃	—	4	2	no	—
75	C₄⋊Dic₅₃	—	4	2	no	—
76	C₄×Dic₅₃	—	4	2	no	—
77	#11719917	—	4	2	no	—
78	#11719917	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₁₆×C₅₃	—	2	2	no	—
81	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂³×C₅₃	—	2	1	no	—
86	C₂³×C₅₃	#135	1	1	yes	characteristic
87	Q₈×C₅₃	—	2	1	no	—
88	Q₈×C₅₃	#107	1	2	yes	—
89	D₄×C₅₃	—	2	2	no	—
90	D₄×C₅₃	—	2	2	no	—
91	D₄×C₅₃	—	2	1	no	—
92	D₄×C₅₃	#107	1	2	yes	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
102	C₂×Dic₅₃	—	8	2	no	—
103	C₈×C₅₃	—	2	2	no	—
104	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₈×C₅₃	#117	1	1	yes	characteristic
106	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
108	#88322	—	8	2	no	—
109	#12841	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	4	1	no	—
112	C₂²×C₅₃	—	4	2	no	—
113	C₂²×C₅₃	#427	1	1	yes	characteristic
114	C₂²×C₅₃	#466	1	1	yes	characteristic
115	C₂²×C₅₃	#467	1	1	yes	characteristic
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	#380	1	1	yes	characteristic
118	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
119	C₄×C₅₃	#466	1	1	yes	characteristic
120	C₄×C₅₃	#467	1	1	yes	characteristic
121	Dic₅₃	—	16	2	no	—
122	C₂×#504	D₅₃	1	1	yes	characteristic
123	C₂×#368	—	53	1	no	—
124	#2158	—	53	1	no	—
125	#1814	—	53	2	no	—
126	#1389	—	53	2	no	—
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	—	2	1	no	—
129	C₂×C₅₃	—	2	1	no	—
130	C₂×C₅₃	—	4	2	no	—
131	C₂×C₅₃	#1808	1	1	yes	characteristic
132	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
133	#504	—	2	1	no	—
134	#504	—	2	1	no	—
135	#504	Dic₅₃	1	2	yes	complemented
136	#504	Dic₅₃	1	2	yes	complemented
137	C₂×M₃₂	—	2	1	no	—
138	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
139	C₂²×C₁₆	—	2	1	no	—
140	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
141	C₂×#108	—	53	2	no	—
142	#427	—	53	2	no	—
143	#421	—	106	2	no	—
144	#368	—	106	1	no	—
145	#368	—	106	1	no	—
146	#367	—	106	1	no	—
147	#366	—	106	1	no	—
148	#344	—	106	2	no	—
149	C₅₃	#12841	1	1	yes	characteristic, complemented, Sylow
150	C₂×#41	D₂₁₂	1	1	yes	characteristic
151	#131	—	2	1	no	—
152	#131	—	2	1	no	—
153	#131	Dic₁₀₆	1	2	yes	—
154	#131	C₂×Dic₅₃	1	2	yes	—
155	C₂×M₁₆	—	2	1	no	—
156	C₂×M₁₆	—	106	2	no	—
157	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
160	#117	—	106	2	no	—
161	M₃₂	—	2	1	no	—
162	M₃₂	—	2	2	no	—
163	M₃₂	—	2	2	no	—
164	M₃₂	C₂×Dic₅₃	1	2	yes	—
165	C₂×C₁₆	—	2	1	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
169	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
170	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
172	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
174	#108	—	106	2	no	—
175	#108	—	106	2	no	—
176	#107	—	106	2	no	—
177	#106	—	106	2	no	—
178	#104	—	212	4	no	—
179	#41	—	2	1	no	—
180	#41	—	2	1	no	—
181	#41	D₄₂₄	1	2	yes	—
182	#41	C₄⋊Dic₅₃	1	2	yes	—
183	C₂×Q₈	Dic₂₁₂	1	1	yes	characteristic
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	Dic₂₁₂	1	1	yes	characteristic
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
188	M₁₆	—	2	1	no	—
189	M₁₆	—	2	2	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	212	2	no	—
192	M₁₆	—	212	2	no	—
193	M₁₆	C₄⋊Dic₅₃	1	2	yes	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	212	2	no	—
198	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
199	C₂×C₈	#11719945	1	1	yes	characteristic
200	C₂×C₈	#11719945	1	1	yes	characteristic
201	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
202	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
203	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
204	C₄⋊C₄	—	212	2	no	—
205	C₂²⋊C₄	—	212	2	no	—
206	C₄²	—	212	2	no	—
207	C₁₆	—	2	2	no	—
208	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
209	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
210	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
211	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	C₂³	#13133239	1	1	yes	characteristic
214	Q₈	—	2	1	no	—
215	Q₈	#13133151	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	#13133151	1	2	yes	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	424	2	no	—
224	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
225	C₂×C₄	#13133191	1	1	yes	characteristic
226	C₂×C₄	#13133191	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
228	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
229	C₂×C₄	#13133239	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	424	2	no	—
232	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
233	C₈	#13133176	1	1	yes	characteristic
234	C₈	#13133196	1	1	yes	characteristic
235	C₈	#13133196	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	2	no	—
239	C₂²	#16810661	1	1	yes	characteristic
240	C₂²	#16810859	1	1	yes	characteristic
241	C₂²	#16810864	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	848	2	no	—
244	C₄	#16810483	1	1	yes	characteristic
245	C₄	C₂×#13133151	1	1	yes	characteristic
246	C₄	#16810859	1	1	yes	characteristic
247	C₄	#16810864	1	1	yes	characteristic
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	2	1	no	—
251	C₂	—	4	2	no	—
252	C₂	#20274744	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16810859
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#504	C₂
socle	C₂×C₅₃	#1808
prime radical	C₂×#41×C₅₃	D₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	#12841
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#16810453	C₂	semidirect
2. #20276499	C₂	semidirect
4. #20272830	C₂	semidirect
17. C₅₃×#504	C₄	semidirect
18. C₅₃×#504	C₄	semidirect
135. #504	Dic₅₃	semidirect
136. #504	Dic₅₃	semidirect
149. C₅₃	#12841	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810453	2
#20276499	2
#20274769	2
#20272830	2
C₂×C₅₃×#504	2
#12841	53