Subgroups — #27537513

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20287851	C₂	1	1	yes	maximal, characteristic
2	#20283633	C₂	1	2	yes	maximal, complemented
3	#20276602	C₂	1	4	yes	maximal, complemented
4	#20276597	C₂	1	4	yes	maximal, complemented
5	#20276510	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16810431	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133239	C₂²	1	2	yes	—
9	#16811389	—	2	2	no	—
10	#16811066	C₄	1	4	yes	complemented
11	#16811061	C₄	1	4	yes	complemented
12	#16811055	C₂²	1	4	yes	—
13	#16810746	—	2	4	no	—
14	#16810741	C₂²	1	8	yes	complemented
15	C₂×#13133153	C₂²	1	2	yes	—
16	#16810641	—	2	2	no	—
17	#16810453	C₂²	1	4	yes	complemented
18	#16810431	C₂²	1	4	yes	complemented
19	#16810431	C₂²	1	4	yes	complemented
20	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	C₅₃×#405	C₂²	1	1	yes	characteristic
23	C₅₃×#346	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₅₃	C₂²	1	2	yes	complemented
25	C₂×#11719945	—	2	2	no	—
26	#13133239	—	2	2	no	—
27	#13133239	—	2	2	no	—
28	#13133239	C₂×C₄	1	4	yes	—
29	#13133239	C₂³	1	4	yes	—
30	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
31	C₂×#11719913	—	2	2	no	—
32	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
33	#13133225	—	4	4	no	—
34	#13133223	—	2	8	no	—
35	C₂×#11719917	—	2	2	no	—
36	M₁₆×D₅₃	—	4	4	no	—
37	Dic₄₂₄	C₂×C₄	1	4	yes	—
38	#13133162	C₂×C₄	1	8	yes	—
39	D₈₄₈	C₂×C₄	1	4	yes	—
40	#13133153	—	2	2	no	—
41	#13133153	C₂³	1	4	yes	—
42	#13133145	—	2	8	no	—
43	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
45	#117×C₅₃	D₄	1	1	yes	characteristic
46	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
47	C₂×C₁₆×C₅₃	—	2	2	no	—
48	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
50	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
51	#97×C₅₃	D₄	1	2	yes	—
52	C₄×C₈×C₅₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₅₃	—	4	2	no	—
55	#11719945	—	2	4	no	—
56	#11719945	—	2	4	no	—
57	#11719945	—	4	2	no	—
58	#11719945	—	4	2	no	—
59	C₂×D₂₁₂	—	2	2	no	—
60	C₂×C₄×D₅₃	—	4	2	no	—
61	C₂×Dic₁₀₆	—	2	2	no	—
62	M₁₆×C₅₃	D₈	1	4	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₃	#41	1	2	yes	—
67	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₅₃	—	2	1	no	—
69	C₂²⋊C₄×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	C₂×D₄	1	2	yes	—
71	#11719917	—	2	4	no	—
72	#11719917	—	4	2	no	—
73	C₂×#88322	—	4	2	no	—
74	Dic₂₁₂	—	2	2	no	—
75	Dic₂₁₂	C₂²×C₄	1	4	yes	—
76	D₄₂₄	—	2	2	no	—
77	D₄₂₄	C₂²×C₄	1	4	yes	—
78	#11719913	—	2	4	no	—
79	#11719913	—	2	4	no	—
80	#11719912	—	4	8	no	—
81	C₈×D₅₃	—	4	8	no	—
82	C₁₆×C₅₃	—	2	1	no	—
83	C₁₆×C₅₃	—	2	1	no	—
84	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
85	C₂³×C₅₃	#135	1	1	yes	characteristic
86	C₂²×D₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
92	C₂²⋊D₅₃	—	4	4	no	—
93	C₂²⋊D₅₃	—	4	4	no	—
94	C₂×Dic₅₃	—	4	2	no	—
95	D₂₁₂	—	2	4	no	—
96	D₂₁₂	—	4	2	no	—
97	C₄×D₅₃	—	4	4	no	—
98	C₄×D₅₃	—	4	4	no	—
99	Dic₁₀₆	—	2	4	no	—
100	Dic₁₀₆	—	4	2	no	—
101	C₈×C₅₃	C₄×D₄	1	2	yes	—
102	C₈×C₅₃	C₂×D₈	1	4	yes	—
103	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₅₃	C₂×#41	1	1	yes	characteristic
105	#88322	—	4	4	no	—
106	#33464	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	—	4	1	no	—
108	C₂²×C₅₃	C₄×D₈	1	2	yes	—
109	C₂²×C₅₃	C₂×#135	1	1	yes	characteristic
110	C₂×D₅₃	—	4	4	no	—
111	C₂×D₅₃	—	16	2	no	—
112	C₄×C₅₃	—	4	2	no	—
113	C₄×C₅₃	C₄×D₈	1	2	yes	—
114	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
115	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
116	Dic₅₃	—	4	4	no	—
117	#3414	—	53	1	no	—
118	#2957	—	53	2	no	—
119	#2179	—	53	4	no	—
120	#2178	—	53	4	no	—
121	#2161	—	53	1	no	—
122	C₂×#359	—	53	2	no	—
123	#2108	D₅₃	1	1	yes	characteristic, complemented
124	C₂×C₅₃	—	2	1	no	—
125	C₂×C₅₃	—	2	2	no	—
126	C₂×C₅₃	C₂×C₄×D₈	1	1	yes	characteristic
127	D₅₃	—	8	4	no	—
128	C₂×#135	—	53	2	no	—
129	#568	—	106	2	no	—
130	#510	—	53	4	no	—
131	#509	—	53	4	no	—
132	#508	—	53	4	no	—
133	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
134	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
135	#444	—	106	4	no	—
136	#443	—	53	8	no	—
137	C₂×#108	—	53	2	no	—
138	#421	—	106	2	no	—
139	#405	C₂×D₅₃	1	1	yes	characteristic
140	#368	—	53	4	no	—
141	#359	—	53	4	no	—
142	#359	—	53	4	no	—
143	#346	C₂×D₅₃	1	2	yes	complemented
144	C₄×C₁₆	C₂×D₅₃	1	2	yes	complemented
145	C₅₃	#33464	1	1	yes	characteristic, complemented, Sylow
146	C₂×#41	—	106	2	no	—
147	#135	—	53	4	no	—
148	#135	—	53	4	no	—
149	#135	—	106	2	no	—
150	#135	—	106	2	no	—
151	C₂×Q₁₆	—	53	2	no	—
152	C₂×SD₁₆	—	106	2	no	—
153	C₂×D₈	—	53	2	no	—
154	#131	—	106	8	no	—
155	#131	—	212	4	no	—
156	C₂×M₁₆	—	106	2	no	—
157	C₂×M₁₆	—	212	4	no	—
158	C₂×M₁₆	D₂₁₂	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
160	#117	D₂₁₂	1	1	yes	characteristic
161	Q₃₂	—	53	4	no	—
162	SD₃₂	—	53	8	no	—
163	D₁₆	—	53	4	no	—
164	M₃₂	C₄×D₅₃	1	4	yes	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
167	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
168	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
169	#108	—	53	4	no	—
170	#108	—	106	2	no	—
171	#104	—	106	8	no	—
172	#97	D₂₁₂	1	2	yes	—
173	C₄×C₈	C₂²×D₅₃	1	2	yes	—
174	#41	—	106	4	no	—
175	#41	—	106	4	no	—
176	#41	—	212	2	no	—
177	#41	—	212	2	no	—
178	C₂×Q₈	—	106	2	no	—
179	C₂×D₄	—	106	2	no	—
180	C₂×D₄	—	212	2	no	—
181	C₂²×C₄	—	212	2	no	—
182	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
183	Q₁₆	—	53	4	no	—
184	Q₁₆	—	106	2	no	—
185	SD₁₆	—	106	4	no	—
186	SD₁₆	—	106	4	no	—
187	D₈	—	53	4	no	—
188	D₈	—	106	2	no	—
189	M₁₆	—	106	4	no	—
190	M₁₆	—	212	2	no	—
191	M₁₆	—	212	8	no	—
192	M₁₆	D₄₂₄	1	4	yes	—
193	C₂×C₈	—	212	2	no	—
194	C₂×C₈	—	212	8	no	—
195	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
196	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
197	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
198	C₂×C₈	#11719945	1	2	yes	—
199	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	C₂×D₂₁₂	1	2	yes	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	2	1	no	—
205	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
206	C₂³	—	212	2	no	—
207	C₂³	#13133239	1	1	yes	characteristic
208	Q₈	—	106	4	no	—
209	Q₈	—	212	2	no	—
210	D₄	—	106	4	no	—
211	D₄	—	212	4	no	—
212	D₄	—	212	4	no	—
213	D₄	—	212	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	212	2	no	—
216	C₂×C₄	—	212	4	no	—
217	C₂×C₄	—	212	4	no	—
218	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
219	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
220	C₂×C₄	#13133239	1	1	yes	characteristic
221	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
222	C₈	—	212	4	no	—
223	C₈	C₄×D₂₁₂	1	2	yes	—
224	C₈	C₂×D₄₂₄	1	4	yes	—
225	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
226	C₈	C₂×#11719945	1	1	yes	characteristic
227	C₂²	—	4	1	no	—
228	C₂²	—	848	2	no	—
229	C₂²	—	212	4	no	—
230	C₂²	C₄×D₄₂₄	1	2	yes	—
231	C₂²	C₂×#13133239	1	1	yes	characteristic
232	C₄	—	4	2	no	—
233	C₄	—	212	4	no	—
234	C₄	C₄×D₄₂₄	1	2	yes	—
235	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
236	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
237	C₂	—	2	1	no	—
238	C₂	—	2	2	no	—
239	C₂	—	424	4	no	—
240	C₂	C₂×C₄×D₄₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₅₃×#2108	C₂
socle	C₂×C₅₃	C₂×C₄×D₈
prime radical	#20287851	C₂
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2108	D₅₃
hypocenter	C₅₃	#33464
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #20283633	C₂	semidirect
3. #20276602	C₂	semidirect
4. #20276597	C₂	semidirect
5. #20276510	C₂	semidirect
6. C₂×#16810431	C₂	semidirect
7. C₅₃×#2108	C₂	semidirect
10. #16811066	C₄	semidirect
11. #16811061	C₄	semidirect
14. #16810741	C₂²	semidirect
17. #16810453	C₂²	semidirect
18. #16810431	C₂²	semidirect
19. #16810431	C₂²	semidirect
23. C₅₃×#346	C₂²	semidirect
24. C₄×C₁₆×C₅₃	C₂²	semidirect
123. #2108	D₅₃	semidirect
143. #346	C₂×D₅₃	semidirect
144. C₄×C₁₆	C₂×D₅₃	semidirect
145. C₅₃	#33464	semidirect

Maximal subgroups

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

subgroup	index
#20287851	2
#20283633	2
#20276602	2
#20276597	2
#20276510	2
C₂×#16810431	2
C₅₃×#2108	2
#33464	53