Subgroups — #27302333

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810430	C₂	1	4	yes	maximal, complemented
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274887	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133153	—	2	1	no	—
7	C₂×#13133153	C₂²	1	2	yes	—
8	#16810647	—	2	1	no	—
9	#16810430	—	2	8	no	—
10	C₄×#11719908	—	2	1	no	—
11	C₄×#11719908	—	2	1	no	—
12	C₂×#135×C₅₃	—	2	1	no	—
13	C₂×#135×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#532	—	2	1	no	—
15	C₅₃×#495	—	2	1	no	—
16	C₅₃×#495	C₄	1	2	yes	—
17	C₅₃×#494	C₄	1	1	yes	characteristic
18	C₅₃×#493	C₄	1	1	yes	characteristic
19	C₅₃×#492	—	2	1	no	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	D₄	1	4	yes	complemented
25	#13133153	—	2	4	no	—
26	#13133153	—	4	2	no	—
27	C₄⋊#88322	—	4	2	no	—
28	C₂×#41×C₅₃	—	2	2	no	—
29	#135×C₅₃	—	2	4	no	—
30	#135×C₅₃	—	2	4	no	—
31	#135×C₅₃	—	4	2	no	—
32	#135×C₅₃	—	4	2	no	—
33	C₂×Q₁₆×C₅₃	—	2	1	no	—
34	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×SD₁₆×C₅₃	—	2	2	no	—
36	C₂×SD₁₆×C₅₃	—	2	2	no	—
37	C₂×D₈×C₅₃	—	2	1	no	—
38	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₅₃	—	2	1	no	—
42	#127×C₅₃	—	2	1	no	—
43	#124×C₅₃	—	2	2	no	—
44	#121×C₅₃	—	4	2	no	—
45	C₄×D₄×C₅₃	—	4	2	no	—
46	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
51	#41×C₅₃	—	4	2	no	—
52	#41×C₅₃	—	4	2	no	—
53	#41×C₅₃	—	4	2	no	—
54	#41×C₅₃	—	4	2	no	—
55	C₂×Q₈×C₅₃	—	2	2	no	—
56	C₂×D₄×C₅₃	—	2	2	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	—	4	2	no	—
60	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	Q₁₆×C₅₃	—	2	4	no	—
63	Q₁₆×C₅₃	—	4	2	no	—
64	SD₁₆×C₅₃	—	4	4	no	—
65	SD₁₆×C₅₃	—	4	4	no	—
66	D₈×C₅₃	—	2	4	no	—
67	D₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	D₈	1	2	yes	—
71	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
72	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	#41	1	2	yes	—
76	C₄⋊C₄×C₅₃	—	4	2	no	—
77	C₂²⋊C₄×C₅₃	—	4	4	no	—
78	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₄²×C₅₃	D₈	1	2	yes	—
81	#11719917	—	4	2	no	—
82	#11719917	—	4	2	no	—
83	C₂×#88322	—	4	2	no	—
84	#11719908	—	2	4	no	—
85	C₂³×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
87	Q₈×C₅₃	—	4	2	no	—
88	Q₈×C₅₃	—	4	2	no	—
89	D₄×C₅₃	—	4	2	no	—
90	D₄×C₅₃	—	4	2	no	—
91	D₄×C₅₃	—	8	2	no	—
92	D₄×C₅₃	—	8	2	no	—
93	D₄×C₅₃	—	8	4	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	8	2	no	—
97	C₂×C₄×C₅₃	—	8	2	no	—
98	C₂×C₄×C₅₃	#102	1	2	yes	—
99	C₂×C₄×C₅₃	#102	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
106	C₈×C₅₃	#102	1	2	yes	—
107	C₈×C₅₃	#102	1	2	yes	—
108	C₈×C₅₃	C₄×D₄	1	2	yes	—
109	C₈×C₅₃	#121	1	2	yes	—
110	#88322	—	8	2	no	—
111	#13208	—	53	1	no	maximal, Sylow
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	8	4	no	—
114	C₂²×C₅₃	—	8	2	no	—
115	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
116	C₂²×C₅₃	#486	1	1	yes	characteristic
117	C₂²×C₅₃	#492	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	8	2	no	—
120	C₄×C₅₃	#359	1	4	yes	complemented
121	C₄×C₅₃	#390	1	1	yes	characteristic
122	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
123	C₄×C₅₃	#486	1	1	yes	characteristic
124	C₄×C₅₃	#493	1	1	yes	characteristic
125	#3147	D₅₃	1	1	yes	characteristic
126	C₂×#359	—	53	4	no	—
127	C₂×C₄×C₁₆	—	53	1	no	—
128	#1848	—	53	1	no	—
129	C₂×C₅₃	—	2	1	no	—
130	C₂×C₅₃	—	2	1	no	—
131	C₂×C₅₃	—	16	2	no	—
132	C₂×C₅₃	#1957	1	1	yes	characteristic
133	C₂×C₅₃	C₂×#359	1	2	yes	—
134	C₂×#135	—	2	1	no	—
135	C₂×#135	C₂×D₅₃	1	2	yes	—
136	#532	—	2	1	no	—
137	C₂²×C₁₆	—	53	2	no	—
138	#495	—	2	1	no	—
139	#495	Dic₅₃	1	2	yes	—
140	#494	Dic₅₃	1	1	yes	characteristic
141	#493	Dic₅₃	1	1	yes	characteristic
142	#492	—	2	1	no	—
143	C₂×#108	—	53	2	no	—
144	C₂×#108	—	106	1	no	—
145	#423	—	106	1	no	—
146	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
147	#359	—	106	8	no	—
148	C₄×C₁₆	—	106	1	no	—
149	C₄×C₁₆	—	106	1	no	—
150	C₅₃	#13208	1	1	yes	characteristic, complemented, Sylow
151	C₂×#41	—	2	2	no	—
152	#135	—	2	4	no	—
153	#135	—	2	4	no	—
154	#135	—	4	2	no	—
155	#135	—	4	2	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
162	C₂×M₁₆	—	106	2	no	—
163	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
165	C₄⋊Q₈	—	2	1	no	—
166	#127	—	2	1	no	—
167	#124	—	2	2	no	—
168	#121	—	4	2	no	—
169	C₄×D₄	—	4	2	no	—
170	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
171	C₂×C₁₆	—	53	4	no	—
172	C₂×C₁₆	—	106	2	no	—
173	C₂×C₁₆	—	106	2	no	—
174	#108	—	106	4	no	—
175	#108	—	212	2	no	—
176	C₄⋊C₈	—	212	2	no	—
177	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
178	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
180	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
181	#41	—	4	2	no	—
182	#41	—	4	2	no	—
183	#41	—	4	2	no	—
184	#41	—	4	2	no	—
185	C₂×Q₈	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	#11719923	1	2	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
192	Q₁₆	—	2	4	no	—
193	Q₁₆	—	4	2	no	—
194	SD₁₆	—	4	4	no	—
195	SD₁₆	—	4	4	no	—
196	D₈	—	2	4	no	—
197	D₈	—	4	2	no	—
198	M₁₆	—	212	2	no	—
199	M₁₆	—	212	2	no	—
200	C₂×C₈	—	212	2	no	—
201	C₂×C₈	#11719922	1	2	yes	—
202	C₂×C₈	#11719923	1	2	yes	—
203	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
204	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
205	C₂×C₈	D₄×D₅₃	1	2	yes	—
206	C₂×C₈	#11719947	1	2	yes	—
207	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
208	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
209	C₄⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₄²	#11719922	1	2	yes	—
212	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
213	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
214	C₁₆	—	106	4	no	—
215	C₂³	—	4	2	no	—
216	C₂³	C₂×#11719923	1	1	yes	characteristic
217	Q₈	—	4	2	no	—
218	Q₈	—	4	2	no	—
219	D₄	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	8	2	no	—
222	D₄	—	8	2	no	—
223	D₄	—	8	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	#13133138	1	2	yes	—
229	C₂×C₄	#13133138	1	2	yes	—
230	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
231	C₂×C₄	#13133204	1	1	yes	characteristic
232	C₂×C₄	#13133213	1	1	yes	characteristic
233	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
234	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
235	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
236	C₈	—	424	2	no	—
237	C₈	#13133138	1	2	yes	—
238	C₈	#13133138	1	2	yes	—
239	C₈	D₄×Dic₅₃	1	2	yes	—
240	C₈	#13133188	1	2	yes	—
241	C₂²	—	2	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	—	8	4	no	—
244	C₂²	C₂×#13133138	1	1	yes	characteristic
245	C₂²	#16810969	1	1	yes	characteristic
246	C₂²	#16810995	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	8	2	no	—
249	C₄	#16810430	1	4	yes	complemented
250	C₄	#16810526	1	1	yes	characteristic
251	C₄	C₂×#13133138	1	1	yes	characteristic
252	C₄	#16810969	1	1	yes	characteristic
253	C₄	#16810999	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	16	2	no	—
257	C₂	#20275522	1	1	yes	characteristic
258	C₂	C₂×#16810430	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting, prime radical	C₅₃×#3147	C₂
socle	C₂²×C₅₃	C₂×#102
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#3147	D₅₃
hypocenter	C₅₃	#13208
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#16810430	C₂	semidirect
2. C₂×C₄×#11719908	C₂	semidirect
3. #20274887	C₂	semidirect
24. C₂×#11719908	D₄	semidirect
120. C₄×C₅₃	#359	semidirect
150. C₅₃	#13208	semidirect
249. C₄	#16810430	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810430	2
C₂×C₄×#11719908	2
#20274887	2
C₅₃×#3147	2
#13208	53