Subgroups — #27354839

Order 13568.

590 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	#20283650	C₂	1	2	yes	maximal
2	#20273587	C₂	1	8	yes	maximal, complemented
3	#20273471	C₂	1	1	yes	maximal, characteristic
4	#20273170	C₂	1	2	yes	maximal
5	C₂×#16810345	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#3170	C₂	1	1	yes	maximal, characteristic
7	#16810754	—	2	2	no	—
8	#16810754	C₂²	1	4	yes	—
9	Q₈×#88322	—	2	2	no	—
10	Q₈×#88322	C₂²	1	4	yes	—
11	#16810650	—	2	2	no	—
12	#16810647	C₂²	1	2	yes	—
13	#16810579	—	2	2	no	—
14	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	#16810374	C₂²	1	4	yes	—
17	Q₈⋊#88322	C₂²	1	8	yes	—
18	#16810345	C₂²	1	4	yes	—
19	C₅₃×#533	C₂²	1	2	yes	—
20	C₅₃×#500	C₄	1	1	yes	characteristic
21	C₅₃×#494	C₄	1	1	yes	characteristic
22	C₅₃×#487	C₄	1	2	yes	—
23	C₅₃×#467	C₄	1	4	yes	—
24	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	—
25	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#11719917	—	2	2	no	—
27	C₂²×#88322	—	2	1	no	—
28	C₂²×#88322	D₄	1	2	yes	—
29	C₄⋊#88322	—	2	2	no	—
30	C₄⋊#88322	—	2	4	no	—
31	C₄⋊#88322	—	2	4	no	—
32	C₄⋊#88322	C₂³	1	4	yes	—
33	#13133128	—	2	4	no	—
34	C₄×#88322	—	2	1	no	—
35	C₄×#88322	—	2	4	no	—
36	C₄×#88322	D₄	1	1	yes	characteristic
37	C₄×#88322	D₄	1	1	yes	characteristic
38	C₄×#88322	C₂³	1	2	yes	—
39	C₄×#88322	C₂³	1	2	yes	—
40	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
42	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
43	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
44	#125×C₅₃	—	2	2	no	—
45	C₂²⋊Q₈×C₅₃	—	2	2	no	—
46	C₂²⋊Q₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₄×Q₈×C₅₃	—	2	2	no	—
48	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
49	#117×C₅₃	—	2	2	no	—
50	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
51	#107×C₅₃	C₂×C₄	1	4	yes	—
52	#103×C₅₃	C₂×C₄	1	8	yes	—
53	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂×Q₈×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
61	Q₁₆×C₅₃	—	4	4	no	—
62	C₂×C₈×C₅₃	—	4	2	no	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
65	C₄⋊C₄×C₅₃	—	2	2	no	—
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₄⋊C₄×C₅₃	—	2	2	no	—
68	C₄⋊C₄×C₅₃	—	2	4	no	—
69	C₄⋊C₄×C₅₃	—	2	4	no	—
70	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
71	C₂²⋊C₄×C₅₃	—	2	4	no	—
72	C₄²×C₅₃	—	2	4	no	—
73	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	#41	1	1	yes	characteristic
76	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
77	#11719917	—	4	4	no	—
78	C₂×#88322	—	2	4	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	2	2	no	—
81	C₂×#88322	—	2	4	no	—
82	C₂×#88322	C₂×D₄	1	2	yes	—
83	C₂×#88322	C₂×D₄	1	2	yes	—
84	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
85	Q₈×C₅₃	—	2	8	no	—
86	Q₈×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#137	1	4	yes	—
95	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
98	C₈×C₅₃	—	2	2	no	—
99	C₈×C₅₃	—	2	2	no	—
100	#88322	—	2	4	no	—
101	#88322	—	2	2	no	—
102	#88322	—	2	2	no	—
103	#88322	—	4	4	no	—
104	#17980	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	—	2	1	no	—
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	#441	1	2	yes	—
108	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	4	4	no	—
112	C₄×C₅₃	—	4	4	no	—
113	C₄×C₅₃	#441	1	2	yes	—
114	C₄×C₅₃	#443	1	2	yes	—
115	C₄×C₅₃	#568	1	2	yes	—
116	C₄×C₅₃	C₂×#137	1	2	yes	—
117	#3170	D₅₃	1	1	yes	characteristic
118	#2960	—	53	2	no	—
119	#1587	—	53	8	no	—
120	#1566	—	53	1	no	—
121	#1493	—	53	2	no	—
122	C₂×#322	—	53	1	no	—
123	C₂×C₅₃	—	2	2	no	—
124	C₂×C₅₃	#2932	1	1	yes	characteristic
125	C₂×C₅₃	C₂×#441	1	1	yes	characteristic
126	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
127	#533	C₂×D₅₃	1	2	yes	—
128	#500	Dic₅₃	1	1	yes	characteristic
129	#494	Dic₅₃	1	1	yes	characteristic
130	#487	Dic₅₃	1	2	yes	—
131	#467	Dic₅₃	1	4	yes	—
132	#446	—	53	4	no	—
133	#446	—	106	2	no	—
134	C₈×Q₈	—	53	4	no	—
135	C₈×Q₈	—	106	2	no	—
136	C₄×Q₁₆	C₂×D₅₃	1	4	yes	—
137	#424	—	106	2	no	—
138	#423	—	53	2	no	—
139	#405	—	106	2	no	—
140	C₂×C₄×C₈	—	53	1	no	—
141	C₂×C₄×C₈	—	53	1	no	—
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	#334	—	53	4	no	—
144	Q₈⋊C₈	—	53	8	no	—
145	#322	—	53	4	no	—
146	C₅₃	#17980	1	1	yes	characteristic, complemented, Sylow
147	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	—
148	C₂×M₁₆	—	106	2	no	—
149	C₂²×C₈	—	53	2	no	—
150	C₂²×C₈	—	106	1	no	—
151	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
152	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
153	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
154	#125	—	2	2	no	—
155	C₂²⋊Q₈	—	2	2	no	—
156	C₂²⋊Q₈	C₂×Dic₅₃	1	4	yes	—
157	C₄×Q₈	—	2	2	no	—
158	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
159	#117	—	2	2	no	—
160	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
161	#107	C₂×Dic₅₃	1	4	yes	—
162	C₄⋊C₈	—	53	4	no	—
163	C₄⋊C₈	—	106	4	no	—
164	C₄⋊C₈	—	106	4	no	—
165	C₄⋊C₈	—	106	2	no	—
166	#103	C₂×Dic₅₃	1	8	yes	—
167	#97	—	106	4	no	—
168	C₄×C₈	—	53	1	no	—
169	C₄×C₈	—	53	1	no	—
170	C₄×C₈	—	53	2	no	—
171	C₄×C₈	—	53	2	no	—
172	C₄×C₈	—	106	4	no	—
173	C₄×C₈	—	106	1	no	—
174	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
175	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
176	C₄×C₈	C₂²×D₅₃	1	2	yes	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
179	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
180	C₂²×C₄	#11719947	1	1	yes	characteristic
181	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
182	Q₁₆	—	4	4	no	—
183	M₁₆	—	212	4	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	53	2	no	—
186	C₂×C₈	—	53	2	no	—
187	C₂×C₈	—	106	4	no	—
188	C₂×C₈	—	106	4	no	—
189	C₂×C₈	—	106	2	no	—
190	C₂×C₈	—	106	4	no	—
191	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
192	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
199	C₂²⋊C₄	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	D₄×D₅₃	1	1	yes	characteristic
202	C₄²	#11719947	1	1	yes	characteristic
203	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
204	C₄²	C₂³×D₅₃	1	1	yes	characteristic
205	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
206	Q₈	—	2	8	no	—
207	Q₈	—	4	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
214	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
215	C₂×C₄	#13133249	1	4	yes	—
216	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
217	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	—	106	4	no	—
222	C₈	—	106	2	no	—
223	C₈	—	106	2	no	—
224	C₈	—	212	4	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#16810730	1	2	yes	—
228	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	4	no	—
232	C₄	—	4	4	no	—
233	C₄	#16810730	1	2	yes	—
234	C₄	#16810740	1	2	yes	—
235	C₄	#16811390	1	2	yes	—
236	C₄	C₂×#13133249	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	#20283433	1	1	yes	characteristic
239	C₂	C₂×#16810730	1	1	yes	characteristic
240	C₂	C₂×#16810740	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3170	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#3170	D₅₃
hypocenter	C₅₃	#17980
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #20273587	C₂	semidirect
146. C₅₃	#17980	semidirect

Maximal subgroups

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

subgroup	index
#20283650	2
#20273587	2
#20273471	2
#20273170	2
C₂×#16810345	2
C₅₃×#3170	2
#17980	53