Subgroups — #27354815

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 248 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	#20273566	C₂	1	8	yes	maximal, complemented
3	#20273474	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	#16810376	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	1	yes	characteristic
29	C₂²×#88322	D₄	1	1	yes	characteristic
30	C₄⋊#88322	—	2	2	no	—
31	C₄⋊#88322	—	2	4	no	—
32	C₄⋊#88322	—	2	4	no	—
33	C₄⋊#88322	C₂³	1	4	yes	—
34	#13133128	—	2	4	no	—
35	C₄×#88322	—	2	1	no	—
36	C₄×#88322	—	2	4	no	—
37	C₄×#88322	D₄	1	2	yes	—
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	2	no	—
79	C₂×#88322	—	2	4	no	—
80	C₂×#88322	—	2	2	no	—
81	C₂×#88322	—	4	2	no	—
82	C₂×#88322	Q₁₆	1	4	yes	complemented
83	C₂×#88322	C₂×D₄	1	2	yes	—
84	C₂×#88322	C₂×D₄	1	2	yes	—
85	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
86	Q₈×C₅₃	—	2	8	no	—
87	Q₈×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
96	C₂×C₄×C₅₃	#135	1	2	yes	—
97	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	2	2	no	—
102	#88322	—	2	2	no	—
103	#88322	—	4	2	no	—
104	#88322	—	4	4	no	—
105	#88322	C₂×Q₁₆	1	4	yes	—
106	#17977	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	C₄×Q₁₆	1	2	yes	—
110	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	—	4	4	no	—
114	C₄×C₅₃	—	4	4	no	—
115	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
116	C₄×C₅₃	#444	1	2	yes	—
117	C₄×C₅₃	#568	1	2	yes	—
118	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
119	C₄×C₅₃	C₂×#135	1	1	yes	characteristic
120	#3170	D₅₃	1	1	yes	characteristic
121	#2960	—	53	2	no	—
122	#1584	—	53	8	no	—
123	#1567	—	53	1	no	—
124	#1493	—	53	2	no	—
125	C₂×#322	—	53	1	no	—
126	C₂×C₅₃	—	2	2	no	—
127	C₂×C₅₃	#2932	1	1	yes	characteristic
128	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
129	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
130	#533	C₂×D₅₃	1	2	yes	—
131	#500	Dic₅₃	1	1	yes	characteristic
132	#494	Dic₅₃	1	1	yes	characteristic
133	#487	Dic₅₃	1	2	yes	—
134	#467	Dic₅₃	1	4	yes	—
135	#446	—	53	4	no	—
136	#446	—	106	2	no	—
137	C₈×Q₈	—	53	4	no	—
138	C₈×Q₈	—	106	2	no	—
139	C₄×Q₁₆	C₂×D₅₃	1	4	yes	—
140	#424	—	106	2	no	—
141	#423	—	53	2	no	—
142	#405	—	106	2	no	—
143	C₂×C₄×C₈	—	53	1	no	—
144	C₂×C₄×C₈	—	53	1	no	—
145	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
146	#335	—	53	4	no	—
147	Q₈⋊C₈	—	53	8	no	—
148	#322	—	53	4	no	—
149	C₅₃	#17977	1	1	yes	characteristic, complemented, Sylow
150	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	—
151	C₂×M₁₆	—	106	2	no	—
152	C₂²×C₈	—	53	1	no	—
153	C₂²×C₈	—	53	1	no	—
154	C₂²×C₈	—	106	1	no	—
155	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
156	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
157	C₄⋊Q₈	C₂×Dic₅₃	1	2	yes	—
158	#125	—	2	2	no	—
159	C₂²⋊Q₈	—	2	2	no	—
160	C₂²⋊Q₈	C₂×Dic₅₃	1	4	yes	—
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	C₂²×D₅₃	1	4	yes	—
163	#117	—	2	2	no	—
164	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
165	#107	C₂×Dic₅₃	1	4	yes	—
166	C₄⋊C₈	—	53	4	no	—
167	C₄⋊C₈	—	106	4	no	—
168	C₄⋊C₈	—	106	4	no	—
169	C₄⋊C₈	—	106	2	no	—
170	#103	C₂×Dic₅₃	1	8	yes	—
171	#97	—	106	4	no	—
172	C₄×C₈	—	53	2	no	—
173	C₄×C₈	—	53	2	no	—
174	C₄×C₈	—	53	2	no	—
175	C₄×C₈	—	106	4	no	—
176	C₄×C₈	—	106	1	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	2	yes	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	C₂²×Dic₅₃	1	4	yes	—
182	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
183	C₂²×C₄	#11719947	1	1	yes	characteristic
184	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
185	Q₁₆	—	4	4	no	—
186	M₁₆	—	212	4	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	53	4	no	—
189	C₂×C₈	—	53	2	no	—
190	C₂×C₈	—	53	2	no	—
191	C₂×C₈	—	106	2	no	—
192	C₂×C₈	—	106	4	no	—
193	C₂×C₈	—	106	2	no	—
194	C₂×C₈	—	212	2	no	—
195	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
196	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	C₂²×Dic₅₃	1	4	yes	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	D₄×D₅₃	1	1	yes	characteristic
206	C₄²	#11719947	1	1	yes	characteristic
207	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
208	C₄²	C₂³×D₅₃	1	1	yes	characteristic
209	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
210	Q₈	—	2	8	no	—
211	Q₈	—	4	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
217	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
218	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
219	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
220	C₂×C₄	#13133240	1	2	yes	—
221	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	53	4	no	—
227	C₈	—	106	2	no	—
228	C₈	—	212	2	no	—
229	C₈	—	212	4	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	Q₁₆×Dic₅₃	1	2	yes	—
233	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	4	no	—
237	C₄	—	4	4	no	—
238	C₄	Q₁₆×Dic₅₃	1	2	yes	—
239	C₄	#16810745	1	2	yes	—
240	C₄	#16811390	1	2	yes	—
241	C₄	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
242	C₄	C₂×#13133240	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#20283433	1	1	yes	characteristic
245	C₂	C₂×Q₁₆×Dic₅₃	1	1	yes	characteristic
246	C₂	C₂×#16810745	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	#17977
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. #20273566	C₂	semidirect
82. C₂×#88322	Q₁₆	semidirect
149. C₅₃	#17977	semidirect

Maximal subgroups

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

subgroup	index
#20283650	2
#20273566	2
#20273474	2
#20273170	2
C₂×#16810345	2
C₅₃×#3170	2
#17977	53