Subgroups — #27353115

Order 13568.

710 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	2	yes	maximal
2	#20273482	C₂	1	8	yes	maximal, complemented
3	#20273463	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810680	—	2	2	no	—
6	#16810680	C₂²	1	4	yes	—
7	#16810676	—	2	2	no	—
8	#16810676	C₂²	1	4	yes	—
9	#16810647	C₂²	1	4	yes	—
10	#16810587	—	2	2	no	—
11	C₄×#11719917	—	2	2	no	—
12	#16810376	C₂²	1	8	yes	complemented
13	#16810374	C₂²	1	8	yes	complemented
14	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₅₃	C₄	1	4	yes	—
17	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
18	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
19	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂×#11719917	—	2	4	no	—
22	C₄⋊#88322	—	2	4	no	—
23	C₄⋊#88322	C₂³	1	8	yes	—
24	C₂²⋊#88322	—	2	4	no	—
25	C₂²⋊#88322	—	2	4	no	—
26	#13133128	—	2	4	no	—
27	C₄×#88322	—	2	4	no	—
28	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
31	C₂×M₁₆×C₅₃	—	2	8	no	—
32	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	D₄	1	2	yes	—
38	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
41	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	#97×C₅₃	C₂×C₄	1	4	yes	—
44	#97×C₅₃	C₂×C₄	1	4	yes	—
45	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₅₃	C₂³	1	4	yes	—
47	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	—	2	4	no	—
49	C₂²×C₄×C₅₃	—	4	2	no	—
50	C₂²×C₄×C₅₃	—	4	2	no	—
51	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	2	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	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
63	M₁₆×C₅₃	—	2	16	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	M₁₆	1	8	yes	complemented
66	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
68	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
69	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
71	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
73	C₄²×C₅₃	C₂×D₄	1	2	yes	—
74	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
76	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
77	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
78	#11719917	—	4	8	no	—
79	C₂×#88322	—	2	8	no	—
80	C₂³×C₅₃	—	2	2	no	—
81	C₂³×C₅₃	—	4	1	no	—
82	C₂³×C₅₃	—	4	1	no	—
83	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
84	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	4	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
103	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
106	C₈×C₅₃	—	2	4	no	—
107	C₈×C₅₃	C₂×M₁₆	1	8	yes	—
108	#88322	—	4	8	no	—
109	#17795	—	53	1	no	maximal, Sylow
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	2	no	—
113	C₂²×C₅₃	—	2	2	no	—
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	4	4	no	—
116	C₂²×C₅₃	#423	1	2	yes	—
117	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	2	1	no	—
120	C₄×C₅₃	—	2	2	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	#423	1	2	yes	—
124	C₄×C₅₃	#430	1	2	yes	—
125	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
126	C₄×C₅₃	#575	1	2	yes	—
127	#2920	—	53	2	no	—
128	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
129	#1569	—	53	8	no	—
130	#1564	—	53	4	no	—
131	C₂×C₅₃	—	2	2	no	—
132	C₂×C₅₃	—	4	2	no	—
133	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
134	C₂×C₅₃	#2911	1	1	yes	characteristic
135	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
136	C₂²×M₁₆	Dic₅₃	1	2	yes	—
137	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
138	#432	—	53	4	no	—
139	#432	—	106	2	no	—
140	#431	—	53	4	no	—
141	#431	—	106	2	no	—
142	#423	—	53	4	no	—
143	#407	—	106	2	no	—
144	C₄×M₁₆	—	106	2	no	—
145	C₄×M₁₆	Dic₅₃	1	4	yes	—
146	C₄×M₁₆	C₂×D₅₃	1	4	yes	—
147	C₂×#97	Dic₅₃	1	1	yes	characteristic
148	C₂×#97	Dic₅₃	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
150	#335	—	53	8	no	—
151	#334	—	53	8	no	—
152	C₅₃	#17795	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
154	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
155	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
156	C₂×M₁₆	—	2	8	no	—
157	C₂×M₁₆	—	106	4	no	—
158	C₂×M₁₆	C₂×Dic₅₃	1	8	yes	—
159	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
160	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
161	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
162	C₂×C₄²	Dic₁₀₆	1	2	yes	—
163	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
164	C₂×C₄²	D₂₁₂	1	2	yes	—
165	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
167	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
168	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
169	C₄⋊C₈	—	53	8	no	—
170	C₄⋊C₈	—	106	4	no	—
171	C₂²⋊C₈	—	106	4	no	—
172	C₂²⋊C₈	—	106	4	no	—
173	#97	—	106	4	no	—
174	#97	C₂×Dic₅₃	1	4	yes	—
175	#97	C₂×Dic₅₃	1	4	yes	—
176	C₄×C₈	—	106	4	no	—
177	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
178	C₄×C₈	C₂²×D₅₃	1	4	yes	—
179	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
184	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
185	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
186	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
187	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
189	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
190	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
191	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
193	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
195	M₁₆	—	2	16	no	—
196	M₁₆	—	212	8	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	106	8	no	—
199	C₂×C₈	#11719917	1	8	yes	complemented
200	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
201	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
202	C₄²	C₄⋊Dic₅₃	1	2	yes	—
203	C₄²	C₄⋊Dic₅₃	1	2	yes	—
204	C₄²	C₄⋊Dic₅₃	1	2	yes	—
205	C₄²	C₂×Dic₁₀₆	1	2	yes	—
206	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
207	C₄²	C₂×D₂₁₂	1	2	yes	—
208	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
209	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
210	C₄²	C₂²×Dic₅₃	1	2	yes	—
211	C₄²	C₂³×D₅₃	1	1	yes	characteristic
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
216	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	1	no	—
228	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
231	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
232	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
234	C₂×C₄	C₂×#11719917	1	4	yes	—
235	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
236	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
237	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	212	8	no	—
240	C₈	C₂×#11719917	1	8	yes	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	4	no	—
247	C₂²	#16810647	1	2	yes	—
248	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	#16810647	1	2	yes	—
255	C₄	#16810672	1	2	yes	—
256	C₄	C₂²×#11719917	1	2	yes	—
257	C₄	#16811423	1	2	yes	—
258	C₂	—	2	2	no	—
259	C₂	—	4	2	no	—
260	C₂	C₂×#16810647	1	1	yes	characteristic
261	C₂	#20283305	1	1	yes	characteristic
262	C₂	C₂×#16810672	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#17795
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #20273482	C₂	semidirect
3. #20273463	C₂	semidirect
12. #16810376	C₂²	semidirect
13. #16810374	C₂²	semidirect
65. C₂×C₈×C₅₃	M₁₆	semidirect
152. C₅₃	#17795	semidirect
199. C₂×C₈	#11719917	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
#20273482	2
#20273463	2
C₂×C₄×M₁₆×C₅₃	2
#17795	53