Subgroups — #27353076

Order 13568.

778 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	2	yes	maximal
2	#20273467	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	#16810676	—	2	2	no	—
6	#16810676	—	2	2	no	—
7	#16810676	C₂²	1	4	yes	—
8	#16810676	C₂²	1	4	yes	—
9	C₂×C₄⋊#88322	C₂²	1	4	yes	—
10	C₂×C₂²⋊#88322	—	2	2	no	—
11	C₂×C₄×#88322	—	2	2	no	—
12	#16810376	C₂²	1	8	yes	complemented
13	#16810374	C₂²	1	8	yes	complemented
14	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
15	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
16	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₅₃	C₄	1	4	yes	—
18	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
19	C₂×#97×C₅₃	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
21	C₂²×#88322	—	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	C₄×#88322	—	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	4	no	—
32	C₂×M₁₆×C₅₃	C₈	1	8	yes	complemented
33	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₅₃	D₄	1	2	yes	—
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
43	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
44	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
45	#97×C₅₃	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₅₃	C₂³	1	4	yes	—
48	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
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	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
65	M₁₆×C₅₃	—	2	8	no	—
66	M₁₆×C₅₃	C₂×C₈	1	16	yes	complemented
67	C₂×C₈×C₅₃	—	2	4	no	—
68	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	complemented
69	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
71	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
72	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
73	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
74	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
76	C₄²×C₅₃	C₂×D₄	1	2	yes	—
77	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
79	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
80	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂×#88322	—	2	8	no	—
82	C₂×#88322	—	4	8	no	—
83	C₂³×C₅₃	—	4	1	no	—
84	C₂³×C₅₃	—	4	1	no	—
85	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
86	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
87	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	1	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	C₄⋊C₈	1	8	yes	—
97	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
98	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
106	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
108	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
111	C₈×C₅₃	—	2	4	no	—
112	C₈×C₅₃	C₂²×C₈	1	8	yes	—
113	#88322	—	4	8	no	—
114	#17787	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	2	no	—
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	4	4	no	—
120	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
121	C₂²×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
122	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	2	no	—
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
128	C₄×C₅₃	C₂×C₄⋊C₈	1	4	yes	—
129	C₄×C₅₃	#430	1	2	yes	—
130	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
131	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
132	C₄×C₅₃	#575	1	2	yes	—
133	C₂×#431	—	53	2	no	—
134	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
135	#1565	—	53	8	no	—
136	#1564	—	53	4	no	—
137	C₂×C₅₃	—	2	2	no	—
138	C₂×C₅₃	—	2	4	no	—
139	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
140	C₂×C₅₃	#2911	1	1	yes	characteristic
141	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
142	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
143	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
144	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
145	#431	—	53	4	no	—
146	#431	—	53	4	no	—
147	#431	—	106	2	no	—
148	#431	—	106	2	no	—
149	C₂×C₄⋊C₈	—	53	4	no	—
150	C₂×C₂²⋊C₈	—	106	2	no	—
151	C₄×M₁₆	Dic₅₃	1	4	yes	—
152	C₄×M₁₆	C₂×D₅₃	1	4	yes	—
153	C₂×#97	Dic₅₃	1	2	yes	—
154	C₂×C₄×C₈	—	106	2	no	—
155	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
156	#335	—	53	8	no	—
157	#334	—	53	8	no	—
158	C₅₃	#17787	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
160	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
161	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	#88322	1	8	yes	complemented
164	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
165	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
166	C₂²×C₈	—	106	4	no	—
167	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
168	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
169	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
170	C₂×C₄²	Dic₁₀₆	1	2	yes	—
171	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
172	C₂×C₄²	D₂₁₂	1	2	yes	—
173	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
174	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
175	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
176	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
177	C₄⋊C₈	—	53	8	no	—
178	C₄⋊C₈	—	106	4	no	—
179	C₂²⋊C₈	—	106	4	no	—
180	C₂²⋊C₈	—	106	4	no	—
181	#97	C₂×Dic₅₃	1	8	yes	—
182	C₄×C₈	—	106	4	no	—
183	C₄×C₈	—	106	4	no	—
184	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
185	C₄×C₈	C₂²×D₅₃	1	4	yes	—
186	C₂⁴	C₄⋊Dic₅₃	1	1	yes	characteristic
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₂×#88322	1	4	yes	—
190	C₂²×C₄	#11719917	1	4	yes	—
191	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
192	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
193	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
194	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
197	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
198	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
199	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
201	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
203	M₁₆	—	2	8	no	—
204	M₁₆	C₂×#88322	1	16	yes	complemented
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	106	8	no	—
207	C₂×C₈	—	212	8	no	—
208	C₂×C₈	C₂×#88322	1	8	yes	complemented
209	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
210	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
211	C₄²	C₄⋊Dic₅₃	1	2	yes	—
212	C₄²	C₄⋊Dic₅₃	1	2	yes	—
213	C₄²	C₄⋊Dic₅₃	1	2	yes	—
214	C₄²	C₂×Dic₁₀₆	1	2	yes	—
215	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
216	C₄²	C₂×D₂₁₂	1	2	yes	—
217	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
218	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
219	C₄²	C₂²×Dic₅₃	1	2	yes	—
220	C₄²	C₂³×D₅₃	1	1	yes	characteristic
221	C₂³	—	4	1	no	—
222	C₂³	—	4	1	no	—
223	C₂³	C₄⋊#88322	1	4	yes	—
224	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
225	C₂³	C₂×C₄⋊Dic₅₃	1	2	yes	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	C₄⋊#88322	1	4	yes	—
235	C₂×C₄	C₄⋊#88322	1	8	yes	—
236	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
237	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
238	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
239	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
240	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂²×#88322	1	2	yes	—
243	C₂×C₄	C₂²×#88322	1	4	yes	—
244	C₂×C₄	C₂×#11719917	1	2	yes	—
245	C₂×C₄	C₂×#11719917	1	4	yes	—
246	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
248	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
249	C₈	—	2	4	no	—
250	C₈	—	212	8	no	—
251	C₈	C₂²×#88322	1	8	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₂×C₄⋊#88322	1	2	yes	—
258	C₂²	C₂×C₄⋊#88322	1	4	yes	—
259	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	1	no	—
264	C₄	C₂×C₄⋊#88322	1	4	yes	—
265	C₄	C₂×C₄⋊#88322	1	2	yes	—
266	C₄	#16810672	1	2	yes	—
267	C₄	C₂³×#88322	1	1	yes	characteristic
268	C₄	C₂²×#11719917	1	1	yes	characteristic
269	C₄	#16811423	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	—	2	2	no	—
272	C₂	C₂²×C₄⋊#88322	1	1	yes	characteristic
273	C₂	#20283305	1	1	yes	characteristic
274	C₂	C₂×#16810672	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₁₂
commutator	C₄×C₅₃	C₂³×C₈
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	#17787
LCS	G ▷ 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. #20273467	C₂	semidirect
3. #20273463	C₂	semidirect
12. #16810376	C₂²	semidirect
13. #16810374	C₂²	semidirect
32. C₂×M₁₆×C₅₃	C₈	semidirect
66. M₁₆×C₅₃	C₂×C₈	semidirect
68. C₂×C₈×C₅₃	C₂×C₈	semidirect
158. C₅₃	#17787	semidirect
163. C₂×M₁₆	#88322	semidirect
204. M₁₆	C₂×#88322	semidirect
208. C₂×C₈	C₂×#88322	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
#20273467	2
#20273463	2
C₂×C₄×M₁₆×C₅₃	2
#17787	53