Subgroups — #27537183

Order 13568.

598 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#13133223	C₂	1	1	yes	maximal, characteristic
2	#20276520	C₂	1	4	yes	maximal, complemented
3	#20276512	C₂	1	4	yes	maximal, complemented
4	#20276473	C₂	1	2	yes	maximal, complemented
5	#20276328	C₂	1	2	yes	maximal
6	C₄×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13133223	C₄	1	1	yes	characteristic
9	C₂×#13133223	C₂²	1	1	yes	characteristic
10	C₄×#11719945	C₄	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	2	yes	—
12	#16810578	C₄	1	1	yes	characteristic
13	#16810578	C₂²	1	1	yes	characteristic
14	#16810577	C₄	1	2	yes	—
15	#16810577	C₂²	1	2	yes	—
16	C₄×#11719917	C₄	1	1	yes	characteristic
17	C₄×#11719912	C₄	1	2	yes	—
18	C₄×C₈×D₅₃	C₂²	1	2	yes	—
19	#16810441	C₂²	1	8	yes	—
20	#16810402	C₂²	1	8	yes	—
21	#16810399	C₂²	1	2	yes	—
22	C₄×#11719908	C₂²	1	2	yes	—
23	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
25	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
26	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
27	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
28	#13133223	—	2	4	no	—
29	#13133223	—	2	4	no	—
30	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
31	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
32	C₂×#11719912	C₂×C₄	1	2	yes	—
33	C₂×#11719912	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₅₃	C₂³	1	2	yes	—
36	C₄×Dic₁₀₆	C₈	1	2	yes	—
37	C₄×D₂₁₂	C₈	1	2	yes	—
38	C₄×C₂²⋊D₅₃	C₈	1	4	yes	—
39	#13133176	C₂×C₄	1	1	yes	characteristic
40	#13133174	C₂×C₄	1	2	yes	—
41	C₄²×D₅₃	C₂×C₄	1	2	yes	—
42	#13133157	—	2	4	no	—
43	C₂×#11719908	C₂³	1	4	yes	—
44	#13133128	C₂×C₄	1	2	yes	—
45	#13133127	C₂×C₄	1	2	yes	—
46	#13133127	C₂×C₄	1	2	yes	—
47	C₄×#88322	C₂×C₄	1	2	yes	—
48	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
49	C₈×Dic₅₃	C₂³	1	2	yes	—
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₅₃	—	2	4	no	—
54	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
55	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
60	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
62	#11719945	—	2	2	no	—
63	#11719945	—	2	2	no	—
64	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
65	C₂×C₄×D₅₃	C₂×C₈	1	4	yes	—
66	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
67	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
68	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	#41	1	4	yes	—
73	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
74	C₄²×C₅₃	C₂×C₈	1	2	yes	—
75	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
78	#11719921	C₂×C₈	1	4	yes	—
79	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
80	#11719919	C₂×C₈	1	4	yes	—
81	C₄×Dic₅₃	C₂×C₈	1	4	yes	—
82	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
83	#11719917	—	2	2	no	—
84	#11719917	—	2	2	no	—
85	C₂×#88322	C₂²×C₄	1	2	yes	—
86	C₂×#88322	C₂²×C₄	1	2	yes	—
87	#11719912	—	2	4	no	—
88	#11719912	—	2	4	no	—
89	C₈×D₅₃	—	2	4	no	—
90	C₈×D₅₃	—	2	4	no	—
91	C₁₆×C₅₃	—	2	4	no	—
92	#11719908	—	2	4	no	—
93	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
94	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₅₃	—	2	4	no	—
96	C₂×C₄×C₅₃	#117	1	2	yes	—
97	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
102	C₂²⋊D₅₃	—	2	4	no	—
103	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
104	C₂×Dic₅₃	C₂²×C₈	1	4	yes	—
105	D₂₁₂	—	2	2	no	—
106	C₄×D₅₃	—	2	2	no	—
107	C₄×D₅₃	—	2	2	no	—
108	C₄×D₅₃	—	2	8	no	—
109	Dic₁₀₆	—	2	2	no	—
110	C₈×C₅₃	#117	1	2	yes	—
111	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
112	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
113	C₈×C₅₃	C₂×#41	1	2	yes	—
114	#88322	—	2	2	no	—
115	#88322	—	2	2	no	—
116	#33420	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	2	2	no	—
118	C₂²×C₅₃	#431	1	2	yes	—
119	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
120	C₂×D₅₃	—	2	4	no	—
121	C₂×D₅₃	—	2	2	no	—
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	#431	1	2	yes	—
124	C₄×C₅₃	#504	1	4	yes	—
125	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
126	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
127	Dic₅₃	—	2	4	no	—
128	Dic₅₃	—	2	2	no	—
129	C₄×#131	—	53	1	no	—
130	#2164	—	53	4	no	—
131	#2162	—	53	4	no	—
132	#2151	—	53	2	no	—
133	#2113	—	53	2	no	—
134	C₄×M₃₂	—	53	1	no	—
135	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic, complemented
136	C₂×C₅₃	—	2	2	no	—
137	C₂×C₅₃	C₂×#431	1	1	yes	characteristic
138	C₂×C₅₃	C₂×#504	1	2	yes	—
139	D₅₃	—	4	2	no	—
140	C₂×#131	—	53	1	no	—
141	C₂×#131	—	53	1	no	—
142	C₄×#41	—	53	1	no	—
143	C₂×M₃₂	—	53	2	no	—
144	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
145	#405	—	53	1	no	—
146	#405	—	53	1	no	—
147	#405	—	53	2	no	—
148	#405	—	53	2	no	—
149	C₄×M₁₆	—	53	1	no	—
150	C₄×M₁₆	—	53	2	no	—
151	C₂×C₄×C₈	—	53	2	no	—
152	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
153	C₄⋊C₁₆	—	53	8	no	—
154	C₂²⋊C₁₆	—	53	8	no	—
155	#346	—	53	2	no	—
156	C₄×C₁₆	—	53	2	no	—
157	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
158	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
159	C₅₃	#33420	1	1	yes	characteristic, complemented, Sylow
160	C₂×#41	—	53	1	no	—
161	#131	—	106	4	no	—
162	#131	—	106	4	no	—
163	C₂×M₁₆	—	53	1	no	—
164	C₂×M₁₆	—	53	1	no	—
165	C₂×M₁₆	—	53	2	no	—
166	C₂×M₁₆	—	53	2	no	—
167	C₂²×C₈	—	53	2	no	—
168	C₂²×C₈	—	53	2	no	—
169	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
170	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
171	C₄×Q₈	—	53	2	no	—
172	C₄×D₄	—	53	4	no	—
173	C₄×D₄	—	53	2	no	—
174	#117	—	53	1	no	—
175	#117	—	53	2	no	—
176	C₂×C₄²	—	53	2	no	—
177	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
178	M₃₂	—	106	4	no	—
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	—	53	4	no	—
181	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
182	#97	—	53	2	no	—
183	#97	—	53	2	no	—
184	#97	—	53	2	no	—
185	C₄×C₈	—	53	2	no	—
186	C₄×C₈	—	53	2	no	—
187	C₄×C₈	—	53	2	no	—
188	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
189	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
192	#41	—	106	2	no	—
193	#41	—	106	2	no	—
194	C₂×Q₈	—	53	1	no	—
195	C₂×D₄	—	53	2	no	—
196	C₂×D₄	—	53	1	no	—
197	C₂²×C₄	—	53	4	no	—
198	C₂²×C₄	—	53	2	no	—
199	C₂²×C₄	C₈×D₅₃	1	2	yes	—
200	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
201	M₁₆	—	106	2	no	—
202	M₁₆	—	106	2	no	—
203	M₁₆	—	106	4	no	—
204	M₁₆	—	106	4	no	—
205	C₂×C₈	—	53	2	no	—
206	C₂×C₈	—	53	2	no	—
207	C₂×C₈	—	106	4	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	#11719912	1	4	yes	—
210	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
211	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
212	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
213	C₂×C₈	#11719945	1	4	yes	—
214	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
215	C₄⋊C₄	—	53	4	no	—
216	C₄⋊C₄	—	53	2	no	—
217	C₂²⋊C₄	—	53	2	no	—
218	C₂²⋊C₄	—	53	4	no	—
219	C₄²	—	53	4	no	—
220	C₄²	—	53	2	no	—
221	C₄²	C₈×D₅₃	1	2	yes	—
222	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
223	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
224	C₁₆	—	2	4	no	—
225	C₁₆	—	106	4	no	—
226	C₂³	—	53	2	no	—
227	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
228	Q₈	—	106	2	no	—
229	D₄	—	106	4	no	—
230	D₄	—	106	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	53	4	no	—
233	C₂×C₄	—	53	2	no	—
234	C₂×C₄	—	106	8	no	—
235	C₂×C₄	—	106	2	no	—
236	C₂×C₄	—	106	2	no	—
237	C₂×C₄	#13133174	1	2	yes	—
238	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
239	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
240	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂×#11719912	1	2	yes	—
242	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
243	C₈	—	106	2	no	—
244	C₈	—	106	2	no	—
245	C₈	#13133174	1	2	yes	—
246	C₈	C₂×#11719912	1	2	yes	—
247	C₈	C₂×#11719912	1	2	yes	—
248	C₈	C₂×#11719945	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	106	4	no	—
251	C₂²	—	106	2	no	—
252	C₂²	#16810674	1	2	yes	—
253	C₂²	C₂²×C₈×D₅₃	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	106	4	no	—
256	C₄	—	106	2	no	—
257	C₄	#16810674	1	2	yes	—
258	C₄	#16811039	1	4	yes	—
259	C₄	C₂×#13133174	1	1	yes	characteristic
260	C₄	C₂²×#11719912	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	212	2	no	—
263	C₂	C₂×#16810674	1	1	yes	characteristic
264	C₂	C₂×#16811039	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	#33420
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276520	C₂	semidirect
3. #20276512	C₂	semidirect
4. #20276473	C₂	semidirect
6. C₄×#13133157	C₂	semidirect
7. C₂×C₄×C₁₆×C₅₃	C₂	semidirect
135. C₂×C₄×C₁₆	D₅₃	semidirect
159. C₅₃	#33420	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133223	2
#20276520	2
#20276512	2
#20276473	2
#20276328	2
C₄×#13133157	2
C₂×C₄×C₁₆×C₅₃	2
#33420	53