Subgroups — #27283246

Order 13568.

430 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285301	C₂	1	1	yes	maximal, characteristic
2	#20275658	C₂	1	2	yes	maximal
3	#20273213	C₂	1	2	yes	maximal
4	C₈×#11719917	C₂	1	1	yes	maximal, characteristic
5	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811180	C₂²	1	2	yes	—
7	#16811028	C₄	1	1	yes	characteristic, complemented
8	#16811003	C₄	1	1	yes	characteristic, complemented
9	#16810974	C₄	1	2	yes	complemented
10	#16810864	—	2	1	no	—
11	#16810864	—	2	1	no	—
12	C₄×Dic₂₁₂	—	2	1	no	—
13	C₄×Dic₂₁₂	—	2	1	no	—
14	#16810676	C₂²	1	1	yes	characteristic
15	C₄×#11719917	C₂²	1	1	yes	characteristic
16	#16810383	—	2	4	no	—
17	#16810359	—	2	4	no	—
18	#16810345	—	2	1	no	—
19	C₈×#88322	—	2	1	no	—
20	C₂×#107×C₅₃	—	2	1	no	—
21	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
23	C₅₃×#340	—	2	1	no	—
24	C₂×Dic₂₁₂	—	2	1	no	—
25	C₂×Dic₂₁₂	—	2	1	no	—
26	C₂×#11719917	D₄	1	2	yes	—
27	#13133217	C₂×C₄	1	2	yes	—
28	#13133217	C₂×C₄	1	2	yes	—
29	#13133209	—	2	2	no	—
30	#13133196	—	2	2	no	—
31	#13133196	—	2	2	no	—
32	C₄×Dic₁₀₆	—	2	2	no	—
33	C₄×Dic₁₀₆	—	2	2	no	—
34	#13133176	—	2	2	no	—
35	#13133151	—	2	1	no	—
36	#13133151	—	2	1	no	—
37	C₄⋊#88322	—	2	1	no	—
38	#13133142	—	2	2	no	—
39	#13133142	—	2	2	no	—
40	C₂²⋊#88322	D₄	1	2	yes	—
41	#13133128	—	2	1	no	—
42	C₄×#88322	—	2	1	no	—
43	C₄×#88322	—	2	1	no	—
44	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
47	C₂×C₄²×C₅₃	—	2	2	no	—
48	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
49	#107×C₅₃	—	2	4	no	—
50	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
54	#95×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	4	2	no	—
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₁₀₆	—	2	2	no	—
60	C₂×Dic₁₀₆	—	2	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	4	1	no	—
63	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄⋊C₄×C₅₃	—	2	4	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	—	4	4	no	—
70	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₅₃	#41	1	1	yes	characteristic
73	C₄²×C₅₃	#41	1	1	yes	characteristic
74	C₂²⋊Dic₅₃	—	4	2	no	—
75	C₄⋊Dic₅₃	—	2	2	no	—
76	C₄⋊Dic₅₃	—	2	2	no	—
77	C₄⋊Dic₅₃	—	2	2	no	—
78	C₄⋊Dic₅₃	—	2	2	no	—
79	#11719919	—	4	2	no	—
80	#11719919	—	4	2	no	—
81	C₄×Dic₅₃	—	4	2	no	—
82	#11719917	—	2	2	no	—
83	#11719917	Q₁₆	1	4	yes	complemented
84	C₂×#88322	—	2	2	no	—
85	C₂×#88322	—	2	2	no	—
86	Dic₂₁₂	—	4	2	no	—
87	Dic₂₁₂	—	4	2	no	—
88	C₂³×C₅₃	#127	1	1	yes	characteristic
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	4	1	no	—
92	C₂×C₄×C₅₃	—	4	4	no	—
93	C₂×C₄×C₅₃	—	8	2	no	—
94	C₂×C₄×C₅₃	—	8	2	no	—
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
102	C₂×C₄×C₅₃	#135	1	2	yes	—
103	C₂×Dic₅₃	—	4	2	no	—
104	C₂×Dic₅₃	—	4	2	no	—
105	Dic₁₀₆	—	4	2	no	—
106	Dic₁₀₆	—	4	2	no	—
107	Dic₁₀₆	—	8	2	no	—
108	C₈×C₅₃	—	2	1	no	—
109	C₈×C₅₃	—	2	1	no	—
110	C₈×C₅₃	—	2	1	no	—
111	C₈×C₅₃	#104	1	2	yes	—
112	#88322	—	2	4	no	—
113	#88322	—	4	2	no	—
114	#9718	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	#390	1	1	yes	characteristic
118	C₂²×C₅₃	#494	1	1	yes	characteristic
119	C₂²×C₅₃	#495	1	1	yes	characteristic
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	2	1	no	—
122	C₄×C₅₃	—	2	1	no	—
123	C₄×C₅₃	—	8	4	no	—
124	C₄×C₅₃	C₂×#104	1	1	yes	characteristic
125	C₄×C₅₃	#421	1	1	yes	characteristic
126	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
127	C₄×C₅₃	#467	1	2	yes	—
128	Dic₅₃	—	8	2	no	—
129	Dic₅₃	—	8	2	no	—
130	#3170	—	53	1	no	—
131	#1986	—	53	2	no	—
132	C₄×#107	D₅₃	1	1	yes	characteristic
133	#1503	—	53	2	no	—
134	C₈×M₁₆	—	53	1	no	—
135	C₂×C₅₃	—	2	1	no	—
136	C₂×C₅₃	—	2	1	no	—
137	C₂×C₅₃	#1969	1	1	yes	characteristic
138	C₂×C₅₃	#1976	1	1	yes	characteristic
139	C₂×C₅₃	#1980	1	1	yes	characteristic
140	#533	—	53	2	no	—
141	#500	—	53	1	no	—
142	#494	—	53	1	no	—
143	#487	—	53	2	no	—
144	#467	—	106	1	no	—
145	#467	—	106	1	no	—
146	C₄×Q₁₆	—	106	1	no	—
147	C₄×Q₁₆	—	106	1	no	—
148	#431	—	53	1	no	—
149	C₂×#107	—	2	1	no	—
150	C₄×M₁₆	—	53	1	no	—
151	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
152	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
153	#340	—	2	1	no	—
154	#339	—	106	4	no	—
155	#328	—	106	4	no	—
156	#322	—	106	1	no	—
157	C₈²	—	106	1	no	—
158	C₅₃	#9718	1	1	yes	characteristic, complemented, Sylow
159	C₂×Q₁₆	—	106	1	no	—
160	C₂×Q₁₆	—	106	1	no	—
161	C₂×M₁₆	—	53	2	no	—
162	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
164	C₄⋊Q₈	—	53	2	no	—
165	C₄⋊Q₈	—	53	2	no	—
166	#125	—	106	2	no	—
167	C₂²⋊Q₈	—	106	2	no	—
168	C₂²⋊Q₈	—	106	2	no	—
169	C₄×Q₈	—	106	2	no	—
170	C₄×Q₈	—	106	2	no	—
171	#117	—	106	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
175	#107	—	2	4	no	—
176	#107	—	106	1	no	—
177	#107	—	106	1	no	—
178	C₄⋊C₈	—	106	1	no	—
179	#103	—	106	2	no	—
180	#103	—	106	2	no	—
181	C₂²⋊C₈	—	53	2	no	—
182	#97	—	106	1	no	—
183	C₄×C₈	—	106	1	no	—
184	C₄×C₈	—	106	1	no	—
185	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
186	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
187	C₄×C₈	D₂₁₂	1	1	yes	characteristic
188	C₄×C₈	D₂₁₂	1	1	yes	characteristic
189	#95	—	2	2	no	—
190	C₂×Q₈	—	106	2	no	—
191	C₂×Q₈	—	106	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
194	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
196	Q₁₆	—	212	2	no	—
197	Q₁₆	—	212	2	no	—
198	M₁₆	—	53	4	no	—
199	M₁₆	—	106	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	106	2	no	—
203	C₂×C₈	—	106	2	no	—
204	C₂×C₈	#11719921	1	1	yes	characteristic
205	C₂×C₈	#11719921	1	1	yes	characteristic
206	C₂×C₈	#11719921	1	1	yes	characteristic
207	C₂×C₈	#11719921	1	1	yes	characteristic
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	106	2	no	—
211	C₄⋊C₄	—	106	2	no	—
212	C₄⋊C₄	—	106	2	no	—
213	C₄⋊C₄	—	106	2	no	—
214	C₄⋊C₄	—	212	2	no	—
215	C₄⋊C₄	—	212	2	no	—
216	C₂²⋊C₄	—	212	2	no	—
217	C₄²	—	4	4	no	—
218	C₄²	—	212	2	no	—
219	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
220	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
221	C₄²	#11719949	1	1	yes	characteristic
222	C₄²	#11719949	1	1	yes	characteristic
223	C₂³	#13133213	1	1	yes	characteristic
224	Q₈	—	424	2	no	—
225	Q₈	—	212	2	no	—
226	Q₈	—	212	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	212	2	no	—
234	C₂×C₄	—	212	2	no	—
235	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
236	C₂×C₄	#13133179	1	1	yes	characteristic
237	C₂×C₄	#13133179	1	1	yes	characteristic
238	C₂×C₄	#13133192	1	1	yes	characteristic
239	C₂×C₄	#13133192	1	1	yes	characteristic
240	C₂×C₄	#13133206	1	1	yes	characteristic
241	C₂×C₄	Q₁₆×D₅₃	1	2	yes	—
242	C₂×C₄	#13133236	1	2	yes	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	106	4	no	—
247	C₈	—	212	2	no	—
248	C₈	#13133145	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	#16810524	1	1	yes	characteristic
252	C₂²	#16811001	1	1	yes	characteristic
253	C₂²	#16811006	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	8	4	no	—
258	C₄	—	424	2	no	—
259	C₄	—	424	2	no	—
260	C₄	C₂×#13133145	1	1	yes	characteristic
261	C₄	#16810641	1	1	yes	characteristic
262	C₄	#16810717	1	2	yes	—
263	C₄	#16810865	1	2	yes	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	#20275574	1	1	yes	characteristic
267	C₂	#20275611	1	1	yes	characteristic
268	C₂	#20275633	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#107×C₅₃	C₂
socle	C₂²×C₅₃	#390
prime radical	C₂³×C₅₃	#127
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#107	D₅₃
hypocenter	C₅₃	#9718
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
7. #16811028	C₄	semidirect
8. #16811003	C₄	semidirect
9. #16810974	C₄	semidirect
83. #11719917	Q₁₆	semidirect
158. C₅₃	#9718	semidirect

Maximal subgroups

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

subgroup	index
#20285301	2
#20275658	2
#20273213	2
C₈×#11719917	2
C₄×#107×C₅₃	2
#9718	53