Subgroups — #27300174

Order 13568.

410 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810440	C₂	1	2	yes	maximal
3	#20274887	C₂	1	1	yes	maximal, characteristic
4	#20272824	C₂	1	2	yes	maximal
5	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₁₂	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₀₆	—	2	1	no	—
8	C₄×Dic₂₁₂	—	2	1	no	—
9	C₄×Dic₂₁₂	—	2	1	no	—
10	C₄×Dic₂₁₂	C₄	1	2	yes	—
11	C₄×Dic₂₁₂	C₄	1	2	yes	—
12	C₂×#13133153	—	2	1	no	—
13	C₂×#13133153	C₂²	1	2	yes	—
14	C₂×#13133151	C₂²	1	1	yes	characteristic
15	#16810647	—	2	1	no	—
16	C₂×#13133142	—	2	1	no	—
17	#16810440	—	2	4	no	—
18	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₅₃×C₄⋊C₁₆	—	2	1	no	—
21	C₅₃×C₄⋊C₁₆	—	2	1	no	—
22	C₂²×Dic₁₀₆	—	2	1	no	—
23	C₂×Dic₂₁₂	—	2	1	no	—
24	C₂×Dic₂₁₂	—	2	1	no	—
25	C₂×Dic₂₁₂	—	2	1	no	—
26	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
27	C₂×Dic₂₁₂	D₄	1	2	yes	—
28	C₂×Dic₂₁₂	Q₈	1	2	yes	—
29	C₂×#11719917	—	2	2	no	—
30	C₄×Dic₁₀₆	—	2	2	no	—
31	C₄×Dic₁₀₆	—	2	2	no	—
32	C₄×Dic₁₀₆	—	4	1	no	—
33	C₄×Dic₁₀₆	—	4	1	no	—
34	C₂×C₄⋊Dic₅₃	—	2	1	no	—
35	C₂×#11719919	—	4	1	no	—
36	C₂×C₄×Dic₅₃	—	4	1	no	—
37	#13133153	—	2	2	no	—
38	#13133153	—	2	2	no	—
39	#13133153	—	4	2	no	—
40	#13133151	—	2	1	no	—
41	#13133151	C₂×C₄	1	2	yes	—
42	C₄⋊#88322	—	4	1	no	—
43	C₄⋊#88322	—	4	1	no	—
44	#13133142	—	2	2	no	—
45	#13133142	—	2	2	no	—
46	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
49	M₃₂×C₅₃	—	2	2	no	—
50	M₃₂×C₅₃	—	2	2	no	—
51	C₂×C₁₆×C₅₃	—	2	2	no	—
52	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₅₃	—	4	1	no	—
60	C₂²×Dic₅₃	—	4	1	no	—
61	C₂×Dic₁₀₆	—	2	2	no	—
62	C₂×Dic₁₀₆	—	2	2	no	—
63	C₂×Dic₁₀₆	—	2	2	no	—
64	C₂×Dic₁₀₆	—	4	1	no	—
65	C₂×Dic₁₀₆	—	4	1	no	—
66	C₂×Dic₁₀₆	—	4	1	no	—
67	C₂×C₈×C₅₃	—	2	1	no	—
68	C₂×C₈×C₅₃	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
77	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₄²×C₅₃	D₈	1	1	yes	characteristic
80	C₄²×C₅₃	D₈	1	1	yes	characteristic
81	C₄⋊Dic₅₃	—	2	2	no	—
82	C₄⋊Dic₅₃	—	4	1	no	—
83	#11719919	—	4	2	no	—
84	#11719919	—	4	2	no	—
85	C₄×Dic₅₃	—	4	2	no	—
86	C₄×Dic₅₃	—	4	2	no	—
87	#11719917	—	4	2	no	—
88	#11719917	—	4	2	no	—
89	C₂×#88322	—	4	2	no	—
90	Dic₂₁₂	—	2	2	no	—
91	Dic₂₁₂	—	2	2	no	—
92	Dic₂₁₂	—	2	2	no	—
93	Dic₂₁₂	C₄⋊C₄	1	4	yes	—
94	C₁₆×C₅₃	—	4	2	no	—
95	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	4	1	no	—
98	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
100	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
101	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
108	C₂×Dic₅₃	—	4	2	no	—
109	C₂×Dic₅₃	—	4	2	no	—
110	C₂×Dic₅₃	—	4	2	no	—
111	C₂×Dic₅₃	—	4	2	no	—
112	C₂×Dic₅₃	—	8	2	no	—
113	Dic₁₀₆	—	2	4	no	—
114	Dic₁₀₆	—	4	2	no	—
115	Dic₁₀₆	—	4	2	no	—
116	Dic₁₀₆	—	4	2	no	—
117	C₈×C₅₃	—	2	1	no	—
118	C₈×C₅₃	—	2	1	no	—
119	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₈×C₅₃	#117	1	1	yes	characteristic
121	C₈×C₅₃	#121	1	1	yes	characteristic
122	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
123	#88322	—	8	2	no	—
124	#12819	—	53	1	no	maximal, Sylow
125	C₂²×C₅₃	—	2	1	no	—
126	C₂²×C₅₃	—	2	1	no	—
127	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
128	C₂²×C₅₃	#461	1	1	yes	characteristic
129	C₂²×C₅₃	#476	1	1	yes	characteristic
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	#362	1	2	yes	complemented
134	C₄×C₅₃	#380	1	1	yes	characteristic
135	C₄×C₅₃	#417	1	1	yes	characteristic
136	C₄×C₅₃	#459	1	1	yes	characteristic
137	C₄×C₅₃	#474	1	1	yes	characteristic
138	Dic₅₃	—	4	4	no	—
139	Dic₅₃	—	8	2	no	—
140	C₂×C₄×Q₁₆	—	53	1	no	—
141	#2150	D₅₃	1	1	yes	characteristic
142	C₂×#362	—	53	2	no	—
143	#1848	—	53	1	no	—
144	#1387	—	53	2	no	—
145	C₂×C₅₃	—	2	1	no	—
146	C₂×C₅₃	—	2	1	no	—
147	C₂×C₅₃	#1802	1	1	yes	characteristic
148	C₂×C₅₃	C₂×#362	1	1	yes	characteristic
149	C₂×C₅₃	#2148	1	1	yes	characteristic
150	C₂²×Q₁₆	—	53	1	no	—
151	C₂×C₄×Q₈	—	106	1	no	—
152	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
153	C₄×Q₁₆	—	53	2	no	—
154	C₄×Q₁₆	—	53	2	no	—
155	C₄×Q₁₆	—	106	1	no	—
156	C₄×Q₁₆	—	106	1	no	—
157	C₂×#108	—	53	2	no	—
158	C₂×#108	—	106	1	no	—
159	C₂×#107	—	53	1	no	—
160	#423	—	106	1	no	—
161	C₂×#103	—	106	1	no	—
162	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
163	C₄⋊C₁₆	—	2	1	no	—
164	C₄⋊C₁₆	—	2	1	no	—
165	#362	—	106	4	no	—
166	C₅₃	#12819	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₈	—	106	1	no	—
168	C₂×Q₁₆	—	53	2	no	—
169	C₂×Q₁₆	—	53	2	no	—
170	C₂×Q₁₆	—	53	2	no	—
171	C₂×Q₁₆	—	106	1	no	—
172	C₂×Q₁₆	—	106	1	no	—
173	C₂×Q₁₆	—	106	1	no	—
174	C₂×M₁₆	—	106	2	no	—
175	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
177	C₄×Q₈	—	106	2	no	—
178	C₄×Q₈	—	106	2	no	—
179	C₄×Q₈	—	212	1	no	—
180	C₄×Q₈	—	212	1	no	—
181	C₂×C₄⋊C₄	—	106	1	no	—
182	C₂×C₄⋊C₄	—	212	1	no	—
183	C₂×C₄²	—	212	1	no	—
184	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
185	M₃₂	—	2	2	no	—
186	M₃₂	—	2	2	no	—
187	C₂×C₁₆	—	2	2	no	—
188	#108	—	106	2	no	—
189	#108	—	106	2	no	—
190	#108	—	212	2	no	—
191	#107	—	53	2	no	—
192	#107	—	106	1	no	—
193	C₄⋊C₈	—	212	1	no	—
194	C₄⋊C₈	—	212	1	no	—
195	#103	—	106	2	no	—
196	#103	—	106	2	no	—
197	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
198	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
201	C₂×Q₈	—	106	2	no	—
202	C₂×Q₈	—	106	2	no	—
203	C₂×Q₈	—	106	2	no	—
204	C₂×Q₈	—	212	1	no	—
205	C₂×Q₈	—	212	1	no	—
206	C₂×Q₈	—	212	1	no	—
207	C₂²×C₄	—	212	1	no	—
208	C₂²×C₄	—	212	1	no	—
209	C₂²×C₄	#11719913	1	1	yes	characteristic
210	C₂²×C₄	#11719913	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
212	Q₁₆	—	53	4	no	—
213	Q₁₆	—	106	2	no	—
214	Q₁₆	—	106	2	no	—
215	Q₁₆	—	106	2	no	—
216	M₁₆	—	212	2	no	—
217	M₁₆	—	212	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	212	2	no	—
221	C₂×C₈	#11719921	1	1	yes	characteristic
222	C₂×C₈	#11719921	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
224	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
225	C₂×C₈	#11719947	1	1	yes	characteristic
226	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
227	C₂×C₈	#11719949	1	1	yes	characteristic
228	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
229	C₄⋊C₄	—	106	2	no	—
230	C₄⋊C₄	—	212	2	no	—
231	C₄⋊C₄	—	212	2	no	—
232	C₄⋊C₄	—	212	1	no	—
233	C₄²	—	212	2	no	—
234	C₄²	—	212	2	no	—
235	C₄²	D₄₂₄	1	1	yes	characteristic
236	C₄²	D₄₂₄	1	1	yes	characteristic
237	C₄²	#11719921	1	1	yes	characteristic
238	C₄²	#11719921	1	1	yes	characteristic
239	C₁₆	—	4	2	no	—
240	C₂³	C₂×#11719913	1	1	yes	characteristic
241	Q₈	—	106	4	no	—
242	Q₈	—	212	2	no	—
243	Q₈	—	212	2	no	—
244	Q₈	—	212	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	424	2	no	—
248	C₂×C₄	—	212	2	no	—
249	C₂×C₄	—	212	2	no	—
250	C₂×C₄	—	212	2	no	—
251	C₂×C₄	—	212	2	no	—
252	C₂×C₄	#13133139	1	1	yes	characteristic
253	C₂×C₄	#13133139	1	1	yes	characteristic
254	C₂×C₄	#13133139	1	1	yes	characteristic
255	C₂×C₄	#13133139	1	1	yes	characteristic
256	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
257	C₂×C₄	#13133192	1	1	yes	characteristic
258	C₂×C₄	#13133197	1	1	yes	characteristic
259	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
260	C₂×C₄	#13133245	1	1	yes	characteristic
261	C₂×C₄	#13133250	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	424	2	no	—
265	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
266	C₈	#13133175	1	1	yes	characteristic
267	C₈	#13133188	1	1	yes	characteristic
268	C₈	#13133193	1	1	yes	characteristic
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×#13133139	1	1	yes	characteristic
272	C₂²	#16810829	1	1	yes	characteristic
273	C₂²	#16810914	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	424	2	no	—
278	C₄	—	212	4	no	—
279	C₄	#16810440	1	2	yes	complemented
280	C₄	#16810480	1	1	yes	characteristic
281	C₄	#16810625	1	1	yes	characteristic
282	C₄	#16810819	1	1	yes	characteristic
283	C₄	#16810904	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	#20274716	1	1	yes	characteristic
287	C₂	C₂×#16810440	1	1	yes	characteristic
288	C₂	#20276465	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133139
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#12819
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
133. C₄×C₅₃	#362	semidirect
166. C₅₃	#12819	semidirect
279. C₄	#16810440	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₁₂	2
C₂×#16810440	2
#20274887	2
#20272824	2
C₅₃×#2150	2
#12819	53