Subgroups — #27300139

Order 13568.

446 conjugacy classes of subgroups.

The subgroup lattice has 278 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×#16810428	C₂	1	2	yes	maximal
3	#20274875	C₂	1	1	yes	maximal, characteristic
4	#20272807	C₂	1	2	yes	maximal
5	C₂×C₅₃×C₄⋊C₁₆	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	2	no	—
9	C₄×Dic₂₁₂	C₄	1	4	yes	complemented
10	C₂×#13133151	—	2	1	no	—
11	C₂×#13133151	C₂²	1	2	yes	—
12	C₂×#13133151	C₂²	1	1	yes	characteristic
13	C₂×#13133142	—	2	1	no	—
14	#16810493	—	2	1	no	—
15	#16810428	—	2	4	no	—
16	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×C₄⋊C₁₆	—	2	2	no	—
19	C₂²×Dic₁₀₆	—	2	1	no	—
20	C₂×Dic₂₁₂	—	2	1	no	—
21	C₂×Dic₂₁₂	—	2	1	no	—
22	C₂×Dic₂₁₂	—	2	1	no	—
23	C₂×Dic₂₁₂	C₂×C₄	1	2	yes	—
24	C₂×Dic₂₁₂	D₄	1	2	yes	—
25	C₂×Dic₂₁₂	Q₈	1	2	yes	—
26	C₄×Dic₁₀₆	—	2	4	no	—
27	C₄×Dic₁₀₆	—	4	2	no	—
28	C₂×C₄⋊Dic₅₃	—	2	1	no	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	C₂×C₄⋊Dic₅₃	—	4	1	no	—
31	C₂×C₄⋊Dic₅₃	—	4	1	no	—
32	C₂×#11719919	—	4	1	no	—
33	C₂×C₄×Dic₅₃	—	4	1	no	—
34	#13133151	—	2	1	no	—
35	#13133151	—	2	2	no	—
36	#13133151	—	2	2	no	—
37	#13133151	—	4	2	no	—
38	#13133151	C₂×C₄	1	2	yes	—
39	#13133142	—	2	4	no	—
40	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₅₃	—	2	2	no	—
44	C₂×C₁₆×C₅₃	—	2	2	no	—
45	C₂×C₁₆×C₅₃	—	2	2	no	—
46	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₅₃	D₄	1	2	yes	—
48	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	—	4	1	no	—
52	C₂²×Dic₅₃	—	4	1	no	—
53	C₂²×Dic₅₃	—	4	2	no	—
54	C₂×Dic₁₀₆	—	2	2	no	—
55	C₂×Dic₁₀₆	—	2	2	no	—
56	C₂×Dic₁₀₆	—	2	2	no	—
57	C₂×Dic₁₀₆	—	4	1	no	—
58	C₂×Dic₁₀₆	—	4	1	no	—
59	C₂×Dic₁₀₆	—	4	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	1	no	—
62	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
68	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
69	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
70	C₄²×C₅₃	SD₁₆	1	2	yes	—
71	C₄⋊Dic₅₃	—	2	2	no	—
72	C₄⋊Dic₅₃	—	4	1	no	—
73	C₄⋊Dic₅₃	—	4	2	no	—
74	C₄⋊Dic₅₃	—	4	2	no	—
75	C₄⋊Dic₅₃	—	8	2	no	—
76	C₄⋊Dic₅₃	—	8	2	no	—
77	#11719919	—	4	4	no	—
78	C₄×Dic₅₃	—	4	4	no	—
79	Dic₂₁₂	—	2	2	no	—
80	Dic₂₁₂	—	2	2	no	—
81	Dic₂₁₂	—	2	2	no	—
82	Dic₂₁₂	C₄⋊C₄	1	4	yes	complemented
83	C₁₆×C₅₃	—	4	2	no	—
84	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	#102	1	2	yes	—
88	C₂×C₄×C₅₃	#102	1	2	yes	—
89	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
90	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
97	C₂×Dic₅₃	—	4	2	no	—
98	C₂×Dic₅₃	—	4	2	no	—
99	C₂×Dic₅₃	—	4	2	no	—
100	C₂×Dic₅₃	—	4	2	no	—
101	C₂×Dic₅₃	—	8	1	no	—
102	C₂×Dic₅₃	—	8	1	no	—
103	C₂×Dic₅₃	—	8	2	no	—
104	C₂×Dic₅₃	—	8	2	no	—
105	C₂×Dic₅₃	—	8	2	no	—
106	Dic₁₀₆	—	2	4	no	—
107	Dic₁₀₆	—	4	2	no	—
108	Dic₁₀₆	—	4	2	no	—
109	Dic₁₀₆	—	4	2	no	—
110	C₈×C₅₃	—	2	2	no	—
111	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈×C₅₃	#117	1	1	yes	characteristic
113	C₈×C₅₃	#121	1	1	yes	characteristic
114	C₈×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
115	#12812	—	53	1	no	maximal, Sylow
116	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
117	C₂²×C₅₃	#459	1	1	yes	characteristic
118	C₂²×C₅₃	#474	1	1	yes	characteristic
119	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
120	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
121	C₂²×C₅₃	#509	1	1	yes	characteristic
122	C₂²×C₅₃	#510	1	1	yes	characteristic
123	C₄×C₅₃	—	2	2	no	—
124	C₄×C₅₃	#358	1	4	yes	complemented
125	C₄×C₅₃	#380	1	1	yes	characteristic
126	C₄×C₅₃	#417	1	1	yes	characteristic
127	C₄×C₅₃	#461	1	1	yes	characteristic
128	C₄×C₅₃	#476	1	1	yes	characteristic
129	Dic₅₃	—	4	4	no	—
130	Dic₅₃	—	8	2	no	—
131	Dic₅₃	—	16	2	no	—
132	C₂×C₄×Q₁₆	—	53	1	no	—
133	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
134	C₂×#358	—	53	2	no	—
135	#1845	—	53	1	no	—
136	#1380	—	53	2	no	—
137	C₂×C₅₃	#1802	1	1	yes	characteristic
138	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
139	C₂×C₅₃	#2139	1	1	yes	characteristic
140	C₂×C₅₃	#2207	1	1	yes	characteristic
141	C₂×C₅₃	#2209	1	1	yes	characteristic
142	C₂×C₅₃	#2225	1	1	yes	characteristic
143	C₂×C₅₃	#2227	1	1	yes	characteristic
144	C₂²×Q₁₆	—	53	1	no	—
145	C₂×C₄×Q₈	—	106	1	no	—
146	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
147	C₄×Q₁₆	—	53	4	no	—
148	C₄×Q₁₆	—	106	2	no	—
149	C₂×#107	—	53	1	no	—
150	C₂×#107	—	53	2	no	—
151	C₂×#107	—	106	1	no	—
152	C₂×#103	—	106	1	no	—
153	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
154	#384	—	106	1	no	—
155	C₄⋊C₁₆	—	2	2	no	—
156	#358	—	106	4	no	—
157	C₅₃	#12812	1	1	yes	characteristic, complemented, Sylow
158	C₂²×Q₈	—	106	1	no	—
159	C₂×Q₁₆	—	53	2	no	—
160	C₂×Q₁₆	—	53	2	no	—
161	C₂×Q₁₆	—	53	2	no	—
162	C₂×Q₁₆	—	106	1	no	—
163	C₂×Q₁₆	—	106	1	no	—
164	C₂×Q₁₆	—	106	1	no	—
165	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
167	C₄×Q₈	—	106	4	no	—
168	C₄×Q₈	—	212	2	no	—
169	C₂×C₄⋊C₄	—	106	1	no	—
170	C₂×C₄⋊C₄	—	106	2	no	—
171	C₂×C₄⋊C₄	—	212	1	no	—
172	C₂×C₄⋊C₄	—	212	1	no	—
173	C₂×C₄⋊C₄	—	212	1	no	—
174	C₂×C₄²	—	212	1	no	—
175	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	2	2	no	—
179	#107	—	53	2	no	—
180	#107	—	106	1	no	—
181	#107	—	106	2	no	—
182	#107	—	106	2	no	—
183	#107	—	212	2	no	—
184	#103	—	106	4	no	—
185	C₄×C₈	C₄×D₅₃	1	2	yes	—
186	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
187	C₂×Q₈	—	106	2	no	—
188	C₂×Q₈	—	106	2	no	—
189	C₂×Q₈	—	106	2	no	—
190	C₂×Q₈	—	212	1	no	—
191	C₂×Q₈	—	212	1	no	—
192	C₂×Q₈	—	212	1	no	—
193	C₂²×C₄	—	212	2	no	—
194	C₂²×C₄	—	212	1	no	—
195	C₂²×C₄	—	212	1	no	—
196	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
197	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
199	Q₁₆	—	53	4	no	—
200	Q₁₆	—	106	2	no	—
201	Q₁₆	—	106	2	no	—
202	Q₁₆	—	106	2	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	#11719921	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
207	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
208	C₂×C₈	#11719947	1	1	yes	characteristic
209	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
210	C₂×C₈	#11719949	1	1	yes	characteristic
211	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
212	C₄⋊C₄	—	106	2	no	—
213	C₄⋊C₄	—	424	2	no	—
214	C₄⋊C₄	—	424	2	no	—
215	C₄⋊C₄	—	212	2	no	—
216	C₄⋊C₄	—	212	2	no	—
217	C₄⋊C₄	—	212	4	no	—
218	C₄⋊C₄	—	212	1	no	—
219	C₄²	—	212	4	no	—
220	C₄²	#11719913	1	2	yes	—
221	C₄²	#11719921	1	2	yes	—
222	C₁₆	—	4	2	no	—
223	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
224	Q₈	—	106	4	no	—
225	Q₈	—	212	2	no	—
226	Q₈	—	212	2	no	—
227	Q₈	—	212	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	424	2	no	—
231	C₂×C₄	—	424	2	no	—
232	C₂×C₄	—	424	2	no	—
233	C₂×C₄	—	424	1	no	—
234	C₂×C₄	—	424	1	no	—
235	C₂×C₄	—	212	2	no	—
236	C₂×C₄	—	212	2	no	—
237	C₂×C₄	—	212	2	no	—
238	C₂×C₄	—	212	2	no	—
239	C₂×C₄	#13133139	1	2	yes	—
240	C₂×C₄	#13133139	1	2	yes	—
241	C₂×C₄	#13133162	1	2	yes	—
242	C₂×C₄	Dic₄₂₄	1	2	yes	—
243	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
244	C₂×C₄	#13133192	1	1	yes	characteristic
245	C₂×C₄	#13133197	1	1	yes	characteristic
246	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
247	C₂×C₄	#13133245	1	1	yes	characteristic
248	C₂×C₄	#13133250	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
251	C₈	#13133175	1	1	yes	characteristic
252	C₈	#13133188	1	1	yes	characteristic
253	C₈	#13133193	1	1	yes	characteristic
254	C₂²	C₂×#13133139	1	1	yes	characteristic
255	C₂²	#16810819	1	1	yes	characteristic
256	C₂²	#16810904	1	1	yes	characteristic
257	C₂²	C₂×#13133162	1	1	yes	characteristic
258	C₂²	C₂×Dic₄₂₄	1	1	yes	characteristic
259	C₂²	#16811061	1	1	yes	characteristic
260	C₂²	#16811066	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	848	2	no	—
263	C₄	—	424	2	no	—
264	C₄	—	212	4	no	—
265	C₄	#16810428	1	4	yes	complemented
266	C₄	#16810480	1	1	yes	characteristic
267	C₄	#16810625	1	1	yes	characteristic
268	C₄	#16810829	1	1	yes	characteristic
269	C₄	#16810914	1	1	yes	characteristic
270	C₂	#20274716	1	1	yes	characteristic
271	C₂	C₂×#16810428	1	1	yes	characteristic
272	C₂	#20276428	1	1	yes	characteristic
273	C₂	#20276756	1	1	yes	characteristic
274	C₂	#20276766	1	1	yes	characteristic
275	C₂	#20276858	1	1	yes	characteristic
276	C₂	#20276868	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#12812
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×Dic₂₁₂	C₄	semidirect
82. Dic₂₁₂	C₄⋊C₄	semidirect
124. C₄×C₅₃	#358	semidirect
157. C₅₃	#12812	semidirect
265. C₄	#16810428	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×#16810428	2
#20274875	2
#20272807	2
C₂×C₅₃×C₄⋊C₁₆	2
#12812	53