Subgroups — #27300138

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⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810427	C₂	1	2	yes	maximal
3	#20274875	C₂	1	1	yes	maximal, characteristic
4	#20272806	C₂	1	2	yes	maximal
5	C₂×C₄×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₂×#13133151	—	2	1	no	—
8	C₂×#13133151	C₂²	1	2	yes	—
9	#16810493	—	2	1	no	—
10	C₄⋊#11719908	—	2	2	no	—
11	#16810427	—	2	4	no	—
12	C₂²×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₅₃	—	2	1	no	—
14	C₄×Q₁₆×C₅₃	—	2	2	no	—
15	C₄×Q₁₆×C₅₃	C₄	1	4	yes	complemented
16	C₂×#107×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#103×C₅₃	—	2	1	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄⋊Dic₅₃	—	4	1	no	—
21	C₂×C₄⋊Dic₅₃	—	4	1	no	—
22	C₂×#11719908	—	2	2	no	—
23	C₂×#11719908	—	2	2	no	—
24	C₂×#11719908	—	2	2	no	—
25	#13133151	—	2	2	no	—
26	#13133151	—	2	2	no	—
27	#13133151	—	4	2	no	—
28	C₂²×Q₈×C₅₃	—	2	1	no	—
29	C₂×Q₁₆×C₅₃	—	2	1	no	—
30	C₂×Q₁₆×C₅₃	—	2	1	no	—
31	C₂×Q₁₆×C₅₃	—	2	1	no	—
32	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×Q₁₆×C₅₃	D₄	1	2	yes	—
34	C₂×Q₁₆×C₅₃	Q₈	1	2	yes	—
35	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₅₃	—	2	4	no	—
38	C₄×Q₈×C₅₃	—	4	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
41	C₂×C₄²×C₅₃	—	4	1	no	—
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	#107×C₅₃	—	2	1	no	—
44	#107×C₅₃	C₂×C₄	1	2	yes	—
45	#103×C₅₃	—	2	4	no	—
46	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₄×C₈×C₅₃	D₄	1	2	yes	—
48	C₂×Q₈×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	—	4	1	no	—
52	C₂×Q₈×C₅₃	—	4	1	no	—
53	C₂×Q₈×C₅₃	—	4	1	no	—
54	C₂²×C₄×C₅₃	—	4	1	no	—
55	C₂²×C₄×C₅₃	—	4	1	no	—
56	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₅₃	—	4	2	no	—
60	Q₁₆×C₅₃	—	2	2	no	—
61	Q₁₆×C₅₃	—	2	2	no	—
62	Q₁₆×C₅₃	—	2	2	no	—
63	Q₁₆×C₅₃	C₄⋊C₄	1	4	yes	complemented
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
72	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
73	C₄⋊C₄×C₅₃	—	2	2	no	—
74	C₄⋊C₄×C₅₃	—	4	4	no	—
75	C₄⋊C₄×C₅₃	—	4	1	no	—
76	C₄²×C₅₃	—	4	4	no	—
77	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₅₃	SD₁₆	1	2	yes	—
79	C₄⋊Dic₅₃	—	4	2	no	—
80	C₄⋊Dic₅₃	—	4	2	no	—
81	C₄⋊Dic₅₃	—	8	2	no	—
82	C₄⋊Dic₅₃	—	8	2	no	—
83	#11719908	—	4	2	no	—
84	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₅₃	—	2	4	no	—
86	Q₈×C₅₃	—	4	2	no	—
87	Q₈×C₅₃	—	4	2	no	—
88	Q₈×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	2	1	no	—
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	4	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	8	1	no	—
96	C₂×C₄×C₅₃	—	8	1	no	—
97	C₂×C₄×C₅₃	#102	1	2	yes	—
98	C₂×C₄×C₅₃	#102	1	2	yes	—
99	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
100	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
101	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
107	C₂×Dic₅₃	—	8	2	no	—
108	C₂×Dic₅₃	—	8	2	no	—
109	C₂×Dic₅₃	—	8	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	—	4	4	no	—
125	C₄×C₅₃	—	8	2	no	—
126	C₄×C₅₃	#358	1	4	yes	complemented
127	C₄×C₅₃	#380	1	1	yes	characteristic
128	C₄×C₅₃	#417	1	1	yes	characteristic
129	C₄×C₅₃	#461	1	1	yes	characteristic
130	C₄×C₅₃	#476	1	1	yes	characteristic
131	Dic₅₃	—	16	2	no	—
132	C₂×C₄×Q₁₆	D₅₃	1	1	yes	characteristic
133	C₂×C₄⋊C₁₆	—	53	1	no	—
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	C₂×D₅₃	1	1	yes	characteristic
145	C₂×C₄×Q₈	—	2	1	no	—
146	C₂²×C₁₆	—	53	2	no	—
147	C₄×Q₁₆	—	2	2	no	—
148	C₄×Q₁₆	Dic₅₃	1	4	yes	complemented
149	C₂×#107	—	53	2	no	—
150	C₂×#107	—	106	1	no	—
151	C₂×#107	C₂×D₅₃	1	1	yes	characteristic
152	C₂×#103	—	2	1	no	—
153	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
154	#384	—	106	1	no	—
155	C₄⋊C₁₆	—	106	2	no	—
156	#358	—	106	4	no	—
157	C₅₃	#12812	1	1	yes	characteristic, complemented, Sylow
158	C₂²×Q₈	—	2	1	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	—	2	1	no	—
162	C₂×Q₁₆	Dic₁₀₆	1	2	yes	—
163	C₂×Q₁₆	D₂₁₂	1	2	yes	—
164	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
165	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
166	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
167	C₄×Q₈	—	2	4	no	—
168	C₄×Q₈	—	4	2	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	4	1	no	—
171	C₂×C₄⋊C₄	—	106	2	no	—
172	C₂×C₄⋊C₄	—	212	1	no	—
173	C₂×C₄⋊C₄	—	212	1	no	—
174	C₂×C₄²	—	4	1	no	—
175	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
176	C₂×C₁₆	—	106	2	no	—
177	C₂×C₁₆	—	106	2	no	—
178	C₂×C₁₆	—	106	2	no	—
179	#107	—	2	1	no	—
180	#107	—	106	2	no	—
181	#107	—	106	2	no	—
182	#107	—	212	2	no	—
183	#107	C₂×Dic₅₃	1	2	yes	—
184	#103	—	2	4	no	—
185	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
186	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	4	1	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂×Q₈	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	212	2	no	—
196	C₂²×C₄	#11719922	1	1	yes	characteristic
197	C₂²×C₄	#11719922	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
199	Q₁₆	—	2	2	no	—
200	Q₁₆	—	2	2	no	—
201	Q₁₆	—	2	2	no	—
202	Q₁₆	C₄⋊Dic₅₃	1	4	yes	complemented
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
206	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
207	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
208	C₂×C₈	#11719945	1	1	yes	characteristic
209	C₂×C₈	#11719945	1	1	yes	characteristic
210	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
211	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	424	2	no	—
216	C₄⋊C₄	—	424	2	no	—
217	C₄⋊C₄	—	212	2	no	—
218	C₄⋊C₄	—	212	2	no	—
219	C₄²	—	4	4	no	—
220	C₄²	#11719923	1	2	yes	—
221	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
222	C₁₆	—	212	2	no	—
223	C₂³	C₂×#11719922	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	Q₈	—	4	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	8	1	no	—
236	C₂×C₄	—	424	2	no	—
237	C₂×C₄	—	424	2	no	—
238	C₂×C₄	—	424	2	no	—
239	C₂×C₄	#13133138	1	2	yes	—
240	C₂×C₄	#13133138	1	2	yes	—
241	C₂×C₄	#13133160	1	2	yes	—
242	C₂×C₄	#13133163	1	2	yes	—
243	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
244	C₂×C₄	#13133191	1	1	yes	characteristic
245	C₂×C₄	#13133196	1	1	yes	characteristic
246	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
247	C₂×C₄	#13133242	1	1	yes	characteristic
248	C₂×C₄	#13133247	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
251	C₈	#13133176	1	1	yes	characteristic
252	C₈	#13133191	1	1	yes	characteristic
253	C₈	#13133196	1	1	yes	characteristic
254	C₂²	C₂×#13133138	1	1	yes	characteristic
255	C₂²	#16810818	1	1	yes	characteristic
256	C₂²	#16810903	1	1	yes	characteristic
257	C₂²	C₂×#13133160	1	1	yes	characteristic
258	C₂²	C₂×#13133163	1	1	yes	characteristic
259	C₂²	#16811058	1	1	yes	characteristic
260	C₂²	#16811063	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	8	2	no	—
264	C₄	—	848	2	no	—
265	C₄	#16810427	1	4	yes	complemented
266	C₄	#16810483	1	1	yes	characteristic
267	C₄	#16810624	1	1	yes	characteristic
268	C₄	#16810828	1	1	yes	characteristic
269	C₄	#16810913	1	1	yes	characteristic
270	C₂	#20274715	1	1	yes	characteristic
271	C₂	C₂×#16810427	1	1	yes	characteristic
272	C₂	#20276427	1	1	yes	characteristic
273	C₂	#20276755	1	1	yes	characteristic
274	C₂	#20276765	1	1	yes	characteristic
275	C₂	#20276857	1	1	yes	characteristic
276	C₂	#20276867	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₄×Q₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₅₃
hypocenter	C₅₃	#12812
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
15. C₄×Q₁₆×C₅₃	C₄	semidirect
63. Q₁₆×C₅₃	C₄⋊C₄	semidirect
126. C₄×C₅₃	#358	semidirect
148. C₄×Q₁₆	Dic₅₃	semidirect
157. C₅₃	#12812	semidirect
202. Q₁₆	C₄⋊Dic₅₃	semidirect
265. C₄	#16810427	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₂×#16810427	2
#20274875	2
#20272806	2
C₂×C₄×Q₁₆×C₅₃	2
#12812	53