Subgroups — #27295324

Order 13568.

418 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276471	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272758	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#495	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	—	2	1	no	—
6	#16810676	—	2	1	no	—
7	C₄⋊#11719908	—	2	1	no	—
8	C₄⋊#11719908	C₂²	1	2	yes	complemented
9	#16810376	—	2	1	no	—
10	#16810374	C₂²	1	2	yes	complemented
11	C₂²×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₂²×SD₁₆×C₅₃	—	2	1	no	—
13	C₂²×D₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×#124×C₅₃	—	2	1	no	—
15	C₅₃×#495	—	2	1	no	—
16	C₅₃×#495	—	2	1	no	—
17	C₅₃×#495	C₄	1	2	yes	—
18	C₅₃×#495	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	#13133157	—	4	2	no	—
21	C₂×#11719908	—	2	2	no	—
22	C₄⋊#88322	—	2	2	no	—
23	C₂²⋊#88322	—	4	1	no	—
24	C₄×#88322	—	4	1	no	—
25	C₂²×Q₈×C₅₃	—	2	1	no	—
26	C₂²×D₄×C₅₃	—	2	1	no	—
27	C₂×Q₁₆×C₅₃	—	2	1	no	—
28	C₂×Q₁₆×C₅₃	—	4	2	no	—
29	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×SD₁₆×C₅₃	—	2	2	no	—
31	C₂×SD₁₆×C₅₃	—	2	2	no	—
32	C₂×SD₁₆×C₅₃	—	4	2	no	—
33	C₂×SD₁₆×C₅₃	—	4	2	no	—
34	C₂×D₈×C₅₃	—	2	1	no	—
35	C₂×D₈×C₅₃	—	4	2	no	—
36	C₂×D₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	#124×C₅₃	—	2	2	no	—
40	#124×C₅₃	—	2	2	no	—
41	#124×C₅₃	—	4	1	no	—
42	#124×C₅₃	—	4	1	no	—
43	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
44	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
45	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
50	C₂⁴×C₅₃	—	4	1	no	—
51	C₂×Q₈×C₅₃	—	2	2	no	—
52	C₂×Q₈×C₅₃	—	4	1	no	—
53	C₂×Q₈×C₅₃	—	4	2	no	—
54	C₂×Q₈×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	2	2	no	—
56	C₂×D₄×C₅₃	—	4	2	no	—
57	C₂×D₄×C₅₃	—	4	2	no	—
58	C₂×D₄×C₅₃	—	4	1	no	—
59	C₂²×C₄×C₅₃	—	4	1	no	—
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
63	Q₁₆×C₅₃	—	4	2	no	—
64	Q₁₆×C₅₃	—	4	2	no	—
65	SD₁₆×C₅₃	—	4	4	no	—
66	SD₁₆×C₅₃	—	4	2	no	—
67	SD₁₆×C₅₃	—	4	2	no	—
68	D₈×C₅₃	—	4	2	no	—
69	D₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₃	D₈	1	2	yes	—
74	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
75	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₅₃	—	4	2	no	—
78	C₂²⋊C₄×C₅₃	—	4	2	no	—
79	C₂²⋊C₄×C₅₃	—	4	2	no	—
80	C₂²⋊C₄×C₅₃	—	4	2	no	—
81	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₄²×C₅₃	D₈	1	1	yes	characteristic
83	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
84	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
85	C₂×#88322	—	4	2	no	—
86	#11719908	—	4	2	no	—
87	C₂³×C₅₃	—	4	2	no	—
88	C₂³×C₅₃	—	8	2	no	—
89	C₂³×C₅₃	—	8	2	no	—
90	C₂³×C₅₃	—	8	2	no	—
91	C₂³×C₅₃	#102	1	1	yes	characteristic
92	Q₈×C₅₃	—	4	2	no	—
93	Q₈×C₅₃	—	4	2	no	—
94	Q₈×C₅₃	—	8	2	no	—
95	D₄×C₅₃	—	4	2	no	—
96	D₄×C₅₃	—	4	2	no	—
97	D₄×C₅₃	—	8	2	no	—
98	C₂×C₄×C₅₃	—	4	1	no	—
99	C₂×C₄×C₅₃	—	4	1	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	8	2	no	—
102	C₂×C₄×C₅₃	#100	1	2	yes	—
103	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
104	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#102	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₈×C₅₃	—	2	1	no	—
111	C₈×C₅₃	—	2	1	no	—
112	C₈×C₅₃	#102	1	2	yes	—
113	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
114	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
115	#88322	—	8	2	no	—
116	#11926	—	53	1	no	maximal, Sylow
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	—	8	2	no	—
120	C₂²×C₅₃	—	8	2	no	—
121	C₂²×C₅₃	—	8	2	no	—
122	C₂²×C₅₃	—	16	4	no	—
123	C₂²×C₅₃	#331	1	2	yes	—
124	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
125	C₄×C₅₃	—	2	1	no	—
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	—	2	1	no	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	8	2	no	—
130	C₄×C₅₃	#331	1	2	yes	—
131	C₄×C₅₃	C₂×#100	1	1	yes	characteristic
132	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
133	C₂×#495	D₅₃	1	1	yes	characteristic
134	#2150	—	53	1	no	—
135	#1564	—	53	1	no	—
136	#1360	—	53	4	no	—
137	C₂×C₅₃	—	2	2	no	—
138	C₂×C₅₃	—	16	2	no	—
139	C₂×C₅₃	C₂×#331	1	1	yes	characteristic
140	C₂×C₅₃	#2143	1	1	yes	characteristic
141	C₂×C₅₃	#2145	1	1	yes	characteristic
142	C₂²×Q₁₆	Dic₅₃	1	1	yes	characteristic
143	C₂²×SD₁₆	—	2	1	no	—
144	C₂²×D₈	Dic₅₃	1	1	yes	characteristic
145	C₂×#124	—	2	1	no	—
146	C₂×M₃₂	—	106	1	no	—
147	#495	—	2	1	no	—
148	#495	—	2	1	no	—
149	#495	Dic₅₃	1	2	yes	—
150	#495	C₂×D₅₃	1	2	yes	—
151	#431	—	106	1	no	—
152	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
153	C₄⋊C₁₆	—	53	2	no	—
154	C₄⋊C₁₆	—	106	1	no	—
155	#335	—	106	1	no	—
156	#334	—	53	2	no	—
157	C₅₃	#11926	1	1	yes	characteristic, complemented, Sylow
158	C₂²×Q₈	—	2	1	no	—
159	C₂²×D₄	—	2	1	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	—	4	2	no	—
162	C₂×Q₁₆	C₂×Dic₅₃	1	2	yes	—
163	C₂×SD₁₆	—	2	2	no	—
164	C₂×SD₁₆	—	2	2	no	—
165	C₂×SD₁₆	—	4	2	no	—
166	C₂×SD₁₆	—	4	2	no	—
167	C₂×D₈	—	2	1	no	—
168	C₂×D₈	—	4	2	no	—
169	C₂×D₈	C₂×Dic₅₃	1	2	yes	—
170	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
172	#124	—	2	2	no	—
173	#124	—	2	2	no	—
174	#124	—	4	1	no	—
175	#124	—	4	1	no	—
176	C₂×C₂²⋊C₄	—	4	1	no	—
177	C₂×C₂²⋊C₄	—	4	1	no	—
178	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
179	M₃₂	—	212	2	no	—
180	C₂×C₁₆	—	106	2	no	—
181	C₄⋊C₈	—	106	2	no	—
182	C₂²⋊C₈	—	212	1	no	—
183	C₄×C₈	—	212	1	no	—
184	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
188	C₂⁴	—	4	1	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	4	1	no	—
191	C₂×Q₈	—	4	2	no	—
192	C₂×Q₈	—	4	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	#11719922	1	1	yes	characteristic
199	C₂²×C₄	#11719923	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
201	Q₁₆	—	4	2	no	—
202	Q₁₆	—	4	2	no	—
203	SD₁₆	—	4	4	no	—
204	SD₁₆	—	4	2	no	—
205	SD₁₆	—	4	2	no	—
206	D₈	—	4	2	no	—
207	D₈	—	4	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	212	2	no	—
210	C₂×C₈	#11719922	1	2	yes	—
211	C₂×C₈	#11719923	1	2	yes	—
212	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
214	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
215	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₄²	#11719922	1	1	yes	characteristic
221	C₄²	#11719923	1	1	yes	characteristic
222	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
223	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
224	C₁₆	—	212	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	#13133138	1	1	yes	characteristic
230	Q₈	—	4	2	no	—
231	Q₈	—	4	2	no	—
232	Q₈	—	8	2	no	—
233	D₄	—	4	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	8	2	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	#13133134	1	2	yes	—
241	C₂×C₄	#13133138	1	1	yes	characteristic
242	C₂×C₄	#13133138	1	1	yes	characteristic
243	C₂×C₄	#13133138	1	2	yes	—
244	C₂×C₄	#13133138	1	1	yes	characteristic
245	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
246	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
247	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	424	2	no	—
251	C₈	#13133138	1	2	yes	—
252	C₈	C₂×#11719922	1	1	yes	characteristic
253	C₈	C₂×#11719923	1	1	yes	characteristic
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	16	4	no	—
260	C₂²	#16810367	1	2	yes	—
261	C₂²	C₂×#13133138	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	8	2	no	—
267	C₄	#16810367	1	2	yes	—
268	C₄	C₂×#13133134	1	1	yes	characteristic
269	C₄	C₂×#13133138	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	—	16	2	no	—
272	C₂	C₂×#16810367	1	1	yes	characteristic
273	C₂	#20276443	1	1	yes	characteristic
274	C₂	#20276451	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#495	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂²×D₈×C₅₃	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#495	D₅₃
hypocenter	C₅₃	#11926
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
1. #20276471	C₂	semidirect
2. #20273463	C₂	semidirect
3. #20272758	C₂	semidirect
8. C₄⋊#11719908	C₂²	semidirect
10. #16810374	C₂²	semidirect
157. C₅₃	#11926	semidirect

Maximal subgroups

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

subgroup	index
#20276471	2
#20273463	2
#20272758	2
C₂×C₅₃×#495	2
#11926	53