Subgroups — #28317000

Order 13568.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291504	C₂	1	2	yes	maximal
2	#20290354	C₂	1	4	yes	maximal, complemented
3	#20290303	C₂	1	4	yes	maximal, complemented
4	#20290273	C₂	1	1	yes	maximal, characteristic
5	#20290065	C₂	1	2	yes	maximal
6	C₈×D₅₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#20285302	C₂	1	1	yes	maximal, characteristic
8	#16811674	—	2	2	no	—
9	#16811673	C₂²	1	4	yes	—
10	#16811672	C₂²	1	4	yes	—
11	Q₈×#88324	—	2	2	no	—
12	#16811630	—	2	2	no	—
13	#16811627	C₂²	1	2	yes	—
14	#16811588	—	2	2	no	—
15	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
17	#16811486	C₂²	1	4	yes	—
18	#16811473	C₂²	1	8	yes	—
19	C₈×#88324	C₂²	1	2	yes	—
20	C₈×#88324	C₂²	1	2	yes	—
21	#16811178	C₂²	1	2	yes	—
22	#16811003	C₄	1	1	yes	characteristic
23	#16810975	C₄	1	2	yes	—
24	#16810865	C₄	1	4	yes	—
25	#16810717	C₂²	1	4	yes	—
26	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#500	C₄	1	1	yes	characteristic
28	C₂×#11719936	—	2	2	no	—
29	C₂×D₅₃⋊C₈	—	2	1	no	—
30	C₂×D₅₃⋊C₈	D₄	1	1	yes	characteristic
31	C₂×D₅₃⋊C₈	D₄	1	1	yes	characteristic
32	#13133284	—	2	4	no	—
33	C₄⋊#88324	—	2	2	no	—
34	C₄⋊#88324	—	2	4	no	—
35	C₄⋊#88324	C₂³	1	4	yes	—
36	#13133272	—	2	4	no	—
37	C₄×#88324	—	2	1	no	—
38	C₄×#88324	—	2	4	no	—
39	C₄×#88324	D₄	1	1	yes	characteristic
40	C₄×#88324	D₄	1	1	yes	characteristic
41	C₄×#88324	C₂³	1	2	yes	—
42	C₄×#88324	C₂³	1	1	yes	characteristic
43	C₄×#88324	C₂³	1	1	yes	characteristic
44	C₂×Dic₂₁₂	C₂×C₄	1	4	yes	—
45	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
46	#13133217	C₂×C₄	1	2	yes	—
47	#13133208	—	2	2	no	—
48	#13133197	C₂×C₄	1	4	yes	—
49	#13133193	—	2	2	no	—
50	Q₈×Dic₅₃	—	2	2	no	—
51	#13133182	C₂³	1	4	yes	—
52	#13133175	—	2	2	no	—
53	C₄²×D₅₃	C₂³	1	1	yes	characteristic
54	#13133140	C₂×C₄	1	8	yes	—
55	C₄×#88322	C₂×C₄	1	1	yes	characteristic
56	C₈×Dic₅₃	C₂³	1	2	yes	—
57	C₄⋊Q₈×C₅₃	C₂×C₄	1	2	yes	—
58	#107×C₅₃	C₂×C₄	1	4	yes	—
59	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×Q₈×C₅₃	—	2	2	no	—
61	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
64	C₂×Dic₁₀₆	C₂²×C₄	1	4	yes	—
65	C₂×#88324	—	2	2	no	—
66	C₂×#88324	—	2	4	no	—
67	C₂×#88324	C₂×D₄	1	1	yes	characteristic
68	C₂×#88324	C₂×D₄	1	1	yes	characteristic
69	C₂×#88324	C₂×D₄	1	1	yes	characteristic
70	C₂×#88324	C₂×D₄	1	1	yes	characteristic
71	#11719936	—	4	4	no	—
72	D₅₃⋊C₈	—	2	4	no	—
73	D₅₃⋊C₈	—	2	2	no	—
74	D₅₃⋊C₈	Q₁₆	1	4	yes	complemented
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₄⋊C₄×C₅₃	—	2	2	no	—
77	C₄⋊C₄×C₅₃	C₂²×C₄	1	4	yes	—
78	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	#11719921	—	2	4	no	—
80	C₄⋊Dic₅₃	—	2	2	no	—
81	C₄⋊Dic₅₃	—	2	2	no	—
82	C₄⋊Dic₅₃	—	2	4	no	—
83	#11719919	—	2	4	no	—
84	C₄×Dic₅₃	—	2	4	no	—
85	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
86	C₄×Dic₅₃	#41	1	1	yes	characteristic
87	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
88	C₂×#88322	C₂²×C₄	1	2	yes	—
89	Dic₂₁₂	—	2	8	no	—
90	C₈×D₅₃	—	2	4	no	—
91	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
92	Q₈×C₅₃	—	4	4	no	—
93	C₂×C₄×C₅₃	—	2	4	no	—
94	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₅₃	—	2	4	no	—
98	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
99	C₂×Dic₅₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×Dic₅₃	C₂×#41	1	1	yes	characteristic
101	C₄×D₅₃	—	2	2	no	—
102	C₄×D₅₃	—	2	2	no	—
103	C₄×D₅₃	C₂×Q₁₆	1	2	yes	—
104	C₄×D₅₃	#135	1	2	yes	—
105	Dic₁₀₆	—	2	8	no	—
106	#88324	—	2	4	no	—
107	#88324	—	2	1	no	—
108	#88324	—	2	1	no	—
109	#88324	—	4	4	no	—
110	#88324	C₂×Q₁₆	1	2	yes	—
111	#88324	C₂×Q₁₆	1	2	yes	—
112	C₈×C₅₃	#131	1	4	yes	—
113	#88322	—	2	2	no	—
114	#17958	—	53	1	no	maximal, Sylow
115	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
116	C₂×D₅₃	—	2	1	no	—
117	C₂×D₅₃	—	2	1	no	—
118	C₂×D₅₃	C₄×Q₁₆	1	2	yes	—
119	C₄×C₅₃	—	4	4	no	—
120	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
121	C₄×C₅₃	#443	1	2	yes	—
122	C₄×C₅₃	C₂×#131	1	2	yes	—
123	Dic₅₃	—	2	1	no	—
124	Dic₅₃	—	2	1	no	—
125	Dic₅₃	—	4	4	no	—
126	Dic₅₃	C₂²×Q₁₆	1	1	yes	characteristic
127	Dic₅₃	C₂×#135	1	1	yes	characteristic
128	#3170	—	53	1	no	—
129	#2960	—	53	2	no	—
130	#1591	—	53	4	no	—
131	C₈×Q₁₆	—	53	4	no	—
132	#1567	—	53	1	no	—
133	#1493	—	53	2	no	—
134	C₂×C₈²	—	53	1	no	—
135	C₂×C₅₃	#2931	1	1	yes	characteristic
136	C₂×C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
137	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
138	D₅₃	—	2	2	no	—
139	#533	—	53	2	no	—
140	#500	#6157	1	1	yes	characteristic
141	#494	—	53	1	no	—
142	#487	—	53	2	no	—
143	#467	—	53	4	no	—
144	#446	—	53	4	no	—
145	#446	—	106	2	no	—
146	C₈×Q₈	—	53	4	no	—
147	C₈×Q₈	—	106	2	no	—
148	C₄×Q₁₆	—	53	4	no	—
149	#424	—	106	2	no	—
150	#423	—	53	2	no	—
151	#405	—	106	2	no	—
152	C₂×C₄×C₈	—	53	1	no	—
153	C₂×C₄×C₈	—	53	1	no	—
154	C₂×C₄×C₈	—	53	1	no	—
155	#335	—	53	4	no	—
156	Q₈⋊C₈	—	53	8	no	—
157	C₈²	—	53	2	no	—
158	C₈²	—	53	2	no	—
159	C₅₃	#17958	1	1	yes	characteristic, complemented, Sylow
160	C₂×Q₁₆	—	53	4	no	—
161	C₂×M₁₆	—	106	2	no	—
162	C₂²×C₈	—	53	1	no	—
163	C₂²×C₈	—	53	1	no	—
164	C₂²×C₈	—	53	2	no	—
165	C₂²×C₈	—	106	1	no	—
166	C₄⋊Q₈	—	53	2	no	—
167	C₄⋊Q₈	C₂×#6157	1	2	yes	—
168	#125	—	106	2	no	—
169	C₂²⋊Q₈	—	53	4	no	—
170	C₂²⋊Q₈	—	106	2	no	—
171	C₄×Q₈	—	53	4	no	—
172	C₄×Q₈	—	106	2	no	—
173	#117	—	106	2	no	—
174	C₂×C₄²	—	53	1	no	—
175	#107	C₂×#6157	1	4	yes	—
176	C₄⋊C₈	—	53	4	no	—
177	C₄⋊C₈	—	106	2	no	—
178	C₄⋊C₈	—	106	4	no	—
179	C₄⋊C₈	—	106	4	no	—
180	#103	—	53	8	no	—
181	#97	—	106	4	no	—
182	C₄×C₈	—	53	1	no	—
183	C₄×C₈	—	53	1	no	—
184	C₄×C₈	—	53	1	no	—
185	C₄×C₈	—	53	1	no	—
186	C₄×C₈	—	53	2	no	—
187	C₄×C₈	—	53	2	no	—
188	C₄×C₈	—	53	1	no	—
189	C₄×C₈	—	106	1	no	—
190	C₄×C₈	—	106	4	no	—
191	C₄×C₈	C₂×#6157	1	1	yes	characteristic
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	53	4	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	—	53	1	no	—
197	Q₁₆	—	106	8	no	—
198	M₁₆	—	212	4	no	—
199	C₂×C₈	—	53	1	no	—
200	C₂×C₈	—	53	1	no	—
201	C₂×C₈	—	53	4	no	—
202	C₂×C₈	—	53	1	no	—
203	C₂×C₈	—	53	1	no	—
204	C₂×C₈	—	53	2	no	—
205	C₂×C₈	—	106	2	no	—
206	C₂×C₈	—	106	2	no	—
207	C₂×C₈	—	106	4	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	—	106	4	no	—
210	C₂×C₈	C₂²×#6157	1	2	yes	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	106	4	no	—
213	C₄⋊C₄	—	106	4	no	—
214	C₄⋊C₄	—	106	2	no	—
215	C₄⋊C₄	—	106	2	no	—
216	C₄⋊C₄	C₂²×#6157	1	4	yes	—
217	C₂²⋊C₄	—	106	4	no	—
218	C₄²	—	53	1	no	—
219	C₄²	—	53	1	no	—
220	C₄²	—	53	1	no	—
221	C₄²	—	106	4	no	—
222	C₄²	C₂²×#6157	1	1	yes	characteristic
223	C₂³	—	53	1	no	—
224	Q₈	—	4	4	no	—
225	Q₈	—	106	8	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	53	1	no	—
228	C₂×C₄	—	53	2	no	—
229	C₂×C₄	—	53	2	no	—
230	C₂×C₄	—	53	1	no	—
231	C₂×C₄	—	53	1	no	—
232	C₂×C₄	—	106	2	no	—
233	C₂×C₄	—	106	2	no	—
234	C₂×C₄	—	106	4	no	—
235	C₂×C₄	D₄×#6157	1	1	yes	characteristic
236	C₂×C₄	D₄×#6157	1	1	yes	characteristic
237	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
238	C₈	—	53	2	no	—
239	C₈	—	53	2	no	—
240	C₈	—	106	1	no	—
241	C₈	—	106	1	no	—
242	C₈	—	106	4	no	—
243	C₈	—	106	2	no	—
244	C₈	—	212	4	no	—
245	C₈	#13133305	1	4	yes	—
246	C₂²	—	53	2	no	—
247	C₂²	—	106	1	no	—
248	C₂²	—	106	1	no	—
249	C₂²	C₂×D₄×#6157	1	1	yes	characteristic
250	C₄	—	4	4	no	—
251	C₄	—	53	1	no	—
252	C₄	—	53	1	no	—
253	C₄	—	106	1	no	—
254	C₄	—	106	1	no	—
255	C₄	—	212	4	no	—
256	C₄	Q₁₆×#6157	1	2	yes	—
257	C₄	#16811665	1	2	yes	—
258	C₄	C₂×#13133305	1	2	yes	—
259	C₂	—	106	2	no	—
260	C₂	#20291440	1	1	yes	characteristic
261	C₂	C₂×Q₁₆×#6157	1	1	yes	characteristic
262	C₂	C₂×#16811665	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#6157
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₅₃×#500	C₄
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂²×D₅₃	C₄×D₄
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	#500	#6157
hypocenter	C₅₃	#17958
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
2. #20290354	C₂	semidirect
3. #20290303	C₂	semidirect
74. D₅₃⋊C₈	Q₁₆	semidirect
159. C₅₃	#17958	semidirect

Maximal subgroups

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

subgroup	index
#20291504	2
#20290354	2
#20290303	2
#20290273	2
#20290065	2
C₈×D₅₃⋊C₈	2
#20285302	2
#17958	53