Subgroups — #27302677

Order 13568.

358 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20285229	C₂	1	1	yes	maximal, characteristic
2	#20276471	C₂	1	1	yes	maximal, characteristic
3	#20272822	C₂	1	4	yes	maximal
4	C₄×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16811180	—	2	1	no	—
6	C₂×#13133157	C₂²	1	2	yes	—
7	#16811028	C₄	1	1	yes	characteristic, complemented
8	#16811028	C₄	1	1	yes	characteristic, complemented
9	#16811025	—	2	1	no	—
10	#16811025	C₄	1	2	yes	complemented
11	#16811021	—	2	1	no	—
12	#16810661	—	2	1	no	—
13	#16810661	C₂²	1	2	yes	—
14	C₄⋊#11719908	—	2	1	no	—
15	C₄⋊#11719908	—	2	1	no	—
16	C₂×#106×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
19	C₅₃×#340	—	2	1	no	—
20	#13133217	—	2	1	no	—
21	#13133217	—	2	1	no	—
22	#13133209	—	2	2	no	—
23	#13133196	—	4	1	no	—
24	#13133196	—	4	1	no	—
25	C₄×Dic₁₀₆	—	4	1	no	—
26	C₄×Dic₁₀₆	—	4	1	no	—
27	#13133176	—	2	2	no	—
28	#13133157	—	2	4	no	—
29	C₂×#11719908	—	2	2	no	—
30	#13133151	—	2	1	no	—
31	#13133151	—	2	1	no	—
32	#13133151	C₂×C₄	1	2	yes	—
33	#13133151	C₂×C₄	1	2	yes	—
34	#13133149	—	2	2	no	—
35	#13133149	—	2	2	no	—
36	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
39	C₂×C₄²×C₅₃	—	4	1	no	—
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	#106×C₅₃	—	2	2	no	—
42	#106×C₅₃	Q₈	1	4	yes	complemented
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
47	#95×C₅₃	—	4	1	no	—
48	C₂²×C₄×C₅₃	—	4	2	no	—
49	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₁₀₆	—	4	1	no	—
53	C₂×Dic₁₀₆	—	4	1	no	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₅₃	#41	1	2	yes	—
61	C₄⋊C₄×C₅₃	—	2	4	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄²×C₅₃	—	8	2	no	—
64	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
67	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
68	C₂²⋊Dic₅₃	—	4	2	no	—
69	C₄⋊Dic₅₃	—	2	2	no	—
70	C₄⋊Dic₅₃	—	2	2	no	—
71	C₄⋊Dic₅₃	—	4	1	no	—
72	C₄⋊Dic₅₃	—	4	1	no	—
73	#11719919	—	4	2	no	—
74	#11719919	—	4	2	no	—
75	C₄×Dic₅₃	—	4	2	no	—
76	#11719908	—	4	2	no	—
77	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	4	1	no	—
80	C₂×C₄×C₅₃	—	4	4	no	—
81	C₂×C₄×C₅₃	—	8	2	no	—
82	C₂×C₄×C₅₃	—	8	2	no	—
83	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#137	1	2	yes	—
91	C₂×Dic₅₃	—	4	2	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	Dic₁₀₆	—	8	2	no	—
94	Dic₁₀₆	—	8	2	no	—
95	C₈×C₅₃	—	2	1	no	—
96	C₈×C₅₃	—	2	1	no	—
97	C₈×C₅₃	C₄×Q₈	1	2	yes	—
98	C₈×C₅₃	#124	1	1	yes	characteristic
99	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
100	#13278	—	53	1	no	maximal, Sylow
101	C₂²×C₅₃	—	2	1	no	—
102	C₂²×C₅₃	—	2	1	no	—
103	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
104	C₂²×C₅₃	#474	1	2	yes	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	8	4	no	—
109	C₄×C₅₃	#360	1	2	yes	—
110	C₄×C₅₃	#391	1	1	yes	characteristic
111	C₄×C₅₃	#419	1	1	yes	characteristic
112	C₄×C₅₃	#475	1	2	yes	—
113	Dic₅₃	—	8	2	no	—
114	Dic₅₃	—	8	2	no	—
115	#3160	—	53	1	no	—
116	#2150	—	53	1	no	—
117	C₄×#106	D₅₃	1	1	yes	characteristic
118	#1386	—	53	4	no	—
119	C₂×C₅₃	—	2	2	no	—
120	C₂×C₅₃	#1988	1	1	yes	characteristic
121	C₂×C₅₃	C₂×#360	1	1	yes	characteristic
122	C₂×C₅₃	#2145	1	1	yes	characteristic
123	#533	—	106	1	no	—
124	C₂×M₃₂	—	53	2	no	—
125	#500	—	53	1	no	—
126	#500	—	53	1	no	—
127	#499	—	53	2	no	—
128	#499	—	106	1	no	—
129	#498	—	106	1	no	—
130	#427	—	53	2	no	—
131	#427	—	106	1	no	—
132	C₂×#106	C₂×D₅₃	1	2	yes	—
133	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
134	C₄×C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₁₆	—	106	1	no	—
136	C₄⋊C₁₆	—	106	1	no	—
137	#340	—	2	1	no	—
138	C₅₃	#13278	1	1	yes	characteristic, complemented, Sylow
139	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
141	C₄⋊Q₈	—	106	1	no	—
142	C₄⋊Q₈	—	106	1	no	—
143	#125	—	106	2	no	—
144	C₂²⋊Q₈	—	212	1	no	—
145	C₂²⋊Q₈	—	212	1	no	—
146	C₄×Q₈	—	212	1	no	—
147	C₄×Q₈	—	212	1	no	—
148	#117	—	106	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄²	—	4	1	no	—
151	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
152	M₃₂	—	106	4	no	—
153	C₂×C₁₆	—	106	2	no	—
154	#107	—	53	2	no	—
155	#107	—	53	2	no	—
156	#107	—	106	1	no	—
157	#107	—	106	1	no	—
158	#106	—	2	2	no	—
159	#106	—	106	2	no	—
160	#106	—	106	2	no	—
161	#106	Dic₁₀₆	1	4	yes	complemented
162	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
163	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
164	C₄×C₈	D₂₁₂	1	1	yes	characteristic
165	C₄×C₈	D₂₁₂	1	1	yes	characteristic
166	#95	—	4	1	no	—
167	C₂×Q₈	—	212	1	no	—
168	C₂×Q₈	—	212	1	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	#11719922	1	1	yes	characteristic
171	C₂²×C₄	#11719922	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	#11719921	1	1	yes	characteristic
175	C₂×C₈	#11719921	1	1	yes	characteristic
176	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
177	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
178	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
179	C₂×C₈	#11719945	1	2	yes	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	106	2	no	—
183	C₄⋊C₄	—	106	2	no	—
184	C₄⋊C₄	—	212	2	no	—
185	C₄⋊C₄	—	212	2	no	—
186	C₄⋊C₄	—	212	1	no	—
187	C₄⋊C₄	—	212	1	no	—
188	C₂²⋊C₄	—	212	2	no	—
189	C₄²	—	8	2	no	—
190	C₄²	—	212	2	no	—
191	C₄²	#11719921	1	1	yes	characteristic
192	C₄²	#11719921	1	1	yes	characteristic
193	C₄²	#11719924	1	1	yes	characteristic
194	C₄²	#11719924	1	1	yes	characteristic
195	C₁₆	—	212	2	no	—
196	C₂³	C₂×#11719922	1	1	yes	characteristic
197	Q₈	—	424	2	no	—
198	Q₈	—	424	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	8	2	no	—
203	C₂×C₄	—	8	2	no	—
204	C₂×C₄	—	212	2	no	—
205	C₂×C₄	—	212	2	no	—
206	C₂×C₄	#13133137	1	1	yes	characteristic
207	C₂×C₄	#13133137	1	1	yes	characteristic
208	C₂×C₄	#13133137	1	1	yes	characteristic
209	C₂×C₄	#13133137	1	1	yes	characteristic
210	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
211	C₂×C₄	#13133196	1	2	yes	—
212	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
213	C₂×C₄	#13133247	1	2	yes	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	C₄×Dic₁₀₆	1	2	yes	—
217	C₈	#13133205	1	1	yes	characteristic
218	C₈	#13133217	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₂×#13133137	1	1	yes	characteristic
222	C₂²	#16810903	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	4	no	—
227	C₄	—	424	2	no	—
228	C₄	—	424	2	no	—
229	C₄	#16810432	1	2	yes	—
230	C₄	#16810529	1	1	yes	characteristic
231	C₄	#16810631	1	1	yes	characteristic
232	C₄	#16810908	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	#20275667	1	1	yes	characteristic
235	C₂	C₂×#16810432	1	1	yes	characteristic
236	C₂	#20276450	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133137
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×#106×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×#106	D₅₃
hypocenter	C₅₃	#13278
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
7. #16811028	C₄	semidirect
8. #16811028	C₄	semidirect
10. #16811025	C₄	semidirect
42. #106×C₅₃	Q₈	semidirect
138. C₅₃	#13278	semidirect
161. #106	Dic₁₀₆	semidirect

Maximal subgroups

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

subgroup	index
#20285229	2
#20276471	2
#20272822	2
C₄×#106×C₅₃	2
#13278	53