Subgroups — #27295393

Order 13568.

310 conjugacy classes of subgroups.

The subgroup lattice has 232 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	#20272774	C₂	1	2	yes	maximal, complemented
4	#20272771	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#335	C₂	1	1	yes	maximal, characteristic
6	#16811180	—	2	1	no	—
7	C₂×#13133157	—	2	1	no	—
8	#16811028	C₂²	1	1	yes	characteristic
9	#16811028	C₂²	1	1	yes	characteristic
10	#16811025	—	2	1	no	—
11	#16811025	C₄	1	2	yes	—
12	#16811021	—	2	1	no	—
13	#16810661	—	2	1	no	—
14	#16810661	C₄	1	2	yes	—
15	C₄⋊#11719908	—	2	1	no	—
16	C₄⋊#11719908	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#335	—	2	1	no	—
20	C₅₃×#335	C₂²	1	2	yes	—
21	#13133217	—	2	1	no	—
22	#13133217	—	2	1	no	—
23	#13133209	—	2	2	no	—
24	#13133196	—	4	1	no	—
25	#13133196	—	4	1	no	—
26	C₄×Dic₁₀₆	—	4	1	no	—
27	C₄×Dic₁₀₆	—	4	1	no	—
28	#13133176	—	2	2	no	—
29	#13133157	—	4	2	no	—
30	C₂×#11719908	—	2	2	no	—
31	#13133151	—	2	1	no	—
32	#13133151	—	2	1	no	—
33	#13133151	C₂×C₄	1	2	yes	—
34	#13133151	C₂×C₄	1	2	yes	—
35	#13133149	—	2	2	no	—
36	#13133149	—	2	2	no	—
37	C₂²×C₈×C₅₃	—	4	1	no	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	4	1	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
50	C₂×Dic₁₀₆	—	4	1	no	—
51	C₂×Dic₁₀₆	—	4	1	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	4	2	no	—
54	C₂×C₈×C₅₃	—	8	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	SD₁₆	1	2	yes	—
58	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
59	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₅₃	D₈	1	1	yes	characteristic
63	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²⋊Dic₅₃	—	4	2	no	—
66	C₄⋊Dic₅₃	—	2	2	no	—
67	C₄⋊Dic₅₃	—	2	2	no	—
68	C₄⋊Dic₅₃	—	4	1	no	—
69	C₄⋊Dic₅₃	—	4	1	no	—
70	#11719919	—	4	2	no	—
71	#11719919	—	4	2	no	—
72	C₄×Dic₅₃	—	4	2	no	—
73	#11719908	—	4	2	no	—
74	C₂³×C₅₃	#102	1	1	yes	characteristic
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	4	1	no	—
78	C₂×C₄×C₅₃	#101	1	2	yes	—
79	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#103	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×Dic₅₃	—	4	2	no	—
87	C₂×Dic₅₃	—	4	2	no	—
88	Dic₁₀₆	—	8	2	no	—
89	Dic₁₀₆	—	8	1	no	—
90	Dic₁₀₆	—	8	1	no	—
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	—	2	1	no	—
93	C₈×C₅₃	—	8	2	no	—
94	C₈×C₅₃	#103	1	2	yes	—
95	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
97	#11942	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	#332	1	2	yes	—
101	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	#332	1	2	yes	—
106	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
107	C₄×C₅₃	C₂×#103	1	1	yes	characteristic
108	C₄×C₅₃	#509	1	1	yes	characteristic
109	C₄×C₅₃	#510	1	1	yes	characteristic
110	Dic₅₃	—	8	2	no	—
111	Dic₅₃	—	8	2	no	—
112	#3160	—	53	1	no	—
113	#2150	—	53	1	no	—
114	C₂×#335	D₅₃	1	1	yes	characteristic
115	#1365	—	53	2	no	—
116	#1364	—	53	2	no	—
117	C₂×C₅₃	—	2	2	no	—
118	C₂×C₅₃	C₂×#332	1	1	yes	characteristic
119	C₂×C₅₃	#2139	1	1	yes	characteristic
120	C₂×C₅₃	#2148	1	1	yes	characteristic
121	#533	—	106	1	no	—
122	C₂×M₃₂	—	106	1	no	—
123	#500	—	53	1	no	—
124	#500	—	53	1	no	—
125	#499	—	53	2	no	—
126	#499	—	106	1	no	—
127	#498	—	106	1	no	—
128	#427	—	53	2	no	—
129	#427	—	106	1	no	—
130	C₂×C₄⋊C₈	—	2	1	no	—
131	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
132	C₄⋊C₁₆	—	53	2	no	—
133	C₄⋊C₁₆	—	106	1	no	—
134	#335	—	2	1	no	—
135	#335	C₂×D₅₃	1	2	yes	—
136	C₅₃	#11942	1	1	yes	characteristic, complemented, Sylow
137	C₂²×C₈	—	4	1	no	—
138	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
139	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
140	C₄⋊Q₈	—	106	1	no	—
141	C₄⋊Q₈	—	106	1	no	—
142	#125	—	106	2	no	—
143	C₂²⋊Q₈	—	212	1	no	—
144	C₂²⋊Q₈	—	212	1	no	—
145	C₄×Q₈	—	212	1	no	—
146	C₄×Q₈	—	212	1	no	—
147	#117	—	106	2	no	—
148	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
149	M₃₂	—	212	2	no	—
150	C₂×C₁₆	—	106	2	no	—
151	#107	—	53	2	no	—
152	#107	—	53	2	no	—
153	#107	—	106	1	no	—
154	#107	—	106	1	no	—
155	#106	—	106	2	no	—
156	#106	—	106	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	4	1	no	—
159	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
160	C₄×C₈	D₂₁₂	1	1	yes	characteristic
161	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
162	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
163	C₂×Q₈	—	212	1	no	—
164	C₂×Q₈	—	212	1	no	—
165	C₂²×C₄	#11719921	1	1	yes	characteristic
166	C₂²×C₄	#11719922	1	1	yes	characteristic
167	C₂²×C₄	#11719924	1	1	yes	characteristic
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	8	2	no	—
171	C₂×C₈	#11719913	1	2	yes	—
172	C₂×C₈	Dic₂₁₂	1	2	yes	—
173	C₂×C₈	#11719921	1	1	yes	characteristic
174	C₂×C₈	#11719921	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
176	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
177	C₄⋊C₄	—	106	2	no	—
178	C₄⋊C₄	—	106	2	no	—
179	C₄⋊C₄	—	212	2	no	—
180	C₄⋊C₄	—	212	2	no	—
181	C₄⋊C₄	—	212	1	no	—
182	C₄⋊C₄	—	212	1	no	—
183	C₂²⋊C₄	—	212	2	no	—
184	C₄²	—	212	2	no	—
185	C₄²	#11719921	1	1	yes	characteristic
186	C₄²	#11719922	1	1	yes	characteristic
187	C₄²	#11719924	1	1	yes	characteristic
188	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
189	C₁₆	—	212	2	no	—
190	C₂³	#13133137	1	1	yes	characteristic
191	Q₈	—	424	2	no	—
192	Q₈	—	424	1	no	—
193	Q₈	—	424	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	212	2	no	—
198	C₂×C₄	—	212	2	no	—
199	C₂×C₄	#13133135	1	2	yes	—
200	C₂×C₄	#13133137	1	1	yes	characteristic
201	C₂×C₄	#13133137	1	1	yes	characteristic
202	C₂×C₄	#13133137	1	1	yes	characteristic
203	C₂×C₄	#13133142	1	2	yes	—
204	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
205	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
206	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	8	2	no	—
210	C₈	#13133142	1	2	yes	—
211	C₈	C₂×#11719913	1	1	yes	characteristic
212	C₈	C₂×Dic₂₁₂	1	1	yes	characteristic
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	#16810370	1	2	yes	—
216	C₂²	C₂×#13133137	1	1	yes	characteristic
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	424	2	no	—
221	C₄	—	424	2	no	—
222	C₄	#16810370	1	2	yes	—
223	C₄	C₂×#13133135	1	1	yes	characteristic
224	C₄	C₂×#13133142	1	1	yes	characteristic
225	C₄	#16811059	1	1	yes	characteristic
226	C₄	#16811064	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₂×#16810370	1	1	yes	characteristic
229	C₂	#20276425	1	1	yes	characteristic
230	C₂	#20276462	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13133137
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#335	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#335	D₅₃
hypocenter	C₅₃	#11942
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #20272774	C₂	semidirect
4. #20272771	C₂	semidirect
136. C₅₃	#11942	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
#20272774	2
#20272771	2
C₂×C₅₃×#335	2
#11942	53