Subgroups — #27302618

Order 13568.

338 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283649	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133157	C₂	1	1	yes	maximal, characteristic
3	#20272803	C₂	1	4	yes	maximal
4	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#16811180	C₄	1	1	yes	characteristic
6	C₂×#13133157	C₂²	1	2	yes	—
7	#16810753	—	2	1	no	—
8	#16810753	—	2	1	no	—
9	C₈×Dic₁₀₆	—	2	1	no	—
10	C₈×Dic₁₀₆	—	2	1	no	—
11	#16810649	C₄	1	2	yes	—
12	#16810647	C₄	1	1	yes	characteristic
13	#16810578	C₂²	1	2	yes	—
14	#16810399	—	2	1	no	—
15	C₄×#11719908	—	2	1	no	—
16	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×C₄⋊C₁₆	—	2	1	no	—
19	C₅₃×C₄⋊C₁₆	—	2	1	no	—
20	C₂×#11719917	C₂×C₄	1	2	yes	—
21	#13133217	C₈	1	1	yes	characteristic
22	#13133217	C₈	1	1	yes	characteristic
23	#13133209	C₈	1	2	yes	—
24	#13133196	—	2	1	no	—
25	#13133196	—	2	1	no	—
26	C₄×Dic₁₀₆	—	2	1	no	—
27	C₄×Dic₁₀₆	—	2	1	no	—
28	#13133176	C₂×C₄	1	2	yes	—
29	#13133157	—	2	2	no	—
30	#13133157	Q₈	1	4	yes	complemented
31	C₂×#11719908	—	2	2	no	—
32	C₄⋊#88322	—	2	1	no	—
33	C₄⋊#88322	—	2	1	no	—
34	#13133146	—	2	2	no	—
35	#13133146	—	2	2	no	—
36	#13133127	—	2	2	no	—
37	C₈×Dic₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₃	—	2	2	no	—
42	C₂×C₁₆×C₅₃	—	2	4	no	—
43	C₄×C₈×C₅₃	D₄	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	D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₁₀₆	—	2	1	no	—
51	C₂×Dic₁₀₆	—	2	1	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
58	C₂×C₈×C₅₃	#41	1	2	yes	—
59	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
63	C₂²⋊Dic₅₃	—	2	2	no	—
64	C₄⋊Dic₅₃	—	2	1	no	—
65	C₄⋊Dic₅₃	—	2	1	no	—
66	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
67	C₄⋊Dic₅₃	C₂×C₈	1	2	yes	—
68	#11719919	—	2	2	no	—
69	#11719919	—	2	2	no	—
70	C₄×Dic₅₃	—	2	2	no	—
71	#11719917	—	2	4	no	—
72	C₂×#88322	—	2	2	no	—
73	C₁₆×C₅₃	—	4	2	no	—
74	#11719908	—	2	4	no	—
75	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	4	1	no	—
78	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
84	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#131	1	2	yes	—
86	C₂×Dic₅₃	—	2	2	no	—
87	C₂×Dic₅₃	—	2	2	no	—
88	Dic₁₀₆	—	4	2	no	—
89	Dic₁₀₆	—	4	2	no	—
90	C₈×C₅₃	—	2	1	no	—
91	C₈×C₅₃	—	2	1	no	—
92	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
93	C₈×C₅₃	#124	1	1	yes	characteristic
94	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
95	#88322	—	4	2	no	—
96	#13265	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	—	2	1	no	—
98	C₂²×C₅₃	—	2	1	no	—
99	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₂²×C₅₃	#446	1	2	yes	—
101	C₄×C₅₃	—	2	1	no	—
102	C₄×C₅₃	—	2	1	no	—
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	#350	1	2	yes	—
105	C₄×C₅₃	#391	1	1	yes	characteristic
106	C₄×C₅₃	#408	1	1	yes	characteristic
107	C₄×C₅₃	C₈×Q₈	1	2	yes	—
108	Dic₅₃	—	4	2	no	—
109	Dic₅₃	—	4	2	no	—
110	#2960	—	53	1	no	—
111	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
112	C₄×M₃₂	—	53	1	no	—
113	#1379	—	53	4	no	—
114	C₂×C₅₃	—	2	2	no	—
115	C₂×C₅₃	#1984	1	1	yes	characteristic
116	C₂×C₅₃	C₂×#350	1	1	yes	characteristic
117	C₂×C₅₃	#2117	1	1	yes	characteristic
118	#533	—	53	1	no	—
119	C₂×M₃₂	—	53	2	no	—
120	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
121	#446	—	106	1	no	—
122	#446	—	106	1	no	—
123	C₈×Q₈	—	106	1	no	—
124	C₈×Q₈	—	106	1	no	—
125	#424	—	53	2	no	—
126	#423	—	53	1	no	—
127	#405	—	53	2	no	—
128	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
129	C₄⋊C₁₆	—	2	1	no	—
130	C₄⋊C₁₆	—	2	1	no	—
131	#346	—	106	1	no	—
132	C₄×C₁₆	—	106	1	no	—
133	C₅₃	#13265	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	53	2	no	—
135	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
137	C₄⋊Q₈	—	53	1	no	—
138	C₄⋊Q₈	—	53	1	no	—
139	#125	—	53	2	no	—
140	C₂²⋊Q₈	—	106	1	no	—
141	C₂²⋊Q₈	—	106	1	no	—
142	C₄×Q₈	—	106	1	no	—
143	C₄×Q₈	—	106	1	no	—
144	#117	—	53	2	no	—
145	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
146	M₃₂	—	53	4	no	—
147	M₃₂	—	106	2	no	—
148	C₂×C₁₆	—	2	4	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	106	2	no	—
151	C₄⋊C₈	—	106	2	no	—
152	C₄⋊C₈	—	106	2	no	—
153	C₄⋊C₈	—	106	1	no	—
154	C₄⋊C₈	—	106	1	no	—
155	#97	—	106	2	no	—
156	C₄×C₈	—	106	2	no	—
157	C₄×C₈	D₂₁₂	1	1	yes	characteristic
158	C₄×C₈	D₂₁₂	1	1	yes	characteristic
159	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
160	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
161	C₂×Q₈	—	106	1	no	—
162	C₂×Q₈	—	106	1	no	—
163	C₂²×C₄	#11719912	1	1	yes	characteristic
164	C₂²×C₄	#11719912	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
166	M₁₆	—	106	4	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	106	2	no	—
169	C₂×C₈	#11719921	1	1	yes	characteristic
170	C₂×C₈	#11719921	1	1	yes	characteristic
171	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
172	C₂×C₈	#11719947	1	2	yes	—
173	C₂×C₈	Q₈×D₅₃	1	2	yes	—
174	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
175	C₄⋊C₄	—	53	2	no	—
176	C₄⋊C₄	—	53	2	no	—
177	C₄⋊C₄	—	106	2	no	—
178	C₄⋊C₄	—	106	2	no	—
179	C₄⋊C₄	—	106	1	no	—
180	C₄⋊C₄	—	106	1	no	—
181	C₂²⋊C₄	—	106	2	no	—
182	C₄²	—	106	2	no	—
183	C₄²	C₈×D₅₃	1	1	yes	characteristic
184	C₄²	C₈×D₅₃	1	1	yes	characteristic
185	C₄²	#11719921	1	1	yes	characteristic
186	C₄²	#11719921	1	1	yes	characteristic
187	C₁₆	—	4	2	no	—
188	C₁₆	—	106	4	no	—
189	C₂³	C₂×#11719912	1	1	yes	characteristic
190	Q₈	—	212	2	no	—
191	Q₈	—	212	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	106	2	no	—
195	C₂×C₄	—	106	2	no	—
196	C₂×C₄	#13133129	1	1	yes	characteristic
197	C₂×C₄	#13133129	1	1	yes	characteristic
198	C₂×C₄	#13133129	1	1	yes	characteristic
199	C₂×C₄	#13133129	1	1	yes	characteristic
200	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
201	C₂×C₄	#13133182	1	2	yes	—
202	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
203	C₂×C₄	#13133225	1	2	yes	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	212	2	no	—
207	C₈	#13133197	1	2	yes	—
208	C₈	#13133204	1	1	yes	characteristic
209	C₈	#13133215	1	1	yes	characteristic
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂×#13133129	1	1	yes	characteristic
213	C₂²	#16810750	1	2	yes	—
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	212	2	no	—
218	C₄	—	212	2	no	—
219	C₄	#16810406	1	2	yes	—
220	C₄	#16810528	1	1	yes	characteristic
221	C₄	#16810591	1	1	yes	characteristic
222	C₄	#16810747	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	#20275650	1	1	yes	characteristic
225	C₂	C₂×#16810406	1	1	yes	characteristic
226	C₂	#20276342	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#13265
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
30. #13133157	Q₈	semidirect
133. C₅₃	#13265	semidirect

Maximal subgroups

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

subgroup	index
#20283649	2
C₄×#13133157	2
#20272803	2
C₂×C₅₃×C₄⋊C₁₆	2
#13265	53