Subgroups — #27295526

Order 13568.

330 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276514	C₂	1	1	yes	maximal, characteristic
2	#20273474	C₂	1	1	yes	maximal, characteristic
3	#20272773	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
5	#16810647	—	2	1	no	—
6	C₄⋊#11719908	C₂²	1	2	yes	—
7	C₂²⋊#11719908	—	2	1	no	—
8	C₄×#11719908	—	2	1	no	—
9	#16810376	—	2	1	no	—
10	#16810376	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
12	C₅₃×#500	—	2	1	no	—
13	C₅₃×#500	—	2	1	no	—
14	C₅₃×#500	C₄	1	2	yes	—
15	C₅₃×#500	C₂²	1	2	yes	—
16	C₂×#107×C₅₃	—	2	1	no	—
17	C₂×#107×C₅₃	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×#11719917	—	4	1	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₄⋊#88322	—	2	2	no	—
22	C₄⋊#88322	—	4	1	no	—
23	C₂²×Q₈×C₅₃	—	4	1	no	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₄⋊Q₈×C₅₃	—	2	2	no	—
27	C₄⋊Q₈×C₅₃	—	2	2	no	—
28	C₄⋊Q₈×C₅₃	—	4	1	no	—
29	C₄⋊Q₈×C₅₃	—	4	1	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
32	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
33	#107×C₅₃	—	2	2	no	—
34	#107×C₅₃	—	2	2	no	—
35	#107×C₅₃	—	2	2	no	—
36	#107×C₅₃	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×Q₈×C₅₃	—	4	2	no	—
42	C₂×Q₈×C₅₃	—	4	2	no	—
43	C₂×Q₈×C₅₃	—	8	4	no	—
44	C₂²×C₄×C₅₃	—	4	2	no	—
45	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₅₃	—	2	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₅₃	—	2	4	no	—
55	C₄⋊C₄×C₅₃	—	4	2	no	—
56	C₄⋊C₄×C₅₃	—	4	2	no	—
57	C₄⋊C₄×C₅₃	—	4	2	no	—
58	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₄²×C₅₃	D₈	1	1	yes	characteristic
60	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
62	#11719917	—	8	2	no	—
63	C₂×#88322	—	4	2	no	—
64	#11719908	—	4	2	no	—
65	C₂³×C₅₃	#102	1	1	yes	characteristic
66	Q₈×C₅₃	—	8	4	no	—
67	Q₈×C₅₃	—	8	4	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	4	4	no	—
72	C₂×C₄×C₅₃	—	8	4	no	—
73	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
76	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
77	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₈×C₅₃	—	2	1	no	—
82	C₈×C₅₃	—	2	1	no	—
83	C₈×C₅₃	#101	1	2	yes	—
84	C₈×C₅₃	#137	1	2	yes	—
85	#88322	—	8	2	no	—
86	#11972	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	—	2	1	no	—
88	C₂²×C₅₃	—	2	1	no	—
89	C₂²×C₅₃	#358	1	2	yes	—
90	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
91	C₄×C₅₃	—	2	1	no	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	—	8	4	no	—
94	C₄×C₅₃	#358	1	2	yes	—
95	C₄×C₅₃	#361	1	2	yes	—
96	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
97	C₄×C₅₃	#419	1	1	yes	characteristic
98	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
99	C₄×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
100	C₂×#500	D₅₃	1	1	yes	characteristic
101	#2162	—	53	1	no	—
102	#1567	—	53	1	no	—
103	#1365	—	53	4	no	—
104	C₂×C₅₃	—	2	2	no	—
105	C₂×C₅₃	#1546	1	1	yes	characteristic
106	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
107	C₂×C₅₃	C₂×#361	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	—	2	1	no	—
109	#500	—	2	1	no	—
110	#500	—	2	1	no	—
111	#500	Dic₅₃	1	2	yes	—
112	#500	C₂×D₅₃	1	2	yes	—
113	C₂×#107	—	2	1	no	—
114	C₂×#107	Dic₅₃	1	2	yes	—
115	#423	—	106	1	no	—
116	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
117	C₄⋊C₁₆	—	53	2	no	—
118	C₂²⋊C₁₆	—	106	1	no	—
119	C₄×C₁₆	—	106	1	no	—
120	#335	—	53	2	no	—
121	#335	—	106	1	no	—
122	C₅₃	#11972	1	1	yes	characteristic, complemented, Sylow
123	C₂²×Q₈	—	4	1	no	—
124	C₂×M₁₆	—	212	1	no	—
125	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	2	no	—
128	C₄⋊Q₈	—	2	2	no	—
129	C₄⋊Q₈	—	4	1	no	—
130	C₄⋊Q₈	—	4	1	no	—
131	C₂×C₄⋊C₄	—	2	2	no	—
132	C₂×C₄⋊C₄	—	4	1	no	—
133	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
134	C₂×C₁₆	—	106	2	no	—
135	#107	—	2	2	no	—
136	#107	—	2	2	no	—
137	#107	—	2	2	no	—
138	#107	C₂×Dic₅₃	1	4	yes	—
139	C₄⋊C₈	—	106	2	no	—
140	C₄⋊C₈	—	212	1	no	—
141	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
143	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
144	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
145	C₂×Q₈	—	4	2	no	—
146	C₂×Q₈	—	4	2	no	—
147	C₂×Q₈	—	8	4	no	—
148	C₂²×C₄	—	4	2	no	—
149	C₂²×C₄	#11719922	1	1	yes	characteristic
150	C₂²×C₄	#11719923	1	1	yes	characteristic
151	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
152	M₁₆	—	424	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	212	2	no	—
156	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
158	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
159	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄²	#11719922	1	1	yes	characteristic
165	C₄²	#11719923	1	1	yes	characteristic
166	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
167	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
168	C₁₆	—	212	2	no	—
169	C₂³	#13133138	1	1	yes	characteristic
170	Q₈	—	8	4	no	—
171	Q₈	—	8	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	8	4	no	—
177	C₂×C₄	#13133138	1	1	yes	characteristic
178	C₂×C₄	#13133138	1	1	yes	characteristic
179	C₂×C₄	#13133138	1	1	yes	characteristic
180	C₂×C₄	#13133160	1	2	yes	—
181	C₂×C₄	#13133163	1	2	yes	—
182	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
183	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
184	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	424	2	no	—
188	C₈	#13133136	1	2	yes	—
189	C₈	#13133248	1	2	yes	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	#16810427	1	2	yes	—
193	C₂²	C₂×#13133138	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	8	4	no	—
197	C₄	#16810427	1	2	yes	—
198	C₄	#16810436	1	2	yes	—
199	C₄	C₂×#13133136	1	1	yes	characteristic
200	C₄	#16810632	1	1	yes	characteristic
201	C₄	C₂×#13133160	1	1	yes	characteristic
202	C₄	C₂×#13133163	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	#20273387	1	1	yes	characteristic
205	C₂	C₂×#16810427	1	1	yes	characteristic
206	C₂	C₂×#16810436	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×#500	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	#102
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	#11972
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #20272773	C₂	semidirect
122. C₅₃	#11972	semidirect

Maximal subgroups

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

subgroup	index
#20276514	2
#20273474	2
#20272773	2
C₂×C₅₃×#500	2
#11972	53