Subgroups — #27302669

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133151	C₂	1	1	yes	maximal, characteristic
3	#20272806	C₂	1	4	yes	maximal
4	C₂×C₅₃×#500	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133151	C₂²	1	2	yes	—
7	C₄×C₄⋊Dic₅₃	—	2	1	no	—
8	C₄⋊#11719908	—	2	2	no	—
9	#16810386	—	2	1	no	—
10	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
11	C₅₃×#500	—	2	2	no	—
12	C₅₃×#500	C₄	1	4	yes	complemented
13	C₂×#107×C₅₃	—	2	1	no	—
14	C₂×#107×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₅₃	—	2	2	no	—
17	C₂×C₄×Dic₅₃	—	4	1	no	—
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	—	2	4	no	—
20	#13133151	—	2	2	no	—
21	#13133151	Q₈	1	4	yes	complemented
22	#13133124	—	4	1	no	—
23	C₂²×Q₈×C₅₃	—	4	1	no	—
24	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊Q₈×C₅₃	—	2	4	no	—
27	C₄⋊Q₈×C₅₃	—	4	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	#107×C₅₃	—	2	4	no	—
32	#107×C₅₃	—	2	2	no	—
33	#107×C₅₃	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₂×Q₈×C₅₃	—	4	4	no	—
37	C₂×Q₈×C₅₃	—	8	4	no	—
38	C₂²×C₄×C₅₃	—	4	2	no	—
39	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
45	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
48	C₂×C₈×C₅₃	#41	1	2	yes	—
49	C₄⋊C₄×C₅₃	—	2	4	no	—
50	C₄⋊C₄×C₅₃	—	4	4	no	—
51	C₄⋊C₄×C₅₃	—	4	2	no	—
52	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₅₃	SD₁₆	1	2	yes	—
54	C₄⋊Dic₅₃	—	2	4	no	—
55	C₄⋊Dic₅₃	—	4	2	no	—
56	C₄×Dic₅₃	—	8	2	no	—
57	#11719908	—	4	2	no	—
58	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
59	Q₈×C₅₃	—	8	8	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	4	4	no	—
62	C₂×C₄×C₅₃	—	8	4	no	—
63	C₂×C₄×C₅₃	#102	1	2	yes	—
64	C₂×C₄×C₅₃	#102	1	2	yes	—
65	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
66	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
67	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
69	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#137	1	2	yes	—
71	C₂×Dic₅₃	—	4	4	no	—
72	C₂×Dic₅₃	—	8	2	no	—
73	C₂×Dic₅₃	—	8	2	no	—
74	C₈×C₅₃	—	2	2	no	—
75	C₈×C₅₃	C₄×Q₈	1	2	yes	—
76	C₈×C₅₃	#124	1	1	yes	characteristic
77	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
78	#13276	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
80	C₂²×C₅₃	#474	1	2	yes	—
81	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
82	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
83	C₂²×C₅₃	#509	1	2	yes	—
84	C₄×C₅₃	—	2	2	no	—
85	C₄×C₅₃	—	8	4	no	—
86	C₄×C₅₃	#358	1	4	yes	—
87	C₄×C₅₃	#391	1	1	yes	characteristic
88	C₄×C₅₃	#419	1	1	yes	characteristic
89	C₄×C₅₃	#475	1	2	yes	—
90	Dic₅₃	—	8	4	no	—
91	C₂×#500	D₅₃	1	1	yes	characteristic
92	C₂×C₄⋊C₁₆	—	53	1	no	—
93	C₄×#107	—	53	1	no	—
94	#1380	—	53	4	no	—
95	C₂×C₅₃	#1988	1	1	yes	characteristic
96	C₂×C₅₃	C₂×#358	1	1	yes	characteristic
97	C₂×C₅₃	#2140	1	1	yes	characteristic
98	C₂×C₅₃	#2226	1	2	yes	—
99	C₂×C₅₃	#2227	1	2	yes	—
100	C₂×C₄⋊Q₈	—	2	1	no	—
101	C₂²×C₁₆	—	53	2	no	—
102	#500	—	2	2	no	—
103	#500	Dic₅₃	1	4	yes	complemented
104	C₂×#107	—	2	1	no	—
105	C₂×#107	—	53	2	no	—
106	C₂×#107	C₂×D₅₃	1	2	yes	—
107	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
108	C₄×C₄⋊C₄	—	106	1	no	—
109	C₄⋊C₁₆	—	106	2	no	—
110	#340	—	106	1	no	—
111	C₅₃	#13276	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₈	—	4	1	no	—
113	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
115	C₄⋊Q₈	—	2	4	no	—
116	C₄⋊Q₈	—	4	2	no	—
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄⋊C₄	—	106	2	no	—
120	C₂×C₄²	—	212	1	no	—
121	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
122	C₂×C₁₆	—	106	4	no	—
123	C₂×C₁₆	—	106	2	no	—
124	#107	—	2	2	no	—
125	#107	—	2	4	no	—
126	#107	—	53	4	no	—
127	#107	—	106	2	no	—
128	#107	C₂×Dic₅₃	1	4	yes	—
129	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
130	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
131	#95	—	212	1	no	—
132	C₂×Q₈	—	4	4	no	—
133	C₂×Q₈	—	8	4	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	212	2	no	—
136	C₂²×C₄	#11719922	1	1	yes	characteristic
137	C₂²×C₄	#11719922	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
141	C₂×C₈	Q₈×D₅₃	1	2	yes	—
142	C₂×C₈	#11719949	1	2	yes	—
143	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
144	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	106	4	no	—
149	C₄⋊C₄	—	212	2	no	—
150	C₄²	—	424	2	no	—
151	C₄²	#11719923	1	2	yes	—
152	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
153	C₁₆	—	212	2	no	—
154	C₂³	C₂×#11719922	1	1	yes	characteristic
155	Q₈	—	8	8	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	4	4	no	—
158	C₂×C₄	—	8	4	no	—
159	C₂×C₄	—	424	2	no	—
160	C₂×C₄	—	424	2	no	—
161	C₂×C₄	—	212	4	no	—
162	C₂×C₄	#13133138	1	2	yes	—
163	C₂×C₄	#13133138	1	2	yes	—
164	C₂×C₄	#13133160	1	2	yes	—
165	C₂×C₄	#13133163	1	2	yes	—
166	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
167	C₂×C₄	#13133193	1	2	yes	—
168	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
169	C₂×C₄	#13133248	1	2	yes	—
170	C₈	—	2	2	no	—
171	C₈	Q₈×Dic₅₃	1	2	yes	—
172	C₈	#13133206	1	1	yes	characteristic
173	C₈	#13133215	1	1	yes	characteristic
174	C₂²	C₂×#13133138	1	1	yes	characteristic
175	C₂²	#16810900	1	2	yes	—
176	C₂²	C₂×#13133160	1	1	yes	characteristic
177	C₂²	C₂×#13133163	1	1	yes	characteristic
178	C₂²	#16811059	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	8	4	no	—
181	C₄	—	424	4	no	—
182	C₄	#16810427	1	4	yes	—
183	C₄	#16810530	1	1	yes	characteristic
184	C₄	#16810632	1	1	yes	characteristic
185	C₄	#16810905	1	2	yes	—
186	C₂	#20275668	1	1	yes	characteristic
187	C₂	C₂×#16810427	1	1	yes	characteristic
188	C₂	#20276431	1	1	yes	characteristic
189	C₂	#20276859	1	2	yes	—
190	C₂	#20276864	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#500	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#500	D₅₃
hypocenter	C₅₃	#13276
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
12. C₅₃×#500	C₄	semidirect
21. #13133151	Q₈	semidirect
103. #500	Dic₅₃	semidirect
111. C₅₃	#13276	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₄×#13133151	2
#20272806	2
C₂×C₅₃×#500	2
#13276	53