Subgroups — #27302311

Order 13568.

510 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810427	C₂	1	4	yes	maximal
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	#20274875	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133151	—	2	1	no	—
7	C₂×#13133151	C₂²	1	2	yes	—
8	#16810493	—	2	1	no	—
9	#16810427	—	2	8	no	—
10	C₄×#11719908	—	2	2	no	—
11	C₂²×Q₁₆×C₅₃	—	2	1	no	—
12	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₅₃	—	2	1	no	—
14	C₅₃×#494	—	2	2	no	—
15	C₅₃×#494	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅₃	—	2	2	no	—
18	C₂×C₄⋊Dic₅₃	—	4	2	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	D₄	1	4	yes	—
22	#13133151	—	2	4	no	—
23	#13133151	—	4	2	no	—
24	C₂²×Q₈×C₅₃	—	2	2	no	—
25	C₂×Q₁₆×C₅₃	—	2	4	no	—
26	C₂×Q₁₆×C₅₃	—	2	2	no	—
27	C₂×Q₁₆×C₅₃	—	2	4	no	—
28	C₂×Q₁₆×C₅₃	—	2	4	no	—
29	C₂×Q₁₆×C₅₃	—	4	2	no	—
30	C₂×Q₁₆×C₅₃	—	4	2	no	—
31	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₅₃	—	2	4	no	—
35	C₄⋊Q₈×C₅₃	—	4	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
37	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
38	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₂×Q₈×C₅₃	—	2	4	no	—
41	C₂×Q₈×C₅₃	—	4	2	no	—
42	C₂×Q₈×C₅₃	—	4	2	no	—
43	C₂×Q₈×C₅₃	—	4	2	no	—
44	C₂×Q₈×C₅₃	—	4	2	no	—
45	C₂×Q₈×C₅₃	—	4	2	no	—
46	C₂²×C₄×C₅₃	—	4	2	no	—
47	C₂²×C₄×C₅₃	D₈	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₅₃	—	4	2	no	—
50	Q₁₆×C₅₃	—	2	8	no	—
51	Q₁₆×C₅₃	—	4	8	no	—
52	Q₁₆×C₅₃	—	4	4	no	—
53	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	D₈	1	2	yes	—
55	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	#41	1	2	yes	—
60	C₄⋊C₄×C₅₃	—	4	8	no	—
61	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₅₃	SD₁₆	1	2	yes	—
63	C₄⋊Dic₅₃	—	4	2	no	—
64	C₄⋊Dic₅₃	—	4	2	no	—
65	C₄⋊Dic₅₃	—	8	4	no	—
66	#11719908	—	2	4	no	—
67	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₅₃	—	4	4	no	—
69	Q₈×C₅₃	—	4	4	no	—
70	Q₈×C₅₃	—	8	2	no	—
71	Q₈×C₅₃	—	8	2	no	—
72	C₂×C₄×C₅₃	—	4	4	no	—
73	C₂×C₄×C₅₃	—	8	2	no	—
74	C₂×C₄×C₅₃	—	8	2	no	—
75	C₂×C₄×C₅₃	#102	1	4	yes	—
76	C₂×C₄×C₅₃	SD₃₂	1	4	yes	—
77	C₂×C₄×C₅₃	Q₃₂	1	4	yes	—
78	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×Dic₅₃	—	8	2	no	—
85	C₂×Dic₅₃	—	8	4	no	—
86	C₈×C₅₃	#102	1	4	yes	—
87	C₈×C₅₃	C₄×D₄	1	2	yes	—
88	C₈×C₅₃	#121	1	2	yes	—
89	#13203	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
91	C₂²×C₅₃	#486	1	1	yes	characteristic
92	C₂²×C₅₃	#493	1	1	yes	characteristic
93	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
94	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
95	C₄×C₅₃	—	8	4	no	—
96	C₄×C₅₃	#358	1	8	yes	complemented
97	C₄×C₅₃	#390	1	1	yes	characteristic
98	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
99	C₄×C₅₃	#486	1	1	yes	characteristic
100	C₄×C₅₃	#492	1	1	yes	characteristic
101	Dic₅₃	—	16	2	no	—
102	C₂×#494	D₅₃	1	1	yes	characteristic
103	C₂×#358	—	53	4	no	—
104	C₂×C₄×C₁₆	—	53	1	no	—
105	#1845	—	53	1	no	—
106	C₂×C₅₃	#1957	1	1	yes	characteristic
107	C₂×C₅₃	C₂×#358	1	2	yes	—
108	C₂×C₅₃	#2241	1	2	yes	—
109	C₂×C₅₃	#2247	1	1	yes	characteristic
110	C₂×C₅₃	#2248	1	1	yes	characteristic
111	C₂²×Q₁₆	—	2	1	no	—
112	C₂²×Q₁₆	C₂×D₅₃	1	2	yes	—
113	C₂×C₄⋊Q₈	—	2	1	no	—
114	C₂²×C₁₆	—	53	2	no	—
115	#494	—	2	2	no	—
116	#494	Dic₅₃	1	4	yes	complemented
117	C₂×#107	—	53	2	no	—
118	C₂×#107	—	106	1	no	—
119	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
120	#384	—	106	1	no	—
121	#358	—	106	8	no	—
122	C₄×C₁₆	—	106	2	no	—
123	C₅₃	#13203	1	1	yes	characteristic, complemented, Sylow
124	C₂²×Q₈	—	2	2	no	—
125	C₂×Q₁₆	—	2	4	no	—
126	C₂×Q₁₆	—	2	4	no	—
127	C₂×Q₁₆	—	2	2	no	—
128	C₂×Q₁₆	—	2	4	no	—
129	C₂×Q₁₆	—	4	2	no	—
130	C₂×Q₁₆	—	4	2	no	—
131	C₂×Q₁₆	C₂×Dic₅₃	1	4	yes	—
132	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
133	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
134	C₄⋊Q₈	—	2	4	no	—
135	C₄⋊Q₈	—	4	2	no	—
136	C₂×C₄⋊C₄	—	4	2	no	—
137	C₂×C₄⋊C₄	—	106	2	no	—
138	C₂×C₄⋊C₄	—	212	2	no	—
139	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
140	C₂×C₁₆	—	53	4	no	—
141	C₂×C₁₆	—	106	2	no	—
142	C₂×C₁₆	—	106	2	no	—
143	#107	—	106	4	no	—
144	#107	—	212	2	no	—
145	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
146	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	—	212	2	no	—
155	C₂²×C₄	#11719922	1	2	yes	—
156	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
157	Q₁₆	—	2	8	no	—
158	Q₁₆	—	4	8	no	—
159	Q₁₆	—	4	4	no	—
160	C₂×C₈	#11719922	1	2	yes	—
161	C₂×C₈	#11719923	1	2	yes	—
162	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
163	C₂×C₈	D₄×D₅₃	1	2	yes	—
164	C₂×C₈	#11719947	1	2	yes	—
165	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
166	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
167	C₄⋊C₄	—	4	8	no	—
168	C₄⋊C₄	—	424	4	no	—
169	C₄⋊C₄	—	212	2	no	—
170	C₄⋊C₄	—	212	2	no	—
171	C₄²	#11719923	1	2	yes	—
172	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
173	C₁₆	—	106	4	no	—
174	C₂³	C₂×#11719922	1	1	yes	characteristic
175	Q₈	—	4	4	no	—
176	Q₈	—	4	4	no	—
177	Q₈	—	8	2	no	—
178	Q₈	—	8	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	424	4	no	—
183	C₂×C₄	—	424	2	no	—
184	C₂×C₄	#13133138	1	4	yes	—
185	C₂×C₄	#13133160	1	4	yes	—
186	C₂×C₄	#13133163	1	4	yes	—
187	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
188	C₂×C₄	#13133204	1	1	yes	characteristic
189	C₂×C₄	#13133213	1	1	yes	characteristic
190	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
191	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
192	C₂×C₄	C₂×#11719923	1	1	yes	characteristic
193	C₈	#13133138	1	4	yes	—
194	C₈	D₄×Dic₅₃	1	2	yes	—
195	C₈	#13133188	1	2	yes	—
196	C₂²	C₂×#13133138	1	1	yes	characteristic
197	C₂²	#16810969	1	1	yes	characteristic
198	C₂²	#16810999	1	1	yes	characteristic
199	C₂²	C₂×#13133160	1	2	yes	—
200	C₂²	C₂×#13133163	1	2	yes	—
201	C₄	—	8	4	no	—
202	C₄	—	848	2	no	—
203	C₄	#16810427	1	8	yes	complemented
204	C₄	#16810526	1	1	yes	characteristic
205	C₄	C₂×#13133138	1	1	yes	characteristic
206	C₄	#16810969	1	1	yes	characteristic
207	C₄	#16810995	1	1	yes	characteristic
208	C₂	#20275522	1	1	yes	characteristic
209	C₂	C₂×#16810427	1	2	yes	—
210	C₂	#20276942	1	2	yes	—
211	C₂	#20276967	1	1	yes	characteristic
212	C₂	#20276972	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#494	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#494	D₅₃
hypocenter	C₅₃	#13203
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
15. C₅₃×#494	C₄	semidirect
96. C₄×C₅₃	#358	semidirect
116. #494	Dic₅₃	semidirect
123. C₅₃	#13203	semidirect
203. C₄	#16810427	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810427	2
C₂×C₄×#11719908	2
#20274875	2
C₂×C₅₃×#494	2
#13203	53