Subgroups — #27300262

Order 13568.

438 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	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810450	C₂	1	2	yes	maximal, complemented
3	#20274744	C₂	1	1	yes	maximal, characteristic
4	#20272806	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1808	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	—	2	4	no	—
7	C₂²×#11719908	C₄	1	4	yes	complemented
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	C₂²×#11719908	C₂²	1	1	yes	characteristic
10	C₂×#13133151	C₂²	1	2	yes	—
11	#16810483	—	2	1	no	—
12	#16810450	—	2	4	no	—
13	#16810386	—	2	1	no	—
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#107×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#380	—	2	1	no	—
17	C₅₃×#340	—	2	1	no	—
18	C₂×C₄⋊Dic₅₃	—	2	2	no	—
19	C₂×C₂²⋊Dic₅₃	—	4	1	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	—	2	8	no	—
22	C₂×#11719908	C₂×C₄	1	2	yes	—
23	C₂×#11719908	C₂×C₄	1	2	yes	—
24	C₂×#11719908	D₄	1	4	yes	—
25	#13133151	—	2	2	no	—
26	#13133151	—	2	2	no	—
27	#13133124	—	4	1	no	—
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	—	2	2	no	—
30	C₂²×C₈×C₅₃	—	2	2	no	—
31	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
34	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₅₃	—	4	1	no	—
37	#107×C₅₃	—	2	2	no	—
38	#107×C₅₃	—	2	2	no	—
39	#95×C₅₃	—	4	1	no	—
40	C₂⁴×C₅₃	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	4	2	no	—
44	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×Dic₅₃	—	4	2	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	2	2	no	—
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	4	no	—
53	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₅₃	#41	1	2	yes	—
58	C₄⋊C₄×C₅₃	—	4	2	no	—
59	C₄⋊C₄×C₅₃	—	4	2	no	—
60	C₂²⋊C₄×C₅₃	—	8	2	no	—
61	C₂²⋊Dic₅₃	—	8	2	no	—
62	C₄⋊Dic₅₃	—	4	2	no	—
63	C₄⋊Dic₅₃	—	4	2	no	—
64	#11719908	—	2	2	no	—
65	#11719908	C₂²⋊C₄	1	4	yes	complemented
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	2	no	—
68	C₂³×C₅₃	#107	1	2	yes	—
69	C₂³×C₅₃	SD₃₂	1	2	yes	—
70	C₂³×C₅₃	SD₃₂	1	2	yes	—
71	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	—	2	2	no	—
73	C₂×C₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	4	no	—
76	C₂×C₄×C₅₃	—	8	2	no	—
77	C₂×C₄×C₅₃	—	8	2	no	—
78	C₂×C₄×C₅₃	—	8	2	no	—
79	C₂×C₄×C₅₃	#107	1	2	yes	—
80	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#121	1	2	yes	—
82	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#135	1	2	yes	—
84	C₂×Dic₅₃	—	8	2	no	—
85	C₂×Dic₅₃	—	8	2	no	—
86	C₂×Dic₅₃	—	8	2	no	—
87	C₈×C₅₃	—	2	2	no	—
88	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₈×C₅₃	#117	1	1	yes	characteristic
90	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
91	#12835	—	53	1	no	maximal, Sylow
92	C₂²×C₅₃	—	2	2	no	—
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	—	2	4	no	—
95	C₂²×C₅₃	—	2	4	no	—
96	C₂²×C₅₃	#367	1	4	yes	complemented
97	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
98	C₂²×C₅₃	#466	1	2	yes	—
99	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
100	C₂²×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
101	C₂²×C₅₃	#508	1	2	yes	—
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	16	2	no	—
104	C₄×C₅₃	#380	1	1	yes	characteristic
105	C₄×C₅₃	#427	1	1	yes	characteristic
106	C₄×C₅₃	#467	1	2	yes	—
107	Dic₅₃	—	16	2	no	—
108	C₂³×C₁₆	—	53	1	no	—
109	C₂×#367	—	53	2	no	—
110	#1808	—	53	1	no	—
111	#1808	D₅₃	1	1	yes	characteristic
112	#1380	—	53	2	no	—
113	C₂×C₅₃	—	2	4	no	—
114	C₂×C₅₃	#1808	1	1	yes	characteristic
115	C₂×C₅₃	C₂×#367	1	1	yes	characteristic
116	C₂×C₅₃	#2158	1	1	yes	characteristic
117	C₂×C₅₃	#2217	1	2	yes	—
118	C₂×C₅₃	#2217	1	2	yes	—
119	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
120	C₂²×C₁₆	—	53	1	no	—
121	C₂²×C₁₆	—	53	4	no	—
122	C₂²×C₁₆	—	53	1	no	—
123	C₂²×C₁₆	—	106	4	no	—
124	C₂×#107	—	53	2	no	—
125	C₂×#107	C₂×D₅₃	1	2	yes	—
126	#380	—	2	1	no	—
127	#380	—	106	1	no	—
128	#367	—	106	4	no	—
129	#340	—	2	1	no	—
130	#340	—	106	1	no	—
131	C₅₃	#12835	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	Dic₁₀₆	1	2	yes	—
136	C₂²×C₈	C₄×D₅₃	1	2	yes	—
137	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
138	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	106	2	no	—
141	C₂×C₂²⋊C₄	—	4	1	no	—
142	C₂×C₂²⋊C₄	—	212	1	no	—
143	C₂×C₁₆	—	53	2	no	—
144	C₂×C₁₆	—	53	4	no	—
145	C₂×C₁₆	—	53	2	no	—
146	C₂×C₁₆	—	106	8	no	—
147	C₂×C₁₆	—	106	2	no	—
148	#107	—	2	2	no	—
149	#107	—	2	2	no	—
150	#107	—	106	2	no	—
151	#107	—	106	2	no	—
152	#95	—	4	1	no	—
153	#95	—	212	1	no	—
154	C₂⁴	#11719922	1	1	yes	characteristic
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	212	2	no	—
159	C₂²×C₄	#11719919	1	2	yes	—
160	C₂²×C₄	#11719922	1	1	yes	characteristic
161	C₂²×C₄	#11719925	1	2	yes	—
162	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	#11719919	1	2	yes	—
168	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
169	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
170	C₂×C₈	D₄×D₅₃	1	2	yes	—
171	C₂×C₈	#11719947	1	2	yes	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	212	2	no	—
175	C₄⋊C₄	—	212	2	no	—
176	C₂²⋊C₄	—	8	2	no	—
177	C₂²⋊C₄	—	424	2	no	—
178	C₁₆	—	53	4	no	—
179	C₁₆	—	106	2	no	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	2	no	—
182	C₂³	#13133150	1	2	yes	—
183	C₂³	#13133160	1	2	yes	—
184	C₂³	#13133161	1	2	yes	—
185	C₂³	C₂×#11719922	1	1	yes	characteristic
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	2	no	—
193	C₂×C₄	—	424	2	no	—
194	C₂×C₄	—	424	2	no	—
195	C₂×C₄	—	424	2	no	—
196	C₂×C₄	#13133150	1	2	yes	—
197	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
198	C₂×C₄	#13133189	1	2	yes	—
199	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
200	C₂×C₄	#13133237	1	2	yes	—
201	C₈	—	2	2	no	—
202	C₈	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
203	C₈	#13133173	1	1	yes	characteristic
204	C₈	#13133194	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	4	no	—
208	C₂²	—	2	4	no	—
209	C₂²	#16810450	1	4	yes	complemented
210	C₂²	C₂×#13133150	1	1	yes	characteristic
211	C₂²	#16810857	1	2	yes	—
212	C₂²	C₂×#13133160	1	1	yes	characteristic
213	C₂²	C₂×#13133161	1	1	yes	characteristic
214	C₂²	#16811053	1	2	yes	—
215	C₄	—	2	2	no	—
216	C₄	—	16	2	no	—
217	C₄	—	848	2	no	—
218	C₄	#16810481	1	1	yes	characteristic
219	C₄	#16810659	1	1	yes	characteristic
220	C₄	#16810862	1	2	yes	—
221	C₂	—	2	4	no	—
222	C₂	#20274742	1	1	yes	characteristic
223	C₂	C₂×#16810450	1	1	yes	characteristic
224	C₂	#20276497	1	1	yes	characteristic
225	C₂	#20276809	1	2	yes	—
226	C₂	#20276813	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1808	C₂
socle	C₂³×C₅₃	C₂×D₈
prime radical	C₂⁴×C₅₃	D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1808	D₅₃
hypocenter	C₅₃	#12835
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1808

Complements

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

N	H ≅ G/N	type
2. C₂×#16810450	C₂	semidirect
4. #20272806	C₂	semidirect
7. C₂²×#11719908	C₄	semidirect
65. #11719908	C₂²⋊C₄	semidirect
96. C₂²×C₅₃	#367	semidirect
131. C₅₃	#12835	semidirect
209. C₂²	#16810450	semidirect

Maximal subgroups

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

subgroup	index
C₂³×#11719908	2
C₂×#16810450	2
#20274744	2
#20272806	2
C₅₃×#1808	2
#12835	53