Subgroups — #27300324

Order 13568.

430 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810405	C₂	1	2	yes	maximal
2	#20272797	C₂	1	2	yes	maximal, complemented
3	#20272795	C₂	1	2	yes	maximal, complemented
4	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	4	yes	—
6	#16810405	—	2	4	no	—
7	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
10	C₅₃×#405	—	2	4	no	—
11	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	#13133157	—	2	4	no	—
16	#13133157	—	2	4	no	—
17	C₂×#11719908	—	2	4	no	—
18	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₅₃	—	2	2	no	—
20	C₂×M₁₆×C₅₃	—	2	2	no	—
21	C₂×M₁₆×C₅₃	D₄	1	4	yes	—
22	C₂²×C₈×C₅₃	—	2	2	no	—
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	—	2	2	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	#117×C₅₃	—	2	2	no	—
30	#117×C₅₃	C₈	1	4	yes	—
31	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#97×C₅₃	—	2	2	no	—
36	#97×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	—	2	1	no	—
41	C₂²×C₄×C₅₃	—	2	1	no	—
42	C₂²×C₄×C₅₃	—	2	1	no	—
43	C₂²×C₄×C₅₃	—	2	1	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₅₃	—	2	8	no	—
51	C₂×C₈×C₅₃	—	2	8	no	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	—	2	1	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
60	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
61	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₅₃	#41	1	2	yes	—
63	C₂×C₈×C₅₃	#41	1	2	yes	—
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₂²⋊C₄×C₅₃	—	2	4	no	—
66	C₄²×C₅₃	—	2	1	no	—
67	C₄²×C₅₃	—	2	1	no	—
68	C₄²×C₅₃	C₂×C₈	1	2	yes	—
69	C₄²×C₅₃	C₂×C₈	1	2	yes	—
70	#11719908	—	4	4	no	—
71	C₂³×C₅₃	—	2	1	no	—
72	C₂³×C₅₃	—	2	1	no	—
73	C₂³×C₅₃	—	4	2	no	—
74	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
75	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
93	C₈×C₅₃	—	2	2	no	—
94	C₈×C₅₃	—	2	2	no	—
95	C₈×C₅₃	—	2	2	no	—
96	C₈×C₅₃	#123	1	4	yes	—
97	#12850	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	—	2	2	no	—
99	C₂²×C₅₃	—	2	2	no	—
100	C₂²×C₅₃	—	4	2	no	—
101	C₂²×C₅₃	—	4	2	no	—
102	C₂²×C₅₃	—	4	2	no	—
103	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₂²×C₅₃	#432	1	1	yes	characteristic
105	C₂²×C₅₃	#432	1	1	yes	characteristic
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	4	2	no	—
108	C₄×C₅₃	—	4	2	no	—
109	C₄×C₅₃	#381	1	1	yes	characteristic
110	C₄×C₅₃	#407	1	1	yes	characteristic
111	C₄×C₅₃	#431	1	1	yes	characteristic
112	C₄×C₅₃	#431	1	1	yes	characteristic
113	C₂×#405	D₅₃	1	1	yes	characteristic
114	C₂×#349	—	53	2	no	—
115	#1376	—	53	2	no	—
116	#1375	—	53	2	no	—
117	C₂×C₅₃	—	2	1	no	—
118	C₂×C₅₃	—	2	1	no	—
119	C₂×C₅₃	—	4	2	no	—
120	C₂×C₅₃	#1817	1	1	yes	characteristic
121	C₂×C₅₃	#2166	1	2	yes	—
122	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
123	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
124	C₂×#117	Dic₅₃	1	1	yes	characteristic
125	C₂×M₃₂	—	53	4	no	—
126	#405	—	2	4	no	—
127	C₂×#97	Dic₅₃	1	1	yes	characteristic
128	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
129	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
130	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
131	#349	—	106	4	no	—
132	C₅₃	#12850	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
134	C₂×M₁₆	—	2	2	no	—
135	C₂×M₁₆	—	2	2	no	—
136	C₂×M₁₆	C₂²⋊D₅₃	1	4	yes	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
141	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
142	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
143	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
144	#117	—	2	2	no	—
145	#117	#88322	1	4	yes	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₂²⋊C₄	—	2	1	no	—
148	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
149	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
150	M₃₂	—	106	4	no	—
151	M₃₂	—	106	4	no	—
152	C₂×C₁₆	—	106	4	no	—
153	#97	—	2	2	no	—
154	#97	—	2	2	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₂⁴	#11719917	1	1	yes	characteristic
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	C₂×#88322	1	2	yes	—
165	C₂²×C₄	#11719917	1	1	yes	characteristic
166	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
167	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
168	M₁₆	—	2	8	no	—
169	C₂×C₈	—	2	8	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	#11719917	1	2	yes	—
177	C₂×C₈	#11719917	1	2	yes	—
178	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
179	C₂×C₈	#11719945	1	2	yes	—
180	C₂×C₈	#11719945	1	2	yes	—
181	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
182	C₄⋊C₄	—	2	4	no	—
183	C₂²⋊C₄	—	2	4	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	C₂×#88322	1	2	yes	—
187	C₄²	C₂×#88322	1	2	yes	—
188	C₁₆	—	212	4	no	—
189	C₂³	—	2	1	no	—
190	C₂³	—	2	1	no	—
191	C₂³	—	4	2	no	—
192	C₂³	C₂²⋊#88322	1	2	yes	—
193	C₂³	C₂×#11719917	1	1	yes	characteristic
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	C₂²⋊#88322	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
206	C₂×C₄	#13133176	1	1	yes	characteristic
207	C₂×C₄	#13133176	1	1	yes	characteristic
208	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
209	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
210	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₈	—	2	2	no	—
214	C₈	#13133200	1	4	yes	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
221	C₂²	#16810680	1	1	yes	characteristic
222	C₂²	#16810680	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	4	2	no	—
225	C₄	—	4	2	no	—
226	C₄	#16810486	1	1	yes	characteristic
227	C₄	#16810587	1	1	yes	characteristic
228	C₄	#16810676	1	1	yes	characteristic
229	C₄	#16810676	1	1	yes	characteristic
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	—	4	2	no	—
233	C₂	#20274783	1	1	yes	characteristic
234	C₂	#20276531	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	#432
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#12850
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #20272797	C₂	semidirect
3. #20272795	C₂	semidirect
132. C₅₃	#12850	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810405	2
#20272797	2
#20272795	2
C₂×C₅₃×#405	2
#12850	53