Subgroups — #27299123

Order 13568.

590 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11719908	C₂	1	2	yes	maximal
2	C₂×#16810399	C₂	1	2	yes	maximal, complemented
3	#20272793	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	4	yes	—
6	C₂²⋊#11719908	C₄	1	8	yes	complemented
7	#16810399	—	2	4	no	—
8	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	—	2	4	no	—
12	C₂×C₄×C₈×C₅₃	C₄	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	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#11719908	—	2	4	no	—
17	C₂×#11719908	—	2	4	no	—
18	C₂×#11719908	C₂×C₄	1	8	yes	complemented
19	C₂³×C₄×C₅₃	C₈	1	2	yes	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	—	2	2	no	—
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	C₈	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	—	2	2	no	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	C₈	1	2	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	—	2	2	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	C₄×C₈×C₅₃	—	2	2	no	—
39	C₄×C₈×C₅₃	D₄	1	4	yes	—
40	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	—	2	4	no	—
42	C₂²×C₄×C₅₃	—	2	4	no	—
43	C₂²×C₄×C₅₃	—	2	4	no	—
44	C₂²×C₄×C₅₃	—	2	1	no	—
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	—	2	8	no	—
59	C₂×C₈×C₅₃	—	2	8	no	—
60	C₂×C₈×C₅₃	C₄²	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
63	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
64	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
65	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₅₃	#41	1	2	yes	—
68	C₄²×C₅₃	—	2	4	no	—
69	C₄²×C₅₃	—	2	1	no	—
70	C₄²×C₅₃	—	2	1	no	—
71	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₅₃	M₁₆	1	2	yes	—
73	#11719908	—	2	8	no	—
74	C₂³×C₅₃	—	2	1	no	—
75	C₂³×C₅₃	—	2	1	no	—
76	C₂³×C₅₃	—	2	2	no	—
77	C₂³×C₅₃	C₄×C₈	1	2	yes	—
78	C₂³×C₅₃	M₃₂	1	4	yes	—
79	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	16	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	2	2	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	2	no	—
91	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
92	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
93	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
94	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₈×C₅₃	—	2	2	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	C₂²⋊C₈	1	4	yes	—
103	C₈×C₅₃	C₄×D₄	1	4	yes	—
104	#12599	—	53	1	no	maximal, Sylow
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	4	no	—
107	C₂²×C₅₃	—	2	4	no	—
108	C₂²×C₅₃	—	2	2	no	—
109	C₂²×C₅₃	#346	1	4	yes	complemented
110	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
111	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
112	C₂²×C₅₃	#431	1	1	yes	characteristic
113	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
114	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
115	C₄×C₅₃	—	2	4	no	—
116	C₄×C₅₃	—	2	4	no	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
120	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄×C₅₃	#431	1	1	yes	characteristic
122	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
123	C₂×C₂²⋊C₁₆	—	53	2	no	—
124	C₂×#346	—	53	2	no	—
125	#1374	—	53	2	no	—
126	C₂×C₅₃	—	2	4	no	—
127	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
128	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
129	C₂×C₅₃	C₄×M₃₂	1	1	yes	characteristic
130	C₂×C₅₃	#2112	1	2	yes	—
131	C₂×C₅₃	#2163	1	2	yes	—
132	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
133	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
134	C₂²×C₄²	Dic₅₃	1	1	yes	characteristic
135	C₂²×C₁₆	—	53	4	no	—
136	C₂×C₄×C₈	—	2	4	no	—
137	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
138	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
141	C₂²⋊C₁₆	—	53	8	no	—
142	#346	—	106	4	no	—
143	C₅₃	#12599	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₄	#88322	1	2	yes	—
145	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
146	C₂²×C₈	—	2	2	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	#88322	1	4	yes	—
151	C₂²×C₈	C₄×D₅₃	1	4	yes	—
152	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
153	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	#88322	1	2	yes	—
159	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
160	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
161	C₂×C₁₆	—	53	8	no	—
162	C₂×C₁₆	—	106	4	no	—
163	C₂×C₁₆	—	106	4	no	—
164	C₄×C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	C₂²⋊D₅₃	1	4	yes	—
168	C₂⁴	C₂×#88322	1	1	yes	characteristic
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
177	C₂²×C₄	C₂×#88322	1	2	yes	—
178	C₂²×C₄	C₂×#88322	1	2	yes	—
179	C₂²×C₄	C₂×#88322	1	2	yes	—
180	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	8	no	—
187	C₂×C₈	—	2	8	no	—
188	C₂×C₈	C₂×#88322	1	2	yes	—
189	C₂×C₈	C₂×#88322	1	2	yes	—
190	C₂×C₈	#11719917	1	2	yes	—
191	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
192	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
193	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
194	C₂×C₈	#11719945	1	2	yes	—
195	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	#11719917	1	2	yes	—
200	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
201	C₁₆	—	106	8	no	—
202	C₂³	—	2	2	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	C₄×#88322	1	2	yes	—
206	C₂³	#13133157	1	4	yes	—
207	C₂³	C₂²×#88322	1	1	yes	characteristic
208	C₂×C₄	—	2	16	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	C₄×#88322	1	2	yes	—
220	C₂×C₄	C₂²⋊#88322	1	4	yes	—
221	C₂×C₄	#13133157	1	4	yes	—
222	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
224	C₂×C₄	#13133176	1	1	yes	characteristic
225	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
226	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
227	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	C₂²⋊#88322	1	4	yes	—
231	C₈	C₄×C₂²⋊D₅₃	1	4	yes	—
232	C₂²	—	2	4	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	4	no	—
236	C₂²	#16810399	1	4	yes	complemented
237	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
238	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
239	C₂²	#16810676	1	1	yes	characteristic
240	C₂²	C₂×#13133157	1	2	yes	—
241	C₂²	C₂×#13133157	1	2	yes	—
242	C₄	—	2	2	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	4	no	—
245	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
246	C₄	C₄×#11719917	1	1	yes	characteristic
247	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
248	C₄	#16810676	1	1	yes	characteristic
249	C₂	—	2	4	no	—
250	C₂	C₄×C₂²⋊#88322	1	1	yes	characteristic
251	C₂	C₂×#16810399	1	1	yes	characteristic
252	C₂	C₄×#13133157	1	1	yes	characteristic
253	C₂	#20276326	1	2	yes	—
254	C₂	#20276518	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	#12599
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16810399	C₂	semidirect
3. #20272793	C₂	semidirect
6. C₂²⋊#11719908	C₄	semidirect
18. C₂×#11719908	C₂×C₄	semidirect
109. C₂²×C₅₃	#346	semidirect
143. C₅₃	#12599	semidirect
236. C₂²	#16810399	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#11719908	2
C₂×#16810399	2
#20272793	2
C₂²×C₄×C₈×C₅₃	2
#12599	53