Subgroups — #27302252

Order 13568.

522 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	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊#11719908	C₂	1	4	yes	maximal, complemented
3	C₂×#16810399	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂²×#11719908	C₂²	1	2	yes	—
7	C₄⋊#11719908	—	2	2	no	—
8	C₂²⋊#11719908	—	2	8	no	—
9	#16810399	—	2	2	no	—
10	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂×C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
12	C₈×D₄×C₅₃	—	2	2	no	—
13	C₈×D₄×C₅₃	—	2	2	no	—
14	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#11719908	—	2	2	no	—
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	—	2	4	no	—
20	C₂×#11719908	—	2	2	no	—
21	C₂×#11719908	D₄	1	4	yes	complemented
22	C₂²×D₄×C₅₃	C₈	1	1	yes	characteristic
23	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	—	2	4	no	—
25	C₂²×C₈×C₅₃	—	2	2	no	—
26	C₂²×C₈×C₅₃	—	2	2	no	—
27	C₂²×C₈×C₅₃	—	2	4	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
31	C₄×D₄×C₅₃	—	2	2	no	—
32	C₄×D₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	C₂²⋊C₈×C₅₃	—	2	4	no	—
38	C₄×C₈×C₅₃	D₄	1	2	yes	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	C₂⁴×C₅₃	C₂×C₈	1	2	yes	—
41	C₂×D₄×C₅₃	—	2	4	no	—
42	C₂×D₄×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	—	2	2	no	—
44	C₂²×C₄×C₅₃	—	2	4	no	—
45	C₂²×C₄×C₅₃	—	2	2	no	—
46	C₂²×C₄×C₅₃	—	2	4	no	—
47	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₅₃	—	2	2	no	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	2	4	no	—
54	C₂×C₈×C₅₃	—	4	4	no	—
55	C₂×C₈×C₅₃	—	4	2	no	—
56	C₂×C₈×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	#41	1	2	yes	—
62	C₄⋊C₄×C₅₃	—	2	2	no	—
63	C₂²⋊C₄×C₅₃	—	2	4	no	—
64	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₅₃	M₁₆	1	2	yes	—
66	#11719908	—	2	4	no	—
67	#11719908	—	4	2	no	—
68	C₂³×C₅₃	—	2	2	no	—
69	C₂³×C₅₃	—	2	4	no	—
70	C₂³×C₅₃	—	2	2	no	—
71	C₂³×C₅₃	—	2	4	no	—
72	C₂³×C₅₃	M₃₂	1	4	yes	—
73	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
74	D₄×C₅₃	—	2	8	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	2	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	4	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	4	4	no	—
82	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
83	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
84	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#131	1	2	yes	—
88	C₈×C₅₃	—	2	2	no	—
89	C₈×C₅₃	—	4	2	no	—
90	C₈×C₅₃	#121	1	2	yes	—
91	C₈×C₅₃	#124	1	1	yes	characteristic
92	C₈×C₅₃	#127	1	1	yes	characteristic
93	#13190	—	53	1	no	maximal, Sylow
94	C₂²×C₅₃	—	2	4	no	—
95	C₂²×C₅₃	—	2	8	no	—
96	C₂²×C₅₃	—	4	2	no	—
97	C₂²×C₅₃	—	4	4	no	—
98	C₂²×C₅₃	—	4	2	no	—
99	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
100	C₂²×C₅₃	C₈×D₄	1	2	yes	—
101	C₂²×C₅₃	C₂×M₃₂	1	2	yes	—
102	C₂²×C₅₃	#504	1	2	yes	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	—	2	2	no	—
105	C₄×C₅₃	—	4	2	no	—
106	C₄×C₅₃	#390	1	1	yes	characteristic
107	C₄×C₅₃	#408	1	1	yes	characteristic
108	C₄×C₅₃	#436	1	2	yes	—
109	C₂×C₈×D₄	D₅₃	1	1	yes	characteristic
110	C₂×C₄⋊C₁₆	—	53	1	no	—
111	C₂×C₂²⋊C₁₆	—	53	4	no	—
112	C₂×#346	—	53	1	no	—
113	C₂×C₅₃	—	4	4	no	—
114	C₂×C₅₃	#1954	1	1	yes	characteristic
115	C₂×C₅₃	#2112	1	1	yes	characteristic
116	C₂×C₅₃	#2113	1	1	yes	characteristic
117	C₂×C₅₃	#2168	1	4	yes	—
118	C₂³×C₈	C₂×D₅₃	1	2	yes	—
119	C₂×C₄×D₄	Dic₅₃	1	1	yes	characteristic
120	C₂²×C₁₆	—	53	2	no	—
121	C₂²×C₁₆	—	53	2	no	—
122	C₈×D₄	—	2	2	no	—
123	C₈×D₄	—	2	2	no	—
124	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
125	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
126	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
127	C₄⋊C₁₆	—	106	2	no	—
128	C₂²⋊C₁₆	—	106	8	no	—
129	#346	—	106	2	no	—
130	C₅₃	#13190	1	1	yes	characteristic, complemented, Sylow
131	C₂²×D₄	#88322	1	1	yes	characteristic
132	C₂³×C₄	C₂×Dic₅₃	1	2	yes	—
133	C₂²×C₈	—	2	4	no	—
134	C₂²×C₈	—	2	4	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
138	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
140	C₄×D₄	—	2	2	no	—
141	C₄×D₄	—	2	2	no	—
142	C₂×C₄⋊C₄	#88322	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	#88322	1	2	yes	—
144	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
145	C₂×C₁₆	—	53	4	no	—
146	C₂×C₁₆	—	106	2	no	—
147	C₂×C₁₆	—	106	2	no	—
148	C₂×C₁₆	—	106	4	no	—
149	C₂×C₁₆	—	106	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₂²⋊C₈	—	2	4	no	—
152	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
153	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
154	C₂⁴	C₂×#88322	1	2	yes	—
155	C₂×D₄	—	2	4	no	—
156	C₂×D₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	C₂×#88322	1	2	yes	—
162	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
163	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
164	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	4	4	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
172	C₂×C₈	D₄×D₅₃	1	2	yes	—
173	C₂×C₈	#11719947	1	2	yes	—
174	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
175	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
176	C₄⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	4	no	—
178	C₄²	#11719917	1	2	yes	—
179	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
180	C₁₆	—	106	4	no	—
181	C₁₆	—	212	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	4	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	4	no	—
186	C₂³	#13133157	1	4	yes	—
187	C₂³	C₂²×#88322	1	1	yes	characteristic
188	D₄	—	2	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂²⋊#88322	1	2	yes	—
197	C₂×C₄	C₂²⋊#88322	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
199	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
200	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
201	C₂×C₄	#13133224	1	2	yes	—
202	C₈	—	2	2	no	—
203	C₈	—	4	2	no	—
204	C₈	#13133188	1	2	yes	—
205	C₈	#13133204	1	1	yes	characteristic
206	C₈	#13133213	1	1	yes	characteristic
207	C₂²	—	2	8	no	—
208	C₂²	—	2	4	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
213	C₂²	D₄×#88322	1	2	yes	—
214	C₂²	C₂×#13133157	1	2	yes	—
215	C₂²	#16811040	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	4	2	no	—
219	C₄	#16810526	1	1	yes	characteristic
220	C₄	#16810590	1	1	yes	characteristic
221	C₄	#16810701	1	2	yes	—
222	C₂	—	4	4	no	—
223	C₂	#20275510	1	1	yes	characteristic
224	C₂	#20276326	1	1	yes	characteristic
225	C₂	#20276329	1	1	yes	characteristic
226	C₂	#20276541	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₈
Fitting	C₂×C₈×D₄×C₅₃	C₂
prime radical	C₂²×D₄×C₅₃	C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₈×D₄	D₅₃
hypocenter	C₅₃	#13190
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#11719908	C₂	semidirect
2. C₂×C₂²⋊#11719908	C₂	semidirect
3. C₂×#16810399	C₂	semidirect
21. C₂×#11719908	D₄	semidirect
130. C₅₃	#13190	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
C₂×C₂²⋊#11719908	2
C₂×#16810399	2
C₂×C₈×D₄×C₅₃	2
#13190	53