Subgroups — #27275223

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₄²×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810344	C₂	1	2	yes	maximal
4	#20272542	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	C₄	1	2	yes	—
7	C₂×C₄×#88322	C₄	1	2	yes	—
8	C₂×C₄×#88322	C₄	1	2	yes	—
9	C₂×C₄×#88322	C₂²	1	2	yes	—
10	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
12	#16810378	C₄	1	2	yes	—
13	#16810344	—	2	4	no	—
14	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
16	C₄³×C₅₃	C₂²	1	1	yes	characteristic
17	C₅₃×#336	C₄	1	2	yes	—
18	C₂²×#88322	C₂×C₄	1	4	yes	—
19	C₂²×#88322	C₂×C₄	1	2	yes	—
20	C₂²×#88322	C₂×C₄	1	2	yes	—
21	C₄⋊#88322	—	2	4	no	—
22	C₄×#88322	—	2	4	no	—
23	C₄×#88322	—	2	2	no	—
24	C₄×#88322	—	2	1	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	2	no	—
27	C₄×#88322	—	2	4	no	—
28	C₄×#88322	D₄	1	2	yes	—
29	C₄×#88322	Q₈	1	2	yes	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₈×C₅₃	C₈	1	8	yes	complemented
39	C₄×C₈×C₅₃	—	2	2	no	—
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	—	2	4	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	—
51	C₄²×C₅₃	—	2	2	no	—
52	C₄²×C₅₃	—	2	2	no	—
53	C₄²×C₅₃	—	2	2	no	—
54	C₄²×C₅₃	C₂×C₈	1	4	yes	—
55	C₄²×C₅₃	M₁₆	1	4	yes	—
56	C₄²×C₅₃	M₁₆	1	4	yes	—
57	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₅₃	#41	1	1	yes	characteristic
60	C₄²×C₅₃	#41	1	1	yes	characteristic
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	—	2	2	no	—
65	C₂×#88322	—	2	4	no	—
66	C₂×#88322	—	2	4	no	—
67	C₂×#88322	—	2	4	no	—
68	C₂×#88322	C₄⋊C₄	1	4	yes	—
69	C₂×#88322	M₁₆	1	4	yes	—
70	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
78	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
79	C₂×C₄×C₅₃	#97	1	2	yes	—
80	C₂×C₄×C₅₃	#97	1	2	yes	—
81	C₂×C₄×C₅₃	#97	1	2	yes	—
82	C₂×C₄×C₅₃	#97	1	2	yes	—
83	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₅₃	#131	1	2	yes	—
94	C₂×C₄×C₅₃	#131	1	2	yes	—
95	C₈×C₅₃	—	2	8	no	—
96	#88322	—	2	4	no	—
97	#88322	—	2	4	no	—
98	#88322	—	4	4	no	—
99	#8261	—	53	1	no	maximal, Sylow
100	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
103	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
104	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
105	C₂²×C₅₃	#405	1	1	yes	characteristic
106	C₂²×C₅₃	#405	1	1	yes	characteristic
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	2	2	no	—
109	C₄×C₅₃	—	2	2	no	—
110	C₄×C₅₃	#322	1	4	yes	complemented
111	C₄×C₅₃	#423	1	2	yes	—
112	C₄×C₅₃	#432	1	2	yes	—
113	C₄×C₅₃	C₈×D₄	1	2	yes	—
114	C₄×C₅₃	#436	1	2	yes	—
115	C₄×C₅₃	C₈×Q₈	1	2	yes	—
116	C₄×C₅₃	#446	1	2	yes	—
117	C₄×C₄⋊C₈	—	53	1	no	—
118	C₄×C₄⋊C₈	D₅₃	1	1	yes	characteristic
119	C₄²×C₈	—	53	1	no	—
120	C₂×#322	—	53	2	no	—
121	#1273	—	53	2	no	—
122	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
123	C₂×C₅₃	C₈×M₁₆	1	1	yes	characteristic
124	C₂×C₅₃	#1451	1	1	yes	characteristic
125	C₂×C₅₃	#1452	1	1	yes	characteristic
126	C₂×C₅₃	#1767	1	1	yes	characteristic
127	C₂×C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₅₃	#1770	1	1	yes	characteristic
129	C₂×C₄⋊C₈	—	53	2	no	—
130	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
131	C₂×C₄×C₈	—	53	1	no	—
132	C₂×C₄×C₈	—	53	2	no	—
133	C₂×C₄×C₈	—	53	2	no	—
134	C₂×C₄×C₈	—	53	2	no	—
135	C₂×C₄×C₈	—	53	1	no	—
136	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
137	C₄³	C₂×D₅₃	1	1	yes	characteristic
138	#336	—	53	2	no	—
139	#336	Dic₅₃	1	2	yes	—
140	#322	—	106	4	no	—
141	C₅₃	#8261	1	1	yes	characteristic, complemented, Sylow
142	C₂²×C₈	—	53	2	no	—
143	C₂²×C₈	—	53	2	no	—
144	C₂²×C₈	—	53	4	no	—
145	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
146	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
147	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
148	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
149	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
150	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
152	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
153	C₄⋊C₈	—	106	4	no	—
154	C₄⋊C₈	#88322	1	8	yes	complemented
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	53	2	no	—
158	C₄×C₈	—	53	2	no	—
159	C₄×C₈	—	106	4	no	—
160	C₄×C₈	—	106	2	no	—
161	C₄×C₈	—	106	1	no	—
162	C₄×C₈	—	106	1	no	—
163	C₄×C₈	—	106	2	no	—
164	C₄×C₈	—	106	4	no	—
165	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
167	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
168	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
171	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	53	4	no	—
175	C₂×C₈	—	53	4	no	—
176	C₂×C₈	—	106	2	no	—
177	C₂×C₈	—	106	2	no	—
178	C₂×C₈	—	106	2	no	—
179	C₂×C₈	—	106	2	no	—
180	C₂×C₈	—	106	4	no	—
181	C₂×C₈	—	106	4	no	—
182	C₂×C₈	—	106	4	no	—
183	C₂×C₈	C₂×#88322	1	8	yes	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	2	no	—
187	C₄²	#11719912	1	4	yes	—
188	C₄²	#11719912	1	4	yes	—
189	C₄²	C₂×#88322	1	4	yes	—
190	C₄²	D₄×D₅₃	1	1	yes	characteristic
191	C₄²	#11719947	1	1	yes	characteristic
192	C₄²	Q₈×D₅₃	1	1	yes	characteristic
193	C₄²	#11719949	1	1	yes	characteristic
194	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	C₄×#88322	1	2	yes	—
202	C₂×C₄	C₄×#88322	1	2	yes	—
203	C₂×C₄	#13133127	1	2	yes	—
204	C₂×C₄	#13133127	1	2	yes	—
205	C₂×C₄	#13133127	1	2	yes	—
206	C₂×C₄	#13133127	1	2	yes	—
207	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
208	C₂×C₄	#13133175	1	1	yes	characteristic
209	C₂×C₄	#13133179	1	1	yes	characteristic
210	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
211	C₂×C₄	#13133182	1	1	yes	characteristic
212	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	C₂²×#88322	1	2	yes	—
214	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
215	C₂×C₄	C₂×#11719912	1	2	yes	—
216	C₂×C₄	C₂×#11719912	1	2	yes	—
217	C₂×C₄	#13133224	1	2	yes	—
218	C₂×C₄	#13133225	1	2	yes	—
219	C₈	—	2	8	no	—
220	C₈	—	106	4	no	—
221	C₈	—	106	4	no	—
222	C₈	—	212	4	no	—
223	C₂²	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
224	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
225	C₂²	C₂×#13133127	1	1	yes	characteristic
226	C₂²	C₂×#13133127	1	1	yes	characteristic
227	C₂²	M₁₆×Dic₅₃	1	1	yes	characteristic
228	C₂²	#16810579	1	1	yes	characteristic
229	C₂²	#16810581	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	#16810344	1	4	yes	complemented
234	C₄	#16810645	1	2	yes	—
235	C₄	#16810679	1	2	yes	—
236	C₄	D₄×#88322	1	2	yes	—
237	C₄	#16810700	1	2	yes	—
238	C₄	Q₈×#88322	1	2	yes	—
239	C₄	#16810750	1	2	yes	—
240	C₂	C₂×#16810344	1	1	yes	characteristic
241	C₂	M₁₆×#88322	1	1	yes	characteristic
242	C₂	#20273005	1	1	yes	characteristic
243	C₂	#20273008	1	1	yes	characteristic
244	C₂	#20274570	1	1	yes	characteristic
245	C₂	C₄⋊C₄×#88322	1	1	yes	characteristic
246	C₂	#20274585	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₄×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#8261
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
38. C₄⋊C₈×C₅₃	C₈	semidirect
110. C₄×C₅₃	#322	semidirect
141. C₅₃	#8261	semidirect
154. C₄⋊C₈	#88322	semidirect
233. C₄	#16810344	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#88322	2
C₄²×#88322	2
C₂×#16810344	2
#20272542	2
C₄×C₄⋊C₈×C₅₃	2
#8261	53