Subgroups — #27274570

Order 13568.

642 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	C₄×C₂²⋊#88322	C₂	1	2	yes	maximal
2	C₂×#16810344	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272542	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂²⋊#88322	C₄	1	4	yes	—
7	C₂×C₄×#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	2	yes	—
9	#16810378	C₄	1	4	yes	—
10	#16810344	—	2	2	no	—
11	C₈×#88322	—	2	2	no	—
12	C₂²×M₁₆×C₅₃	C₄	1	2	yes	—
13	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×M₁₆×C₅₃	—	2	2	no	—
15	C₄×M₁₆×C₅₃	—	2	2	no	—
16	C₂×#97×C₅₃	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×#88322	C₂×C₄	1	4	yes	—
20	C₂²×#88322	C₂×C₄	1	4	yes	—
21	C₂²⋊#88322	—	2	8	no	—
22	C₄×#88322	—	2	2	no	—
23	C₄×#88322	—	2	2	no	—
24	C₄×#88322	—	2	2	no	—
25	C₄×#88322	—	2	2	no	—
26	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₅₃	—	2	4	no	—
29	C₂×M₁₆×C₅₃	—	2	4	no	—
30	C₂×M₁₆×C₅₃	C₈	1	8	yes	complemented
31	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	—	2	1	no	—
34	C₂×C₄²×C₅₃	—	2	1	no	—
35	C₂×C₄²×C₅₃	—	2	2	no	—
36	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	#97×C₅₃	—	2	4	no	—
40	C₄×C₈×C₅₃	—	2	1	no	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	D₄	1	2	yes	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	4	no	—
51	C₂²×C₄×C₅₃	—	2	4	no	—
52	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
53	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
57	M₁₆×C₅₃	—	2	16	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
61	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
62	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
63	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
64	C₄²×C₅₃	—	2	1	no	—
65	C₄²×C₅₃	—	2	1	no	—
66	C₄²×C₅₃	—	2	2	no	—
67	C₄²×C₅₃	—	2	2	no	—
68	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₅₃	#41	1	2	yes	—
71	C₂×#88322	—	2	4	no	—
72	C₂×#88322	—	2	4	no	—
73	C₂×#88322	—	2	4	no	—
74	C₂×#88322	—	2	4	no	—
75	C₂×#88322	—	2	4	no	—
76	C₂×#88322	M₁₆	1	8	yes	complemented
77	C₂³×C₅₃	—	2	1	no	—
78	C₂³×C₅₃	—	2	2	no	—
79	C₂³×C₅₃	—	2	1	no	—
80	C₂³×C₅₃	—	2	2	no	—
81	C₂³×C₅₃	C₄×C₈	1	2	yes	—
82	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₅₃	—	2	1	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	2	4	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	—	4	4	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
93	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
99	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₅₃	#131	1	4	yes	—
101	C₈×C₅₃	—	2	4	no	—
102	C₈×C₅₃	C₂²⋊C₈	1	8	yes	complemented
103	#88322	—	2	8	no	—
104	#88322	—	4	4	no	—
105	#8146	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	4	no	—
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	—	4	1	no	—
109	C₂²×C₅₃	—	4	1	no	—
110	C₂²×C₅₃	—	4	2	no	—
111	C₂²×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
113	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
114	C₂²×C₅₃	C₄×M₁₆	1	1	yes	characteristic
115	C₂²×C₅₃	#405	1	2	yes	—
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	—	4	2	no	—
118	C₄×C₅₃	C₂×C₂²⋊C₈	1	2	yes	—
119	C₄×C₅₃	#431	1	2	yes	—
120	C₄×C₅₃	#435	1	4	yes	—
121	C₄×C₅₃	#436	1	4	yes	—
122	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
123	C₄×C₂²⋊C₈	—	53	2	no	—
124	C₂×#322	—	53	1	no	—
125	C₂×C₈²	—	53	1	no	—
126	#1273	—	53	2	no	—
127	C₂×C₅₃	—	4	2	no	—
128	C₂×C₅₃	C₈×M₁₆	1	2	yes	—
129	C₂×C₅₃	#1453	1	1	yes	characteristic
130	C₂×C₅₃	#1455	1	1	yes	characteristic
131	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
132	C₂×C₅₃	#1753	1	2	yes	—
133	C₂²×M₁₆	Dic₅₃	1	2	yes	—
134	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
135	C₂×C₂²⋊C₈	—	53	4	no	—
136	C₄×M₁₆	—	2	2	no	—
137	C₄×M₁₆	—	2	2	no	—
138	C₂×#97	Dic₅₃	1	2	yes	—
139	C₂×C₄×C₈	—	53	2	no	—
140	C₂×C₄×C₈	—	53	2	no	—
141	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
143	#336	—	53	4	no	—
144	#322	—	106	2	no	—
145	C₈²	—	106	2	no	—
146	C₅₃	#8146	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	C₄×D₅₃	1	2	yes	—
148	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
149	C₂×M₁₆	—	2	4	no	—
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	#88322	1	8	yes	complemented
152	C₂²×C₈	—	53	4	no	—
153	C₂²×C₈	—	53	4	no	—
154	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
155	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	—	2	1	no	—
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	C₄×D₅₃	1	2	yes	—
160	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
162	C₂²⋊C₈	—	106	8	no	—
163	#97	—	2	4	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	106	2	no	—
167	C₄×C₈	—	106	2	no	—
168	C₄×C₈	—	106	2	no	—
169	C₄×C₈	—	106	2	no	—
170	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
171	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
172	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	C₂×#88322	1	4	yes	—
181	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
182	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
185	M₁₆	—	2	16	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	53	8	no	—
189	C₂×C₈	—	106	4	no	—
190	C₂×C₈	—	106	4	no	—
191	C₂×C₈	—	106	4	no	—
192	C₂×C₈	—	106	4	no	—
193	C₂×C₈	—	106	4	no	—
194	C₂×C₈	C₂×#88322	1	4	yes	—
195	C₂×C₈	C₂×#88322	1	4	yes	—
196	C₂×C₈	#11719917	1	4	yes	—
197	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	#11719945	1	2	yes	—
203	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
205	C₂³	—	2	2	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	1	no	—
209	C₂³	C₄×#88322	1	2	yes	—
210	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	C₄×#88322	1	2	yes	—
221	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
222	C₂×C₄	#13133176	1	1	yes	characteristic
223	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
224	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
225	C₂×C₄	C₂²×#88322	1	2	yes	—
226	C₂×C₄	M₁₆×D₅₃	1	4	yes	—
227	C₂×C₄	C₂×#11719917	1	2	yes	—
228	C₂×C₄	#13133225	1	4	yes	—
229	C₈	—	2	4	no	—
230	C₈	—	106	8	no	—
231	C₈	—	212	4	no	—
232	C₈	C₂²⋊#88322	1	8	yes	complemented
233	C₂²	—	2	2	no	—
234	C₂²	—	2	4	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
239	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
240	C₂²	M₁₆×Dic₅₃	1	2	yes	—
241	C₂²	C₄×#11719917	1	1	yes	characteristic
242	C₂²	#16810581	1	2	yes	—
243	C₄	—	2	2	no	—
244	C₄	—	4	2	no	—
245	C₄	C₂×C₂²⋊#88322	1	2	yes	—
246	C₄	#16810676	1	2	yes	—
247	C₄	#16810692	1	4	yes	—
248	C₄	#16810699	1	4	yes	—
249	C₂	—	4	2	no	—
250	C₂	M₁₆×#88322	1	2	yes	—
251	C₂	#20273012	1	1	yes	characteristic
252	C₂	#20273018	1	1	yes	characteristic
253	C₂	C₄×C₂²⋊#88322	1	1	yes	characteristic
254	C₂	#20274496	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	#8146
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16810344	C₂	semidirect
3. C₂×C₈×#88322	C₂	semidirect
4. #20272542	C₂	semidirect
30. C₂×M₁₆×C₅₃	C₈	semidirect
76. C₂×#88322	M₁₆	semidirect
102. C₈×C₅₃	C₂²⋊C₈	semidirect
146. C₅₃	#8146	semidirect
151. C₂×M₁₆	#88322	semidirect
232. C₈	C₂²⋊#88322	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊#88322	2
C₂×#16810344	2
C₂×C₈×#88322	2
#20272542	2
C₂×C₄×M₁₆×C₅₃	2
#8146	53