Subgroups — #27299136

Order 13568.

490 conjugacy classes of subgroups.

The subgroup lattice has 254 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	C₄×#13133157	C₂	1	2	yes	maximal, complemented
3	#20272799	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	C₄	1	8	yes	complemented
7	#16810399	—	2	2	no	—
8	C₄×#11719908	—	2	2	no	—
9	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
12	C₅₃×#405	—	2	2	no	—
13	C₅₃×#405	—	2	2	no	—
14	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
15	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	#13133157	—	2	4	no	—
19	#13133157	C₂×C₄	1	8	yes	—
20	C₂×#11719908	—	2	4	no	—
21	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₅₃	—	2	2	no	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₅₃	—	2	1	no	—
26	C₂²×C₈×C₅₃	—	2	1	no	—
27	C₂²×C₈×C₅₃	—	2	2	no	—
28	C₂²×C₈×C₅₃	C₈	1	4	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	#117×C₅₃	—	2	2	no	—
34	#117×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
36	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
37	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
39	#97×C₅₃	—	2	2	no	—
40	#97×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	—	2	1	no	—
42	C₄×C₈×C₅₃	—	2	1	no	—
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	—	2	1	no	—
47	C₂²×C₄×C₅₃	—	2	1	no	—
48	C₂²×C₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
51	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₅₃	—	2	8	no	—
57	C₂×C₈×C₅₃	—	2	1	no	—
58	C₂×C₈×C₅₃	—	2	1	no	—
59	C₂×C₈×C₅₃	—	2	1	no	—
60	C₂×C₈×C₅₃	—	2	1	no	—
61	C₂×C₈×C₅₃	—	2	4	no	—
62	C₂×C₈×C₅₃	—	2	4	no	—
63	C₂×C₈×C₅₃	C₄²	1	2	yes	—
64	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
69	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₅₃	#41	1	2	yes	—
72	C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₂²⋊C₄×C₅₃	—	2	4	no	—
74	C₄²×C₅₃	—	2	1	no	—
75	C₄²×C₅₃	—	2	1	no	—
76	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₅₃	M₁₆	1	2	yes	—
78	#11719908	—	2	8	no	—
79	C₂³×C₅₃	—	2	1	no	—
80	C₂³×C₅₃	—	2	1	no	—
81	C₂³×C₅₃	—	4	2	no	—
82	C₂³×C₅₃	C₄×C₈	1	2	yes	—
83	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	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	—	4	2	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
94	C₂×C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
95	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	—	2	2	no	—
103	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
104	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
105	C₈×C₅₃	C₄×D₄	1	4	yes	—
106	#12602	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	—	2	2	no	—
109	C₂²×C₅₃	—	4	2	no	—
110	C₂²×C₅₃	—	4	2	no	—
111	C₂²×C₅₃	—	4	2	no	—
112	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
113	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₅₃	#431	1	1	yes	characteristic
115	C₄×C₅₃	—	2	4	no	—
116	C₄×C₅₃	—	2	2	no	—
117	C₄×C₅₃	—	4	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	C₂×#405	D₅₃	1	1	yes	characteristic
123	C₂×#349	—	53	2	no	—
124	C₄×M₃₂	—	53	2	no	—
125	#1377	—	53	2	no	—
126	C₂×C₅₃	—	2	1	no	—
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	—	4	2	no	—
129	C₂×C₅₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₅₃	#2115	1	2	yes	—
131	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
132	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
133	C₂×#117	Dic₅₃	1	1	yes	characteristic
134	C₂×M₃₂	—	53	4	no	—
135	#405	—	2	2	no	—
136	#405	—	2	2	no	—
137	C₂×#97	Dic₅₃	1	1	yes	characteristic
138	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
139	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
141	#349	—	53	8	no	—
142	#346	—	106	2	no	—
143	C₄×C₁₆	—	106	2	no	—
144	C₅₃	#12602	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂×M₁₆	C₄×D₅₃	1	4	yes	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	—	2	1	no	—
151	C₂²×C₈	—	2	1	no	—
152	C₂²×C₈	#88322	1	4	yes	—
153	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
155	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
157	#117	—	2	2	no	—
158	#117	—	2	2	no	—
159	C₂×C₄⋊C₄	#88322	1	2	yes	—
160	C₂×C₂²⋊C₄	#88322	1	2	yes	—
161	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
162	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
163	M₃₂	—	53	8	no	—
164	M₃₂	—	106	4	no	—
165	C₂×C₁₆	—	106	4	no	—
166	#97	—	2	2	no	—
167	#97	—	2	2	no	—
168	C₄×C₈	—	2	1	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
171	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
172	C₂⁴	C₂×#88322	1	1	yes	characteristic
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
178	C₂²×C₄	C₂×#88322	1	2	yes	—
179	C₂²×C₄	C₂×#88322	1	2	yes	—
180	C₂²×C₄	C₂×#88322	1	2	yes	—
181	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
182	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
183	M₁₆	—	2	8	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	C₂×#88322	1	2	yes	—
191	C₂×C₈	C₂×#88322	1	2	yes	—
192	C₂×C₈	#11719917	1	2	yes	—
193	C₂×C₈	C₄×Dic₅₃	1	2	yes	—
194	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
195	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
196	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
197	C₂×C₈	#11719945	1	2	yes	—
198	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
199	C₄⋊C₄	—	2	4	no	—
200	C₂²⋊C₄	—	2	4	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	#11719917	1	2	yes	—
204	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
205	C₁₆	—	106	8	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	4	2	no	—
209	C₂³	C₄×#88322	1	2	yes	—
210	C₂³	C₂²×#88322	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₄×#88322	1	2	yes	—
221	C₂×C₄	C₂²⋊#88322	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	2	yes	—
231	C₈	C₂²⋊#88322	1	2	yes	—
232	C₈	C₄×C₂²⋊D₅₃	1	4	yes	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	C₂×C₄×#88322	1	1	yes	characteristic
239	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
240	C₂²	#16810676	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	2	4	no	—
243	C₄	—	4	2	no	—
244	C₄	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
245	C₄	C₄×#11719917	1	1	yes	characteristic
246	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
247	C₄	#16810676	1	1	yes	characteristic
248	C₂	—	2	1	no	—
249	C₂	—	2	1	no	—
250	C₂	—	4	2	no	—
251	C₂	C₄×C₂²⋊#88322	1	1	yes	characteristic
252	C₂	#20276336	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#405	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#12602
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. C₄×#13133157	C₂	semidirect
3. #20272799	C₂	semidirect
6. #16810405	C₄	semidirect
144. C₅₃	#12602	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
C₄×#13133157	2
#20272799	2
C₂×C₅₃×#405	2
#12602	53