Subgroups — #28247793

Order 13568.

2564 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811390	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811390	C₂	1	1	yes	maximal, characteristic
3	#20283324	C₂	1	16	yes	maximal, complemented
4	C₂×#16810671	C₂	1	12	yes	maximal, complemented
5	C₂×C₅₃×#568	C₂	1	1	yes	maximal, characteristic
6	#16811390	—	2	4	no	—
7	#16811390	C₄	1	8	yes	—
8	#16811390	C₄	1	8	yes	—
9	#16811390	C₂²	1	8	yes	—
10	C₂×#13133224	—	2	4	no	—
11	C₂×#13133224	C₂²	1	8	yes	—
12	C₂²×#11719917	C₂²	1	6	yes	—
13	C₂²×#11719917	C₂²	1	6	yes	—
14	#16810671	C₂²	1	16	yes	—
15	#16810671	C₂²	1	48	yes	complemented
16	C₂×#13133152	C₂²	1	48	yes	complemented
17	C₂²×#41×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#568	—	2	4	no	—
19	C₅₃×#568	C₂²	1	8	yes	—
20	C₂×#131×C₅₃	—	2	4	no	—
21	C₂²×M₁₆×C₅₃	C₂²	1	6	yes	—
22	#13133224	—	2	32	no	—
23	#13133224	C₂×C₄	1	64	yes	—
24	C₂×#11719917	—	2	6	no	—
25	C₂×#11719917	—	2	12	no	—
26	C₂×#11719917	C₂×C₄	1	24	yes	—
27	C₂×#11719917	C₂×C₄	1	24	yes	—
28	C₂×#11719917	C₂³	1	4	yes	—
29	C₂×#11719917	C₂³	1	24	yes	—
30	C₂×#11719917	C₂³	1	4	yes	—
31	C₂×#11719917	C₂³	1	12	yes	—
32	C₂²×#88322	—	2	6	no	—
33	C₂²×#88322	C₂³	1	12	yes	—
34	#13133152	C₂³	1	64	yes	—
35	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
36	C₂×#41×C₅₃	C₂×C₄	1	4	yes	—
37	C₂×#41×C₅₃	D₄	1	4	yes	—
38	C₂×#41×C₅₃	D₄	1	4	yes	—
39	C₂×#41×C₅₃	Q₈	1	4	yes	—
40	C₂×#41×C₅₃	C₂³	1	4	yes	—
41	C₂²×Q₈×C₅₃	Q₈	1	1	yes	characteristic
42	C₂²×D₄×C₅₃	Q₈	1	3	yes	—
43	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
44	#131×C₅₃	—	2	32	no	—
45	C₂×M₁₆×C₅₃	—	2	6	no	—
46	C₂×M₁₆×C₅₃	—	2	12	no	—
47	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
48	C₂×M₁₆×C₅₃	C₂³	1	24	yes	—
49	C₂²×C₈×C₅₃	—	2	6	no	—
50	C₂⁴×C₅₃	C₂×Q₈	1	3	yes	—
51	#41×C₅₃	C₄⋊C₄	1	16	yes	—
52	#41×C₅₃	C₄⋊C₄	1	16	yes	—
53	#41×C₅₃	C₂²×C₄	1	16	yes	—
54	#41×C₅₃	C₂×D₄	1	16	yes	—
55	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
56	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
57	C₂×Q₈×C₅₃	C₂×Q₈	1	4	yes	—
58	C₂×D₄×C₅₃	C₄⋊C₄	1	12	yes	—
59	C₂×D₄×C₅₃	C₄⋊C₄	1	12	yes	—
60	C₂×D₄×C₅₃	C₂×Q₈	1	12	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
64	C₂²×C₄×C₅₃	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
66	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
67	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
69	M₁₆×C₅₃	—	2	4	no	—
70	M₁₆×C₅₃	—	2	24	no	—
71	M₁₆×C₅₃	C₂⁴	1	8	yes	—
72	C₂×C₈×C₅₃	—	2	4	no	—
73	C₂×C₈×C₅₃	—	2	24	no	—
74	#11719917	—	2	4	no	—
75	#11719917	—	2	24	no	—
76	#11719917	C₂²×C₄	1	8	yes	—
77	#11719917	C₂²×C₄	1	8	yes	—
78	#11719917	C₂²×C₄	1	48	yes	—
79	#11719917	C₂⁴	1	8	yes	—
80	C₂×#88322	—	2	4	no	—
81	C₂×#88322	—	2	24	no	—
82	C₂×#88322	C₂²×C₄	1	48	yes	—
83	C₂×#88322	C₂⁴	1	8	yes	—
84	C₂³×C₅₃	—	8	3	no	—
85	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
86	C₂³×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
87	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
88	C₂³×C₅₃	C₂²×Q₈	1	6	yes	—
89	Q₈×C₅₃	C₂×C₄⋊C₄	1	16	yes	—
90	D₄×C₅₃	C₂×C₄⋊C₄	1	48	yes	—
91	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
92	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	6	yes	—
93	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
94	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
95	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
98	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×D₄	1	12	yes	—
101	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
103	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
104	C₈×C₅₃	—	2	8	no	—
105	#88322	—	2	8	no	—
106	#88322	C₂³×C₄	1	16	yes	—
107	#60113	—	53	1	no	maximal, Sylow
108	C₂²×C₅₃	—	4	1	no	—
109	C₂²×C₅₃	—	4	6	no	—
110	C₂²×C₅₃	—	4	6	no	—
111	C₂²×C₅₃	—	4	6	no	—
112	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
115	C₂²×C₅₃	C₂³×Q₈	1	1	yes	characteristic
116	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₅₃	C₂²×C₄⋊C₄	1	12	yes	—
119	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
120	C₄×C₅₃	C₂³×D₄	1	1	yes	characteristic
121	C₂×#568	—	53	1	no	—
122	C₂×#568	—	53	1	no	—
123	C₂×#568	D₅₃	1	1	yes	characteristic
124	#2916	—	53	16	no	—
125	C₂×#430	—	53	12	no	—
126	C₂×C₅₃	—	2	1	no	—
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	—	2	1	no	—
129	C₂×C₅₃	—	2	12	no	—
130	C₂×C₅₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×#41	C₂×D₅₃	1	1	yes	characteristic
132	#568	—	2	4	no	—
133	#568	—	53	8	no	—
134	#568	—	53	8	no	—
135	#568	—	53	8	no	—
136	#568	—	106	4	no	—
137	#568	C₂×D₅₃	1	8	yes	—
138	C₂×#131	—	2	4	no	—
139	C₂×#131	—	53	8	no	—
140	C₂×#131	—	106	4	no	—
141	C₂²×M₁₆	—	53	6	no	—
142	C₂²×M₁₆	—	53	6	no	—
143	C₂²×M₁₆	C₂×D₅₃	1	6	yes	—
144	#430	—	53	16	no	—
145	#430	—	53	48	no	—
146	C₂×#108	—	53	48	no	—
147	C₅₃	#60113	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	Dic₁₀₆	1	4	yes	—
149	C₂×#41	C₄×D₅₃	1	4	yes	—
150	C₂×#41	C₄×D₅₃	1	4	yes	—
151	C₂×#41	C₂²⋊D₅₃	1	4	yes	—
152	C₂×#41	C₂²⋊D₅₃	1	4	yes	—
153	C₂×#41	C₂²×D₅₃	1	4	yes	—
154	C₂²×Q₈	Dic₁₀₆	1	1	yes	characteristic
155	C₂²×D₄	Dic₁₀₆	1	3	yes	—
156	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
157	#131	—	2	32	no	—
158	#131	—	53	64	no	—
159	#131	—	106	32	no	—
160	C₂×M₁₆	—	2	6	no	—
161	C₂×M₁₆	—	2	12	no	—
162	C₂×M₁₆	—	53	24	no	—
163	C₂×M₁₆	—	53	24	no	—
164	C₂×M₁₆	—	53	4	no	—
165	C₂×M₁₆	—	53	12	no	—
166	C₂×M₁₆	—	53	4	no	—
167	C₂×M₁₆	—	53	24	no	—
168	C₂×M₁₆	—	106	6	no	—
169	C₂×M₁₆	—	106	12	no	—
170	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
171	C₂×M₁₆	C₂²×D₅₃	1	24	yes	—
172	C₂²×C₈	—	2	6	no	—
173	C₂²×C₈	—	53	12	no	—
174	C₂²×C₈	—	106	6	no	—
175	#108	—	53	64	no	—
176	C₂⁴	C₂×Dic₁₀₆	1	3	yes	—
177	#41	#11719919	1	16	yes	—
178	#41	#11719919	1	16	yes	—
179	#41	C₂×C₄×D₅₃	1	16	yes	—
180	#41	C₂×C₂²⋊D₅₃	1	16	yes	—
181	C₂×Q₈	#11719919	1	4	yes	—
182	C₂×Q₈	#11719919	1	4	yes	—
183	C₂×Q₈	C₂×Dic₁₀₆	1	4	yes	—
184	C₂×D₄	#11719919	1	12	yes	—
185	C₂×D₄	#11719919	1	12	yes	—
186	C₂×D₄	C₂×Dic₁₀₆	1	12	yes	—
187	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
188	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
189	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
190	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
191	C₂²×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
192	C₂²×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
193	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
194	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
195	M₁₆	—	2	4	no	—
196	M₁₆	—	2	24	no	—
197	M₁₆	—	53	8	no	—
198	M₁₆	—	53	8	no	—
199	M₁₆	—	53	48	no	—
200	M₁₆	—	53	8	no	—
201	M₁₆	—	106	4	no	—
202	M₁₆	—	106	24	no	—
203	M₁₆	C₂³×D₅₃	1	8	yes	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	24	no	—
206	C₂×C₈	—	53	48	no	—
207	C₂×C₈	—	53	8	no	—
208	C₂×C₈	—	106	4	no	—
209	C₂×C₈	—	106	24	no	—
210	C₂³	—	8	3	no	—
211	C₂³	C₂×#11719919	1	6	yes	—
212	C₂³	C₂×#11719919	1	6	yes	—
213	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
214	C₂³	C₂²×Dic₁₀₆	1	6	yes	—
215	Q₈	C₂×#11719919	1	16	yes	—
216	D₄	C₂×#11719919	1	48	yes	—
217	C₂×C₄	C₂×#11719919	1	6	yes	—
218	C₂×C₄	C₂×#11719919	1	6	yes	—
219	C₂×C₄	C₂×#11719919	1	12	yes	—
220	C₂×C₄	C₂×#11719919	1	12	yes	—
221	C₂×C₄	C₂²×C₄×D₅₃	1	12	yes	—
222	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₂²⋊D₅₃	1	12	yes	—
226	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
227	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
228	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
229	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
230	C₈	—	2	8	no	—
231	C₈	—	53	16	no	—
232	C₈	—	106	8	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	6	no	—
235	C₂²	—	4	6	no	—
236	C₂²	—	4	6	no	—
237	C₂²	C₂²×#11719919	1	1	yes	characteristic
238	C₂²	C₂²×#11719919	1	1	yes	characteristic
239	C₂²	C₂²×#11719919	1	12	yes	—
240	C₂²	C₂³×Dic₁₀₆	1	1	yes	characteristic
241	C₄	C₂²×#11719919	1	1	yes	characteristic
242	C₄	C₂²×#11719919	1	1	yes	characteristic
243	C₄	C₂²×#11719919	1	12	yes	—
244	C₄	C₂³×C₄×D₅₃	1	1	yes	characteristic
245	C₄	C₂³×C₂²⋊D₅₃	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	1	no	—
248	C₂	—	2	1	no	—
249	C₂	—	2	12	no	—
250	C₂	C₂³×#11719919	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂³×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂×C₅₃×#568	C₂
socle	C₂×C₅₃	C₂³×C₄⋊C₄
prime radical	C₂²×#41×C₅₃	C₂²
cosocle	C₂×C₄×C₅₃	C₂⁵
hypercenter	C₂×#568	D₅₃
hypocenter	C₅₃	#60113
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
3. #20283324	C₂	semidirect
4. C₂×#16810671	C₂	semidirect
15. #16810671	C₂²	semidirect
16. C₂×#13133152	C₂²	semidirect
147. C₅₃	#60113	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811390	2
#20283324	2
C₂×#16810671	2
C₂×C₅₃×#568	2
#60113	53