Subgroups — #28303532

Order 13568.

382 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290632	C₂	1	2	yes	maximal
2	#20289545	C₂	1	2	yes	maximal
3	#20289545	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88324	C₂²	1	4	yes	—
6	#16811489	—	2	2	no	—
7	#16811489	—	2	2	no	—
8	C₂×C₄×#88322	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂²×#88324	—	2	4	no	—
16	C₄⋊#88324	—	2	4	no	—
17	C₄⋊#88324	—	2	4	no	—
18	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
24	C₂×C₄×Dic₅₃	Q₈	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
26	C₄×#88322	—	2	2	no	—
27	C₈×Dic₅₃	—	2	2	no	—
28	C₈×Dic₅₃	—	2	2	no	—
29	C₈×Dic₅₃	—	2	2	no	—
30	C₈×Dic₅₃	C₈	1	4	yes	complemented
31	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×#88324	—	4	4	no	—
43	C₂×#88324	—	4	4	no	—
44	C₂×#88324	—	4	4	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
51	C₄²×C₅₃	—	2	1	no	—
52	C₄²×C₅₃	—	2	1	no	—
53	C₄×Dic₅₃	—	2	2	no	—
54	C₄×Dic₅₃	—	2	2	no	—
55	C₄×Dic₅₃	—	2	2	no	—
56	C₄×Dic₅₃	—	2	2	no	—
57	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
58	C₄×Dic₅₃	M₁₆	1	2	yes	—
59	C₄×Dic₅₃	#41	1	2	yes	—
60	C₄×Dic₅₃	#41	1	2	yes	—
61	C₂×#88322	—	2	1	no	—
62	C₂×#88322	—	2	1	no	—
63	C₂×#88322	—	2	1	no	—
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	C₂×C₈	1	2	yes	—
66	C₂×#88322	M₁₆	1	2	yes	—
67	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	#108	1	2	yes	—
73	C₂×C₄×C₅₃	#108	1	2	yes	—
74	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×Dic₅₃	—	2	1	no	—
81	C₂×Dic₅₃	—	2	1	no	—
82	C₂×Dic₅₃	—	2	1	no	—
83	C₂×Dic₅₃	—	2	1	no	—
84	C₂×Dic₅₃	—	2	1	no	—
85	C₂×Dic₅₃	—	2	1	no	—
86	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
87	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
88	C₂×Dic₅₃	#124	1	2	yes	—
89	C₂×Dic₅₃	#125	1	2	yes	—
90	C₂×Dic₅₃	#135	1	2	yes	—
91	C₂×Dic₅₃	#135	1	2	yes	—
92	#88324	—	8	4	no	—
93	C₈×C₅₃	—	2	2	no	—
94	C₈×C₅₃	—	2	2	no	—
95	#88322	—	2	2	no	—
96	#88322	—	2	2	no	—
97	#8785	—	53	1	no	maximal, Sylow
98	C₂²×C₅₃	#382	1	1	yes	characteristic
99	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₂²×C₅₃	#423	1	1	yes	characteristic
101	C₂²×C₅₃	#427	1	1	yes	characteristic
102	C₂²×C₅₃	#427	1	1	yes	characteristic
103	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
104	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	#431	1	1	yes	characteristic
108	C₄×C₅₃	#431	1	1	yes	characteristic
109	C₄×C₅₃	#432	1	1	yes	characteristic
110	C₄×C₅₃	#432	1	1	yes	characteristic
111	Dic₅₃	—	2	2	no	—
112	Dic₅₃	—	2	2	no	—
113	Dic₅₃	—	2	2	no	—
114	Dic₅₃	#488	1	4	yes	—
115	#1831	—	53	2	no	—
116	C₄²×C₈	—	53	1	no	—
117	#1277	—	53	2	no	—
118	#1277	—	53	2	no	—
119	C₂×C₅₃	#1564	1	1	yes	characteristic
120	C₂×C₅₃	#1564	1	1	yes	characteristic
121	C₂×C₅₃	#1568	1	1	yes	characteristic
122	C₂×C₅₃	#1568	1	1	yes	characteristic
123	C₂×C₅₃	#1831	1	1	yes	characteristic
124	C₂×C₅₃	#1835	1	1	yes	characteristic
125	C₂×C₅₃	#1837	1	1	yes	characteristic
126	C₂×C₄⋊C₈	—	53	4	no	—
127	C₂×C₄×C₈	—	53	1	no	—
128	C₂×C₄×C₈	—	53	1	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	C₂×C₄×C₈	—	53	1	no	—
131	C₂×C₄×C₈	—	53	1	no	—
132	C₂×C₄×C₈	#6157	1	1	yes	characteristic
133	C₄³	—	53	1	no	—
134	#336	—	106	2	no	—
135	#336	—	106	2	no	—
136	C₅₃	#8785	1	1	yes	characteristic, complemented, Sylow
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	—	53	1	no	—
139	C₂²×C₈	—	106	4	no	—
140	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
141	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
142	C₂×C₄²	—	53	1	no	—
143	C₂×C₄²	—	53	1	no	—
144	C₂×C₄²	—	53	1	no	—
145	C₂×C₄²	—	53	1	no	—
146	C₂×C₄²	—	53	1	no	—
147	C₂×C₄²	—	53	1	no	—
148	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
149	C₄⋊C₈	—	106	4	no	—
150	C₄⋊C₈	—	106	4	no	—
151	C₄×C₈	—	2	2	no	—
152	C₄×C₈	—	53	4	no	—
153	C₄×C₈	—	106	2	no	—
154	C₄×C₈	—	106	2	no	—
155	C₄×C₈	—	106	2	no	—
156	C₄×C₈	—	106	2	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	—	53	1	no	—
161	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
162	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
163	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	—	53	2	no	—
170	C₂×C₈	—	106	1	no	—
171	C₂×C₈	—	106	1	no	—
172	C₂×C₈	—	106	1	no	—
173	C₂×C₈	—	106	1	no	—
174	C₂×C₈	—	212	4	no	—
175	C₂×C₈	—	212	4	no	—
176	C₂×C₈	—	212	4	no	—
177	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
178	C₂×C₈	#11719940	1	2	yes	—
179	C₄²	—	2	1	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	53	2	no	—
182	C₄²	—	53	2	no	—
183	C₄²	—	53	2	no	—
184	C₄²	—	53	2	no	—
185	C₄²	—	106	2	no	—
186	C₄²	—	106	2	no	—
187	C₄²	—	106	2	no	—
188	C₄²	—	106	2	no	—
189	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	53	2	no	—
198	C₂×C₄	—	53	2	no	—
199	C₂×C₄	—	53	2	no	—
200	C₂×C₄	—	106	1	no	—
201	C₂×C₄	—	106	1	no	—
202	C₂×C₄	—	106	1	no	—
203	C₂×C₄	—	106	1	no	—
204	C₂×C₄	—	106	1	no	—
205	C₂×C₄	—	106	1	no	—
206	C₂×C₄	#13133287	1	2	yes	—
207	C₂×C₄	#13133288	1	2	yes	—
208	C₂×C₄	#13133296	1	1	yes	characteristic
209	C₂×C₄	#13133296	1	1	yes	characteristic
210	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
211	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
212	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
213	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	—	106	2	no	—
217	C₈	—	106	2	no	—
218	C₈	—	424	4	no	—
219	C₂²	#16811567	1	1	yes	characteristic
220	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
221	C₂²	#16811626	1	1	yes	characteristic
222	C₂²	#16811633	1	1	yes	characteristic
223	C₂²	#16811633	1	1	yes	characteristic
224	C₂²	C₂×#13133287	1	1	yes	characteristic
225	C₂²	C₂×#13133288	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	53	4	no	—
229	C₄	—	106	2	no	—
230	C₄	—	106	2	no	—
231	C₄	—	106	2	no	—
232	C₄	#16811640	1	1	yes	characteristic
233	C₄	#16811641	1	1	yes	characteristic
234	C₄	#16811643	1	1	yes	characteristic
235	C₄	#16811646	1	1	yes	characteristic
236	C₂	#20290263	1	1	yes	characteristic
237	C₂	#20290265	1	1	yes	characteristic
238	C₂	#20290275	1	1	yes	characteristic
239	C₂	#20290277	1	1	yes	characteristic
240	C₂	#20290631	1	1	yes	characteristic
241	C₂	#20290637	1	1	yes	characteristic
242	C₂	#20290639	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#8785
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
30. C₈×Dic₅₃	C₈	semidirect
136. C₅₃	#8785	semidirect

Maximal subgroups

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

subgroup	index
#20290632	2
#20289545	2
C₄×C₈×Dic₅₃	2
#8785	53