Subgroups — #28302099

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290501	C₂	1	2	yes	maximal
2	C₂×#16811486	C₂	1	1	yes	maximal, characteristic
3	C₂×#16811484	C₂	1	1	yes	maximal, characteristic
4	#20289545	C₂	1	2	yes	maximal
5	#97×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88324	C₂²	1	2	yes	—
7	C₂×C₄⋊#88324	C₂²	1	2	yes	—
8	C₂×#13133272	—	2	2	no	—
9	#16811489	—	2	2	no	—
10	#16811486	C₄	1	4	yes	complemented
11	#16811484	C₄	1	4	yes	complemented
12	C₂×#13133128	C₄	1	1	yes	characteristic
13	C₂×#13133127	C₄	1	1	yes	characteristic
14	C₂×#13133127	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
18	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
19	C₂²×#88324	—	2	2	no	—
20	C₂²×#88324	—	2	2	no	—
21	C₄⋊#88324	—	2	2	no	—
22	C₄⋊#88324	—	2	2	no	—
23	C₄⋊#88324	C₂×C₄	1	4	yes	—
24	C₄⋊#88324	C₂×C₄	1	4	yes	—
25	#13133272	—	4	4	no	—
26	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
30	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
31	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
32	#13133128	—	2	2	no	—
33	#13133127	—	2	2	no	—
34	#13133127	—	2	2	no	—
35	C₈×Dic₅₃	—	2	1	no	—
36	C₈×Dic₅₃	—	2	1	no	—
37	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
38	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
42	#97×C₅₃	—	2	2	no	—
43	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
46	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×#88324	—	2	4	no	—
49	C₂×#88324	—	2	4	no	—
50	C₂×#88324	—	4	2	no	—
51	C₂×#88324	—	4	2	no	—
52	C₂×#88324	—	4	2	no	—
53	C₂×#88324	—	4	2	no	—
54	C₂×C₈×C₅₃	—	2	1	no	—
55	C₂×C₈×C₅₃	—	2	1	no	—
56	C₂×C₈×C₅₃	C₄²	1	2	yes	—
57	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
58	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
59	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
60	C₄²×C₅₃	—	2	1	no	—
61	C₄²×C₅₃	—	2	1	no	—
62	C₄×Dic₅₃	—	2	1	no	—
63	C₄×Dic₅₃	—	2	1	no	—
64	C₄×Dic₅₃	—	2	2	no	—
65	C₄×Dic₅₃	—	2	2	no	—
66	C₄×Dic₅₃	—	4	2	no	—
67	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₄×Dic₅₃	#41	1	1	yes	characteristic
70	C₄×Dic₅₃	#41	1	1	yes	characteristic
71	C₂×#88322	—	2	1	no	—
72	C₂×#88322	—	2	1	no	—
73	C₂×#88322	—	2	1	no	—
74	C₂×#88322	—	2	1	no	—
75	C₂×#88322	C₄²	1	2	yes	—
76	C₂×#88322	M₁₆	1	2	yes	—
77	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×Dic₅₃	—	2	2	no	—
88	C₂×Dic₅₃	—	2	2	no	—
89	C₂×Dic₅₃	—	2	2	no	—
90	C₂×Dic₅₃	—	2	2	no	—
91	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
92	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
93	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
94	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
95	C₂×Dic₅₃	AGL(1,ℤ/8)	1	2	yes	—
96	C₂×Dic₅₃	#137	1	2	yes	—
97	#88324	—	4	4	no	—
98	#88324	—	4	4	no	—
99	C₈×C₅₃	—	2	2	no	—
100	C₈×C₅₃	#97	1	4	yes	complemented
101	#88322	—	2	2	no	—
102	#88322	—	2	2	no	—
103	#8276	—	53	1	no	maximal, Sylow
104	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₅₃	#423	1	1	yes	characteristic
106	C₂²×C₅₃	#423	1	1	yes	characteristic
107	C₂²×C₅₃	#428	1	2	yes	—
108	C₂²×C₅₃	#430	1	2	yes	—
109	C₄×C₅₃	—	4	2	no	—
110	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
111	C₄×C₅₃	C₄×M₁₆	1	1	yes	characteristic
112	C₄×C₅₃	#432	1	1	yes	characteristic
113	C₄×C₅₃	#432	1	1	yes	characteristic
114	Dic₅₃	—	2	1	no	—
115	Dic₅₃	—	2	1	no	—
116	Dic₅₃	—	4	2	no	—
117	Dic₅₃	#440	1	2	yes	—
118	Dic₅₃	#441	1	1	yes	characteristic
119	Dic₅₃	#442	1	1	yes	characteristic
120	#1767	—	53	2	no	—
121	C₄×#97	—	53	1	no	—
122	C₂×#335	—	53	1	no	—
123	C₂×#334	—	53	1	no	—
124	#1277	—	53	2	no	—
125	C₂×C₅₃	#1569	1	2	yes	—
126	C₂×C₅₃	#1569	1	2	yes	—
127	C₂×C₅₃	#1767	1	1	yes	characteristic
128	C₂×C₅₃	#1776	1	1	yes	characteristic
129	C₂×C₅₃	#1778	1	1	yes	characteristic
130	C₂×C₄⋊C₈	—	53	2	no	—
131	C₂×C₄⋊C₈	—	53	2	no	—
132	C₂×#97	—	53	1	no	—
133	C₂×#97	—	53	1	no	—
134	C₂×#97	—	53	1	no	—
135	C₂×#97	—	106	2	no	—
136	C₂×#97	#6157	1	1	yes	characteristic
137	C₂×C₄×C₈	—	53	1	no	—
138	C₂×C₄×C₈	—	53	1	no	—
139	C₄³	—	53	1	no	—
140	#336	—	106	2	no	—
141	#335	—	53	4	no	—
142	#334	—	53	4	no	—
143	C₅₃	#8276	1	1	yes	characteristic, complemented, Sylow
144	C₂²×C₈	—	53	1	no	—
145	C₂²×C₈	—	53	1	no	—
146	C₂²×C₈	—	106	2	no	—
147	C₂²×C₈	—	106	2	no	—
148	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
149	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
150	C₂×C₄²	—	53	1	no	—
151	C₂×C₄²	—	53	2	no	—
152	C₂×C₄²	—	53	1	no	—
153	C₂×C₄²	—	53	2	no	—
154	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
155	C₄⋊C₈	—	53	4	no	—
156	C₄⋊C₈	—	53	4	no	—
157	C₄⋊C₈	—	106	2	no	—
158	C₄⋊C₈	—	106	2	no	—
159	#97	—	2	2	no	—
160	#97	—	106	2	no	—
161	#97	—	106	2	no	—
162	#97	—	106	2	no	—
163	#97	—	212	4	no	—
164	C₄×C₈	—	53	2	no	—
165	C₄×C₈	—	53	2	no	—
166	C₄×C₈	—	106	1	no	—
167	C₄×C₈	—	106	1	no	—
168	C₂²×C₄	—	53	1	no	—
169	C₂²×C₄	—	53	1	no	—
170	C₂²×C₄	—	53	2	no	—
171	C₂²×C₄	C₄⋊#6157	1	2	yes	—
172	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	53	2	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	106	1	no	—
178	C₂×C₈	—	106	1	no	—
179	C₂×C₈	—	106	1	no	—
180	C₂×C₈	—	106	1	no	—
181	C₂×C₈	—	106	4	no	—
182	C₂×C₈	—	106	4	no	—
183	C₂×C₈	—	212	2	no	—
184	C₂×C₈	—	212	2	no	—
185	C₂×C₈	—	212	2	no	—
186	C₂×C₈	—	212	2	no	—
187	C₂×C₈	#11719936	1	2	yes	—
188	C₂×C₈	#11719936	1	2	yes	—
189	C₂×C₈	C₄×#6157	1	2	yes	—
190	C₂×C₈	#11719940	1	2	yes	—
191	C₄²	—	2	1	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	53	1	no	—
194	C₄²	—	53	1	no	—
195	C₄²	—	53	1	no	—
196	C₄²	—	53	1	no	—
197	C₄²	—	106	2	no	—
198	C₄²	—	106	2	no	—
199	C₄²	—	106	1	no	—
200	C₄²	—	106	1	no	—
201	C₄²	—	212	2	no	—
202	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	53	1	no	—
207	C₂×C₄	—	53	1	no	—
208	C₂×C₄	—	53	2	no	—
209	C₂×C₄	—	53	2	no	—
210	C₂×C₄	—	53	1	no	—
211	C₂×C₄	—	53	1	no	—
212	C₂×C₄	—	106	2	no	—
213	C₂×C₄	—	106	2	no	—
214	C₂×C₄	—	106	2	no	—
215	C₂×C₄	—	106	2	no	—
216	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
217	C₂×C₄	#13133296	1	1	yes	characteristic
218	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
219	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
220	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
221	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	106	2	no	—
224	C₈	—	106	2	no	—
225	C₈	—	212	4	no	—
226	C₈	—	212	4	no	—
227	C₈	#13133272	1	4	yes	complemented
228	C₂²	C₄×C₄⋊#6157	1	1	yes	characteristic
229	C₂²	#16811626	1	1	yes	characteristic
230	C₂²	#16811627	1	1	yes	characteristic
231	C₂²	#16811634	1	2	yes	—
232	C₂²	#16811638	1	2	yes	—
233	C₄	—	4	2	no	—
234	C₄	—	53	1	no	—
235	C₄	—	53	2	no	—
236	C₄	—	53	1	no	—
237	C₄	—	106	1	no	—
238	C₄	—	106	1	no	—
239	C₄	—	212	2	no	—
240	C₄	C₂×#13133272	1	1	yes	characteristic
241	C₄	C₄×#11719936	1	1	yes	characteristic
242	C₄	#16811644	1	1	yes	characteristic
243	C₄	#16811645	1	1	yes	characteristic
244	C₂	#20290279	1	2	yes	—
245	C₂	#20290280	1	2	yes	—
246	C₂	#20290501	1	1	yes	characteristic
247	C₂	#20290529	1	1	yes	characteristic
248	C₂	#20290535	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₄⋊#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#97×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#97	#6157
hypocenter	C₅₃	#8276
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
10. #16811486	C₄	semidirect
11. #16811484	C₄	semidirect
100. C₈×C₅₃	#97	semidirect
143. C₅₃	#8276	semidirect
227. C₈	#13133272	semidirect

Maximal subgroups

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

subgroup	index
#20290501	2
C₂×#16811486	2
C₂×#16811484	2
#20289545	2
#97×Dic₅₃	2
#8276	53