Subgroups — #28302124

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20290511	C₂	1	2	yes	maximal
2	C₂×#16811484	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
4	#20289551	C₂	1	2	yes	maximal
5	#106×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88324	C₂²	1	2	yes	—
7	C₂×#13133272	—	2	2	no	—
8	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
10	#16811489	—	2	2	no	—
11	#16811484	C₄	1	4	yes	complemented
12	C₈×#88324	C₄	1	4	yes	complemented
13	C₂×#13133149	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₅₃	C₂²	1	2	yes	—
16	#16810385	C₄	1	2	yes	—
17	C₂×#106×C₅₃	C₄	1	1	yes	characteristic
18	C₂²×#88324	—	2	1	no	—
19	C₂²×#88324	—	2	2	no	—
20	C₂²×#88324	D₄	1	1	yes	characteristic
21	C₂²×#88324	D₄	1	1	yes	characteristic
22	C₄⋊#88324	—	2	2	no	—
23	C₄⋊#88324	C₂×C₄	1	4	yes	—
24	#13133272	—	4	4	no	—
25	C₄×#88324	—	2	1	no	—
26	C₄×#88324	C₂×C₄	1	2	yes	—
27	C₄×#88324	C₂×C₄	1	2	yes	—
28	C₄×#88324	Q₈	1	2	yes	—
29	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₅₃	—	2	2	no	—
32	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
33	#13133149	—	2	2	no	—
34	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
35	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
36	#13133124	—	2	2	no	—
37	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
39	#106×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	2	2	no	—
43	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₅₃	#41	1	1	yes	characteristic
45	C₂×#88324	—	2	4	no	—
46	C₂×#88324	—	2	2	no	—
47	C₂×#88324	—	2	2	no	—
48	C₂×#88324	—	2	4	no	—
49	C₂×#88324	—	4	2	no	—
50	C₂×#88324	—	4	2	no	—
51	C₂×#88324	C₄⋊C₄	1	2	yes	—
52	C₂×#88324	C₄⋊C₄	1	2	yes	—
53	C₂×#88324	SD₁₆	1	4	yes	—
54	C₂×C₈×C₅₃	C₄²	1	2	yes	—
55	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
56	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
57	C₄⋊C₄×C₅₃	—	2	2	no	—
58	C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₄⋊Dic₅₃	—	2	2	no	—
60	C₄⋊Dic₅₃	—	2	2	no	—
61	C₄×Dic₅₃	—	4	2	no	—
62	C₄×Dic₅₃	—	4	2	no	—
63	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₄×Dic₅₃	#41	1	1	yes	characteristic
67	C₂×#88322	—	2	1	no	—
68	C₂×#88322	—	2	1	no	—
69	C₂×#88322	C₄²	1	2	yes	—
70	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	—	4	2	no	—
74	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
80	C₂×Dic₅₃	—	2	2	no	—
81	C₂×Dic₅₃	—	4	2	no	—
82	C₂×Dic₅₃	—	4	2	no	—
83	C₂×Dic₅₃	—	4	2	no	—
84	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×Dic₅₃	#117	1	1	yes	characteristic
86	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
87	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
88	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
89	C₂×Dic₅₃	#135	1	2	yes	—
90	#88324	—	2	4	no	—
91	#88324	—	2	2	no	—
92	#88324	—	4	4	no	—
93	#88324	#106	1	4	yes	complemented
94	C₈×C₅₃	#97	1	4	yes	complemented
95	#88322	—	2	2	no	—
96	#8283	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₅₃	#436	1	1	yes	characteristic
99	C₂²×C₅₃	#436	1	1	yes	characteristic
100	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
101	C₂²×C₅₃	#443	1	2	yes	—
102	C₄×C₅₃	—	8	2	no	—
103	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
104	C₄×C₅₃	#405	1	1	yes	characteristic
105	C₄×C₅₃	#446	1	1	yes	characteristic
106	C₄×C₅₃	#446	1	1	yes	characteristic
107	Dic₅₃	—	2	1	no	—
108	Dic₅₃	—	2	1	no	—
109	Dic₅₃	—	8	2	no	—
110	Dic₅₃	C₂×#106	1	1	yes	characteristic
111	Dic₅₃	#427	1	1	yes	characteristic
112	Dic₅₃	C₄×SD₁₆	1	2	yes	—
113	C₄×#106	—	53	1	no	—
114	#1770	—	53	2	no	—
115	C₂×#334	—	53	1	no	—
116	C₂×C₈²	—	53	1	no	—
117	#1279	—	53	2	no	—
118	C₂×C₅₃	#1590	1	2	yes	—
119	C₂×C₅₃	#1590	1	2	yes	—
120	C₂×C₅₃	#1770	1	1	yes	characteristic
121	C₂×C₅₃	#1772	1	1	yes	characteristic
122	C₂×C₅₃	C₄×#106	1	1	yes	characteristic
123	C₂×#106	—	53	1	no	—
124	C₂×#106	#6157	1	1	yes	characteristic
125	C₂×C₄⋊C₈	—	53	2	no	—
126	C₂×#97	—	106	2	no	—
127	C₂×C₄×C₈	—	53	1	no	—
128	C₂×C₄×C₈	—	53	1	no	—
129	C₂×C₄×C₈	—	53	1	no	—
130	C₄×C₄⋊C₄	—	53	2	no	—
131	#340	—	53	2	no	—
132	#336	—	106	2	no	—
133	#334	—	53	4	no	—
134	C₈²	—	53	4	no	—
135	C₅₃	#8283	1	1	yes	characteristic, complemented, Sylow
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	—	53	1	no	—
139	C₂²×C₈	—	106	1	no	—
140	C₂²×C₈	—	106	2	no	—
141	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	53	2	no	—
143	C₂×C₄⋊C₄	C₂×#6157	1	2	yes	—
144	C₂×C₄²	—	53	1	no	—
145	C₂×C₄²	—	106	2	no	—
146	#106	—	2	2	no	—
147	#106	—	106	2	no	—
148	C₄⋊C₈	—	53	4	no	—
149	C₄⋊C₈	—	106	2	no	—
150	#97	—	212	4	no	—
151	C₄×C₈	—	53	2	no	—
152	C₄×C₈	—	53	2	no	—
153	C₄×C₈	—	53	2	no	—
154	C₄×C₈	—	53	2	no	—
155	C₄×C₈	—	53	2	no	—
156	C₄×C₈	—	106	1	no	—
157	#95	—	106	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	53	1	no	—
160	C₂²×C₄	—	53	1	no	—
161	C₂²×C₄	—	106	2	no	—
162	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
163	C₂×C₈	—	53	2	no	—
164	C₂×C₈	—	53	4	no	—
165	C₂×C₈	—	53	2	no	—
166	C₂×C₈	—	53	2	no	—
167	C₂×C₈	—	106	2	no	—
168	C₂×C₈	—	106	2	no	—
169	C₂×C₈	—	106	4	no	—
170	C₂×C₈	—	106	1	no	—
171	C₂×C₈	—	106	1	no	—
172	C₂×C₈	—	106	4	no	—
173	C₂×C₈	—	212	2	no	—
174	C₂×C₈	—	212	2	no	—
175	C₂×C₈	#11719936	1	2	yes	—
176	C₂×C₈	C₄×#6157	1	2	yes	—
177	C₂×C₈	#11719940	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	106	2	no	—
181	C₄⋊C₄	—	106	2	no	—
182	C₄²	—	53	1	no	—
183	C₄²	—	53	1	no	—
184	C₄²	—	53	1	no	—
185	C₄²	—	53	1	no	—
186	C₄²	—	212	2	no	—
187	C₄²	—	212	2	no	—
188	C₂³	D₄×#6157	1	1	yes	characteristic
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	2	no	—
195	C₂×C₄	—	53	2	no	—
196	C₂×C₄	—	53	1	no	—
197	C₂×C₄	—	53	1	no	—
198	C₂×C₄	—	106	2	no	—
199	C₂×C₄	—	212	2	no	—
200	C₂×C₄	—	212	2	no	—
201	C₂×C₄	—	212	2	no	—
202	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
203	C₂×C₄	Q₈×#6157	1	1	yes	characteristic
204	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
205	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
206	C₂×C₄	#13133304	1	1	yes	characteristic
207	C₂×C₄	#13133305	1	1	yes	characteristic
208	C₈	—	53	4	no	—
209	C₈	—	106	2	no	—
210	C₈	—	106	4	no	—
211	C₈	—	106	2	no	—
212	C₈	—	212	4	no	—
213	C₈	#13133272	1	4	yes	complemented
214	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
215	C₂²	#16811657	1	1	yes	characteristic
216	C₂²	#16811658	1	1	yes	characteristic
217	C₂²	SD₁₆×#6157	1	2	yes	—
218	C₂²	#16811666	1	2	yes	—
219	C₄	—	8	2	no	—
220	C₄	—	53	1	no	—
221	C₄	—	53	1	no	—
222	C₄	—	53	2	no	—
223	C₄	—	106	1	no	—
224	C₄	—	106	1	no	—
225	C₄	—	424	2	no	—
226	C₄	C₂×#13133272	1	1	yes	characteristic
227	C₄	#16811588	1	1	yes	characteristic
228	C₄	#16811673	1	1	yes	characteristic
229	C₄	#16811674	1	1	yes	characteristic
230	C₂	#20290350	1	2	yes	—
231	C₂	#20290351	1	2	yes	—
232	C₂	#20290511	1	1	yes	characteristic
233	C₂	#20290518	1	1	yes	characteristic
234	C₂	#106×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₂×#106×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×#106	#6157
hypocenter	C₅₃	#8283
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
11. #16811484	C₄	semidirect
12. C₈×#88324	C₄	semidirect
93. #88324	#106	semidirect
94. C₈×C₅₃	#97	semidirect
135. C₅₃	#8283	semidirect
213. C₈	#13133272	semidirect

Maximal subgroups

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

subgroup	index
#20290511	2
C₂×#16811484	2
C₂×C₈×#88324	2
#20289551	2
#106×Dic₅₃	2
#8283	53