Subgroups — #28302119

Order 13568.

414 conjugacy classes of subgroups.

The subgroup lattice has 234 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×#16811487	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
4	#20289551	C₂	1	2	yes	maximal
5	#20274607	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	#16811487	C₄	1	4	yes	complemented
12	C₈×#88324	C₄	1	4	yes	complemented
13	C₂×#13133150	C₄	1	2	yes	—
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×Dic₅₃	C₂²	1	2	yes	—
16	#16810386	C₄	1	1	yes	characteristic
17	C₅₃×#340	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	#13133150	—	2	4	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	C₂²×C₄×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×Dic₅₃	—	2	2	no	—
42	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅₃	#41	1	1	yes	characteristic
44	C₂×#88324	—	2	4	no	—
45	C₂×#88324	—	2	2	no	—
46	C₂×#88324	—	2	1	no	—
47	C₂×#88324	—	2	1	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	D₈	1	2	yes	—
54	C₂×#88324	Q₁₆	1	2	yes	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	C₄²	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	C₄²	1	2	yes	—
68	C₂×#88322	M₁₆	1	4	yes	—
69	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	—	4	2	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	—	4	2	no	—
73	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₅₃	#131	1	2	yes	—
77	C₂×Dic₅₃	—	2	2	no	—
78	C₂×Dic₅₃	—	4	2	no	—
79	C₂×Dic₅₃	—	4	2	no	—
80	C₂×Dic₅₃	—	4	2	no	—
81	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×Dic₅₃	#117	1	1	yes	characteristic
83	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
84	C₂×Dic₅₃	C₄×Q₈	1	1	yes	characteristic
85	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
86	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×Dic₅₃	#135	1	1	yes	characteristic
88	C₂×Dic₅₃	#135	1	1	yes	characteristic
89	#88324	—	2	4	no	—
90	#88324	—	2	2	no	—
91	#88324	—	4	4	no	—
92	#88324	#107	1	4	yes	complemented
93	C₈×C₅₃	—	2	2	no	—
94	#88322	#97	1	4	yes	complemented
95	#8282	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₅₃	#436	1	2	yes	—
98	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
99	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
100	C₂²×C₅₃	#443	1	2	yes	—
101	C₄×C₅₃	—	8	2	no	—
102	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
103	C₄×C₅₃	#405	1	1	yes	characteristic
104	C₄×C₅₃	#446	1	2	yes	—
105	Dic₅₃	—	2	1	no	—
106	Dic₅₃	—	2	1	no	—
107	Dic₅₃	—	8	2	no	—
108	Dic₅₃	C₂×#107	1	1	yes	characteristic
109	Dic₅₃	#427	1	1	yes	characteristic
110	Dic₅₃	C₄×D₈	1	1	yes	characteristic
111	Dic₅₃	C₄×Q₁₆	1	1	yes	characteristic
112	C₄×#107	—	53	1	no	—
113	#1770	—	53	2	no	—
114	C₂×#335	—	53	1	no	—
115	C₂×C₈²	—	53	1	no	—
116	#1279	—	53	2	no	—
117	C₂×C₅₃	#1589	1	2	yes	—
118	C₂×C₅₃	#1591	1	2	yes	—
119	C₂×C₅₃	#1770	1	1	yes	characteristic
120	C₂×C₅₃	#1772	1	1	yes	characteristic
121	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
122	C₂×#107	—	53	2	no	—
123	C₂×C₄⋊C₈	—	53	2	no	—
124	C₂×#97	—	106	2	no	—
125	C₂×C₄×C₈	—	53	1	no	—
126	C₂×C₄×C₈	—	53	1	no	—
127	C₂×C₄×C₈	—	53	1	no	—
128	C₄×C₄⋊C₄	—	53	2	no	—
129	#340	—	53	1	no	—
130	#340	#6157	1	1	yes	characteristic
131	#336	—	106	2	no	—
132	#335	—	53	4	no	—
133	C₈²	—	53	4	no	—
134	C₅₃	#8282	1	1	yes	characteristic, complemented, Sylow
135	C₂²×C₈	—	53	1	no	—
136	C₂²×C₈	—	53	1	no	—
137	C₂²×C₈	—	53	1	no	—
138	C₂²×C₈	—	106	1	no	—
139	C₂²×C₈	—	106	2	no	—
140	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	53	2	no	—
142	C₂×C₄⋊C₄	C₂×#6157	1	2	yes	—
143	C₂×C₄²	—	53	1	no	—
144	C₂×C₄²	—	106	2	no	—
145	#107	—	106	4	no	—
146	C₄⋊C₈	—	53	4	no	—
147	C₄⋊C₈	—	106	2	no	—
148	#97	—	212	4	no	—
149	C₄×C₈	—	53	2	no	—
150	C₄×C₈	—	53	2	no	—
151	C₄×C₈	—	53	2	no	—
152	C₄×C₈	—	53	2	no	—
153	C₄×C₈	—	53	2	no	—
154	C₄×C₈	—	106	1	no	—
155	#95	—	106	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	53	1	no	—
158	C₂²×C₄	—	53	1	no	—
159	C₂²×C₄	—	106	2	no	—
160	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	53	2	no	—
163	C₂×C₈	—	53	2	no	—
164	C₂×C₈	—	53	2	no	—
165	C₂×C₈	—	53	2	no	—
166	C₂×C₈	—	53	4	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	106	2	no	—
169	C₂×C₈	—	106	1	no	—
170	C₂×C₈	—	106	1	no	—
171	C₂×C₈	—	106	4	no	—
172	C₂×C₈	—	106	4	no	—
173	C₂×C₈	—	212	2	no	—
174	C₂×C₈	—	212	2	no	—
175	C₂×C₈	C₄×#6157	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	106	2	no	—
179	C₄⋊C₄	—	106	2	no	—
180	C₄²	—	53	1	no	—
181	C₄²	—	53	1	no	—
182	C₄²	—	53	1	no	—
183	C₄²	—	53	1	no	—
184	C₄²	—	212	2	no	—
185	C₄²	—	212	2	no	—
186	C₂³	D₄×#6157	1	1	yes	characteristic
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	53	1	no	—
191	C₂×C₄	—	53	1	no	—
192	C₂×C₄	—	53	1	no	—
193	C₂×C₄	—	53	1	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	1	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	D₅₃⋊M₁₆	1	2	yes	—
205	C₂×C₄	#13133304	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	53	4	no	—
208	C₈	—	53	4	no	—
209	C₈	—	106	2	no	—
210	C₈	—	106	4	no	—
211	C₈	—	212	4	no	—
212	C₂²	C₄⋊C₄×#6157	1	1	yes	characteristic
213	C₂²	#16811656	1	2	yes	—
214	C₂²	D₈×#6157	1	1	yes	characteristic
215	C₂²	Q₁₆×#6157	1	1	yes	characteristic
216	C₂²	#16811666	1	2	yes	—
217	C₄	—	8	2	no	—
218	C₄	—	53	1	no	—
219	C₄	—	53	1	no	—
220	C₄	—	53	1	no	—
221	C₄	—	53	1	no	—
222	C₄	—	106	1	no	—
223	C₄	—	106	1	no	—
224	C₄	—	424	2	no	—
225	C₄	#16811582	1	1	yes	characteristic
226	C₄	#16811587	1	1	yes	characteristic
227	C₄	#16811675	1	2	yes	—
228	C₂	#20290346	1	2	yes	—
229	C₂	#20290353	1	2	yes	—
230	C₂	#20290510	1	1	yes	characteristic
231	C₂	#20290518	1	1	yes	characteristic
232	C₂	#107×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×#340	C₄
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	#340	#6157
hypocenter	C₅₃	#8282
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

Complements

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

N	H ≅ G/N	type
11. #16811487	C₄	semidirect
12. C₈×#88324	C₄	semidirect
92. #88324	#107	semidirect
94. #88322	#97	semidirect
134. C₅₃	#8282	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×#16811487	2
C₂×C₈×#88324	2
#20289551	2
#20274607	2
#8282	53