Subgroups — #28303586

Order 13568.

422 conjugacy classes of subgroups.

The subgroup lattice has 220 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	C₂×#16811486	C₂	1	2	yes	maximal
3	C₂×#16811485	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	4	no	—
7	#16811486	—	2	4	no	—
8	#16811485	—	2	4	no	—
9	C₂×C₄×#88322	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
16	C₂²×#88324	—	2	4	no	—
17	C₄⋊#88324	—	2	4	no	—
18	C₄⋊#88324	—	2	4	no	—
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₅₃	D₄	1	2	yes	—
23	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
24	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
25	C₄×#88322	—	2	2	no	—
26	C₈×Dic₅₃	—	2	2	no	—
27	C₈×Dic₅₃	C₈	1	4	yes	complemented
28	C₈×Dic₅₃	D₄	1	2	yes	—
29	C₈×Dic₅₃	D₄	1	2	yes	—
30	C₈×Dic₅₃	Q₈	1	2	yes	—
31	C₈×Dic₅₃	Q₈	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₅₃	—	2	2	no	—
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
39	C₂²×Dic₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×#88324	—	4	4	no	—
42	C₂×#88324	—	4	8	no	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
49	C₄²×C₅₃	—	2	1	no	—
50	C₄²×C₅₃	—	2	1	no	—
51	C₄×Dic₅₃	—	2	4	no	—
52	C₄×Dic₅₃	—	2	4	no	—
53	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
54	C₄×Dic₅₃	M₁₆	1	2	yes	—
55	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
56	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
57	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
58	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×#88322	—	2	1	no	—
60	C₂×#88322	—	2	1	no	—
61	C₂×#88322	C₄⋊C₄	1	2	yes	—
62	C₂×#88322	C₄⋊C₄	1	2	yes	—
63	C₂×#88322	C₂×C₈	1	2	yes	—
64	C₂×#88322	M₁₆	1	2	yes	—
65	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	#108	1	4	yes	—
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₅₃	—	2	2	no	—
76	C₂×Dic₅₃	—	2	2	no	—
77	C₂×Dic₅₃	—	2	2	no	—
78	C₂×Dic₅₃	C₄⋊C₈	1	4	yes	—
79	C₂×Dic₅₃	#127	1	1	yes	characteristic
80	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
81	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×Dic₅₃	#135	1	4	yes	—
84	#88324	—	8	4	no	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
87	#88322	—	2	2	no	—
88	#88322	C₄⋊C₈	1	4	yes	—
89	#8807	—	53	1	no	maximal, Sylow
90	C₂²×C₅₃	#384	1	1	yes	characteristic
91	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₅₃	#423	1	1	yes	characteristic
93	C₂²×C₅₃	#427	1	2	yes	—
94	C₂²×C₅₃	C₂×#108	1	2	yes	—
95	C₄×C₅₃	—	2	4	no	—
96	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
97	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₅₃	#423	1	1	yes	characteristic
99	C₄×C₅₃	#423	1	1	yes	characteristic
100	Dic₅₃	—	2	4	no	—
101	Dic₅₃	—	2	1	no	—
102	Dic₅₃	—	2	1	no	—
103	Dic₅₃	#495	1	2	yes	—
104	Dic₅₃	#499	1	2	yes	—
105	#1831	—	53	2	no	—
106	C₄²×C₈	—	53	1	no	—
107	C₂×#335	—	53	2	no	—
108	C₂×#334	—	53	2	no	—
109	C₂×C₅₃	#1564	1	2	yes	—
110	C₂×C₅₃	#1568	1	2	yes	—
111	C₂×C₅₃	#1842	1	1	yes	characteristic
112	C₂×C₅₃	#1846	1	1	yes	characteristic
113	C₂×C₅₃	#1848	1	1	yes	characteristic
114	C₂×C₄⋊C₈	—	53	4	no	—
115	C₂×C₄×C₈	—	53	1	no	—
116	C₂×C₄×C₈	—	53	1	no	—
117	C₂×C₄×C₈	—	53	1	no	—
118	C₂×C₄×C₈	—	53	1	no	—
119	C₂×C₄×C₈	—	53	1	no	—
120	C₂×C₄×C₈	#6157	1	1	yes	characteristic
121	C₄³	—	53	1	no	—
122	#336	—	106	4	no	—
123	#335	—	106	4	no	—
124	#334	—	106	4	no	—
125	C₅₃	#8807	1	1	yes	characteristic, complemented, Sylow
126	C₂²×C₈	—	53	1	no	—
127	C₂²×C₈	—	53	1	no	—
128	C₂²×C₈	—	106	4	no	—
129	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
130	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
131	C₂×C₄²	—	53	1	no	—
132	C₂×C₄²	—	53	1	no	—
133	C₂×C₄²	—	53	2	no	—
134	C₂×C₄²	—	53	2	no	—
135	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
136	C₄⋊C₈	—	106	4	no	—
137	C₄⋊C₈	—	106	4	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	—	53	2	no	—
140	C₄×C₈	—	53	2	no	—
141	C₄×C₈	—	53	2	no	—
142	C₄×C₈	—	53	2	no	—
143	C₄×C₈	—	53	4	no	—
144	C₄×C₈	—	106	2	no	—
145	C₄×C₈	—	106	2	no	—
146	C₂²×C₄	—	53	1	no	—
147	C₂²×C₄	—	53	2	no	—
148	C₂²×C₄	—	53	1	no	—
149	C₂²×C₄	C₄⋊#6157	1	2	yes	—
150	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	53	2	no	—
154	C₂×C₈	—	53	2	no	—
155	C₂×C₈	—	53	2	no	—
156	C₂×C₈	—	53	2	no	—
157	C₂×C₈	—	106	1	no	—
158	C₂×C₈	—	106	1	no	—
159	C₂×C₈	—	212	4	no	—
160	C₂×C₈	—	212	8	no	—
161	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
162	C₂×C₈	C₄⋊#6157	1	2	yes	—
163	C₂×C₈	C₄⋊#6157	1	2	yes	—
164	C₂×C₈	#11719940	1	2	yes	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	53	1	no	—
168	C₄²	—	53	1	no	—
169	C₄²	—	53	1	no	—
170	C₄²	—	53	1	no	—
171	C₄²	—	53	2	no	—
172	C₄²	—	53	2	no	—
173	C₄²	—	106	4	no	—
174	C₄²	—	106	4	no	—
175	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	53	1	no	—
179	C₂×C₄	—	53	1	no	—
180	C₂×C₄	—	53	4	no	—
181	C₂×C₄	—	53	1	no	—
182	C₂×C₄	—	53	1	no	—
183	C₂×C₄	—	53	4	no	—
184	C₂×C₄	—	106	2	no	—
185	C₂×C₄	—	106	2	no	—
186	C₂×C₄	—	106	2	no	—
187	C₂×C₄	#13133288	1	4	yes	—
188	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
189	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
190	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
191	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
192	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
193	C₂×C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	53	4	no	—
196	C₈	—	106	2	no	—
197	C₈	—	424	4	no	—
198	C₈	#13133284	1	4	yes	—
199	C₂²	#16811569	1	1	yes	characteristic
200	C₂²	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
201	C₂²	#16811626	1	1	yes	characteristic
202	C₂²	#16811633	1	2	yes	—
203	C₂²	C₂×#13133288	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	53	2	no	—
206	C₄	—	53	2	no	—
207	C₄	—	106	1	no	—
208	C₄	—	106	1	no	—
209	C₄	—	106	4	no	—
210	C₄	D₅₃⋊(C₄⋊C₈)	1	1	yes	characteristic
211	C₄	C₂×#13133284	1	1	yes	characteristic
212	C₄	#16811626	1	1	yes	characteristic
213	C₄	#16811628	1	1	yes	characteristic
214	C₂	#20290265	1	2	yes	—
215	C₂	#20290277	1	2	yes	—
216	C₂	#20290648	1	1	yes	characteristic
217	C₂	#20290652	1	1	yes	characteristic
218	C₂	#20290656	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	#8807
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
27. C₈×Dic₅₃	C₈	semidirect
125. C₅₃	#8807	semidirect

Maximal subgroups

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

subgroup	index
#20290632	2
C₂×#16811486	2
C₂×#16811485	2
C₄×C₈×Dic₅₃	2
#8807	53