Subgroups — #27277837

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	#20274832	C₂	1	2	yes	maximal
2	C₂×#16810376	C₂	1	2	yes	maximal
3	C₂×#16810374	C₂	1	2	yes	maximal
4	C₄²×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#88322	C₂²	1	4	yes	—
6	#16810378	—	2	4	no	—
7	#16810376	—	2	4	no	—
8	#16810374	—	2	4	no	—
9	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄³×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×#88322	—	2	4	no	—
17	C₄⋊#88322	—	2	4	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	D₄	1	2	yes	—
26	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₄×C₈×C₅₃	—	2	2	no	—
29	C₄×C₈×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	C₈	1	4	yes	complemented
32	C₄×C₈×C₅₃	D₄	1	2	yes	—
33	C₄×C₈×C₅₃	D₄	1	2	yes	—
34	C₄×C₈×C₅₃	Q₈	1	2	yes	—
35	C₄×C₈×C₅₃	Q₈	1	2	yes	—
36	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	—	2	1	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	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
52	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
53	C₄²×C₅₃	—	2	4	no	—
54	C₄²×C₅₃	—	2	1	no	—
55	C₄²×C₅₃	—	2	1	no	—
56	C₄²×C₅₃	—	2	4	no	—
57	C₄²×C₅₃	C₂×C₈	1	2	yes	—
58	C₄²×C₅₃	M₁₆	1	2	yes	—
59	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₂×#88322	—	4	4	no	—
64	C₂×#88322	—	4	8	no	—
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	—	2	2	no	—
69	C₂×C₄×C₅₃	—	2	2	no	—
70	C₂×C₄×C₅₃	—	2	2	no	—
71	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
72	C₂×C₄×C₅₃	#108	1	4	yes	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#135	1	4	yes	—
84	C₈×C₅₃	—	2	2	no	—
85	C₈×C₅₃	—	2	2	no	—
86	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
87	C₈×C₅₃	C₄⋊C₈	1	4	yes	—
88	#88322	—	8	4	no	—
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	—	2	4	no	—
97	C₄×C₅₃	—	2	1	no	—
98	C₄×C₅₃	—	2	1	no	—
99	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
101	C₄×C₅₃	#423	1	1	yes	characteristic
102	C₄×C₅₃	#423	1	1	yes	characteristic
103	C₄×C₅₃	#495	1	2	yes	—
104	C₄×C₅₃	#499	1	2	yes	—
105	#1831	—	53	2	no	—
106	C₄²×C₈	D₅₃	1	1	yes	characteristic
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	Dic₅₃	1	1	yes	characteristic
116	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
117	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
118	C₂×C₄×C₈	Dic₅₃	1	1	yes	characteristic
119	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
121	C₄³	C₂×D₅₃	1	1	yes	characteristic
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	—	106	4	no	—
127	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
128	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
129	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
130	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
131	C₂×C₄²	Dic₁₀₆	1	2	yes	—
132	C₂×C₄²	D₂₁₂	1	2	yes	—
133	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
134	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
135	C₂×C₄²	C₂²×D₅₃	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	—	2	2	no	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	#88322	1	4	yes	complemented
142	C₄×C₈	Dic₁₀₆	1	2	yes	—
143	C₄×C₈	Dic₁₀₆	1	2	yes	—
144	C₄×C₈	D₂₁₂	1	2	yes	—
145	C₄×C₈	D₂₁₂	1	2	yes	—
146	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
147	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
148	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
150	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	1	no	—
155	C₂×C₈	—	212	4	no	—
156	C₂×C₈	—	212	8	no	—
157	C₂×C₈	C₂×#88322	1	2	yes	—
158	C₂×C₈	C₂×#88322	1	2	yes	—
159	C₂×C₈	#11719917	1	2	yes	—
160	C₂×C₈	#11719917	1	2	yes	—
161	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
162	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
163	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
164	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	4	no	—
169	C₄²	C₂×#88322	1	2	yes	—
170	C₄²	#11719917	1	2	yes	—
171	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
172	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
173	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
174	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
175	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	C₄⋊#88322	1	4	yes	—
182	C₂×C₄	#13133153	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
185	C₂×C₄	#13133214	1	1	yes	characteristic
186	C₂×C₄	#13133217	1	1	yes	characteristic
187	C₂×C₄	#13133217	1	1	yes	characteristic
188	C₂×C₄	#13133217	1	1	yes	characteristic
189	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
190	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
191	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
192	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
193	C₂×C₄	#13133239	1	4	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	2	2	no	—
196	C₈	—	424	4	no	—
197	C₈	C₄⋊#88322	1	4	yes	—
198	C₈	C₄⋊#88322	1	4	yes	—
199	C₂²	#16810493	1	1	yes	characteristic
200	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
201	C₂²	#16810647	1	1	yes	characteristic
202	C₂²	#16810661	1	2	yes	—
203	C₂²	C₂×#13133153	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	2	4	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
209	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
210	C₄	#16810647	1	1	yes	characteristic
211	C₄	#16810647	1	1	yes	characteristic
212	C₄	#16811007	1	2	yes	—
213	C₄	#16811025	1	2	yes	—
214	C₂	#20273463	1	2	yes	—
215	C₂	#20273478	1	2	yes	—
216	C₂	#20274867	1	1	yes	characteristic
217	C₂	#20274879	1	1	yes	characteristic
218	C₂	#20274887	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×C₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×C₈	D₅₃
hypocenter	C₅₃	#8807
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
31. C₄×C₈×C₅₃	C₈	semidirect
125. C₅₃	#8807	semidirect
141. C₄×C₈	#88322	semidirect

Maximal subgroups

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

subgroup	index
#20274832	2
C₂×#16810376	2
C₂×#16810374	2
C₄²×C₈×C₅₃	2
#8807	53