Subgroups — #27270996

Order 13568.

286 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	—	2	1	no	—
5	C₂²×#11719908	—	2	1	no	—
6	C₄⋊#11719908	—	2	1	no	—
7	C₄⋊#11719908	C₄	1	2	yes	complemented
8	C₄×#11719908	—	2	1	no	—
9	C₄×#11719908	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	—	2	1	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
14	C₅₃×#336	—	2	1	no	—
15	C₂×#11719908	—	2	2	no	—
16	C₂×#11719908	—	2	2	no	—
17	C₂×#11719908	—	2	2	no	—
18	C₂×#11719908	—	2	2	no	—
19	C₂×#11719908	—	4	1	no	—
20	C₂×#11719908	—	4	1	no	—
21	C₂²×C₈×C₅₃	—	2	1	no	—
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
25	C₂×C₄²×C₅₃	—	2	1	no	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	—	2	1	no	—
28	C₄⋊C₈×C₅₃	C₈	1	2	yes	—
29	C₄×C₈×C₅₃	—	4	1	no	—
30	C₄×C₈×C₅₃	—	4	1	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
35	#95×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	—	2	2	no	—
37	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₅₃	—	2	1	no	—
41	C₂×C₈×C₅₃	—	2	1	no	—
42	C₂×C₈×C₅₃	—	2	2	no	—
43	C₂×C₈×C₅₃	—	4	1	no	—
44	C₂×C₈×C₅₃	—	4	1	no	—
45	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	C₁₆	1	4	yes	complemented
55	C₄²×C₅₃	—	4	1	no	—
56	C₄²×C₅₃	—	4	1	no	—
57	C₄²×C₅₃	C₄²	1	1	yes	characteristic
58	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
61	#11719908	—	2	4	no	—
62	#11719908	—	4	2	no	—
63	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	2	no	—
65	C₂×C₄×C₅₃	—	2	4	no	—
66	C₂×C₄×C₅₃	—	4	2	no	—
67	C₂×C₄×C₅₃	—	4	2	no	—
68	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
73	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
74	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
75	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
76	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
79	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
80	C₈×C₅₃	—	2	1	no	—
81	C₈×C₅₃	—	2	1	no	—
82	C₈×C₅₃	—	4	2	no	—
83	C₈×C₅₃	#102	1	1	yes	characteristic
84	C₈×C₅₃	#103	1	1	yes	characteristic
85	C₈×C₅₃	#106	1	1	yes	characteristic
86	C₈×C₅₃	#107	1	1	yes	characteristic
87	#7167	—	53	1	no	maximal, Sylow
88	C₂²×C₅₃	D₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
90	C₂²×C₅₃	#336	1	1	yes	characteristic
91	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
92	C₂²×C₅₃	#350	1	2	yes	—
93	C₄×C₅₃	—	2	1	no	—
94	C₄×C₅₃	—	2	1	no	—
95	C₄×C₅₃	—	4	4	no	—
96	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
97	C₄×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
98	C₄×C₅₃	#339	1	1	yes	characteristic
99	C₄×C₅₃	#340	1	1	yes	characteristic
100	C₄×C₅₃	C₄×C₁₆	1	1	yes	characteristic
101	C₄×C₅₃	#346	1	1	yes	characteristic
102	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
103	C₂×C₄⋊C₁₆	—	53	1	no	—
104	C₂×C₄×C₁₆	—	53	1	no	—
105	C₈×C₄⋊C₄	D₅₃	1	1	yes	characteristic
106	C₂×C₅₃	#1279	1	1	yes	characteristic
107	C₂×C₅₃	D₄⋊C₁₆	1	2	yes	—
108	C₂×C₅₃	Q₈⋊C₁₆	1	2	yes	—
109	C₂×C₅₃	#1374	1	1	yes	characteristic
110	C₂×C₅₃	#1379	1	1	yes	characteristic
111	C₂²×C₁₆	—	106	1	no	—
112	C₂²×C₁₆	—	106	1	no	—
113	C₂×C₄⋊C₈	Dic₅₃	1	1	yes	characteristic
114	C₂×C₄×C₈	—	2	1	no	—
115	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
116	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
117	C₄⋊C₁₆	—	53	2	no	—
118	C₄⋊C₁₆	—	106	1	no	—
119	C₄×C₁₆	—	53	2	no	—
120	C₄×C₁₆	—	106	1	no	—
121	#336	—	2	1	no	—
122	C₅₃	#7167	1	1	yes	characteristic, complemented, Sylow
123	C₂²×C₈	—	2	1	no	—
124	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
126	C₂×C₄⋊C₄	#88322	1	2	yes	—
127	C₂×C₄²	—	2	1	no	—
128	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
129	C₂×C₁₆	—	106	2	no	—
130	C₂×C₁₆	—	106	2	no	—
131	C₂×C₁₆	—	106	2	no	—
132	C₂×C₁₆	—	106	2	no	—
133	C₂×C₁₆	—	212	1	no	—
134	C₂×C₁₆	—	212	1	no	—
135	C₄⋊C₈	—	2	1	no	—
136	C₄⋊C₈	#88322	1	2	yes	—
137	C₄×C₈	—	4	1	no	—
138	C₄×C₈	—	4	1	no	—
139	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
140	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
141	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
142	C₄×C₈	D₂₁₂	1	1	yes	characteristic
143	#95	—	2	1	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
146	C₂²×C₄	#11719917	1	1	yes	characteristic
147	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	#11719919	1	1	yes	characteristic
154	C₂×C₈	#11719919	1	1	yes	characteristic
155	C₂×C₈	#11719921	1	1	yes	characteristic
156	C₂×C₈	#11719921	1	1	yes	characteristic
157	C₂×C₈	#11719922	1	1	yes	characteristic
158	C₂×C₈	#11719923	1	1	yes	characteristic
159	C₂×C₈	#11719924	1	1	yes	characteristic
160	C₂×C₈	#11719925	1	1	yes	characteristic
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	#11719908	1	4	yes	complemented
163	C₄²	—	4	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	C₂×#88322	1	1	yes	characteristic
166	C₄²	#11719917	1	1	yes	characteristic
167	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
168	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
169	C₁₆	—	106	4	no	—
170	C₁₆	—	212	2	no	—
171	C₂³	C₂²⋊#88322	1	1	yes	characteristic
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	#13133124	1	1	yes	characteristic
177	C₂×C₄	C₄×#88322	1	1	yes	characteristic
178	C₂×C₄	#13133128	1	1	yes	characteristic
179	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
180	C₂×C₄	#13133138	1	1	yes	characteristic
181	C₂×C₄	#13133141	1	1	yes	characteristic
182	C₂×C₄	#13133144	1	1	yes	characteristic
183	C₂×C₄	#13133144	1	1	yes	characteristic
184	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
185	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
186	C₂×C₄	C₂×#11719908	1	2	yes	—
187	C₂×C₄	#13133157	1	2	yes	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	4	2	no	—
191	C₈	#13133137	1	1	yes	characteristic
192	C₈	#13133140	1	1	yes	characteristic
193	C₈	#13133148	1	1	yes	characteristic
194	C₈	#13133150	1	1	yes	characteristic
195	C₂²	D₄⋊#88322	1	1	yes	characteristic
196	C₂²	Q₈⋊#88322	1	1	yes	characteristic
197	C₂²	#16810378	1	1	yes	characteristic
198	C₂²	C₂²⋊#11719908	1	2	yes	—
199	C₂²	#16810407	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	4	no	—
203	C₄	D₄⋊#88322	1	1	yes	characteristic
204	C₄	Q₈⋊#88322	1	1	yes	characteristic
205	C₄	#16810382	1	1	yes	characteristic
206	C₄	#16810385	1	1	yes	characteristic
207	C₄	C₄×#11719908	1	1	yes	characteristic
208	C₄	#16810399	1	1	yes	characteristic
209	C₄	C₄⋊#11719908	1	2	yes	—
210	C₂	#20272556	1	1	yes	characteristic
211	C₂	D₄⋊#11719908	1	2	yes	—
212	C₂	Q₈⋊#11719908	1	2	yes	—
213	C₂	#20272793	1	1	yes	characteristic
214	C₂	#20272804	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₄×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₅₃
Fitting	C₈×C₄⋊C₄×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₅₃	C₂²
hypercenter	C₈×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#7167
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₄⋊#11719908	C₄	semidirect
9. C₄×#11719908	C₄	semidirect
54. C₄⋊C₄×C₅₃	C₁₆	semidirect
122. C₅₃	#7167	semidirect
162. C₄⋊C₄	#11719908	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11719908	2
C₂×C₄×#11719908	2
C₈×C₄⋊C₄×C₅₃	2
#7167	53