Subgroups — C₄×#16810446 #27299476

Order 13568.

330 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810446	C₂	1	2	yes	maximal
2	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133149	C₂	1	1	yes	maximal, characteristic
4	#20272814	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133153	C₂²	1	2	yes	—
7	C₂×#13133149	C₂²	1	2	yes	—
8	C₄×#11719917	—	2	1	no	—
9	C₄×C₄⋊Dic₅₃	—	2	1	no	—
10	#16810446	C₄	1	8	yes	direct product factor
11	#16810388	—	2	1	no	—
12	#16810386	—	2	1	no	—
13	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#346	C₄	1	2	yes	complemented
17	C₄×C₁₆×C₅₃	C₄	1	2	yes	complemented
18	C₂×#11719917	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	—	2	2	no	—
20	C₂×C₄×Dic₅₃	—	4	1	no	—
21	#13133153	—	2	2	no	—
22	#13133153	C₂×C₄	1	4	yes	—
23	#13133149	—	2	2	no	—
24	#13133149	C₂×C₄	1	4	yes	—
25	#13133128	—	4	1	no	—
26	C₄×#88322	—	4	1	no	—
27	#13133124	—	4	1	no	—
28	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	M₃₂×C₅₃	—	2	2	no	—
32	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
38	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×Dic₅₃	—	4	2	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	#41	1	2	yes	—
50	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
53	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
54	C₄⋊Dic₅₃	—	2	4	no	—
55	C₄⋊Dic₅₃	—	4	2	no	—
56	C₄×Dic₅₃	—	8	2	no	—
57	#11719917	—	2	4	no	—
58	#11719917	—	4	2	no	—
59	C₂×#88322	—	4	2	no	—
60	C₁₆×C₅₃	—	2	2	no	—
61	C₁₆×C₅₃	C₄²	1	4	yes	complemented
62	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	2	no	—
64	C₂×C₄×C₅₃	—	4	1	no	—
65	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
66	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
67	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
68	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×C₄×C₅₃	#135	1	2	yes	—
73	C₂×Dic₅₃	—	4	4	no	—
74	C₂×Dic₅₃	—	8	2	no	—
75	C₂×Dic₅₃	—	8	2	no	—
76	C₈×C₅₃	—	2	1	no	—
77	C₈×C₅₃	—	2	1	no	—
78	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
79	C₈×C₅₃	#117	1	1	yes	characteristic
80	C₈×C₅₃	C₄×Q₈	1	2	yes	—
81	#88322	—	4	4	no	—
82	C₄×#365	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	—	2	1	no	—
84	C₂²×C₅₃	—	2	1	no	—
85	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
86	C₂²×C₅₃	C₄×SD₁₆	1	2	yes	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	—	2	1	no	—
90	C₄×C₅₃	#365	1	2	yes	complemented
91	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₅₃	#427	1	1	yes	characteristic
93	C₄×C₅₃	C₄×SD₁₆	1	2	yes	—
94	Dic₅₃	—	8	4	no	—
95	C₂×#365	—	53	2	no	—
96	C₄×M₃₂	D₅₃	1	1	yes	characteristic
97	C₄×#108	—	53	1	no	—
98	C₄×#106	—	53	1	no	—
99	#1383	—	53	2	no	—
100	C₂×C₅₃	—	2	2	no	—
101	C₂×C₅₃	C₄×#106	1	1	yes	characteristic
102	C₂×C₅₃	C₂×#365	1	1	yes	characteristic
103	C₂×C₅₃	#2155	1	1	yes	characteristic
104	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
105	C₂×M₃₂	C₂×D₅₃	1	1	yes	characteristic
106	C₂×#108	—	53	2	no	—
107	C₂×#106	—	53	2	no	—
108	C₄×M₁₆	—	106	1	no	—
109	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
110	C₄×C₄⋊C₄	—	106	1	no	—
111	#365	—	53	8	no	—
112	#346	Dic₅₃	1	2	yes	complemented
113	C₄×C₁₆	Dic₅₃	1	2	yes	complemented
114	#341	—	106	1	no	—
115	#340	—	106	1	no	—
116	C₅₃	C₄×#365	1	1	yes	characteristic, complemented, Sylow
117	C₂×M₁₆	—	106	2	no	—
118	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	106	2	no	—
121	C₂×C₄²	—	212	1	no	—
122	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
123	M₃₂	—	2	2	no	—
124	M₃₂	C₄×D₅₃	1	4	yes	—
125	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
126	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
127	#108	—	53	4	no	—
128	#108	—	106	2	no	—
129	#106	—	53	4	no	—
130	#106	—	106	2	no	—
131	#97	—	212	1	no	—
132	C₄×C₈	—	212	1	no	—
133	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
134	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
135	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
136	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
137	#95	—	212	1	no	—
138	C₂²×C₄	—	212	2	no	—
139	C₂²×C₄	#11719913	1	1	yes	characteristic
140	C₂²×C₄	#11719913	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
142	M₁₆	—	106	4	no	—
143	M₁₆	—	212	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	212	2	no	—
146	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
147	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
148	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
150	C₂×C₈	#11719945	1	2	yes	—
151	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
152	C₄⋊C₄	—	106	4	no	—
153	C₄⋊C₄	—	212	2	no	—
154	C₄²	—	424	2	no	—
155	C₄²	#11719913	1	1	yes	characteristic
156	C₄²	#11719913	1	1	yes	characteristic
157	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
158	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
159	C₁₆	—	2	2	no	—
160	C₁₆	C₄×Dic₅₃	1	4	yes	complemented
161	C₂³	C₂×#11719913	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	424	2	no	—
165	C₂×C₄	—	424	2	no	—
166	C₂×C₄	—	212	4	no	—
167	C₂×C₄	#13133149	1	1	yes	characteristic
168	C₂×C₄	#13133149	1	1	yes	characteristic
169	C₂×C₄	#13133149	1	1	yes	characteristic
170	C₂×C₄	#13133149	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
172	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
173	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
174	C₂×C₄	#13133239	1	2	yes	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	212	4	no	—
178	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
179	C₈	#13133176	1	1	yes	characteristic
180	C₈	C₄×Dic₁₀₆	1	2	yes	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	C₂×#13133149	1	1	yes	characteristic
184	C₂²	C₄×#11719913	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	424	4	no	—
189	C₄	#16810446	1	2	yes	direct product factor
190	C₄	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
191	C₄	#16810661	1	1	yes	characteristic
192	C₄	C₄×#11719913	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	C₄×#13133149	1	1	yes	characteristic
195	C₂	C₂×#16810446	1	1	yes	characteristic
196	C₂	#20276488	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133149
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×M₃₂	D₅₃
hypocenter	C₅₃	C₄×#365
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. #16810446	C₄	direct (N × H)
16. C₅₃×#346	C₄	semidirect
17. C₄×C₁₆×C₅₃	C₄	semidirect
61. C₁₆×C₅₃	C₄²	semidirect
90. C₄×C₅₃	#365	semidirect
112. #346	Dic₅₃	semidirect
113. C₄×C₁₆	Dic₅₃	semidirect
116. C₅₃	C₄×#365	semidirect
160. C₁₆	C₄×Dic₅₃	semidirect
189. C₄	#16810446	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16810446	2
C₄×#13133153	2
C₄×#13133149	2
#20272814	2
C₄×M₃₂×C₅₃	2
C₄×#365	53

Subgroups — C4×#16810446 #27299476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810446 #27299476