Subgroups — C₄×#16810448 #27299492

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810448	C₂	1	2	yes	maximal
2	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic
3	C₄×#13133151	C₂	1	1	yes	maximal, characteristic
4	#20272806	C₂	1	2	yes	maximal
5	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂²×#11719908	C₂²	1	1	yes	characteristic
8	C₂×#13133151	C₂²	1	2	yes	—
9	C₄×C₄⋊Dic₅₃	—	2	1	no	—
10	#16810448	C₄	1	8	yes	direct product factor
11	C₄×#11719908	C₄	1	4	yes	complemented
12	#16810386	—	2	1	no	—
13	C₂×#107×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₅₃×#340	—	2	1	no	—
17	C₂×C₄⋊Dic₅₃	—	2	2	no	—
18	C₂×C₄×Dic₅₃	—	4	1	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₂×#11719908	C₂×C₄	1	2	yes	—
21	C₂×#11719908	C₂×C₄	1	2	yes	—
22	C₂×#11719908	C₂×C₄	1	4	yes	—
23	#13133151	—	2	2	no	—
24	#13133151	C₂×C₄	1	4	yes	—
25	#13133124	—	4	1	no	—
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄²×C₅₃	—	4	1	no	—
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	#107×C₅₃	—	2	2	no	—
32	#107×C₅₃	C₂×C₄	1	4	yes	—
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₅₃	Q₈	1	2	yes	—
35	#95×C₅₃	—	4	1	no	—
36	C₂²×C₄×C₅₃	—	4	2	no	—
37	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	2	2	no	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₅₃	#41	1	2	yes	—
47	C₄⋊C₄×C₅₃	—	2	4	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄²×C₅₃	—	8	2	no	—
50	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
51	C₄²×C₅₃	Q₁₆	1	2	yes	—
52	C₄⋊Dic₅₃	—	2	4	no	—
53	C₄⋊Dic₅₃	—	4	2	no	—
54	C₄×Dic₅₃	—	8	2	no	—
55	#11719908	—	2	2	no	—
56	#11719908	C₄²	1	4	yes	complemented
57	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	4	4	no	—
60	C₂×C₄×C₅₃	—	8	2	no	—
61	C₂×C₄×C₅₃	—	8	2	no	—
62	C₂×C₄×C₅₃	#107	1	2	yes	—
63	C₂×C₄×C₅₃	#107	1	2	yes	—
64	C₂×C₄×C₅₃	D₁₆	1	2	yes	—
65	C₂×C₄×C₅₃	Q₃₂	1	2	yes	—
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
68	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₃	#135	1	2	yes	—
70	C₂×Dic₅₃	—	4	4	no	—
71	C₂×Dic₅₃	—	8	2	no	—
72	C₂×Dic₅₃	—	8	2	no	—
73	C₈×C₅₃	—	2	2	no	—
74	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
75	C₈×C₅₃	#117	1	1	yes	characteristic
76	C₈×C₅₃	C₄×Q₈	1	2	yes	—
77	C₄×#366	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
79	C₂²×C₅₃	C₄×D₈	1	2	yes	—
80	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
81	C₂²×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
82	C₂²×C₅₃	#508	1	2	yes	—
83	C₄×C₅₃	—	2	2	no	—
84	C₄×C₅₃	—	8	4	no	—
85	C₄×C₅₃	#366	1	4	yes	complemented
86	C₄×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₄×C₅₃	#427	1	1	yes	characteristic
88	C₄×C₅₃	C₄×Q₁₆	1	2	yes	—
89	Dic₅₃	—	8	4	no	—
90	C₂×#366	—	53	2	no	—
91	C₂×C₄×C₁₆	—	53	1	no	—
92	C₄×#107	—	53	1	no	—
93	C₄×#107	D₅₃	1	1	yes	characteristic
94	#1380	—	53	2	no	—
95	C₂×C₅₃	C₄×#107	1	1	yes	characteristic
96	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
97	C₂×C₅₃	#2158	1	1	yes	characteristic
98	C₂×C₅₃	C₄×D₁₆	1	2	yes	—
99	C₂×C₅₃	C₄×Q₃₂	1	2	yes	—
100	C₂²×C₁₆	—	53	1	no	—
101	C₂²×C₁₆	—	53	1	no	—
102	C₂×#107	—	53	2	no	—
103	C₂×#107	C₂×D₅₃	1	2	yes	—
104	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
105	C₄×C₄⋊C₄	—	2	1	no	—
106	C₄×C₄⋊C₄	—	106	1	no	—
107	#366	—	53	8	no	—
108	C₄×C₁₆	—	53	4	no	—
109	#340	—	2	1	no	—
110	#340	—	106	1	no	—
111	C₅₃	C₄×#366	1	1	yes	characteristic, complemented, Sylow
112	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	106	2	no	—
116	C₂×C₄²	—	4	1	no	—
117	C₂×C₄²	—	212	1	no	—
118	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
119	C₂×C₁₆	—	53	2	no	—
120	C₂×C₁₆	—	53	4	no	—
121	C₂×C₁₆	—	53	2	no	—
122	C₂×C₁₆	—	106	2	no	—
123	#107	—	2	2	no	—
124	#107	—	53	4	no	—
125	#107	—	106	2	no	—
126	#107	C₄×D₅₃	1	4	yes	—
127	C₄×C₈	Dic₁₀₆	1	2	yes	—
128	C₄×C₈	C₄×D₅₃	1	2	yes	—
129	#95	—	4	1	no	—
130	#95	—	212	1	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	212	2	no	—
133	C₂²×C₄	#11719922	1	1	yes	characteristic
134	C₂²×C₄	#11719922	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	#11719919	1	2	yes	—
138	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
140	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
141	C₂×C₈	#11719945	1	2	yes	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	106	4	no	—
145	C₄⋊C₄	—	212	2	no	—
146	C₄²	—	8	2	no	—
147	C₄²	—	424	2	no	—
148	C₄²	#11719919	1	2	yes	—
149	C₄²	#11719925	1	2	yes	—
150	C₁₆	—	53	4	no	—
151	C₁₆	—	106	2	no	—
152	C₂³	C₂×#11719922	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	424	2	no	—
158	C₂×C₄	—	424	2	no	—
159	C₂×C₄	—	212	4	no	—
160	C₂×C₄	#13133150	1	2	yes	—
161	C₂×C₄	#13133150	1	2	yes	—
162	C₂×C₄	#13133158	1	2	yes	—
163	C₂×C₄	#13133163	1	2	yes	—
164	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
165	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
166	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
167	C₂×C₄	#13133237	1	2	yes	—
168	C₈	—	2	2	no	—
169	C₈	C₄²×D₅₃	1	1	yes	characteristic
170	C₈	#13133174	1	1	yes	characteristic
171	C₈	C₄×Dic₁₀₆	1	2	yes	—
172	C₂²	C₂×#13133150	1	1	yes	characteristic
173	C₂²	C₄×#11719922	1	2	yes	—
174	C₂²	C₂×#13133158	1	1	yes	characteristic
175	C₂²	C₂×#13133163	1	1	yes	characteristic
176	C₂²	#16811053	1	2	yes	—
177	C₄	—	2	2	no	—
178	C₄	—	8	4	no	—
179	C₄	—	424	4	no	—
180	C₄	#16810448	1	4	yes	direct product factor
181	C₄	C₄×#11719919	1	1	yes	characteristic
182	C₄	#16810659	1	1	yes	characteristic
183	C₄	C₄×#11719925	1	2	yes	—
184	C₂	C₄×#13133150	1	1	yes	characteristic
185	C₂	C₂×#16810448	1	1	yes	characteristic
186	C₂	#20276497	1	1	yes	characteristic
187	C₂	C₄×#13133158	1	2	yes	—
188	C₂	C₄×#13133163	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#11719922
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₄×#107×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₄×#107	D₅₃
hypocenter	C₅₃	C₄×#366
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
10. #16810448	C₄	direct (N × H)
11. C₄×#11719908	C₄	semidirect
56. #11719908	C₄²	semidirect
85. C₄×C₅₃	#366	semidirect
111. C₅₃	C₄×#366	semidirect
180. C₄	#16810448	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×#16810448	2
C₂×C₄×#11719908	2
C₄×#13133151	2
#20272806	2
C₄×#107×C₅₃	2
C₄×#366	53

Subgroups — C4×#16810448 #27299492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810448 #27299492