Subgroups — C₄×#16810381 #27289259

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719917	C₂	1	2	yes	maximal
2	C₂×#16810381	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	C₂²	1	4	yes	—
5	C₄×#11719917	—	2	4	no	—
6	C₄×#11719917	—	2	4	no	—
7	C₂×#13133128	C₄	1	4	yes	—
8	C₂×C₄×#88322	C₄	1	4	yes	—
9	#16810381	C₄	1	16	yes	direct product factor
10	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×M₁₆×C₅₃	—	2	4	no	—
13	C₂×#97×C₅₃	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
15	C₂×#11719917	—	2	4	no	—
16	C₂×#11719917	—	2	4	no	—
17	C₂×#11719917	—	2	8	no	—
18	C₂×#11719917	C₂×C₄	1	16	yes	—
19	C₂²×#88322	C₂×C₄	1	8	yes	—
20	#13133128	—	2	8	no	—
21	C₄×#88322	—	2	8	no	—
22	C₂³×C₄×C₅₃	Q₈	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×M₁₆×C₅₃	—	2	4	no	—
25	C₂×M₁₆×C₅₃	—	2	4	no	—
26	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₅₃	D₄	1	2	yes	—
33	C₂×C₄²×C₅₃	D₄	1	4	yes	—
34	#97×C₅₃	—	2	4	no	—
35	C₄×C₈×C₅₃	—	2	4	no	—
36	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	4	4	no	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅₃	#41	1	4	yes	—
48	M₁₆×C₅₃	—	2	16	no	—
49	C₂×C₈×C₅₃	—	2	4	no	—
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	C₄²	1	8	yes	—
52	C₄²×C₅₃	C₄²	1	2	yes	—
53	C₄²×C₅₃	C₄²	1	1	yes	characteristic
54	C₄²×C₅₃	C₄²	1	1	yes	characteristic
55	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₅₃	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
59	#11719917	—	2	16	no	—
60	#11719917	—	2	16	no	—
61	C₂×#88322	—	2	8	no	—
62	C₂×#88322	—	2	8	no	—
63	C₂×#88322	C₄²	1	16	yes	—
64	C₂³×C₅₃	—	4	2	no	—
65	C₂³×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
66	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂³×C₅₃	C₄×Q₈	1	4	yes	—
68	C₂×C₄×C₅₃	—	2	8	no	—
69	C₂×C₄×C₅₃	—	2	4	no	—
70	C₂×C₄×C₅₃	—	4	4	no	—
71	C₂×C₄×C₅₃	—	4	2	no	—
72	C₂×C₄×C₅₃	#95	1	4	yes	—
73	C₂×C₄×C₅₃	#95	1	2	yes	—
74	C₂×C₄×C₅₃	#95	1	2	yes	—
75	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
76	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
78	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#117	1	4	yes	—
81	C₂×C₄×C₅₃	#117	1	2	yes	—
82	C₂×C₄×C₅₃	C₄×D₄	1	8	yes	—
83	C₂×C₄×C₅₃	C₄×D₄	1	4	yes	—
84	C₈×C₅₃	—	2	8	no	—
85	#88322	—	2	16	no	—
86	C₄×#338	—	53	1	no	maximal, Sylow
87	C₂²×C₅₃	—	2	4	no	—
88	C₂²×C₅₃	—	2	2	no	—
89	C₂²×C₅₃	—	4	4	no	—
90	C₂²×C₅₃	C₂×#95	1	1	yes	characteristic
91	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
92	C₂²×C₅₃	C₄×C₄⋊C₄	1	4	yes	—
93	C₄×C₅₃	—	2	4	no	—
94	C₄×C₅₃	—	2	2	no	—
95	C₄×C₅₃	#338	1	4	yes	complemented
96	C₄×C₅₃	C₄³	1	1	yes	characteristic
97	C₄×C₅₃	#376	1	1	yes	characteristic
98	C₄×C₅₃	C₄×C₂²⋊C₄	1	2	yes	—
99	C₄×C₅₃	C₄×C₂²⋊C₄	1	4	yes	—
100	C₂×C₄×M₁₆	—	53	2	no	—
101	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
102	C₂×#338	—	53	4	no	—
103	C₂×C₅₃	—	2	2	no	—
104	C₂×C₅₃	—	2	4	no	—
105	C₂×C₅₃	C₄×#95	1	1	yes	characteristic
106	C₂×C₅₃	C₂×#338	1	2	yes	—
107	C₂²×M₁₆	—	53	4	no	—
108	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
109	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
110	C₄×M₁₆	—	2	4	no	—
111	C₄×M₁₆	—	106	4	no	—
112	C₄×M₁₆	—	106	4	no	—
113	C₂×#97	—	53	4	no	—
114	C₂×#97	Dic₅₃	1	2	yes	—
115	C₂×C₄×C₈	—	53	4	no	—
116	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
117	#338	—	53	16	no	—
118	C₅₃	C₄×#338	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₄	Dic₁₀₆	1	2	yes	—
120	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
121	C₂×M₁₆	—	2	4	no	—
122	C₂×M₁₆	—	2	4	no	—
123	C₂×M₁₆	—	53	16	no	—
124	C₂×M₁₆	—	106	4	no	—
125	C₂×M₁₆	—	106	8	no	—
126	C₂×M₁₆	—	106	4	no	—
127	C₂×M₁₆	C₄×D₅₃	1	8	yes	—
128	C₂²×C₈	—	53	8	no	—
129	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
130	C₂×C₄²	C₄×D₅₃	1	2	yes	—
131	C₂×C₄²	C₄×D₅₃	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	4	yes	—
136	#97	—	2	4	no	—
137	#97	—	106	8	no	—
138	C₄×C₈	—	2	4	no	—
139	C₄×C₈	—	106	8	no	—
140	C₂⁴	C₂×Dic₁₀₆	1	1	yes	characteristic
141	C₂²×C₄	—	4	4	no	—
142	C₂²×C₄	#11719919	1	4	yes	—
143	C₂²×C₄	#11719919	1	4	yes	—
144	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
145	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
148	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
149	C₂²×C₄	#11719945	1	4	yes	—
150	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
152	M₁₆	—	2	16	no	—
153	M₁₆	—	106	16	no	—
154	M₁₆	—	106	16	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	53	16	no	—
158	C₂×C₈	—	106	8	no	—
159	C₂×C₈	—	106	8	no	—
160	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
161	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
162	C₄²	C₄×Dic₅₃	1	2	yes	—
163	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
164	C₄²	#11719921	1	4	yes	—
165	C₄²	#11719921	1	4	yes	—
166	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
167	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
168	C₂³	—	4	2	no	—
169	C₂³	C₂×#11719919	1	2	yes	—
170	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
171	C₂³	C₄×Dic₁₀₆	1	4	yes	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	#13133124	1	4	yes	—
177	C₂×C₄	#13133124	1	2	yes	—
178	C₂×C₄	#13133124	1	2	yes	—
179	C₂×C₄	C₄²×D₅₃	1	4	yes	—
180	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
182	C₂×C₄	C₂×#11719921	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
184	C₂×C₄	#13133174	1	4	yes	—
185	C₂×C₄	#13133176	1	2	yes	—
186	C₂×C₄	C₄×C₂²⋊D₅₃	1	8	yes	—
187	C₂×C₄	C₄×D₂₁₂	1	4	yes	—
188	C₈	—	2	8	no	—
189	C₈	—	106	16	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	2	no	—
192	C₂²	—	4	4	no	—
193	C₂²	C₂×#13133124	1	1	yes	characteristic
194	C₂²	C₄×C₄⋊Dic₅₃	1	2	yes	—
195	C₂²	C₄×#11719919	1	4	yes	—
196	C₄	—	2	4	no	—
197	C₄	—	2	2	no	—
198	C₄	#16810381	1	4	yes	direct product factor
199	C₄	C₄²×Dic₅₃	1	1	yes	characteristic
200	C₄	#16810469	1	1	yes	characteristic
201	C₄	C₄×C₂²⋊Dic₅₃	1	2	yes	—
202	C₄	C₄×#11719921	1	4	yes	—
203	C₂	—	2	2	no	—
204	C₂	—	2	4	no	—
205	C₂	C₄×#13133124	1	1	yes	characteristic
206	C₂	C₂×#16810381	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄×Dic₅₃
commutator	C₄×C₅₃	C₄³
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#95
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	C₄×#338
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
9. #16810381	C₄	direct (N × H)
95. C₄×C₅₃	#338	semidirect
118. C₅₃	C₄×#338	semidirect
198. C₄	#16810381	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×C₄×#11719917	2
C₂×#16810381	2
C₂×C₄×M₁₆×C₅₃	2
C₄×#338	53

Subgroups — C4×#16810381 #27289259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16810381 #27289259