Subgroups — C₂²×#16811020 #28253680

Order 13568.

2398 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	C₂²×#13133216	C₂	1	2	yes	maximal
2	C₂×#16811020	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133149	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13133148	C₂	1	2	yes	maximal
5	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₀₆	—	2	2	no	—
8	C₂³×#88322	C₂²	1	1	yes	characteristic
9	C₂×#13133216	—	2	12	no	—
10	C₂×#13133216	C₂²	1	24	yes	—
11	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂²×#11719919	—	2	2	no	—
13	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16811020	C₂²	1	64	yes	direct product factor
15	C₂×#13133149	C₂²	1	12	yes	—
16	C₂×#13133148	C₂²	1	24	yes	—
17	C₂×C₈×Dic₅₃	C₂²	1	12	yes	—
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×#106×C₅₃	C₂²	1	12	yes	—
21	C₂²×Dic₁₀₆	—	2	24	no	—
22	C₂²×Dic₁₀₆	—	4	16	no	—
23	C₂³×Dic₅₃	—	2	2	no	—
24	C₂³×Dic₅₃	D₄	1	2	yes	—
25	C₂²×#88322	Q₈	1	8	yes	complemented
26	C₂²×#88322	C₂³	1	6	yes	—
27	#13133216	—	2	48	no	—
28	#13133216	C₂³	1	32	yes	—
29	C₂×C₄⋊Dic₅₃	—	2	6	no	—
30	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
31	C₂×#11719919	—	2	24	no	—
32	C₂×C₄×Dic₅₃	D₄	1	6	yes	—
33	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
34	#13133149	C₂³	1	16	yes	—
35	#13133148	C₂³	1	32	yes	—
36	C₈×Dic₅₃	C₂³	1	16	yes	—
37	C₂³×C₄×C₅₃	—	2	2	no	—
38	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	Q₈	1	8	yes	complemented
40	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
41	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
42	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
43	#106×C₅₃	C₂³	1	16	yes	—
44	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	12	no	—
46	C₂²×C₄×C₅₃	—	4	8	no	—
47	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
48	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
50	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
51	C₂²×Dic₅₃	—	2	12	no	—
52	C₂²×Dic₅₃	—	4	8	no	—
53	C₂²×Dic₅₃	SD₁₆	1	16	yes	—
54	C₂²×Dic₅₃	C₂×D₄	1	12	yes	—
55	C₂×Dic₁₀₆	—	2	32	no	—
56	C₂×Dic₁₀₆	—	4	96	no	—
57	C₂×C₈×C₅₃	C₂×Q₈	1	24	yes	complemented
58	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
59	C₄⋊C₄×C₅₃	—	2	12	no	—
60	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
61	C₄⋊Dic₅₃	—	2	12	no	—
62	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
63	#11719919	—	2	32	no	—
64	C₄×Dic₅₃	C₂×D₄	1	12	yes	—
65	C₄×Dic₅₃	C₂⁴	1	4	yes	—
66	C₂×#88322	C₂×Q₈	1	24	yes	complemented
67	C₂×#88322	C₂⁴	1	4	yes	—
68	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
69	C₂³×C₅₃	C₂×SD₁₆	1	8	yes	—
70	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
71	C₂×C₄×C₅₃	—	2	8	no	—
72	C₂×C₄×C₅₃	—	4	24	no	—
73	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
74	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
75	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
76	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
77	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
78	C₂×Dic₅₃	—	2	8	no	—
79	C₂×Dic₅₃	—	4	24	no	—
80	C₂×Dic₅₃	C₂×SD₁₆	1	48	yes	—
81	C₂×Dic₅₃	C₂²×D₄	1	8	yes	—
82	Dic₁₀₆	—	4	64	no	—
83	C₈×C₅₃	C₂²×Q₈	1	8	yes	complemented
84	#88322	C₂²×Q₈	1	8	yes	complemented
85	C₂²×#498	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	#498	1	16	yes	complemented
87	C₂²×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
88	C₂²×C₅₃	C₂²×SD₁₆	1	12	yes	—
89	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
90	C₄×C₅₃	—	4	8	no	—
91	C₄×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
92	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
93	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
94	Dic₅₃	—	4	8	no	—
95	Dic₅₃	C₂²×SD₁₆	1	16	yes	—
96	C₂²×C₄⋊Q₈	—	53	2	no	—
97	C₂×#498	—	53	24	no	—
98	C₂²×#106	—	53	1	no	—
99	C₂²×#106	—	53	2	no	—
100	C₂²×#106	D₅₃	1	1	yes	characteristic
101	C₂²×C₄×C₈	—	53	1	no	—
102	C₂×C₅₃	C₂×#498	1	12	yes	complemented
103	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₅₃	C₂³×SD₁₆	1	2	yes	—
105	C₂³×Q₈	—	106	2	no	—
106	C₂³×C₈	—	53	1	no	—
107	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	—	53	24	no	—
109	C₂×C₄⋊Q₈	—	106	12	no	—
110	C₂²×C₄⋊C₄	—	53	2	no	—
111	C₂²×C₄⋊C₄	—	106	2	no	—
112	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
113	C₂²×C₄²	—	53	1	no	—
114	#498	—	53	64	no	—
115	C₂×#106	—	53	12	no	—
116	C₂×#106	—	53	24	no	—
117	C₂×#106	C₂×D₅₃	1	12	yes	—
118	C₂×C₄×C₈	—	53	12	no	—
119	C₅₃	C₂²×#498	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₈	—	106	24	no	—
121	C₂²×Q₈	—	212	16	no	—
122	C₂³×C₄	—	2	2	no	—
123	C₂³×C₄	—	53	2	no	—
124	C₂³×C₄	—	106	2	no	—
125	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
126	C₂²×C₈	—	53	8	no	—
127	C₂²×C₈	—	53	6	no	—
128	C₂²×C₈	Dic₁₀₆	1	8	yes	complemented
129	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
130	C₄⋊Q₈	—	53	32	no	—
131	C₄⋊Q₈	—	106	48	no	—
132	C₂×C₄⋊C₄	—	2	6	no	—
133	C₂×C₄⋊C₄	—	53	12	no	—
134	C₂×C₄⋊C₄	—	106	24	no	—
135	C₂×C₄⋊C₄	—	106	6	no	—
136	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
137	C₂×C₄²	—	53	6	no	—
138	C₂×C₄²	—	53	6	no	—
139	#106	—	53	16	no	—
140	#106	—	53	32	no	—
141	#106	C₂²×D₅₃	1	16	yes	—
142	C₄×C₈	—	53	16	no	—
143	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
144	C₂×Q₈	—	106	32	no	—
145	C₂×Q₈	—	212	96	no	—
146	C₂²×C₄	—	2	12	no	—
147	C₂²×C₄	—	4	8	no	—
148	C₂²×C₄	—	53	16	no	—
149	C₂²×C₄	—	53	12	no	—
150	C₂²×C₄	—	106	12	no	—
151	C₂²×C₄	—	212	8	no	—
152	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
153	C₂²×C₄	D₄×D₅₃	1	3	yes	—
154	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
155	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
156	C₂×C₈	—	53	24	no	—
157	C₂×C₈	—	53	4	no	—
158	C₂×C₈	C₂×Dic₁₀₆	1	24	yes	complemented
159	C₂×C₈	C₂³×D₅₃	1	4	yes	—
160	C₄⋊C₄	—	2	12	no	—
161	C₄⋊C₄	—	53	8	no	—
162	C₄⋊C₄	—	106	32	no	—
163	C₄⋊C₄	—	106	12	no	—
164	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
165	C₄²	—	53	12	no	—
166	C₄²	—	53	4	no	—
167	C₂³	#13133216	1	4	yes	—
168	C₂³	SD₁₆×D₅₃	1	8	yes	—
169	C₂³	C₂×D₄×D₅₃	1	3	yes	—
170	Q₈	—	212	64	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	4	24	no	—
173	C₂×C₄	—	53	48	no	—
174	C₂×C₄	—	53	8	no	—
175	C₂×C₄	—	106	8	no	—
176	C₂×C₄	—	212	24	no	—
177	C₂×C₄	#13133216	1	12	yes	—
178	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
179	C₂×C₄	C₂×Q₈×D₅₃	1	6	yes	—
180	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
181	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
182	C₈	—	53	8	no	—
183	C₈	C₂²×Dic₁₀₆	1	8	yes	complemented
184	C₂²	#16811020	1	16	yes	direct product factor
185	C₂²	C₂×#13133216	1	6	yes	—
186	C₂²	C₂×SD₁₆×D₅₃	1	12	yes	—
187	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
188	C₄	—	4	8	no	—
189	C₄	—	53	16	no	—
190	C₄	—	212	8	no	—
191	C₄	C₂×#13133216	1	6	yes	—
192	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
193	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
194	C₂	C₂×#16811020	1	12	yes	direct product factor
195	C₂	C₂²×#13133216	1	1	yes	characteristic
196	C₂	C₂²×SD₁₆×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×#106×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
hypercenter	C₂²×#106	D₅₃
hypocenter	C₅₃	C₂²×#498
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
2. C₂×#16811020	C₂	direct (N × H)
14. #16811020	C₂²	direct (N × H)
25. C₂²×#88322	Q₈	semidirect
39. C₂²×C₈×C₅₃	Q₈	semidirect
57. C₂×C₈×C₅₃	C₂×Q₈	semidirect
66. C₂×#88322	C₂×Q₈	semidirect
83. C₈×C₅₃	C₂²×Q₈	semidirect
84. #88322	C₂²×Q₈	semidirect
86. C₂²×C₅₃	#498	semidirect
102. C₂×C₅₃	C₂×#498	semidirect
119. C₅₃	C₂²×#498	semidirect
128. C₂²×C₈	Dic₁₀₆	semidirect
158. C₂×C₈	C₂×Dic₁₀₆	semidirect
183. C₈	C₂²×Dic₁₀₆	semidirect
184. C₂²	#16811020	direct (N × H)
194. C₂	C₂×#16811020	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133216	2
C₂×#16811020	2
C₂²×#13133149	2
C₂²×#13133148	2
C₂²×C₈×Dic₅₃	2
C₂²×#106×C₅₃	2
C₂²×#498	53

Subgroups — C22×#16811020 #28253680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811020 #28253680