Subgroups — C₂²×#16811682 #28330790

Order 13568.

2118 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811682	C₂	1	24	yes	maximal, direct product factor
2	C₂²×#13133292	C₂	1	2	yes	maximal, complemented
3	C₂²×#13133291	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₅₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂³×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13133225	C₂	1	1	yes	maximal, characteristic
7	#16811682	C₂²	1	64	yes	direct product factor
8	C₂×#13133292	C₂²	1	24	yes	complemented
9	C₂×#13133291	C₂²	1	12	yes	complemented
10	C₂×D₅₃⋊C₁₆	C₂²	1	24	yes	complemented
11	C₂²×#11719910	C₂²	1	1	yes	characteristic
12	C₂²×#11719910	C₂²	1	1	yes	characteristic
13	C₂²×#11719910	C₂²	1	2	yes	—
14	C₂²×#11719910	C₂²	1	12	yes	complemented
15	C₂²×#11719945	C₄	1	1	yes	characteristic
16	C₂×#13133225	C₄	1	12	yes	—
17	C₂×#13133225	C₂²	1	12	yes	—
18	C₂²×#11719912	C₄	1	2	yes	—
19	C₂³×#88322	C₂²	1	1	yes	characteristic
20	C₂²×C₈×D₅₃	C₂²	1	2	yes	—
21	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
22	#13133292	C₂³	1	32	yes	complemented
23	#13133291	C₂³	1	16	yes	complemented
24	D₅₃⋊C₁₆	C₂³	1	32	yes	complemented
25	C₂×#11719910	C₂³	1	6	yes	—
26	C₂×#11719910	C₂³	1	6	yes	—
27	C₂×#11719910	C₂³	1	12	yes	—
28	C₂×#11719910	C₂³	1	16	yes	complemented
29	C₂×#11719945	C₈	1	12	yes	—
30	C₂×#11719945	C₂×C₄	1	12	yes	—
31	C₂²×Dic₁₀₆	C₈	1	1	yes	characteristic
32	C₂²×D₂₁₂	C₈	1	1	yes	characteristic
33	C₂²×C₂²⋊D₅₃	C₈	1	2	yes	—
34	C₂²×C₄×D₅₃	C₂×C₄	1	2	yes	—
35	#13133225	C₂×C₄	1	48	yes	—
36	#13133225	C₂³	1	16	yes	—
37	C₂×#11719912	C₂×C₄	1	24	yes	—
38	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
39	C₂²×#88322	C₂×C₄	1	6	yes	—
40	C₂²×#88322	C₂³	1	1	yes	characteristic
41	C₂²×#88322	C₂³	1	6	yes	—
42	C₂×C₈×D₅₃	C₂×C₄	1	12	yes	—
43	C₂×C₈×D₅₃	C₂³	1	12	yes	—
44	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×M₁₆×C₅₃	C₂×C₄	1	12	yes	—
46	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂³×D₅₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×C₈	1	6	yes	—
51	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
52	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊D₅₃	C₂×C₈	1	24	yes	—
54	C₂²×Dic₅₃	C₂×C₈	1	2	yes	—
55	#11719945	C₂×C₈	1	48	yes	—
56	#11719945	C₂²×C₄	1	16	yes	—
57	C₂×D₂₁₂	C₂×C₈	1	12	yes	—
58	C₂×C₄×D₅₃	C₂×C₈	1	12	yes	—
59	C₂×C₄×D₅₃	C₂²×C₄	1	12	yes	—
60	C₂×Dic₁₀₆	C₂×C₈	1	12	yes	—
61	M₁₆×C₅₃	C₂²×C₄	1	16	yes	—
62	C₂×C₈×C₅₃	C₂²×C₄	1	12	yes	—
63	C₂×#88322	C₂²×C₄	1	3	yes	—
64	C₂×#88322	C₂²×C₄	1	6	yes	—
65	C₂×#88322	C₂²×C₄	1	12	yes	—
66	C₂×#88322	C₂⁴	1	3	yes	—
67	C₂×#88322	C₂⁴	1	4	yes	—
68	#11719912	C₂²×C₄	1	32	yes	—
69	C₈×D₅₃	C₂²×C₄	1	24	yes	—
70	C₈×D₅₃	C₂⁴	1	8	yes	—
71	#11719910	C₂⁴	1	4	yes	—
72	#11719910	C₂⁴	1	4	yes	—
73	#11719910	C₂⁴	1	8	yes	—
74	C₂³×C₅₃	—	2	4	no	—
75	C₂³×C₅₃	C₂²×C₈	1	6	yes	—
76	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
77	C₂²×D₅₃	—	2	8	no	—
78	C₂²×D₅₃	C₂²×C₈	1	12	yes	—
79	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
80	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
81	C₂×C₄×C₅₃	C₂²×C₈	1	12	yes	—
82	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
83	C₂×C₄×C₅₃	C₂³×C₄	1	4	yes	—
84	C₂²⋊D₅₃	C₂²×C₈	1	32	yes	—
85	C₂×Dic₅₃	C₂²×C₈	1	12	yes	—
86	D₂₁₂	C₂²×C₈	1	16	yes	—
87	C₄×D₅₃	C₂²×C₈	1	24	yes	—
88	C₄×D₅₃	C₂³×C₄	1	8	yes	—
89	Dic₁₀₆	C₂²×C₈	1	16	yes	—
90	C₈×C₅₃	C₂³×C₄	1	8	yes	—
91	#88322	C₂³×C₄	1	3	yes	—
92	#88322	C₂³×C₄	1	4	yes	—
93	#88322	C₂⁵	1	1	yes	characteristic
94	C₂²×#504	—	53	1	no	maximal, Sylow
95	C₂²×C₅₃	—	2	12	no	—
96	C₂²×C₅₃	#504	1	4	yes	complemented
97	C₂²×C₅₃	C₂³×C₈	1	4	yes	—
98	C₂²×C₅₃	C₂³×C₈	1	3	yes	—
99	C₂×D₅₃	—	2	24	no	—
100	C₂×D₅₃	C₂³×C₈	1	8	yes	—
101	C₄×C₅₃	C₂³×C₈	1	4	yes	—
102	C₄×C₅₃	C₂³×C₈	1	3	yes	—
103	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
104	Dic₅₃	C₂³×C₈	1	8	yes	—
105	C₂²×#131	—	53	1	no	—
106	C₂×#504	—	53	24	no	—
107	C₂²×M₃₂	—	53	1	no	—
108	C₂²×M₃₂	—	53	2	no	—
109	C₂³×C₁₆	—	53	1	no	—
110	C₂³×C₁₆	—	53	2	no	—
111	C₂×C₅₃	—	2	4	no	—
112	C₂×C₅₃	C₂×#504	1	6	yes	complemented
113	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
114	D₅₃	—	2	8	no	—
115	C₂²×#41	—	53	1	no	—
116	C₂×#131	—	53	12	no	—
117	C₂×#131	—	53	12	no	—
118	C₂²×M₁₆	—	53	2	no	—
119	C₂²×M₁₆	#6157	1	1	yes	characteristic
120	C₂³×C₈	—	53	1	no	—
121	C₂³×C₈	—	53	2	no	—
122	#504	—	53	64	no	—
123	C₂×M₃₂	—	53	12	no	—
124	C₂×M₃₂	—	53	24	no	—
125	C₂²×C₁₆	—	53	1	no	—
126	C₂²×C₁₆	—	53	1	no	—
127	C₂²×C₁₆	—	53	2	no	—
128	C₂²×C₁₆	—	53	12	no	—
129	C₂²×C₁₆	—	53	24	no	—
130	C₅₃	C₂²×#504	1	1	yes	characteristic, complemented, Sylow
131	C₂×#41	—	53	12	no	—
132	C₂×#41	—	53	12	no	—
133	C₂²×Q₈	—	53	1	no	—
134	C₂²×D₄	—	53	2	no	—
135	C₂²×D₄	—	53	1	no	—
136	C₂³×C₄	—	53	2	no	—
137	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
138	#131	—	53	48	no	—
139	#131	—	53	16	no	—
140	C₂×M₁₆	—	53	24	no	—
141	C₂×M₁₆	C₂×#6157	1	12	yes	—
142	C₂²×C₈	—	53	6	no	—
143	C₂²×C₈	—	53	6	no	—
144	C₂²×C₈	—	53	1	no	—
145	C₂²×C₈	—	53	1	no	—
146	C₂²×C₈	—	53	12	no	—
147	C₂²×C₈	—	53	12	no	—
148	C₂²×C₈	C₂×#6157	1	2	yes	—
149	M₃₂	—	53	16	no	—
150	M₃₂	—	53	32	no	—
151	C₂×C₁₆	—	53	6	no	—
152	C₂×C₁₆	—	53	6	no	—
153	C₂×C₁₆	—	53	12	no	—
154	C₂×C₁₆	—	53	16	no	—
155	C₂×C₁₆	—	53	32	no	—
156	C₂⁴	—	53	2	no	—
157	C₂⁴	D₅₃⋊C₈	1	1	yes	characteristic
158	#41	—	53	48	no	—
159	#41	—	53	16	no	—
160	C₂×Q₈	—	53	12	no	—
161	C₂×D₄	—	53	24	no	—
162	C₂×D₄	—	53	12	no	—
163	C₂²×C₄	—	53	2	no	—
164	C₂²×C₄	—	53	12	no	—
165	C₂²×C₄	—	53	12	no	—
166	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
167	C₂²×C₄	D₅₃⋊C₈	1	6	yes	—
168	C₂²×C₄	C₂²×#6157	1	6	yes	—
169	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
170	M₁₆	—	53	32	no	—
171	M₁₆	C₂²×#6157	1	16	yes	—
172	C₂×C₈	—	53	3	no	—
173	C₂×C₈	—	53	4	no	—
174	C₂×C₈	—	53	3	no	—
175	C₂×C₈	—	53	6	no	—
176	C₂×C₈	—	53	12	no	—
177	C₂×C₈	—	53	8	no	—
178	C₂×C₈	—	53	24	no	—
179	C₂×C₈	C₂²×#6157	1	12	yes	—
180	C₁₆	—	53	4	no	—
181	C₁₆	—	53	4	no	—
182	C₁₆	—	53	8	no	—
183	C₂³	—	2	4	no	—
184	C₂³	—	53	12	no	—
185	C₂³	—	106	8	no	—
186	C₂³	C₂×D₅₃⋊C₈	1	6	yes	—
187	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
188	Q₈	—	53	16	no	—
189	D₄	—	53	32	no	—
190	D₄	—	53	16	no	—
191	C₂×C₄	—	53	12	no	—
192	C₂×C₄	—	53	24	no	—
193	C₂×C₄	—	53	8	no	—
194	C₂×C₄	C₂×D₅₃⋊C₈	1	3	yes	—
195	C₂×C₄	C₂×D₅₃⋊C₈	1	6	yes	—
196	C₂×C₄	C₂×D₅₃⋊C₈	1	12	yes	—
197	C₂×C₄	C₂³×#6157	1	3	yes	—
198	C₂×C₄	C₂³×#6157	1	4	yes	—
199	C₈	—	53	3	no	—
200	C₈	—	53	4	no	—
201	C₈	—	53	1	no	—
202	C₈	C₂³×#6157	1	8	yes	—
203	C₂²	—	2	12	no	—
204	C₂²	—	53	8	no	—
205	C₂²	—	106	24	no	—
206	C₂²	#16811682	1	4	yes	direct product factor
207	C₂²	C₂²×D₅₃⋊C₈	1	4	yes	—
208	C₂²	C₂²×D₅₃⋊C₈	1	3	yes	—
209	C₄	—	53	8	no	—
210	C₄	C₂²×D₅₃⋊C₈	1	3	yes	—
211	C₄	C₂²×D₅₃⋊C₈	1	4	yes	—
212	C₄	C₂⁴×#6157	1	1	yes	characteristic
213	C₂	—	2	4	no	—
214	C₂	—	106	8	no	—
215	C₂	C₂×#16811682	1	6	yes	direct product factor
216	C₂	C₂³×D₅₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#6157
Fitting	C₂²×M₁₆×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
prime radical	C₂²×#11719945	C₄
cosocle	#88322	C₂⁵
hypercenter	C₂²×M₁₆	#6157
hypocenter	C₅₃	C₂²×#504
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16811682	C₂	direct (N × H)
2. C₂²×#13133292	C₂	semidirect
3. C₂²×#13133291	C₂	semidirect
4. C₂²×D₅₃⋊C₁₆	C₂	semidirect
5. C₂³×#11719910	C₂	semidirect
7. #16811682	C₂²	direct (N × H)
8. C₂×#13133292	C₂²	semidirect
9. C₂×#13133291	C₂²	semidirect
10. C₂×D₅₃⋊C₁₆	C₂²	semidirect
14. C₂²×#11719910	C₂²	semidirect
22. #13133292	C₂³	semidirect
23. #13133291	C₂³	semidirect
24. D₅₃⋊C₁₆	C₂³	semidirect
28. C₂×#11719910	C₂³	semidirect
96. C₂²×C₅₃	#504	semidirect
112. C₂×C₅₃	C₂×#504	semidirect
130. C₅₃	C₂²×#504	semidirect
206. C₂²	#16811682	direct (N × H)
215. C₂	C₂×#16811682	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×#16811682	2
C₂²×#13133292	2
C₂²×#13133291	2
C₂²×D₅₃⋊C₁₆	2
C₂³×#11719910	2
C₂²×#13133225	2
C₂²×#504	53

Subgroups — C22×#16811682 #28330790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811682 #28330790