Subgroups — C₂²×#16811681 #28330788

Order 13568.

2118 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811681	C₂	1	24	yes	maximal, direct product factor
2	C₂²×#13133291	C₂	1	3	yes	maximal, complemented
3	C₂³×#11719910	C₂	1	3	yes	maximal, complemented
4	C₂²×#13133224	C₂	1	1	yes	maximal, characteristic
5	#16811681	C₂²	1	64	yes	direct product factor
6	C₂×#13133291	C₂²	1	36	yes	complemented
7	C₂²×#11719910	C₂²	1	1	yes	characteristic
8	C₂²×#11719910	C₂²	1	3	yes	—
9	C₂²×#11719910	C₂²	1	36	yes	complemented
10	C₂×#13133224	C₄	1	12	yes	—
11	C₂×#13133224	C₂²	1	12	yes	—
12	C₂²×#11719917	C₄	1	3	yes	—
13	C₂³×#88322	C₂²	1	3	yes	—
14	C₂²×#41×C₅₃	C₄	1	1	yes	characteristic
15	#13133291	C₂³	1	48	yes	complemented
16	C₂×#11719910	C₂³	1	6	yes	—
17	C₂×#11719910	C₂³	1	18	yes	—
18	C₂×#11719910	C₂³	1	48	yes	complemented
19	#13133224	C₂×C₄	1	48	yes	—
20	#13133224	C₂³	1	16	yes	—
21	C₂×#11719917	C₂×C₄	1	36	yes	—
22	C₂²×#88322	C₂×C₄	1	3	yes	—
23	C₂²×#88322	C₂×C₄	1	18	yes	—
24	C₂²×#88322	C₂³	1	1	yes	characteristic
25	C₂²×#88322	C₂³	1	18	yes	—
26	C₂×#41×C₅₃	C₈	1	12	yes	—
27	C₂×#41×C₅₃	C₂×C₄	1	12	yes	—
28	C₂²×Q₈×C₅₃	C₈	1	1	yes	characteristic
29	C₂²×D₄×C₅₃	C₈	1	3	yes	—
30	C₂³×C₄×C₅₃	C₂×C₄	1	3	yes	—
31	C₂⁴×C₅₃	C₂×C₈	1	3	yes	—
32	#41×C₅₃	C₂×C₈	1	48	yes	—
33	#41×C₅₃	C₂²×C₄	1	16	yes	—
34	C₂×Q₈×C₅₃	C₂×C₈	1	12	yes	—
35	C₂×D₄×C₅₃	C₂×C₈	1	36	yes	—
36	C₂²×C₄×C₅₃	C₂×C₈	1	3	yes	—
37	C₂²×C₄×C₅₃	C₂×C₈	1	18	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	18	yes	—
39	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	#11719917	C₂²×C₄	1	48	yes	—
41	C₂×#88322	C₂²×C₄	1	3	yes	—
42	C₂×#88322	C₂²×C₄	1	18	yes	—
43	C₂×#88322	C₂²×C₄	1	36	yes	—
44	C₂×#88322	C₂⁴	1	3	yes	—
45	C₂×#88322	C₂⁴	1	12	yes	—
46	#11719910	C₂⁴	1	4	yes	—
47	#11719910	C₂⁴	1	12	yes	—
48	C₂³×C₅₃	—	2	12	no	—
49	C₂³×C₅₃	C₂²×C₈	1	18	yes	—
50	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
51	Q₈×C₅₃	C₂²×C₈	1	16	yes	—
52	D₄×C₅₃	C₂²×C₈	1	48	yes	—
53	C₂×C₄×C₅₃	C₂²×C₈	1	3	yes	—
54	C₂×C₄×C₅₃	C₂²×C₈	1	18	yes	—
55	C₂×C₄×C₅₃	C₂²×C₈	1	36	yes	—
56	C₂×C₄×C₅₃	C₂³×C₄	1	3	yes	—
57	C₂×C₄×C₅₃	C₂³×C₄	1	12	yes	—
58	#88322	C₂³×C₄	1	3	yes	—
59	#88322	C₂³×C₄	1	12	yes	—
60	#88322	C₂⁵	1	1	yes	characteristic
61	C₂²×#504	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	—	2	36	no	—
63	C₂²×C₅₃	#504	1	4	yes	complemented
64	C₂²×C₅₃	C₂³×C₈	1	12	yes	—
65	C₂²×C₅₃	C₂³×C₈	1	3	yes	—
66	C₄×C₅₃	C₂³×C₈	1	3	yes	—
67	C₄×C₅₃	C₂³×C₈	1	12	yes	—
68	C₄×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
69	C₂²×#131	—	53	1	no	—
70	C₂×#504	—	53	24	no	—
71	C₂²×M₃₂	—	53	3	no	—
72	C₂³×C₁₆	—	53	3	no	—
73	C₂×C₅₃	—	2	12	no	—
74	C₂×C₅₃	C₂×#504	1	6	yes	complemented
75	C₂×C₅₃	C₂⁴×C₈	1	1	yes	characteristic
76	C₂²×#41	#6157	1	1	yes	characteristic
77	C₂×#131	—	53	12	no	—
78	C₂×#131	—	53	12	no	—
79	C₂²×M₁₆	—	53	3	no	—
80	C₂³×C₈	—	53	3	no	—
81	#504	—	53	64	no	—
82	C₂×M₃₂	—	53	36	no	—
83	C₂²×C₁₆	—	53	1	no	—
84	C₂²×C₁₆	—	53	3	no	—
85	C₂²×C₁₆	—	53	36	no	—
86	C₅₃	C₂²×#504	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	#88324	1	12	yes	—
88	C₂×#41	C₂×#6157	1	12	yes	—
89	C₂²×Q₈	#88324	1	1	yes	characteristic
90	C₂²×D₄	#88324	1	3	yes	—
91	C₂³×C₄	C₂×#6157	1	3	yes	—
92	#131	—	53	48	no	—
93	#131	—	53	16	no	—
94	C₂×M₁₆	—	53	36	no	—
95	C₂²×C₈	—	53	18	no	—
96	C₂²×C₈	—	53	18	no	—
97	C₂²×C₈	—	53	1	no	—
98	C₂²×C₈	—	53	3	no	—
99	M₃₂	—	53	48	no	—
100	C₂×C₁₆	—	53	6	no	—
101	C₂×C₁₆	—	53	18	no	—
102	C₂×C₁₆	—	53	48	no	—
103	C₂⁴	C₂×#88324	1	3	yes	—
104	#41	C₂×#88324	1	48	yes	—
105	#41	C₂²×#6157	1	16	yes	—
106	C₂×Q₈	C₂×#88324	1	12	yes	—
107	C₂×D₄	C₂×#88324	1	36	yes	—
108	C₂²×C₄	C₂×#88324	1	3	yes	—
109	C₂²×C₄	C₂×#88324	1	18	yes	—
110	C₂²×C₄	C₂²×#6157	1	18	yes	—
111	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
112	M₁₆	—	53	48	no	—
113	C₂×C₈	—	53	3	no	—
114	C₂×C₈	—	53	12	no	—
115	C₂×C₈	—	53	3	no	—
116	C₂×C₈	—	53	18	no	—
117	C₂×C₈	—	53	36	no	—
118	C₁₆	—	53	4	no	—
119	C₁₆	—	53	12	no	—
120	C₂³	—	2	12	no	—
121	C₂³	C₂²×#88324	1	18	yes	—
122	C₂³	C₂²×#88324	1	1	yes	characteristic
123	Q₈	C₂²×#88324	1	16	yes	—
124	D₄	C₂²×#88324	1	48	yes	—
125	C₂×C₄	C₂²×#88324	1	3	yes	—
126	C₂×C₄	C₂²×#88324	1	18	yes	—
127	C₂×C₄	C₂²×#88324	1	36	yes	—
128	C₂×C₄	C₂³×#6157	1	3	yes	—
129	C₂×C₄	C₂³×#6157	1	12	yes	—
130	C₈	—	53	3	no	—
131	C₈	—	53	12	no	—
132	C₈	—	53	1	no	—
133	C₂²	—	2	36	no	—
134	C₂²	#16811681	1	4	yes	direct product factor
135	C₂²	C₂³×#88324	1	12	yes	—
136	C₂²	C₂³×#88324	1	3	yes	—
137	C₄	C₂³×#88324	1	3	yes	—
138	C₄	C₂³×#88324	1	12	yes	—
139	C₄	C₂⁴×#6157	1	1	yes	characteristic
140	C₂	—	2	12	no	—
141	C₂	C₂×#16811681	1	6	yes	direct product factor
142	C₂	C₂⁴×#88324	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#6157
Fitting, prime radical	C₂²×#41×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₈
cosocle	#88322	C₂⁵
hypercenter	C₂²×#41	#6157
hypocenter	C₅₃	C₂²×#504
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#16811681	C₂	direct (N × H)
2. C₂²×#13133291	C₂	semidirect
3. C₂³×#11719910	C₂	semidirect
5. #16811681	C₂²	direct (N × H)
6. C₂×#13133291	C₂²	semidirect
9. C₂²×#11719910	C₂²	semidirect
15. #13133291	C₂³	semidirect
18. C₂×#11719910	C₂³	semidirect
63. C₂²×C₅₃	#504	semidirect
74. C₂×C₅₃	C₂×#504	semidirect
86. C₅₃	C₂²×#504	semidirect
134. C₂²	#16811681	direct (N × H)
141. C₂	C₂×#16811681	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×#16811681	2
C₂²×#13133291	2
C₂³×#11719910	2
C₂²×#13133224	2
C₂²×#504	53

Subgroups — C22×#16811681 #28330788

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811681 #28330788