Subgroups — C₂²×#16811025 #28253686

Order 13568.

1966 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133209	C₂	1	2	yes	maximal
2	C₂×#16811025	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133151	C₂	1	2	yes	maximal
4	C₂²×#13133149	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133209	—	2	12	no	—
7	C₂×#13133209	C₂²	1	24	yes	—
8	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
9	C₂²×#11719919	—	2	4	no	—
10	#16811025	C₂²	1	64	yes	direct product factor
11	C₂×#13133151	C₂²	1	24	yes	—
12	C₂×#13133149	C₂²	1	24	yes	—
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	12	yes	—
17	C₂³×Dic₅₃	—	2	4	no	—
18	#13133209	—	2	48	no	—
19	#13133209	C₂³	1	32	yes	—
20	C₂×C₄⋊Dic₅₃	—	2	12	no	—
21	C₂×C₄⋊Dic₅₃	C₂³	1	24	yes	—
22	C₂×#11719919	—	2	48	no	—
23	#13133151	C₂³	1	32	yes	—
24	#13133149	C₂³	1	32	yes	—
25	C₂³×C₄×C₅₃	D₄	1	2	yes	—
26	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	Q₈	1	8	yes	—
28	C₂²×C₈×C₅₃	Q₈	1	8	yes	—
29	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
30	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
31	C₂×C₄²×C₅₃	D₄	1	6	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
33	C₄×C₈×C₅₃	C₂³	1	16	yes	—
34	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	—	2	8	no	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	12	yes	—
37	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
38	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
39	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
40	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
41	C₂²×Dic₅₃	—	2	24	no	—
42	C₂²×Dic₅₃	—	4	16	no	—
43	C₂×C₈×C₅₃	C₂×Q₈	1	24	yes	—
44	C₂×C₈×C₅₃	C₂×Q₈	1	24	yes	—
45	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
46	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
47	C₄²×C₅₃	C₂×D₄	1	12	yes	—
48	C₄²×C₅₃	C₂⁴	1	4	yes	—
49	C₄⋊Dic₅₃	—	2	24	no	—
50	C₄⋊Dic₅₃	C₂⁴	1	16	yes	—
51	#11719919	—	2	64	no	—
52	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
53	C₂³×C₅₃	#135	1	8	yes	—
54	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
55	C₂×C₄×C₅₃	—	2	24	no	—
56	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
57	C₂×C₄×C₅₃	C₂²×D₄	1	8	yes	—
58	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
59	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
60	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
61	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
62	C₂×Dic₅₃	—	2	16	no	—
63	C₂×Dic₅₃	—	4	48	no	—
64	C₈×C₅₃	C₂²×Q₈	1	8	yes	—
65	C₈×C₅₃	C₂²×Q₈	1	8	yes	—
66	C₂²×#499	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	#499	1	16	yes	complemented
68	C₂²×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
69	C₂²×C₅₃	C₂×#135	1	12	yes	—
70	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
71	C₄×C₅₃	—	2	8	no	—
72	C₄×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
73	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
74	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
75	Dic₅₃	—	4	16	no	—
76	C₂²×#125	—	53	2	no	—
77	C₂×#499	—	53	24	no	—
78	C₂²×#107	—	53	2	no	—
79	C₂²×#106	—	53	2	no	—
80	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
81	C₂×C₅₃	C₂×#499	1	12	yes	complemented
82	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₅₃	C₂²×#135	1	2	yes	—
84	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
85	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₂×#125	—	53	24	no	—
87	C₂×#125	—	106	12	no	—
88	C₂²×C₄⋊C₄	—	53	4	no	—
89	C₂²×C₄⋊C₄	—	106	4	no	—
90	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
91	#499	—	53	64	no	—
92	C₂×#107	—	53	24	no	—
93	C₂×#106	—	53	24	no	—
94	C₂×C₄×C₈	C₂×D₅₃	1	12	yes	—
95	C₅₃	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
96	C₂³×C₄	—	106	4	no	—
97	C₂³×C₄	D₂₁₂	1	2	yes	—
98	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	C₂²×C₈	Dic₁₀₆	1	8	yes	—
100	C₂²×C₈	Dic₁₀₆	1	8	yes	—
101	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
102	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
103	#125	—	53	32	no	—
104	#125	—	106	48	no	—
105	C₂×C₄⋊C₄	—	53	24	no	—
106	C₂×C₄⋊C₄	—	106	12	no	—
107	C₂×C₄⋊C₄	—	106	48	no	—
108	C₂×C₄²	D₂₁₂	1	6	yes	—
109	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
110	#107	—	53	32	no	—
111	#106	—	53	32	no	—
112	C₄×C₈	C₂²×D₅₃	1	16	yes	—
113	C₂⁴	C₂×D₂₁₂	1	1	yes	characteristic
114	C₂²×C₄	—	2	8	no	—
115	C₂²×C₄	—	106	24	no	—
116	C₂²×C₄	—	212	16	no	—
117	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
118	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
119	C₂²×C₄	C₂×D₂₁₂	1	12	yes	—
120	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
121	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
122	C₂×C₈	C₂×Dic₁₀₆	1	24	yes	—
123	C₂×C₈	C₂×Dic₁₀₆	1	24	yes	—
124	C₂×C₈	C₂³×D₅₃	1	4	yes	—
125	C₂×C₈	C₂³×D₅₃	1	4	yes	—
126	C₄⋊C₄	—	53	16	no	—
127	C₄⋊C₄	—	106	24	no	—
128	C₄⋊C₄	—	106	64	no	—
129	C₄²	C₂×D₂₁₂	1	12	yes	—
130	C₄²	C₂³×D₅₃	1	4	yes	—
131	C₂³	#13133217	1	4	yes	—
132	C₂³	#13133239	1	8	yes	—
133	C₂³	C₂²×D₂₁₂	1	3	yes	—
134	C₂×C₄	—	2	24	no	—
135	C₂×C₄	—	106	16	no	—
136	C₂×C₄	—	212	48	no	—
137	C₂×C₄	#13133217	1	12	yes	—
138	C₂×C₄	C₂²×D₂₁₂	1	8	yes	—
139	C₂×C₄	C₂²×D₂₁₂	1	3	yes	—
140	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
141	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
142	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
143	C₈	C₂²×Dic₁₀₆	1	8	yes	—
144	C₈	C₂²×Dic₁₀₆	1	8	yes	—
145	C₂²	#16811025	1	16	yes	direct product factor
146	C₂²	C₂×#13133217	1	6	yes	—
147	C₂²	C₂×#13133239	1	12	yes	—
148	C₂²	C₂³×D₂₁₂	1	1	yes	characteristic
149	C₄	—	2	8	no	—
150	C₄	—	212	16	no	—
151	C₄	C₂×#13133217	1	6	yes	—
152	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
153	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
154	C₂	C₂×#16811025	1	12	yes	direct product factor
155	C₂	C₂²×#13133217	1	1	yes	characteristic
156	C₂	C₂²×#13133239	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₁₂
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂²×#499
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16811025	C₂	direct (N × H)
10. #16811025	C₂²	direct (N × H)
67. C₂²×C₅₃	#499	semidirect
81. C₂×C₅₃	C₂×#499	semidirect
95. C₅₃	C₂²×#499	semidirect
145. C₂²	#16811025	direct (N × H)
154. C₂	C₂×#16811025	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×#13133209	2
C₂×#16811025	2
C₂²×#13133151	2
C₂²×#13133149	2
C₂²×C₄×C₈×C₅₃	2
C₂²×#499	53

Subgroups — C22×#16811025 #28253686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811025 #28253686