Subgroups — C₂×#20276990 #27539569

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	2	yes	maximal
2	#20276990	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810449	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133217	—	2	2	no	—
6	#16811028	—	2	4	no	—
7	#16811028	C₂²	1	8	yes	—
8	C₂×#13133151	—	2	2	no	—
9	C₂×#13133151	C₂²	1	4	yes	—
10	#16810449	C₂²	1	16	yes	—
11	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₅₃	C₂²	1	4	yes	—
14	C₂²×Dic₁₀₆	—	4	2	no	—
15	#13133217	—	2	8	no	—
16	#13133217	—	4	4	no	—
17	C₂×C₄⋊Dic₅₃	—	2	4	no	—
18	C₂×C₄⋊Dic₅₃	—	4	2	no	—
19	#13133151	—	2	4	no	—
20	#13133151	—	2	8	no	—
21	#13133151	C₂³	1	8	yes	—
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	Q₈	1	8	yes	complemented
26	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
27	C₄×C₈×C₅₃	D₄	1	2	yes	—
28	C₄×C₈×C₅₃	C₂³	1	2	yes	—
29	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×Dic₅₃	—	4	4	no	—
33	C₂×Dic₁₀₆	—	4	8	no	—
34	C₂×Dic₁₀₆	—	8	8	no	—
35	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
36	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
39	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
40	C₄²×C₅₃	D₈	1	2	yes	—
41	C₄²×C₅₃	C₂×D₄	1	2	yes	—
42	C₄⋊Dic₅₃	—	2	8	no	—
43	C₄⋊Dic₅₃	—	4	4	no	—
44	C₄⋊Dic₅₃	—	4	8	no	—
45	C₁₆×C₅₃	C₂×Q₈	1	8	yes	complemented
46	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	D₁₆	1	4	yes	—
48	C₂×C₄×C₅₃	Q₃₂	1	4	yes	—
49	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
53	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
54	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
55	C₂×Dic₅₃	—	4	8	no	—
56	C₂×Dic₅₃	—	8	8	no	—
57	Dic₁₀₆	—	8	16	no	—
58	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
59	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
60	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
61	C₂×#2252	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	#500	1	2	yes	—
63	C₂²×C₅₃	C₂×D₁₆	1	2	yes	—
64	C₂²×C₅₃	C₂×Q₃₂	1	2	yes	—
65	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
66	C₄×C₅₃	#500	1	2	yes	—
67	C₄×C₅₃	C₂×D₁₆	1	4	yes	—
68	C₄×C₅₃	C₂×Q₃₂	1	4	yes	—
69	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
70	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
71	Dic₅₃	—	8	8	no	—
72	C₂×#500	—	53	2	no	—
73	#2252	—	53	8	no	—
74	C₂×#366	—	53	4	no	—
75	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
76	C₂×C₅₃	#2252	1	4	yes	complemented
77	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
78	C₂×C₅₃	C₂²×D₁₆	1	1	yes	characteristic
79	C₂×C₅₃	C₂²×Q₃₂	1	1	yes	characteristic
80	C₂×C₄⋊Q₈	—	106	2	no	—
81	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
82	#500	—	53	8	no	—
83	#500	—	106	4	no	—
84	C₂×#107	—	53	4	no	—
85	C₂×#107	—	106	2	no	—
86	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
87	#366	—	53	16	no	—
88	C₄×C₁₆	C₂×D₅₃	1	4	yes	—
89	C₅₃	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	212	2	no	—
91	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
93	C₄⋊Q₈	—	106	8	no	—
94	C₄⋊Q₈	—	212	4	no	—
95	C₂×C₄⋊C₄	—	106	4	no	—
96	C₂×C₄⋊C₄	—	212	2	no	—
97	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₀₆	1	8	yes	complemented
99	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
100	#107	—	53	8	no	—
101	#107	—	106	4	no	—
102	#107	—	106	8	no	—
103	C₄×C₈	D₂₁₂	1	2	yes	—
104	C₄×C₈	C₂²×D₅₃	1	2	yes	—
105	C₂×Q₈	—	424	8	no	—
106	C₂×Q₈	—	212	8	no	—
107	C₂²×C₄	—	212	4	no	—
108	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
109	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
110	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
111	C₂×C₈	Dic₂₁₂	1	4	yes	—
112	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
113	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
114	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
115	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
116	C₄⋊C₄	—	106	8	no	—
117	C₄⋊C₄	—	212	4	no	—
118	C₄⋊C₄	—	212	8	no	—
119	C₄²	D₄₂₄	1	2	yes	—
120	C₄²	C₂×D₂₁₂	1	2	yes	—
121	C₁₆	C₂×Dic₁₀₆	1	8	yes	complemented
122	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
123	Q₈	—	424	16	no	—
124	C₂×C₄	—	424	8	no	—
125	C₂×C₄	—	212	8	no	—
126	C₂×C₄	D₈₄₈	1	4	yes	—
127	C₂×C₄	Dic₄₂₄	1	4	yes	—
128	C₂×C₄	#13133217	1	2	yes	—
129	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
130	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
131	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
132	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
133	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
134	C₈	#13133217	1	2	yes	—
135	C₈	C₂×Dic₂₁₂	1	4	yes	—
136	C₈	C₂²×Dic₁₀₆	1	2	yes	—
137	C₂²	#16811028	1	2	yes	—
138	C₂²	C₂×D₈₄₈	1	2	yes	—
139	C₂²	C₂×Dic₄₂₄	1	2	yes	—
140	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
141	C₄	—	424	8	no	—
142	C₄	#16811028	1	2	yes	—
143	C₄	C₂×D₈₄₈	1	4	yes	—
144	C₄	C₂×Dic₄₂₄	1	4	yes	—
145	C₄	C₂×#13133217	1	1	yes	characteristic
146	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
147	C₂	#20276990	1	4	yes	direct product factor
148	C₂	C₂×#16811028	1	1	yes	characteristic
149	C₂	C₂²×D₈₄₈	1	1	yes	characteristic
150	C₂	C₂²×Dic₄₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	C₂×#2252
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20276990	C₂	direct (N × H)
25. C₂×C₁₆×C₅₃	Q₈	semidirect
45. C₁₆×C₅₃	C₂×Q₈	semidirect
76. C₂×C₅₃	#2252	semidirect
89. C₅₃	C₂×#2252	semidirect
98. C₂×C₁₆	Dic₁₀₆	semidirect
121. C₁₆	C₂×Dic₁₀₆	semidirect
147. C₂	#20276990	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×#16811028	2
#20276990	2
C₂×#16810449	2
C₂×C₄×C₁₆×C₅₃	2
C₂×#2252	53

Subgroups — C2×#20276990 #27539569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276990 #27539569