Subgroups — C₂×#20272912 #27304581

Order 13568.

214 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810457	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20272912	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×M₆₄	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₄	1	1	yes	characteristic
6	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
7	#16810457	—	2	1	no	—
8	#16810457	C₂²	1	2	yes	—
9	C₂×#13133123	—	2	1	no	—
10	C₁₆×Dic₅₃	C₄	1	2	yes	—
11	C₁₆×Dic₅₃	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×M₆₄	—	2	1	no	—
14	C₅₃×M₆₄	C₂²	1	2	yes	—
15	C₂×C₃₂×C₅₃	—	2	1	no	—
16	C₂²×#88322	C₈	1	1	yes	characteristic
17	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
18	C₂×#11719908	—	2	1	no	—
19	C₂×#11719908	—	2	1	no	—
20	C₂×#11719908	C₂×C₄	1	2	yes	—
21	C₈×Dic₅₃	C₈	1	2	yes	—
22	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
23	#13133123	—	2	2	no	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	Q₈	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
31	C₃₂×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×Dic₅₃	—	2	1	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
38	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₄×Dic₅₃	—	2	1	no	—
41	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
42	C₂×#88322	C₁₆	1	2	yes	—
43	C₂×#88322	C₁₆	1	2	yes	—
44	C₂×#88322	C₂×C₈	1	2	yes	—
45	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
46	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₅₃	C₂×Q₈	1	1	yes	characteristic
49	#11719908	—	2	2	no	—
50	C₂³×C₅₃	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
56	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	4	no	—
58	C₂×Dic₅₃	—	2	2	no	—
59	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
61	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
63	#88322	C₂×C₁₆	1	4	yes	—
64	C₂×#1422	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	—	2	2	no	—
66	C₂²×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
67	C₂²×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
68	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
69	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
70	C₄×C₅₃	C₄⋊C₁₆	1	1	yes	characteristic
71	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
73	Dic₅₃	—	2	4	no	—
74	C₂×M₆₄	—	53	1	no	—
75	C₂×M₆₄	D₅₃	1	1	yes	characteristic
76	C₂×C₄×C₁₆	—	53	1	no	—
77	#1422	—	53	4	no	—
78	C₂×C₅₃	—	2	1	no	—
79	C₂×C₅₃	—	2	1	no	—
80	C₂×C₅₃	#1422	1	2	yes	complemented
81	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
82	C₂²×C₁₆	—	53	1	no	—
83	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
84	C₂×C₄×C₈	—	53	1	no	—
85	M₆₄	—	2	1	no	—
86	M₆₄	—	53	2	no	—
87	M₆₄	—	106	1	no	—
88	M₆₄	C₂×D₅₃	1	2	yes	—
89	C₂×C₃₂	—	2	1	no	—
90	C₂×C₃₂	—	106	1	no	—
91	C₄×C₁₆	—	53	2	no	—
92	C₄×C₁₆	—	53	2	no	—
93	C₅₃	C₂×#1422	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	53	1	no	—
95	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
96	C₂×C₄²	—	53	1	no	—
97	C₂×C₁₆	—	53	2	no	—
98	C₂×C₁₆	—	106	1	no	—
99	C₂×C₁₆	—	106	1	no	—
100	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
101	C₂×C₁₆	Dic₁₀₆	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
103	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
105	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
106	C₄×C₈	—	53	2	no	—
107	C₄×C₈	—	53	2	no	—
108	C₃₂	—	2	2	no	—
109	C₃₂	—	106	2	no	—
110	C₂²×C₄	—	106	1	no	—
111	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
112	C₂×C₈	—	53	2	no	—
113	C₂×C₈	—	53	2	no	—
114	C₂×C₈	—	53	2	no	—
115	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
116	C₂×C₈	#11719912	1	1	yes	characteristic
117	C₂×C₈	#11719912	1	1	yes	characteristic
118	C₂×C₈	#11719919	1	1	yes	characteristic
119	C₂×C₈	#11719919	1	1	yes	characteristic
120	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
121	C₄²	—	53	2	no	—
122	C₄²	—	106	1	no	—
123	C₁₆	—	106	2	no	—
124	C₁₆	#11719919	1	1	yes	characteristic
125	C₁₆	#11719919	1	1	yes	characteristic
126	C₁₆	C₂×Dic₁₀₆	1	1	yes	characteristic
127	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
128	C₂³	#13133156	1	1	yes	characteristic
129	C₂×C₄	—	106	4	no	—
130	C₂×C₄	—	106	2	no	—
131	C₂×C₄	#13133146	1	1	yes	characteristic
132	C₂×C₄	#13133146	1	1	yes	characteristic
133	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
134	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
135	C₂×C₄	#13133156	1	1	yes	characteristic
136	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
137	C₈	—	53	4	no	—
138	C₈	#13133146	1	1	yes	characteristic
139	C₈	#13133146	1	1	yes	characteristic
140	C₈	C₂×#11719919	1	1	yes	characteristic
141	C₈	C₂×#11719912	1	1	yes	characteristic
142	C₂²	—	2	2	no	—
143	C₂²	#16810441	1	1	yes	characteristic
144	C₂²	#16810441	1	1	yes	characteristic
145	C₂²	C₂×#13133156	1	1	yes	characteristic
146	C₄	—	106	4	no	—
147	C₄	#16810441	1	1	yes	characteristic
148	C₄	#16810441	1	1	yes	characteristic
149	C₄	C₂×#13133146	1	1	yes	characteristic
150	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	2	1	no	—
153	C₂	#20272912	1	2	yes	direct product factor
154	C₂	C₂×#16810441	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂×C₅₃×M₆₄	C₂
socle	C₂²×C₅₃	C₄⋊C₁₆
prime radical	C₂³×C₅₃	M₃₂
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×M₆₄	D₅₃
hypocenter	C₅₃	C₂×#1422
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
3. #20272912	C₂	direct (N × H)
80. C₂×C₅₃	#1422	semidirect
93. C₅₃	C₂×#1422	semidirect
153. C₂	#20272912	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×#16810457	2
C₂×C₁₆×Dic₅₃	2
#20272912	2
C₂×C₅₃×M₆₄	2
C₂×#1422	53

Subgroups — C2×#20272912 #27304581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272912 #27304581