Subgroups — C₂×#20272791 #27295685

Order 13568.

230 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810444	C₂	1	2	yes	maximal
2	#20272791	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133157	—	2	2	no	—
5	#16810444	—	2	2	no	—
6	#16810444	C₂²	1	4	yes	—
7	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
8	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₅₃	C₄	1	2	yes	—
11	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
12	#13133157	—	2	4	no	—
13	#13133157	—	4	2	no	—
14	C₂×#11719908	—	4	2	no	—
15	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	—	2	1	no	—
19	C₂×C₁₆×C₅₃	—	2	1	no	—
20	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
21	C₂×C₁₆×C₅₃	C₈	1	2	yes	—
22	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₄×C₈×C₅₃	Q₈	1	1	yes	characteristic
27	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	—	2	1	no	—
29	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
35	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
38	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
39	C₂×C₈×C₅₃	Q₁₆	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₅₃	—	2	1	no	—
43	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₁₆×C₅₃	—	2	2	no	—
46	C₁₆×C₅₃	C₂×C₈	1	4	yes	—
47	#11719908	—	4	4	no	—
48	C₂³×C₅₃	#108	1	1	yes	characteristic
49	C₂×C₄×C₅₃	—	2	4	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	—	2	1	no	—
52	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
59	C₈×C₅₃	C₄⋊C₈	1	1	yes	characteristic
60	C₈×C₅₃	#107	1	1	yes	characteristic
61	C₈×C₅₃	#107	1	1	yes	characteristic
62	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
63	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
64	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
65	C₈×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×#1373	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	—	2	2	no	—
68	C₂²×C₅₃	#335	1	1	yes	characteristic
69	C₂²×C₅₃	#335	1	1	yes	characteristic
70	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
71	C₄×C₅₃	—	2	2	no	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	#335	1	1	yes	characteristic
74	C₄×C₅₃	#335	1	1	yes	characteristic
75	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
76	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
77	C₂×#364	—	53	2	no	—
78	C₂×C₄×C₁₆	D₅₃	1	1	yes	characteristic
79	#1373	—	53	4	no	—
80	C₂×C₅₃	—	2	1	no	—
81	C₂×C₅₃	—	2	1	no	—
82	C₂×C₅₃	#1373	1	2	yes	complemented
83	C₂×C₅₃	C₂×#335	1	1	yes	characteristic
84	C₂×M₃₂	—	106	2	no	—
85	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
86	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
88	#364	—	53	4	no	—
89	#364	—	106	2	no	—
90	C₄×C₁₆	Dic₅₃	1	2	yes	—
91	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
92	C₅₃	C₂×#1373	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
94	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
95	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
96	M₃₂	—	106	4	no	—
97	M₃₂	—	212	2	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	212	2	no	—
101	C₂×C₁₆	#88322	1	2	yes	—
102	C₂×C₁₆	#88322	1	2	yes	—
103	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
104	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
105	C₄×C₈	Dic₁₀₆	1	1	yes	characteristic
106	C₄×C₈	D₂₁₂	1	1	yes	characteristic
107	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
111	C₂×C₈	D₄₂₄	1	1	yes	characteristic
112	C₂×C₈	D₄₂₄	1	1	yes	characteristic
113	C₂×C₈	Dic₂₁₂	1	1	yes	characteristic
114	C₂×C₈	Dic₂₁₂	1	1	yes	characteristic
115	C₂×C₈	C₂×#88322	1	1	yes	characteristic
116	C₂×C₈	C₂×#88322	1	1	yes	characteristic
117	C₂×C₈	#11719917	1	1	yes	characteristic
118	C₂×C₈	#11719917	1	1	yes	characteristic
119	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
120	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
121	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
122	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
123	C₄²	—	2	1	no	—
124	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
125	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
126	C₁₆	—	2	2	no	—
127	C₁₆	—	212	4	no	—
128	C₁₆	C₂×#88322	1	4	yes	—
129	C₂³	#13133153	1	1	yes	characteristic
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
134	C₂×C₄	C₄⋊#88322	1	1	yes	characteristic
135	C₂×C₄	#13133151	1	1	yes	characteristic
136	C₂×C₄	#13133151	1	1	yes	characteristic
137	C₂×C₄	#13133153	1	1	yes	characteristic
138	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
139	C₈	C₄⋊#88322	1	1	yes	characteristic
140	C₈	C₄⋊#88322	1	1	yes	characteristic
141	C₈	#13133151	1	1	yes	characteristic
142	C₈	#13133151	1	1	yes	characteristic
143	C₈	C₂²×#88322	1	1	yes	characteristic
144	C₈	C₂×#11719917	1	1	yes	characteristic
145	C₈	C₂×D₄₂₄	1	1	yes	characteristic
146	C₈	C₂×Dic₂₁₂	1	1	yes	characteristic
147	C₂²	—	2	2	no	—
148	C₂²	#16810376	1	1	yes	characteristic
149	C₂²	#16810376	1	1	yes	characteristic
150	C₂²	C₂×#13133153	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	#16810376	1	1	yes	characteristic
154	C₄	#16810376	1	1	yes	characteristic
155	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
156	C₄	C₂×#13133151	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	#20272791	1	2	yes	direct product factor
160	C₂	C₂×#16810376	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄₂₄
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#335
prime radical	C₂³×C₅₃	#108
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₃
hypocenter	C₅₃	C₂×#1373
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #20272791	C₂	direct (N × H)
82. C₂×C₅₃	#1373	semidirect
92. C₅₃	C₂×#1373	semidirect
159. C₂	#20272791	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×#16810444	2
#20272791	2
C₂×C₄×C₁₆×C₅₃	2
C₂×#1373	53

Subgroups — C2×#20272791 #27295685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272791 #27295685