Subgroups — C₂×#20289627 #28307749

Order 13568.

366 conjugacy classes of subgroups.

The subgroup lattice has 138 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11719910	C₂	1	2	yes	maximal
2	#20289627	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719910	C₄	1	4	yes	—
5	C₄×#11719910	C₄	1	4	yes	—
6	C₄×#11719910	C₂²	1	4	yes	—
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₈×#88322	C₄	1	2	yes	—
10	C₈×#88322	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#11719910	C₈	1	16	yes	complemented
13	C₂×#11719910	C₂×C₄	1	8	yes	—
14	C₂²×#88322	C₈	1	2	yes	—
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₄×#88322	C₈	1	2	yes	—
17	C₄×#88322	C₂×C₄	1	1	yes	characteristic
18	C₄×#88322	C₂×C₄	1	2	yes	—
19	C₄×#88322	C₂×C₄	1	2	yes	—
20	C₄×#88322	C₂³	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₈	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	C₈	1	2	yes	—
24	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₅₃	—	2	4	no	—
29	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
30	C₄²×C₅₃	C₄²	1	1	yes	characteristic
31	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×#88322	—	2	4	no	—
35	C₂×#88322	C₄²	1	2	yes	—
36	C₂×#88322	C₂×C₈	1	4	yes	—
37	C₂×#88322	C₂×C₈	1	8	yes	—
38	C₂×#88322	C₂²×C₄	1	2	yes	—
39	#11719910	C₂×C₈	1	16	yes	complemented
40	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
42	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
46	C₂×C₄×C₅₃	M₃₂	1	4	yes	—
47	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₅₃	—	2	4	no	—
51	#88322	—	2	4	no	—
52	#88322	C₄×C₈	1	4	yes	—
53	#88322	C₂²×C₈	1	4	yes	—
54	C₂×#1311	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	C₈²	1	2	yes	—
56	C₂²×C₅₃	#346	1	2	yes	—
57	C₂²×C₅₃	#346	1	2	yes	—
58	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
59	C₄×C₅₃	C₈²	1	2	yes	—
60	C₄×C₅₃	#346	1	2	yes	—
61	C₄×C₅₃	#346	1	2	yes	—
62	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
63	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
64	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
65	C₂×C₄×C₁₆	—	53	2	no	—
66	C₂×C₈²	—	53	1	no	—
67	#1311	—	53	4	no	—
68	C₂×C₅₃	#1311	1	4	yes	complemented
69	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
70	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
71	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
72	C₂²×C₁₆	—	53	4	no	—
73	C₂×C₄×C₈	—	53	1	no	—
74	C₂×C₄×C₈	—	53	1	no	—
75	C₂×C₄×C₈	#6157	1	1	yes	characteristic
76	C₄×C₁₆	—	53	4	no	—
77	C₄×C₁₆	—	53	4	no	—
78	C₈²	—	53	2	no	—
79	C₈²	—	53	2	no	—
80	C₅₃	C₂×#1311	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	53	2	no	—
82	C₂²×C₈	—	53	2	no	—
83	C₂²×C₈	#88324	1	2	yes	—
84	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
85	C₂×C₁₆	—	53	8	no	—
86	C₂×C₁₆	—	53	16	no	—
87	C₄×C₈	—	53	2	no	—
88	C₄×C₈	—	53	1	no	—
89	C₄×C₈	—	53	2	no	—
90	C₄×C₈	—	53	1	no	—
91	C₄×C₈	—	53	2	no	—
92	C₄×C₈	#88324	1	2	yes	—
93	C₄×C₈	C₂×#6157	1	2	yes	—
94	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
95	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
96	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	53	8	no	—
99	C₂×C₈	—	53	2	no	—
100	C₂×C₈	—	53	2	no	—
101	C₂×C₈	—	53	4	no	—
102	C₂×C₈	—	106	4	no	—
103	C₂×C₈	C₂×#88324	1	4	yes	—
104	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
105	C₄²	C₄×#6157	1	1	yes	characteristic
106	C₄²	C₂×#88324	1	1	yes	characteristic
107	C₄²	C₂²×#6157	1	1	yes	characteristic
108	C₁₆	—	53	16	no	—
109	C₂³	C₄×#88324	1	1	yes	characteristic
110	C₂×C₄	C₈×#6157	1	4	yes	—
111	C₂×C₄	C₄×#88324	1	1	yes	characteristic
112	C₂×C₄	C₄×#88324	1	1	yes	characteristic
113	C₂×C₄	C₄×#88324	1	1	yes	characteristic
114	C₂×C₄	#13133291	1	4	yes	—
115	C₂×C₄	#13133292	1	4	yes	—
116	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
117	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
118	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	53	4	no	—
121	C₈	—	53	4	no	—
122	C₈	—	106	4	no	—
123	C₂²	C₈×#88324	1	2	yes	—
124	C₂²	#16811511	1	2	yes	—
125	C₂²	#16811512	1	2	yes	—
126	C₂²	C₂×C₄×#88324	1	1	yes	characteristic
127	C₄	C₈×#88324	1	2	yes	—
128	C₄	#16811511	1	2	yes	—
129	C₄	#16811512	1	2	yes	—
130	C₄	C₂×C₈×#6157	1	2	yes	—
131	C₄	C₂×#13133291	1	2	yes	—
132	C₄	C₂×#13133292	1	2	yes	—
133	C₂	#20289627	1	4	yes	direct product factor
134	C₂	C₂×C₈×#88324	1	1	yes	characteristic
135	C₂	C₂×#16811511	1	1	yes	characteristic
136	C₂	C₂×#16811512	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×#6157
commutator	C₂×C₅₃	C₂×C₈²
Frattini	C₄²	C₂²×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₄×#88322	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₂×#1311
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20289627	C₂	direct (N × H)
12. C₂×#11719910	C₈	semidirect
39. #11719910	C₂×C₈	semidirect
68. C₂×C₅₃	#1311	semidirect
80. C₅₃	C₂×#1311	semidirect
133. C₂	#20289627	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×C₄×#11719910	2
#20289627	2
C₂×C₈×#88322	2
C₂×#1311	53

Subgroups — C2×#20289627 #28307749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20289627 #28307749