Subgroups — C₂×#26022546 #36712537

Order 22784.

470 conjugacy classes of subgroups.

The subgroup lattice has 118 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022546	C₂	1	8	yes	maximal, direct product factor
2	C₂×#19151432	C₂	1	3	yes	maximal, complemented
3	C₂²×#15667918	C₂	1	3	yes	maximal, complemented
4	C₂×#504×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151432	C₂²	1	12	yes	complemented
6	C₂×#15667918	C₂²	1	1	yes	characteristic
7	C₂×#15667918	C₂²	1	3	yes	—
8	C₂×#15667918	C₂²	1	12	yes	complemented
9	C₂×#131×C₈₉	C₄	1	1	yes	characteristic
10	#504×C₈₉	C₄	1	4	yes	—
11	#504×C₈₉	C₂²	1	4	yes	—
12	C₂×M₃₂×C₈₉	C₄	1	3	yes	—
13	C₂²×C₁₆×C₈₉	C₂²	1	3	yes	—
14	#15667918	C₂³	1	2	yes	—
15	#15667918	C₂³	1	6	yes	—
16	C₂×#41×C₈₉	C₈	1	1	yes	characteristic
17	#131×C₈₉	C₈	1	4	yes	—
18	#131×C₈₉	C₂×C₄	1	4	yes	—
19	C₂×M₁₆×C₈₉	C₈	1	3	yes	—
20	C₂²×C₈×C₈₉	C₂×C₄	1	3	yes	—
21	M₃₂×C₈₉	C₂×C₄	1	12	yes	—
22	C₂×C₁₆×C₈₉	C₂×C₄	1	3	yes	—
23	C₂×C₁₆×C₈₉	C₂×C₄	1	6	yes	—
24	C₂×C₁₆×C₈₉	C₂³	1	6	yes	—
25	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
26	#41×C₈₉	C₁₆	1	4	yes	—
27	#41×C₈₉	C₂×C₈	1	4	yes	—
28	C₂×Q₈×C₈₉	C₁₆	1	1	yes	characteristic
29	C₂×D₄×C₈₉	C₁₆	1	3	yes	—
30	C₂²×C₄×C₈₉	C₂×C₈	1	3	yes	—
31	M₁₆×C₈₉	C₂×C₈	1	12	yes	—
32	C₂×C₈×C₈₉	C₂×C₈	1	3	yes	—
33	C₂×C₈×C₈₉	C₂×C₈	1	6	yes	—
34	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
36	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
37	C₁₆×C₈₉	C₂²×C₄	1	6	yes	—
38	C₁₆×C₈₉	C₂⁴	1	1	yes	characteristic
39	C₂³×C₈₉	C₂×C₁₆	1	3	yes	—
40	Q₈×C₈₉	C₂×C₁₆	1	4	yes	—
41	D₄×C₈₉	C₂×C₁₆	1	12	yes	—
42	C₂×C₄×C₈₉	C₂×C₁₆	1	3	yes	—
43	C₂×C₄×C₈₉	C₂×C₁₆	1	6	yes	—
44	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂²×C₈	1	6	yes	—
46	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
47	C₈×C₈₉	C₂²×C₈	1	6	yes	—
48	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
49	C₂²×C₈₉	—	2	6	no	—
50	C₂²×C₈₉	C₂²×C₁₆	1	6	yes	—
51	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
52	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
53	C₄×C₈₉	C₂²×C₁₆	1	6	yes	—
54	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
55	C₂×#2258	—	89	1	no	maximal, Sylow
56	C₂×C₈₉	—	2	6	no	—
57	C₂×C₈₉	#2258	1	2	yes	complemented
58	C₂×C₈₉	C₂³×C₁₆	1	1	yes	characteristic
59	C₂×#504	D₈₉	1	1	yes	characteristic
60	#2258	—	89	8	no	—
61	C₂×M₆₄	—	89	3	no	—
62	C₂²×C₃₂	—	89	3	no	—
63	C₈₉	C₂×#2258	1	1	yes	characteristic, complemented, Sylow
64	C₂×#131	Dic₈₉	1	1	yes	characteristic
65	#504	Dic₈₉	1	4	yes	—
66	#504	C₂×D₈₉	1	4	yes	—
67	C₂×M₃₂	Dic₈₉	1	3	yes	—
68	C₂²×C₁₆	C₂×D₈₉	1	3	yes	—
69	M₆₄	—	89	12	no	—
70	C₂×C₃₂	—	89	12	no	—
71	C₂×C₃₂	—	89	1	no	—
72	C₂×C₃₂	—	89	3	no	—
73	C₂×#41	#10623494	1	1	yes	characteristic
74	#131	#10623494	1	4	yes	—
75	#131	C₂×Dic₈₉	1	4	yes	—
76	C₂×M₁₆	#10623494	1	3	yes	—
77	C₂²×C₈	C₂×Dic₈₉	1	3	yes	—
78	M₃₂	C₂×Dic₈₉	1	12	yes	—
79	C₂×C₁₆	C₂×Dic₈₉	1	3	yes	—
80	C₂×C₁₆	C₂×Dic₈₉	1	6	yes	—
81	C₂×C₁₆	C₂²×D₈₉	1	6	yes	—
82	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
83	C₃₂	—	89	2	no	—
84	C₃₂	—	89	6	no	—
85	#41	#13083549	1	4	yes	—
86	#41	C₂×#10623494	1	4	yes	—
87	C₂×Q₈	#13083549	1	1	yes	characteristic
88	C₂×D₄	#13083549	1	3	yes	—
89	C₂²×C₄	C₂×#10623494	1	3	yes	—
90	M₁₆	C₂×#10623494	1	12	yes	—
91	C₂×C₈	C₂×#10623494	1	3	yes	—
92	C₂×C₈	C₂×#10623494	1	6	yes	—
93	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
94	C₂×C₈	C₂²×Dic₈₉	1	6	yes	—
95	C₁₆	C₂²×Dic₈₉	1	1	yes	characteristic
96	C₁₆	C₂²×Dic₈₉	1	6	yes	—
97	C₁₆	C₂³×D₈₉	1	1	yes	characteristic
98	C₂³	C₂×#13083549	1	3	yes	—
99	Q₈	C₂×#13083549	1	4	yes	—
100	D₄	C₂×#13083549	1	12	yes	—
101	C₂×C₄	C₂×#13083549	1	3	yes	—
102	C₂×C₄	C₂×#13083549	1	6	yes	—
103	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
104	C₂×C₄	C₂²×#10623494	1	6	yes	—
105	C₈	C₂²×#10623494	1	1	yes	characteristic
106	C₈	C₂²×#10623494	1	6	yes	—
107	C₈	C₂³×Dic₈₉	1	1	yes	characteristic
108	C₂²	—	2	6	no	—
109	C₂²	C₂²×#13083549	1	6	yes	—
110	C₂²	C₂²×#13083549	1	1	yes	characteristic
111	C₄	C₂²×#13083549	1	1	yes	characteristic
112	C₄	C₂²×#13083549	1	6	yes	—
113	C₄	C₂³×#10623494	1	1	yes	characteristic
114	C₂	—	2	6	no	—
115	C₂	#26022546	1	2	yes	direct product factor
116	C₂	C₂³×#13083549	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₈₉
commutator	C₂×C₈₉	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₈₉
Fitting	C₂×#504×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×C₁₆
prime radical	C₂×#41×C₈₉	C₈
cosocle	C₁₆×C₈₉	C₂⁴
hypercenter	C₂×#504	D₈₉
hypocenter	C₈₉	C₂×#2258
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. #26022546	C₂	direct (N × H)
2. C₂×#19151432	C₂	semidirect
3. C₂²×#15667918	C₂	semidirect
5. #19151432	C₂²	semidirect
8. C₂×#15667918	C₂²	semidirect
57. C₂×C₈₉	#2258	semidirect
63. C₈₉	C₂×#2258	semidirect
115. C₂	#26022546	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26022546	2
C₂×#19151432	2
C₂²×#15667918	2
C₂×#504×C₈₉	2
C₂×#2258	89

Subgroups — C2×#26022546 #36712537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022546 #36712537