Subgroups — C₂²×#19151444 #36712598

Order 22784.

5150 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	C₂³×C₄×Dic₈₉	C₂	1	2	yes	maximal
2	C₂×#19151444	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#15667919	C₂	1	4	yes	maximal
4	C₂³×C₄²×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×Dic₈₉	C₂²	1	4	yes	—
6	C₂²×C₄×Dic₈₉	C₄	1	32	yes	complemented
7	C₂²×C₄×Dic₈₉	C₂²	1	24	yes	—
8	#19151444	C₂²	1	64	yes	direct product factor
9	C₂×#15667919	C₂²	1	48	yes	—
10	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₈₉	C₄	1	16	yes	complemented
13	C₂²×C₄²×C₈₉	C₂²	1	12	yes	—
14	C₂³×Dic₈₉	—	2	32	no	—
15	C₂³×Dic₈₉	C₂×C₄	1	32	yes	—
16	C₂³×Dic₈₉	C₂³	1	24	yes	—
17	C₂×C₄×Dic₈₉	C₂×C₄	1	192	yes	complemented
18	C₂×C₄×Dic₈₉	C₂³	1	32	yes	—
19	#15667919	C₂³	1	64	yes	—
20	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₈₉	—	2	16	no	—
22	C₂³×C₄×C₈₉	C₂×C₄	1	16	yes	—
23	C₂³×C₄×C₈₉	C₂×C₄	1	16	yes	—
24	C₂³×C₄×C₈₉	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₈₉	C₂³	1	12	yes	—
26	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
27	C₂×C₄²×C₈₉	C₂×C₄	1	96	yes	complemented
28	C₂×C₄²×C₈₉	C₂³	1	16	yes	—
29	C₂²×Dic₈₉	—	2	128	no	—
30	C₂²×Dic₈₉	—	2	96	no	—
31	C₂²×Dic₈₉	C₂²×C₄	1	96	yes	—
32	C₂²×Dic₈₉	C₂⁴	1	16	yes	—
33	C₄×Dic₈₉	C₂²×C₄	1	128	yes	complemented
34	C₂⁴×C₈₉	C₂²×C₄	1	8	yes	—
35	C₂⁴×C₈₉	C₂²×C₄	1	4	yes	—
36	C₂⁴×C₈₉	#41	1	16	yes	—
37	C₂⁴×C₈₉	C₂⁴	1	3	yes	—
38	C₂²×C₄×C₈₉	—	2	64	no	—
39	C₂²×C₄×C₈₉	—	2	48	no	—
40	C₂²×C₄×C₈₉	C₄²	1	64	yes	—
41	C₂²×C₄×C₈₉	C₂²×C₄	1	48	yes	—
42	C₂²×C₄×C₈₉	C₂²×C₄	1	48	yes	—
43	C₂²×C₄×C₈₉	C₂²×C₄	1	24	yes	—
44	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
45	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
46	C₄²×C₈₉	C₂²×C₄	1	64	yes	complemented
47	C₂³×C₈₉	C₂×C₄²	1	16	yes	—
48	C₂³×C₈₉	#117	1	64	yes	—
49	C₂³×C₈₉	#117	1	32	yes	—
50	C₂³×C₈₉	C₂³×C₄	1	12	yes	—
51	C₂³×C₈₉	C₂³×C₄	1	6	yes	—
52	C₂³×C₈₉	C₂×#41	1	24	yes	—
53	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
54	C₂×C₄×C₈₉	—	2	96	no	—
55	C₂×C₄×C₈₉	—	2	16	no	—
56	C₂×C₄×C₈₉	C₂×C₄²	1	96	yes	—
57	C₂×C₄×C₈₉	C₂³×C₄	1	16	yes	—
58	C₂×C₄×C₈₉	C₂³×C₄	1	16	yes	—
59	C₂×C₄×C₈₉	C₂³×C₄	1	8	yes	—
60	C₂×Dic₈₉	—	2	192	no	—
61	C₂×Dic₈₉	—	2	32	no	—
62	C₂×Dic₈₉	C₂³×C₄	1	32	yes	—
63	C₂²×C₈₉	#376	1	64	yes	complemented
64	C₂²×C₈₉	C₂²×C₄²	1	12	yes	—
65	C₂²×C₈₉	C₂×#117	1	48	yes	—
66	C₂²×C₈₉	C₂×#117	1	24	yes	—
67	C₂²×C₈₉	C₂⁴×C₄	1	2	yes	—
68	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
69	C₂²×C₈₉	C₂²×#41	1	4	yes	—
70	C₄×C₈₉	—	2	16	no	—
71	C₄×C₈₉	C₂²×C₄²	1	16	yes	—
72	Dic₈₉	—	2	32	no	—
73	C₂²×#376	—	89	1	no	maximal, Sylow
74	C₂×C₈₉	C₂×#376	1	24	yes	complemented
75	C₂×C₈₉	C₂³×C₄²	1	1	yes	characteristic
76	C₂×C₈₉	C₂²×#117	1	4	yes	—
77	C₂×C₈₉	C₂²×#117	1	2	yes	—
78	C₂³×C₄²	—	89	2	no	—
79	C₂³×C₄²	D₈₉	1	1	yes	characteristic
80	C₂×#376	—	89	24	no	—
81	C₂²×#95	—	89	4	no	—
82	C₈₉	C₂²×#376	1	1	yes	characteristic, complemented, Sylow
83	C₂⁴×C₄	—	89	4	no	—
84	C₂⁴×C₄	C₂×D₈₉	1	2	yes	—
85	C₂⁴×C₄	C₂×D₈₉	1	1	yes	characteristic
86	C₂²×C₄²	—	89	32	no	—
87	C₂²×C₄²	—	89	24	no	—
88	C₂²×C₄²	Dic₈₉	1	16	yes	complemented
89	C₂²×C₄²	C₂×D₈₉	1	12	yes	—
90	#376	—	89	64	no	—
91	C₂×#95	—	89	48	no	—
92	C₂⁵	C₂²×D₈₉	1	1	yes	characteristic
93	C₂³×C₄	—	2	16	no	—
94	C₂³×C₄	—	89	32	no	—
95	C₂³×C₄	—	89	24	no	—
96	C₂³×C₄	—	178	32	no	—
97	C₂³×C₄	C₄×D₈₉	1	16	yes	—
98	C₂³×C₄	C₂×Dic₈₉	1	16	yes	—
99	C₂³×C₄	C₂×Dic₈₉	1	8	yes	—
100	C₂³×C₄	C₂²×D₈₉	1	12	yes	—
101	C₂³×C₄	C₂²×D₈₉	1	6	yes	—
102	C₂×C₄²	—	89	192	no	—
103	C₂×C₄²	—	89	32	no	—
104	C₂×C₄²	C₂×Dic₈₉	1	96	yes	complemented
105	C₂×C₄²	C₂²×D₈₉	1	16	yes	—
106	#95	—	89	64	no	—
107	C₂⁴	C₂×C₄×D₈₉	1	8	yes	—
108	C₂⁴	C₂²×Dic₈₉	1	4	yes	—
109	C₂⁴	#13083574	1	16	yes	—
110	C₂⁴	C₂³×D₈₉	1	3	yes	—
111	C₂²×C₄	—	2	64	no	—
112	C₂²×C₄	—	2	48	no	—
113	C₂²×C₄	—	89	96	no	—
114	C₂²×C₄	—	89	16	no	—
115	C₂²×C₄	—	178	128	no	—
116	C₂²×C₄	—	178	96	no	—
117	C₂²×C₄	C₄×Dic₈₉	1	64	yes	—
118	C₂²×C₄	C₂×C₄×D₈₉	1	48	yes	—
119	C₂²×C₄	C₂²×Dic₈₉	1	48	yes	—
120	C₂²×C₄	C₂²×Dic₈₉	1	24	yes	—
121	C₂²×C₄	C₂³×D₈₉	1	8	yes	—
122	C₂²×C₄	C₂³×D₈₉	1	4	yes	—
123	C₄²	—	89	128	no	—
124	C₄²	C₂²×Dic₈₉	1	64	yes	complemented
125	C₂³	C₂×C₄×Dic₈₉	1	16	yes	—
126	C₂³	#15667969	1	64	yes	—
127	C₂³	#15667971	1	32	yes	—
128	C₂³	C₂²×C₄×D₈₉	1	12	yes	—
129	C₂³	C₂³×Dic₈₉	1	6	yes	—
130	C₂³	C₂×#13083574	1	24	yes	—
131	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
132	C₂×C₄	—	2	16	no	—
133	C₂×C₄	—	2	96	no	—
134	C₂×C₄	—	89	32	no	—
135	C₂×C₄	—	178	32	no	—
136	C₂×C₄	—	178	192	no	—
137	C₂×C₄	C₂×C₄×Dic₈₉	1	96	yes	—
138	C₂×C₄	C₂²×C₄×D₈₉	1	16	yes	—
139	C₂×C₄	C₂³×Dic₈₉	1	16	yes	—
140	C₂×C₄	C₂³×Dic₈₉	1	8	yes	—
141	C₂²	#19151444	1	64	yes	direct product factor
142	C₂²	C₂²×C₄×Dic₈₉	1	12	yes	—
143	C₂²	C₂×#15667969	1	48	yes	—
144	C₂²	C₂×#15667971	1	24	yes	—
145	C₂²	C₂³×C₄×D₈₉	1	2	yes	—
146	C₂²	C₂⁴×Dic₈₉	1	1	yes	characteristic
147	C₂²	C₂²×#13083574	1	4	yes	—
148	C₄	—	2	16	no	—
149	C₄	—	178	32	no	—
150	C₄	C₂²×C₄×Dic₈₉	1	16	yes	—
151	C₂	C₂×#19151444	1	24	yes	direct product factor
152	C₂	C₂³×C₄×Dic₈₉	1	1	yes	characteristic
153	C₂	C₂²×#15667969	1	4	yes	—
154	C₂	C₂²×#15667971	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₈₉
commutator	C₂×C₈₉	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂³×C₄²×C₈₉	C₂
socle, prime radical	C₂⁵×C₈₉	C₂³
cosocle	C₂³×C₈₉	C₂⁵
hypercenter	C₂³×C₄²	D₈₉
hypocenter	C₈₉	C₂²×#376
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#19151444	C₂	direct (N × H)
6. C₂²×C₄×Dic₈₉	C₄	semidirect
8. #19151444	C₂²	direct (N × H)
12. C₂²×C₄²×C₈₉	C₄	semidirect
17. C₂×C₄×Dic₈₉	C₂×C₄	semidirect
27. C₂×C₄²×C₈₉	C₂×C₄	semidirect
33. C₄×Dic₈₉	C₂²×C₄	semidirect
46. C₄²×C₈₉	C₂²×C₄	semidirect
63. C₂²×C₈₉	#376	semidirect
74. C₂×C₈₉	C₂×#376	semidirect
82. C₈₉	C₂²×#376	semidirect
88. C₂²×C₄²	Dic₈₉	semidirect
104. C₂×C₄²	C₂×Dic₈₉	semidirect
124. C₄²	C₂²×Dic₈₉	semidirect
141. C₂²	#19151444	direct (N × H)
151. C₂	C₂×#19151444	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×Dic₈₉	2
C₂×#19151444	2
C₂²×#15667919	2
C₂³×C₄²×C₈₉	2
C₂²×#376	89

Subgroups — C22×#19151444 #36712598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19151444 #36712598