Subgroups — #36469427

Order 22784.

414 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13083549	C₂	1	2	yes	maximal
2	#26018330	C₂	1	4	yes	maximal
3	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13083549	C₂²	1	4	yes	—
5	C₄×#13083549	C₄	1	8	yes	complemented
6	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₄³×C₈₉	C₄	1	1	yes	characteristic
11	C₂×#13083549	—	2	8	no	—
12	C₂×#13083549	C₂×C₄	1	8	yes	—
13	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₈₉	C₈	1	2	yes	—
17	C₂×C₄²×C₈₉	C₈	1	2	yes	—
18	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₈₉	—	2	4	no	—
21	C₄×C₈×C₈₉	—	2	4	no	—
22	C₄×C₈×C₈₉	C₈	1	4	yes	—
23	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
24	#13083549	—	2	8	no	—
25	C₂²×C₄×C₈₉	C₂×C₈	1	4	yes	—
26	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
27	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₈₉	—	2	2	no	—
30	C₂×C₈×C₈₉	—	2	2	no	—
31	C₂×C₈×C₈₉	C₄²	1	2	yes	—
32	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
33	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
34	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
35	C₂×C₈×C₈₉	#41	1	4	yes	—
36	C₄²×C₈₉	—	2	4	no	—
37	C₄²×C₈₉	—	2	2	no	—
38	C₄²×C₈₉	—	2	2	no	—
39	C₄²×C₈₉	C₁₆	1	8	yes	complemented
40	C₄²×C₈₉	C₄²	1	2	yes	—
41	C₄²×C₈₉	C₂×C₈	1	2	yes	—
42	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
43	C₂×C₄×C₈₉	—	2	4	no	—
44	C₂×C₄×C₈₉	—	2	4	no	—
45	C₂×C₄×C₈₉	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₈₉	C₄×C₈	1	2	yes	—
47	C₂×C₄×C₈₉	C₂×C₁₆	1	8	yes	—
48	C₂×C₄×C₈₉	C₂×C₁₆	1	4	yes	—
49	C₂×C₄×C₈₉	M₃₂	1	4	yes	—
50	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₈₉	#117	1	2	yes	—
52	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
54	C₈×C₈₉	—	2	4	no	—
55	C₈×C₈₉	#97	1	4	yes	—
56	C₈×C₈₉	#117	1	4	yes	—
57	C₂²×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
58	C₂²×C₈₉	#431	1	2	yes	—
59	C₂²×C₈₉	C₂²×C₁₆	1	2	yes	—
60	C₂²×C₈₉	C₂×M₃₂	1	2	yes	—
61	C₄×C₈₉	—	2	8	no	—
62	C₄×C₈₉	C₄×C₁₆	1	4	yes	—
63	C₄×C₈₉	#346	1	4	yes	—
64	C₄×C₈₉	#376	1	1	yes	characteristic
65	C₄×C₈₉	C₂×#97	1	1	yes	characteristic
66	C₄×C₈₉	#431	1	2	yes	—
67	#12590	—	89	1	no	maximal, Sylow
68	C₂×C₈₉	#1744	1	1	yes	characteristic
69	C₂×C₈₉	C₂×C₄×C₁₆	1	1	yes	characteristic
70	C₂×C₈₉	C₂×#346	1	1	yes	characteristic
71	C₂×C₈₉	#2162	1	4	yes	—
72	C₂×C₄×C₁₆	—	89	2	no	—
73	C₄²×C₈	D₈₉	1	1	yes	characteristic
74	#1374	—	89	4	no	—
75	C₈₉	#12590	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₁₆	—	89	4	no	—
77	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
78	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
79	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
80	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
81	C₄³	Dic₈₉	1	1	yes	characteristic
82	C₄×C₁₆	—	89	8	no	—
83	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
84	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
85	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
86	C₂×C₄²	#10623494	1	2	yes	—
87	C₂×C₄²	#10623494	1	2	yes	—
88	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
89	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
90	C₂×C₁₆	—	89	8	no	—
91	C₂×C₁₆	—	178	8	no	—
92	C₄×C₈	—	2	4	no	—
93	C₄×C₈	—	2	4	no	—
94	C₄×C₈	#10623494	1	4	yes	—
95	C₄×C₈	C₄×D₈₉	1	4	yes	—
96	C₂²×C₄	C₂×#10623494	1	4	yes	—
97	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
98	C₂²×C₄	C₂×#10623494	1	1	yes	characteristic
99	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	C₄×Dic₈₉	1	2	yes	—
103	C₂×C₈	C₂×#10623494	1	4	yes	—
104	C₂×C₈	#13083558	1	4	yes	—
105	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
106	C₂×C₈	#13083574	1	4	yes	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	—	2	4	no	—
110	C₄²	#13083549	1	8	yes	complemented
111	C₄²	C₄×Dic₈₉	1	2	yes	—
112	C₄²	C₂×#10623494	1	2	yes	—
113	C₁₆	—	178	8	no	—
114	C₂³	C₂²×#10623494	1	1	yes	characteristic
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	C₄×#10623494	1	2	yes	—
118	C₂×C₄	C₄×#10623494	1	2	yes	—
119	C₂×C₄	C₂×#13083549	1	8	yes	—
120	C₂×C₄	C₂×#13083549	1	4	yes	—
121	C₂×C₄	#15667952	1	4	yes	—
122	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
123	C₂×C₄	#15667971	1	2	yes	—
124	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
125	C₂×C₄	C₂×#13083558	1	2	yes	—
126	C₈	—	2	4	no	—
127	C₈	#15667923	1	4	yes	—
128	C₈	#15667969	1	4	yes	—
129	C₂²	C₂×C₄×#10623494	1	1	yes	characteristic
130	C₂²	#19151651	1	2	yes	—
131	C₂²	C₂²×#13083549	1	2	yes	—
132	C₂²	C₂×#15667952	1	2	yes	—
133	C₄	—	2	8	no	—
134	C₄	C₄×#13083549	1	4	yes	—
135	C₄	#19151374	1	4	yes	—
136	C₄	#19151444	1	1	yes	characteristic
137	C₄	C₂×#15667923	1	1	yes	characteristic
138	C₄	#19151651	1	2	yes	—
139	C₂	#26019998	1	1	yes	characteristic
140	C₂	C₂×C₄×#13083549	1	1	yes	characteristic
141	C₂	C₂×#19151374	1	1	yes	characteristic
142	C₂	#26022051	1	4	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₈₉
commutator	C₂×C₈₉	C₂×C₄×C₁₆
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₄²×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×C₈
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	#12590
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
5. C₄×#13083549	C₄	semidirect
39. C₄²×C₈₉	C₁₆	semidirect
75. C₈₉	#12590	semidirect
110. C₄²	#13083549	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#13083549	2
#26018330	2
C₄²×C₈×C₈₉	2
#12590	89