Subgroups — #36474609

Order 22784.

238 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₂×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26021847	C₂	1	1	yes	maximal, characteristic
4	#26018385	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#15667918	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1397	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15667951	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
10	#19151552	C₄	1	1	yes	characteristic
11	#19151431	C₄	1	4	yes	complemented
12	C₂×#15667918	C₂²	1	1	yes	characteristic
13	C₂×#15667918	C₂²	1	1	yes	characteristic
14	C₃₂×D₈₉	C₄	1	4	yes	complemented
15	#19151374	C₄	1	1	yes	characteristic
16	#19151373	C₄	1	1	yes	characteristic
17	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
20	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#13083557	C₈	1	1	yes	characteristic
22	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
23	#15667969	C₈	1	1	yes	characteristic
24	#15667951	C₈	1	4	yes	—
25	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
26	C₂×#13083549	C₂×C₄	1	1	yes	characteristic
27	C₁₆×D₈₉	C₂×C₄	1	2	yes	—
28	C₁₆×D₈₉	C₂×C₄	1	2	yes	—
29	#15667923	C₈	1	1	yes	characteristic
30	#15667922	C₈	1	1	yes	characteristic
31	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
32	#15667918	C₂×C₄	1	2	yes	—
33	#15667918	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
36	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
37	C₃₂×C₈₉	C₂×C₄	1	2	yes	—
38	C₃₂×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
40	#13083557	C₁₆	1	4	yes	—
41	C₂×#10623494	C₂×C₈	1	1	yes	characteristic
42	C₂×#10623494	C₂×C₈	1	1	yes	characteristic
43	C₈×D₈₉	C₄²	1	2	yes	—
44	C₈×D₈₉	C₂×C₈	1	2	yes	—
45	#13083553	C₁₆	1	2	yes	—
46	#13083551	C₁₆	1	2	yes	—
47	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
48	#13083549	C₄²	1	2	yes	—
49	#13083549	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
53	C₁₆×C₈₉	C₂×C₈	1	2	yes	—
54	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
55	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₂²×D₈₉	C₂×C₁₆	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
58	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
59	C₂×Dic₈₉	C₂×C₁₆	1	2	yes	—
60	C₂×Dic₈₉	C₂×C₁₆	1	1	yes	characteristic
61	C₄×D₈₉	C₄×C₈	1	2	yes	—
62	C₄×D₈₉	C₂×C₁₆	1	2	yes	—
63	C₈×C₈₉	C₂×C₁₆	1	2	yes	—
64	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
65	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
66	#10623494	C₄×C₈	1	2	yes	—
67	#10623494	C₂×C₁₆	1	2	yes	—
68	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
69	C₂×D₈₉	—	2	2	no	—
70	C₂×D₈₉	C₄×C₁₆	1	2	yes	—
71	C₄×C₈₉	—	2	2	no	—
72	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
73	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
74	Dic₈₉	—	2	2	no	—
75	Dic₈₉	C₄×C₁₆	1	2	yes	—
76	#13549	—	89	1	no	maximal, Sylow
77	C₂×C₈₉	C₂×C₄×C₁₆	1	1	yes	characteristic
78	C₂×C₈₉	#2258	1	2	yes	—
79	D₈₉	—	2	2	no	—
80	C₂×M₆₄	—	89	1	no	—
81	C₂²×C₃₂	—	89	1	no	—
82	#2108	—	89	1	no	—
83	#1397	—	89	1	no	—
84	#1397	D₈₉	1	1	yes	characteristic, complemented
85	C₄×C₃₂	—	89	1	no	—
86	C₄×C₃₂	—	89	1	no	—
87	C₈₉	#13549	1	1	yes	characteristic, complemented, Sylow
88	C₂×M₃₂	—	89	1	no	—
89	C₂²×C₁₆	—	89	1	no	—
90	#405	—	89	1	no	—
91	M₆₄	—	89	4	no	—
92	C₂×C₃₂	—	89	1	no	—
93	C₂×C₃₂	—	89	1	no	—
94	C₂×C₃₂	—	89	4	no	—
95	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
96	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
97	#346	—	89	1	no	—
98	#346	—	89	1	no	—
99	C₄×C₁₆	—	89	1	no	—
100	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
101	C₂×M₁₆	—	89	1	no	—
102	C₂²×C₈	—	89	1	no	—
103	#117	—	89	1	no	—
104	M₃₂	—	89	4	no	—
105	C₂×C₁₆	—	89	1	no	—
106	C₂×C₁₆	—	89	1	no	—
107	C₂×C₁₆	—	89	2	no	—
108	C₂×C₁₆	—	89	2	no	—
109	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
110	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
111	#97	—	89	1	no	—
112	#97	—	89	1	no	—
113	C₄×C₈	—	89	1	no	—
114	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
115	C₃₂	—	89	2	no	—
116	C₃₂	—	89	2	no	—
117	C₃₂	C₄×D₈₉	1	2	yes	—
118	C₃₂	C₄×D₈₉	1	2	yes	—
119	C₂²×C₄	—	89	1	no	—
120	M₁₆	—	89	4	no	—
121	C₂×C₈	—	89	1	no	—
122	C₂×C₈	—	89	1	no	—
123	C₂×C₈	—	89	2	no	—
124	C₂×C₈	—	89	2	no	—
125	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
126	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
127	C₄⋊C₄	—	89	2	no	—
128	C₂²⋊C₄	—	89	2	no	—
129	C₄²	—	89	1	no	—
130	C₄²	C₈×D₈₉	1	1	yes	characteristic
131	C₁₆	—	89	2	no	—
132	C₁₆	—	89	2	no	—
133	C₁₆	C₈×D₈₉	1	2	yes	—
134	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
135	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
136	C₂³	—	89	1	no	—
137	C₂×C₄	—	89	1	no	—
138	C₂×C₄	—	89	2	no	—
139	C₂×C₄	—	89	2	no	—
140	C₂×C₄	—	89	2	no	—
141	C₂×C₄	C₁₆×D₈₉	1	2	yes	—
142	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
143	C₈	—	89	2	no	—
144	C₈	—	89	2	no	—
145	C₈	C₁₆×D₈₉	1	2	yes	—
146	C₈	C₄²×D₈₉	1	1	yes	characteristic
147	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
148	C₂²	—	89	2	no	—
149	C₂²	—	178	2	no	—
150	C₂²	C₂×C₁₆×D₈₉	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	89	2	no	—
153	C₄	—	178	2	no	—
154	C₄	C₄×C₈×D₈₉	1	1	yes	characteristic
155	C₄	C₂×C₁₆×D₈₉	1	1	yes	characteristic
156	C₂	—	178	2	no	—
157	C₂	C₄×C₁₆×D₈₉	1	1	yes	characteristic
158	C₂	#26022547	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₈₉
commutator	C₂×C₈₉	C₂×C₄×C₁₆
Fitting	C₈₉×#1397	C₂
socle	C₂²×C₈₉	C₂²×C₁₆
prime radical	C₂²×D₈₉	C₂×C₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	#1397	D₈₉
hypocenter	C₈₉	#13549
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ #1397

Complements

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

N	H ≅ G/N	type
4. #26018385	C₂	semidirect
5. C₄×#15667918	C₂	semidirect
6. C₃₂×Dic₈₉	C₂	semidirect
7. C₈₉×#1397	C₂	semidirect
11. #19151431	C₄	semidirect
14. C₃₂×D₈₉	C₄	semidirect
84. #1397	D₈₉	semidirect
87. C₈₉	#13549	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151431	2
C₂×C₃₂×D₈₉	2
#26021847	2
#26018385	2
C₄×#15667918	2
C₃₂×Dic₈₉	2
C₈₉×#1397	2
#13549	89