Subgroups — #36443790

Order 22784.

558 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	#26019998	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667923	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151320	C₂	1	2	yes	maximal
4	C₂×#19151319	C₂	1	2	yes	maximal
5	C₄²×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667923	C₄	1	4	yes	—
7	C₂×C₄×#10623494	C₂²	1	2	yes	—
8	C₂×C₄×#10623494	C₂²	1	2	yes	—
9	#19151353	C₄	1	4	yes	—
10	#19151320	C₄	1	8	yes	complemented
11	#19151319	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₈₉	C₄	1	4	yes	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄³×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×#10623494	C₂×C₄	1	4	yes	—
17	C₂²×#10623494	C₂×C₄	1	4	yes	—
18	#15667923	—	2	8	no	—
19	C₄×#10623494	C₂×C₄	1	4	yes	—
20	C₄×#10623494	C₂×C₄	1	4	yes	—
21	C₄×#10623494	C₂×C₄	1	4	yes	—
22	C₄×#10623494	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₈₉	—	2	8	no	—
30	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
32	C₂×#10623494	—	2	8	no	—
33	C₂×#10623494	—	2	8	no	—
34	C₂×#10623494	C₄²	1	8	yes	—
35	C₂×#10623494	C₄²	1	8	yes	—
36	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
37	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₉	—	2	4	no	—
41	C₂×C₈×C₈₉	C₄²	1	4	yes	—
42	C₂×C₈×C₈₉	C₄²	1	4	yes	—
43	C₂×C₈×C₈₉	M₁₆	1	8	yes	—
44	C₄²×C₈₉	—	2	4	no	—
45	C₄²×C₈₉	—	2	4	no	—
46	C₄²×C₈₉	M₁₆	1	8	yes	—
47	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
48	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
49	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₄×C₈₉	—	2	4	no	—
51	C₂×C₄×C₈₉	—	2	4	no	—
52	C₂×C₄×C₈₉	#97	1	8	yes	—
53	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
54	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
55	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
58	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
59	C₂×C₄×C₈₉	#131	1	4	yes	—
60	C₈×C₈₉	—	2	4	no	—
61	C₈×C₈₉	#97	1	8	yes	complemented
62	#10623494	—	2	8	no	—
63	#10623494	—	2	8	no	—
64	C₂²×C₈₉	C₄³	1	1	yes	characteristic
65	C₂²×C₈₉	C₂×#97	1	2	yes	—
66	C₂²×C₈₉	C₄×M₁₆	1	2	yes	—
67	C₂²×C₈₉	#405	1	2	yes	—
68	C₄×C₈₉	—	2	8	no	—
69	C₄×C₈₉	C₂×#97	1	2	yes	—
70	C₄×C₈₉	C₄×M₁₆	1	4	yes	—
71	C₄×C₈₉	C₄×M₁₆	1	2	yes	—
72	C₄×C₈₉	#405	1	4	yes	—
73	#7842	—	89	1	no	maximal, Sylow
74	C₂×C₈₉	#1451	1	4	yes	—
75	C₂×C₈₉	C₄×#97	1	1	yes	characteristic
76	C₂×C₈₉	C₄×#97	1	1	yes	characteristic
77	C₂×C₈₉	#1726	1	1	yes	characteristic
78	#1744	—	89	1	no	—
79	C₄×#97	—	89	1	no	—
80	C₄²×C₈	D₈₉	1	1	yes	characteristic
81	C₂×#322	—	89	2	no	—
82	C₂×#322	—	89	2	no	—
83	C₈₉	#7842	1	1	yes	characteristic, complemented, Sylow
84	C₂×#97	—	89	4	no	—
85	C₂×C₄×C₈	—	89	2	no	—
86	C₂×C₄×C₈	—	89	2	no	—
87	C₂×C₄×C₈	Dic₈₉	1	4	yes	—
88	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
90	C₄³	C₂×D₈₉	1	1	yes	characteristic
91	#336	—	89	4	no	—
92	#322	—	89	8	no	—
93	#322	—	89	8	no	—
94	C₂²×C₈	—	89	4	no	—
95	C₂²×C₈	—	89	4	no	—
96	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
97	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
98	C₂×C₄²	C₄×D₈₉	1	2	yes	—
99	C₂×C₄²	C₄×D₈₉	1	2	yes	—
100	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
101	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
102	#97	—	178	8	no	—
103	C₄×C₈	—	2	8	no	—
104	C₄×C₈	—	89	4	no	—
105	C₄×C₈	—	89	4	no	—
106	C₄×C₈	—	89	4	no	—
107	C₄×C₈	—	89	4	no	—
108	C₄×C₈	C₄×D₈₉	1	4	yes	—
109	C₄×C₈	C₄×D₈₉	1	4	yes	—
110	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
111	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
113	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	89	8	no	—
116	C₂×C₈	—	89	8	no	—
117	C₂×C₈	—	178	8	no	—
118	C₂×C₈	—	178	8	no	—
119	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
120	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
121	C₂×C₈	#13083558	1	8	yes	—
122	C₄²	—	2	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	#13083557	1	8	yes	—
125	C₄²	C₂×C₄×D₈₉	1	2	yes	—
126	C₄²	C₂×C₄×D₈₉	1	2	yes	—
127	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	#15667922	1	8	yes	—
131	C₂×C₄	C₄²×D₈₉	1	2	yes	—
132	C₂×C₄	C₄²×D₈₉	1	2	yes	—
133	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
135	C₂×C₄	C₂×#13083557	1	4	yes	—
136	C₂×C₄	C₂×#13083558	1	4	yes	—
137	C₂×C₄	#15668018	1	4	yes	—
138	C₈	—	2	4	no	—
139	C₈	—	178	8	no	—
140	C₈	—	178	8	no	—
141	C₈	#15667923	1	8	yes	complemented
142	C₂²	C₄²×Dic₈₉	1	1	yes	characteristic
143	C₂²	C₂×#15667922	1	2	yes	—
144	C₂²	C₄×#13083558	1	2	yes	—
145	C₂²	#19151553	1	2	yes	—
146	C₄	—	2	8	no	—
147	C₄	C₂×#15667923	1	2	yes	—
148	C₄	C₄×#13083557	1	4	yes	—
149	C₄	C₄×#13083558	1	2	yes	—
150	C₄	#19151552	1	4	yes	—
151	C₂	#26018541	1	4	yes	—
152	C₂	C₄×#15667922	1	1	yes	characteristic
153	C₂	C₄×#15667923	1	1	yes	characteristic
154	C₂	#26019938	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₈₉
commutator	C₂²×C₈₉	C₄³
Fitting	C₄²×C₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₄²×C₈	D₈₉
hypocenter	C₈₉	#7842
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
10. #19151320	C₄	semidirect
11. #19151319	C₄	semidirect
61. C₈×C₈₉	#97	semidirect
83. C₈₉	#7842	semidirect
141. C₈	#15667923	semidirect

Maximal subgroups

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

subgroup	index
#26019998	2
C₄×#15667923	2
C₂×#19151320	2
C₂×#19151319	2
C₄²×C₈×C₈₉	2
#7842	89