Subgroups — #36443801

Order 22784.

486 conjugacy classes of subgroups.

The subgroup lattice has 204 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	#26019998	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151320	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151320	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151319	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1744	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
9	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
10	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
12	#19151353	C₄	1	4	yes	—
13	#19151353	C₄	1	4	yes	—
14	#19151320	C₄	1	4	yes	complemented
15	#19151320	C₄	1	4	yes	complemented
16	#19151319	C₄	1	4	yes	complemented
17	C₈×#10623494	C₄	1	4	yes	complemented
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₄³×C₈₉	C₂²	1	1	yes	characteristic
21	#336×C₈₉	C₄	1	4	yes	—
22	C₂²×#10623494	C₂×C₄	1	2	yes	—
23	C₂²×#10623494	C₂×C₄	1	2	yes	—
24	C₂²×#10623494	C₂×C₄	1	2	yes	—
25	C₂²×#10623494	C₂×C₄	1	2	yes	—
26	C₄×#10623494	C₂×C₄	1	2	yes	—
27	C₄×#10623494	C₂×C₄	1	2	yes	—
28	C₄×#10623494	C₂×C₄	1	2	yes	—
29	C₄×#10623494	C₂×C₄	1	2	yes	—
30	C₄×#10623494	C₂×C₄	1	2	yes	—
31	C₄×#10623494	C₂×C₄	1	2	yes	—
32	C₄×#10623494	C₂×C₄	1	2	yes	—
33	C₄×#10623494	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
44	C₂×#10623494	—	2	4	no	—
45	C₂×#10623494	—	2	4	no	—
46	C₂×#10623494	—	2	4	no	—
47	C₂×#10623494	—	2	4	no	—
48	C₂×#10623494	C₄²	1	4	yes	—
49	C₂×#10623494	C₄²	1	4	yes	—
50	C₂×#10623494	C₄²	1	4	yes	—
51	C₂×#10623494	C₄²	1	4	yes	—
52	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
53	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₈₉	—	2	4	no	—
57	C₂×C₈×C₈₉	—	2	4	no	—
58	C₂×C₈×C₈₉	C₄²	1	4	yes	—
59	C₂×C₈×C₈₉	C₄²	1	4	yes	—
60	C₄²×C₈₉	—	2	4	no	—
61	C₄²×C₈₉	—	2	4	no	—
62	C₄²×C₈₉	—	2	4	no	—
63	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₈₉	—	2	4	no	—
69	C₂×C₄×C₈₉	—	2	4	no	—
70	C₂×C₄×C₈₉	—	2	4	no	—
71	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₈₉	#131	1	4	yes	—
78	C₂×C₄×C₈₉	#131	1	4	yes	—
79	C₂×C₄×C₈₉	#131	1	4	yes	—
80	C₈×C₈₉	—	2	4	no	—
81	C₈×C₈₉	—	2	4	no	—
82	#10623494	—	2	4	no	—
83	#10623494	—	2	4	no	—
84	#10623494	—	2	4	no	—
85	#10623494	—	2	4	no	—
86	C₂²×C₈₉	C₄³	1	1	yes	characteristic
87	C₂²×C₈₉	#405	1	2	yes	—
88	C₂²×C₈₉	#405	1	2	yes	—
89	C₂²×C₈₉	#405	1	2	yes	—
90	C₄×C₈₉	—	4	4	no	—
91	C₄×C₈₉	#405	1	2	yes	—
92	C₄×C₈₉	#405	1	2	yes	—
93	C₄×C₈₉	#405	1	2	yes	—
94	C₄×C₈₉	#405	1	2	yes	—
95	C₄×C₈₉	#405	1	2	yes	—
96	C₄×C₈₉	#405	1	2	yes	—
97	#7845	—	89	1	no	maximal, Sylow
98	C₂×C₈₉	#1454	1	4	yes	—
99	C₂×C₈₉	#1726	1	1	yes	characteristic
100	C₂×C₈₉	#1726	1	1	yes	characteristic
101	C₂×C₈₉	#1726	1	1	yes	characteristic
102	#1744	—	89	1	no	—
103	#1744	—	89	1	no	—
104	#1744	D₈₉	1	1	yes	characteristic
105	C₂×#322	—	89	1	no	—
106	C₂×#322	—	89	1	no	—
107	C₂×#322	—	89	1	no	—
108	C₂×C₈²	—	89	1	no	—
109	C₈₉	#7845	1	1	yes	characteristic, complemented, Sylow
110	C₂×C₄×C₈	—	89	1	no	—
111	C₂×C₄×C₈	—	89	1	no	—
112	C₂×C₄×C₈	—	89	1	no	—
113	C₂×C₄×C₈	—	89	1	no	—
114	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
116	C₄³	C₂×D₈₉	1	1	yes	characteristic
117	#336	—	89	4	no	—
118	#336	—	89	4	no	—
119	#336	Dic₈₉	1	4	yes	—
120	#322	—	89	4	no	—
121	#322	—	89	4	no	—
122	#322	—	89	4	no	—
123	C₈²	—	89	4	no	—
124	C₂²×C₈	—	89	2	no	—
125	C₂²×C₈	—	89	2	no	—
126	C₂²×C₈	—	89	2	no	—
127	C₂²×C₈	—	89	2	no	—
128	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
129	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
130	C₂×C₄²	C₄×D₈₉	1	2	yes	—
131	C₂×C₄²	C₄×D₈₉	1	2	yes	—
132	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
133	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
134	C₄×C₈	—	89	2	no	—
135	C₄×C₈	—	89	2	no	—
136	C₄×C₈	—	89	2	no	—
137	C₄×C₈	—	89	2	no	—
138	C₄×C₈	—	89	2	no	—
139	C₄×C₈	—	89	2	no	—
140	C₄×C₈	—	89	2	no	—
141	C₄×C₈	—	89	2	no	—
142	C₄×C₈	C₄×D₈₉	1	2	yes	—
143	C₄×C₈	C₄×D₈₉	1	2	yes	—
144	C₄×C₈	C₄×D₈₉	1	2	yes	—
145	C₄×C₈	C₄×D₈₉	1	2	yes	—
146	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
147	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
149	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	89	4	no	—
153	C₂×C₈	—	89	4	no	—
154	C₂×C₈	—	89	4	no	—
155	C₂×C₈	—	89	4	no	—
156	C₂×C₈	—	178	4	no	—
157	C₂×C₈	—	178	4	no	—
158	C₂×C₈	—	178	4	no	—
159	C₂×C₈	—	178	4	no	—
160	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
161	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
162	C₄²	—	2	4	no	—
163	C₄²	—	2	4	no	—
164	C₄²	—	2	4	no	—
165	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
166	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
167	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
168	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
169	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
174	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
175	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
176	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
179	C₂×C₄	#15668018	1	4	yes	—
180	C₂×C₄	#15668019	1	4	yes	—
181	C₂×C₄	#15668020	1	4	yes	—
182	C₈	—	2	4	no	—
183	C₈	—	2	4	no	—
184	C₈	—	178	4	no	—
185	C₈	—	178	4	no	—
186	C₈	—	178	4	no	—
187	C₈	—	178	4	no	—
188	C₂²	C₄²×Dic₈₉	1	1	yes	characteristic
189	C₂²	#19151553	1	2	yes	—
190	C₂²	#19151554	1	2	yes	—
191	C₂²	#19151556	1	2	yes	—
192	C₄	—	4	4	no	—
193	C₄	#19151552	1	2	yes	—
194	C₄	#19151552	1	2	yes	—
195	C₄	#19151554	1	2	yes	—
196	C₄	#19151554	1	2	yes	—
197	C₄	#19151555	1	2	yes	—
198	C₄	#19151555	1	2	yes	—
199	C₂	#26018552	1	4	yes	—
200	C₂	#26019938	1	1	yes	characteristic
201	C₂	#26019939	1	1	yes	characteristic
202	C₂	#26019940	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₈₉
commutator	C₂²×C₈₉	C₄³
Fitting	C₈₉×#1744	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1744	D₈₉
hypocenter	C₈₉	#7845
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ #1744

Complements

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

N	H ≅ G/N	type
14. #19151320	C₄	semidirect
15. #19151320	C₄	semidirect
16. #19151319	C₄	semidirect
17. C₈×#10623494	C₄	semidirect
109. C₈₉	#7845	semidirect

Maximal subgroups

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

subgroup	index
#26019998	2
C₂×#19151320	2
C₂×#19151319	2
C₂×C₈×#10623494	2
C₈₉×#1744	2
#7845	89