Finite Group Databasebeta

← #36458929

Subgroups — #36458929

Order 22784.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26028524C211yesmaximal, characteristic
2#26020421C211yesmaximal, characteristic
3#26020386C212yesmaximal
4#26018134C212yesmaximal
5C89×#1295C211yesmaximal, characteristic
6C2×#15667945—21no—
7C2×#15667943—21no—
8C2×#15667923C2211yescharacteristic
9#19151540—22no—
10#19151537—21no—
11#19151519—22no—
12#19151518—21no—
13#19151468C2211yescharacteristic
14#19151464C2212yes—
15#19151361—22no—
16#19151361—22no—
17C2×#97×C89C2212yes—
18#382×C89C2211yescharacteristic
19C22×Dic178—21no—
20C22×#10623494—21no—
21C2×C4⋊Dic89—21no—
22C2×C4⋊Dic89—22no—
23C2×C4⋊Dic89—21no—
24C2×C4⋊Dic89D412yes—
25C2×C4⋊Dic89D412yes—
26C2×#13083553—22no—
27#15667945—42no—
28#15667943—42no—
29#15667923—21no—
30#15667923Q812yescomplemented
31#15667919—21no—
32#15667919—22no—
33#15667919—21no—
34#15667919—22no—
35#15667919—22no—
36C22×C8×C89—22no—
37C2×C4⋊C4×C89D411yescharacteristic
38C2×C4⋊C4×C89D411yescharacteristic
39C2×C42×C89C2311yescharacteristic
40#97×C89—22no—
41#97×C89Q814yescomplemented
42#95×C89—22no—
43C2×Dic178—81no—
44C2×Dic178—82no—
45C22×Dic89—22no—
46C22×Dic89—22no—
47C22×Dic89—41no—
48C2×#10623494—22no—
49C2×#10623494—22no—
50C2×#10623494—22no—
51C4⋊Dic89—42no—
52C4⋊Dic89—42no—
53C4⋊Dic89—41no—
54C4⋊Dic89—41no—
55C4⋊Dic89—44no—
56C4⋊Dic89—81no—
57#13083553—82no—
58C22×C4×C89—21no—
59C22×C4×C89—21no—
60C22×C4×C89C2×D412yes—
61C22×C4×C89C2×D411yescharacteristic
62C2×C8×C89—24no—
63C2×C8×C89—42no—
64C2×C8×C89—42no—
65C4⋊C4×C89—41no—
66C4⋊C4×C89—41no—
67C42×C89C2×Q811yescharacteristic
68C42×C89C2×Q812yes—
69C42×C89#4111yescharacteristic
70C23×C89#12011yescharacteristic
71C2×C4×C89—21no—
72C2×C4×C89—22no—
73C2×C4×C89—22no—
74C2×C4×C89—21no—
75C2×C4×C89—41no—
76C2×C4×C89—41no—
77C2×C4×C89—42no—
78C2×C4×C89—42no—
79C2×C4×C89C22⋊Q812yes—
80C2×C4×C89C22⋊Q811yescharacteristic
81C2×C4×C89C4⋊Q812yes—
82C2×C4×C89C4⋊Q811yescharacteristic
83C2×Dic89—42no—
84C2×Dic89—42no—
85C2×Dic89—42no—
86C2×Dic89—42no—
87C2×Dic89—42no—
88C2×Dic89—42no—
89C2×Dic89—81no—
90C2×Dic89—82no—
91Dic178—84no—
92C8×C89—44no—
93#10623494—24no—
94C22×C89#39511yescharacteristic
95C22×C89#45311yescharacteristic
96C22×C89#45412yes—
97C22×C89#45512yes—
98C22×C89#45611yescharacteristic
99C4×C89—21no—
100C4×C89—21no—
101C4×C89—22no—
102C4×C89—22no—
103C4×C89—81no—
104C4×C89—81no—
105Dic89—82no—
106Dic89—82no—
107Dic89—161no—
108#10584—891nomaximal, Sylow
109C2×C89#165711yescharacteristic
110C2×C89#165911yescharacteristic
111C2×C89#166412yes—
112C2×C89#206011yescharacteristic
113C2×C89#206112yes—
114#2844—891no—
115#1847—891no—
116#1836—892no—
117#1295—892no—
118#1295D8911yescharacteristic
119C89#1058411yescharacteristic, complemented, Sylow
120C2×#107—1781no—
121C2×#106—1781no—
122C2×#97—891no—
123C2×#97C2×D8912yes—
124#400—1782no—
125#400—1781no—
126#395—1782no—
127#395—1781no—
128#384—891no—
129#382—892no—
130#382C2×D8911yescharacteristic
131#340—1782no—
132#340—1782no—
133C22×Q8—1781no—
134C22×C8—22no—
135C22×C8—1781no—
136C2×C4⋊C4—892no—
137C2×C4⋊C4—892no—
138C2×C4⋊C4—1782no—
139C2×C4⋊C4—1781no—
140C2×C4⋊C4—1782no—
141C2×C4⋊C4—1781no—
142C2×C4⋊C4D35611yescharacteristic
143C2×C4⋊C4D35611yescharacteristic
144C2×C42C22×D8911yescharacteristic
145#107—3562no—
146#106—3562no—
147#97—22no—
148#97—892no—
149#97—1781no—
150#97Dic17814yescomplemented
151#95—22no—
152#95—1782no—
153#95—1781no—
154#95—1781no—
155#95—1782no—
156#95—1782no—
157C2×Q8—7121no—
158C2×Q8—7122no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—3561no—
162C22×C4—1782no—
163C22×C4—1782no—
164C22×C4D4×D8912yes—
165C22×C4C2×D35611yescharacteristic
166C2×C8—24no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—1782no—
170C2×C8—1782no—
171C2×C8—1782no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—3562no—
175C4⋊C4—3562no—
176C4⋊C4—3561no—
177C4⋊C4—3561no—
178C4⋊C4—3564no—
179C4⋊C4—7122no—
180C4⋊C4—7121no—
181C42Q8×D8911yescharacteristic
182C42C2×Dic17812yes—
183C42#1308357311yescharacteristic
184C23#1566798211yescharacteristic
185Q8—7124no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—3562no—
195C2×C4—3562no—
196C2×C4—3562no—
197C2×C4—3562no—
198C2×C4—3562no—
199C2×C4—3562no—
200C2×C4—7121no—
201C2×C4—7122no—
202C2×C4#1566798912yes—
203C2×C4#1566799211yescharacteristic
204C2×C4#1566801112yes—
205C2×C4#1566801211yescharacteristic
206C8—44no—
207C8—1784no—
208C22#1915151911yescharacteristic
209C22#1915176811yescharacteristic
210C22#1915177212yes—
211C22#1915177712yes—
212C22#1915178311yescharacteristic
213C4—21no—
214C4—21no—
215C4—22no—
216C4—22no—
217C4—81no—
218C4—81no—
219C4—14241no—
220C4—7122no—
221C4—7122no—
222C2#2601955111yescharacteristic
223C2#2601955911yescharacteristic
224C2#2601958812yes—
225C2#2602159811yescharacteristic
226C2#2602160112yes—
subgroupG/H
centerC23#15667982
commutator, cosocleC2×C42×C89C23
FrattiniC2×C42C22×D89
FittingC89×#1295C2
socle, prime radicalC23×C89#120
hypercenter#1295D89
hypocenterC89#10584
LCSG ▷ C2×C42×C89 ▷ C22×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
30. #15667923Q8semidirect
41. #97×C89Q8semidirect
119. C89#10584semidirect
150. #97Dic178semidirect

Maximal subgroups

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

subgroupindex
#260285242
#260204212
#260203862
#260181342
C89×#12952
#1058489