Finite Group Databasebeta

← #36443559

Subgroups — #36443559

Order 22784.

366 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 87 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#26021795C211yesmaximal, characteristic
2#26021723C212yesmaximal
3#26018078C211yesmaximal, characteristic
4#26018075C212yesmaximal
5C89×#2809C211yesmaximal, characteristic
6C2×#15667945—22no—
7C2×#15667943—21no—
8C2×#15667936—21no—
9#19151468C2211yescharacteristic
10#19151360—22no—
11#19151353C2211yescharacteristic
12#19151353C2212yes—
13#400×C89—21no—
14#398×C89—22no—
15#391×C89—21no—
16#391×C89C2211yescharacteristic
17#389×C89C2212yes—
18#387×C89—22no—
19C22×#10623494—22no—
20C22×#10623494—21no—
21C2×C4⋊Dic89—22no—
22C2×C4⋊Dic89—21no—
23#15667945—44no—
24#15667943—42no—
25#15667936—42no—
26C22×Q8×C89D411yescharacteristic
27C2×C4⋊C4×C89—22no—
28C2×C4⋊C4×C89—21no—
29C2×C4⋊C4×C89—21no—
30C2×C4⋊C4×C89D412yes—
31C2×C4⋊C4×C89D412yes—
32C2×C4⋊C4×C89D411yescharacteristic
33C2×C42×C89—21no—
34C2×C42×C89C2311yescharacteristic
35#95×C89—21no—
36#95×C89—22no—
37#95×C89—22no—
38#95×C89—22no—
39#95×C89—22no—
40C22×Dic89—41no—
41C2×#10623494—24no—
42C2×#10623494—22no—
43C2×#10623494—42no—
44C2×#10623494—41no—
45C2×#10623494—42no—
46C2×#10623494—41no—
47C4⋊Dic89—41no—
48C4⋊Dic89—41no—
49C4⋊Dic89—42no—
50C4⋊Dic89—42no—
51C2×Q8×C89—21no—
52C2×Q8×C89—41no—
53C2×Q8×C89—41no—
54C2×Q8×C89Q1612yes—
55C22×C4×C89—21no—
56C22×C4×C89—21no—
57C22×C4×C89—22no—
58C22×C4×C89—22no—
59C22×C4×C89C2×D411yescharacteristic
60C22×C4×C89C2×D412yes—
61C4⋊C4×C89—22no—
62C4⋊C4×C89—22no—
63C4⋊C4×C89—42no—
64C4⋊C4×C89—41no—
65C4⋊C4×C89—42no—
66C4⋊C4×C89—42no—
67C4⋊C4×C89—81no—
68C4⋊C4×C89—81no—
69C42×C89—41no—
70C42×C89—81no—
71C23×C89#12011yescharacteristic
72Q8×C89—42no—
73Q8×C89—81no—
74C2×C4×C89—21no—
75C2×C4×C89—22no—
76C2×C4×C89—22no—
77C2×C4×C89—21no—
78C2×C4×C89—22no—
79C2×C4×C89—41no—
80C2×C4×C89—41no—
81C2×C4×C89—42no—
82C2×C4×C89—42no—
83C2×C4×C89—42no—
84C2×C4×C89—41no—
85C2×C4×C89—42no—
86C2×C4×C89—42no—
87C2×C4×C89—42no—
88C2×C4×C89—42no—
89C2×C4×C89C2×Q1611yescharacteristic
90C2×C4×C89#13512yes—
91C2×C4×C89AGL(1,ℤ/8)12yes—
92C2×C4×C89#13711yescharacteristic
93C2×Dic89—82no—
94C2×Dic89—81no—
95#10623494—44no—
96#10623494—42no—
97C22×C89#44912yes—
98C22×C89#45111yescharacteristic
99C22×C89#45311yescharacteristic
100C22×C89#45612yes—
101C22×C89#45811yescharacteristic
102C4×C89—22no—
103C4×C89—22no—
104C4×C89—21no—
105C4×C89—21no—
106C4×C89—42no—
107C4×C89—82no—
108C4×C89—82no—
109C4×C89—81no—
110Dic89—161no—
111#7802—891nomaximal, Sylow
112C2×C89#160912yes—
113C2×C89#161411yescharacteristic
114C2×C89#162411yescharacteristic
115C2×C89#164812yes—
116C2×C89#166011yescharacteristic
117#2809D8911yescharacteristic
118#2096—891no—
119#2083—892no—
120#1272—891no—
121#1271—892no—
122C89#780211yescharacteristic, complemented, Sylow
123C2×#107—1782no—
124C2×#106—1781no—
125C2×#103—1781no—
126#400—21no—
127#398—22no—
128#391—21no—
129#391C2×D8911yescharacteristic
130#389C2×D8912yes—
131#387—22no—
132#384—891no—
133#340—1782no—
134#336—892no—
135#336—891no—
136C22×Q8C22⋊D8911yescharacteristic
137C22×C8—1782no—
138C22×C8—1781no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—1782no—
143C2×C4⋊C4—1781no—
144C2×C4⋊C4C22⋊D8912yes—
145C2×C4⋊C4C22⋊D8912yes—
146C2×C4⋊C4C22⋊D8911yescharacteristic
147C2×C42—21no—
148C2×C42C22×D8911yescharacteristic
149#107—3564no—
150#106—3562no—
151#103—3562no—
152#95—21no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95—22no—
157C2×Q8—21no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8#1308356412yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—3561no—
166C22×C4C2×C22⋊D8911yescharacteristic
167C22×C4C2×C22⋊D8912yes—
168C2×C8—3562no—
169C2×C8—3561no—
170C2×C8—3562no—
171C2×C8—3561no—
172C2×C8—1782no—
173C2×C8—1784no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—41no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—3561no—
183C4⋊C4—3561no—
184C4⋊C4—3562no—
185C4⋊C4—3562no—
186C42—41no—
187C42—81no—
188C23#1566798111yescharacteristic
189Q8—42no—
190Q8—81no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—7122no—
207C2×C4—7121no—
208C2×C4C2×#1308356411yescharacteristic
209C2×C4#1566803212yes—
210C2×C4#1566803712yes—
211C2×C4#1566804311yescharacteristic
212C8—3564no—
213C8—3562no—
214C22#1915174312yes—
215C22#1915175411yescharacteristic
216C22#1915176611yescharacteristic
217C22#1915178112yes—
218C22#1915178811yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—82no—
225C4—82no—
226C4—81no—
227C4—14241no—
228C2#2601924912yes—
229C2#2601927711yescharacteristic
230C2#2601933511yescharacteristic
231C2#2601949412yes—
232C2#2601956311yescharacteristic
subgroupG/H
centerC23#15667981
commutator, cosocleC2×C42×C89C23
FrattiniC2×C42C22×D89
FittingC89×#2809C2
socle, prime radicalC23×C89#120
hypercenter#2809D89
hypocenterC89#7802
LCSG ▷ C2×C42×C89 ▷ C23×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2809

Complements

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

NH ≅ G/Ntype
122. C89#7802semidirect

Maximal subgroups

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

subgroupindex
#260217952
#260217232
#260180782
#260180752
C89×#28092
#780289