Finite Group Databasebeta

← #32983541

Subgroups — #32983541

Order 18688.

508 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 95 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#22620388C211yesmaximal, characteristic
2#22613112C211yesmaximal, characteristic, complemented
3#22608479C212yesmaximal
4#22607797C212yesmaximal, complemented
5C73×#1558C211yesmaximal, characteristic, complemented
6C2×#14556622—22no—
7#17989137C2211yescharacteristic
8#17988979—22no—
9#17988966C412yescomplemented
10#17988961C412yescomplemented
11C4×#11924012C2211yescharacteristic
12#17988494—22no—
13#17988473—24no—
14#17988455C2212yes—
15#17988322—24no—
16C2×C4⋊Q8×C73C2211yescharacteristic
17#432×C73C2212yes—
18#333×C73—22no—
19C2×#11924028—21no—
20#14556622—44no—
21#14556622—44no—
22C2×#11924018—24no—
23C2×#11924015—24no—
24C2×#11924012—21no—
25#14556591C2×C411yescharacteristic
26#14556588C2×C411yescharacteristic
27#14556579—21no—
28#14556565—22no—
29C4×D292—22no—
30C2×C4⋊Dic73—22no—
31C2×C4×Dic73—22no—
32#14556502—21no—
33C4×#10596831C2×C412yes—
34#14556498—24no—
35#14556498—22no—
36#14556498—22no—
37#14556498D414yes—
38C22×Q8×C73D412yes—
39C4⋊Q8×C73—22no—
40C4⋊Q8×C73C2×C412yes—
41C4⋊Q8×C73D412yes—
42C2×C4⋊C4×C73—22no—
43C2×C42×C73C2311yescharacteristic
44C4⋊C8×C73—22no—
45C22⋊C8×C73—22no—
46#97×C73—22no—
47#11924028—42no—
48#11924028—42no—
49C2×Dic146—21no—
50C2×D292—21no—
51C2×D292—22no—
52C2×C22⋊D73—41no—
53C22×Dic73—24no—
54C2×C4×D73—41no—
55#11924018—48no—
56#11924015—48no—
57#11924012—24no—
58C2×#10596831—22no—
59C4⋊Dic73—22no—
60C4⋊Dic73—82no—
61#11924005—42no—
62C4×Dic73—82no—
63C2×Q8×C73—22no—
64C2×Q8×C73—42no—
65C2×Q8×C73—42no—
66C2×Q8×C73C22⋊C414yes—
67C22×C4×C73—22no—
68C22×C4×C73C2×D412yes—
69C22×C4×C73C2×D411yescharacteristic
70C2×C8×C73—42no—
71C4⋊C4×C73—24no—
72C4⋊C4×C73—24no—
73C42×C73C22×C411yescharacteristic
74C42×C73C2×D411yescharacteristic
75C42×C73#4112yes—
76C23×C73#12011yescharacteristic
77C22×D73—41no—
78Q8×C73—44no—
79Q8×C73—82no—
80C2×C4×C73—22no—
81C2×C4×C73—24no—
82C2×C4×C73—42no—
83C2×C4×C73—42no—
84C2×C4×C73—42no—
85C2×C4×C73C2×C22⋊C411yescharacteristic
86C2×C4×C73C4×D412yes—
87C2×C4×C73#12112yes—
88C2×C4×C73#12311yescharacteristic
89C22⋊D73—82no—
90C2×Dic73—48no—
91C2×Dic73—41no—
92C2×Dic73—44no—
93D292—42no—
94D292—82no—
95C4×D73—81no—
96C4×D73—81no—
97Dic146—42no—
98C8×C73—82no—
99#10596831—24no—
100C22×C73—22no—
101C22×C73#38611yescharacteristic
102C22×C73#45312yes—
103C2×D73—81no—
104C2×D73—82no—
105C4×C73—21no—
106C4×C73—21no—
107C4×C73—22no—
108C4×C73—22no—
109C4×C73—44no—
110Dic73—84no—
111Dic73—81no—
112#9429—731nomaximal, Sylow
113C2×C73—22no—
114C2×C73#188711yescharacteristic
115C2×C73#190512yes—
116D73—161no—
117#3152—731no—
118#2381—731no—
119#1558D7311yescharacteristic, complemented
120#1514—732no—
121#1288—732no—
122C73#942911yescharacteristic, complemented, Sylow
123C2×#137—1462no—
124#532—731no—
125C2×C4⋊Q8C2×D7311yescharacteristic
126#497—1462no—
127#494—732no—
128#492—732no—
129#432C2×D7312yes—
130C4×M16—731no—
131#391—1462no—
132#387—1464no—
133#383—732no—
134#333—22no—
135#327—1464no—
136C2×#41—1461no—
137C22×Q8D29212yes—
138#137—2924no—
139#137—2924no—
140C2×Q16—1464no—
141C2×SD16—1464no—
142C2×M16—1461no—
143C4⋊Q8—22no—
144C4⋊Q8—731no—
145C4⋊Q8C4×D7312yes—
146C4⋊Q8C22⋊D7312yes—
147#127—731no—
148#124—1461no—
149#121—1462no—
150C4×D4—1462no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—1462no—
153C2×C42—1462no—
154C2×C42C22×D7311yescharacteristic
155C4⋊C8—22no—
156C22⋊C8—22no—
157#97—22no—
158#97—1461no—
159C4×C8—732no—
160#95—734no—
161#95—1464no—
162#95—1462no—
163#95—1462no—
164#41—2922no—
165#41—2922no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—1461no—
170C2×Q8#1192400514yes—
171C2×D4—2921no—
172C2×D4—1461no—
173C2×D4—1462no—
174C22×C4—22no—
175C22×C4—2921no—
176C22×C4—1464no—
177C22×C4D4×D7312yes—
178C22×C4C2×D29211yescharacteristic
179Q16—2928no—
180SD16—2928no—
181M16—1464no—
182C2×C8—42no—
183C2×C8—1462no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—5842no—
187C4⋊C4—1462no—
188C22⋊C4—2922no—
189C42—5842no—
190C42C2×C4×D7311yescharacteristic
191C42C2×C22⋊D7311yescharacteristic
192C42#1192402712yes—
193C23—2921no—
194C23#1455656111yescharacteristic
195Q8—44no—
196Q8—82no—
197Q8—2922no—
198D4—2922no—
199D4—5842no—
200D4—5842no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—2928no—
207C2×C4—2921no—
208C2×C4—2924no—
209C2×C4—5841no—
210C2×C4—5841no—
211C2×C4C2×#1192400511yescharacteristic
212C2×C4#1455655112yes—
213C2×C4#1455656312yes—
214C2×C4#1455657511yescharacteristic
215C8—82no—
216C8—1464no—
217C22—22no—
218C22—5841no—
219C22—5842no—
220C22#1798846411yescharacteristic
221C22#1798875712yes—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—5844no—
228C4—5841no—
229C2—22no—
230C2—11681no—
231C2#2261032711yescharacteristic
232C2#2261042312yes—
subgroupG/H
centerC22#17988464
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1558C2
socleC22×C73#386
prime radical#17989137C22
cosocleC2×C42×C73C23
hypercenter#1558D73
hypocenterC73#9429
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1558

Complements

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

NH ≅ G/Ntype
2. #22613112C2semidirect
4. #22607797C2semidirect
5. C73×#1558C2semidirect
9. #17988966C4semidirect
10. #17988961C4semidirect
119. #1558D73semidirect
122. C73#9429semidirect

Maximal subgroups

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

subgroupindex
#226203882
#226131122
#226084792
#226077972
C73×#15582
#942973