Finite Group Databasebeta

← #34032952

Subgroups — #34032952

Order 18688.

322 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 65 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#22627043C212yesmaximal
2#22625599C211yesmaximal, characteristic
3#17989700—22no—
4#17989673C414yes—
5#17989672—22no—
6C22×#11924036C411yescharacteristic
7C2×#14556670C2211yescharacteristic
8#17989452—22no—
9C2×M16×D73C411yescharacteristic
10C4⋊#10596842—42no—
11C22⋊#10596842—24no—
12C4×#10596842—42no—
13C2×#11924036—21no—
14C2×#11924036—41no—
15C2×#11924036—41no—
16C2×#11924036C812yescomplemented
17C22×#10596833—21no—
18#14556670—41no—
19#14556670—41no—
20C2×C4⋊#64408C2×C412yes—
21C2×C22⋊#64408D411yescharacteristic
22C2×C22⋊#64408Q811yescharacteristic
23C2×C4×#64408D412yes—
24C22×C4×D73C2×C411yescharacteristic
25C2×#11924012C811yescharacteristic, complemented
26C2×#11924011—21no—
27M16×D73—41no—
28M16×D73—41no—
29C2×C8×D73—21no—
30C2×M16×C73C811yescharacteristic, complemented
31C2×#10596842—44no—
32C22×#64408C22⋊C412yes—
33C22×#64408C4⋊C412yes—
34#11924036—41no—
35#11924036—41no—
36#11924036—42no—
37C2×#10596833—22no—
38C2×#10596833—42no—
39C4⋊#64408—42no—
40C22⋊#64408—41no—
41C22⋊#64408—41no—
42C4×#64408—22no—
43C4×#64408—22no—
44C23×D73C4⋊C411yescharacteristic
45C22×Dic73M1611yescharacteristic
46C2×C4×D73—21no—
47C2×C4×D73—21no—
48C2×C4×D73—21no—
49C2×C4×D73—21no—
50C2×C4×D73C4211yescharacteristic
51C2×C4×D73C22⋊C411yescharacteristic
52C2×C4×D73C2×C811yescharacteristic
53C2×C4×D73M1611yescharacteristic
54#11924012—41no—
55#11924011—42no—
56C2×#10596831—21no—
57C8×D73—41no—
58C8×D73—41no—
59C22×C4×C73C2×C811yescharacteristic
60M16×C73—41no—
61C2×C8×C73—21no—
62C23×C73C4⋊C811yescharacteristic
63C22×D73—21no—
64C22×D73—21no—
65C22×D73—42no—
66C22×D73#9511yescharacteristic
67C22×D73C4⋊C811yescharacteristic
68C2×#64408—22no—
69C2×#64408—22no—
70C2×#64408—42no—
71C2×#64408—42no—
72#10596842—84no—
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73C4×C811yescharacteristic
76C2×C4×C73C22⋊C811yescharacteristic
77C2×Dic73—21no—
78C2×Dic73—21no—
79C2×Dic73#9711yescharacteristic
80C2×Dic73C22⋊C811yescharacteristic
81C4×D73—21no—
82C4×D73—21no—
83C4×D73—21no—
84C4×D73—21no—
85C4×D73—21no—
86C4×D73—21no—
87C4×D73#9911yescharacteristic
88C4×D73#9911yescharacteristic
89C4×D73#10011yescharacteristic
90C4×D73#10111yescharacteristic
91#10596833—42no—
92C8×C73—41no—
93#10596831—41no—
94C22×C73—21no—
95C22×C73—41no—
96C22×C73#33611yescharacteristic
97C2×D73—22no—
98C2×D73—21no—
99C2×D73—42no—
100C2×D73—42no—
101C2×D73—41no—
102C2×D73#34211yescharacteristic
103C2×D73#34311yescharacteristic
104#64408—42no—
105#64408—42no—
106C4×C73—21no—
107C4×C73#32311yescharacteristic
108C4×C73#32411yescharacteristic
109Dic73—21no—
110Dic73#32311yescharacteristic
111Dic73#32411yescharacteristic
112#7190—731nomaximal, Sylow
113C2×C73—21no—
114C2×C73—41no—
115C2×C73#128011yescharacteristic
116D73—21no—
117D73—41no—
118D73#139512yes—
119#1729—731no—
120#1458—732no—
121C73#719011yescharacteristic, complemented, Sylow
122C22×M16—731no—
123C23×C8—731no—
124C2×#117—731no—
125#431—1462no—
126#336—1462no—
127#324—734no—
128#323—1462no—
129C23×C4—731no—
130C2×M16—2921no—
131C2×M16—2921no—
132C2×M16—731no—
133C2×M16—1461no—
134C2×M16#1059684211yescharacteristic, complemented
135C22×C8—2921no—
136C22×C8—2921no—
137C22×C8—732no—
138C22×C8—1461no—
139C22×C8—1461no—
140C22×C8—1461no—
141#117—2921no—
142#117—2921no—
143C2×C4⋊C4—732no—
144C2×C22⋊C4—731no—
145C2×C22⋊C4—731no—
146C2×C42—732no—
147C4⋊C8—2922no—
148C22⋊C8—1464no—
149C4×C8—2922no—
150C24—731no—
151C22×C4—732no—
152C22×C4—732no—
153C22×C4—731no—
154C22×C4—731no—
155C22×C4—731no—
156C22×C4—731no—
157C22×C4—731no—
158C22×C4—1461no—
159C22×C4—1461no—
160C22×C4—1461no—
161C22×C4—1461no—
162C22×C4C2×#1059684211yescharacteristic
163M16—41no—
164M16—2922no—
165M16—2921no—
166C2×C8—21no—
167C2×C8—2921no—
168C2×C8—2921no—
169C2×C8—2921no—
170C2×C8—2921no—
171C2×C8—2922no—
172C2×C8—2922no—
173C2×C8—2924no—
174C2×C8—1462no—
175C2×C8—1461no—
176C4⋊C4—2922no—
177C22⋊C4—2921no—
178C22⋊C4—2921no—
179C42—1462no—
180C42—1462no—
181C23—2922no—
182C23—731no—
183C23—731no—
184C23—1461no—
185C23—1461no—
186C23C4⋊#1059684211yescharacteristic
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—2922no—
190C2×C4—2922no—
191C2×C4—731no—
192C2×C4—731no—
193C2×C4—731no—
194C2×C4—731no—
195C2×C4—731no—
196C2×C4—731no—
197C2×C4—1461no—
198C2×C4—1461no—
199C2×C4—1461no—
200C2×C4—1461no—
201C2×C4—1462no—
202C2×C4—1462no—
203C2×C4—1461no—
204C2×C4—1461no—
205C2×C4—1461no—
206C2×C4—1461no—
207C2×C4C4×#1059684211yescharacteristic
208C2×C4C22⋊#1059684211yescharacteristic
209C8—41no—
210C8—2922no—
211C8—2921no—
212C8—5844no—
213C22—21no—
214C22—41no—
215C22—2921no—
216C22—2922no—
217C22—2922no—
218C22—731no—
219C22—731no—
220C22—1462no—
221C22—1461no—
222C22(C2×#10596842)⋊C411yescharacteristic
223C4—21no—
224C4—2922no—
225C4—2922no—
226C4—731no—
227C4—731no—
228C4—1461no—
229C4#1798967211yescharacteristic
230C4#1798967311yescharacteristic
231C2—21no—
232C2—41no—
233C2—2921no—
234C2—732no—
235C2—1461no—
236C2#2262694011yescharacteristic
subgroupG/H
centerC4#17989672
commutatorC2×C4×C73C4×C8
FrattiniC22×C4C2×#10596842
FittingC2×M16×C73C8
socleC2×C73#1280
prime radicalC23×D73C4⋊C4
cosocleC2×#14556670C22
hypercenterC2×M16#10596842
hypocenterC73#7190
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
16. C2×#11924036C8semidirect
25. C2×#11924012C8semidirect
30. C2×M16×C73C8semidirect
121. C73#7190semidirect
134. C2×M16#10596842semidirect

Maximal subgroups

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

subgroupindex
#226270432
#226255992
#719073