Finite Group Databasebeta

← #34032947

Subgroups — #34032947

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#22625596C211yesmaximal, characteristic
3#17989700—22no—
4#17989673—22no—
5#17989672C414yes—
6C2×D73⋊M16C411yescharacteristic
7C2×#14556670C2211yescharacteristic
8#17989452—22no—
9C22×C8×D73C411yescharacteristic
10C4⋊#10596842—42no—
11C22⋊#10596842—24no—
12C4×#10596842—42no—
13C2×#11924038—21no—
14D73⋊M16—41no—
15D73⋊M16—41no—
16C2×#11924037C812yescomplemented
17C2×#11924036—21no—
18#14556670—41no—
19#14556670—41no—
20C2×C4⋊#64408D411yescharacteristic
21C2×C4⋊#64408Q811yescharacteristic
22C2×C22⋊#64408C2×C412yes—
23C2×C4×#64408D412yes—
24C22×C4×D73C2×C411yescharacteristic
25C22×#10596831C811yescharacteristic, complemented
26C2×C8×D73—21no—
27C2×C8×D73—21no—
28C2×C8×D73—41no—
29C2×C8×D73—41no—
30C22×C8×C73C811yescharacteristic, complemented
31C2×#10596842—44no—
32C22×#64408C22⋊C412yes—
33C22×#64408C4⋊C412yes—
34#11924038—42no—
35#11924037—42no—
36#11924036—41no—
37#11924036—41no—
38C2×#10596833—22no—
39C4⋊#64408—41no—
40C4⋊#64408—41no—
41C22⋊#64408—42no—
42C4×#64408—22no—
43C4×#64408—22no—
44C23×D73C4211yescharacteristic
45C22×Dic73C2×C811yescharacteristic
46C2×C4×D73—21no—
47C2×C4×D73—21no—
48C2×C4×D73—21no—
49C2×C4×D73—21no—
50C2×C4×D73C22⋊C411yescharacteristic
51C2×C4×D73C4⋊C411yescharacteristic
52C2×C4×D73M1611yescharacteristic
53C2×C4×D73M1611yescharacteristic
54C2×#10596831—21no—
55C2×#10596831—41no—
56C8×D73—42no—
57C8×D73—41no—
58C8×D73—41no—
59C22×C4×C73C2×C811yescharacteristic
60C2×C8×C73—21no—
61C2×C8×C73—41no—
62C23×C73C4×C811yescharacteristic
63C22×D73—21no—
64C22×D73—21no—
65C22×D73—41no—
66C22×D73—41no—
67C22×D73#9511yescharacteristic
68C22×D73#9711yescharacteristic
69C2×#64408—22no—
70C2×#64408—22no—
71C2×#64408—42no—
72C2×#64408—42no—
73#10596842—84no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73C22⋊C811yescharacteristic
77C2×C4×C73C4⋊C811yescharacteristic
78C2×Dic73—21no—
79C2×Dic73—21no—
80C2×Dic73C22⋊C811yescharacteristic
81C2×Dic73C4⋊C811yescharacteristic
82C4×D73—21no—
83C4×D73—21no—
84C4×D73—21no—
85C4×D73—21no—
86C4×D73—21no—
87C4×D73—21no—
88C4×D73#9911yescharacteristic
89C4×D73#9911yescharacteristic
90C4×D73#10011yescharacteristic
91C4×D73#10111yescharacteristic
92#10596833—42no—
93C8×C73—41no—
94#10596831—41no—
95C22×C73—21no—
96C22×C73—41no—
97C22×C73#33611yescharacteristic
98C2×D73—21no—
99C2×D73—21no—
100C2×D73—21no—
101C2×D73—42no—
102C2×D73—41no—
103C2×D73—41no—
104C2×D73—41no—
105C2×D73#34211yescharacteristic
106C2×D73#34311yescharacteristic
107#64408—42no—
108#64408—42no—
109C4×C73—21no—
110C4×C73#32311yescharacteristic
111C4×C73#32411yescharacteristic
112Dic73—21no—
113Dic73#32311yescharacteristic
114Dic73#32411yescharacteristic
115#7188—731nomaximal, Sylow
116C2×C73—21no—
117C2×C73—41no—
118C2×C73#128011yescharacteristic
119D73—21no—
120D73—41no—
121D73#139111yescharacteristic
122D73#139211yescharacteristic
123#1729—731no—
124#1458—732no—
125C73#718811yescharacteristic, complemented, Sylow
126C22×M16—731no—
127C23×C8—731no—
128C2×#117—731no—
129#431—1462no—
130#336—1462no—
131#324—1462no—
132#323—734no—
133C23×C4—731no—
134C2×M16—2921no—
135C2×M16—2921no—
136C2×M16—732no—
137C2×M16—1461no—
138C22×C8—2921no—
139C22×C8—2921no—
140C22×C8—731no—
141C22×C8—1461no—
142C22×C8—1461no—
143C22×C8—1461no—
144C22×C8#1059684211yescharacteristic, complemented
145#117—2921no—
146#117—2921no—
147C2×C4⋊C4—731no—
148C2×C4⋊C4—731no—
149C2×C22⋊C4—732no—
150C2×C42—732no—
151C4⋊C8—2922no—
152C22⋊C8—1464no—
153C4×C8—2922no—
154C24—731no—
155C22×C4—732no—
156C22×C4—732no—
157C22×C4—731no—
158C22×C4—731no—
159C22×C4—731no—
160C22×C4—731no—
161C22×C4—731no—
162C22×C4—1461no—
163C22×C4—1461no—
164C22×C4—1461no—
165C22×C4—1461no—
166C22×C4C2×#1059684211yescharacteristic
167M16—2922no—
168M16—2922no—
169C2×C8—21no—
170C2×C8—41no—
171C2×C8—2921no—
172C2×C8—2921no—
173C2×C8—2922no—
174C2×C8—2921no—
175C2×C8—2921no—
176C2×C8—2921no—
177C2×C8—2924no—
178C2×C8—1462no—
179C2×C8—1461no—
180C4⋊C4—2921no—
181C4⋊C4—2921no—
182C22⋊C4—2922no—
183C42—1462no—
184C42—1462no—
185C23—2921no—
186C23—2921no—
187C23—731no—
188C23—731no—
189C23—1461no—
190C23—1461no—
191C23C4×#1059684211yescharacteristic
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—2922no—
195C2×C4—2922no—
196C2×C4—731no—
197C2×C4—731no—
198C2×C4—731no—
199C2×C4—731no—
200C2×C4—731no—
201C2×C4—731no—
202C2×C4—1461no—
203C2×C4—1461no—
204C2×C4—1461no—
205C2×C4—1461no—
206C2×C4—1462no—
207C2×C4—1462no—
208C2×C4—1461no—
209C2×C4—1461no—
210C2×C4—1461no—
211C2×C4—1461no—
212C2×C4C22⋊#1059684211yescharacteristic
213C2×C4C4⋊#1059684211yescharacteristic
214C8—41no—
215C8—2922no—
216C8—2921no—
217C8—5844no—
218C22—21no—
219C22—41no—
220C22—2921no—
221C22—2921no—
222C22—2921no—
223C22—2922no—
224C22—731no—
225C22—731no—
226C22—1461no—
227C22—1461no—
228C22—1461no—
229C22(C2×#10596842)⋊C411yescharacteristic
230C4—21no—
231C4—2922no—
232C4—2922no—
233C4—731no—
234C4—731no—
235C4—1461no—
236C4#1798967211yescharacteristic
237C4#1798967311yescharacteristic
238C2—21no—
239C2—41no—
240C2—2921no—
241C2—731no—
242C2—731no—
243C2—1461no—
244C2#2262694011yescharacteristic
subgroupG/H
centerC4#17989672
commutatorC23×C73C4×C8
FrattiniC22×C4C2×#10596842
FittingC22×C8×C73C8
socleC2×C73#1280
prime radicalC23×D73C42
cosocleC2×#14556670C22
hypercenterC22×C8#10596842
hypocenterC73#7188
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
16. C2×#11924037C8semidirect
25. C22×#10596831C8semidirect
30. C22×C8×C73C8semidirect
125. C73#7188semidirect
144. C22×C8#10596842semidirect

Maximal subgroups

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

subgroupindex
#226270432
#226255962
#718873