Finite Group Databasebeta

← #34010086

Subgroups — #34010086

Order 18688.

578 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 88 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#22625179C212yesmaximal
2#22624873C214yesmaximal, complemented
3C2×#17988457C211yesmaximal, characteristic
4#17989557—22no—
5#17989432—22no—
6#17989432C2214yes—
7C22×C4⋊Dic73—22no—
8C22×C4⋊Dic73C411yescharacteristic
9C22×C4⋊Dic73C2211yescharacteristic
10#17988457—22no—
11#17988457C412yes—
12#17988457C2212yes—
13C22×C42×C73C411yescharacteristic
14C2×#11924037—42no—
15C22⋊#10596833—24no—
16C22⋊#10596833—42no—
17C23×Dic73—21no—
18C23×Dic73D412yes—
19C2×C4⋊Dic73—24no—
20C2×C4⋊Dic73—24no—
21C2×C4⋊Dic73—21no—
22C2×C4⋊Dic73—41no—
23C2×C4⋊Dic73—41no—
24C2×C4⋊Dic73—48no—
25C2×C4⋊Dic73—41no—
26C2×C4⋊Dic73—41no—
27C2×C4⋊Dic73C2×C412yes—
28C2×C4⋊Dic73C2×C411yescharacteristic
29C2×C4⋊Dic73D412yes—
30C2×C4⋊Dic73C2311yescharacteristic
31C23×C4×C73—21no—
32C23×C4×C73C2×C411yescharacteristic
33C2×C42×C73—21no—
34C2×C42×C73—24no—
35C2×C42×C73C2×C412yes—
36#11924037—84no—
37C2×#10596833—44no—
38C22×Dic73—22no—
39C22×Dic73—22no—
40C22×Dic73—22no—
41C22×Dic73—44no—
42C22×Dic73—44no—
43C22×Dic73—41no—
44C22×Dic73—41no—
45C22×Dic73C22⋊C412yes—
46C22×Dic73C2×D412yes—
47C4⋊Dic73—41no—
48C4⋊Dic73—42no—
49C4⋊Dic73—41no—
50C4⋊Dic73—48no—
51C4⋊Dic73—48no—
52C4⋊Dic73—42no—
53C4⋊Dic73—42no—
54C24×C73C22⋊C411yescharacteristic
55C22×C4×C73—24no—
56C22×C4×C73—24no—
57C22×C4×C73—24no—
58C22×C4×C73—21no—
59C22×C4×C73—21no—
60C22×C4×C73—22no—
61C22×C4×C73—22no—
62C22×C4×C73—22no—
63C22×C4×C73C22⋊C411yescharacteristic
64C22×C4×C73C22×C411yescharacteristic
65C42×C73—24no—
66C42×C73—22no—
67C42×C73M1614yescomplemented
68C23×C73—22no—
69C23×C73—22no—
70C23×C73—21no—
71C23×C73—21no—
72C23×C73#9911yescharacteristic
73C23×C73#9911yescharacteristic
74C23×C73C2×C22⋊C411yescharacteristic
75C2×C4×C73—216no—
76C2×C4×C73—24no—
77C2×C4×C73—24no—
78C2×C4×C73—24no—
79C2×C4×C73—24no—
80C2×C4×C73—21no—
81C2×C4×C73—24no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73#10112yes—
86C2×C4×C73C2×M1612yes—
87C2×Dic73—22no—
88C2×Dic73—22no—
89C2×Dic73—44no—
90C2×Dic73—42no—
91C2×Dic73—42no—
92C2×Dic73—44no—
93C2×Dic73—42no—
94C2×Dic73—82no—
95C2×Dic73—82no—
96C2×Dic73—82no—
97#10596833—84no—
98C22×C73—24no—
99C22×C73—24no—
100C22×C73—21no—
101C22×C73—21no—
102C22×C73—21no—
103C22×C73—21no—
104C22×C73—21no—
105C22×C73—21no—
106C22×C73#35312yes—
107C22×C73#35612yes—
108C22×C73#40711yescharacteristic
109C22×C73C2×#9911yescharacteristic
110C22×C73C2×#10111yescharacteristic
111C4×C73—28no—
112C4×C73—22no—
113C4×C73—22no—
114Dic73—42no—
115Dic73—42no—
116Dic73—82no—
117#11512—731nomaximal, Sylow
118C2×C73—21no—
119C2×C73—22no—
120C2×C73—22no—
121C2×C73—21no—
122C2×C73#146311yescharacteristic
123C2×C73C2×#35311yescharacteristic
124C2×C73C2×#35611yescharacteristic
125C2×#384—731no—
126#1463—732no—
127#1325—734no—
128C73#1151211yescharacteristic, complemented, Sylow
129C22×C4⋊C4—731no—
130C22×C4⋊C4—731no—
131C22×C4⋊C4—1462no—
132C22×C42#6440811yescharacteristic
133#407—1462no—
134#384—732no—
135#384—732no—
136#384—1462no—
137#324—734no—
138#324—1462no—
139C23×C4—21no—
140C23×C4—732no—
141C23×C4—1461no—
142C23×C4C2×#6440811yescharacteristic
143C2×M16—2922no—
144C2×C4⋊C4—2921no—
145C2×C4⋊C4—2921no—
146C2×C4⋊C4—2928no—
147C2×C4⋊C4—2921no—
148C2×C4⋊C4—2921no—
149C2×C4⋊C4—732no—
150C2×C4⋊C4—731no—
151C2×C4⋊C4—731no—
152C2×C4⋊C4—732no—
153C2×C4⋊C4—1464no—
154C2×C4⋊C4—1464no—
155C2×C4⋊C4—1461no—
156C2×C42—21no—
157C2×C42—24no—
158C2×C42C2×#6440812yes—
159C22⋊C8—2922no—
160C22⋊C8—1464no—
161C24C22⋊#6440811yescharacteristic
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—2924no—
171C22×C4—2924no—
172C22×C4—2921no—
173C22×C4—2921no—
174C22×C4—732no—
175C22×C4—732no—
176C22×C4—1462no—
177C22×C4—1462no—
178C22×C4—1462no—
179C22×C4C22⋊#6440811yescharacteristic
180C22×C4C22×#6440811yescharacteristic
181M16—5844no—
182C2×C8—2924no—
183C4⋊C4—2922no—
184C4⋊C4—2928no—
185C4⋊C4—2928no—
186C4⋊C4—2922no—
187C4⋊C4—2922no—
188C4⋊C4—2921no—
189C4⋊C4—2921no—
190C42—24no—
191C42—22no—
192C42#1192403714yescomplemented
193C23—22no—
194C23—22no—
195C23—21no—
196C23—21no—
197C23#1455665011yescharacteristic
198C23#1455665011yescharacteristic
199C23C2×C22⋊#6440811yescharacteristic
200C2×C4—24no—
201C2×C4—216no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—2924no—
211C2×C4—2922no—
212C2×C4—2922no—
213C2×C4—2924no—
214C2×C4—2922no—
215C2×C4—5842no—
216C2×C4—5842no—
217C2×C4—5842no—
218C2×C4—1462no—
219C2×C4—1462no—
220C2×C4#1455665212yes—
221C2×C4C2×#1192403712yes—
222C8—5844no—
223C22—24no—
224C22—24no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22#1798949212yes—
232C22#1798949812yes—
233C22#1798955711yescharacteristic
234C22C2×#1455665011yescharacteristic
235C22C2×#1455665211yescharacteristic
236C4—28no—
237C4—22no—
238C4—22no—
239C4—2922no—
240C4—2922no—
241C4—5842no—
242C2—21no—
243C2—22no—
244C2—22no—
245C2—21no—
246C2#2262517911yescharacteristic
247C2C2×#1798949211yescharacteristic
248C2C2×#1798949811yescharacteristic
subgroupG/H
centerC22C2×#14556650
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC22×C42×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC22×C42#64408
hypocenterC73#11512
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #22624873C2semidirect
67. C42×C73M16semidirect
128. C73#11512semidirect
192. C42#11924037semidirect

Maximal subgroups

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

subgroupindex
#226251792
#226248732
C2×#179884572
#1151273