Finite Group Databasebeta

← #34013086

Subgroups — #34013086

Order 18688.

500 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×#17989495C212yesmaximal
2#22625797C212yesmaximal, complemented
3#22625029C212yesmaximal, complemented
4C4×#14556578C211yesmaximal, characteristic
5#17989590—22no—
6C2×#14556652C2214yes—
7#17989495—24no—
8#17989467—22no—
9C2×#14556578C2211yescharacteristic
10C2×C4×Dic146C2211yescharacteristic
11#17988492C2211yescharacteristic
12#17988490C411yescharacteristic
13#17988481—22no—
14#17988452C411yescharacteristic
15C4×C22⋊Dic73—22no—
16C42×Dic73C411yescharacteristic
17C2×C4×D4×C73C411yescharacteristic
18C2×#11924037—24no—
19#14556658—42no—
20C4⋊#10596833—42no—
21#14556652—24no—
22#14556652—24no—
23C22×Dic146C2311yescharacteristic
24#14556578—22no—
25#14556578D412yes—
26#14556578Q812yes—
27C4×Dic146—41no—
28C4×Dic146—41no—
29C2×C4⋊Dic73C2×C411yescharacteristic
30C2×#11924007D412yes—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73—22no—
33C2×C4×Dic73—21no—
34C2×C4×Dic73—21no—
35C2×C4×Dic73C2×C411yescharacteristic
36C2×C4×Dic73C2×C411yescharacteristic
37C2×C4×Dic73D412yes—
38#14556498—22no—
39#14556498—22no—
40C22×D4×C73C2×C411yescharacteristic
41C23×C4×C73—21no—
42C4×D4×C73—41no—
43C4×D4×C73—41no—
44C2×C4⋊C4×C73C2×C411yescharacteristic
45C2×C22⋊C4×C73—21no—
46C2×C42×C73C2×C411yescharacteristic
47#11924037—44no—
48#11924037—44no—
49C2×#10596833—44no—
50C2×Dic146—41no—
51C2×Dic146—41no—
52C2×Dic146C2×D411yescharacteristic
53C2×Dic146C2×Q811yescharacteristic
54C2×Dic146#4112yes—
55C22×Dic73—21no—
56C22×Dic73—21no—
57C22×Dic73C22⋊C412yes—
58C22×Dic73C2×D412yes—
59C4⋊Dic73—41no—
60#11924007—42no—
61C22⋊Dic73—24no—
62C22⋊Dic73—24no—
63C22⋊Dic73—44no—
64C4×Dic73—22no—
65C4×Dic73—22no—
66C4×Dic73—21no—
67C4×Dic73—42no—
68C4×Dic73—42no—
69C4×Dic73—41no—
70C4×Dic73C4⋊C412yes—
71C24×C73—21no—
72C2×D4×C73—21no—
73C2×D4×C73—41no—
74C2×D4×C73—41no—
75C2×D4×C73C4⋊C412yes—
76C22×C4×C73—21no—
77C22×C4×C73—44no—
78C22×C4×C73—41no—
79C22×C4×C73—41no—
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22×C411yescharacteristic
83C4⋊C4×C73—41no—
84C22⋊C4×C73—42no—
85C42×C73—21no—
86C42×C73—21no—
87C23×C73—22no—
88C23×C73—42no—
89C23×C73—42no—
90C23×C73—41no—
91C23×C73—41no—
92C23×C73C2×C22⋊C411yescharacteristic
93D4×C73—42no—
94D4×C73—42no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—22no—
98C2×C4×C73—21no—
99C2×C4×C73—41no—
100C2×C4×C73—42no—
101C2×C4×C73—41no—
102C2×C4×C73—82no—
103C2×C4×C73—82no—
104C2×C4×C73#9911yescharacteristic
105C2×C4×C73#9911yescharacteristic
106C2×C4×C73C2×C4⋊C411yescharacteristic
107C2×C4×C73#11711yescharacteristic
108C2×Dic73—22no—
109C2×Dic73—22no—
110C2×Dic73—22no—
111C2×Dic73—22no—
112C2×Dic73—22no—
113C2×Dic73—41no—
114C2×Dic73—41no—
115C2×Dic73—41no—
116C2×Dic73—41no—
117C2×Dic73—42no—
118C2×Dic73#12112yes—
119C2×Dic73C22⋊Q812yes—
120Dic146—41no—
121Dic146—41no—
122Dic146—42no—
123#10596833—84no—
124C22×C73—21no—
125C22×C73—21no—
126C22×C73—44no—
127C22×C73—42no—
128C22×C73—82no—
129C22×C73—81no—
130C22×C73—81no—
131C22×C73#38011yescharacteristic
132C22×C73C2×#9911yescharacteristic
133C22×C73#41011yescharacteristic
134C4×C73—21no—
135C4×C73—21no—
136C4×C73—21no—
137C4×C73—41no—
138C4×C73—81no—
139C4×C73#35512yescomplemented
140Dic73—22no—
141Dic73—22no—
142Dic73—42no—
143Dic73—42no—
144Dic73—42no—
145#12801—731nomaximal, Sylow
146C2×C73—21no—
147C2×C73—21no—
148C2×C73—82no—
149C2×C73#179911yescharacteristic
150C2×C73C2×#35511yescharacteristic
151C2×C73#213211yescharacteristic
152C4×#124—731no—
153C2×#355—732no—
154#1801—732no—
155#1394—732no—
156C73#1280111yescharacteristic, complemented, Sylow
157C2×#124—731no—
158C2×C4×Q8—731no—
159C2×C4×D4#6440811yescharacteristic
160#423—1462no—
161C2×#101—734no—
162#391—731no—
163#390—731no—
164#388—1462no—
165#382—731no—
166C4×C22⋊C4—1462no—
167C43—731no—
168#355—1464no—
169#343—1462no—
170C22×Q8—731no—
171C22×D4C2×#6440811yescharacteristic
172C23×C4—21no—
173C2×M16—1464no—
174#124—732no—
175#124—732no—
176#124—1462no—
177C4×Q8—2921no—
178C4×Q8—2921no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—732no—
182C2×C4⋊C4—731no—
183C2×C4⋊C4C2×#6440811yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—1462no—
186C2×C22⋊C4—1462no—
187C2×C42—731no—
188C2×C42—731no—
189C2×C42—732no—
190C2×C42—1461no—
191C2×C42—1461no—
192C2×C42C2×#6440811yescharacteristic
193C4⋊C8—2922no—
194C4⋊C8—2922no—
195#101—1464no—
196#101—1464no—
197#95—1462no—
198#95—1462no—
199C24—21no—
200C2×Q8—2921no—
201C2×Q8—2921no—
202C2×Q8—731no—
203C2×Q8—731no—
204C2×Q8—732no—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C4⋊#6440812yes—
209C22×C4—21no—
210C22×C4—44no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—732no—
214C22×C4—732no—
215C22×C4—1461no—
216C22×C4—1461no—
217C22×C4C22⋊#6440811yescharacteristic
218C22×C4C22⋊#6440811yescharacteristic
219C22×C4C22×#6440811yescharacteristic
220M16—2924no—
221M16—2924no—
222C2×C8—2924no—
223C4⋊C4—41no—
224C4⋊C4—2922no—
225C4⋊C4—2921no—
226C22⋊C4—42no—
227C22⋊C4—2924no—
228C22⋊C4—1464no—
229C22⋊C4—1464no—
230C42—21no—
231C42—21no—
232C42—2921no—
233C42—2922no—
234C42—2922no—
235C42—732no—
236C42—1462no—
237C42—1462no—
238C42—1461no—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23C2×C22⋊#6440811yescharacteristic
245Q8—2921no—
246Q8—2921no—
247Q8—2922no—
248D4—42no—
249D4—42no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—2921no—
260C2×C4—2921no—
261C2×C4—2921no—
262C2×C4—2921no—
263C2×C4—2922no—
264C2×C4—732no—
265C2×C4—732no—
266C2×C4—1462no—
267C2×C4—1462no—
268C2×C4—1462no—
269C2×C4—1462no—
270C2×C4—1462no—
271C2×C4#1455664911yescharacteristic
272C2×C4#1455664911yescharacteristic
273C2×C4C2×C4⋊#6440811yescharacteristic
274C2×C4#1455667011yescharacteristic
275C8—5844no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22#1798952911yescharacteristic
284C22C2×#1455664911yescharacteristic
285C22#1798956611yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—41no—
290C4—81no—
291C4—2922no—
292C4—2922no—
293C4—2922no—
294C4—1462no—
295C4—1462no—
296C4#1798949512yescomplemented
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2262579111yescharacteristic
301C2C2×#1798949511yescharacteristic
302C2#2262622211yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4×D4×C73C4
socleC22×C73C2×#99
prime radicalC22×D4×C73C2×C4
cosocleC22×Dic146C23
hypercenterC2×C4×D4#64408
hypocenterC73#12801
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #22625797C2semidirect
3. #22625029C2semidirect
139. C4×C73#355semidirect
156. C73#12801semidirect
296. C4#17989495semidirect

Maximal subgroups

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

subgroupindex
C2×#179894952
#226257972
#226250292
C4×#145565782
#1280173