Finite Group Databasebeta

← #34005278

Subgroups — #34005278

Order 18688.

622 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 124 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×#17989430C212yesmaximal
2#22624774C214yesmaximal, complemented
3#22612284C211yesmaximal, characteristic
4C2×C22⋊#10596833C2214yes—
5#17989430—22no—
6#17989430—22no—
7C22×C22⋊Dic73C411yescharacteristic
8C22×C22⋊Dic73C2211yescharacteristic
9C22⋊C4×Dic73—22no—
10C22⋊C4×Dic73C412yes—
11C22⋊C4×Dic73C2212yes—
12C4×C22⋊Dic73—22no—
13C22×C22⋊C4×C73C411yescharacteristic
14C22×#10596833—24no—
15C22⋊#10596833—24no—
16C22⋊#10596833—24no—
17C23×Dic73—21no—
18C23×Dic73C2×C411yescharacteristic
19C23×Dic73C2311yescharacteristic
20C2×C22⋊Dic73—24no—
21C2×C22⋊Dic73—24no—
22C2×C22⋊Dic73—21no—
23C2×C22⋊Dic73C2×C412yes—
24C2×C22⋊Dic73D412yes—
25C2×C22⋊Dic73D412yes—
26C2×C4×Dic73—22no—
27C2×C4×Dic73—22no—
28#14556498—22no—
29#14556498—22no—
30C25×C73C2×C411yescharacteristic
31C23×C4×C73—21no—
32C2×C22⋊C4×C73—21no—
33C2×C22⋊C4×C73—22no—
34C2×C22⋊C4×C73C814yescomplemented
35C2×C22⋊C4×C73C2×C412yes—
36C2×#10596833—44no—
37C2×#10596833—84no—
38C22×Dic73—22no—
39C22×Dic73—24no—
40C22×Dic73—22no—
41C22×Dic73—81no—
42C22×Dic73—81no—
43C22×Dic73—81no—
44C22×Dic73—81no—
45C22×Dic73C22⋊C412yes—
46C22×Dic73C2×D412yes—
47C22×Dic73#4114yes—
48C22⋊Dic73—24no—
49C22⋊Dic73—22no—
50C22⋊Dic73—24no—
51C22⋊Dic73—24no—
52C22⋊Dic73M1614yes—
53C4×Dic73—82no—
54C4×Dic73—82no—
55C24×C73—22no—
56C24×C73—22no—
57C24×C73—81no—
58C24×C73—81no—
59C24×C73C22⋊C411yescharacteristic
60C24×C73C22⋊C411yescharacteristic
61C24×C73C2×C812yes—
62C24×C73M1612yes—
63C24×C73C22×C411yescharacteristic
64C22×C4×C73—22no—
65C22×C4×C73—24no—
66C22×C4×C73—81no—
67C22×C4×C73—81no—
68C22⋊C4×C73—24no—
69C22⋊C4×C73—22no—
70C22⋊C4×C73C2×C814yes—
71C23×C73—21no—
72C23×C73—21no—
73C23×C73—21no—
74C23×C73—21no—
75C23×C73—24no—
76C23×C73—24no—
77C23×C73—41no—
78C23×C73—41no—
79C23×C73—41no—
80C23×C73—41no—
81C23×C73—41no—
82C23×C73—41no—
83C23×C73—82no—
84C23×C73—82no—
85C23×C73—82no—
86C23×C73—82no—
87C23×C73—84no—
88C23×C73C22⋊C812yes—
89C23×C73C22⋊C812yes—
90C23×C73C2×C22⋊C411yescharacteristic
91C23×C73#11712yes—
92C23×C73C22×C811yescharacteristic
93C23×C73C2×M1611yescharacteristic
94C23×C73C2×M1612yes—
95C2×C4×C73—24no—
96C2×C4×C73—42no—
97C2×C4×C73—42no—
98C2×C4×C73—84no—
99C2×Dic73—22no—
100C2×Dic73—24no—
101C2×Dic73—42no—
102C2×Dic73—44no—
103C2×Dic73—42no—
104C2×Dic73—42no—
105C2×Dic73—42no—
106C2×Dic73—82no—
107C2×Dic73—84no—
108C2×Dic73#12314yes—
109#10596833—84no—
110C22×C73—21no—
111C22×C73—22no—
112C22×C73—21no—
113C22×C73—22no—
114C22×C73—22no—
115C22×C73—22no—
116C22×C73—41no—
117C22×C73—41no—
118C22×C73—42no—
119C22×C73—42no—
120C22×C73—42no—
121C22×C73—42no—
122C22×C73—42no—
123C22×C73—41no—
124C22×C73—41no—
125C22×C73—41no—
126C22×C73—41no—
127C22×C73—84no—
128C22×C73—84no—
129C22×C73#38111yescharacteristic
130C22×C73C2×C22⋊C811yescharacteristic
131C22×C73#40711yescharacteristic
132C22×C73#43112yes—
133C22×C73#43212yes—
134C4×C73—44no—
135Dic73—24no—
136Dic73—44no—
137Dic73—42no—
138#8734—731nomaximal, Sylow
139C2×C73—21no—
140C2×C73—21no—
141C2×C73—22no—
142C2×C73—22no—
143C2×C73—42no—
144C2×C73—42no—
145C2×C73#181711yescharacteristic
146C2×C73#182811yescharacteristic
147C2×C73#182911yescharacteristic
148#2273—731no—
149C2×#323—732no—
150#1282—734no—
151C73#873411yescharacteristic, complemented, Sylow
152C22×C22⋊C4—731no—
153C22×C22⋊C4—731no—
154C22×C22⋊C4#6440811yescharacteristic
155C2×C22⋊C8—734no—
156C4×C22⋊C4—732no—
157C4×C22⋊C4—732no—
158C4×C22⋊C4—1462no—
159C4×C22⋊C4—1462no—
160#323—1462no—
161#323—1462no—
162C25C2×#6440811yescharacteristic
163C23×C4—21no—
164C23×C4—731no—
165C23×C4—731no—
166C23×C4—1461no—
167C22×C8—1464no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—732no—
171C2×C22⋊C4—732no—
172C2×C22⋊C4—732no—
173C2×C22⋊C4—1464no—
174C2×C22⋊C4—1464no—
175C2×C22⋊C4—1461no—
176C2×C22⋊C4#1059683314yescomplemented
177C2×C22⋊C4C2×#6440812yes—
178C2×C42—1462no—
179C2×C42—1462no—
180C22⋊C8—1464no—
181C22⋊C8—1464no—
182#95—1462no—
183#95—1462no—
184C24—22no—
185C24—22no—
186C24—81no—
187C24—81no—
188C24C22⋊#6440811yescharacteristic
189C24C22⋊#6440811yescharacteristic
190C24C2×#1059683312yes—
191C24#1192403712yes—
192C24C22×#6440811yescharacteristic
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4—5841no—
198C22×C4—5841no—
199C22×C4—5841no—
200C22×C4—5841no—
201C22×C4—732no—
202C22×C4—734no—
203C22×C4—732no—
204C22×C4—1462no—
205C22×C4—1464no—
206C22×C4—1462no—
207C2×C8—2924no—
208C2×C8—5844no—
209C22⋊C4—22no—
210C22⋊C4—24no—
211C22⋊C4—734no—
212C22⋊C4—1464no—
213C22⋊C4—1464no—
214C22⋊C4—1464no—
215C22⋊C4—1462no—
216C22⋊C4C2×#1059683314yes—
217C42—5842no—
218C42—5842no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—24no—
224C23—24no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—82no—
232C23—82no—
233C23—82no—
234C23—82no—
235C23—84no—
236C23C22⋊#1059683312yes—
237C23C22⋊#1059683312yes—
238C23C2×C22⋊#6440811yescharacteristic
239C23#1455667012yes—
240C23C22×#1059683311yescharacteristic
241C23C2×#1192403711yescharacteristic
242C23D73⋊M1612yes—
243C2×C4—24no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—84no—
247C2×C4—2922no—
248C2×C4—2922no—
249C2×C4—2922no—
250C2×C4—2922no—
251C2×C4—2924no—
252C2×C4—5842no—
253C2×C4—5844no—
254C2×C4—734no—
255C2×C4—1462no—
256C2×C4—1464no—
257C8—5844no—
258C22—21no—
259C22—22no—
260C22—21no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—84no—
276C22—84no—
277C22#1798953011yescharacteristic
278C22C2×C22⋊#1059683311yescharacteristic
279C22#1798955911yescharacteristic
280C22#1798960312yes—
281C22#1798960712yes—
282C4—44no—
283C4—2924no—
284C4—2922no—
285C4—1464no—
286C2—21no—
287C2—21no—
288C2—22no—
289C2—22no—
290C2—42no—
291C2—42no—
292C2#2262582111yescharacteristic
293C2#2262584211yescharacteristic
294C2#2262584411yescharacteristic
subgroupG/H
centerC22#17989530
commutatorC23×C73C22×C8
FrattiniC24C22×#64408
FittingC22×C22⋊C4×C73C4
socleC22×C73#381
prime radicalC25×C73C2×C4
cosocleC23×Dic73C23
hypercenterC22×C22⋊C4#64408
hypocenterC73#8734
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #22624774C2semidirect
34. C2×C22⋊C4×C73C8semidirect
151. C73#8734semidirect
176. C2×C22⋊C4#10596833semidirect

Maximal subgroups

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

subgroupindex
C2×#179894302
#226247742
#226122842
#873473