Finite Group Databasebeta

← #32983814

Subgroups — #32983814

Order 18688.

614 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 107 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#22612529C211yesmaximal, characteristic
2#22610030C212yesmaximal, complemented
3#22608258C211yesmaximal, characteristic
4#22607783C212yesmaximal, complemented
5C73×#2482C211yesmaximal, characteristic
6#17988551C2212yes—
7#17988480—24no—
8#17988447—21no—
9#17988443—21no—
10C4×C22⋊Dic73C2212yes—
11#17988346—24no—
12#17988311—22no—
13C2×#117×C73C2211yescharacteristic
14C22×C22⋊C4×C73C2211yescharacteristic
15#399×C73C412yescomplemented
16#397×C73C412yescomplemented
17#388×C73—22no—
18#385×C73—21no—
19#381×C73C2211yescharacteristic
20#379×C73—21no—
21C2×#11924012—22no—
22C2×C4⋊Dic73—22no—
23C2×C22⋊Dic73—22no—
24C2×C22⋊Dic73—22no—
25C2×C22⋊Dic73—22no—
26C2×C4×Dic73D414yes—
27C22⋊#10596831—22no—
28C22⋊#10596831—22no—
29#14556498—22no—
30#14556498—22no—
31C25×C73D411yescharacteristic
32C23×C4×C73D411yescharacteristic
33C23×C4×C73C2311yescharacteristic
34#117×C73—22no—
35#117×C73—22no—
36C2×C4⋊C4×C73—22no—
37C2×C4⋊C4×C73C2×C411yescharacteristic
38C2×C4⋊C4×C73C2×C411yescharacteristic
39C2×C22⋊C4×C73—22no—
40C2×C22⋊C4×C73—22no—
41C2×C22⋊C4×C73—22no—
42C2×C22⋊C4×C73—21no—
43C2×C22⋊C4×C73—21no—
44C2×C22⋊C4×C73—22no—
45C2×C22⋊C4×C73C2×C412yes—
46C2×C22⋊C4×C73D412yes—
47C2×C42×C73—21no—
48#95×C73—22no—
49#95×C73—21no—
50#95×C73C2×C412yes—
51C22×Dic73—24no—
52#11924012—44no—
53C2×#10596831—42no—
54C4⋊Dic73—82no—
55C22⋊Dic73—44no—
56C22⋊Dic73—82no—
57C22⋊Dic73—82no—
58C4×Dic73—44no—
59C24×C73—21no—
60C24×C73—21no—
61C24×C73—42no—
62C24×C73—42no—
63C24×C73—81no—
64C24×C73C22⋊C412yes—
65C24×C73C2×D411yescharacteristic
66C22×C4×C73—21no—
67C22×C4×C73—21no—
68C22×C4×C73—22no—
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73—42no—
72C22×C4×C73—81no—
73C22×C4×C73C22⋊C412yes—
74C22×C4×C73C22×C411yescharacteristic
75C22×C4×C73C2×D412yes—
76C22×C4×C73C2×D411yescharacteristic
77C22×C4×C73#4112yes—
78C4⋊C4×C73—22no—
79C4⋊C4×C73—22no—
80C4⋊C4×C73—82no—
81C22⋊C4×C73—24no—
82C22⋊C4×C73—24no—
83C22⋊C4×C73—42no—
84C22⋊C4×C73—44no—
85C22⋊C4×C73—41no—
86C22⋊C4×C73—41no—
87C22⋊C4×C73—82no—
88C42×C73—42no—
89C23×C73—22no—
90C23×C73—22no—
91C23×C73—22no—
92C23×C73—22no—
93C23×C73—22no—
94C23×C73—41no—
95C23×C73—42no—
96C23×C73—42no—
97C23×C73—42no—
98C23×C73—42no—
99C23×C73—42no—
100C23×C73—42no—
101C23×C73—81no—
102C23×C73—81no—
103C23×C73—88no—
104C23×C73C2×C22⋊C411yescharacteristic
105C23×C73#12011yescharacteristic
106C23×C73#12311yescharacteristic
107C2×C4×C73—24no—
108C2×C4×C73—22no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—42no—
112C2×C4×C73—42no—
113C2×C4×C73—42no—
114C2×C4×C73—42no—
115C2×C4×C73—42no—
116C2×C4×C73—42no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—81no—
122C2×C4×C73C4×D412yes—
123C2×C4×C73#12112yes—
124C2×Dic73—44no—
125C2×Dic73—44no—
126C2×Dic73—44no—
127#10596831—82no—
128C22×C73—21no—
129C22×C73—21no—
130C22×C73—21no—
131C22×C73—21no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73—44no—
136C22×C73—44no—
137C22×C73—42no—
138C22×C73—42no—
139C22×C73—42no—
140C22×C73—82no—
141C22×C73—84no—
142C22×C73—84no—
143C22×C73#38611yescharacteristic
144C22×C73#42112yes—
145C22×C73#45812yes—
146C4×C73—22no—
147C4×C73—22no—
148C4×C73—42no—
149C4×C73—82no—
150C4×C73—81no—
151C4×C73—81no—
152Dic73—84no—
153#9481—731nomaximal, Sylow
154C2×C73—22no—
155C2×C73—22no—
156C2×C73—41no—
157C2×C73—82no—
158C2×C73#189011yescharacteristic
159C2×C73#189711yescharacteristic
160C2×C73#191111yescharacteristic
161#2482D7311yescharacteristic
162#2312—731no—
163#1826—732no—
164#1462—731no—
165#1284—732no—
166C73#948111yescharacteristic, complemented, Sylow
167C2×#117C2×D7311yescharacteristic
168C22×C22⋊C4C2×D7311yescharacteristic
169#407—732no—
170#399Dic7312yescomplemented
171#397Dic7312yescomplemented
172#388—22no—
173#388—1464no—
174#385—21no—
175#381C2×D7311yescharacteristic
176#380—1461no—
177#379—21no—
178#379—1461no—
179C4×C22⋊C4—732no—
180#339—1464no—
181#323—1462no—
182C25C22⋊D7311yescharacteristic
183C23×C4C22⋊D7311yescharacteristic
184C23×C4C22×D7311yescharacteristic
185C2×M16—1462no—
186#117—22no—
187#117—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—1462no—
190C2×C4⋊C4C2×Dic7311yescharacteristic
191C2×C4⋊C4C2×Dic7311yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—1462no—
199C2×C22⋊C4—1462no—
200C2×C22⋊C4—1462no—
201C2×C22⋊C4C2×Dic7312yes—
202C2×C22⋊C4C22⋊D7312yes—
203C2×C42—21no—
204C2×C42—734no—
205C22⋊C8—1462no—
206C22⋊C8—1462no—
207#95—22no—
208#95—21no—
209#95—1462no—
210#95—1462no—
211#95C2×Dic7312yes—
212C24—21no—
213C24—21no—
214C24—42no—
215C24—42no—
216C24—81no—
217C24C22⋊Dic7312yes—
218C24C2×C22⋊D7311yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—42no—
225C22×C4—81no—
226C22×C4—1464no—
227C22×C4C22⋊Dic7312yes—
228C22×C4C22×Dic7311yescharacteristic
229C22×C4D4×D7312yes—
230C22×C4C2×C22⋊D7311yescharacteristic
231C22×C4#1192402712yes—
232M16—2924no—
233C2×C8—2922no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—82no—
237C4⋊C4—5842no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—42no—
241C22⋊C4—44no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—82no—
245C22⋊C4—2924no—
246C22⋊C4—5842no—
247C22⋊C4—5842no—
248C42—42no—
249C42—2924no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—81no—
263C23—81no—
264C23—88no—
265C23C2×C22⋊Dic7311yescharacteristic
266C23#1455655911yescharacteristic
267C23#1455657211yescharacteristic
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—2924no—
284C2×C4—2924no—
285C2×C4—2924no—
286C2×C4D4×Dic7312yes—
287C2×C4#1455656312yes—
288C8—5842no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—44no—
297C22—44no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—84no—
303C22—84no—
304C22#1798846711yescharacteristic
305C22#1798860412yes—
306C22#1798877612yes—
307C4—22no—
308C4—22no—
309C4—42no—
310C4—82no—
311C4—81no—
312C4—81no—
313C4—5844no—
314C2—22no—
315C2—22no—
316C2—41no—
317C2—82no—
318C2#2261034511yescharacteristic
319C2#2261038011yescharacteristic
320C2#2261046411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2482C2
socleC22×C73#386
prime radicalC25×C73D4
cosocleC23×C4×C73C23
hypercenter#2482D73
hypocenterC73#9481
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2482

Complements

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

NH ≅ G/Ntype
2. #22610030C2semidirect
4. #22607783C2semidirect
15. #399×C73C4semidirect
16. #397×C73C4semidirect
166. C73#9481semidirect
170. #399Dic73semidirect
171. #397Dic73semidirect

Maximal subgroups

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

subgroupindex
#226125292
#226100302
#226082582
#226077832
C73×#24822
#948173