Finite Group Databasebeta

← #33001978

Subgroups — #33001978

Order 18688.

524 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 93 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#22612641C211yesmaximal, characteristic
2C2×#17988381C212yesmaximal
3#22609917C212yesmaximal
4#22608056C211yesmaximal, characteristic
5C73×#1393C211yesmaximal, characteristic
6C2×#14556578C2211yescharacteristic
7C2×#14556556C411yescharacteristic, complemented
8C2×D4×Dic73C2211yescharacteristic
9C2×#14556506C2212yes—
10#17988493C411yescharacteristic, complemented
11#17988481—22no—
12#17988459—22no—
13#17988456C411yescharacteristic, complemented
14#17988445—22no—
15#17988433C411yescharacteristic, complemented
16#17988381—24no—
17#17988353—22no—
18C2×#99×C73C2212yes—
19#390×C73C2211yescharacteristic
20C22×Dic146C2×C411yescharacteristic
21C23×Dic73D412yes—
22#14556578—22no—
23#14556578D412yes—
24#14556578Q812yes—
25#14556556—42no—
26D4×Dic73—42no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924007—21no—
29C2×#11924007—22no—
30C2×#11924007—42no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73—22no—
33C2×C22⋊Dic73D412yes—
34C2×C4×Dic73—21no—
35C2×C4×Dic73C2×C411yescharacteristic
36C2×C4×Dic73C2×C411yescharacteristic
37#14556506—22no—
38#14556506—22no—
39#14556498—21no—
40#14556498—21no—
41#14556498—22no—
42#14556498—42no—
43C22×D4×C73C2311yescharacteristic
44C2×C4⋊C4×C73C2×C411yescharacteristic
45C2×C22⋊C4×C73—22no—
46C2×C22⋊C4×C73—22no—
47C2×C42×C73C2×C411yescharacteristic
48#99×C73—22no—
49#99×C73—22no—
50C2×Dic146—21no—
51C2×Dic146—41no—
52C2×Dic146—41no—
53C2×Dic146C4⋊C412yescomplemented
54C22×Dic73—21no—
55C22×Dic73—21no—
56C22×Dic73—22no—
57C22×Dic73—22no—
58C22×Dic73—42no—
59C22×Dic73—82no—
60C22×Dic73C22⋊C411yescharacteristic
61C22×Dic73C22⋊C412yes—
62C22×Dic73C22⋊C411yescharacteristic
63C4⋊Dic73—41no—
64#11924007—42no—
65#11924007—82no—
66#11924007—84no—
67C22⋊Dic73—24no—
68C22⋊Dic73—24no—
69C22⋊Dic73—42no—
70C22⋊Dic73—42no—
71C22⋊Dic73—42no—
72C4×Dic73—21no—
73C4×Dic73—22no—
74C4×Dic73—42no—
75C4×Dic73C4⋊C412yescomplemented
76C24×C73C2×D412yes—
77C2×D4×C73—41no—
78C2×D4×C73—41no—
79C2×D4×C73C2×D411yescharacteristic
80C2×D4×C73C2×Q811yescharacteristic
81C2×D4×C73#4112yes—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—42no—
85C22×C4×C73C22×C411yescharacteristic
86C4⋊C4×C73—41no—
87C22⋊C4×C73—42no—
88C22⋊C4×C73—42no—
89C22⋊C4×C73—82no—
90C42×C73—41no—
91C23×C73—22no—
92C23×C73—22no—
93C23×C73—42no—
94C23×C73—42no—
95C23×C73C2×C22⋊C411yescharacteristic
96C23×C73#12112yes—
97C23×C73C22⋊Q812yes—
98D4×C73—41no—
99D4×C73—41no—
100D4×C73—42no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—41no—
104C2×C4×C73—42no—
105C2×C4×C73—41no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—82no—
109C2×C4×C73—82no—
110C2×C4×C73—82no—
111C2×C4×C73C2×C4⋊C411yescharacteristic
112C2×C4×C73#11711yescharacteristic
113C2×Dic73—22no—
114C2×Dic73—22no—
115C2×Dic73—21no—
116C2×Dic73—22no—
117C2×Dic73—41no—
118C2×Dic73—44no—
119C2×Dic73—41no—
120C2×Dic73—41no—
121C2×Dic73—41no—
122C2×Dic73—41no—
123C2×Dic73—42no—
124C2×Dic73—41no—
125C2×Dic73—82no—
126C2×Dic73—82no—
127C2×Dic73—82no—
128C2×Dic73—82no—
129C2×Dic73#9912yes—
130Dic146—42no—
131Dic146—42no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73—22no—
135C22×C73—22no—
136C22×C73—42no—
137C22×C73—42no—
138C22×C73—42no—
139C22×C73—42no—
140C22×C73—82no—
141C22×C73#38011yescharacteristic
142C22×C73C2×#9911yescharacteristic
143C22×C73#41011yescharacteristic
144C4×C73—21no—
145C4×C73—21no—
146C4×C73—81no—
147C4×C73—81no—
148C4×C73—162no—
149Dic73—22no—
150Dic73—42no—
151Dic73—42no—
152Dic73—42no—
153Dic73—41no—
154Dic73—162no—
155#12800—731nomaximal, Sylow
156C2×C73—21no—
157C2×C73—21no—
158C2×C73—42no—
159C2×C73—42no—
160C2×C73#179911yescharacteristic
161C2×C73#212711yescharacteristic
162C2×C73#212911yescharacteristic
163#2326—731no—
164C2×#354—732no—
165#1799—732no—
166#1393—731no—
167#1393D7311yescharacteristic
168C73#1280011yescharacteristic, complemented, Sylow
169C2×#124—731no—
170C2×C4×Q8—731no—
171C2×C4×D4—731no—
172C2×#99—732no—
173C2×#99C2×D7312yes—
174#391—731no—
175#390C2×D7311yescharacteristic
176#388—1462no—
177#385—1462no—
178#384—731no—
179#380—1462no—
180#376—731no—
181#354—1464no—
182#342—1462no—
183C22×Q8—731no—
184C22×D4C22×D7311yescharacteristic
185C23×C4—732no—
186#124—732no—
187#124—732no—
188#124—1462no—
189C4×Q8—2922no—
190C4×D4—2922no—
191C2×C4⋊C4—2922no—
192C2×C4⋊C4—731no—
193C2×C4⋊C4—1461no—
194C2×C4⋊C4—1462no—
195C2×C4⋊C4C4×D7311yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—732no—
199C2×C22⋊C4—1462no—
200C2×C22⋊C4—1462no—
201C2×C42—731no—
202C2×C42—731no—
203C2×C42—1461no—
204C2×C42C4×D7311yescharacteristic
205#99—22no—
206#99—22no—
207#99—1462no—
208#99—1462no—
209#95—2922no—
210#95—1461no—
211#95—1461no—
212#95—1462no—
213C24D4×D7312yes—
214C2×Q8—2921no—
215C2×Q8—2921no—
216C2×Q8—732no—
217C2×Q8—1461no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4D4×D7311yescharacteristic
221C2×D4Q8×D7311yescharacteristic
222C2×D4#1192402812yes—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—2922no—
227C22×C4—5842no—
228C22×C4—731no—
229C22×C4—731no—
230C22×C4—732no—
231C22×C4—1462no—
232C22×C4—1461no—
233C22×C4—1461no—
234C22×C4—1462no—
235C22×C4C2×C4×D7311yescharacteristic
236C4⋊C4—41no—
237C4⋊C4—2921no—
238C4⋊C4—2922no—
239C4⋊C4—5842no—
240C4⋊C4—5844no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C22⋊C4—2922no—
245C22⋊C4—2922no—
246C22⋊C4—2922no—
247C22⋊C4—1464no—
248C22⋊C4—1464no—
249C42—41no—
250C42—2922no—
251C42—732no—
252C42—1462no—
253C42—1461no—
254C23—22no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23C22⋊C4×D7311yescharacteristic
259C23#1455656412yes—
260C23#1455656912yes—
261Q8—2922no—
262Q8—2922no—
263D4—41no—
264D4—41no—
265D4—42no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—2921no—
277C2×C4—2921no—
278C2×C4—2921no—
279C2×C4—2921no—
280C2×C4—2924no—
281C2×C4—2922no—
282C2×C4—2921no—
283C2×C4—2921no—
284C2×C4—5842no—
285C2×C4—5842no—
286C2×C4—5842no—
287C2×C4—5842no—
288C2×C4—732no—
289C2×C4—1462no—
290C2×C4—1462no—
291C2×C4—1461no—
292C2×C4—1462no—
293C2×C4C4⋊C4×D7311yescharacteristic
294C2×C4#1455654811yescharacteristic
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22#1798844611yescharacteristic
305C22#99×D7311yescharacteristic
306C22#1798855911yescharacteristic
307C4—21no—
308C4—21no—
309C4—81no—
310C4—81no—
311C4—162no—
312C4—2922no—
313C4—2922no—
314C4—2921no—
315C4—2922no—
316C4—11682no—
317C4—1462no—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—42no—
322C2#2260991811yescharacteristic
323C2#2261159711yescharacteristic
324C2#2261160411yescharacteristic
subgroupG/H
centerC22#99×D73
commutatorC22×C4×C73C22×C4
FrattiniC22×D4C22×D73
FittingC73×#1393C2
socleC22×C73C2×#99
prime radical, cosocleC22×D4×C73C23
hypercenter#1393D73
hypocenterC73#12800
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
7. C2×#14556556C4semidirect
10. #17988493C4semidirect
13. #17988456C4semidirect
15. #17988433C4semidirect
53. C2×Dic146C4⋊C4semidirect
75. C4×Dic73C4⋊C4semidirect
168. C73#12800semidirect

Maximal subgroups

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

subgroupindex
#226126412
C2×#179883812
#226099172
#226080562
C73×#13932
#1280073