Finite Group Databasebeta

← #32997039

Subgroups — #32997039

Order 18688.

476 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 99 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#22614930C211yesmaximal, characteristic
2#22609927C211yesmaximal, characteristic, complemented
3#22607935C214yesmaximal, complemented
4C73×#1461C211yesmaximal, characteristic, complemented
5C2×#14556588C411yescharacteristic
6C2×#14556580—21no—
7#17988611—21no—
8C2×#14556510—21no—
9#17988504C412yes—
10#17988504C2212yes—
11#17988489—21no—
12#17988489—21no—
13#17988488—21no—
14C4×C4⋊Dic73C411yescharacteristic
15#17988356C2212yes—
16C2×C4×Q8×C73C2211yescharacteristic
17#431×C73—21no—
18#324×C73—21no—
19#324×C73C2212yes—
20C22×D292—21no—
21C22×D292—21no—
22C22×D292C2×C412yes—
23C2×#11924012—22no—
24#14556588—41no—
25#14556588—41no—
26#14556580—42no—
27#14556580—42no—
28C2×C4⋊Dic73—21no—
29C2×#11924005—22no—
30C2×#11924005—22no—
31C2×#11924005—22no—
32C2×#11924005—22no—
33C2×C4×Dic73—21no—
34C4⋊#10596831—41no—
35C4⋊#10596831—41no—
36#14556510—42no—
37#14556498C2×C412yes—
38C22×Q8×C73D411yescharacteristic
39C4×Q8×C73—41no—
40C4×Q8×C73—41no—
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C4⋊C4×C73D411yescharacteristic
43C2×C4⋊C4×C73C2311yescharacteristic
44C2×C42×C73C2×C411yescharacteristic
45C2×C42×C73D411yescharacteristic
46C2×C42×C73D411yescharacteristic
47C4⋊C8×C73—41no—
48C22⋊C8×C73—22no—
49C4×C8×C73—41no—
50C23×D73—22no—
51C2×D292—42no—
52C2×D292—42no—
53C2×D292—42no—
54C2×D292—42no—
55C2×D292—42no—
56C2×D292—41no—
57C2×D292—41no—
58C2×D292—42no—
59C2×D292—82no—
60C22×Dic73—22no—
61#11924012—42no—
62#11924012—42no—
63C2×#10596831—42no—
64C4⋊Dic73—42no—
65#11924005—44no—
66#11924005—42no—
67#11924005—42no—
68#11924005—82no—
69#11924005—82no—
70C4×Dic73—42no—
71C2×Q8×C73—41no—
72C2×Q8×C73—41no—
73C2×Q8×C73D812yes—
74C2×Q8×C73SD1612yes—
75C22×C4×C73C22⋊C411yescharacteristic
76C22×C4×C73C22⋊C411yescharacteristic
77C22×C4×C73C22⋊C411yescharacteristic
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22×C411yescharacteristic
80C22×C4×C73C2×D411yescharacteristic
81C22×C4×C73C2×D411yescharacteristic
82C2×C8×C73—42no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73—41no—
85C4⋊C4×C73—41no—
86C42×C73—21no—
87C42×C73—21no—
88C42×C73—21no—
89C42×C73—21no—
90C42×C73—21no—
91C42×C73—21no—
92C23×C73C2×C22⋊C411yescharacteristic
93C22×D73—42no—
94C22×D73—42no—
95C22×D73—42no—
96C22×D73—42no—
97C22×D73—42no—
98C22×D73—42no—
99C22×D73—42no—
100Q8×C73—42no—
101Q8×C73—42no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—22no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73#9911yescharacteristic
116C2×C4×C73#9911yescharacteristic
117C2×C4×C73#10212yes—
118C2×C4×C73#10212yes—
119C2×C4×C73C2×D811yescharacteristic
120C2×C4×C73C2×SD1611yescharacteristic
121C2×Dic73—42no—
122C2×Dic73—42no—
123C2×Dic73—42no—
124D292—82no—
125D292—84no—
126D292—84no—
127D292—82no—
128C8×C73—82no—
129#10596831—82no—
130C22×C73—21no—
131C22×C73—21no—
132C22×C73C2×#9911yescharacteristic
133C22×C73C2×#10211yescharacteristic
134C22×C73#42111yescharacteristic
135C2×D73—82no—
136C2×D73—82no—
137C2×D73—82no—
138C2×D73—82no—
139C2×D73—82no—
140C2×D73—82no—
141C2×D73—82no—
142C4×C73—21no—
143C4×C73—21no—
144C4×C73—22no—
145C4×C73—22no—
146C4×C73—21no—
147C4×C73—41no—
148C4×C73—41no—
149C4×C73—41no—
150C4×C73#35512yes—
151Dic73—82no—
152#11882—731nomaximal, Sylow
153C2×C73—21no—
154C2×C73—21no—
155C2×C73#151911yescharacteristic
156C2×C73#212211yescharacteristic
157C2×C73C2×#35511yescharacteristic
158D73—162no—
159#2548—731no—
160#1801—731no—
161#1461D7311yescharacteristic, complemented
162#1345—734no—
163C73#1188211yescharacteristic, complemented, Sylow
164C2×#127—731no—
165C2×#124—1461no—
166C2×C4×Q8C2×D7311yescharacteristic
167#431—21no—
168#423—1461no—
169C2×#101—1461no—
170#394—732no—
171#394—732no—
172#390—1461no—
173#390—1461no—
174#390—1461no—
175C4×C4⋊C4—731no—
176#343—732no—
177#324—21no—
178#324C2×D7312yes—
179C22×Q8D29211yescharacteristic
180C22×D4—732no—
181C22×D4—1461no—
182C22×D4—1461no—
183C2×M16—1462no—
184#127—2921no—
185#127—2921no—
186#124—2922no—
187#124—2922no—
188C4×Q8—41no—
189C4×Q8—41no—
190C2×C4⋊C4—1461no—
191C2×C4⋊C4C4×D7311yescharacteristic
192C2×C4⋊C4C22⋊D7311yescharacteristic
193C2×C4⋊C4C22×D7311yescharacteristic
194C2×C22⋊C4—1462no—
195C2×C22⋊C4—1462no—
196C2×C22⋊C4—1462no—
197C2×C22⋊C4—1462no—
198C2×C42—1461no—
199C2×C42C4×D7311yescharacteristic
200C2×C42D29211yescharacteristic
201C2×C42C22⋊D7311yescharacteristic
202C4⋊C8—41no—
203C4⋊C8—2921no—
204C4⋊C8—2921no—
205#101—2922no—
206C22⋊C8—22no—
207C4×C8—41no—
208#95—732no—
209C24—1462no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8D58412yes—
213C2×Q8#1192401712yes—
214C2×D4—2922no—
215C2×D4—2922no—
216C2×D4—2922no—
217C2×D4—2922no—
218C2×D4—2922no—
219C2×D4—2921no—
220C2×D4—2921no—
221C2×D4—2922no—
222C2×D4—5842no—
223C22×C4—1462no—
224C22×C4#1192400511yescharacteristic
225C22×C4#1192400511yescharacteristic
226C22×C4#1192400511yescharacteristic
227C22×C4#1192400511yescharacteristic
228C22×C4C2×C4×D7311yescharacteristic
229C22×C4C2×C22⋊D7311yescharacteristic
230C22×C4C2×D29211yescharacteristic
231M16—2922no—
232M16—2922no—
233C2×C8—42no—
234C2×C8—2922no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—2922no—
239C22⋊C4—2924no—
240C22⋊C4—2922no—
241C22⋊C4—2922no—
242C22⋊C4—5842no—
243C22⋊C4—5842no—
244C42—21no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—2922no—
251C23—2922no—
252C23—2922no—
253C23—2922no—
254C23—2922no—
255C23—2922no—
256C23—2922no—
257C23—2922no—
258C23C2×#1192400511yescharacteristic
259Q8—42no—
260Q8—42no—
261D4—5844no—
262D4—5842no—
263D4—5844no—
264D4—5842no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—2922no—
279C2×C4—2922no—
280C2×C4—2922no—
281C2×C4#1455650511yescharacteristic
282C2×C4#1455650511yescharacteristic
283C2×C4#1455651312yes—
284C2×C4#1455651312yes—
285C2×C4C2×D58411yescharacteristic
286C2×C4C2×#1192401711yescharacteristic
287C8—82no—
288C8—5842no—
289C22—21no—
290C22—21no—
291C22—5842no—
292C22—5842no—
293C22—5842no—
294C22—5842no—
295C22—5842no—
296C22—5842no—
297C22—5842no—
298C22C2×#1455650511yescharacteristic
299C22C2×#1455651311yescharacteristic
300C22#1798860311yescharacteristic
301C4—21no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—5842no—
310C4#1798838312yes—
311C2—21no—
312C2—21no—
313C2—11682no—
314C2#2260850011yescharacteristic
315C2#2261158011yescharacteristic
316C2C2×#1798838311yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#1461C2
socleC22×C73C2×#99
prime radicalC2×#14556588C4
cosocleC2×C4⋊C4×C73C23
hypercenter#1461D73
hypocenterC73#11882
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #22609927C2semidirect
3. #22607935C2semidirect
4. C73×#1461C2semidirect
161. #1461D73semidirect
163. C73#11882semidirect

Maximal subgroups

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

subgroupindex
#226149302
#226099272
#226079352
C73×#14612
#1188273