Finite Group Databasebeta

← #32981926

Subgroups — #32981926

Order 18688.

544 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619403C211yesmaximal, characteristic
2#22608490C211yesmaximal, characteristic
3#22608425C211yesmaximal, characteristic
4#22608326C211yesmaximal, characteristic, complemented
5#22607834C211yesmaximal, characteristic, complemented
6#22607789C211yesmaximal, characteristic, complemented
7C73×#2351C211yesmaximal, characteristic, complemented
8C2×#14556622—21no—
9#17989137C2211yescharacteristic
10#17988809—22no—
11#17988797C412yescomplemented
12#17988787C412yescomplemented
13#17988640C2211yescharacteristic
14#17988640C2211yescharacteristic
15#17988611C2211yescharacteristic
16#17988586—21no—
17#17988320—21no—
18#17988320—21no—
19#17988320—22no—
20Q8⋊#10596831—22no—
21C2×C4×Q8×C73—21no—
22C2×C4×Q8×C73C2211yescharacteristic
23#389×C73—21no—
24#389×C73—22no—
25#382×C73—21no—
26#382×C73C2211yescharacteristic
27C4×C4⋊C4×C73—21no—
28C4×C4⋊C4×C73C2211yescharacteristic
29C2×#11924028—21no—
30#14556622—42no—
31#14556622—42no—
32C2×#11924018—22no—
33C2×#11924015—22no—
34C2×#11924012—21no—
35#14556591C2×C411yescharacteristic
36#14556588C2×C411yescharacteristic
37#14556579—21no—
38#14556565—21no—
39#14556565—21no—
40C4×D292—21no—
41C4×D292—21no—
42C4⋊#10596831—21no—
43C4⋊#10596831—21no—
44C4⋊#10596831—21no—
45C4⋊#10596831C2×C412yes—
46#14556514—22no—
47#14556511—22no—
48C22⋊#10596831—21no—
49C22⋊#10596831—21no—
50C4×#10596831—21no—
51C4×#10596831—21no—
52C22×Q8×C73D411yescharacteristic
53C4×Q8×C73—21no—
54C4×Q8×C73—21no—
55C4×Q8×C73—42no—
56C4×Q8×C73—42no—
57C4×Q8×C73C2×C412yes—
58C4×Q8×C73D412yes—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—22no—
61C2×C4⋊C4×C73—22no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C4⋊C4×C73D411yescharacteristic
65C2×C42×C73—21no—
66C2×C42×C73—22no—
67C2×C42×C73—21no—
68C2×C42×C73C2311yescharacteristic
69#95×C73—22no—
70#95×C73—21no—
71#95×C73D411yescharacteristic
72#95×C73D411yescharacteristic
73#95×C73D412yes—
74#11924028—41no—
75#11924028—41no—
76#11924028—41no—
77#11924028—41no—
78C2×Dic146—21no—
79C2×D292—21no—
80C2×D292—21no—
81C2×D292—21no—
82C2×C22⋊D73—41no—
83C2×C4×D73—41no—
84#11924018—44no—
85#11924015—44no—
86#11924012—41no—
87#11924012—41no—
88C2×#10596831—22no—
89C2×#10596831—41no—
90C2×#10596831—41no—
91C4⋊Dic73—21no—
92C4⋊Dic73—21no—
93#11924005—41no—
94#11924005—41no—
95C2×Q8×C73—21no—
96C2×Q8×C73—21no—
97C2×Q8×C73—21no—
98C2×Q8×C73C22⋊C412yes—
99C2×Q8×C73D812yes—
100C2×Q8×C73SD1612yes—
101C22×C4×C73—21no—
102C22×C4×C73—21no—
103C22×C4×C73—21no—
104C22×C4×C73—21no—
105C22×C4×C73—22no—
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73C2×D411yescharacteristic
109C4⋊C4×C73—22no—
110C4⋊C4×C73—22no—
111C4⋊C4×C73—21no—
112C4⋊C4×C73—42no—
113C4⋊C4×C73—44no—
114C4⋊C4×C73—44no—
115C4⋊C4×C73—41no—
116C4⋊C4×C73—41no—
117C4⋊C4×C73C22⋊C412yes—
118C42×C73—22no—
119C42×C73—22no—
120C42×C73—44no—
121C42×C73—41no—
122C42×C73—41no—
123C42×C73C22×C411yescharacteristic
124C42×C73C2×D411yescharacteristic
125C42×C73#4111yescharacteristic
126C42×C73#4111yescharacteristic
127C23×C73#12011yescharacteristic
128C22×D73—41no—
129Q8×C73—22no—
130Q8×C73—22no—
131Q8×C73—22no—
132Q8×C73#10214yes—
133C2×C4×C73—22no—
134C2×C4×C73—22no—
135C2×C4×C73—22no—
136C2×C4×C73—22no—
137C2×C4×C73—22no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—42no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—42no—
145C2×C4×C73—41no—
146C2×C4×C73—42no—
147C2×C4×C73—42no—
148C2×C4×C73—41no—
149C2×C4×C73C2×C22⋊C411yescharacteristic
150C2×C4×C73C4×D411yescharacteristic
151C2×C4×C73C4×D411yescharacteristic
152C2×C4×C73#12111yescharacteristic
153C2×C4×C73#12111yescharacteristic
154C2×C4×C73#12311yescharacteristic
155C2×C4×C73C2×D811yescharacteristic
156C2×C4×C73C2×SD1611yescharacteristic
157C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C73#13711yescharacteristic
159C22⋊D73—81no—
160C22⋊D73—81no—
161C2×Dic73—41no—
162D292—41no—
163D292—41no—
164D292—81no—
165D292—81no—
166C4×D73—81no—
167C4×D73—81no—
168Dic146—41no—
169Dic146—41no—
170#10596831—42no—
171#10596831—81no—
172#10596831—81no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73#38611yescharacteristic
176C22×C73#44711yescharacteristic
177C22×C73#44811yescharacteristic
178C2×D73—81no—
179C2×D73—81no—
180C2×D73—81no—
181C4×C73—24no—
182C4×C73—21no—
183C4×C73—21no—
184C4×C73—21no—
185C4×C73—21no—
186C4×C73—42no—
187C4×C73—42no—
188C4×C73—82no—
189C4×C73—81no—
190C4×C73C2×#10211yescharacteristic
191C4×C73#41711yescharacteristic
192C4×C73#48011yescharacteristic
193C4×C73#48111yescharacteristic
194Dic73—81no—
195#9180—731nomaximal, Sylow
196C2×C73—21no—
197C2×C73—21no—
198C2×C73#187411yescharacteristic
199C2×C73#190511yescharacteristic
200C2×C73#190611yescharacteristic
201D73—161no—
202#3037—731no—
203#2351D7311yescharacteristic, complemented
204#1517—731no—
205#1502—731no—
206#1479—731no—
207#1303—731no—
208#1286—731no—
209C73#918011yescharacteristic, complemented, Sylow
210C2×#137—1461no—
211#532—731no—
212C2×C4×Q8—21no—
213C2×C4×Q8C2×D7311yescharacteristic
214#464—1462no—
215#462—732no—
216#460—732no—
217#431—731no—
218#431—731no—
219#423—731no—
220#417—1461no—
221#389—21no—
222#389—22no—
223#382—21no—
224#382C2×D7311yescharacteristic
225C4×C4⋊C4—21no—
226C4×C4⋊C4C2×D7311yescharacteristic
227#327—1461no—
228#327—1461no—
229#327—1462no—
230Q8⋊C8—1462no—
231C2×#41—1461no—
232C22×Q8C22⋊D7311yescharacteristic
233#137—2922no—
234#137—2922no—
235C2×Q16—1462no—
236C2×SD16—1462no—
237C2×M16—1461no—
238C4⋊Q8—731no—
239#127—731no—
240#124—1461no—
241#121—1461no—
242#121—1461no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×Q8C4×D7312yes—
248C4×Q8D29212yes—
249C4×D4—1461no—
250C4×D4—1461no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22⋊D7311yescharacteristic
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42C22×D7311yescharacteristic
261C4⋊C8—732no—
262C4⋊C8—1461no—
263C4⋊C8—1461no—
264C4⋊C8—1461no—
265#103—1462no—
266#102—1462no—
267C22⋊C8—1461no—
268C22⋊C8—1461no—
269C4×C8—1461no—
270C4×C8—1461no—
271#95—22no—
272#95—21no—
273#95C22⋊D7311yescharacteristic
274#95C22⋊D7311yescharacteristic
275#95C22⋊D7312yes—
276#41—2921no—
277#41—2921no—
278#41—2921no—
279#41—2921no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—1461no—
284C2×Q8#1192400512yes—
285C2×Q8#1192401312yes—
286C2×Q8#1192401512yes—
287C2×D4—2921no—
288C2×D4—1461no—
289C2×D4—1461no—
290C2×D4—1461no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—2921no—
297C22×C4C2×C22⋊D7311yescharacteristic
298C22×C4C2×C22⋊D7311yescharacteristic
299C22×C4C2×C22⋊D7311yescharacteristic
300Q16—2924no—
301SD16—2924no—
302M16—2921no—
303M16—2921no—
304C2×C8—2921no—
305C2×C8—2921no—
306C2×C8—1462no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—1461no—
316C4⋊C4—1461no—
317C4⋊C4#1192400512yes—
318C22⋊C4—2921no—
319C22⋊C4—2921no—
320C42—22no—
321C42—22no—
322C42—44no—
323C42—41no—
324C42—41no—
325C42C2×C4×D7311yescharacteristic
326C42C2×D29211yescharacteristic
327C42#1192402811yescharacteristic
328C42#1192402811yescharacteristic
329C23—2921no—
330C23#1455656011yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—22no—
334Q8—2921no—
335Q8—2921no—
336Q8#1455651114yes—
337D4—2921no—
338D4—2921no—
339D4—5841no—
340D4—5841no—
341D4—5841no—
342D4—5841no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—2921no—
360C2×C4—5841no—
361C2×C4—5841no—
362C2×C4C2×#1192400511yescharacteristic
363C2×C4C4×C22⋊D7311yescharacteristic
364C2×C4C4×C22⋊D7311yescharacteristic
365C2×C4#1455656511yescharacteristic
366C2×C4#1455656511yescharacteristic
367C2×C4#1455657411yescharacteristic
368C2×C4C2×#1192401311yescharacteristic
369C2×C4C2×#1192401511yescharacteristic
370C2×C4#1455661711yescharacteristic
371C2×C4#1455662211yescharacteristic
372C8—2922no—
373C8—5841no—
374C8—5841no—
375C22—21no—
376C22—21no—
377C22—5841no—
378C22—5841no—
379C22—5841no—
380C22#1798846611yescharacteristic
381C22#1798872111yescharacteristic
382C22#1798872611yescharacteristic
383C4—21no—
384C4—21no—
385C4—24no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—42no—
390C4—82no—
391C4—81no—
392C4—5841no—
393C4C2×#1455651111yescharacteristic
394C4#1798858611yescharacteristic
395C4#1798890011yescharacteristic
396C4#1798890511yescharacteristic
397C2—21no—
398C2—21no—
399C2—11681no—
400C2#2261024811yescharacteristic
401C2#2261042511yescharacteristic
402C2#2261043211yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2351C2
socleC22×C73#386
prime radical#17989137C22
cosocleC2×C42×C73C23
hypercenter#2351D73
hypocenterC73#9180
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #22608326C2semidirect
5. #22607834C2semidirect
6. #22607789C2semidirect
7. C73×#2351C2semidirect
11. #17988797C4semidirect
12. #17988787C4semidirect
203. #2351D73semidirect
209. C73#9180semidirect

Maximal subgroups

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

subgroupindex
#226194032
#226084902
#226084252
#226083262
#226078342
#226077892
C73×#23512
#918073