Finite Group Databasebeta

← #33004449

Subgroups — #33004449

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988962C211yesmaximal, characteristic
2C2×#17988405C211yesmaximal, characteristic, complemented
3C2×#17988389C211yesmaximal, characteristic, complemented
4C2×#17988388C211yesmaximal, characteristic, complemented
5C2×#17988387C211yesmaximal, characteristic, complemented
6#107×Dic73C211yesmaximal, characteristic, complemented
7C2×#357×C73C211yesmaximal, characteristic, complemented
8C22×D584C2211yescharacteristic
9C22×#11924013—21no—
10C2×#14556587—21no—
11C22×#11924003C2211yescharacteristic
12#17988962—22no—
13#17988962C414yescomplemented
14C2×#14556525C2211yescharacteristic
15C2×C8×Dic73C2211yescharacteristic
16C4⋊C4×Dic73—21no—
17#17988405—22no—
18#17988389—22no—
19#17988388—22no—
20#17988387—22no—
21#17988349—21no—
22C22×D8×C73C2211yescharacteristic
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25#357×C73—22no—
26C22×D292—21no—
27C22×C22⋊D73—41no—
28C22×C22⋊D73—41no—
29C2×D584—21no—
30C2×D584—21no—
31C2×D584—21no—
32C2×D584—21no—
33C2×D584—21no—
34C2×D584C2×C412yes—
35C2×#11924013—24no—
36C2×#11924013—42no—
37C2×#11924013—42no—
38C22×#10596831D411yescharacteristic
39#14556587—24no—
40#14556587—42no—
41C2×C4⋊Dic73—21no—
42C2×C4×Dic73—41no—
43C2×C4×Dic73D411yescharacteristic
44C2×#11924003—21no—
45C2×#11924003—21no—
46C2×#11924003—21no—
47#14556525—21no—
48#14556525D412yescomplemented
49C8×Dic73C2×C412yes—
50C8×Dic73D412yes—
51#14556498—41no—
52C22×D4×C73—21no—
53C2×D8×C73—21no—
54C2×D8×C73—21no—
55C2×D8×C73—21no—
56C2×D8×C73—21no—
57C2×D8×C73—21no—
58C2×D8×C73C2×C412yes—
59C22×C8×C73C2311yescharacteristic
60C2×C4⋊C4×C73—21no—
61C2×C16×C73—21no—
62C2×C16×C73—21no—
63C2×C16×C73—21no—
64#107×C73—21no—
65#107×C73D412yescomplemented
66C23×D73—41no—
67C2×D292—22no—
68C2×D292—41no—
69C2×D292—41no—
70C2×D292—41no—
71C2×D292—41no—
72C2×D292—41no—
73C2×C22⋊D73—44no—
74C2×C22⋊D73—44no—
75C2×C22⋊D73—82no—
76C2×C22⋊D73—82no—
77C2×C22⋊D73—82no—
78C2×C22⋊D73—82no—
79C22×Dic73—41no—
80C22×Dic73D811yescharacteristic
81C22×Dic73D811yescharacteristic
82D584—22no—
83D584—22no—
84D584—41no—
85D584—41no—
86#11924013—48no—
87C2×#10596831—21no—
88C2×#10596831—21no—
89C2×#10596831C22⋊C412yes—
90C4⋊Dic73—22no—
91C4⋊Dic73—41no—
92C4×Dic73—82no—
93C4×Dic73C22⋊C412yes—
94C4×Dic73SD1612yes—
95#11924003—41no—
96C24×C73—41no—
97C2×D4×C73—22no—
98C2×D4×C73—41no—
99C2×D4×C73—41no—
100C2×D4×C73—41no—
101C2×D4×C73—41no—
102C2×D4×C73—41no—
103C22×C4×C73—41no—
104C22×C4×C73C2×D411yescharacteristic
105D8×C73—22no—
106D8×C73—22no—
107D8×C73—41no—
108D8×C73—41no—
109C2×C8×C73C22×C411yescharacteristic
110C2×C8×C73C2×D411yescharacteristic
111C2×C8×C73C2×D411yescharacteristic
112C2×C8×C73C2×D411yescharacteristic
113C2×C8×C73#4111yescharacteristic
114C2×C8×C73#4111yescharacteristic
115C4⋊C4×C73—22no—
116C4⋊C4×C73—41no—
117C16×C73—41no—
118C23×C73—42no—
119C23×C73—81no—
120C23×C73—81no—
121C23×C73—81no—
122C23×C73—81no—
123C23×C73—81no—
124C23×C73—81no—
125C23×C73C2×D811yescharacteristic
126C22×D73—42no—
127C22×D73—81no—
128C22×D73—81no—
129C22×D73—81no—
130C22×D73—81no—
131C22×D73—81no—
132C22×D73—81no—
133D4×C73—42no—
134D4×C73—42no—
135D4×C73—81no—
136D4×C73—81no—
137C2×C4×C73—42no—
138C2×C4×C73—81no—
139C2×C4×C73—81no—
140C2×C4×C73C2×C22⋊C411yescharacteristic
141C2×C4×C73#12111yescharacteristic
142C2×C4×C73#12111yescharacteristic
143C2×C4×C73C2×SD1611yescharacteristic
144C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
146C22⋊D73—88no—
147C22⋊D73—88no—
148C2×Dic73—21no—
149C2×Dic73—21no—
150C2×Dic73—42no—
151C2×Dic73—81no—
152C2×Dic73—81no—
153C2×Dic73#10212yes—
154C2×Dic73#10212yes—
155C2×Dic73D1612yes—
156C2×Dic73SD3212yes—
157D292—42no—
158D292—42no—
159D292—81no—
160D292—81no—
161C8×C73C4×D411yescharacteristic
162C8×C73C4×D411yescharacteristic
163C8×C73#12411yescharacteristic
164C8×C73#12711yescharacteristic
165#10596831—22no—
166C22×C73—82no—
167C22×C73—82no—
168C22×C73—82no—
169C22×C73—161no—
170C22×C73—161no—
171C22×C73—161no—
172C22×C73—161no—
173C22×C73C2×#10211yescharacteristic
174C22×C73#45911yescharacteristic
175C22×C73#45911yescharacteristic
176C22×C73C2×D1611yescharacteristic
177C22×C73C2×SD3211yescharacteristic
178C22×C73#50911yescharacteristic
179C22×C73#50911yescharacteristic
180C2×D73—82no—
181C2×D73—82no—
182C2×D73—82no—
183C2×D73—161no—
184C2×D73—161no—
185C2×D73—161no—
186C2×D73—161no—
187C4×C73—82no—
188C4×C73#39011yescharacteristic
189C4×C73#41811yescharacteristic
190C4×C73#46011yescharacteristic
191C4×C73#46011yescharacteristic
192Dic73—22no—
193Dic73—82no—
194Dic73#35714yescomplemented
195#13252—731nomaximal, Sylow
196C2×C73—162no—
197C2×C73#198211yescharacteristic
198C2×C73C2×#35711yescharacteristic
199C2×C73#214011yescharacteristic
200C2×C73#220611yescharacteristic
201C2×C73#220611yescharacteristic
202C2×C73#220711yescharacteristic
203C2×C73#220711yescharacteristic
204D73—162no—
205C2×#493—731no—
206C2×C4⋊C16—731no—
207C2×#357—731no—
208C2×#357—731no—
209C2×#357—731no—
210C2×#357D7311yescharacteristic, complemented
211C4×#107—731no—
212C73#1325211yescharacteristic, complemented, Sylow
213C22×D8—731no—
214C22×D8—1461no—
215C22×D8C2×D7311yescharacteristic
216C2×#127—1461no—
217C22×C16—731no—
218C22×C16C2×D7311yescharacteristic
219#493—734no—
220#493—1462no—
221C2×#107—731no—
222C2×#107C2×D7311yescharacteristic
223C2×C4×C8—731no—
224C4×C4⋊C4—1461no—
225C4⋊C16—1462no—
226#357—22no—
227#357—1462no—
228#357—1462no—
229#357—1462no—
230#340—1461no—
231C22×D4—21no—
232C22×D4—2921no—
233C22×D4—2921no—
234C22×D4—1461no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8—21no—
238C2×D8—21no—
239C2×D8—21no—
240C2×D8—2922no—
241C2×D8—2922no—
242C2×D8—732no—
243C2×D8—1464no—
244C2×D8—1461no—
245C2×D8—1461no—
246C2×D8—1461no—
247C2×D8—1461no—
248C2×D8—1461no—
249C2×D8C4×D7312yes—
250C22×C8—731no—
251C22×C8C22×D7311yescharacteristic
252#127—2922no—
253#127—1464no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—1461no—
256C2×C42—2921no—
257C2×C42—731no—
258C2×C16—21no—
259C2×C16—21no—
260C2×C16—21no—
261C2×C16—1461no—
262C2×C16—1461no—
263C2×C16—1461no—
264#107—21no—
265#107—732no—
266#107—1461no—
267#107C22⋊D7312yescomplemented
268C4×C8—732no—
269C4×C8—732no—
270#95—2921no—
271C24—41no—
272C24—2921no—
273C2×D4—22no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—2924no—
280C2×D4—2924no—
281C2×D4—2921no—
282C2×D4—2921no—
283C2×D4—2921no—
284C2×D4—2921no—
285C2×D4—2921no—
286C2×D4—5842no—
287C2×D4—5842no—
288C2×D4—5842no—
289C2×D4—5842no—
290C2×D4—1462no—
291C22×C4—41no—
292C22×C4—2921no—
293C22×C4—731no—
294C22×C4—731no—
295C22×C4D4×D7311yescharacteristic
296D8—22no—
297D8—22no—
298D8—41no—
299D8—41no—
300D8—2928no—
301D8—2921no—
302D8—2921no—
303D8—1462no—
304D8—1462no—
305C2×C8—732no—
306C2×C8—1461no—
307C2×C8—1461no—
308C2×C8C2×C4×D7311yescharacteristic
309C2×C8D4×D7311yescharacteristic
310C2×C8D4×D7311yescharacteristic
311C2×C8C2×C22⋊D7311yescharacteristic
312C2×C8#1192402711yescharacteristic
313C2×C8#1192402811yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—2921no—
317C4⋊C4—1462no—
318C42—5842no—
319C42—732no—
320C42—732no—
321C16—41no—
322C16—2921no—
323C23—42no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—2922no—
331C23—5841no—
332C23—5841no—
333C23—5841no—
334C23—5841no—
335C23—5841no—
336C23—5841no—
337C23D8×D7311yescharacteristic
338D4—42no—
339D4—42no—
340D4—81no—
341D4—81no—
342D4—2922no—
343D4—2922no—
344D4—5848no—
345D4—5848no—
346D4—5841no—
347D4—5841no—
348C2×C4—42no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—2922no—
352C2×C4—5841no—
353C2×C4—5841no—
354C2×C4—732no—
355C2×C4—732no—
356C2×C4—732no—
357C2×C4—732no—
358C2×C4—1461no—
359C2×C4—1461no—
360C2×C4C22⋊C4×D7311yescharacteristic
361C2×C4#1455656311yescharacteristic
362C2×C4#1455656411yescharacteristic
363C2×C4SD16×D7311yescharacteristic
364C2×C4#1455661511yescharacteristic
365C2×C4#1455661811yescharacteristic
366C8—1462no—
367C8#1455655111yescharacteristic
368C8C4×C22⋊D7311yescharacteristic
369C8#1455657711yescharacteristic
370C8#1455658711yescharacteristic
371C22—82no—
372C22—82no—
373C22—82no—
374C22—161no—
375C22—161no—
376C22—161no—
377C22—161no—
378C22—5842no—
379C22—5842no—
380C22—5842no—
381C22—11681no—
382C22—11681no—
383C22—11681no—
384C22—11681no—
385C22#102×D7311yescharacteristic
386C22#1798878011yescharacteristic
387C22#1798878111yescharacteristic
388C22D16×D7311yescharacteristic
389C22SD32×D7311yescharacteristic
390C22#1798902111yescharacteristic
391C22#1798902411yescharacteristic
392C4—82no—
393C4—5842no—
394C4—734no—
395C4—1462no—
396C4#1798848711yescharacteristic
397C4#1798859011yescharacteristic
398C4#1798878511yescharacteristic
399C4#1798878611yescharacteristic
400C2—162no—
401C2—11682no—
402C2#2261085611yescharacteristic
403C2#357×D7311yescharacteristic
404C2#2261164511yescharacteristic
405C2#2261196411yescharacteristic
406C2#2261196511yescharacteristic
407C2#2261196911yescharacteristic
408C2#2261197011yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#357×C73C2
socleC23×C73C2×D8
prime radicalC2×#17988962C2
cosocleC22×C8×C73C23
hypercenterC2×#357D73
hypocenterC73#13252
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#17988405C2semidirect
3. C2×#17988389C2semidirect
4. C2×#17988388C2semidirect
5. C2×#17988387C2semidirect
6. #107×Dic73C2semidirect
7. C2×#357×C73C2semidirect
13. #17988962C4semidirect
48. #14556525D4semidirect
65. #107×C73D4semidirect
194. Dic73#357semidirect
210. C2×#357D73semidirect
212. C73#13252semidirect
267. #107C22⋊D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179889622
C2×#179884052
C2×#179883892
C2×#179883882
C2×#179883872
#107×Dic732
C2×#357×C732
#1325273