Finite Group Databasebeta

← #33002921

Subgroups — #33002921

Order 18688.

626 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988824C211yesmaximal, characteristic
2C2×#107×D73C211yesmaximal, characteristic, complemented
3C2×#17988389C211yesmaximal, characteristic, complemented
4C2×#17988387C211yesmaximal, characteristic, complemented
5C2×#17988366C211yesmaximal, characteristic, complemented
6#22608022C211yesmaximal, characteristic, complemented
7C73×#1380C211yesmaximal, characteristic, complemented
8C22×D584C2211yescharacteristic
9C22×C8×D73C2211yescharacteristic
10C2×#14556566—21no—
11C2×C4⋊C4×D73—21no—
12C22×#11924003C2211yescharacteristic
13#17988824—22no—
14#17988824C414yescomplemented
15C2×#14556525C2211yescharacteristic
16C2×#14556524—21no—
17#107×D73—22no—
18#107×D73—22no—
19C2×#14556511—21no—
20#17988389—22no—
21#17988387—22no—
22#17988366—22no—
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25C2×#107×C73C2211yescharacteristic
26C22×D292—21no—
27C22×D292—41no—
28C22×C4×D73—41no—
29C22×C4×D73D411yescharacteristic
30C2×D584—21no—
31C2×D584—21no—
32C2×D584—21no—
33C2×D584—21no—
34C2×D584—21no—
35C2×D584C2×C412yescomplemented
36C22×#10596831D411yescharacteristic
37C2×C8×D73—21no—
38C2×C8×D73—21no—
39C2×C8×D73—21no—
40C2×C8×D73—21no—
41C2×C8×D73C2×C412yes—
42C2×C8×D73D412yes—
43#14556566—24no—
44#14556566—42no—
45C2×C4⋊Dic73—21no—
46C2×#11924007—41no—
47C4⋊C4×D73—24no—
48C4⋊C4×D73—42no—
49C2×#11924005—41no—
50C2×#11924003—21no—
51C2×#11924003—21no—
52C2×#11924003—21no—
53#14556525—21no—
54#14556525D412yescomplemented
55#14556524—24no—
56#14556511—24no—
57C22×C8×C73C2311yescharacteristic
58C2×C4⋊C4×C73—21no—
59C2×C4⋊C4×C73—21no—
60C2×C16×C73—21no—
61C2×C16×C73—21no—
62C2×C16×C73—21no—
63#107×C73—21no—
64#107×C73—21no—
65#107×C73C2×C412yescomplemented
66#107×C73D412yescomplemented
67C23×D73—41no—
68C23×D73D811yescharacteristic
69C2×D292—22no—
70C2×D292—41no—
71C2×D292—41no—
72C2×D292—41no—
73C2×D292—41no—
74C2×D292—44no—
75C2×D292—41no—
76C2×D292—82no—
77C2×D292—82no—
78C22×Dic73—41no—
79C22×Dic73D811yescharacteristic
80C2×C4×D73—21no—
81C2×C4×D73—21no—
82C2×C4×D73—21no—
83C2×C4×D73—21no—
84C2×C4×D73—44no—
85C2×C4×D73—82no—
86C2×C4×D73—82no—
87C2×C4×D73C22⋊C412yes—
88C2×C4×D73SD1612yes—
89D584—22no—
90D584—22no—
91D584—41no—
92D584—41no—
93C2×#10596831—21no—
94C2×#10596831—21no—
95C2×#10596831C22⋊C412yes—
96C8×D73—22no—
97C8×D73—22no—
98C8×D73—22no—
99C8×D73—22no—
100C4⋊Dic73—22no—
101C4⋊Dic73—41no—
102#11924007—44no—
103#11924005—44no—
104#11924003—41no—
105C22×C4×C73—41no—
106C22×C4×C73—41no—
107C22×C4×C73C2×D411yescharacteristic
108C2×C8×C73C22×C411yescharacteristic
109C2×C8×C73C2×D411yescharacteristic
110C2×C8×C73C2×D411yescharacteristic
111C2×C8×C73C2×D411yescharacteristic
112C2×C8×C73#4111yescharacteristic
113C2×C8×C73#4111yescharacteristic
114C4⋊C4×C73—22no—
115C4⋊C4×C73—22no—
116C4⋊C4×C73—41no—
117C4⋊C4×C73—41no—
118C16×C73—41no—
119C23×C73C2×D811yescharacteristic
120C22×D73—21no—
121C22×D73—21no—
122C22×D73—21no—
123C22×D73—21no—
124C22×D73—42no—
125C22×D73—81no—
126C22×D73—81no—
127C22×D73—81no—
128C22×D73—81no—
129C22×D73—81no—
130C22×D73—81no—
131C22×D73#10212yes—
132C22×D73D1612yes—
133C22×D73SD3212yes—
134C2×C4×C73—42no—
135C2×C4×C73—42no—
136C2×C4×C73—81no—
137C2×C4×C73—81no—
138C2×C4×C73—81no—
139C2×C4×C73—81no—
140C2×C4×C73C2×C22⋊C411yescharacteristic
141C2×C4×C73#12011yescharacteristic
142C2×C4×C73#12311yescharacteristic
143C2×C4×C73C2×SD1611yescharacteristic
144C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C73#13711yescharacteristic
146C2×Dic73—21no—
147C2×Dic73—21no—
148C2×Dic73—42no—
149C2×Dic73—81no—
150C2×Dic73—81no—
151C2×Dic73#10212yes—
152D292—42no—
153D292—42no—
154D292—88no—
155D292—81no—
156D292—81no—
157C4×D73—22no—
158C4×D73—22no—
159C4×D73—22no—
160C4×D73—22no—
161C4×D73—88no—
162C8×C73C4×D411yescharacteristic
163C8×C73C4×D411yescharacteristic
164C8×C73#12111yescharacteristic
165C8×C73#12111yescharacteristic
166#10596831—22no—
167C22×C73C2×#10211yescharacteristic
168C22×C73#44711yescharacteristic
169C22×C73#48011yescharacteristic
170C22×C73C2×D1611yescharacteristic
171C22×C73C2×SD3211yescharacteristic
172C22×C73#50911yescharacteristic
173C22×C73#51011yescharacteristic
174C2×D73—22no—
175C2×D73—22no—
176C2×D73—22no—
177C2×D73—22no—
178C2×D73—22no—
179C2×D73—22no—
180C2×D73—82no—
181C2×D73—82no—
182C2×D73—82no—
183C2×D73—161no—
184C2×D73—161no—
185C2×D73—161no—
186C2×D73—161no—
187C2×D73#35714yescomplemented
188C4×C73—82no—
189C4×C73—82no—
190C4×C73#38611yescharacteristic
191C4×C73#41711yescharacteristic
192C4×C73#44811yescharacteristic
193C4×C73#48111yescharacteristic
194Dic73—22no—
195Dic73—82no—
196#12991—731nomaximal, Sylow
197C2×C73#187411yescharacteristic
198C2×C73C2×#35711yescharacteristic
199C2×C73#213911yescharacteristic
200C2×C73#218411yescharacteristic
201C2×C73#218511yescharacteristic
202C2×C73#223211yescharacteristic
203C2×C73#223311yescharacteristic
204D73—24no—
205D73—162no—
206C2×#466—731no—
207C22×#107—731no—
208C2×#357—731no—
209C2×#357—731no—
210C2×C22⋊C16—731no—
211#1380—731no—
212#1380D7311yescharacteristic, complemented
213C73#1299111yescharacteristic, complemented, Sylow
214C22×D8—731no—
215C23×C8—731no—
216C2×#121—1461no—
217C22×C4⋊C4—1461no—
218C22×C16—731no—
219C22×C16C2×D7311yescharacteristic
220#466—734no—
221#466—1462no—
222C2×#107—731no—
223C2×#107—1462no—
224C2×#107—1461no—
225C2×#107—1462no—
226C2×#107C2×D7311yescharacteristic
227C2×#107C2×D7311yescharacteristic
228C2×#102—1461no—
229#357—1462no—
230#357—1462no—
231C22⋊C16—1462no—
232C22×D4—2921no—
233C22×D4—1461no—
234C23×C4—2921no—
235C23×C4—731no—
236C2×D8—732no—
237C2×D8—1461no—
238C2×D8—1461no—
239C2×D8—1461no—
240C2×D8—1461no—
241C2×D8—1461no—
242C22×C8—731no—
243C22×C8—732no—
244C22×C8—732no—
245C22×C8—1461no—
246C22×C8—1461no—
247C22×C8—1461no—
248C22×C8—1461no—
249C22×C8C22×D7311yescharacteristic
250#121—2922no—
251#121—1464no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—2922no—
255C2×C4⋊C4—2921no—
256C2×C4⋊C4—1464no—
257C2×C4⋊C4—1461no—
258C2×C22⋊C4—2921no—
259C2×C16—21no—
260C2×C16—21no—
261C2×C16—21no—
262C2×C16—1461no—
263C2×C16—1461no—
264C2×C16—1461no—
265#107—21no—
266#107—21no—
267#107—732no—
268#107—1464no—
269#107—1461no—
270#107C4×D7312yescomplemented
271#107D29212yescomplemented
272#102—1464no—
273C24—2921no—
274C24—731no—
275C2×D4—2924no—
276C2×D4—2921no—
277C2×D4—2921no—
278C2×D4—2921no—
279C2×D4—2921no—
280C2×D4—2921no—
281C2×D4—5842no—
282C2×D4—5842no—
283C2×D4—1462no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—2924no—
287C22×C4—2921no—
288C22×C4—5842no—
289C22×C4—5842no—
290C22×C4—731no—
291C22×C4—732no—
292C22×C4—732no—
293C22×C4—1461no—
294C22×C4—1461no—
295C22×C4—1461no—
296C22×C4—1461no—
297C22×C4D4×D7311yescharacteristic
298D8—2921no—
299D8—2921no—
300D8—1462no—
301D8—1462no—
302C2×C8—732no—
303C2×C8—1461no—
304C2×C8—1461no—
305C2×C8—1462no—
306C2×C8—1462no—
307C2×C8—1462no—
308C2×C8—1462no—
309C2×C8C2×C4×D7311yescharacteristic
310C2×C8D4×D7311yescharacteristic
311C2×C8D4×D7311yescharacteristic
312C2×C8C2×D29211yescharacteristic
313C2×C8#1192402811yescharacteristic
314C2×C8#1192402911yescharacteristic
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—2924no—
320C4⋊C4—2921no—
321C4⋊C4—1462no—
322C22⋊C4—2924no—
323C16—41no—
324C16—2921no—
325C23—2922no—
326C23—5841no—
327C23—5841no—
328C23—5841no—
329C23—5841no—
330C23—5841no—
331C23—5841no—
332C23—732no—
333C23—732no—
334C23—732no—
335C23—1461no—
336C23—1461no—
337C23—1461no—
338C23—1461no—
339C23D8×D7311yescharacteristic
340D4—2922no—
341D4—2922no—
342D4—5848no—
343D4—5841no—
344D4—5841no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—2922no—
352C2×C4—5848no—
353C2×C4—5841no—
354C2×C4—5841no—
355C2×C4—732no—
356C2×C4—1461no—
357C2×C4—1461no—
358C2×C4—1462no—
359C2×C4—1462no—
360C2×C4—1462no—
361C2×C4—1462no—
362C2×C4C22⋊C4×D7311yescharacteristic
363C2×C4#1455656111yescharacteristic
364C2×C4#1455657611yescharacteristic
365C2×C4SD16×D7311yescharacteristic
366C2×C4#1455661811yescharacteristic
367C2×C4#1455662311yescharacteristic
368C8—1462no—
369C8#1455655311yescharacteristic
370C8C4×D29211yescharacteristic
371C8#1455656411yescharacteristic
372C8#1455656611yescharacteristic
373C22—5842no—
374C22—5842no—
375C22—5842no—
376C22—734no—
377C22—11681no—
378C22—11681no—
379C22—11681no—
380C22—11681no—
381C22—1462no—
382C22—1462no—
383C22—1462no—
384C22—1462no—
385C22—1462no—
386C22—1462no—
387C22#102×D7311yescharacteristic
388C22#1798872211yescharacteristic
389C22#1798890211yescharacteristic
390C22D16×D7311yescharacteristic
391C22SD32×D7311yescharacteristic
392C22#1798902411yescharacteristic
393C22#1798902911yescharacteristic
394C4—82no—
395C4—82no—
396C4—5842no—
397C4—1462no—
398C4#1798846511yescharacteristic
399C4#1798858711yescharacteristic
400C4#1798872711yescharacteristic
401C4#1798890711yescharacteristic
402C2—11682no—
403C2—1464no—
404C2#2261024711yescharacteristic
405C2#357×D7311yescharacteristic
406C2#2261164211yescharacteristic
407C2#2261184411yescharacteristic
408C2#2261184911yescharacteristic
409C2#2261211311yescharacteristic
410C2#2261211811yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#1380C2
socleC23×C73C2×D8
prime radicalC2×#17988824C2
cosocleC22×C8×C73C23
hypercenter#1380D73
hypocenterC73#12991
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#107×D73C2semidirect
3. C2×#17988389C2semidirect
4. C2×#17988387C2semidirect
5. C2×#17988366C2semidirect
6. #22608022C2semidirect
7. C73×#1380C2semidirect
14. #17988824C4semidirect
35. C2×D584C2×C4semidirect
54. #14556525D4semidirect
65. #107×C73C2×C4semidirect
66. #107×C73D4semidirect
187. C2×D73#357semidirect
212. #1380D73semidirect
213. C73#12991semidirect
270. #107C4×D73semidirect
271. #107D292semidirect

Maximal subgroups

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

subgroupindex
C2×#179888242
C2×#107×D732
C2×#179883892
C2×#179883872
C2×#179883662
#226080222
C73×#13802
#1299173