Finite Group Databasebeta

← #33241079

Subgroups — #33241079

Order 18688.

548 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620501C211yesmaximal, characteristic
2#22612133C211yesmaximal, characteristic, complemented
3#22612116C211yesmaximal, characteristic
4#22612095C212yesmaximal, complemented
5#22611993C212yesmaximal, complemented
6#22611865C211yesmaximal, characteristic, complemented
7#22611848C211yesmaximal, characteristic
8#22611800C212yesmaximal, complemented
9#22611793C212yesmaximal, complemented
10#22611734C211yesmaximal, characteristic, complemented
11C2×C4×Q16×C73C211yesmaximal, characteristic, complemented
12#17989121—21no—
13C2×#14556537C2212yes—
14C2×#14556534C2212yes—
15#17988989C2211yescharacteristic
16#17988971C2211yescharacteristic
17#17988942—21no—
18#17988828C2211yescharacteristic
19#17988823C2211yescharacteristic
20#17988816—21no—
21C4×Dic292C2211yescharacteristic
22C4×#11924017—21no—
23C4×D584C2211yescharacteristic
24#17988414C2212yes—
25#17988412C2212yes—
26C4⋊#11924003C2212yes—
27#17988391C2212yes—
28#17988391C2212yes—
29#17988390C2212yes—
30#17988387C2212yes—
31C22⋊#11924003C2211yescharacteristic
32C22⋊#11924003C2211yescharacteristic
33#17988363C2212yes—
34C22×Q16×C73C2211yescharacteristic
35C2×C4×Q8×C73—21no—
36C4×Q16×C73—21no—
37C4×Q16×C73—21no—
38C4×Q16×C73C2212yes—
39C4×Q16×C73C2212yes—
40C2×#107×C73C2211yescharacteristic
41C2×#103×C73—21no—
42C2×C4×C8×C73C2211yescharacteristic
43C2×Dic292C2311yescharacteristic
44C2×#11924017—21no—
45C2×D584C2311yescharacteristic
46#14556585—41no—
47#14556583—21no—
48#14556579—21no—
49#14556574—41no—
50#14556570—21no—
51#14556565—21no—
52C4×Dic146—21no—
53C4×D292—21no—
54C4×C22⋊D73—41no—
55#14556548—41no—
56#14556537—24no—
57#14556534—24no—
58C2×#11924003C2312yes—
59C2×#11924003C2312yes—
60#14556525C2311yescharacteristic
61#14556525C2311yescharacteristic
62#14556523—21no—
63#14556516—21no—
64#14556516—21no—
65#14556513—21no—
66#14556513—21no—
67C22×Q8×C73—21no—
68C2×Q16×C73—24no—
69C2×Q16×C73—21no—
70C2×Q16×C73C2312yes—
71C22×C8×C73D411yescharacteristic
72C22×C8×C73C2311yescharacteristic
73C4×Q8×C73—22no—
74C4×Q8×C73—22no—
75C4×Q8×C73—41no—
76C4×Q8×C73—41no—
77C2×C4⋊C4×C73—21no—
78C2×C4⋊C4×C73—41no—
79C2×C42×C73—41no—
80C2×C42×C73D411yescharacteristic
81#107×C73—21no—
82#107×C73C2312yes—
83#103×C73—22no—
84#103×C73—22no—
85C4×C8×C73D411yescharacteristic
86C4×C8×C73D411yescharacteristic
87C4×C8×C73C2311yescharacteristic
88C4×C8×C73C2311yescharacteristic
89C2×Dic146—21no—
90C2×D292—21no—
91C2×C22⋊D73—41no—
92C2×C4×D73—41no—
93Dic292—22no—
94#11924017—42no—
95D584—22no—
96C4⋊Dic73—21no—
97C4⋊Dic73—21no—
98#11924007—41no—
99#11924007—41no—
100#11924007—41no—
101#11924007—41no—
102C22⋊Dic73—41no—
103#11924005—41no—
104#11924005—41no—
105#11924005—41no—
106#11924005—41no—
107C4×Dic73—41no—
108#11924003—22no—
109#11924003—22no—
110C2×Q8×C73—24no—
111C2×Q8×C73—22no—
112C2×Q8×C73—42no—
113C2×Q8×C73—41no—
114C22×C4×C73—41no—
115C22×C4×C73—41no—
116C22×C4×C73D811yescharacteristic
117C22×C4×C73D811yescharacteristic
118C22×C4×C73C2×D411yescharacteristic
119Q16×C73—24no—
120Q16×C73—22no—
121Q16×C73#4114yes—
122C2×C8×C73—22no—
123C2×C8×C73—41no—
124C2×C8×C73C2×D411yescharacteristic
125C2×C8×C73C2×D411yescharacteristic
126C2×C8×C73C2×D411yescharacteristic
127C2×C8×C73C2411yescharacteristic
128C4⋊C4×C73—22no—
129C4⋊C4×C73—41no—
130C4⋊C4×C73—42no—
131C4⋊C4×C73—42no—
132C42×C73—42no—
133C42×C73—42no—
134C42×C73D811yescharacteristic
135C42×C73D811yescharacteristic
136C42×C73C2×D411yescharacteristic
137C42×C73C2×D411yescharacteristic
138C23×C73C2×D811yescharacteristic
139C22×D73—41no—
140Q8×C73—24no—
141Q8×C73—44no—
142Q8×C73—42no—
143C2×C4×C73—22no—
144C2×C4×C73—21no—
145C2×C4×C73—21no—
146C2×C4×C73—41no—
147C2×C4×C73—44no—
148C2×C4×C73—42no—
149C2×C4×C73—42no—
150C2×C4×C73—82no—
151C2×C4×C73C2×D811yescharacteristic
152C2×C4×C73C2×D811yescharacteristic
153C2×C4×C73C2×D811yescharacteristic
154C2×C4×C73C2×D811yescharacteristic
155C2×C4×C73C2×D811yescharacteristic
156C2×C4×C73C22×D411yescharacteristic
157C22⋊D73—82no—
158C2×Dic73—41no—
159C2×Dic73—41no—
160C2×Dic73—41no—
161D292—41no—
162D292—41no—
163C4×D73—82no—
164Dic146—41no—
165Dic146—41no—
166C8×C73—21no—
167C8×C73—21no—
168C8×C73—21no—
169C8×C73C2×#4112yes—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73#51012yes—
173C22×C73C22×D811yescharacteristic
174C2×D73—81no—
175C2×D73—81no—
176C2×D73—81no—
177C4×C73—21no—
178C4×C73—21no—
179C4×C73—21no—
180C4×C73—21no—
181C4×C73—21no—
182C4×C73—44no—
183C4×C73—82no—
184C4×C73#52511yescharacteristic
185C4×C73#57511yescharacteristic
186Dic73—81no—
187Dic73—81no—
188Dic73—81no—
189#33677—731nomaximal, Sylow
190C2×C73—22no—
191C2×C73#310011yescharacteristic
192C2×C73C2×#51011yescharacteristic
193C2×C73#341411yescharacteristic
194D73—161no—
195#3168—731no—
196C2×C4×Q16D7311yescharacteristic, complemented
197#2236—731no—
198#2233—731no—
199#2229—732no—
200#2211—732no—
201#2188—731no—
202#2185—731no—
203#2176—732no—
204#2175—732no—
205#2163—731no—
206C73#3367711yescharacteristic, complemented, Sylow
207C22×Q16C2×D7311yescharacteristic
208#529—1461no—
209C2×C4×Q8—21no—
210C2×Q32—732no—
211C2×SD32—732no—
212#499—731no—
213#495—731no—
214#488—1461no—
215#467—731no—
216#466—731no—
217#465—1461no—
218C4×Q16—21no—
219C4×Q16—21no—
220C4×Q16—731no—
221C4×Q16C2×D7312yes—
222C4×Q16C2×D7312yes—
223C4×SD16—1461no—
224C4×D8—731no—
225C2×#107C2×D7311yescharacteristic
226C2×#103—21no—
227C2×C4×C8C2×D7311yescharacteristic
228#367—732no—
229#366—732no—
230C4⋊C16—732no—
231#358—732no—
232#358—732no—
233#358—732no—
234#357—732no—
235C22⋊C16—731no—
236C22⋊C16—731no—
237#346—732no—
238C22×Q8—21no—
239C2×Q16—24no—
240C2×Q16—21no—
241C2×Q16—731no—
242C2×Q16C22×D7312yes—
243C2×SD16—1461no—
244C2×D8—731no—
245C22×C8C22⋊D7311yescharacteristic
246C22×C8C22×D7311yescharacteristic
247#126—2921no—
248#125—1461no—
249#124—1461no—
250#123—2921no—
251C22⋊Q8—1461no—
252#121—1461no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—41no—
256C4×Q8—41no—
257C4×Q8—1461no—
258C4×D4—2921no—
259C4×D4—1461no—
260#117—2921no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—41no—
263C2×C42—41no—
264C2×C42C22⋊D7311yescharacteristic
265Q32—1464no—
266SD32—1464no—
267C2×C16—732no—
268C2×C16—732no—
269#107—21no—
270#107—731no—
271#107—731no—
272#107C22×D7312yes—
273#106—1461no—
274#103—22no—
275#103—22no—
276#103—1461no—
277#103—1461no—
278#102—1461no—
279#102—1461no—
280C4×C8C22⋊D7311yescharacteristic
281C4×C8C22⋊D7311yescharacteristic
282C4×C8C22×D7311yescharacteristic
283C4×C8C22×D7311yescharacteristic
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—1461no—
289C2×D4—2921no—
290C2×D4—1461no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—2921no—
294C22×C4#1192401311yescharacteristic
295C22×C4#1192401311yescharacteristic
296C22×C4C2×C22⋊D7311yescharacteristic
297Q16—24no—
298Q16—22no—
299Q16—1462no—
300Q16#1192402814yes—
301SD16—2922no—
302D8—1462no—
303C2×C8—22no—
304C2×C8—41no—
305C2×C8C2×C22⋊D7311yescharacteristic
306C2×C8C2×C22⋊D7311yescharacteristic
307C2×C8C2×C22⋊D7311yescharacteristic
308C2×C8C23×D7311yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—2921no—
314C4⋊C4—2921no—
315C4⋊C4—2921no—
316C4⋊C4—2921no—
317C4⋊C4—1461no—
318C4⋊C4—1461no—
319C22⋊C4—2921no—
320C22⋊C4—2921no—
321C22⋊C4—2921no—
322C22⋊C4—2921no—
323C22⋊C4—2921no—
324C42—42no—
325C42—42no—
326C42—2921no—
327C42#1192401311yescharacteristic
328C42#1192401311yescharacteristic
329C42C2×C22⋊D7311yescharacteristic
330C42C2×C22⋊D7311yescharacteristic
331C16—1462no—
332C16—1462no—
333C23—2921no—
334C23C2×#1192401311yescharacteristic
335Q8—24no—
336Q8—44no—
337Q8—42no—
338Q8—2921no—
339Q8—2921no—
340D4—2921no—
341D4—2921no—
342D4—5842no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—2921no—
352C2×C4—2921no—
353C2×C4—2921no—
354C2×C4—5842no—
355C2×C4C2×#1192401311yescharacteristic
356C2×C4C2×#1192401311yescharacteristic
357C2×C4C2×#1192401311yescharacteristic
358C2×C4C2×#1192401311yescharacteristic
359C2×C4C2×#1192401311yescharacteristic
360C2×C4C22×C22⋊D7311yescharacteristic
361C8—21no—
362C8—21no—
363C8—21no—
364C8C2×#1192402812yes—
365C22—21no—
366C22—21no—
367C22—5841no—
368C22—5841no—
369C22—5841no—
370C22#1798902812yes—
371C22C22×#1192401311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—44no—
378C4—82no—
379C4—5841no—
380C4—5841no—
381C4—5841no—
382C4#1798909811yescharacteristic
383C4#1798938511yescharacteristic
384C2—22no—
385C2—11681no—
386C2#2261993111yescharacteristic
387C2C2×#1798902811yescharacteristic
388C2#2262306511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC2×C4×Q16×C73C2
socleC22×C73C22×D8
prime radical#17988823C22
hypercenterC2×C4×Q16D73
hypocenterC73#33677
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. #22612133C2semidirect
4. #22612095C2semidirect
5. #22611993C2semidirect
6. #22611865C2semidirect
8. #22611800C2semidirect
9. #22611793C2semidirect
10. #22611734C2semidirect
11. C2×C4×Q16×C73C2semidirect
196. C2×C4×Q16D73semidirect
206. C73#33677semidirect

Maximal subgroups

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

subgroupindex
#226205012
#226121332
#226121162
#226120952
#226119932
#226118652
#226118482
#226118002
#226117932
#226117342
C2×C4×Q16×C732
#3367773