Finite Group Databasebeta

← #33069147

Subgroups — #33069147

Order 18688.

650 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621059C211yesmaximal, characteristic
2#22621052C211yesmaximal, characteristic
3#22618644C211yesmaximal, characteristic
4#22609503C214yesmaximal, complemented
5#22609379C214yesmaximal, complemented
6#22608679C211yesmaximal, characteristic, complemented
7#22608612C211yesmaximal, characteristic, complemented
8#22608603C211yesmaximal, characteristic, complemented
9C2×#487×C73C211yesmaximal, characteristic, complemented
10C2×#14556597—21no—
11#17989137C2211yescharacteristic
12#17988990—21no—
13#17988983—21no—
14#17988935—21no—
15#17988935—21no—
16#17988905—22no—
17#17988900—22no—
18#17988874C2214yes—
19#17988864C2214yes—
20#17988666—21no—
21#17988666C2212yes—
22C8×D292—21no—
23C8×D292C2212yes—
24#17988640C2211yescharacteristic
25#17988640C2211yescharacteristic
26#17988611C2211yescharacteristic
27#17988553—21no—
28#17988340C2212yes—
29#17988338C2212yes—
30#17988336C2214yes—
31#17988333C2214yes—
32C2×C4⋊Q8×C73C2211yescharacteristic
33C2×C4⋊Q8×C73C2211yescharacteristic
34#487×C73—21no—
35#487×C73—21no—
36#487×C73C2212yes—
37#487×C73C2212yes—
38C2×#103×C73—21no—
39C2×#103×C73—21no—
40C2×C4×C8×C73C2211yescharacteristic
41C2×#11924028—21no—
42#14556597—44no—
43C2×#11924012—21no—
44C2×#11924011—22no—
45C2×C8×D73—22no—
46#14556591D411yescharacteristic
47#14556588D411yescharacteristic
48#14556579—21no—
49#14556565—21no—
50#14556565D412yes—
51C4×D292—21no—
52C4×D292C2312yes—
53#14556524—24no—
54#14556522—24no—
55C4⋊#10596831—21no—
56C4⋊#10596831C2312yes—
57C4⋊#10596831C2312yes—
58C4⋊#10596831C2312yes—
59#14556511—24no—
60#14556511—24no—
61C22⋊#10596831—21no—
62C22⋊#10596831—21no—
63#14556503—22no—
64#14556503—22no—
65C4×#10596831—21no—
66C4×#10596831—21no—
67C22×Q8×C73—21no—
68C22×Q8×C73—21no—
69C22×C8×C73D411yescharacteristic
70C22×C8×C73D411yescharacteristic
71C4⋊Q8×C73—21no—
72C4⋊Q8×C73—21no—
73C4⋊Q8×C73—21no—
74C4⋊Q8×C73—21no—
75C4⋊Q8×C73—21no—
76C4⋊Q8×C73—21no—
77C4⋊Q8×C73C2312yes—
78C4⋊Q8×C73C2312yes—
79C2×C4⋊C4×C73—21no—
80C2×C4⋊C4×C73—21no—
81C2×C4⋊C4×C73—21no—
82C2×C4⋊C4×C73—21no—
83C2×C42×C73C2311yescharacteristic
84#103×C73—22no—
85#103×C73—22no—
86#103×C73—22no—
87#103×C73—22no—
88C4×C8×C73D411yescharacteristic
89C4×C8×C73D411yescharacteristic
90C4×C8×C73C2311yescharacteristic
91C4×C8×C73C2311yescharacteristic
92#11924028—42no—
93#11924028—42no—
94C2×Dic146—21no—
95C2×D292—21no—
96C2×D292—21no—
97C2×D292C2×D412yes—
98C2×C22⋊D73—22no—
99C2×C4×D73—22no—
100#11924012—42no—
101#11924011—44no—
102C2×#10596831—22no—
103C2×#10596831—22no—
104C2×#10596831—22no—
105C8×D73—44no—
106C4⋊Dic73—21no—
107C4⋊Dic73C2×D412yes—
108#11924005—22no—
109#11924005—22no—
110C2×Q8×C73—22no—
111C2×Q8×C73—22no—
112C2×Q8×C73—41no—
113C2×Q8×C73—41no—
114C2×Q8×C73—42no—
115C2×Q8×C73—42no—
116C2×Q8×C73—82no—
117C22×C4×C73—41no—
118C22×C4×C73—41no—
119C22×C4×C73C2×D411yescharacteristic
120C22×C4×C73C2×D411yescharacteristic
121C22×C4×C73#4111yescharacteristic
122C2×C8×C73—22no—
123C2×C8×C73—22no—
124C2×C8×C73C2×D411yescharacteristic
125C2×C8×C73C2×D411yescharacteristic
126C2×C8×C73C2×D411yescharacteristic
127C2×C8×C73C2×D411yescharacteristic
128C4⋊C4×C73—22no—
129C4⋊C4×C73—22no—
130C4⋊C4×C73—22no—
131C4⋊C4×C73—22no—
132C4⋊C4×C73—41no—
133C4⋊C4×C73—41no—
134C4⋊C4×C73—41no—
135C4⋊C4×C73—41no—
136C42×C73C2×D411yescharacteristic
137C42×C73C2×D411yescharacteristic
138C42×C73#4111yescharacteristic
139C42×C73C2411yescharacteristic
140C23×C73#12111yescharacteristic
141C22×D73—22no—
142Q8×C73—42no—
143Q8×C73—42no—
144Q8×C73—82no—
145Q8×C73—82no—
146Q8×C73—82no—
147C2×C4×C73—22no—
148C2×C4×C73—22no—
149C2×C4×C73—42no—
150C2×C4×C73—42no—
151C2×C4×C73—82no—
152C2×C4×C73—82no—
153C2×C4×C73#12111yescharacteristic
154C2×C4×C73#12111yescharacteristic
155C2×C4×C73#12111yescharacteristic
156C2×C4×C73#13512yes—
157C2×C4×C73AGL(1,ℤ/8)12yes—
158C2×C4×C73C22×D411yescharacteristic
159C2×C4×C73C22×D411yescharacteristic
160C2×C4×C73C2×#4111yescharacteristic
161C22⋊D73—44no—
162C2×Dic73—22no—
163D292—24no—
164D292—42no—
165D292—42no—
166C4×D73—42no—
167C4×D73—42no—
168Dic146—42no—
169C8×C73—21no—
170C8×C73—21no—
171C8×C73—21no—
172C8×C73—21no—
173#10596831—42no—
174#10596831—42no—
175#10596831—42no—
176C22×C73—21no—
177C22×C73—21no—
178C22×C73#46312yes—
179C22×C73C2×#12111yescharacteristic
180C2×D73—42no—
181C2×D73—42no—
182C2×D73—42no—
183C4×C73—21no—
184C4×C73—21no—
185C4×C73—82no—
186C4×C73—82no—
187C4×C73#46312yes—
188C4×C73#47112yes—
189C4×C73C2×#13511yescharacteristic
190C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
191C4×C73#57812yes—
192Dic73—42no—
193#19199—731nomaximal, Sylow
194C2×C73—22no—
195C2×C73C2×#46311yescharacteristic
196C2×C73#305911yescharacteristic
197C2×C73C2×#47111yescharacteristic
198D73—82no—
199#3227—731no—
200#3226—731no—
201C2×#487D7311yescharacteristic, complemented
202#2931—731no—
203#1703—734no—
204#1683—734no—
205#1564—731no—
206#1548—731no—
207#1546—731no—
208C73#1919911yescharacteristic, complemented, Sylow
209C2×#131—1461no—
210#532—731no—
211C2×C4⋊Q8C2×D7311yescharacteristic
212C2×C4⋊Q8C2×D7311yescharacteristic
213#500—1461no—
214#498—1461no—
215#487—21no—
216#487—21no—
217#487C2×D7312yes—
218#487C2×D7312yes—
219#486—1461no—
220#486—1461no—
221#481—1462no—
222#480—1462no—
223#476—734no—
224#474—734no—
225#436—732no—
226#436—1461no—
227C8×D4—732no—
228C8×D4—1461no—
229#431—731no—
230#431—731no—
231#423—731no—
232C2×#103—21no—
233C2×#103—21no—
234#408—1461no—
235C2×C4×C8C2×D7311yescharacteristic
236#335—732no—
237#334—732no—
238#333—734no—
239#332—734no—
240C2×#41—1461no—
241C22×Q8—21no—
242C22×Q8—21no—
243#131—2924no—
244C2×M16—1461no—
245C2×M16—1462no—
246C22×C8—1462no—
247C22×C8C22⋊D7311yescharacteristic
248C22×C8C22⋊D7311yescharacteristic
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—731no—
256C4⋊Q8C22×D7312yes—
257C4⋊Q8C22×D7312yes—
258#127—731no—
259#124—1461no—
260#121—732no—
261#121—1461no—
262C4×D4—732no—
263C4×D4—1461no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C42C22×D7311yescharacteristic
269#107—1464no—
270#106—1464no—
271C4⋊C8—732no—
272C4⋊C8—732no—
273C4⋊C8—732no—
274C4⋊C8—1461no—
275#103—22no—
276#103—22no—
277#103—22no—
278#103—22no—
279#102—1464no—
280#102—1464no—
281C22⋊C8—1461no—
282C22⋊C8—1461no—
283C22⋊C8—1462no—
284C22⋊C8—1462no—
285C4×C8—1461no—
286C4×C8—1461no—
287C4×C8C22⋊D7311yescharacteristic
288C4×C8C22⋊D7311yescharacteristic
289C4×C8C22×D7311yescharacteristic
290C4×C8C22×D7311yescharacteristic
291#41—2922no—
292#41—2922no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—82no—
300C2×Q8—1461no—
301C2×D4—732no—
302C2×D4—1462no—
303C2×D4—1461no—
304C2×D4—1461no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—1462no—
308C22×C4D4×D7311yescharacteristic
309C22×C4C2×C22⋊D7311yescharacteristic
310C22×C4#1192402711yescharacteristic
311M16—2924no—
312M16—2922no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—2924no—
316C2×C8—1462no—
317C2×C8—1462no—
318C2×C8—1462no—
319C2×C8C2×C22⋊D7311yescharacteristic
320C2×C8C2×C22⋊D7311yescharacteristic
321C2×C8C2×C22⋊D7311yescharacteristic
322C2×C8C2×C22⋊D7311yescharacteristic
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—732no—
332C4⋊C4—1461no—
333C22⋊C4—1462no—
334C22⋊C4—1462no—
335C42D4×D7311yescharacteristic
336C42C2×C22⋊D7311yescharacteristic
337C42#1192402711yescharacteristic
338C42C23×D7311yescharacteristic
339C23—1462no—
340C23#1455656211yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—82no—
344Q8—82no—
345Q8—82no—
346Q8—2922no—
347D4—2924no—
348D4—2922no—
349D4—2922no—
350D4—1464no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—2922no—
358C2×C4—2922no—
359C2×C4—1462no—
360C2×C4#1455656211yescharacteristic
361C2×C4#1455656211yescharacteristic
362C2×C4#1455656211yescharacteristic
363C2×C4#1455661112yes—
364C2×C4#1455661612yes—
365C2×C4C2×D4×D7311yescharacteristic
366C2×C4C22×C22⋊D7311yescharacteristic
367C2×C4C2×#1192402711yescharacteristic
368C8—21no—
369C8—21no—
370C8—21no—
371C8—21no—
372C8—2922no—
373C8—2922no—
374C8—2922no—
375C22—21no—
376C22—21no—
377C22—2922no—
378C22—2922no—
379C22—2922no—
380C22#1798879912yes—
381C22C2×#1455656211yescharacteristic
382C4—21no—
383C4—21no—
384C4—82no—
385C4—82no—
386C4—2922no—
387C4#1798879912yes—
388C4#1798884712yes—
389C4C2×#1455661111yescharacteristic
390C4C2×#1455661611yescharacteristic
391C4#1798940012yes—
392C2—22no—
393C2—5842no—
394C2C2×#1798879911yescharacteristic
395C2#2261958211yescharacteristic
396C2C2×#1798884711yescharacteristic
subgroupG/H
centerC22C2×#14556562
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×#487×C73C2
socleC22×C73C2×#121
prime radical#17989137C22
hypercenterC2×#487D73
hypocenterC73#19199
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
4. #22609503C2semidirect
5. #22609379C2semidirect
6. #22608679C2semidirect
7. #22608612C2semidirect
8. #22608603C2semidirect
9. C2×#487×C73C2semidirect
201. C2×#487D73semidirect
208. C73#19199semidirect

Maximal subgroups

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

subgroupindex
#226210592
#226210522
#226186442
#226095032
#226093792
#226086792
#226086122
#226086032
C2×#487×C732
#1919973