Finite Group Databasebeta

← #33240535

Subgroups — #33240535

Order 18688.

662 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989027C211yesmaximal, characteristic, complemented
2C2×#17989004C211yesmaximal, characteristic, complemented
3#22619586C211yesmaximal, characteristic, complemented
4#22612042C212yesmaximal, complemented
5#22612037C212yesmaximal, complemented
6#22612025C212yesmaximal, complemented
7#22612020C212yesmaximal, complemented
8#22611717C211yesmaximal, characteristic, complemented
9#22611638C211yesmaximal, characteristic, complemented
10C2×#17988390C211yesmaximal, characteristic, complemented
11C73×#3061C211yesmaximal, characteristic
12C22×Dic292C2211yescharacteristic
13#17989158—21no—
14#17989027—24no—
15C2×#14556537C2212yes—
16C2×#14556535C2212yes—
17#17989004—24no—
18C2×#14556531C2212yes—
19C2×#14556531C2211yescharacteristic
20C22×#11924003C2212yes—
21C22×#11924003C2211yescharacteristic
22#17988845—22no—
23#17988828C2212yes—
24#17988828C2212yes—
25#17988625C2211yescharacteristic
26#17988597—21no—
27#17988414C2212yes—
28#17988412C2212yes—
29#17988391C2211yescharacteristic
30#17988391C2211yescharacteristic
31#17988390C2212yes—
32#17988390C2212yes—
33#17988388C2211yescharacteristic
34#17988388C2211yescharacteristic
35C2×#135×C73C2211yescharacteristic
36C2×#131×C73C2211yescharacteristic
37#537×C73—21no—
38#468×C73—22no—
39#467×C73C2212yes—
40#466×C73C2212yes—
41C2×#107×C73C2211yescharacteristic
42#417×C73—21no—
43C22×Dic146—21no—
44C2×Dic292—22no—
45C2×Dic292—21no—
46C2×Dic292—22no—
47C2×Dic292C2312yes—
48#14556589—41no—
49#14556578—41no—
50#14556572—42no—
51#14556570—22no—
52#14556570—22no—
53#14556550—21no—
54#14556537—24no—
55#14556535—24no—
56#14556531—24no—
57#14556531D414yescomplemented
58C2×#11924003—24no—
59C2×#11924003—22no—
60C2×#11924003C2311yescharacteristic
61C2×#11924003C2311yescharacteristic
62C2×#11924003C2312yes—
63#14556525C2311yescharacteristic
64#14556525C2311yescharacteristic
65#14556523—21no—
66#14556516—22no—
67#14556516—22no—
68C2×#41×C73—21no—
69C2×#41×C73D411yescharacteristic
70#135×C73—22no—
71#135×C73—22no—
72C2×Q16×C73C2311yescharacteristic
73C2×SD16×C73—21no—
74C2×D8×C73C2311yescharacteristic
75#131×C73—22no—
76#131×C73—22no—
77C2×M16×C73D411yescharacteristic
78C2×M16×C73D412yes—
79C22×C8×C73C2312yes—
80C22×C8×C73C2311yescharacteristic
81C22⋊Q8×C73—22no—
82C22⋊Q8×C73—41no—
83#121×C73—22no—
84#121×C73—41no—
85#121×C73—42no—
86C2×C4⋊C4×C73—21no—
87#107×C73—21no—
88#107×C73C2312yes—
89#103×C73—22no—
90#102×C73—22no—
91C2×Dic146—22no—
92C2×Dic146—42no—
93C2×Dic146—42no—
94C2×Dic146—41no—
95C22×Dic73—41no—
96Dic292—22no—
97Dic292—22no—
98Dic292—42no—
99C4⋊Dic73—21no—
100C4⋊Dic73—21no—
101#11924007—42no—
102#11924007—42no—
103C22⋊Dic73—41no—
104C22⋊Dic73—42no—
105C22⋊Dic73—42no—
106C4×Dic73—41no—
107#11924003—21no—
108#11924003—21no—
109#11924003C2×D414yes—
110#41×C73—22no—
111#41×C73—22no—
112#41×C73—42no—
113#41×C73—42no—
114C2×Q8×C73—21no—
115C2×Q8×C73D811yescharacteristic
116C2×D4×C73—21no—
117C2×D4×C73—41no—
118C2×D4×C73—42no—
119C2×D4×C73D812yes—
120C2×D4×C73D811yescharacteristic
121C22×C4×C73—41no—
122C22×C4×C73—41no—
123C22×C4×C73C2×D412yes—
124C22×C4×C73C2×D411yescharacteristic
125Q16×C73—21no—
126Q16×C73—21no—
127SD16×C73—42no—
128D8×C73—21no—
129D8×C73—21no—
130M16×C73—22no—
131M16×C73—24no—
132C2×C8×C73—22no—
133C2×C8×C73—22no—
134C2×C8×C73C2×D412yes—
135C2×C8×C73C2×D412yes—
136C2×C8×C73C2×D411yescharacteristic
137C2×C8×C73#4112yes—
138C2×C8×C73C2411yescharacteristic
139C4⋊C4×C73—22no—
140C4⋊C4×C73—41no—
141C4⋊C4×C73—42no—
142C22⋊C4×C73—42no—
143C22⋊C4×C73—42no—
144C23×C73—41no—
145C23×C73C2×D812yes—
146C23×C73C2×D811yescharacteristic
147Q8×C73—22no—
148Q8×C73—41no—
149Q8×C73—41no—
150D4×C73—24no—
151D4×C73—22no—
152D4×C73—41no—
153D4×C73—41no—
154D4×C73—82no—
155D4×C73—162no—
156C2×C4×C73—22no—
157C2×C4×C73—22no—
158C2×C4×C73—42no—
159C2×C4×C73—41no—
160C2×C4×C73—82no—
161C2×C4×C73—161no—
162C2×C4×C73#12112yes—
163C2×C4×C73C2×D812yes—
164C2×C4×C73C2×D811yescharacteristic
165C2×C4×C73#13512yes—
166C2×C4×C73C22×D411yescharacteristic
167C2×Dic73—41no—
168C2×Dic73—41no—
169C2×Dic73—42no—
170C2×Dic73—82no—
171Dic146—42no—
172Dic146—42no—
173Dic146—82no—
174C8×C73—22no—
175C8×C73#12112yes—
176C8×C73C22×D411yescharacteristic
177C8×C73C2×#4111yescharacteristic
178C22×C73—21no—
179C22×C73—21no—
180C22×C73—22no—
181C22×C73—22no—
182C22×C73—22no—
183C22×C73—81no—
184C22×C73—81no—
185C22×C73—81no—
186C22×C73#46612yes—
187C22×C73C22×D811yescharacteristic
188C4×C73—22no—
189C4×C73—82no—
190C4×C73—81no—
191C4×C73#46612yes—
192C4×C73C2×#12111yescharacteristic
193C4×C73C2×#13511yescharacteristic
194Dic73—81no—
195Dic73—81no—
196Dic73—82no—
197#33613—731nomaximal, Sylow
198C2×C73—22no—
199C2×C73—42no—
200C2×C73—161no—
201C2×C73C2×#46611yescharacteristic
202C2×C73#341611yescharacteristic
203C2×C73#341711yescharacteristic
204C2×#510—731no—
205C2×#504—731no—
206#3061D7311yescharacteristic
207#3059—731no—
208#2220—732no—
209#2219—732no—
210#2217—732no—
211#2216—732no—
212#2159—731no—
213#2138—731no—
214C2×#358—731no—
215C73#3361311yescharacteristic, complemented, Sylow
216C2×#135C2×D7311yescharacteristic
217C22×Q16—731no—
218C2×#131C2×D7311yescharacteristic
219#537—21no—
220#536—1461no—
221#510—1464no—
222C2×Q32—732no—
223C2×SD32—732no—
224#504—1464no—
225C2×M32—731no—
226C2×M32—732no—
227C22×C16—731no—
228C22×C16—732no—
229#470—1462no—
230#468—22no—
231#467—732no—
232#467—732no—
233#467C2×D7312yes—
234#466C2×D7312yes—
235#427—731no—
236C2×#107C2×D7311yescharacteristic
237#419—1461no—
238#417—21no—
239#367—732no—
240#366—732no—
241#358—731no—
242#358—731no—
243#358—732no—
244#358—732no—
245#357—731no—
246#357—731no—
247C2×#41—21no—
248C2×#41C22⋊D7311yescharacteristic
249C22×Q8—1461no—
250#135—22no—
251#135—22no—
252C2×Q16—732no—
253C2×Q16—1462no—
254C2×Q16—1461no—
255C2×Q16—1462no—
256C2×Q16C22×D7311yescharacteristic
257C2×SD16—21no—
258C2×D8C22×D7311yescharacteristic
259#131—22no—
260#131—22no—
261C2×M16C22⋊D7311yescharacteristic
262C2×M16C22⋊D7312yes—
263C22×C8C22×D7312yes—
264C22×C8C22×D7311yescharacteristic
265C4⋊Q8—2921no—
266#124—2921no—
267#123—2922no—
268C22⋊Q8—22no—
269C22⋊Q8—41no—
270C22⋊Q8—1462no—
271C22⋊Q8—1462no—
272#121—22no—
273#121—41no—
274#121—42no—
275#117—1461no—
276C2×C4⋊C4—21no—
277Q32—1464no—
278SD32—1464no—
279M32—734no—
280M32—1464no—
281C2×C16—732no—
282C2×C16—731no—
283C2×C16—731no—
284C2×C16—1464no—
285C2×C16—1462no—
286#107—21no—
287#107—731no—
288#107—731no—
289#107C22×D7312yes—
290#106—1461no—
291#103—22no—
292#103—1462no—
293#103—1462no—
294#102—22no—
295#41—22no—
296#41—22no—
297#41—42no—
298#41—42no—
299C2×Q8—21no—
300C2×Q8—2921no—
301C2×Q8—2922no—
302C2×Q8—2922no—
303C2×Q8—1462no—
304C2×Q8#1192401311yescharacteristic
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4#1192401312yes—
309C2×D4#1192401311yescharacteristic
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—2921no—
313C22×C4C2×C22⋊D7312yes—
314C22×C4C2×C22⋊D7311yescharacteristic
315Q16—21no—
316Q16—21no—
317Q16—2922no—
318Q16—1462no—
319Q16—1462no—
320SD16—42no—
321D8—21no—
322D8—21no—
323M16—22no—
324M16—24no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8D4×D7312yes—
328C2×C8C2×C22⋊D7312yes—
329C2×C8C2×C22⋊D7311yescharacteristic
330C2×C8#1192402712yes—
331C2×C8C23×D7311yescharacteristic
332C4⋊C4—22no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—2922no—
336C4⋊C4—2922no—
337C4⋊C4—1461no—
338C4⋊C4—1461no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—2921no—
342C22⋊C4—2922no—
343C22⋊C4—2922no—
344C42—2921no—
345C16—734no—
346C16—1461no—
347C16—1461no—
348C23—41no—
349C23C2×#1192401312yes—
350C23C2×#1192401311yescharacteristic
351Q8—22no—
352Q8—41no—
353Q8—41no—
354Q8—2922no—
355Q8—2922no—
356Q8—5842no—
357D4—24no—
358D4—22no—
359D4—41no—
360D4—41no—
361D4—82no—
362D4—162no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—82no—
368C2×C4—161no—
369C2×C4—2921no—
370C2×C4—2921no—
371C2×C4—2922no—
372C2×C4—5842no—
373C2×C4#1455656312yes—
374C2×C4C2×#1192401312yes—
375C2×C4C2×#1192401311yescharacteristic
376C2×C4#1455661112yes—
377C2×C4C22×C22⋊D7311yescharacteristic
378C8—22no—
379C8#1455656312yes—
380C8C2×D4×D7311yescharacteristic
381C8C2×#1192402711yescharacteristic
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22#1798882112yes—
391C22C22×#1192401311yescharacteristic
392C4—22no—
393C4—82no—
394C4—81no—
395C4—5841no—
396C4—5842no—
397C4—5841no—
398C4#1798882112yes—
399C4C2×#1455656311yescharacteristic
400C4C2×#1455661111yescharacteristic
401C2—22no—
402C2—42no—
403C2—161no—
404C2C2×#1798882111yescharacteristic
405C2#2262307811yescharacteristic
406C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
Fitting, prime radicalC73×#3061C2
socleC22×C73C22×D8
hypercenter#3061D73
hypocenterC73#33613
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3061

Complements

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

NH ≅ G/Ntype
1. C2×#17989027C2semidirect
2. C2×#17989004C2semidirect
3. #22619586C2semidirect
4. #22612042C2semidirect
5. #22612037C2semidirect
6. #22612025C2semidirect
7. #22612020C2semidirect
8. #22611717C2semidirect
9. #22611638C2semidirect
10. C2×#17988390C2semidirect
57. #14556531D4semidirect
215. C73#33613semidirect

Maximal subgroups

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

subgroupindex
C2×#179890272
C2×#179890042
#226195862
#226120422
#226120372
#226120252
#226120202
#226117172
#226116382
C2×#179883902
C73×#30612
#3361373