Finite Group Databasebeta

← #34019428

Subgroups — #34019428

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626777C211yesmaximal, characteristic, complemented
2#22626768C211yesmaximal, characteristic, complemented
3#22625574C212yesmaximal, complemented
4#22625556C212yesmaximal, complemented
5#22625552C212yesmaximal, complemented
6SD16⋊#10596833C212yesmaximal, complemented
7#22625493C211yesmaximal, characteristic, complemented
8#22625262C211yesmaximal, characteristic, complemented
9#22625262C211yesmaximal, characteristic, complemented
10#22625140C211yesmaximal, characteristic, complemented
11#135×Dic73C211yesmaximal, characteristic
12#17989638C2212yes—
13Q8×#10596833C2212yes—
14#17989622—22no—
15#17989622C2212yes—
16#17989618—22no—
17D4×#10596833C2212yes—
18#17989608—21no—
19#17989608C2211yescharacteristic
20#17989608C2211yescharacteristic
21#17989607—21no—
22#17989604C2211yescharacteristic
23C4×#11924037—21no—
24C4×#11924037C2211yescharacteristic
25C2×#14556646—21no—
26#17989450C2212yes—
27#17989449C2212yes—
28Q8⋊#10596833C2212yes—
29Q8⋊#10596833C2212yes—
30D4⋊#10596833C2212yes—
31D4⋊#10596833C2212yes—
32#17989429C2212yes—
33#17989428C2212yes—
34#41×Dic73C2211yescharacteristic
35#41×Dic73C2211yescharacteristic
36Q16×Dic73C411yescharacteristic
37Q16×Dic73C2211yescharacteristic
38SD16×Dic73C411yescharacteristic
39SD16×Dic73C411yescharacteristic
40SD16×Dic73C2211yescharacteristic
41SD16×Dic73C2211yescharacteristic
42D8×Dic73C411yescharacteristic
43D8×Dic73C2211yescharacteristic
44#17988625C411yescharacteristic
45#17988583C411yescharacteristic
46#17988583C411yescharacteristic
47C2×C8×Dic73C2211yescharacteristic
48C2×#135×C73C411yescharacteristic
49C2×#11924037—22no—
50C2×#11924037D412yes—
51C22×#10596833—22no—
52#14556658—22no—
53#14556658—21no—
54C4⋊#10596833—22no—
55C4⋊#10596833C2312yes—
56C4⋊#10596833C2312yes—
57C22⋊#10596833—22no—
58C22⋊#10596833—22no—
59C22⋊#10596833—21no—
60C22⋊#10596833—21no—
61C22⋊#10596833—21no—
62#14556646—22no—
63#14556646—21no—
64#14556646—21no—
65#14556646D411yescharacteristic
66#14556646D411yescharacteristic
67C4×#10596833—22no—
68C4×#10596833C2312yes—
69C4×#10596833C2312yes—
70C22×#10596831C2×C411yescharacteristic
71Q8×Dic73C2×C411yescharacteristic
72Q8×Dic73C2×C411yescharacteristic
73Q8×Dic73C2311yescharacteristic
74Q8×Dic73C2311yescharacteristic
75D4×Dic73—21no—
76D4×Dic73—21no—
77D4×Dic73—21no—
78D4×Dic73—21no—
79D4×Dic73C2×C411yescharacteristic
80D4×Dic73C2×C411yescharacteristic
81D4×Dic73C2311yescharacteristic
82D4×Dic73C2311yescharacteristic
83#14556550—21no—
84#14556550—21no—
85#14556550C2×C411yescharacteristic
86#14556550C2×C411yescharacteristic
87C2×C4×Dic73—21no—
88C2×C4×Dic73—21no—
89C2×C4×Dic73C2311yescharacteristic
90#14556525C2×C411yescharacteristic
91#14556525C2×C411yescharacteristic
92#14556523C2×C411yescharacteristic
93#14556523C2×C411yescharacteristic
94#14556515C2×C411yescharacteristic
95#14556515C2×C411yescharacteristic
96#14556515C2×C411yescharacteristic
97#14556515C2×C411yescharacteristic
98#14556512C2×C411yescharacteristic
99#14556512C2×C411yescharacteristic
100#14556512C2×C411yescharacteristic
101#14556512C2×C411yescharacteristic
102C8×Dic73C2×C411yescharacteristic
103C8×Dic73C2×C411yescharacteristic
104C8×Dic73C2311yescharacteristic
105C8×Dic73C2311yescharacteristic
106C2×#41×C73C2×C411yescharacteristic
107C2×#41×C73C2×C411yescharacteristic
108#135×C73—24no—
109C2×Q16×C73C2×C411yescharacteristic
110C2×SD16×C73C2×C411yescharacteristic
111C2×SD16×C73C2×C411yescharacteristic
112C2×D8×C73C2×C411yescharacteristic
113C22×C8×C73C2×C411yescharacteristic
114#11924037—42no—
115#11924037—44no—
116C2×#10596833—22no—
117C2×#10596833—22no—
118C2×#10596833—22no—
119C2×#10596833—44no—
120C2×#10596833C2×D412yes—
121C2×#10596833C2×D412yes—
122C22×Dic73—21no—
123C22×Dic73—21no—
124C22×Dic73—21no—
125C22×Dic73—21no—
126C22×Dic73C2×D411yescharacteristic
127C22×Dic73#4111yescharacteristic
128C2×#10596831—41no—
129C2×#10596831C22×C411yescharacteristic
130C2×#10596831C22×C411yescharacteristic
131C4⋊Dic73—21no—
132C4⋊Dic73—21no—
133C4⋊Dic73—21no—
134C4⋊Dic73—21no—
135C4⋊Dic73C22×C411yescharacteristic
136C4⋊Dic73C22×C411yescharacteristic
137C4⋊Dic73C22×C411yescharacteristic
138C4⋊Dic73C22×C411yescharacteristic
139C22⋊Dic73—21no—
140C22⋊Dic73—21no—
141C22⋊Dic73—21no—
142C22⋊Dic73—21no—
143C22⋊Dic73—21no—
144C22⋊Dic73—21no—
145C4×Dic73—21no—
146C4×Dic73—21no—
147C4×Dic73—21no—
148C4×Dic73—21no—
149C4×Dic73—21no—
150C4×Dic73—21no—
151C4×Dic73C22×C411yescharacteristic
152C4×Dic73C2×D411yescharacteristic
153C4×Dic73#4111yescharacteristic
154C4×Dic73C2411yescharacteristic
155#41×C73—22no—
156#41×C73—22no—
157#41×C73—41no—
158#41×C73—41no—
159C2×Q8×C73C22×C411yescharacteristic
160C2×Q8×C73C22×C411yescharacteristic
161C2×D4×C73—21no—
162C2×D4×C73—21no—
163C2×D4×C73C22×C411yescharacteristic
164C2×D4×C73C22×C411yescharacteristic
165C22×C4×C73—21no—
166C22×C4×C73—21no—
167C22×C4×C73C22×C411yescharacteristic
168Q16×C73—22no—
169SD16×C73—22no—
170SD16×C73M1614yescomplemented
171D8×C73—22no—
172C2×C8×C73—22no—
173C2×C8×C73C22×C411yescharacteristic
174C2×C8×C73C22×C411yescharacteristic
175C23×C73—21no—
176C23×C73—21no—
177C23×C73C4×D411yescharacteristic
178Q8×C73—21no—
179Q8×C73—21no—
180Q8×C73—21no—
181Q8×C73C2×M1612yes—
182D4×C73—21no—
183D4×C73—21no—
184D4×C73—21no—
185D4×C73—42no—
186D4×C73—42no—
187D4×C73C2×M1612yes—
188C2×C4×C73—21no—
189C2×C4×C73—21no—
190C2×C4×C73—21no—
191C2×C4×C73—21no—
192C2×C4×C73—42no—
193C2×C4×C73—42no—
194C2×C4×C73C4×D411yescharacteristic
195C2×C4×C73C23×C411yescharacteristic
196C2×Dic73—22no—
197C2×Dic73—21no—
198C2×Dic73—21no—
199C2×Dic73—21no—
200C2×Dic73—21no—
201C2×Dic73—41no—
202C2×Dic73—42no—
203C2×Dic73—42no—
204C2×Dic73—81no—
205C2×Dic73—81no—
206C2×Dic73C4×D411yescharacteristic
207C2×Dic73C4×D411yescharacteristic
208C2×Dic73C22×D411yescharacteristic
209C2×Dic73C2×#4111yescharacteristic
210#10596833—22no—
211#10596833—22no—
212#10596833—42no—
213#10596833—42no—
214#10596833—42no—
215C8×C73—21no—
216C8×C73C2×M1612yes—
217#10596831—21no—
218#10596831—21no—
219C22×C73—21no—
220C22×C73—22no—
221C22×C73—21no—
222C22×C73—21no—
223C22×C73—41no—
224C22×C73—41no—
225C22×C73—41no—
226C22×C73—41no—
227C22×C73—41no—
228C22×C73C2×C4×D411yescharacteristic
229C4×C73—21no—
230C4×C73—22no—
231C4×C73—41no—
232C4×C73C22×M1611yescharacteristic
233C4×C73#56811yescharacteristic
234Dic73—21no—
235Dic73—21no—
236Dic73—21no—
237Dic73—41no—
238Dic73—41no—
239Dic73—41no—
240Dic73—41no—
241Dic73#57512yes—
242#18041—731nomaximal, Sylow
243C2×C73—41no—
244C2×C73—42no—
245C2×C73—81no—
246C2×C73D4×M1611yescharacteristic
247C2×C73#294311yescharacteristic
248C2×C73#295711yescharacteristic
249#2971—731no—
250#2970—731no—
251C4×#135—731no—
252#1592—732no—
253#1587—732no—
254#1586—732no—
255SD16⋊C8—732no—
256#1568—731no—
257#1488—731no—
258#1488—731no—
259#1451—731no—
260C73#1804111yescharacteristic, complemented, Sylow
261C2×#135#6440811yescharacteristic
262C4×#41—731no—
263C4×#41—731no—
264#446—732no—
265C8×Q8—732no—
266C4×Q16—731no—
267C4×Q16—731no—
268C4×SD16—731no—
269C4×SD16—731no—
270C4×SD16—731no—
271C4×SD16—731no—
272C4×D8—731no—
273C4×D8—731no—
274#436—732no—
275#436—1462no—
276#435—1462no—
277C8×D4—732no—
278#432—731no—
279#432—731no—
280#432—1461no—
281#432—1461no—
282#431—731no—
283#427—731no—
284#416—731no—
285#416—731no—
286C4×M16—731no—
287C4×M16—1461no—
288C2×#97—1461no—
289C2×C4×C8—731no—
290#335—732no—
291#334—732no—
292Q8⋊C8—732no—
293Q8⋊C8—732no—
294D4⋊C8—732no—
295D4⋊C8—732no—
296#322—732no—
297#322—732no—
298C2×#41C2×#6440811yescharacteristic
299C2×#41C2×#6440811yescharacteristic
300#135—24no—
301C2×Q16C2×#6440811yescharacteristic
302C2×SD16C2×#6440811yescharacteristic
303C2×SD16C2×#6440811yescharacteristic
304C2×D8C2×#6440811yescharacteristic
305C2×M16—732no—
306C2×M16—1462no—
307C22×C8—731no—
308C22×C8—1462no—
309C22×C8C2×#6440811yescharacteristic
310C4×Q8—731no—
311C4×Q8—731no—
312C4×Q8—731no—
313C4×Q8—731no—
314C4×D4—731no—
315C4×D4—731no—
316C4×D4—731no—
317C4×D4—731no—
318C4×D4—1461no—
319C4×D4—1461no—
320C4×D4—1461no—
321C4×D4—1461no—
322#117—731no—
323#117—731no—
324#117—1461no—
325#117—1461no—
326C2×C42—731no—
327C2×C42—1461no—
328C2×C42—1461no—
329#107—731no—
330#107—731no—
331#106—731no—
332#106—731no—
333C4⋊C8—732no—
334C4⋊C8—732no—
335C4⋊C8—1461no—
336C4⋊C8—1462no—
337C4⋊C8—1462no—
338#103—731no—
339#103—731no—
340#103—731no—
341#103—731no—
342#102—731no—
343#102—731no—
344#102—731no—
345#102—731no—
346C22⋊C8—1462no—
347C22⋊C8—1462no—
348C22⋊C8—1461no—
349C22⋊C8—1461no—
350C22⋊C8—1461no—
351#97—731no—
352#97—731no—
353#97—1461no—
354#97—1461no—
355#97—1462no—
356C4×C8—732no—
357C4×C8—731no—
358C4×C8—731no—
359C4×C8—731no—
360C4×C8—731no—
361C4×C8—732no—
362C4×C8—1462no—
363#41—22no—
364#41—22no—
365#41—41no—
366#41—41no—
367C2×Q8C22×#6440811yescharacteristic
368C2×Q8C22×#6440811yescharacteristic
369C2×D4—21no—
370C2×D4—21no—
371C2×D4C22×#6440811yescharacteristic
372C2×D4C22×#6440811yescharacteristic
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—731no—
376C22×C4—731no—
377C22×C4—1461no—
378C22×C4—1461no—
379C22×C4—1461no—
380C22×C4—1461no—
381C22×C4C22×#6440811yescharacteristic
382Q16—22no—
383SD16—22no—
384SD16#1192403714yescomplemented
385D8—22no—
386M16—2924no—
387M16—2922no—
388C2×C8—22no—
389C2×C8—2924no—
390C2×C8—2921no—
391C2×C8—732no—
392C2×C8—732no—
393C2×C8—731no—
394C2×C8—731no—
395C2×C8—1462no—
396C2×C8—1462no—
397C2×C8—1462no—
398C2×C8C22×#6440811yescharacteristic
399C2×C8C22×#6440811yescharacteristic
400C4⋊C4—731no—
401C4⋊C4—731no—
402C4⋊C4—731no—
403C4⋊C4—731no—
404C4⋊C4—1461no—
405C4⋊C4—1461no—
406C4⋊C4—1461no—
407C4⋊C4—1461no—
408C22⋊C4—1461no—
409C22⋊C4—1461no—
410C22⋊C4—1461no—
411C22⋊C4—1461no—
412C22⋊C4—1461no—
413C22⋊C4—1461no—
414C42—731no—
415C42—731no—
416C42—731no—
417C42—731no—
418C42—1461no—
419C42—1461no—
420C42—1461no—
421C42—1461no—
422C42—1461no—
423C42—1461no—
424C23—21no—
425C23—21no—
426C23D4×#6440811yescharacteristic
427Q8—21no—
428Q8—21no—
429Q8—21no—
430Q8C2×#1192403712yes—
431D4—21no—
432D4—21no—
433D4—21no—
434D4—42no—
435D4—42no—
436D4C2×#1192403712yes—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—2922no—
444C2×C4—2922no—
445C2×C4—2921no—
446C2×C4—5841no—
447C2×C4—5841no—
448C2×C4—731no—
449C2×C4—731no—
450C2×C4—731no—
451C2×C4—731no—
452C2×C4—1461no—
453C2×C4—1461no—
454C2×C4—1461no—
455C2×C4—1461no—
456C2×C4—1462no—
457C2×C4D4×#6440811yescharacteristic
458C2×C4C23×#6440811yescharacteristic
459C8—21no—
460C8—2922no—
461C8—2922no—
462C8—2922no—
463C8—1462no—
464C8—1462no—
465C8—1461no—
466C8—1461no—
467C8C2×#1192403712yes—
468C22—21no—
469C22—22no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22C2×D4×#6440811yescharacteristic
478C4—21no—
479C4—22no—
480C4—41no—
481C4—2921no—
482C4—2921no—
483C4—2921no—
484C4—2921no—
485C4—732no—
486C4—1461no—
487C4—1461no—
488C4—1461no—
489C4C22×#1192403711yescharacteristic
490C4#1798966511yescharacteristic
491C2—41no—
492C2—42no—
493C2—81no—
494C2D4×#1192403711yescharacteristic
495C2#2262668511yescharacteristic
496C2#2262670811yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting, prime radicalC2×#135×C73C4
socleC22×C73C2×C4×D4
cosocleC4×Dic73C24
hypercenterC2×#135#64408
hypocenterC73#18041
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #22626777C2semidirect
2. #22626768C2semidirect
3. #22625574C2semidirect
4. #22625556C2semidirect
5. #22625552C2semidirect
6. SD16⋊#10596833C2semidirect
7. #22625493C2semidirect
8. #22625262C2semidirect
9. #22625262C2semidirect
10. #22625140C2semidirect
170. SD16×C73M16semidirect
260. C73#18041semidirect
384. SD16#11924037semidirect

Maximal subgroups

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

subgroupindex
#226267772
#226267682
#226255742
#226255562
#226255522
SD16⋊#105968332
#226254932
#226252622
#226251402
#135×Dic732
#1804173