Finite Group Databasebeta

← #33240472

Subgroups — #33240472

Order 18688.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989030C211yesmaximal, characteristic, complemented
2#22619512C211yesmaximal, characteristic
3C2×#17988828C211yesmaximal, characteristic, complemented
4#22612044C212yesmaximal, complemented
5#22612039C212yesmaximal, complemented
6#22612027C212yesmaximal, complemented
7#22612022C212yesmaximal, complemented
8#22611719C211yesmaximal, characteristic, complemented
9#22611652C211yesmaximal, characteristic, complemented
10#22611644C211yesmaximal, characteristic, complemented
11C22×M32×C73C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13C22×Dic292C2211yescharacteristic
14#17989098—21no—
15C2×#14556570—21no—
16#17989030—24no—
17C2×Dic584C2212yescomplemented
18C2×#14556536C2212yescomplemented
19#17988828—21no—
20#17988828—21no—
21#17988828C2211yescharacteristic
22#17988828C2212yes—
23#17988828C2211yescharacteristic
24#17988828C2212yes—
25#17988823C2211yescharacteristic
26#17988823C2211yescharacteristic
27#17988816—21no—
28#17988816—21no—
29#17988625C2211yescharacteristic
30C2×#14556525C2211yescharacteristic
31#17988584—21no—
32C2×#14556516—21no—
33#17988415C2212yescomplemented
34#17988413C2212yescomplemented
35#17988392C2212yescomplemented
36#17988392C2212yescomplemented
37#17988392C2212yescomplemented
38#17988389C2212yescomplemented
39C23×C8×C73C2211yescharacteristic
40C2×M32×C73—24no—
41C2×M32×C73C2212yes—
42C2×M32×C73C2211yescharacteristic
43C2×M32×C73C2211yescharacteristic
44C22×C16×C73C2212yes—
45C2×#11924028—21no—
46C22×Dic146—21no—
47#14556613—22no—
48#14556613—22no—
49C2×Dic292—22no—
50C2×Dic292—21no—
51C2×Dic292—22no—
52C2×Dic292C2312yes—
53C2×Dic292C2311yescharacteristic
54C2×#11924017—21no—
55C2×D584C2311yescharacteristic
56#14556574—41no—
57#14556570—21no—
58#14556570—21no—
59#14556570—22no—
60#14556570—22no—
61#14556570—41no—
62#14556570—41no—
63#14556565—21no—
64#14556565—21no—
65#14556560—41no—
66C4×C22⋊D73—41no—
67C4×C22⋊D73—41no—
68#14556550—21no—
69C2×C4⋊Dic73—21no—
70C2×#11924007—41no—
71C2×C22⋊Dic73—41no—
72Dic584—24no—
73#14556536—24no—
74#14556525—21no—
75#14556525C2312yes—
76#14556525C2311yescharacteristic
77#14556525C2311yescharacteristic
78#14556523—21no—
79#14556516—21no—
80#14556516—21no—
81#14556516—22no—
82#14556516—22no—
83#14556513—21no—
84#14556513—21no—
85C23×C4×C73D411yescharacteristic
86C22×C8×C73—22no—
87C22×C8×C73—22no—
88C22×C8×C73D411yescharacteristic
89C22×C8×C73D411yescharacteristic
90C22×C8×C73D411yescharacteristic
91C22×C8×C73C2311yescharacteristic
92C22×C8×C73C2311yescharacteristic
93C22×C8×C73C2311yescharacteristic
94M32×C73—22no—
95M32×C73—24no—
96M32×C73D414yescomplemented
97C2×C16×C73—24no—
98C2×C16×C73C2312yes—
99C2×C16×C73C2312yes—
100#11924028—42no—
101#11924028—42no—
102C2×Dic146—21no—
103C2×Dic146—22no—
104C2×Dic146—42no—
105C2×Dic146—42no—
106C2×Dic146—41no—
107C2×D292—21no—
108C2×C22⋊D73—41no—
109C2×C22⋊D73—41no—
110C2×C22⋊D73—41no—
111C22×Dic73—41no—
112C22×Dic73—41no—
113C2×C4×D73—41no—
114Dic292—22no—
115Dic292—21no—
116Dic292—22no—
117Dic292—21no—
118Dic292—42no—
119#11924017—42no—
120D584—21no—
121D584—21no—
122C4⋊Dic73—21no—
123C4⋊Dic73—21no—
124C4⋊Dic73—22no—
125C4⋊Dic73—41no—
126#11924007—42no—
127#11924007—42no—
128#11924007—41no—
129#11924007—41no—
130C22⋊Dic73—42no—
131C22⋊Dic73—42no—
132C22⋊Dic73—41no—
133C22⋊Dic73—41no—
134C22⋊Dic73—41no—
135#11924005—41no—
136#11924005—41no—
137C4×Dic73—41no—
138C24×C73D811yescharacteristic
139C22×C4×C73—22no—
140C22×C4×C73—22no—
141C22×C4×C73D811yescharacteristic
142C22×C4×C73D811yescharacteristic
143C22×C4×C73D811yescharacteristic
144C22×C4×C73C2×D411yescharacteristic
145C22×C4×C73C2×D411yescharacteristic
146C22×C4×C73C2×D411yescharacteristic
147C2×C8×C73—22no—
148C2×C8×C73—22no—
149C2×C8×C73—22no—
150C2×C8×C73—21no—
151C2×C8×C73—21no—
152C2×C8×C73—21no—
153C2×C8×C73—21no—
154C2×C8×C73C2×D412yes—
155C2×C8×C73C2×D411yescharacteristic
156C2×C8×C73C2×D411yescharacteristic
157C2×C8×C73C2×D411yescharacteristic
158C2×C8×C73#4112yes—
159C2×C8×C73C2411yescharacteristic
160C16×C73—22no—
161C16×C73C2×D414yes—
162C23×C73—22no—
163C23×C73—22no—
164C23×C73—41no—
165C23×C73C2×D811yescharacteristic
166C23×C73C2×D811yescharacteristic
167C23×C73C2×D811yescharacteristic
168C22×D73—41no—
169C2×C4×C73—22no—
170C2×C4×C73—21no—
171C2×C4×C73—21no—
172C2×C4×C73—21no—
173C2×C4×C73—21no—
174C2×C4×C73—22no—
175C2×C4×C73—22no—
176C2×C4×C73#12112yes—
177C2×C4×C73C2×D811yescharacteristic
178C2×C4×C73C2×D811yescharacteristic
179C2×C4×C73C2×D811yescharacteristic
180C2×C4×C73#13512yes—
181C2×C4×C73C22×D411yescharacteristic
182C22⋊D73—82no—
183C22⋊D73—82no—
184C22⋊D73—82no—
185C2×Dic73—41no—
186C2×Dic73—41no—
187C2×Dic73—42no—
188C2×Dic73—41no—
189C2×Dic73—42no—
190C2×Dic73—82no—
191C2×Dic73—82no—
192D292—41no—
193D292—41no—
194C4×D73—82no—
195Dic146—41no—
196Dic146—42no—
197Dic146—42no—
198Dic146—41no—
199Dic146—82no—
200C8×C73—21no—
201C8×C73—21no—
202C8×C73#12112yes—
203C8×C73C22×D411yescharacteristic
204C8×C73C2×#4111yescharacteristic
205C22×C73—21no—
206C22×C73—21no—
207C22×C73—21no—
208C22×C73—21no—
209C22×C73—21no—
210C22×C73—21no—
211C22×C73—21no—
212C22×C73—44no—
213C22×C73#46612yes—
214C22×C73#51012yescomplemented
215C22×C73C22×D811yescharacteristic
216C2×D73—81no—
217C2×D73—81no—
218C2×D73—81no—
219C4×C73—21no—
220C4×C73—21no—
221C4×C73#46612yes—
222C4×C73C2×#12111yescharacteristic
223C4×C73C2×#13511yescharacteristic
224Dic73—81no—
225Dic73—82no—
226Dic73—81no—
227Dic73—82no—
228Dic73—81no—
229#33606—731nomaximal, Sylow
230C2×C73—22no—
231C2×C73—22no—
232C2×C73—41no—
233C2×C73C2×#46611yescharacteristic
234C2×C73C2×#51011yescharacteristic
235C2×C73#341411yescharacteristic
236D73—161no—
237C2×#510—731no—
238C22×M32D7311yescharacteristic, complemented
239#3050—731no—
240C2×#467—731no—
241#2220—732no—
242#2219—732no—
243#2217—732no—
244#2216—732no—
245#2159—731no—
246#2141—731no—
247#2139—731no—
248C73#3360611yescharacteristic, complemented, Sylow
249C2×#135—731no—
250C22×Q16—731no—
251C23×C8C2×D7311yescharacteristic
252#525—1461no—
253C2×C22⋊Q8—1461no—
254#510—1464no—
255C2×Q32—732no—
256C2×SD32—732no—
257C2×M32—24no—
258C2×M32C2×D7311yescharacteristic
259C2×M32C2×D7312yes—
260C2×M32C2×D7311yescharacteristic
261C22×C16C2×D7312yes—
262#467—731no—
263#467—732no—
264#467—731no—
265#467—732no—
266#467—1461no—
267#467—1461no—
268#466—731no—
269#466—731no—
270#465—1461no—
271#465—1461no—
272#427—731no—
273C2×#107—731no—
274#416—1461no—
275C2×#103—1461no—
276#367—732no—
277#366—732no—
278#358—732no—
279#358—732no—
280#358—732no—
281#357—732no—
282C2×#41—1461no—
283C22×Q8—1461no—
284C23×C4D29211yescharacteristic
285#135—1462no—
286#135—1462no—
287C2×Q16—732no—
288C2×Q16—731no—
289C2×Q16—1462no—
290C2×Q16—1461no—
291C2×Q16—1462no—
292C2×SD16—1461no—
293C2×D8—731no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8D29211yescharacteristic
297C22×C8D29211yescharacteristic
298C22×C8D29211yescharacteristic
299C22×C8C22×D7311yescharacteristic
300C22×C8C22×D7311yescharacteristic
301C22×C8C22×D7311yescharacteristic
302#123—2921no—
303C22⋊Q8—2921no—
304C22⋊Q8—2921no—
305C22⋊Q8—1462no—
306C22⋊Q8—1462no—
307C22⋊Q8—1461no—
308C22⋊Q8—1461no—
309#121—1461no—
310#121—1461no—
311#120—2921no—
312C4×D4—2921no—
313C4×D4—2921no—
314#117—1461no—
315C2×C4⋊C4—2921no—
316C2×C4⋊C4—1461no—
317C2×C22⋊C4—2921no—
318Q32—1464no—
319SD32—1464no—
320M32—22no—
321M32—24no—
322M32C22⋊D7314yescomplemented
323C2×C16—24no—
324C2×C16C22×D7312yes—
325C2×C16C22×D7312yes—
326#107—732no—
327#107—731no—
328#107—731no—
329#107—1461no—
330#106—1461no—
331#103—1462no—
332#103—1462no—
333#103—1461no—
334#103—1461no—
335#102—1461no—
336#102—1461no—
337C24D58411yescharacteristic
338#41—2922no—
339#41—2922no—
340C2×Q8—2921no—
341C2×Q8—2922no—
342C2×Q8—2922no—
343C2×Q8—1461no—
344C2×Q8—1462no—
345C2×D4—2921no—
346C2×D4—2921no—
347C2×D4—2921no—
348C2×D4—1461no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—2921no—
352C22×C4—2921no—
353C22×C4—2921no—
354C22×C4D58411yescharacteristic
355C22×C4D58411yescharacteristic
356C22×C4D58411yescharacteristic
357C22×C4C2×D29211yescharacteristic
358C22×C4C2×D29211yescharacteristic
359C22×C4C2×D29211yescharacteristic
360Q16—2922no—
361Q16—1462no—
362Q16—1461no—
363Q16—1462no—
364Q16—1461no—
365SD16—2922no—
366D8—1461no—
367D8—1461no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8C2×C22⋊D7312yes—
376C2×C8C2×D29211yescharacteristic
377C2×C8C2×D29211yescharacteristic
378C2×C8C2×D29211yescharacteristic
379C2×C8#1192402812yes—
380C2×C8C23×D7311yescharacteristic
381C4⋊C4—2922no—
382C4⋊C4—2922no—
383C4⋊C4—2921no—
384C4⋊C4—2921no—
385C4⋊C4—2921no—
386C4⋊C4—1461no—
387C4⋊C4—1461no—
388C4⋊C4—1462no—
389C22⋊C4—2922no—
390C22⋊C4—2922no—
391C22⋊C4—2921no—
392C22⋊C4—2921no—
393C22⋊C4—2921no—
394C22⋊C4—2921no—
395C22⋊C4—2921no—
396C42—2921no—
397C16—22no—
398C16C2×C22⋊D7314yes—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—2921no—
403C23C2×D58411yescharacteristic
404C23C2×D58411yescharacteristic
405C23C2×D58411yescharacteristic
406Q8—2921no—
407Q8—2922no—
408Q8—2922no—
409Q8—2921no—
410Q8—5842no—
411D4—2921no—
412D4—2921no—
413D4—5842no—
414D4—5842no—
415D4—5842no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—2921no—
424C2×C4—2921no—
425C2×C4—2922no—
426C2×C4—2921no—
427C2×C4—2922no—
428C2×C4—5842no—
429C2×C4—5842no—
430C2×C4—5842no—
431C2×C4#1455656512yes—
432C2×C4C2×D58411yescharacteristic
433C2×C4C2×D58411yescharacteristic
434C2×C4C2×D58411yescharacteristic
435C2×C4#1455661312yes—
436C2×C4C22×D29211yescharacteristic
437C8—21no—
438C8—21no—
439C8#1455656512yes—
440C8C22×C22⋊D7311yescharacteristic
441C8C2×#1192402811yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—5841no—
451C22—5841no—
452C22—5841no—
453C22#1798882312yes—
454C22#1798903012yescomplemented
455C22C22×D58411yescharacteristic
456C4—21no—
457C4—21no—
458C4—5841no—
459C4—5842no—
460C4—5841no—
461C4—5841no—
462C4—5842no—
463C4#1798882312yes—
464C4C2×#1455656511yescharacteristic
465C4C2×#1455661311yescharacteristic
466C2—22no—
467C2—22no—
468C2—41no—
469C2—11681no—
470C2C2×#1798882311yescharacteristic
471C2C2×#1798903011yescharacteristic
472C2#2262306711yescharacteristic
subgroupG/H
centerC22C22×D584
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC22×M32×C73C2
socleC22×C73C22×D8
prime radical#22619512C2
hypercenterC22×M32D73
hypocenterC73#33606
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C22×M32

Complements

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

NH ≅ G/Ntype
1. C2×#17989030C2semidirect
3. C2×#17988828C2semidirect
4. #22612044C2semidirect
5. #22612039C2semidirect
6. #22612027C2semidirect
7. #22612022C2semidirect
8. #22611719C2semidirect
9. #22611652C2semidirect
10. #22611644C2semidirect
11. C22×M32×C73C2semidirect
17. C2×Dic584C22semidirect
18. C2×#14556536C22semidirect
33. #17988415C22semidirect
34. #17988413C22semidirect
35. #17988392C22semidirect
36. #17988392C22semidirect
37. #17988392C22semidirect
38. #17988389C22semidirect
96. M32×C73D4semidirect
214. C22×C73#510semidirect
238. C22×M32D73semidirect
248. C73#33606semidirect
322. M32C22⋊D73semidirect
454. C22#17989030semidirect

Maximal subgroups

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

subgroupindex
C2×#179890302
#226195122
C2×#179888282
#226120442
#226120392
#226120272
#226120222
#226117192
#226116522
#226116442
C22×M32×C732
#3360673