Finite Group Databasebeta

← #33098291

Subgroups — #33098291

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619719C211yesmaximal, characteristic
2#22618865C211yesmaximal, characteristic, complemented
3#22610875C211yesmaximal, characteristic, complemented
4#22610706C211yesmaximal, characteristic, complemented
5#22610676C212yesmaximal, complemented
6#22610638C212yesmaximal
7#22610576C212yesmaximal
8#22610527C212yesmaximal, complemented
9#22609951C211yesmaximal, characteristic
10C8×C4⋊Dic73C211yesmaximal, characteristic, complemented
11C73×#2922C211yesmaximal, characteristic
12#17989144C2211yescharacteristic
13#17988894C412yescomplemented
14#17988887C412yescomplemented
15#17988882C411yescharacteristic, complemented
16#17988877C411yescharacteristic, complemented
17#17988872C411yescharacteristic, complemented
18#17988867C411yescharacteristic, complemented
19#17988717—21no—
20#17988717—21no—
21C8×Dic146—21no—
22C8×Dic146—21no—
23C2×#14556526C2212yes—
24#17988623C2212yes—
25#17988613C2212yes—
26#17988611C2211yescharacteristic
27C2×C4⋊#10596831C2211yescharacteristic
28C2×#14556518C2212yes—
29#17988583C2212yes—
30#17988542C2212yes—
31C2×C8×Dic73C2212yes—
32C4×C4⋊Dic73C2211yescharacteristic
33#17988351C2212yes—
34#17988348C2212yes—
35#17988348C2212yes—
36#17988346C2212yes—
37#17988341C2212yes—
38C4×#41×C73C2211yescharacteristic
39#433×C73—21no—
40#433×C73—21no—
41#431×C73—21no—
42#431×C73—21no—
43#424×C73C2211yescharacteristic
44#424×C73C2211yescharacteristic
45#408×C73C2211yescharacteristic
46#408×C73C2211yescharacteristic
47C4×M16×C73C2211yescharacteristic
48C2×C4×C8×C73C2211yescharacteristic
49C2×#11924012C2312yes—
50C22×#10596831C2312yes—
51#14556591C2×C411yescharacteristic
52#14556591C2×C411yescharacteristic
53#14556583C2×C412yes—
54#14556570—21no—
55#14556570—21no—
56C4×Dic146—21no—
57C4×Dic146—21no—
58#14556550C2312yes—
59C2×C4⋊Dic73—21no—
60C2×C4×Dic73C2312yes—
61#14556526—42no—
62#14556524C2×C412yes—
63#14556524C2×C412yes—
64#14556522C2×C412yes—
65#14556522C2×C412yes—
66C4⋊#10596831—21no—
67C4⋊#10596831—21no—
68C4⋊#10596831C2×C411yescharacteristic
69C4⋊#10596831C2×C411yescharacteristic
70C4⋊#10596831C2×C412yes—
71#14556520—22no—
72#14556520—22no—
73#14556518—42no—
74#14556515C2×C412yes—
75#14556515C2×C412yes—
76#14556512C2×C412yes—
77#14556512C2×C412yes—
78#14556501—22no—
79C8×Dic73—22no—
80C8×Dic73—42no—
81#14556498—21no—
82C2×#41×C73C2311yescharacteristic
83C2×M16×C73C2311yescharacteristic
84C2×M16×C73C2311yescharacteristic
85C22×C8×C73C2311yescharacteristic
86C22×C8×C73C2311yescharacteristic
87C4×Q8×C73C2×C411yescharacteristic
88C4×Q8×C73C2×C411yescharacteristic
89C4×D4×C73—21no—
90C4×D4×C73—21no—
91C4×D4×C73C2×C411yescharacteristic
92C4×D4×C73C2×C411yescharacteristic
93#117×C73—21no—
94#117×C73C2311yescharacteristic
95C2×C42×C73—21no—
96C2×C42×C73C2311yescharacteristic
97C4⋊C8×C73—21no—
98C4⋊C8×C73—21no—
99C4⋊C8×C73—21no—
100C4⋊C8×C73—21no—
101C22⋊C8×C73—21no—
102C22⋊C8×C73—21no—
103C22⋊C8×C73—21no—
104C22⋊C8×C73—21no—
105#97×C73—21no—
106C4×C8×C73—21no—
107C4×C8×C73D411yescharacteristic
108C4×C8×C73D411yescharacteristic
109C4×C8×C73D411yescharacteristic
110C4×C8×C73D411yescharacteristic
111C2×Dic146—21no—
112C2×Dic146—21no—
113C22×Dic73—22no—
114#11924012—22no—
115#11924012—22no—
116C2×#10596831—22no—
117C2×#10596831—22no—
118C2×#10596831—22no—
119C2×#10596831C22×C412yes—
120C2×#10596831C22×C412yes—
121C4⋊Dic73—24no—
122C4⋊Dic73—21no—
123C4⋊Dic73—21no—
124C4⋊Dic73C22×C412yes—
125C4⋊Dic73C22×C412yes—
126#11924007—22no—
127#11924007—22no—
128C22⋊Dic73—22no—
129C4×Dic73—22no—
130C4×Dic73—22no—
131C4×Dic73—22no—
132#41×C73—21no—
133#41×C73—21no—
134#41×C73—21no—
135#41×C73—21no—
136C2×Q8×C73C22×C411yescharacteristic
137C2×D4×C73—21no—
138C2×D4×C73C22×C411yescharacteristic
139C22×C4×C73—21no—
140C22×C4×C73—21no—
141C22×C4×C73—21no—
142C22×C4×C73#4111yescharacteristic
143C22×C4×C73#4111yescharacteristic
144C22×C4×C73C2411yescharacteristic
145M16×C73—21no—
146M16×C73—21no—
147M16×C73—21no—
148M16×C73—21no—
149C2×C8×C73—21no—
150C2×C8×C73—21no—
151C2×C8×C73—21no—
152C2×C8×C73—21no—
153C2×C8×C73—21no—
154C2×C8×C73—21no—
155C2×C8×C73C2×D411yescharacteristic
156C2×C8×C73C2×D411yescharacteristic
157C2×C8×C73C2×D411yescharacteristic
158C2×C8×C73C2×D411yescharacteristic
159C4⋊C4×C73—21no—
160C4⋊C4×C73—21no—
161C4⋊C4×C73C22×C411yescharacteristic
162C4⋊C4×C73C22×C411yescharacteristic
163C22⋊C4×C73—21no—
164C22⋊C4×C73—21no—
165C22⋊C4×C73—21no—
166C42×C73—21no—
167C42×C73—21no—
168C42×C73—21no—
169C42×C73C22×C411yescharacteristic
170C42×C73C22×C411yescharacteristic
171C42×C73C2×D411yescharacteristic
172C42×C73C2×D411yescharacteristic
173C23×C73—21no—
174C23×C73C2×#4111yescharacteristic
175Q8×C73—21no—
176Q8×C73—21no—
177D4×C73—21no—
178D4×C73—21no—
179D4×C73—41no—
180D4×C73—41no—
181C2×C4×C73—22no—
182C2×C4×C73—21no—
183C2×C4×C73—21no—
184C2×C4×C73—21no—
185C2×C4×C73—41no—
186C2×C4×C73—42no—
187C2×C4×C73—41no—
188C2×C4×C73—41no—
189C2×C4×C73—42no—
190C2×C4×C73C4×D411yescharacteristic
191C2×C4×C73C4×D411yescharacteristic
192C2×C4×C73C4×D411yescharacteristic
193C2×C4×C73C4×D411yescharacteristic
194C2×C4×C73C23×C411yescharacteristic
195C2×C4×C73C22×D411yescharacteristic
196C2×C4×C73H(5,𝔽2)11yescharacteristic
197C2×C4×C732−1+411yescharacteristic
198C2×C4×C732−1+411yescharacteristic
199C2×C4×C732−1+411yescharacteristic
200C2×Dic73—24no—
201C2×Dic73—22no—
202C2×Dic73—22no—
203C2×Dic73—42no—
204C2×Dic73—42no—
205Dic146—42no—
206Dic146—42no—
207C8×C73—21no—
208C8×C73—21no—
209C8×C73—21no—
210C8×C73—21no—
211C8×C73—41no—
212C8×C73—41no—
213#10596831—22no—
214#10596831—22no—
215#10596831—42no—
216C22×C73—21no—
217C22×C73—21no—
218C22×C73—41no—
219C22×C73—41no—
220C22×C73—41no—
221C22×C73C2×C4×D411yescharacteristic
222C22×C73#54111yescharacteristic
223C22×C73#54211yescharacteristic
224C4×C73—21no—
225C4×C73—21no—
226C4×C73—21no—
227C4×C73—41no—
228C4×C73—41no—
229C4×C73—41no—
230C4×C73#44312yes—
231C4×C73#51911yescharacteristic
232C4×C73#52011yescharacteristic
233C4×C73#53611yescharacteristic
234C4×C73#53711yescharacteristic
235Dic73—44no—
236Dic73—42no—
237Dic73—42no—
238#22130—731nomaximal, Sylow
239C2×C73—21no—
240C2×C73—21no—
241C2×C73—81no—
242C2×C73#232311yescharacteristic
243C2×C73C2×#44311yescharacteristic
244C2×C73#295511yescharacteristic
245#3076—731no—
246#2960—731no—
247#2922D7311yescharacteristic
248#1986—731no—
249#1950—731no—
250#1945—732no—
251#1937—732no—
252#1929—732no—
253#1922—732no—
254#1806—731no—
255C8×C4⋊C4—731no—
256C73#2213011yescharacteristic, complemented, Sylow
257#533—731no—
258C4×#41C2×D7311yescharacteristic
259#479—732no—
260#478—732no—
261#477—731no—
262#476—731no—
263#475—731no—
264#474—731no—
265#446—1461no—
266#446—1461no—
267C8×Q8—1461no—
268C8×Q8—1461no—
269#433—21no—
270#433—21no—
271#431—21no—
272#431—21no—
273C2×#108—732no—
274#427—732no—
275#424—732no—
276#424C2×D7311yescharacteristic
277#424C2×D7311yescharacteristic
278#423—731no—
279C2×C4⋊C8—731no—
280C2×#104—732no—
281#416—732no—
282#408C2×D7311yescharacteristic
283#408C2×D7311yescharacteristic
284#405—732no—
285C4×M16C2×D7311yescharacteristic
286C2×C4×C8—732no—
287C2×C4×C8C2×D7311yescharacteristic
288C4×C4⋊C4—731no—
289#341—732no—
290#339—732no—
291#339—732no—
292#339—732no—
293#336—732no—
294C2×#41C22×D7311yescharacteristic
295C2×M16—732no—
296C2×M16C22×D7311yescharacteristic
297C2×M16C22×D7311yescharacteristic
298C22×C8—732no—
299C22×C8C22×D7311yescharacteristic
300C22×C8C22×D7311yescharacteristic
301C4⋊Q8—731no—
302C4⋊Q8—731no—
303#125—732no—
304C22⋊Q8—1461no—
305C22⋊Q8—1461no—
306C4×Q8—1461no—
307C4×Q8—1461no—
308C4×Q8C4×D7311yescharacteristic
309C4×Q8C4×D7311yescharacteristic
310C4×D4—21no—
311C4×D4—21no—
312C4×D4C4×D7311yescharacteristic
313C4×D4C4×D7311yescharacteristic
314#117—21no—
315#117—732no—
316#117C22×D7311yescharacteristic
317C2×C4⋊C4—1461no—
318C2×C42—21no—
319C2×C42—732no—
320C2×C42C22×D7311yescharacteristic
321#108—2922no—
322#107—732no—
323#107—732no—
324#106—732no—
325#106—732no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—731no—
331C4⋊C8—731no—
332C4⋊C8—732no—
333C4⋊C8—1462no—
334C4⋊C8—1462no—
335C4⋊C8—1461no—
336C4⋊C8—1461no—
337#104—2922no—
338#103—732no—
339#103—732no—
340#102—732no—
341#102—732no—
342C22⋊C8—21no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346#97—21no—
347#97—1462no—
348C4×C8—21no—
349C4×C8—2922no—
350C4×C8—1462no—
351C4×C8C22⋊D7311yescharacteristic
352C4×C8C22⋊D7311yescharacteristic
353C4×C8C22⋊D7311yescharacteristic
354C4×C8C22⋊D7311yescharacteristic
355#95—1461no—
356#41—21no—
357#41—21no—
358#41—21no—
359#41—21no—
360C2×Q8—1461no—
361C2×Q8—1461no—
362C2×Q8C2×C4×D7311yescharacteristic
363C2×D4—21no—
364C2×D4C2×C4×D7311yescharacteristic
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—1462no—
369C22×C4#1192402811yescharacteristic
370C22×C4#1192402811yescharacteristic
371C22×C4C23×D7311yescharacteristic
372M16—21no—
373M16—21no—
374M16—21no—
375M16—21no—
376M16—1462no—
377M16—1462no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—732no—
385C2×C8—732no—
386C2×C8—1462no—
387C2×C8—1462no—
388C2×C8—1462no—
389C2×C8C2×C22⋊D7311yescharacteristic
390C2×C8C2×C22⋊D7311yescharacteristic
391C2×C8C2×C22⋊D7311yescharacteristic
392C2×C8C2×C22⋊D7311yescharacteristic
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—732no—
396C4⋊C4—732no—
397C4⋊C4—1464no—
398C4⋊C4—1462no—
399C4⋊C4—1462no—
400C4⋊C4—1461no—
401C4⋊C4—1461no—
402C4⋊C4C2×C4×D7311yescharacteristic
403C4⋊C4C2×C4×D7311yescharacteristic
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—1462no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—1462no—
412C42—1462no—
413C42—1462no—
414C42C2×C4×D7311yescharacteristic
415C42C2×C4×D7311yescharacteristic
416C42C2×C22⋊D7311yescharacteristic
417C42C2×C22⋊D7311yescharacteristic
418C23—21no—
419C23C2×#1192402811yescharacteristic
420Q8—21no—
421Q8—21no—
422Q8—2922no—
423Q8—2922no—
424D4—21no—
425D4—21no—
426D4—41no—
427D4—41no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—2922no—
438C2×C4—2922no—
439C2×C4—1464no—
440C2×C4—1462no—
441C2×C4—1462no—
442C2×C4C4×C22⋊D7311yescharacteristic
443C2×C4C4×C22⋊D7311yescharacteristic
444C2×C4C4×C22⋊D7311yescharacteristic
445C2×C4C4×C22⋊D7311yescharacteristic
446C2×C4C22×C4×D7311yescharacteristic
447C2×C4C22×C22⋊D7311yescharacteristic
448C2×C4#1455663711yescharacteristic
449C2×C4#1455663811yescharacteristic
450C2×C4#1455663811yescharacteristic
451C2×C4#1455663811yescharacteristic
452C8—21no—
453C8—21no—
454C8—21no—
455C8—21no—
456C8—41no—
457C8—41no—
458C8—2922no—
459C8—1462no—
460C8—1462no—
461C22—21no—
462C22—21no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22C2×C4×C22⋊D7311yescharacteristic
467C22#1798919911yescharacteristic
468C22#1798920811yescharacteristic
469C4—21no—
470C4—21no—
471C4—21no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—2924no—
476C4—2922no—
477C4—2922no—
478C4#1798870212yes—
479C4#1798906411yescharacteristic
480C4#1798906711yescharacteristic
481C4#1798915811yescharacteristic
482C4#1798916511yescharacteristic
483C2—21no—
484C2—21no—
485C2—81no—
486C2#2261261311yescharacteristic
487C2C2×#1798870211yescharacteristic
488C2#2261883211yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#2922C2
socleC22×C73C2×C4×D4
prime radicalC2×#41×C73C23
cosocleC22×C4×C73C24
hypercenter#2922D73
hypocenterC73#22130
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2922

Complements

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

NH ≅ G/Ntype
2. #22618865C2semidirect
3. #22610875C2semidirect
4. #22610706C2semidirect
5. #22610676C2semidirect
8. #22610527C2semidirect
10. C8×C4⋊Dic73C2semidirect
13. #17988894C4semidirect
14. #17988887C4semidirect
15. #17988882C4semidirect
16. #17988877C4semidirect
17. #17988872C4semidirect
18. #17988867C4semidirect
256. C73#22130semidirect

Maximal subgroups

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

subgroupindex
#226197192
#226188652
#226108752
#226107062
#226106762
#226106382
#226105762
#226105272
#226099512
C8×C4⋊Dic732
C73×#29222
#2213073