Finite Group Databasebeta

← #32981572

Subgroups — #32981572

Order 18688.

674 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988644C211yesmaximal, characteristic, complemented
2#22618262C211yesmaximal, characteristic
3C2×#17988320C212yesmaximal
4#22607810C212yesmaximal, complemented
5C73×#1434C211yesmaximal, characteristic, complemented
6C2×#14556574—21no—
7C2×#14556565C2211yescharacteristic
8#17988644—24no—
9C2×C4⋊#10596831—21no—
10C2×C22⋊#10596831C2212yes—
11C2×#14556502—21no—
12#17988489C411yescharacteristic, complemented
13#17988489C411yescharacteristic, complemented
14#17988485C411yescharacteristic, complemented
15#17988485C411yescharacteristic, complemented
16#17988478—21no—
17#17988478—21no—
18#17988466—21no—
19#17988447C2211yescharacteristic
20#17988320—24no—
21C22×C42×C73C2211yescharacteristic
22C2×#95×C73—21no—
23C2×#95×C73—21no—
24C2×#95×C73C2212yes—
25C22×D292C2×C411yescharacteristic
26C22×C22⋊D73—21no—
27C22×#10596831—22no—
28#14556574—44no—
29#14556565—22no—
30#14556565—22no—
31C2×C4⋊Dic73C2×C411yescharacteristic
32C2×C4⋊Dic73C2×C411yescharacteristic
33C2×C4⋊Dic73C2×C411yescharacteristic
34C2×#11924007—21no—
35C2×#11924007—21no—
36C2×C22⋊Dic73—21no—
37C2×#11924005—21no—
38C2×#11924005—21no—
39C2×#11924005—21no—
40C2×#11924005—21no—
41C2×#11924005—21no—
42C4⋊#10596831—24no—
43C22⋊#10596831—22no—
44C22⋊#10596831D414yes—
45#14556502—24no—
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49C23×C4×C73—22no—
50C23×C4×C73D411yescharacteristic
51C23×C4×C73D411yescharacteristic
52C23×C4×C73C2311yescharacteristic
53C2×C42×C73—22no—
54C2×C42×C73—22no—
55C2×C42×C73C2×C411yescharacteristic
56C2×C42×C73C2×C411yescharacteristic
57C2×C42×C73D411yescharacteristic
58C2×C42×C73D411yescharacteristic
59#95×C73—24no—
60#95×C73—22no—
61#95×C73—22no—
62#95×C73—42no—
63C23×D73—21no—
64C2×D292—22no—
65C2×D292—41no—
66C2×D292—41no—
67C2×C22⋊D73—42no—
68C2×C22⋊D73—42no—
69C2×C22⋊D73—41no—
70C2×C22⋊D73—41no—
71C22×Dic73—21no—
72C22×Dic73—21no—
73C22×Dic73—21no—
74C2×#10596831—24no—
75C2×#10596831—42no—
76C2×#10596831—42no—
77C4⋊Dic73—22no—
78C4⋊Dic73—41no—
79C4⋊Dic73—41no—
80#11924007—42no—
81#11924007—42no—
82C22⋊Dic73—42no—
83#11924005—42no—
84#11924005—42no—
85#11924005—42no—
86#11924005—42no—
87#11924005—41no—
88#11924005—41no—
89C24×C73C2×D411yescharacteristic
90C22×C4×C73—22no—
91C22×C4×C73—22no—
92C22×C4×C73—21no—
93C22×C4×C73—22no—
94C22×C4×C73—21no—
95C22×C4×C73—22no—
96C22×C4×C73—24no—
97C22×C4×C73—41no—
98C22×C4×C73—41no—
99C22×C4×C73—41no—
100C22×C4×C73—42no—
101C22×C4×C73—42no—
102C22×C4×C73—42no—
103C22×C4×C73—42no—
104C22×C4×C73—42no—
105C22×C4×C73—42no—
106C22×C4×C73C22⋊C411yescharacteristic
107C22×C4×C73C22⋊C411yescharacteristic
108C22×C4×C73C22⋊C411yescharacteristic
109C22×C4×C73C22⋊C411yescharacteristic
110C22×C4×C73C22×C411yescharacteristic
111C22×C4×C73C2×D412yes—
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73#4112yes—
114C42×C73—22no—
115C42×C73—22no—
116C42×C73—21no—
117C42×C73—21no—
118C42×C73—21no—
119C42×C73—21no—
120C23×C73—21no—
121C23×C73—21no—
122C23×C73—21no—
123C23×C73—21no—
124C23×C73C2×C22⋊C411yescharacteristic
125C23×C73#12112yes—
126C23×C73AGL(1,ℤ/8)12yes—
127C23×C73AGL(1,ℤ/8)11yescharacteristic
128C23×C73#13711yescharacteristic
129C22×D73—41no—
130C22×D73—41no—
131C22×D73—41no—
132C22×D73—41no—
133C22×D73—41no—
134C22×D73—41no—
135C22×D73—41no—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—21no—
143C2×C4×C73—21no—
144C2×C4×C73—21no—
145C2×C4×C73—21no—
146C2×C4×C73—21no—
147C2×C4×C73—21no—
148C2×C4×C73—21no—
149C2×C4×C73—21no—
150C2×C4×C73—21no—
151C2×C4×C73—42no—
152C2×C4×C73—42no—
153C2×C4×C73—44no—
154C2×C4×C73—42no—
155C2×C4×C73—42no—
156C2×C4×C73—44no—
157C2×C4×C73—44no—
158C2×C4×C73—44no—
159C2×C4×C73—82no—
160C2×C4×C73—82no—
161C2×C4×C73—82no—
162C2×C4×C73—82no—
163C2×C4×C73#10412yes—
164C2×C4×C73C4×D412yes—
165C2×C4×C73#12011yescharacteristic
166C2×C4×C73#12311yescharacteristic
167C22⋊D73—82no—
168C22⋊D73—82no—
169C2×Dic73—41no—
170C2×Dic73—41no—
171C2×Dic73—41no—
172C2×Dic73—41no—
173C2×Dic73—41no—
174C2×Dic73—41no—
175C2×Dic73—41no—
176C2×Dic73—41no—
177C2×Dic73—41no—
178D292—82no—
179#10596831—44no—
180C22×C73—22no—
181C22×C73—22no—
182C22×C73—22no—
183C22×C73—21no—
184C22×C73—21no—
185C22×C73—21no—
186C22×C73—21no—
187C22×C73—22no—
188C22×C73#38611yescharacteristic
189C22×C73#41011yescharacteristic
190C22×C73#41011yescharacteristic
191C22×C73#41711yescharacteristic
192C22×C73#41811yescharacteristic
193C22×C73C2×#10411yescharacteristic
194C22×C73#42111yescharacteristic
195C22×C73#46812yes—
196C22×C73#46912yes—
197C2×D73—81no—
198C2×D73—81no—
199C2×D73—81no—
200C2×D73—81no—
201C2×D73—81no—
202C2×D73—81no—
203C2×D73—81no—
204C4×C73—22no—
205C4×C73—21no—
206C4×C73—22no—
207C4×C73—21no—
208C4×C73—41no—
209C4×C73—41no—
210C4×C73—41no—
211C4×C73—84no—
212Dic73—81no—
213Dic73—81no—
214Dic73—81no—
215#9118—731nomaximal, Sylow
216C2×C73—22no—
217C2×C73—22no—
218C2×C73#150511yescharacteristic
219C2×C73#150911yescharacteristic
220C2×C73#151211yescharacteristic
221C2×C73#151411yescharacteristic
222C2×C73#187111yescharacteristic
223C2×C73#189511yescharacteristic
224C2×C73#189911yescharacteristic
225D73—161no—
226C2×#432—731no—
227#2854—731no—
228C2×#327—732no—
229#1434D7311yescharacteristic, complemented
230#1294—732no—
231C73#911811yescharacteristic, complemented, Sylow
232C2×#123—1461no—
233C2×#121—731no—
234C22×C42C2×D7311yescharacteristic
235#432—1464no—
236C2×C4⋊C8—1461no—
237C2×C22⋊C8—732no—
238C2×#97—1461no—
239#390—731no—
240#390—731no—
241#389—731no—
242#389—731no—
243#388—1461no—
244#388—1461no—
245#386—1461no—
246#380—731no—
247C2×#95—21no—
248C2×#95—21no—
249C2×#95C2×D7312yes—
250#327—1464no—
251C22×D4—731no—
252C22×D4—1461no—
253C23×C4—22no—
254C23×C4C22⋊D7311yescharacteristic
255C23×C4C22⋊D7311yescharacteristic
256C23×C4C22×D7311yescharacteristic
257C22×C8—1462no—
258#123—2924no—
259#121—1462no—
260#121—1462no—
261C2×C4⋊C4—731no—
262C2×C4⋊C4—731no—
263C2×C4⋊C4—731no—
264C2×C4⋊C4—1461no—
265C2×C4⋊C4—1461no—
266C2×C22⋊C4—1461no—
267C2×C22⋊C4—1461no—
268C2×C22⋊C4—1461no—
269C2×C22⋊C4—1461no—
270C2×C22⋊C4—1461no—
271C2×C22⋊C4—1461no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C4×D7311yescharacteristic
275C2×C42C4×D7311yescharacteristic
276C2×C42D29211yescharacteristic
277C2×C42D29211yescharacteristic
278C4⋊C8—1464no—
279C22⋊C8—734no—
280C22⋊C8—1462no—
281#97—1464no—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—42no—
286#95—1461no—
287#95—1461no—
288#95—1461no—
289C24—1461no—
290C24C2×C22⋊D7311yescharacteristic
291C2×D4—2921no—
292C2×D4—2921no—
293C2×D4—2922no—
294C2×D4—2922no—
295C2×D4—2921no—
296C2×D4—2921no—
297C2×D4—1462no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—1461no—
315C22×C4—1461no—
316C22×C4—1461no—
317C22×C4#1192400511yescharacteristic
318C22×C4#1192400511yescharacteristic
319C22×C4#1192400511yescharacteristic
320C22×C4#1192400511yescharacteristic
321C22×C4C2×C4×D7311yescharacteristic
322C22×C4D4×D7312yes—
323C22×C4C2×D29211yescharacteristic
324C22×C4#1192402712yes—
325C2×C8—2922no—
326C2×C8—2922no—
327C2×C8—1464no—
328C4⋊C4—2921no—
329C4⋊C4—2921no—
330C4⋊C4—2922no—
331C4⋊C4—2922no—
332C4⋊C4—1462no—
333C22⋊C4—2922no—
334C22⋊C4—2922no—
335C22⋊C4—2922no—
336C22⋊C4—2922no—
337C22⋊C4—2922no—
338C22⋊C4—2921no—
339C22⋊C4—2921no—
340C42—22no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—2921no—
351C23—2921no—
352C23—2921no—
353C23—2921no—
354C23—2921no—
355C23—2921no—
356C23—2921no—
357C23C2×#1192400511yescharacteristic
358C23#1455656312yes—
359C23#1455661612yes—
360C23#1455661711yescharacteristic
361C23#1455662211yescharacteristic
362D4—5842no—
363D4—5842no—
364D4—5842no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—2921no—
393C2×C4—2921no—
394C2×C4—2921no—
395C2×C4—2921no—
396C2×C4—2921no—
397C2×C4—2921no—
398C2×C4—2921no—
399C2×C4—2921no—
400C2×C4—2921no—
401C2×C4#1455651912yes—
402C2×C4#1455655112yes—
403C2×C4#1455656111yescharacteristic
404C2×C4#1455657511yescharacteristic
405C8—2924no—
406C22—22no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—5841no—
415C22—5841no—
416C22—5841no—
417C22—5841no—
418C22—5841no—
419C22—5841no—
420C22—5841no—
421C22#1798846411yescharacteristic
422C22#1798856011yescharacteristic
423C22#1798856011yescharacteristic
424C22#1798858611yescharacteristic
425C22#1798859111yescharacteristic
426C22C2×#1455651911yescharacteristic
427C22#1798860511yescharacteristic
428C22#1798883512yes—
429C22#1798883812yes—
430C4—22no—
431C4—21no—
432C4—22no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—84no—
438C4—5841no—
439C4—5841no—
440C4—5841no—
441C2—22no—
442C2—22no—
443C2—11681no—
444C2#2260843811yescharacteristic
445C2#2260845411yescharacteristic
446C2#2260846711yescharacteristic
447C2#2260847711yescharacteristic
448C2#2261023111yescharacteristic
449C2#2261036711yescharacteristic
450C2#2261038711yescharacteristic
subgroupG/H
centerC23C2×#11924005
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#1434C2
socleC23×C73C2×C22⋊C4
prime radicalC2×#14556565C22
cosocleC23×C4×C73C23
hypercenter#1434D73
hypocenterC73#9118
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
1. C2×#17988644C2semidirect
4. #22607810C2semidirect
5. C73×#1434C2semidirect
12. #17988489C4semidirect
13. #17988489C4semidirect
14. #17988485C4semidirect
15. #17988485C4semidirect
229. #1434D73semidirect
231. C73#9118semidirect

Maximal subgroups

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

subgroupindex
C2×#179886442
#226182622
C2×#179883202
#226078102
C73×#14342
#911873