Finite Group Databasebeta

← #32984971

Subgroups — #32984971

Order 18688.

668 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618232C211yesmaximal, characteristic
2C2×#17988325C212yesmaximal, complemented
3C2×#17988322C212yesmaximal
4#22608158C211yesmaximal, characteristic, complemented
5C4×C22⋊C8×C73C211yesmaximal, characteristic, complemented
6C2×#14556579C411yescharacteristic, complemented
7C2×#14556570C2211yescharacteristic
8C2×#14556565C2211yescharacteristic
9#17988493C411yescharacteristic, complemented
10#17988489C411yescharacteristic, complemented
11#17988478—21no—
12#17988478—21no—
13#17988466—21no—
14#17988466—21no—
15#17988453C411yescharacteristic, complemented
16C2×#14556498—21no—
17C2×#14556498—21no—
18C2×#14556498C2212yes—
19#17988325—24no—
20#17988322—24no—
21C22×C42×C73C2211yescharacteristic
22C2×C22⋊C8×C73C2212yes—
23C2×C4×C8×C73—21no—
24#336×C73—21no—
25C22×Dic146C2×C411yescharacteristic
26C22×D292C2×C411yescharacteristic
27C22×C22⋊D73—21no—
28C23×Dic73—22no—
29#14556579—42no—
30#14556570—22no—
31#14556570—22no—
32#14556565—22no—
33#14556565—22no—
34C2×C4⋊Dic73C2×C411yescharacteristic
35C2×C4⋊Dic73C2×C411yescharacteristic
36C2×#11924007—21no—
37C2×C22⋊Dic73—21no—
38C2×#11924005—21no—
39C2×#11924005—21no—
40C2×#11924005—21no—
41C2×#11924005—21no—
42C2×#11924005—21no—
43#14556498—24no—
44#14556498—21no—
45#14556498—21no—
46#14556498—21no—
47#14556498—21no—
48#14556498—22no—
49#14556498—22no—
50#14556498—42no—
51C23×C4×C73D411yescharacteristic
52C23×C4×C73D411yescharacteristic
53C23×C4×C73C2311yescharacteristic
54C22×C8×C73—22no—
55C2×C42×C73—42no—
56C2×C42×C73C2×C411yescharacteristic
57C2×C42×C73C2×C411yescharacteristic
58C2×C42×C73D411yescharacteristic
59C2×C42×C73D411yescharacteristic
60C22⋊C8×C73—22no—
61C22⋊C8×C73D414yes—
62C4×C8×C73—42no—
63C23×D73—21no—
64C2×Dic146—22no—
65C2×Dic146—41no—
66C2×Dic146—41no—
67C2×D292—22no—
68C2×D292—41no—
69C2×D292—41no—
70C2×C22⋊D73—41no—
71C2×C22⋊D73—41no—
72C2×C22⋊D73—42no—
73C2×C22⋊D73—42no—
74C22×Dic73—24no—
75C22×Dic73—21no—
76C22×Dic73—21no—
77C22×Dic73—21no—
78C22×Dic73—42no—
79C22×Dic73—42no—
80C22×Dic73—42no—
81C22×Dic73—42no—
82C22×Dic73—42no—
83C22×Dic73—42no—
84C4⋊Dic73—22no—
85C4⋊Dic73—22no—
86#11924007—42no—
87C22⋊Dic73—41no—
88C22⋊Dic73—41no—
89#11924005—42no—
90#11924005—42no—
91#11924005—42no—
92#11924005—42no—
93#11924005—42no—
94C24×C73C2×D411yescharacteristic
95C22×C4×C73—22no—
96C22×C4×C73—22no—
97C22×C4×C73—22no—
98C22×C4×C73—22no—
99C22×C4×C73—41no—
100C22×C4×C73—41no—
101C22×C4×C73—41no—
102C22×C4×C73—41no—
103C22×C4×C73C22⋊C411yescharacteristic
104C22×C4×C73C22⋊C411yescharacteristic
105C22×C4×C73C22⋊C411yescharacteristic
106C22×C4×C73C22⋊C411yescharacteristic
107C22×C4×C73C22×C411yescharacteristic
108C22×C4×C73C2×D411yescharacteristic
109C22×C4×C73C2×D412yes—
110C22×C4×C73#4112yes—
111C2×C8×C73—24no—
112C2×C8×C73—42no—
113C2×C8×C73—42no—
114C42×C73—22no—
115C42×C73—22no—
116C42×C73—41no—
117C42×C73—41no—
118C23×C73—21no—
119C23×C73—21no—
120C23×C73—21no—
121C23×C73—21no—
122C23×C73C2×C22⋊C411yescharacteristic
123C23×C73#12411yescharacteristic
124C23×C73#12711yescharacteristic
125C23×C73#13512yes—
126C23×C73#13512yes—
127C22×D73—41no—
128C22×D73—41no—
129C22×D73—41no—
130C22×D73—41no—
131C22×D73—41no—
132C22×D73—41no—
133C22×D73—41no—
134C2×C4×C73—21no—
135C2×C4×C73—21no—
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—42no—
149C2×C4×C73—42no—
150C2×C4×C73—44no—
151C2×C4×C73—44no—
152C2×C4×C73#9912yes—
153C2×C4×C73#10412yes—
154C2×C4×C73C4×D412yes—
155C2×C4×C73#12112yes—
156C22⋊D73—82no—
157C22⋊D73—82no—
158C2×Dic73—44no—
159C2×Dic73—44no—
160C2×Dic73—44no—
161C2×Dic73—41no—
162C2×Dic73—41no—
163C2×Dic73—41no—
164C2×Dic73—41no—
165C2×Dic73—41no—
166C2×Dic73—41no—
167C2×Dic73—41no—
168C2×Dic73—41no—
169C2×Dic73—41no—
170C2×Dic73—82no—
171C2×Dic73—82no—
172C2×Dic73—82no—
173C2×Dic73—82no—
174D292—82no—
175Dic146—82no—
176C8×C73—44no—
177C22×C73—21no—
178C22×C73—21no—
179C22×C73—21no—
180C22×C73—21no—
181C22×C73—22no—
182C22×C73—22no—
183C22×C73—22no—
184C22×C73—22no—
185C22×C73#39011yescharacteristic
186C22×C73C2×#9911yescharacteristic
187C22×C73#41011yescharacteristic
188C22×C73#41611yescharacteristic
189C22×C73#41611yescharacteristic
190C22×C73C2×#10411yescharacteristic
191C22×C73#42111yescharacteristic
192C22×C73#48812yes—
193C22×C73#49512yes—
194C2×D73—81no—
195C2×D73—81no—
196C2×D73—81no—
197C2×D73—81no—
198C2×D73—81no—
199C2×D73—81no—
200C2×D73—81no—
201C4×C73—22no—
202C4×C73—22no—
203C4×C73—22no—
204C4×C73—22no—
205C4×C73—41no—
206C4×C73—41no—
207Dic73—84no—
208Dic73—81no—
209Dic73—81no—
210Dic73—81no—
211#9681—731nomaximal, Sylow
212C2×C73—22no—
213C2×C73—22no—
214C2×C73#150011yescharacteristic
215C2×C73#150111yescharacteristic
216C2×C73#151611yescharacteristic
217C2×C73#152111yescharacteristic
218C2×C73#195811yescharacteristic
219C2×C73#196411yescharacteristic
220C2×C73#198011yescharacteristic
221D73—161no—
222#2850—731no—
223C4×C22⋊C8D7311yescharacteristic, complemented
224C2×#328—732no—
225C2×#327—732no—
226#1434—731no—
227C73#968111yescharacteristic, complemented, Sylow
228C2×#124—731no—
229C2×C22⋊Q8—731no—
230C2×#121—731no—
231C22×C42C2×D7311yescharacteristic
232C2×C22⋊C8C2×D7312yes—
233C2×C4×C8—21no—
234#391—731no—
235#390—731no—
236#388—1461no—
237#388—1461no—
238#386—1461no—
239#386—1461no—
240#382—731no—
241C2×#95—732no—
242C2×#95—1461no—
243C2×#95—1461no—
244#336—21no—
245#328—1464no—
246#327—1464no—
247C22×Q8—731no—
248C22×D4—731no—
249C22×D4—1461no—
250C23×C4—1462no—
251C23×C4D29211yescharacteristic
252C23×C4D29211yescharacteristic
253C23×C4C22×D7311yescharacteristic
254C22×C8—22no—
255#124—2922no—
256C22⋊Q8—1462no—
257C22⋊Q8—1462no—
258#121—1462no—
259#121—1462no—
260C2×C4⋊C4—731no—
261C2×C4⋊C4—731no—
262C2×C4⋊C4—1461no—
263C2×C22⋊C4—1461no—
264C2×C22⋊C4—1461no—
265C2×C22⋊C4—1461no—
266C2×C22⋊C4—1461no—
267C2×C22⋊C4—1461no—
268C2×C22⋊C4—1461no—
269C2×C42—42no—
270C2×C42C4×D7311yescharacteristic
271C2×C42C4×D7311yescharacteristic
272C2×C42C22⋊D7311yescharacteristic
273C2×C42C22⋊D7311yescharacteristic
274C22⋊C8—22no—
275C22⋊C8D29214yes—
276C4×C8—42no—
277#95—2922no—
278#95—1464no—
279#95—1462no—
280#95—1462no—
281#95—1461no—
282#95—1461no—
283#95—1461no—
284#95—1461no—
285C24—1461no—
286C24C2×D29211yescharacteristic
287C2×Q8—2921no—
288C2×Q8—2921no—
289C2×Q8—1462no—
290C2×D4—2921no—
291C2×D4—2921no—
292C2×D4—2921no—
293C2×D4—2921no—
294C2×D4—2922no—
295C2×D4—2922no—
296C2×D4—1462no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—2922no—
306C22×C4—2922no—
307C22×C4—2922no—
308C22×C4—2922no—
309C22×C4—2922no—
310C22×C4—2922no—
311C22×C4—1464no—
312C22×C4—1461no—
313C22×C4—1461no—
314C22×C4—1461no—
315C22×C4#1192400511yescharacteristic
316C22×C4#1192400511yescharacteristic
317C22×C4#1192400511yescharacteristic
318C22×C4#1192400511yescharacteristic
319C22×C4C2×C4×D7311yescharacteristic
320C22×C4C2×C22⋊D7311yescharacteristic
321C22×C4C2×D29212yes—
322C22×C4#1192402812yes—
323C2×C8—24no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—2922no—
327C4⋊C4—1462no—
328C4⋊C4—1462no—
329C22⋊C4—2922no—
330C22⋊C4—2922no—
331C22⋊C4—2921no—
332C22⋊C4—2921no—
333C22⋊C4—2922no—
334C22⋊C4—2922no—
335C22⋊C4—2922no—
336C42—22no—
337C42—22no—
338C42—41no—
339C42—41no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—2921no—
345C23—2921no—
346C23—2921no—
347C23—2921no—
348C23—2921no—
349C23—2921no—
350C23—2921no—
351C23C2×#1192400511yescharacteristic
352C23#1455657911yescharacteristic
353C23#1455658811yescharacteristic
354C23#1455661312yes—
355C23#1455661312yes—
356Q8—5842no—
357D4—5842no—
358D4—5842no—
359D4—5842no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
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—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—2924no—
379C2×C4—2924no—
380C2×C4—2924no—
381C2×C4—2921no—
382C2×C4—2921no—
383C2×C4—2921no—
384C2×C4—2921no—
385C2×C4—2921no—
386C2×C4—2921no—
387C2×C4—2921no—
388C2×C4—2921no—
389C2×C4—2921no—
390C2×C4—5842no—
391C2×C4—5842no—
392C2×C4—5842no—
393C2×C4—5842no—
394C2×C4#1455650512yes—
395C2×C4#1455651712yes—
396C2×C4C4×D29212yes—
397C2×C4#1455656512yes—
398C8—44no—
399C22—22no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—22no—
405C22—21no—
406C22—21no—
407C22—5841no—
408C22—5841no—
409C22—5841no—
410C22—5841no—
411C22—5841no—
412C22—5841no—
413C22—5841no—
414C22#1798848911yescharacteristic
415C22C2×#1455650511yescharacteristic
416C22#1798856011yescharacteristic
417C22#1798858411yescharacteristic
418C22#1798858411yescharacteristic
419C22C2×#1455651711yescharacteristic
420C22#1798860311yescharacteristic
421C22#1798894212yes—
422C22#1798897112yes—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—5844no—
430C4—5841no—
431C4—5841no—
432C4—5841no—
433C2—22no—
434C2—22no—
435C2—11681no—
436C2#2260841811yescharacteristic
437C2#2260842311yescharacteristic
438C2#2260848811yescharacteristic
439C2#2260850911yescharacteristic
440C2#2261074211yescharacteristic
441C2#2261076711yescharacteristic
442C2#2261085011yescharacteristic
subgroupG/H
centerC23C2×#11924005
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC4×C22⋊C8×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC2×#14556565C22
cosocleC23×C4×C73C23
hypercenterC4×C22⋊C8D73
hypocenterC73#9681
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#17988325C2semidirect
4. #22608158C2semidirect
5. C4×C22⋊C8×C73C2semidirect
6. C2×#14556579C4semidirect
9. #17988493C4semidirect
10. #17988489C4semidirect
15. #17988453C4semidirect
223. C4×C22⋊C8D73semidirect
227. C73#9681semidirect

Maximal subgroups

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

subgroupindex
#226182322
C2×#179883252
C2×#179883222
#226081582
C4×C22⋊C8×C732
#968173