Finite Group Databasebeta

← #32976980

Subgroups — #32976980

Order 18688.

522 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22609707C211yesmaximal, characteristic, complemented
2C8×C22⋊Dic73C211yesmaximal, characteristic, complemented
3#22608422C211yesmaximal, characteristic
4#22608422C211yesmaximal, characteristic
5#22607818C211yesmaximal, characteristic, complemented
6#22607816C211yesmaximal, characteristic, complemented
7C73×#2929C211yesmaximal, characteristic
8C2×C22⋊#10596831C2211yescharacteristic
9C2×C22⋊#10596831C2211yescharacteristic
10C2×#14556501—21no—
11C2×C8×Dic73—21no—
12C4×C22⋊Dic73C2211yescharacteristic
13C4×C22⋊Dic73C2211yescharacteristic
14#17988341—21no—
15#17988341—21no—
16#17988324C412yescomplemented
17#17988324C412yescomplemented
18#17988321C412yescomplemented
19#17988321C412yescomplemented
20C2×#131×C73C411yescharacteristic, complemented
21C23×C8×C73C2211yescharacteristic
22#525×C73C2211yescharacteristic
23#435×C73—21no—
24#435×C73—21no—
25C8×D4×C73—21no—
26C8×D4×C73—21no—
27#424×C73C411yescharacteristic, complemented
28#408×C73C411yescharacteristic, complemented
29#407×C73C2211yescharacteristic
30#405×C73C411yescharacteristic, complemented
31C22×#10596831—21no—
32C22×#10596831—21no—
33C2×C22⋊Dic73—21no—
34C2×C22⋊Dic73—21no—
35C2×C4×Dic73—21no—
36C2×C4×Dic73—21no—
37C22⋊#10596831—21no—
38C22⋊#10596831—21no—
39C22⋊#10596831C2×C412yes—
40C22⋊#10596831C2×C412yes—
41#14556501—42no—
42C8×Dic73—42no—
43#14556498C2×C412yes—
44#14556498C2×C412yes—
45C2×#41×C73C2×C411yescharacteristic
46C23×C4×C73C2311yescharacteristic
47#131×C73—42no—
48C2×M16×C73—21no—
49C2×M16×C73C2×C411yescharacteristic
50C2×M16×C73C2×C411yescharacteristic
51C22×C8×C73—21no—
52C22×C8×C73—42no—
53C22×C8×C73C2×C411yescharacteristic
54C22×C8×C73C2×C411yescharacteristic
55C22×C8×C73D411yescharacteristic
56C22×C8×C73D411yescharacteristic
57C22×C8×C73D411yescharacteristic
58C22×C8×C73D411yescharacteristic
59#123×C73—21no—
60C22⋊Q8×C73C2×C411yescharacteristic
61C22⋊Q8×C73C2×C411yescharacteristic
62#121×C73C2×C411yescharacteristic
63#121×C73C2×C411yescharacteristic
64#120×C73—21no—
65C4×D4×C73—21no—
66C4×D4×C73—21no—
67#117×C73C2×C411yescharacteristic
68C4⋊C8×C73—21no—
69C4⋊C8×C73—21no—
70C22⋊C8×C73—21no—
71C22⋊C8×C73—21no—
72C22⋊C8×C73—21no—
73C22⋊C8×C73—21no—
74#97×C73—21no—
75C4×C8×C73—21no—
76C22×Dic73—22no—
77C22×Dic73—22no—
78C2×#10596831—22no—
79C2×#10596831—22no—
80C2×#10596831—41no—
81C2×#10596831—41no—
82C2×#10596831—41no—
83C2×#10596831—41no—
84C22⋊Dic73—24no—
85C22⋊Dic73—42no—
86C4×Dic73—41no—
87C4×Dic73—41no—
88C4×Dic73—41no—
89C4×Dic73—41no—
90C24×C73#4111yescharacteristic
91#41×C73—41no—
92#41×C73—41no—
93C2×Q8×C73C4211yescharacteristic
94C2×D4×C73—21no—
95C2×D4×C73—21no—
96C2×D4×C73—21no—
97C2×D4×C73C4211yescharacteristic
98C22×C4×C73—21no—
99C22×C4×C73—41no—
100C22×C4×C73—41no—
101C22×C4×C73C22×C411yescharacteristic
102C22×C4×C73C22×C411yescharacteristic
103C22×C4×C73C22×C411yescharacteristic
104C22×C4×C73C2×D411yescharacteristic
105C22×C4×C73C2×D411yescharacteristic
106C22×C4×C73#4111yescharacteristic
107M16×C73—42no—
108M16×C73—41no—
109M16×C73—41no—
110C2×C8×C73—22no—
111C2×C8×C73—22no—
112C2×C8×C73—22no—
113C2×C8×C73—21no—
114C2×C8×C73—21no—
115C2×C8×C73—41no—
116C2×C8×C73—41no—
117C2×C8×C73—41no—
118C2×C8×C73—42no—
119C2×C8×C73C22⋊C411yescharacteristic
120C2×C8×C73C22⋊C411yescharacteristic
121C2×C8×C73C22⋊C411yescharacteristic
122C2×C8×C73C22⋊C411yescharacteristic
123C4⋊C4×C73—21no—
124C4⋊C4×C73—21no—
125C4⋊C4×C73C4211yescharacteristic
126C4⋊C4×C73C4211yescharacteristic
127C22⋊C4×C73—21no—
128C22⋊C4×C73—21no—
129C22⋊C4×C73—21no—
130C22⋊C4×C73—21no—
131C22⋊C4×C73—21no—
132C42×C73—21no—
133C23×C73—22no—
134C23×C73—22no—
135C23×C73—21no—
136C23×C73—41no—
137C23×C73#11711yescharacteristic
138C23×C73C4×D411yescharacteristic
139C23×C73C4×D411yescharacteristic
140Q8×C73—41no—
141D4×C73—41no—
142D4×C73—42no—
143D4×C73—42no—
144D4×C73—42no—
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—21no—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
159C2×C4×C73C2×C4211yescharacteristic
160C2×C4×C73C2×C22⋊C411yescharacteristic
161C2×C4×C73C4×D411yescharacteristic
162C2×C4×C73C4×D411yescharacteristic
163C2×Dic73—24no—
164C2×Dic73—42no—
165C2×Dic73—42no—
166C2×Dic73—42no—
167C2×Dic73—42no—
168C2×Dic73—42no—
169C8×C73—21no—
170C8×C73—21no—
171C8×C73—21no—
172C8×C73—41no—
173C8×C73—41no—
174C8×C73#9912yes—
175#10596831—42no—
176#10596831—42no—
177C22×C73—21no—
178C22×C73—21no—
179C22×C73—21no—
180C22×C73—21no—
181C22×C73—21no—
182C22×C73—21no—
183C22×C73—21no—
184C22×C73—44no—
185C22×C73—41no—
186C22×C73—41no—
187C22×C73—41no—
188C22×C73C4×C22⋊C411yescharacteristic
189C22×C73#44312yes—
190C22×C73#44412yes—
191C4×C73—21no—
192C4×C73—21no—
193C4×C73—21no—
194C4×C73—41no—
195C4×C73—41no—
196C4×C73—41no—
197C4×C73C2×#9911yescharacteristic
198C4×C73#41011yescharacteristic
199Dic73—44no—
200Dic73—82no—
201#8240—731nomaximal, Sylow
202C2×C73—22no—
203C2×C73—22no—
204C2×C73—41no—
205C2×C73—81no—
206C2×C73C4×#9911yescharacteristic
207C2×C73#176411yescharacteristic
208C2×C73#176511yescharacteristic
209#2929D7311yescharacteristic
210#1752—731no—
211C8×C22⋊C4—731no—
212#1501—731no—
213#1501—731no—
214#1298—731no—
215#1297—731no—
216C73#824011yescharacteristic, complemented, Sylow
217C2×#131Dic7311yescharacteristic, complemented
218C23×C8C2×D7311yescharacteristic
219#525C2×D7311yescharacteristic
220#435—21no—
221#435—21no—
222C8×D4—21no—
223C8×D4—21no—
224#424Dic7311yescharacteristic, complemented
225#408Dic7311yescharacteristic, complemented
226#407C2×D7311yescharacteristic
227C2×C22⋊C8—731no—
228C2×C22⋊C8—731no—
229#405Dic7311yescharacteristic, complemented
230C2×#97—1461no—
231C2×C4×C8—1461no—
232C4×C22⋊C4—731no—
233C4×C22⋊C4—731no—
234#336—1461no—
235#336—1461no—
236#328—732no—
237#328—732no—
238#327—732no—
239#327—732no—
240C2×#41C2×Dic7311yescharacteristic
241C23×C4C22×D7311yescharacteristic
242#131—42no—
243C2×M16—21no—
244C2×M16C2×Dic7311yescharacteristic
245C2×M16C2×Dic7311yescharacteristic
246C22×C8—21no—
247C22×C8—42no—
248C22×C8—1461no—
249C22×C8—1461no—
250C22×C8C2×Dic7311yescharacteristic
251C22×C8C2×Dic7311yescharacteristic
252C22×C8C22⋊D7311yescharacteristic
253C22×C8C22⋊D7311yescharacteristic
254C22×C8C22⋊D7311yescharacteristic
255C22×C8C22⋊D7311yescharacteristic
256#123—21no—
257C22⋊Q8C4×D7311yescharacteristic
258C22⋊Q8C4×D7311yescharacteristic
259#121C4×D7311yescharacteristic
260#121C4×D7311yescharacteristic
261#120—21no—
262C4×D4—21no—
263C4×D4—21no—
264#117C2×Dic7311yescharacteristic
265C2×C22⋊C4—1461no—
266C2×C22⋊C4—1461no—
267C2×C42—1461no—
268C2×C42—1461no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—21no—
274C22⋊C8—21no—
275C22⋊C8—732no—
276C22⋊C8—732no—
277C22⋊C8—1461no—
278C22⋊C8—1461no—
279#97—21no—
280#97—2922no—
281C4×C8—21no—
282C4×C8—2922no—
283#95—732no—
284#95—732no—
285C24#1192402811yescharacteristic
286#41—41no—
287#41—41no—
288C2×Q8C4×Dic7311yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4C4×Dic7311yescharacteristic
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—1462no—
297C22×C4—1462no—
298C22×C4C2×C4×D7311yescharacteristic
299C22×C4C2×C4×D7311yescharacteristic
300C22×C4C22×Dic7311yescharacteristic
301C22×C4C2×C22⋊D7311yescharacteristic
302C22×C4C2×C22⋊D7311yescharacteristic
303C22×C4#1192402811yescharacteristic
304M16—42no—
305M16—41no—
306M16—41no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—2921no—
317C2×C8—2921no—
318C2×C8—2921no—
319C2×C8—2921no—
320C2×C8—1462no—
321C2×C8—1462no—
322C2×C8C22⋊Dic7311yescharacteristic
323C2×C8C22⋊Dic7311yescharacteristic
324C2×C8C22⋊Dic7311yescharacteristic
325C2×C8C22⋊Dic7311yescharacteristic
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4C4×Dic7311yescharacteristic
329C4⋊C4C4×Dic7311yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—2922no—
336C22⋊C4—1464no—
337C42—21no—
338C42—2921no—
339C42—2921no—
340C42—2921no—
341C42—2921no—
342C23—22no—
343C23—22no—
344C23—21no—
345C23—41no—
346C23#1455655011yescharacteristic
347C23C4×C22⋊D7311yescharacteristic
348C23C4×C22⋊D7311yescharacteristic
349Q8—41no—
350D4—41no—
351D4—42no—
352D4—42no—
353D4—42no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—2922no—
369C2×C4—2922no—
370C2×C4—2922no—
371C2×C4—2922no—
372C2×C4—2922no—
373C2×C4—1464no—
374C2×C4C2×C4×Dic7311yescharacteristic
375C2×C4C2×C22⋊Dic7311yescharacteristic
376C2×C4C4×C22⋊D7311yescharacteristic
377C2×C4C4×C22⋊D7311yescharacteristic
378C8—21no—
379C8—21no—
380C8—21no—
381C8—41no—
382C8—41no—
383C8—2922no—
384C8—2922no—
385C8#1455650612yes—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—44no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22C4×C22⋊Dic7311yescharacteristic
398C22#1798870212yes—
399C22#1798870712yes—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—2924no—
407C4—5842no—
408C4C2×#1455650611yescharacteristic
409C4#1798856111yescharacteristic
410C2—22no—
411C2—22no—
412C2—41no—
413C2—81no—
414C2C4×#1455650611yescharacteristic
415C2#2260976611yescharacteristic
416C2#2260977311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊Dic73
commutatorC2×C4×C73C2×C42
FrattiniC23×C4C22×D73
FittingC73×#2929C2
socleC22×C73C4×C22⋊C4
prime radical#525×C73C22
cosocleC23×C4×C73C23
hypercenter#2929D73
hypocenterC73#8240
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #22609707C2semidirect
2. C8×C22⋊Dic73C2semidirect
5. #22607818C2semidirect
6. #22607816C2semidirect
16. #17988324C4semidirect
17. #17988324C4semidirect
18. #17988321C4semidirect
19. #17988321C4semidirect
20. C2×#131×C73C4semidirect
27. #424×C73C4semidirect
28. #408×C73C4semidirect
30. #405×C73C4semidirect
216. C73#8240semidirect
217. C2×#131Dic73semidirect
224. #424Dic73semidirect
225. #408Dic73semidirect
229. #405Dic73semidirect

Maximal subgroups

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

subgroupindex
#226097072
C8×C22⋊Dic732
#226084222
#226078182
#226078162
C73×#29292
#824073