Finite Group Databasebeta

← #2236×D73 #33240772

Subgroups — #2236×D73 #33240772

Order 18688.

630 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#467×D73C211yesmaximal, characteristic
2C2×#107×D73C211yesmaximal, characteristic
3#22612135C212yesmaximal, complemented
4#22612134C211yesmaximal, characteristic, complemented
5#22612133C211yesmaximal, characteristic, complemented
6#22612132C212yesmaximal, complemented
7#22612131C211yesmaximal, characteristic, complemented
8#366×D73C212yesmaximal, complemented
9#358×D73C212yesmaximal, complemented
10C22⋊C16×D73C211yesmaximal, characteristic
11C73×#2236C211yesmaximal, characteristic, complemented
12C2×Q16×D73C2211yescharacteristic
13C22×C8×D73C2211yescharacteristic
14C22⋊Q8×D73—21no—
15C2×C4⋊C4×D73—21no—
16C2×C16×D73C2212yes—
17#17988829C2211yescharacteristic
18#17988828C2211yescharacteristic
19#17988827—21no—
20#17988826—21no—
21#17988825C2211yescharacteristic
22C2×#14556525C2211yescharacteristic
23C2×#14556524—21no—
24#107×D73—21no—
25#107×D73—21no—
26#107×D73C2211yescharacteristic
27#107×D73C2212yes—
28#107×D73C2212yes—
29#103×D73—21no—
30#17988413C2212yescomplemented
31#17988412C2212yescomplemented
32#17988412C2212yescomplemented
33#17988392C2212yescomplemented
34#17988391C2212yescomplemented
35#17988390C2212yescomplemented
36C22⋊#11924003C2211yescharacteristic
37#17988366C2212yes—
38#467×C73C2211yescharacteristic
39C2×#107×C73C2211yescharacteristic
40#366×C73C2212yescomplemented
41#358×C73C2212yescomplemented
42C22⋊C16×C73C2211yescharacteristic
43C2×Q8×D73—21no—
44C22×C4×D73—41no—
45C22×C4×D73D411yescharacteristic
46C2×Dic292C2311yescharacteristic
47C2×#11924018—21no—
48Q16×D73—24no—
49C22×#10596831D411yescharacteristic
50C2×C8×D73—24no—
51C2×C8×D73D411yescharacteristic
52C2×C8×D73D411yescharacteristic
53C2×C8×D73C2311yescharacteristic
54C2×C8×D73C2311yescharacteristic
55#14556571—21no—
56#14556570—21no—
57#14556569—41no—
58#14556568—41no—
59#14556567—21no—
60C2×C4⋊Dic73—21no—
61C2×#11924007—41no—
62C4⋊C4×D73—22no—
63C4⋊C4×D73—22no—
64C4⋊C4×D73—21no—
65C4⋊C4×D73—41no—
66C4⋊C4×D73—41no—
67C4⋊C4×D73—41no—
68C22⋊C4×D73—41no—
69C2×#11924003C2312yes—
70C16×D73—24no—
71#14556525—21no—
72#14556525C2312yes—
73#14556525C2311yescharacteristic
74#14556524—22no—
75#14556524—22no—
76#14556524—21no—
77#14556516—21no—
78#14556515—21no—
79#14556514—21no—
80C2×Q16×C73C2311yescharacteristic
81C22×C8×C73C2311yescharacteristic
82C22⋊Q8×C73—21no—
83C2×C4⋊C4×C73—21no—
84C2×C16×C73C2312yes—
85#107×C73—21no—
86#107×C73C2312yes—
87#107×C73C2311yescharacteristic
88#103×C73—21no—
89C23×D73D811yescharacteristic
90C2×Dic146—21no—
91C2×Dic146—41no—
92Q8×D73—42no—
93Q8×D73—42no—
94C22×Dic73—41no—
95C22×Dic73D811yescharacteristic
96C2×C4×D73—22no—
97C2×C4×D73—22no—
98C2×C4×D73—41no—
99C2×C4×D73—41no—
100C2×C4×D73—42no—
101C2×C4×D73—42no—
102C2×C4×D73—84no—
103C2×C4×D73D811yescharacteristic
104C2×C4×D73D811yescharacteristic
105C2×C4×D73C2×D411yescharacteristic
106C2×C4×D73C2×D411yescharacteristic
107Dic292—22no—
108#11924018—42no—
109C2×#10596831—22no—
110C2×#10596831C2×D411yescharacteristic
111C2×#10596831C2×D411yescharacteristic
112C8×D73—22no—
113C8×D73—22no—
114C8×D73—22no—
115C8×D73#4114yes—
116C4⋊Dic73—21no—
117C4⋊Dic73—22no—
118C4⋊Dic73—41no—
119C4⋊Dic73—41no—
120#11924007—41no—
121#11924007—42no—
122#11924007—42no—
123#11924007—41no—
124#11924007—41no—
125C22⋊Dic73—41no—
126#11924005—41no—
127#11924005—41no—
128#11924003—22no—
129C2×Q8×C73—21no—
130C22×C4×C73—41no—
131C22×C4×C73C2×D411yescharacteristic
132Q16×C73—22no—
133C2×C8×C73—22no—
134C2×C8×C73C2×D411yescharacteristic
135C2×C8×C73C2411yescharacteristic
136C4⋊C4×C73—21no—
137C4⋊C4×C73—22no—
138C4⋊C4×C73—41no—
139C4⋊C4×C73—41no—
140C22⋊C4×C73—41no—
141C16×C73—22no—
142C23×C73C2×D811yescharacteristic
143C22×D73—22no—
144C22×D73—22no—
145C22×D73Q3214yes—
146C22×D73C2×D811yescharacteristic
147C22×D73C2×D811yescharacteristic
148Q8×C73—41no—
149Q8×C73—41no—
150C2×C4×C73—21no—
151C2×C4×C73—21no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—42no—
155C2×C4×C73—82no—
156C2×C4×C73C2×D811yescharacteristic
157C2×C4×C73C22×D411yescharacteristic
158C2×Dic73—22no—
159C2×Dic73—41no—
160C2×Dic73—41no—
161C2×Dic73—42no—
162C2×Dic73—82no—
163C2×Dic73C2×D811yescharacteristic
164C2×Dic73C2×D811yescharacteristic
165C4×D73—22no—
166C4×D73—21no—
167C4×D73—21no—
168C4×D73—22no—
169C4×D73—84no—
170C4×D73—82no—
171C4×D73—82no—
172C4×D73—84no—
173C4×D73#12312yes—
174C4×D73#13712yes—
175Dic146—41no—
176Dic146—41no—
177Dic146—82no—
178C8×C73—21no—
179C8×C73C2×#4112yes—
180#10596831—21no—
181#10596831#12312yes—
182C22×C73—21no—
183C22×C73—21no—
184C22×C73C2×Q3212yes—
185C22×C73C22×D811yescharacteristic
186C2×D73—21no—
187C2×D73—21no—
188C2×D73—28no—
189C2×D73#48012yes—
190C2×D73C2×Q3212yes—
191C2×D73C2×Q3212yes—
192C2×D73#50912yes—
193C4×C73—21no—
194C4×C73—81no—
195C4×C73—82no—
196C4×C73—81no—
197C4×C73C2×#12311yescharacteristic
198C4×C73C2×#13711yescharacteristic
199Dic73—21no—
200Dic73—81no—
201Dic73—82no—
202Dic73—81no—
203Dic73#48012yes—
204C2×#2236—731nomaximal, Sylow
205C2×C73—22no—
206C2×C73C2×#48011yescharacteristic
207C2×C73C22×Q3211yescharacteristic
208C2×C73C2×#50911yescharacteristic
209D73—22no—
210D73#223614yesdirect product factor
211C2×#467—731no—
212C22×#107—731no—
213#2236—731no—
214#2236—731no—
215#2236—732no—
216#2236—732no—
217#2236—731no—
218#2236D7311yescharacteristic, direct product factor
219C2×#366—732no—
220C2×#358—732no—
221C2×C22⋊C16—731no—
222C73C2×#223611yescharacteristic, complemented, Sylow
223C22×Q16—731no—
224C23×C8—731no—
225C2×C22⋊Q8—1461no—
226C22×C4⋊C4—1461no—
227C22×C16—732no—
228#467—731no—
229#467—731no—
230#467—731no—
231#467—1461no—
232#467—1461no—
233#467C2×D7311yescharacteristic
234C2×#107—731no—
235C2×#107—732no—
236C2×#107—732no—
237C2×#107—731no—
238C2×#107—1461no—
239C2×#107—1461no—
240C2×#107—1461no—
241C2×#107C2×D7311yescharacteristic
242C2×#103—1461no—
243#366—732no—
244#366—732no—
245#366—732no—
246#366C2×D7312yescomplemented
247#358—732no—
248#358—732no—
249#358—732no—
250#358C2×D7312yescomplemented
251C22⋊C16—731no—
252C22⋊C16—732no—
253C22⋊C16C2×D7311yescharacteristic
254C22×Q8—1461no—
255C23×C4—2921no—
256C23×C4—731no—
257C2×Q16—731no—
258C2×Q16—1464no—
259C2×Q16—1461no—
260C2×Q16C22×D7311yescharacteristic
261C22×C8—731no—
262C22×C8—731no—
263C22×C8—731no—
264C22×C8—731no—
265C22×C8—731no—
266C22×C8—1464no—
267C22×C8C22×D7311yescharacteristic
268C22⋊Q8—21no—
269C22⋊Q8—2921no—
270C22⋊Q8—2921no—
271C22⋊Q8—1461no—
272C22⋊Q8—1461no—
273C22⋊Q8—1461no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—2921no—
276C2×C4⋊C4—2921no—
277C2×C4⋊C4—2921no—
278C2×C4⋊C4—2921no—
279C2×C4⋊C4—1462no—
280C2×C4⋊C4—1462no—
281C2×C4⋊C4—1461no—
282C2×C4⋊C4—1461no—
283C2×C22⋊C4—2921no—
284C2×C16—732no—
285C2×C16—1464no—
286C2×C16C22×D7312yes—
287#107—21no—
288#107—732no—
289#107—731no—
290#107—1462no—
291#107—1462no—
292#107—1461no—
293#107—1461no—
294#107C22×D7312yes—
295#107C22×D7311yescharacteristic
296#103—21no—
297#103—1461no—
298#103—1461no—
299#103—1461no—
300C24—731no—
301C2×Q8—21no—
302C2×Q8—2921no—
303C2×Q8—2922no—
304C2×Q8—2922no—
305C2×Q8—1461no—
306C22×C4—41no—
307C22×C4—2921no—
308C22×C4—2921no—
309C22×C4—2922no—
310C22×C4—2922no—
311C22×C4—2921no—
312C22×C4—5844no—
313C22×C4—731no—
314C22×C4—731no—
315C22×C4—731no—
316C22×C4—731no—
317C22×C4—731no—
318C22×C4—1462no—
319C22×C4—1462no—
320C22×C4D4×D7311yescharacteristic
321Q16—22no—
322Q16—2922no—
323Q16—1462no—
324C2×C8—22no—
325C2×C8—731no—
326C2×C8—734no—
327C2×C8—731no—
328C2×C8—1462no—
329C2×C8—1462no—
330C2×C8—1462no—
331C2×C8—1462no—
332C2×C8D4×D7311yescharacteristic
333C2×C8C23×D7311yescharacteristic
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—2922no—
339C4⋊C4—2922no—
340C4⋊C4—2921no—
341C4⋊C4—2921no—
342C4⋊C4—2921no—
343C4⋊C4—2921no—
344C4⋊C4—2921no—
345C4⋊C4—1461no—
346C4⋊C4—1462no—
347C22⋊C4—41no—
348C22⋊C4—2921no—
349C22⋊C4—2921no—
350C22⋊C4—2921no—
351C16—22no—
352C16—1462no—
353C23—731no—
354C23—734no—
355C23—731no—
356C23—1462no—
357C23—1462no—
358C23D8×D7311yescharacteristic
359Q8—41no—
360Q8—41no—
361Q8—2921no—
362Q8—2921no—
363Q8—5842no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—2921no—
371C2×C4—2921no—
372C2×C4—2922no—
373C2×C4—5844no—
374C2×C4—5842no—
375C2×C4—5842no—
376C2×C4—5844no—
377C2×C4—5842no—
378C2×C4—731no—
379C2×C4—732no—
380C2×C4—732no—
381C2×C4—731no—
382C2×C4—1462no—
383C2×C4—1462no—
384C2×C4—1462no—
385C2×C4—1461no—
386C2×C4—1461no—
387C2×C4D8×D7311yescharacteristic
388C2×C4C2×D4×D7311yescharacteristic
389C8—21no—
390C8—732no—
391C8—1461no—
392C8#41×D7312yes—
393C22—21no—
394C22—21no—
395C22—732no—
396C22—732no—
397C22—732no—
398C22—732no—
399C22—1468no—
400C22—1461no—
401C22—1461no—
402C22Q32×D7312yes—
403C22C2×D8×D7311yescharacteristic
404C4—21no—
405C4—82no—
406C4—81no—
407C4—81no—
408C4—5842no—
409C4—5841no—
410C4—5841no—
411C4—732no—
412C4—1461no—
413C4#123×D7311yescharacteristic
414C4#137×D7311yescharacteristic
415C2—22no—
416C2—734no—
417C2—1462no—
418C2#480×D7311yescharacteristic
419C2C2×Q32×D7311yescharacteristic
420C2#509×D7311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2236C2
socleC22×C73C22×D8
prime radicalC23×D73D8
hypercenter#2236D73
hypocenterC73C2×#2236
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2236

Complements

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

NH ≅ G/Ntype
3. #22612135C2semidirect
4. #22612134C2semidirect
5. #22612133C2semidirect
6. #22612132C2semidirect
7. #22612131C2semidirect
8. #366×D73C2semidirect
9. #358×D73C2semidirect
11. C73×#2236C2semidirect
30. #17988413C22semidirect
31. #17988412C22semidirect
32. #17988412C22semidirect
33. #17988392C22semidirect
34. #17988391C22semidirect
35. #17988390C22semidirect
40. #366×C73C22semidirect
41. #358×C73C22semidirect
210. D73#2236direct (N × H)
218. #2236D73direct (N × H)
222. C73C2×#2236semidirect
246. #366C2×D73semidirect
250. #358C2×D73semidirect

Maximal subgroups

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

subgroupindex
#467×D732
C2×#107×D732
#226121352
#226121342
#226121332
#226121322
#226121312
#366×D732
#358×D732
C22⋊C16×D732
C73×#22362
C2×#223673