Finite Group Databasebeta

← #2227×D73 #33240609

Subgroups — #2227×D73 #33240609

Order 18688.

574 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#500×D73C211yesmaximal, characteristic
2C4×Q16×D73C211yesmaximal, characteristic
3#22612084C211yesmaximal, characteristic, complemented
4#22612083C211yesmaximal, characteristic, complemented
5#22612082C212yesmaximal, complemented
6#22612081C212yesmaximal, complemented
7#22612080C211yesmaximal, characteristic, complemented
8#366×D73C212yesmaximal
9C4⋊C16×D73C211yesmaximal, characteristic
10#358×D73C212yesmaximal
11C73×#2227C211yesmaximal, characteristic, complemented
12C2×Q16×D73C2211yescharacteristic
13C4⋊Q8×D73—21no—
14C4×Q8×D73—21no—
15C2×C16×D73C2212yes—
16#17988992C2211yescharacteristic
17#17988991—21no—
18#17988991C2212yes—
19#17988990—21no—
20C4×Dic292C2211yescharacteristic
21Q16×Dic73C2211yescharacteristic
22#17988681C2211yescharacteristic
23C4×#11924018—21no—
24#17988679—21no—
25#107×D73—21no—
26#107×D73C2212yes—
27#107×D73C2211yescharacteristic
28#103×D73—21no—
29C4×C8×D73C2211yescharacteristic
30#17988413C2212yes—
31#17988412C2212yes—
32#17988412C2212yes—
33C4⋊#11924003C2211yescharacteristic
34#17988405C2212yes—
35#17988392C2212yes—
36#17988391C2212yes—
37#17988390C2212yes—
38#500×C73C2211yescharacteristic
39C4×Q16×C73C2211yescharacteristic
40#366×C73C2212yes—
41C4⋊C16×C73C2211yescharacteristic
42#358×C73C2212yes—
43C2×Q8×D73—21no—
44C2×Q8×D73—41no—
45C2×Dic292C2311yescharacteristic
46C2×#11924018—21no—
47Q16×D73—22no—
48Q16×D73Q814yescomplemented
49C2×C8×D73D411yescharacteristic
50C2×C8×D73C2311yescharacteristic
51#14556591—21no—
52#14556590—22no—
53#14556590—41no—
54#14556589—41no—
55Q8×Dic73—21no—
56C4×Dic146—21no—
57C4×Dic146—41no—
58#14556556—21no—
59#14556556—41no—
60C4⋊C4×D73—22no—
61C4⋊C4×D73—21no—
62C4⋊C4×D73—41no—
63C4⋊C4×D73—41no—
64C42×D73—41no—
65C42×D73D411yescharacteristic
66C2×#11924003C2312yes—
67C16×D73—24no—
68#14556525—21no—
69#14556525C2312yes—
70#14556525C2311yescharacteristic
71#14556524—22no—
72#14556524—21no—
73#14556524—22no—
74#14556516—21no—
75#14556515—21no—
76#14556514—21no—
77C4×#10596831D411yescharacteristic
78C8×Dic73D411yescharacteristic
79C8×Dic73C2311yescharacteristic
80C2×Q16×C73C2311yescharacteristic
81C4⋊Q8×C73—21no—
82C4×Q8×C73—21no—
83C2×C16×C73C2312yes—
84#107×C73—21no—
85#107×C73C2312yes—
86#107×C73C2311yescharacteristic
87#103×C73—21no—
88C4×C8×C73C2311yescharacteristic
89C2×Dic146—21no—
90C2×Dic146—41no—
91C2×Dic146—41no—
92C2×Dic146—42no—
93Q8×D73—24no—
94Q8×D73—42no—
95Q8×D73—84no—
96C2×C4×D73—41no—
97C2×C4×D73—41no—
98C2×C4×D73—42no—
99C2×C4×D73D811yescharacteristic
100C2×C4×D73D811yescharacteristic
101C2×C4×D73C2×D411yescharacteristic
102Dic292—21no—
103Dic292C2×Q812yescomplemented
104#11924018—24no—
105C2×#10596831C2×D411yescharacteristic
106C2×#10596831C2×D411yescharacteristic
107C8×D73—22no—
108C8×D73C2×Q812yes—
109C8×D73#4112yes—
110C4⋊Dic73—21no—
111C4⋊Dic73—22no—
112C4⋊Dic73—41no—
113C4⋊Dic73—41no—
114#11924007—41no—
115#11924007—41no—
116#11924007—42no—
117#11924007—41no—
118#11924007—42no—
119C4×Dic73—41no—
120C4×Dic73—41no—
121C4×Dic73—41no—
122C4×Dic73D811yescharacteristic
123C4×Dic73D811yescharacteristic
124C4×Dic73C2×D411yescharacteristic
125#11924003—22no—
126C2×Q8×C73—21no—
127C2×Q8×C73—41no—
128Q16×C73—21no—
129Q16×C73C2×Q812yescomplemented
130C2×C8×C73C2×D411yescharacteristic
131C2×C8×C73C2411yescharacteristic
132C4⋊C4×C73—21no—
133C4⋊C4×C73—22no—
134C4⋊C4×C73—41no—
135C4⋊C4×C73—41no—
136C42×C73—41no—
137C42×C73C2×D411yescharacteristic
138C16×C73—22no—
139C22×D73C2×D811yescharacteristic
140Q8×C73—22no—
141Q8×C73—41no—
142Q8×C73—82no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—42no—
146C2×C4×C73C2×D811yescharacteristic
147C2×C4×C73C2×D811yescharacteristic
148C2×C4×C73C22×D411yescharacteristic
149C2×Dic73—41no—
150C2×Dic73—41no—
151C2×Dic73—42no—
152C2×Dic73C2×D811yescharacteristic
153C2×Dic73C2×D811yescharacteristic
154C2×Dic73C2×D811yescharacteristic
155C4×D73—22no—
156C4×D73—44no—
157C4×D73—82no—
158C4×D73—84no—
159C4×D73Q3214yes—
160C4×D73C22⋊Q812yes—
161C4×D73#13712yes—
162Dic146—22no—
163Dic146—41no—
164Dic146—44no—
165Dic146—84no—
166Dic146—82no—
167C8×C73—21no—
168C8×C73C22×Q811yescharacteristic
169C8×C73C2×#4111yescharacteristic
170#10596831—21no—
171#10596831C22⋊Q812yes—
172C22×C73C22×D811yescharacteristic
173C2×D73#47412yes—
174C2×D73C2×Q3212yes—
175C2×D73#50912yes—
176C4×C73—21no—
177C4×C73—42no—
178C4×C73—82no—
179C4×C73—81no—
180C4×C73C2×Q3212yes—
181C4×C73C2×C22⋊Q811yescharacteristic
182C4×C73C2×#13711yescharacteristic
183Dic73—21no—
184Dic73—42no—
185Dic73—82no—
186Dic73—81no—
187Dic73#47412yes—
188Dic73C2×Q3212yes—
189C2×#2227—731nomaximal, Sylow
190C2×C73C2×#47411yescharacteristic
191C2×C73C22×Q3211yescharacteristic
192C2×C73C2×#50911yescharacteristic
193D73#222714yesdirect product factor
194C2×#500—731no—
195C2×C4×Q16—731no—
196#2227—731no—
197#2227—731no—
198#2227—732no—
199#2227—731no—
200#2227—732no—
201#2227D7311yescharacteristic, direct product factor
202C2×#366—732no—
203C2×C4⋊C16—731no—
204C2×#358—732no—
205C73C2×#222711yescharacteristic, complemented, Sylow
206C22×Q16—731no—
207C2×C4⋊Q8—1461no—
208C2×C4×Q8—1461no—
209C22×C16—732no—
210#500—732no—
211#500—731no—
212#500—1461no—
213#500—1461no—
214#500C2×D7311yescharacteristic
215C4×Q16—731no—
216C4×Q16—731no—
217C4×Q16—731no—
218C4×Q16—1461no—
219C4×Q16—1461no—
220C4×Q16C2×D7311yescharacteristic
221C2×#107—732no—
222C2×#107—731no—
223C2×#107—1461no—
224C2×#103—1461no—
225C2×C4×C8—731no—
226#366—732no—
227#366—732no—
228#366—732no—
229#366C2×D7312yes—
230C4⋊C16—732no—
231C4⋊C16—731no—
232C4⋊C16C2×D7311yescharacteristic
233#358—732no—
234#358—732no—
235#358—732no—
236#358C2×D7312yes—
237C22×Q8—2921no—
238C22×Q8—1461no—
239C2×Q16—734no—
240C2×Q16—731no—
241C2×Q16—1462no—
242C2×Q16—1461no—
243C2×Q16C22×D7311yescharacteristic
244C22×C8—731no—
245C22×C8—731no—
246C4⋊Q8—21no—
247C4⋊Q8—2921no—
248C4⋊Q8—2921no—
249C4⋊Q8—1462no—
250C4⋊Q8—1461no—
251C4×Q8—21no—
252C4×Q8—2921no—
253C4×Q8—2921no—
254C4×Q8—1461no—
255C4×Q8—1461no—
256C4×Q8—1461no—
257C2×C4⋊C4—2921no—
258C2×C4⋊C4—2921no—
259C2×C4⋊C4—1461no—
260C2×C4⋊C4—1462no—
261C2×C42—2921no—
262C2×C42—731no—
263C2×C16—732no—
264C2×C16—1464no—
265C2×C16C22×D7312yes—
266#107—21no—
267#107—732no—
268#107—731no—
269#107—1461no—
270#107—1462no—
271#107—1462no—
272#107—1461no—
273#107C22×D7312yes—
274#107C22×D7311yescharacteristic
275#103—21no—
276#103—1461no—
277#103—1461no—
278#103—1461no—
279C4×C8—731no—
280C4×C8—731no—
281C4×C8—731no—
282C4×C8C22×D7311yescharacteristic
283C2×Q8—21no—
284C2×Q8—41no—
285C2×Q8—2921no—
286C2×Q8—2922no—
287C2×Q8—2922no—
288C2×Q8—2921no—
289C2×Q8—5844no—
290C2×Q8—1464no—
291C2×Q8—1461no—
292C22×C4—2921no—
293C22×C4—2921no—
294C22×C4—2922no—
295C22×C4—731no—
296C22×C4—731no—
297C22×C4—731no—
298Q16—21no—
299Q16—732no—
300Q16—1464no—
301Q16—1461no—
302Q16Q8×D7312yescomplemented
303C2×C8—731no—
304C2×C8—732no—
305C2×C8—732no—
306C2×C8—731no—
307C2×C8—1462no—
308C2×C8D4×D7311yescharacteristic
309C2×C8C23×D7311yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—2921no—
315C4⋊C4—2922no—
316C4⋊C4—2921no—
317C4⋊C4—2922no—
318C4⋊C4—2921no—
319C4⋊C4—2921no—
320C4⋊C4—2921no—
321C4⋊C4—1461no—
322C4⋊C4—1462no—
323C42—41no—
324C42—2921no—
325C42—2921no—
326C42—2921no—
327C42—731no—
328C42—731no—
329C42—731no—
330C42D4×D7311yescharacteristic
331C16—22no—
332C16—1462no—
333C23—731no—
334Q8—22no—
335Q8—41no—
336Q8—82no—
337Q8—2924no—
338Q8—2921no—
339Q8—5842no—
340Q8—5844no—
341Q8—1462no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—2924no—
346C2×C4—2921no—
347C2×C4—2921no—
348C2×C4—2922no—
349C2×C4—5844no—
350C2×C4—5842no—
351C2×C4—731no—
352C2×C4—731no—
353C2×C4—732no—
354C2×C4—734no—
355C2×C4—732no—
356C2×C4—731no—
357C2×C4—1462no—
358C2×C4D8×D7311yescharacteristic
359C2×C4D8×D7311yescharacteristic
360C2×C4C2×D4×D7311yescharacteristic
361C8—21no—
362C8—732no—
363C8—1461no—
364C8C2×Q8×D7311yescharacteristic
365C8#41×D7311yescharacteristic
366C22—732no—
367C22—732no—
368C22—732no—
369C22C2×D8×D7311yescharacteristic
370C4—21no—
371C4—42no—
372C4—82no—
373C4—81no—
374C4—2922no—
375C4—5842no—
376C4—5841no—
377C4—732no—
378C4—732no—
379C4—1461no—
380C4Q32×D7312yes—
381C4C22⋊Q8×D7311yescharacteristic
382C4#137×D7311yescharacteristic
383C2—734no—
384C2#474×D7311yescharacteristic
385C2C2×Q32×D7311yescharacteristic
386C2#509×D7311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2227C2
socleC22×C73C22×D8
prime radicalC22×D73C2×D8
hypercenter#2227D73
hypocenterC73C2×#2227
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2227

Complements

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

NH ≅ G/Ntype
3. #22612084C2semidirect
4. #22612083C2semidirect
5. #22612082C2semidirect
6. #22612081C2semidirect
7. #22612080C2semidirect
11. C73×#2227C2semidirect
48. Q16×D73Q8semidirect
103. Dic292C2×Q8semidirect
129. Q16×C73C2×Q8semidirect
193. D73#2227direct (N × H)
201. #2227D73direct (N × H)
205. C73C2×#2227semidirect
302. Q16Q8×D73semidirect

Maximal subgroups

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

subgroupindex
#500×D732
C4×Q16×D732
#226120842
#226120832
#226120822
#226120812
#226120802
#366×D732
C4⋊C16×D732
#358×D732
C73×#22272
C2×#222773