Finite Group Databasebeta

← #33104461

Subgroups — #33104461

Order 18688.

846 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988702C211yesmaximal, characteristic
2#22618802C212yesmaximal
3#22618526C211yesmaximal, characteristic, complemented
4#22618522C211yesmaximal, characteristic, complemented
5#22610715C212yesmaximal, complemented
6#22610167C212yesmaximal, complemented
7#22610162C212yesmaximal, complemented
8#22610092C211yesmaximal, characteristic, complemented
9#22609987C211yesmaximal, characteristic, complemented
10#22609813C211yesmaximal, characteristic, complemented
11C73×#1815C211yesmaximal, characteristic, complemented
12C2×#14556611C2211yescharacteristic
13C2×#14556597C2211yescharacteristic
14#17989070C2212yes—
15#17989068C2212yes—
16#17988702C418yescomplemented
17#17988690—22no—
18#17988684—22no—
19#17988679—22no—
20#17988667—22no—
21C2×#14556526C2211yescharacteristic
22#17988629—22no—
23#17988629C2212yes—
24#17988627C2212yes—
25#17988626—22no—
26#17988625—21no—
27#17988625C2211yescharacteristic
28#17988622—21no—
29C2×#14556524C2212yes—
30C2×#14556523—21no—
31C2×#14556523C2211yescharacteristic
32#106×D73—21no—
33C2×#14556518C2211yescharacteristic
34C2×#14556517C2211yescharacteristic
35#17988589C2212yes—
36#17988545C2212yes—
37C2×C8×Dic73C2211yescharacteristic
38#17988452C2212yes—
39#17988355C2212yes—
40#17988348C2212yes—
41C2×#131×C73C2211yescharacteristic
42C2×#108×C73C2211yescharacteristic
43C2×#106×C73C2211yescharacteristic
44#417×C73C2212yes—
45#343×C73C2212yes—
46C2×#11924028C2311yescharacteristic
47#14556611C2×C414yes—
48#14556611Q814yes—
49C2×#11924018D411yescharacteristic
50C2×#11924016D411yescharacteristic
51C2×#11924015D411yescharacteristic
52C2×#11924013D411yescharacteristic
53#14556597—22no—
54#14556597C2×C414yescomplemented
55C2×#11924012C2311yescharacteristic
56C2×#11924011—21no—
57C22×#10596831C2311yescharacteristic
58C2×C8×D73—21no—
59Q8×Dic73—22no—
60#14556556—22no—
61D4×Dic73—22no—
62D4×Dic73—42no—
63#14556553—22no—
64C4×C22⋊D73—42no—
65#14556550—22no—
66#14556550C2312yes—
67#14556549—22no—
68C2×C4⋊Dic73—22no—
69C2×C4⋊Dic73C2312yes—
70C4⋊C4×D73—22no—
71C2×C4×Dic73C2311yescharacteristic
72#14556526C2×C414yescomplemented
73#14556525—22no—
74#14556525D412yes—
75#14556524—22no—
76#14556524—22no—
77#14556524—22no—
78#14556523—22no—
79#14556523—21no—
80#14556523—21no—
81#14556523D411yescharacteristic
82#14556523D411yescharacteristic
83#14556522—22no—
84#14556518C2×C414yescomplemented
85#14556517C2×C414yescomplemented
86#14556516—22no—
87#14556513—22no—
88#14556501—22no—
89C8×Dic73—22no—
90C8×Dic73C2×C414yescomplemented
91#14556498—42no—
92C2×#41×C73C2311yescharacteristic
93#131×C73—22no—
94#131×C73C2×C414yescomplemented
95C2×M16×C73—21no—
96C2×M16×C73C2312yes—
97C2×M16×C73C2311yescharacteristic
98C22×C8×C73—21no—
99C22×C8×C73C2311yescharacteristic
100C2×C4⋊C4×C73C2312yes—
101#108×C73—22no—
102#106×C73—21no—
103#106×C73—21no—
104#103×C73—22no—
105#102×C73—22no—
106#11924028—21no—
107#11924028—21no—
108#11924028C22×C412yes—
109#11924028C2×Q812yes—
110C2×Dic146C2×D411yescharacteristic
111C2×D292C2×D411yescharacteristic
112C2×C22⋊D73—21no—
113C22×Dic73—21no—
114C22×Dic73—22no—
115C22×Dic73—42no—
116C2×C4×D73—21no—
117C2×C4×D73—42no—
118#11924018C4⋊C414yes—
119#11924016C4⋊C414yes—
120#11924015C4⋊C414yes—
121#11924013C4⋊C414yes—
122#11924012—21no—
123#11924012C22×C412yes—
124#11924011—24no—
125C2×#10596831—21no—
126C2×#10596831C22×C412yes—
127C2×#10596831C2×D411yescharacteristic
128C2×#10596831C2×D411yescharacteristic
129C2×#10596831C2×Q812yes—
130C8×D73—24no—
131C4⋊Dic73—22no—
132C4⋊Dic73—22no—
133C4⋊Dic73—42no—
134C4⋊Dic73C2×D412yes—
135C4⋊Dic73C2×D412yes—
136#11924007—42no—
137C22⋊Dic73—22no—
138C22⋊Dic73—42no—
139#11924005—42no—
140C4×Dic73—22no—
141C4×Dic73—21no—
142C4×Dic73—42no—
143C4×Dic73—42no—
144C4×Dic73C22×C412yes—
145#41×C73—21no—
146#41×C73—21no—
147#41×C73C22×C412yes—
148#41×C73C2×Q812yes—
149C2×Q8×C73C2×D411yescharacteristic
150C2×D4×C73—21no—
151C2×D4×C73C2×D411yescharacteristic
152C22×C4×C73—21no—
153C22×C4×C73—22no—
154C22×C4×C73C2411yescharacteristic
155M16×C73—21no—
156M16×C73—24no—
157M16×C73—22no—
158M16×C73—22no—
159M16×C73C22×C412yes—
160C2×C8×C73—21no—
161C2×C8×C73—24no—
162C2×C8×C73—22no—
163C2×C8×C73C22×C412yes—
164C2×C8×C73C2×D411yescharacteristic
165C2×C8×C73C2×D411yescharacteristic
166C2×C8×C73#4112yes—
167C4⋊C4×C73—22no—
168C4⋊C4×C73—22no—
169C23×C73—21no—
170C23×C73H(5,𝔽2)11yescharacteristic
171C22×D73—21no—
172Q8×C73—21no—
173Q8×C73C2×C4⋊C412yes—
174D4×C73—22no—
175D4×C73—22no—
176D4×C73—21no—
177D4×C73C2×C4⋊C412yes—
178C2×C4×C73—21no—
179C2×C4×C73—22no—
180C2×C4×C73—22no—
181C2×C4×C73—42no—
182C2×C4×C73—82no—
183C2×C4×C73C23×C411yescharacteristic
184C2×C4×C73C22×D411yescharacteristic
185C2×C4×C73C22×D411yescharacteristic
186C2×C4×C73C22×Q811yescharacteristic
187C2×C4×C73C2×#4111yescharacteristic
188C2×C4×C732−1+411yescharacteristic
189C22⋊D73—22no—
190C22⋊D73—22no—
191C2×Dic73—21no—
192C2×Dic73—22no—
193C2×Dic73—22no—
194C2×Dic73—22no—
195C2×Dic73—42no—
196C2×Dic73—42no—
197C2×Dic73—82no—
198C2×Dic73—162no—
199D292—21no—
200D292C2×C4⋊C412yes—
201C4×D73—22no—
202C4×D73—22no—
203C4×D73—162no—
204Dic146—21no—
205Dic146C2×C4⋊C412yes—
206C8×C73—22no—
207C8×C73—42no—
208C8×C73C4×D412yes—
209C8×C73C4×D412yes—
210#10596831—22no—
211#10596831C2×C4⋊C412yes—
212#10596831C2×C4⋊C412yes—
213C22×C73—22no—
214C22×C73—22no—
215C22×C73—42no—
216C22×C73#52011yescharacteristic
217C22×C73D4211yescharacteristic
218C22×C73#55411yescharacteristic
219C2×D73—22no—
220C2×D73—42no—
221C4×C73—22no—
222C4×C73—82no—
223C4×C73C22×C4⋊C411yescharacteristic
224C4×C73C2×C4×D411yescharacteristic
225C4×C73#54711yescharacteristic
226C4×C73D4×Q811yescharacteristic
227Dic73—22no—
228Dic73—42no—
229Dic73—42no—
230Dic73—42no—
231Dic73—82no—
232#22690—731nomaximal, Sylow
233C2×C73—21no—
234C2×C73—21no—
235C2×C73—42no—
236C2×C73D4×C4⋊C411yescharacteristic
237C2×C73#329012yes—
238D73—42no—
239C2×#443—731no—
240#2951—732no—
241#2912—731no—
242#2912—731no—
243#1952—732no—
244#1860—732no—
245#1860—732no—
246#1846—731no—
247#1815—731no—
248#1815D7311yescharacteristic, complemented
249#1772—731no—
250C73#2269011yescharacteristic, complemented, Sylow
251C2×#135—731no—
252C2×#131—731no—
253C2×#131C2×D7311yescharacteristic
254#520—732no—
255#520—732no—
256#443—738no—
257#440—1462no—
258#440—1462no—
259C4×Q16—1462no—
260C4×D8—1462no—
261C2×#108—731no—
262C2×#108C2×D7311yescharacteristic
263#428—732no—
264#428—732no—
265#428—1462no—
266#428—1462no—
267#427—731no—
268#427—1461no—
269#427—1461no—
270C2×#107—732no—
271C2×#106—731no—
272C2×#106—1461no—
273C2×#106—1461no—
274C2×#106C2×D7311yescharacteristic
275C2×#104—731no—
276C2×#104—731no—
277#417—732no—
278#417C2×D7312yes—
279#405—732no—
280C2×C4×C8—731no—
281#382—732no—
282#343—732no—
283#343C2×D7312yes—
284#339—732no—
285C2×#41—731no—
286C2×#41C22×D7311yescharacteristic
287#135—734no—
288#135—734no—
289C2×Q16—731no—
290C2×SD16—731no—
291C2×SD16—731no—
292C2×D8—731no—
293#131—22no—
294#131—734no—
295#131—1462no—
296#131C4×D7314yescomplemented
297C2×M16—21no—
298C2×M16—731no—
299C2×M16—1461no—
300C2×M16C22×D7312yes—
301C2×M16C22×D7311yescharacteristic
302C22×C8—21no—
303C22×C8—731no—
304C22×C8—1461no—
305C22×C8C22×D7311yescharacteristic
306C4×Q8—1462no—
307C4×Q8—1462no—
308C4×D4—2922no—
309C4×D4—2922no—
310C4×D4—1462no—
311C4×D4—1462no—
312#117—732no—
313#117—1462no—
314#117—1462no—
315C2×C4⋊C4—732no—
316C2×C4⋊C4—1462no—
317C2×C4⋊C4—1462no—
318C2×C4⋊C4C22×D7312yes—
319C2×C42—731no—
320#108—22no—
321#108—734no—
322#107—732no—
323#107—1462no—
324#107—1462no—
325#107—1462no—
326#107—1462no—
327#106—21no—
328#106—21no—
329#106—731no—
330#106—731no—
331#106—1462no—
332#106—1462no—
333#106—1461no—
334#106—1461no—
335#104—734no—
336#104—734no—
337#103—22no—
338#103—1462no—
339#102—22no—
340#102—1462no—
341#97—1462no—
342C4×C8—734no—
343C4×C8—1462no—
344#95—2922no—
345#41—21no—
346#41—21no—
347#41—732no—
348#41—732no—
349#41—1461no—
350#41—1461no—
351#41C2×C4×D7312yes—
352#41Q8×D7312yes—
353C2×Q8—731no—
354C2×Q8D4×D7311yescharacteristic
355C2×D4—21no—
356C2×D4—731no—
357C2×D4—1461no—
358C2×D4D4×D7311yescharacteristic
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—2922no—
362C22×C4—2922no—
363C22×C4—1461no—
364C22×C4—1462no—
365C22×C4—1461no—
366C22×C4C23×D7311yescharacteristic
367Q16—734no—
368SD16—734no—
369SD16—734no—
370D8—734no—
371M16—21no—
372M16—24no—
373M16—22no—
374M16—22no—
375M16—732no—
376M16—1461no—
377M16—1464no—
378M16C2×C4×D7312yes—
379C2×C8—21no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—732no—
383C2×C8—731no—
384C2×C8—731no—
385C2×C8—732no—
386C2×C8—1461no—
387C2×C8—1464no—
388C2×C8C2×C4×D7312yes—
389C2×C8D4×D7311yescharacteristic
390C2×C8D4×D7311yescharacteristic
391C2×C8#1192402712yes—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—2922no—
395C4⋊C4—2922no—
396C4⋊C4—732no—
397C4⋊C4—732no—
398C4⋊C4—1462no—
399C4⋊C4—1462no—
400C22⋊C4—2922no—
401C22⋊C4—2922no—
402C22⋊C4—1462no—
403C42—2922no—
404C42—2922no—
405C42—732no—
406C42—1462no—
407C42—1461no—
408C23—21no—
409C23—1461no—
410C23#1455663711yescharacteristic
411Q8—21no—
412Q8—732no—
413Q8—1461no—
414Q8C4⋊C4×D7312yes—
415D4—22no—
416D4—22no—
417D4—21no—
418D4—732no—
419D4—1462no—
420D4—1462no—
421D4—1461no—
422D4C4⋊C4×D7312yes—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—42no—
427C2×C4—82no—
428C2×C4—2922no—
429C2×C4—2922no—
430C2×C4—5842no—
431C2×C4—11682no—
432C2×C4—11682no—
433C2×C4—1462no—
434C2×C4—1462no—
435C2×C4—1462no—
436C2×C4—1461no—
437C2×C4—1462no—
438C2×C4—1462no—
439C2×C4C22×C4×D7311yescharacteristic
440C2×C4C2×D4×D7311yescharacteristic
441C2×C4C2×D4×D7311yescharacteristic
442C2×C4C2×Q8×D7311yescharacteristic
443C2×C4C2×#1192402711yescharacteristic
444C2×C4#1455663811yescharacteristic
445C8—22no—
446C8—42no—
447C8—732no—
448C8—732no—
449C8—1462no—
450C8#1455655112yes—
451C8#1455655112yes—
452C22—22no—
453C22—22no—
454C22—42no—
455C22—2922no—
456C22—1462no—
457C22#1798906711yescharacteristic
458C22#1798922111yescharacteristic
459C22#1798929511yescharacteristic
460C4—22no—
461C4—82no—
462C4—2922no—
463C4—2922no—
464C4—2922no—
465C4—5842no—
466C4—1462no—
467C4C2×C4⋊C4×D7311yescharacteristic
468C4C2×#1455655111yescharacteristic
469C4#1798923711yescharacteristic
470C4#1798925011yescharacteristic
471C2—21no—
472C2—21no—
473C2—42no—
474C2—2922no—
475C2#2261282711yescharacteristic
476C2#2262171412yes—
subgroupG/H
centerC22#17989221
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1815C2
socleC22×C73D42
prime radicalC2×#14556611C22
cosocleC22×C4×C73C24
hypercenter#1815D73
hypocenterC73#22690
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1815

Complements

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

NH ≅ G/Ntype
3. #22618526C2semidirect
4. #22618522C2semidirect
5. #22610715C2semidirect
6. #22610167C2semidirect
7. #22610162C2semidirect
8. #22610092C2semidirect
9. #22609987C2semidirect
10. #22609813C2semidirect
11. C73×#1815C2semidirect
16. #17988702C4semidirect
54. #14556597C2×C4semidirect
72. #14556526C2×C4semidirect
84. #14556518C2×C4semidirect
85. #14556517C2×C4semidirect
90. C8×Dic73C2×C4semidirect
94. #131×C73C2×C4semidirect
248. #1815D73semidirect
250. C73#22690semidirect
296. #131C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179887022
#226188022
#226185262
#226185222
#226107152
#226101672
#226101622
#226100922
#226099872
#226098132
C73×#18152
#2269073