Finite Group Databasebeta

← #34021317

Subgroups — #34021317

Order 18688.

608 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22625861C212yesmaximal, complemented
2#22625851C212yesmaximal, complemented
3#22625841C212yesmaximal
4#22625815C211yesmaximal, characteristic
5#22625813C211yesmaximal, characteristic
6#22625737C212yesmaximal, complemented
7#22625687C212yesmaximal
8#22625652C212yesmaximal, complemented
9#22618358C211yesmaximal, characteristic
10D73⋊#108C2212yes—
11#17989597C2212yes—
12#17989573C2212yes—
13#17989566C2212yes—
14#17989550C2214yes—
15#17989469C2214yes—
16#17989458C2214yes—
17#17989456C2212yes—
18#17989455C2212yes—
19#17989454C2214yes—
20C2×#14556599C2211yescharacteristic
21C2×#14556597C2211yescharacteristic
22#17989055C2211yescharacteristic
23#17988544C411yescharacteristic, complemented
24#17988544C2211yescharacteristic
25#17988543C411yescharacteristic, complemented
26#17988541C411yescharacteristic, complemented
27#17988541C2211yescharacteristic
28M16×Dic73C411yescharacteristic, complemented
29C4×#11924011C411yescharacteristic, complemented
30C4×#11924011C2211yescharacteristic
31C2×#14556501C411yescharacteristic, complemented
32#97×D73C411yescharacteristic, complemented
33#97×D73C2211yescharacteristic
34#405×C73C411yescharacteristic, complemented
35D73⋊M16C2314yes—
36#14556670C2314yes—
37#14556662—22no—
38#14556661—22no—
39#14556660—22no—
40#14556659—22no—
41#14556652—22no—
42#14556651—22no—
43#14556650—22no—
44#14556649—22no—
45#14556645—24no—
46C8×#64408—24no—
47C2×#11924028C2311yescharacteristic
48#14556599—24no—
49#14556597—22no—
50#14556597—22no—
51C2×#11924012C2×C411yescharacteristic
52C2×#11924011C2×C411yescharacteristic
53C2×#11924011C2×C411yescharacteristic
54C2×#11924011C2311yescharacteristic
55C2×#11924011C2311yescharacteristic
56C22×#10596831C2×C411yescharacteristic
57C2×C8×D73C2×C411yescharacteristic
58C2×C8×D73C2×C411yescharacteristic
59C2×C8×D73C2311yescharacteristic
60C2×C8×D73C2311yescharacteristic
61#14556556—21no—
62#14556553—21no—
63#14556551—21no—
64#14556551—21no—
65#14556548C2×C411yescharacteristic
66#14556548C2311yescharacteristic
67C2×C4×Dic73C2×C411yescharacteristic
68C42×D73C2×C411yescharacteristic
69C42×D73C2311yescharacteristic
70#14556502C2×C411yescharacteristic
71#14556502C2×C411yescharacteristic
72#14556501C2×C411yescharacteristic
73#14556501C2×C411yescharacteristic
74#14556501C2×C411yescharacteristic
75#14556501C2×C411yescharacteristic
76#14556501C2×C411yescharacteristic
77#14556501C2×C411yescharacteristic
78C4×#10596831C2×C411yescharacteristic
79C4×#10596831C2×C411yescharacteristic
80C8×Dic73C2×C411yescharacteristic
81C8×Dic73C2×C411yescharacteristic
82C2×M16×C73C2×C411yescharacteristic
83C22×C8×C73C2×C411yescharacteristic
84#117×C73C2×C411yescharacteristic
85#97×C73C2×C411yescharacteristic
86#97×C73C2×C411yescharacteristic
87C4×C8×C73C2×C411yescharacteristic
88C4×C8×C73C2×C411yescharacteristic
89#11924038—24no—
90#11924037—24no—
91#11924036—24no—
92C4⋊#64408—24no—
93C22⋊#64408—24no—
94C4×#64408—24no—
95#11924028—22no—
96#11924028—22no—
97C2×Dic146#4111yescharacteristic
98C2×D292#4111yescharacteristic
99C2×C22⋊D73#4111yescharacteristic
100C2×C22⋊D73#4111yescharacteristic
101C22×Dic73—21no—
102C2×C4×D73—21no—
103C2×C4×D73—21no—
104C2×C4×D73C22×C411yescharacteristic
105C2×C4×D73C2411yescharacteristic
106#11924012—22no—
107#11924011—22no—
108#11924011—22no—
109#11924011—22no—
110#11924011—22no—
111C2×#10596831—22no—
112C2×#10596831C22×C411yescharacteristic
113C2×#10596831C22×C411yescharacteristic
114C2×#10596831C22×C411yescharacteristic
115C2×#10596831C22×C411yescharacteristic
116C8×D73—21no—
117C8×D73—21no—
118C8×D73—22no—
119C8×D73#4112yes—
120C8×D73#4112yes—
121C8×D73#4114yes—
122#11924007—21no—
123#11924007—21no—
124#11924005—21no—
125#11924005—21no—
126C4×Dic73—21no—
127C4×Dic73—21no—
128C4×Dic73—21no—
129C4×Dic73C22×C411yescharacteristic
130C4×Dic73C22×C411yescharacteristic
131C22×C4×C73C22×C411yescharacteristic
132M16×C73—22no—
133C2×C8×C73—21no—
134C2×C8×C73—21no—
135C2×C8×C73C22×C411yescharacteristic
136C2×C8×C73C22×C411yescharacteristic
137C2×C8×C73C22×C411yescharacteristic
138C2×C8×C73C22×C411yescharacteristic
139C4⋊C4×C73—21no—
140C22⋊C4×C73—21no—
141C42×C73C22×C411yescharacteristic
142C42×C73C22×C411yescharacteristic
143C23×C73#11711yescharacteristic
144C22×D73#11711yescharacteristic
145C22×D73C2×#4111yescharacteristic
146C2×#64408—44no—
147C2×C4×C73—21no—
148C2×C4×C73—21no—
149C2×C4×C73—21no—
150C2×C4×C73—21no—
151C2×C4×C73#11711yescharacteristic
152C2×C4×C73C23×C411yescharacteristic
153C22⋊D73—22no—
154C22⋊D73—22no—
155C2×Dic73—21no—
156C2×Dic73—21no—
157C2×Dic73—81no—
158C2×Dic73#11711yescharacteristic
159C2×Dic73C2×#4111yescharacteristic
160D292—22no—
161C4×D73—21no—
162C4×D73—21no—
163C4×D73—81no—
164C4×D73—81no—
165C4×D73C2×#4112yes—
166C4×D73C2×#4112yes—
167Dic146—22no—
168#10596833—44no—
169C8×C73—21no—
170C8×C73—21no—
171C8×C73#11711yescharacteristic
172C8×C73#11711yescharacteristic
173C8×C73#11712yes—
174#10596831—21no—
175#10596831—21no—
176#10596831#11712yes—
177#10596831#11712yes—
178C22×C73—21no—
179C22×C73—41no—
180C22×C73C2×#11711yescharacteristic
181C2×D73—21no—
182C2×D73—41no—
183C2×D73—41no—
184C2×D73#52912yes—
185#64408—84no—
186C4×C73—21no—
187C4×C73—41no—
188C4×C73—41no—
189C4×C73C2×#11711yescharacteristic
190C4×C73C2×#11711yescharacteristic
191Dic73—21no—
192Dic73—41no—
193Dic73—41no—
194Dic73#52912yes—
195#20788—731nomaximal, Sylow
196C2×C73—21no—
197C2×C73—41no—
198C2×C73#228311yescharacteristic
199D73—22no—
200D73—41no—
201#2875—731no—
202#1841—732no—
203#1833—732no—
204#1827—732no—
205#1813—731no—
206#1812—731no—
207#1773—732no—
208#1757—732no—
209#1747—732no—
210C73#2078811yescharacteristic, complemented, Sylow
211C2×#131—731no—
212C2×#131—731no—
213C4×#41—731no—
214C2×#108—732no—
215#427—732no—
216#413—732no—
217#410—732no—
218#405—731no—
219#405—731no—
220#405—731no—
221#405—731no—
222#405—731no—
223#405—734no—
224#405#6440811yescharacteristic, complemented
225C4×M16—731no—
226C4×M16—731no—
227C4×M16—731no—
228C2×#97—731no—
229C2×#97—731no—
230C2×#97—731no—
231#344—734no—
232#339—734no—
233#338—732no—
234H(3,ℤ/4)—732no—
235H(3,ℤ/4)—734no—
236C2×#41—731no—
237#131—1462no—
238#131—1462no—
239#131—1464no—
240C2×M16—731no—
241C2×M16—731no—
242C2×M16—731no—
243C2×M16—731no—
244C2×M16—731no—
245C2×M16—734no—
246C2×M16C2×#6440811yescharacteristic
247C22×C8—731no—
248C22×C8—731no—
249C22×C8—731no—
250C22×C8—731no—
251C22×C8—731no—
252C22×C8C2×#6440811yescharacteristic
253C4×Q8—1461no—
254C4×D4—1461no—
255C4×D4—1461no—
256C4×D4—1461no—
257#117—731no—
258#117—731no—
259#117—734no—
260#117C2×#6440811yescharacteristic
261C2×C42—731no—
262C2×C42—731no—
263C2×C42—731no—
264#108—1462no—
265#108—1462no—
266#107—1462no—
267#106—1462no—
268#101—1462no—
269#100—1462no—
270#99—1462no—
271#99—1462no—
272#97—731no—
273#97—731no—
274#97—731no—
275#97—731no—
276#97—731no—
277#97—731no—
278#97—731no—
279#97—731no—
280#97—1464no—
281#97C2×#6440811yescharacteristic
282#97C2×#6440811yescharacteristic
283C4×C8—731no—
284C4×C8—731no—
285C4×C8—731no—
286C4×C8—731no—
287C4×C8—1464no—
288C4×C8C2×#6440811yescharacteristic
289C4×C8C2×#6440811yescharacteristic
290#41—1462no—
291#41—1462no—
292C2×Q8—731no—
293C2×D4—731no—
294C2×D4—731no—
295C2×D4—731no—
296C22×C4—731no—
297C22×C4—731no—
298C22×C4—1461no—
299C22×C4—1461no—
300C22×C4—1461no—
301C22×C4C22×#6440811yescharacteristic
302M16—22no—
303M16—1462no—
304M16—1462no—
305M16—1462no—
306M16—1464no—
307M16—1464no—
308M16—1462no—
309M16—1462no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—731no—
313C2×C8—731no—
314C2×C8—734no—
315C2×C8—732no—
316C2×C8—732no—
317C2×C8—731no—
318C2×C8—731no—
319C2×C8—1462no—
320C2×C8—1464no—
321C2×C8—1462no—
322C2×C8—1461no—
323C2×C8—1461no—
324C2×C8C22×#6440811yescharacteristic
325C2×C8C22×#6440811yescharacteristic
326C2×C8C22×#6440811yescharacteristic
327C2×C8C22×#6440811yescharacteristic
328C4⋊C4—21no—
329C4⋊C4—1461no—
330C4⋊C4—1461no—
331C4⋊C4—1464no—
332C22⋊C4—21no—
333C22⋊C4—1461no—
334C22⋊C4—1461no—
335C22⋊C4—1464no—
336C42—731no—
337C42—731no—
338C42—1461no—
339C42—1461no—
340C42—1464no—
341C42—1461no—
342C42C22×#6440811yescharacteristic
343C42C22×#6440811yescharacteristic
344C23—731no—
345C23—731no—
346C23#1455667011yescharacteristic
347Q8—1462no—
348D4—1462no—
349D4—1462no—
350D4—1462no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—2924no—
356C2×C4—5841no—
357C2×C4—5841no—
358C2×C4—5841no—
359C2×C4—731no—
360C2×C4—732no—
361C2×C4—732no—
362C2×C4—731no—
363C2×C4—1461no—
364C2×C4—1461no—
365C2×C4—1461no—
366C2×C4—1461no—
367C2×C4#1455667011yescharacteristic
368C2×C4C23×#6440811yescharacteristic
369C8—21no—
370C8—21no—
371C8—2924no—
372C8—732no—
373C8—732no—
374C8—1461no—
375C8—1461no—
376C8#1455667011yescharacteristic
377C8#1455667011yescharacteristic
378C8#1455667012yes—
379C22—21no—
380C22—41no—
381C22—2921no—
382C22—2921no—
383C22—732no—
384C22—1461no—
385C22C2×#1455667011yescharacteristic
386C4—21no—
387C4—41no—
388C4—41no—
389C4—2921no—
390C4—2921no—
391C4—5844no—
392C4—732no—
393C4—1461no—
394C4C2×#1455667011yescharacteristic
395C4C2×#1455667011yescharacteristic
396C2—21no—
397C2—41no—
398C2—2921no—
399C2—1462no—
400C2#2262637811yescharacteristic
subgroupG/H
centerC4C2×#14556670
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#405×C73C4
socleC2×C73#2283
prime radicalC2×#11924028C23
cosocleC2×C4×D73C24
hypercenter#405#64408
hypocenterC73#20788
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C4×C8 ◁ #405

Complements

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

NH ≅ G/Ntype
1. #22625861C2semidirect
2. #22625851C2semidirect
6. #22625737C2semidirect
8. #22625652C2semidirect
23. #17988544C4semidirect
25. #17988543C4semidirect
26. #17988541C4semidirect
28. M16×Dic73C4semidirect
29. C4×#11924011C4semidirect
31. C2×#14556501C4semidirect
32. #97×D73C4semidirect
34. #405×C73C4semidirect
210. C73#20788semidirect
224. #405#64408semidirect

Maximal subgroups

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

subgroupindex
#226258612
#226258512
#226258412
#226258152
#226258132
#226257372
#226256872
#226256522
#226183582
#2078873