Finite Group Databasebeta

← #34021312

Subgroups — #34021312

Order 18688.

616 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22625862C212yesmaximal, complemented
2#22625851C212yesmaximal, complemented
3#22625841C212yesmaximal
4#22625816C211yesmaximal, characteristic
5#22625814C211yesmaximal, characteristic
6#22625733C212yesmaximal, complemented
7#22625686C212yesmaximal
8#22625652C212yesmaximal, complemented
9#22618353C211yesmaximal, 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
28#17988539C411yescharacteristic, complemented
29#17988539C2211yescharacteristic
30M16×Dic73C411yescharacteristic, complemented
31C2×C8×Dic73C411yescharacteristic, complemented
32C4×C8×D73C411yescharacteristic, complemented
33C4×C8×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—22no—
49#14556599—22no—
50#14556597—22no—
51#14556597—22no—
52C2×#11924012C2×C411yescharacteristic
53C2×#11924011C2×C411yescharacteristic
54C2×#11924011C2×C411yescharacteristic
55C2×#11924011C2311yescharacteristic
56C2×#11924011C2311yescharacteristic
57C22×#10596831C2×C411yescharacteristic
58C2×C8×D73C2×C411yescharacteristic
59C2×C8×D73C2×C411yescharacteristic
60C2×C8×D73C2311yescharacteristic
61C2×C8×D73C2311yescharacteristic
62#14556556—21no—
63#14556553—21no—
64#14556551—21no—
65#14556551—21no—
66#14556548C2×C411yescharacteristic
67#14556548C2311yescharacteristic
68C2×C4×Dic73C2×C411yescharacteristic
69C42×D73C2×C411yescharacteristic
70C42×D73C2311yescharacteristic
71#14556502C2×C411yescharacteristic
72#14556502C2×C411yescharacteristic
73#14556501C2×C411yescharacteristic
74#14556501C2×C411yescharacteristic
75C4×#10596831C2×C411yescharacteristic
76C4×#10596831C2×C411yescharacteristic
77C8×Dic73C2×C411yescharacteristic
78C8×Dic73C2×C411yescharacteristic
79C8×Dic73C2×C411yescharacteristic
80C8×Dic73C2×C411yescharacteristic
81C8×Dic73C2×C411yescharacteristic
82C8×Dic73C2×C411yescharacteristic
83C2×M16×C73C2×C411yescharacteristic
84C22×C8×C73C2×C411yescharacteristic
85#117×C73C2×C411yescharacteristic
86#97×C73C2×C411yescharacteristic
87#97×C73C2×C411yescharacteristic
88C4×C8×C73C2×C411yescharacteristic
89C4×C8×C73C2×C411yescharacteristic
90#11924038—24no—
91#11924037—24no—
92#11924036—24no—
93C4⋊#64408—24no—
94C22⋊#64408—24no—
95C4×#64408—24no—
96#11924028—22no—
97#11924028—22no—
98C2×Dic146#4111yescharacteristic
99C2×D292#4111yescharacteristic
100C2×C22⋊D73#4111yescharacteristic
101C2×C22⋊D73#4111yescharacteristic
102C22×Dic73—21no—
103C2×C4×D73—21no—
104C2×C4×D73—21no—
105C2×C4×D73C22×C411yescharacteristic
106C2×C4×D73C2411yescharacteristic
107#11924012—22no—
108#11924011—22no—
109#11924011—22no—
110#11924011—22no—
111#11924011—22no—
112C2×#10596831—21no—
113C2×#10596831—21no—
114C2×#10596831C22×C411yescharacteristic
115C2×#10596831C22×C411yescharacteristic
116C2×#10596831C22×C411yescharacteristic
117C2×#10596831C22×C411yescharacteristic
118C8×D73—21no—
119C8×D73—21no—
120C8×D73—21no—
121C8×D73—21no—
122C8×D73#4112yes—
123C8×D73#4112yes—
124C8×D73#4112yes—
125C8×D73#4112yes—
126#11924007—21no—
127#11924007—21no—
128#11924005—21no—
129#11924005—21no—
130C4×Dic73—21no—
131C4×Dic73—21no—
132C4×Dic73—21no—
133C4×Dic73C22×C411yescharacteristic
134C4×Dic73C22×C411yescharacteristic
135C22×C4×C73C22×C411yescharacteristic
136M16×C73—22no—
137C2×C8×C73—21no—
138C2×C8×C73—21no—
139C2×C8×C73C22×C411yescharacteristic
140C2×C8×C73C22×C411yescharacteristic
141C2×C8×C73C22×C411yescharacteristic
142C2×C8×C73C22×C411yescharacteristic
143C4⋊C4×C73—21no—
144C22⋊C4×C73—21no—
145C42×C73C22×C411yescharacteristic
146C42×C73C22×C411yescharacteristic
147C23×C73#11711yescharacteristic
148C22×D73#11711yescharacteristic
149C22×D73C2×#4111yescharacteristic
150C2×#64408—44no—
151C2×C4×C73—21no—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73#11711yescharacteristic
156C2×C4×C73C23×C411yescharacteristic
157C22⋊D73—22no—
158C22⋊D73—22no—
159C2×Dic73—21no—
160C2×Dic73—21no—
161C2×Dic73—42no—
162C2×Dic73#11711yescharacteristic
163C2×Dic73C2×#4111yescharacteristic
164D292—22no—
165C4×D73—21no—
166C4×D73—21no—
167C4×D73—42no—
168C4×D73—42no—
169C4×D73C2×#4112yes—
170C4×D73C2×#4112yes—
171Dic146—22no—
172#10596833—44no—
173C8×C73—21no—
174C8×C73—21no—
175C8×C73#11711yescharacteristic
176C8×C73#11712yes—
177C8×C73#11711yescharacteristic
178#10596831—21no—
179#10596831—21no—
180#10596831#11711yescharacteristic
181#10596831#11711yescharacteristic
182#10596831#11712yes—
183C22×C73—21no—
184C22×C73—41no—
185C22×C73C2×#11711yescharacteristic
186C2×D73—21no—
187C2×D73—41no—
188C2×D73—41no—
189C2×D73#52912yes—
190#64408—84no—
191C4×C73—21no—
192C4×C73—41no—
193C4×C73—41no—
194C4×C73C2×#11711yescharacteristic
195C4×C73C2×#11711yescharacteristic
196Dic73—22no—
197Dic73—21no—
198Dic73—41no—
199Dic73#52912yes—
200#20787—731nomaximal, Sylow
201C2×C73—21no—
202C2×C73—41no—
203C2×C73#228311yescharacteristic
204D73—22no—
205D73—41no—
206C4×#131—731no—
207#1841—732no—
208#1833—732no—
209#1827—732no—
210#1813—731no—
211#1812—731no—
212#1772—732no—
213#1757—732no—
214#1747—732no—
215C73#2078711yescharacteristic, complemented, Sylow
216C2×#131—731no—
217C2×#131—731no—
218C4×#41—731no—
219C2×#108—732no—
220#427—732no—
221#413—732no—
222#410—732no—
223#405—731no—
224#405—731no—
225#405—731no—
226#405—731no—
227#405—731no—
228#405—734no—
229#405#6440811yescharacteristic, complemented
230C4×M16—731no—
231C4×M16—731no—
232C4×M16—731no—
233C2×C4×C8—731no—
234C2×C4×C8—731no—
235C2×C4×C8—731no—
236#344—734no—
237#339—734no—
238#338—732no—
239H(3,ℤ/4)—732no—
240H(3,ℤ/4)—734no—
241C2×#41—731no—
242#131—1462no—
243#131—1462no—
244#131—1462no—
245#131—1462no—
246C2×M16—731no—
247C2×M16—731no—
248C2×M16—731no—
249C2×M16—731no—
250C2×M16—731no—
251C2×M16—734no—
252C2×M16C2×#6440811yescharacteristic
253C22×C8—731no—
254C22×C8—731no—
255C22×C8—731no—
256C22×C8—731no—
257C22×C8—731no—
258C22×C8C2×#6440811yescharacteristic
259C4×Q8—1461no—
260C4×D4—1461no—
261C4×D4—1461no—
262C4×D4—1461no—
263#117—731no—
264#117—731no—
265#117—734no—
266#117C2×#6440811yescharacteristic
267C2×C42—731no—
268C2×C42—731no—
269C2×C42—731no—
270#108—1462no—
271#108—1462no—
272#107—1462no—
273#106—1462no—
274#101—1462no—
275#100—1462no—
276#99—1462no—
277#99—1462no—
278#97—731no—
279#97—731no—
280#97—731no—
281#97—731no—
282#97—1464no—
283#97C2×#6440811yescharacteristic
284#97C2×#6440811yescharacteristic
285C4×C8—731no—
286C4×C8—731no—
287C4×C8—731no—
288C4×C8—731no—
289C4×C8—731no—
290C4×C8—731no—
291C4×C8—731no—
292C4×C8—731no—
293C4×C8—1464no—
294C4×C8C2×#6440811yescharacteristic
295C4×C8C2×#6440811yescharacteristic
296#41—1462no—
297#41—1462no—
298C2×Q8—731no—
299C2×D4—731no—
300C2×D4—731no—
301C2×D4—731no—
302C22×C4—731no—
303C22×C4—731no—
304C22×C4—1461no—
305C22×C4—1461no—
306C22×C4—1461no—
307C22×C4C22×#6440811yescharacteristic
308M16—22no—
309M16—1462no—
310M16—1462no—
311M16—1462no—
312M16—1464no—
313M16—1464no—
314M16—1462no—
315M16—1462no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—731no—
319C2×C8—731no—
320C2×C8—732no—
321C2×C8—732no—
322C2×C8—732no—
323C2×C8—732no—
324C2×C8—731no—
325C2×C8—731no—
326C2×C8—1461no—
327C2×C8—1461no—
328C2×C8—1464no—
329C2×C8—1461no—
330C2×C8—1461no—
331C2×C8—1461no—
332C2×C8—1461no—
333C2×C8C22×#6440811yescharacteristic
334C2×C8C22×#6440811yescharacteristic
335C2×C8C22×#6440811yescharacteristic
336C2×C8C22×#6440811yescharacteristic
337C4⋊C4—21no—
338C4⋊C4—1461no—
339C4⋊C4—1461no—
340C4⋊C4—1464no—
341C22⋊C4—21no—
342C22⋊C4—1461no—
343C22⋊C4—1461no—
344C22⋊C4—1464no—
345C42—731no—
346C42—731no—
347C42—1461no—
348C42—1461no—
349C42—1464no—
350C42—1461no—
351C42C22×#6440811yescharacteristic
352C42C22×#6440811yescharacteristic
353C23—731no—
354C23—731no—
355C23#1455667011yescharacteristic
356Q8—1462no—
357D4—1462no—
358D4—1462no—
359D4—1462no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—2922no—
365C2×C4—2922no—
366C2×C4—2922no—
367C2×C4—2924no—
368C2×C4—731no—
369C2×C4—732no—
370C2×C4—732no—
371C2×C4—731no—
372C2×C4—1461no—
373C2×C4—1461no—
374C2×C4—1461no—
375C2×C4—1461no—
376C2×C4#1455667011yescharacteristic
377C2×C4C23×#6440811yescharacteristic
378C8—21no—
379C8—21no—
380C8—2924no—
381C8—731no—
382C8—731no—
383C8—732no—
384C8—1461no—
385C8—1461no—
386C8#1455667011yescharacteristic
387C8#1455667012yes—
388C8#1455667011yescharacteristic
389C22—21no—
390C22—41no—
391C22—2921no—
392C22—2921no—
393C22—732no—
394C22—1461no—
395C22C2×#1455667011yescharacteristic
396C4—21no—
397C4—41no—
398C4—41no—
399C4—2921no—
400C4—5844no—
401C4—732no—
402C4—1462no—
403C4—1461no—
404C4C2×#1455667011yescharacteristic
405C4C2×#1455667011yescharacteristic
406C2—21no—
407C2—41no—
408C2—2921no—
409C2—1462no—
410C2#2262637811yescharacteristic
subgroupG/H
centerC8#14556670
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#405×C73C4
socleC2×C73#2283
prime radicalC2×#11924028C23
cosocleC2×C4×D73C24
hypercenter#405#64408
hypocenterC73#20787
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C8 ◁ C4×C8 ◁ #405

Complements

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

NH ≅ G/Ntype
1. #22625862C2semidirect
2. #22625851C2semidirect
6. #22625733C2semidirect
8. #22625652C2semidirect
23. #17988544C4semidirect
25. #17988543C4semidirect
26. #17988541C4semidirect
28. #17988539C4semidirect
30. M16×Dic73C4semidirect
31. C2×C8×Dic73C4semidirect
32. C4×C8×D73C4semidirect
34. #405×C73C4semidirect
215. C73#20787semidirect
229. #405#64408semidirect

Maximal subgroups

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

subgroupindex
#226258622
#226258512
#226258412
#226258162
#226258142
#226257332
#226256862
#226256522
#226183532
#2078773