Finite Group Databasebeta

← #32982178

Subgroups — #32982178

Order 18688.

546 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988878C211yesmaximal, characteristic
2C2×#17988666C211yesmaximal, characteristic
3#22610892C211yesmaximal, characteristic, complemented
4#22609932C211yesmaximal, characteristic
5C2×#17988335C211yesmaximal, characteristic, complemented
6#22607815C211yesmaximal, characteristic, complemented
7C73×#1277C211yesmaximal, characteristic, complemented
8C22×#11924011—21no—
9C2×#14556591C2211yescharacteristic
10C2×C4×D292C2211yescharacteristic
11#17988878—22no—
12#17988878C414yescomplemented
13#17988666—22no—
14#17988666—22no—
15C2×#14556523—21no—
16C2×C4⋊#10596831C2211yescharacteristic
17C2×#14556513—21no—
18C2×#14556513—21no—
19C2×#14556503—21no—
20#17988457C2211yescharacteristic
21#17988350—21no—
22#17988350—21no—
23#17988350—21no—
24#17988335—22no—
25C2×C4⋊C8×C73C2211yescharacteristic
26C2×C4⋊C8×C73C2211yescharacteristic
27C2×C4×C8×C73C2211yescharacteristic
28C22×Dic146—21no—
29C22×D292D411yescharacteristic
30C22×C4×D73—21no—
31C2×#11924011—24no—
32C2×#11924011—42no—
33C2×#11924011—42no—
34C22×#10596831—21no—
35#14556591—21no—
36#14556591—21no—
37#14556591—21no—
38#14556591C2×C412yes—
39C4×D292—21no—
40C4×D292—21no—
41C4×D292C2×C412yes—
42C4×D292D412yes—
43C2×C4⋊Dic73—21no—
44C2×C4⋊Dic73—21no—
45C2×C4⋊Dic73—21no—
46C2×C4⋊Dic73—21no—
47C2×C4⋊Dic73—21no—
48C2×C4⋊Dic73D411yescharacteristic
49C2×#11924005—21no—
50#14556523—24no—
51C4⋊#10596831—21no—
52C4⋊#10596831D412yes—
53#14556513—24no—
54#14556513—42no—
55#14556503—24no—
56C22×C8×C73—21no—
57C22×C8×C73—21no—
58C22×C8×C73D411yescharacteristic
59C22×C8×C73D411yescharacteristic
60C2×C42×C73C2311yescharacteristic
61C4⋊C8×C73—21no—
62C4⋊C8×C73—21no—
63C4⋊C8×C73—21no—
64C4⋊C8×C73C2×C412yes—
65C4×C8×C73—21no—
66C4×C8×C73—21no—
67C23×D73—21no—
68C2×Dic146—22no—
69C2×Dic146—41no—
70C2×Dic146—81no—
71C2×Dic146—81no—
72C2×D292—21no—
73C2×D292—21no—
74C2×D292—21no—
75C2×D292—21no—
76C2×D292—21no—
77C2×D292C22⋊C412yes—
78C22×Dic73—21no—
79C22×Dic73—41no—
80C22×Dic73—41no—
81C2×C4×D73—22no—
82C2×C4×D73—22no—
83C2×C4×D73—41no—
84C2×C4×D73—41no—
85C2×C4×D73—41no—
86C2×C4×D73—41no—
87#11924011—48no—
88C2×#10596831—22no—
89C2×#10596831—41no—
90C2×#10596831—41no—
91C4⋊Dic73—21no—
92C4⋊Dic73—22no—
93C4⋊Dic73—22no—
94C4⋊Dic73—41no—
95C4⋊Dic73—41no—
96C4⋊Dic73—41no—
97C4⋊Dic73—41no—
98C4⋊Dic73—41no—
99C4⋊Dic73—41no—
100C4⋊Dic73—81no—
101C4⋊Dic73C22⋊C412yes—
102#11924005—22no—
103#11924005—22no—
104C22×C4×C73C2×D411yescharacteristic
105C22×C4×C73C2×D411yescharacteristic
106C22×C4×C73#4111yescharacteristic
107C2×C8×C73—21no—
108C2×C8×C73—21no—
109C2×C8×C73—22no—
110C2×C8×C73—21no—
111C2×C8×C73—21no—
112C2×C8×C73—21no—
113C2×C8×C73—41no—
114C2×C8×C73—41no—
115C2×C8×C73—41no—
116C2×C8×C73—41no—
117C2×C8×C73—41no—
118C2×C8×C73SD1612yes—
119C42×C73C22×C411yescharacteristic
120C42×C73C2×D411yescharacteristic
121C42×C73C2×D411yescharacteristic
122C42×C73#4111yescharacteristic
123C23×C73#12111yescharacteristic
124C22×D73—22no—
125C22×D73—41no—
126C22×D73—41no—
127C22×D73—41no—
128C22×D73—41no—
129C22×D73—41no—
130C22×D73—41no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73C2×C22⋊C411yescharacteristic
134C2×C4×C73C4×D411yescharacteristic
135C2×C4×C73C4×D411yescharacteristic
136C2×C4×C73#12011yescharacteristic
137C2×C4×C73#12111yescharacteristic
138C2×C4×C73#12311yescharacteristic
139C2×C4×C73C2×SD1611yescharacteristic
140C2×C4×C73#13511yescharacteristic
141C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C73#13711yescharacteristic
144C2×C4×C73#13711yescharacteristic
145C2×C4×C73#13711yescharacteristic
146C2×C4×C73#13711yescharacteristic
147C2×Dic73—22no—
148C2×Dic73—41no—
149C2×Dic73—41no—
150C2×Dic73—42no—
151C2×Dic73—81no—
152C2×Dic73—81no—
153C2×Dic73—81no—
154C2×Dic73—81no—
155C2×Dic73—81no—
156D292—22no—
157D292—22no—
158D292—41no—
159D292—41no—
160C4×D73—42no—
161C4×D73—42no—
162C4×D73—42no—
163C4×D73—42no—
164Dic146—82no—
165Dic146—82no—
166C8×C73—22no—
167C8×C73—42no—
168C8×C73—41no—
169C8×C73—81no—
170#10596831—42no—
171C22×C73#38611yescharacteristic
172C22×C73#46111yescharacteristic
173C22×C73#46311yescharacteristic
174C22×C73#46811yescharacteristic
175C22×C73#47011yescharacteristic
176C22×C73#47111yescharacteristic
177C22×C73#47211yescharacteristic
178C2×D73—42no—
179C2×D73—42no—
180C2×D73—42no—
181C2×D73—81no—
182C2×D73—81no—
183C2×D73—81no—
184C2×D73—81no—
185C4×C73—21no—
186C4×C73—21no—
187C4×C73#41711yescharacteristic
188C4×C73#41911yescharacteristic
189C4×C73C4×SD1611yescharacteristic
190C4×C73#44011yescharacteristic
191C4×C73#45011yescharacteristic
192C4×C73#45211yescharacteristic
193C4×C73#46811yescharacteristic
194C4×C73#47011yescharacteristic
195Dic73—42no—
196Dic73—82no—
197Dic73—161no—
198#9223—731nomaximal, Sylow
199C2×C73#169111yescharacteristic
200C2×C73#169511yescharacteristic
201C2×C73#169711yescharacteristic
202C2×C73#170311yescharacteristic
203C2×C73#187711yescharacteristic
204C2×C73#189411yescharacteristic
205C2×C73#190711yescharacteristic
206D73—82no—
207C2×#476—731no—
208C2×#436—731no—
209#1990—731no—
210#1802—731no—
211C2×#333—731no—
212#1296—731no—
213#1277D7311yescharacteristic, complemented
214C73#922311yescharacteristic, complemented, Sylow
215C22×M16—1461no—
216C2×C4⋊Q8—731no—
217C2×C4×D4—731no—
218#476—734no—
219#476—1462no—
220#436—1462no—
221#436—1462no—
222C2×#106—1461no—
223C2×C4⋊C8—731no—
224C2×C4⋊C8C2×D7311yescharacteristic
225C2×C4⋊C8C2×D7311yescharacteristic
226C2×#102—1461no—
227C2×#102—1461no—
228C2×C22⋊C8—1461no—
229C2×C4×C8C2×D7311yescharacteristic
230#384—731no—
231#340—1461no—
232#340—1461no—
233#340—1461no—
234#333—1462no—
235C22×Q8—1461no—
236C22×D4—731no—
237C23×C4—1461no—
238C2×M16—2922no—
239C2×M16—2922no—
240C2×M16—1464no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—1461no—
244C22×C8D29211yescharacteristic
245C22×C8D29211yescharacteristic
246C4⋊Q8—732no—
247C4⋊Q8—1461no—
248C4⋊Q8—1461no—
249C4⋊Q8—1461no—
250C4×D4—732no—
251C4×D4—732no—
252C4×D4—1461no—
253C4×D4—1461no—
254C2×C4⋊C4—731no—
255C2×C4⋊C4—1461no—
256C2×C4⋊C4—1461no—
257C2×C4⋊C4—1461no—
258C2×C4⋊C4—1461no—
259C2×C4⋊C4—1461no—
260C2×C22⋊C4—1461no—
261C2×C42C22×D7311yescharacteristic
262#106—1464no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—732no—
267C4⋊C8—1461no—
268C4⋊C8C4×D7312yes—
269#102—2922no—
270#102—1464no—
271C22⋊C8—1464no—
272C4×C8—21no—
273C4×C8—21no—
274C24—1461no—
275C2×Q8—2921no—
276C2×Q8—5841no—
277C2×Q8—5841no—
278C2×Q8—1462no—
279C2×D4—732no—
280C2×D4—1461no—
281C2×D4—1461no—
282C2×D4—1461no—
283C2×D4—1461no—
284C2×D4—1461no—
285C22×C4—2921no—
286C22×C4—2921no—
287C22×C4—2921no—
288C22×C4—2921no—
289C22×C4—2921no—
290C22×C4—2921no—
291C22×C4—1461no—
292C22×C4—1462no—
293C22×C4—1462no—
294C22×C4D4×D7311yescharacteristic
295C22×C4C2×D29211yescharacteristic
296C22×C4#1192402911yescharacteristic
297M16—2928no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—2921no—
310C2×C8—2921no—
311C2×C8—1462no—
312C2×C8#1192401712yes—
313C4⋊C4—2921no—
314C4⋊C4—2921no—
315C4⋊C4—2921no—
316C4⋊C4—2921no—
317C4⋊C4—2921no—
318C4⋊C4—2921no—
319C4⋊C4—5841no—
320C4⋊C4—732no—
321C4⋊C4—1462no—
322C4⋊C4—1462no—
323C4⋊C4—1461no—
324C22⋊C4—1462no—
325C22⋊C4—1462no—
326C42C2×C4×D7311yescharacteristic
327C42D4×D7311yescharacteristic
328C42D4×D7311yescharacteristic
329C42#1192402811yescharacteristic
330C23—2921no—
331C23—2921no—
332C23—2921no—
333C23—2921no—
334C23—2921no—
335C23—2921no—
336C23—1462no—
337C23#1455656611yescharacteristic
338Q8—5842no—
339Q8—5842no—
340D4—2921no—
341D4—2921no—
342D4—1462no—
343D4—1462no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—2921no—
347C2×C4—2921no—
348C2×C4—2922no—
349C2×C4—2922no—
350C2×C4—2922no—
351C2×C4—2922no—
352C2×C4—2922no—
353C2×C4—5841no—
354C2×C4—5841no—
355C2×C4—5841no—
356C2×C4—5841no—
357C2×C4—5841no—
358C2×C4—1462no—
359C2×C4C22⋊C4×D7311yescharacteristic
360C2×C4#1455655311yescharacteristic
361C2×C4C4×D29211yescharacteristic
362C2×C4#1455656111yescharacteristic
363C2×C4#1455656411yescharacteristic
364C2×C4#1455657611yescharacteristic
365C2×C4C2×#1192401711yescharacteristic
366C2×C4#1455661311yescharacteristic
367C2×C4#1455661511yescharacteristic
368C2×C4#1455661911yescharacteristic
369C2×C4#1455662011yescharacteristic
370C2×C4#1455662011yescharacteristic
371C2×C4#1455662311yescharacteristic
372C2×C4#1455662411yescharacteristic
373C8—22no—
374C8—41no—
375C8—42no—
376C8—81no—
377C8—2922no—
378C22—2922no—
379C22—2922no—
380C22—2922no—
381C22—5841no—
382C22—5841no—
383C22—5841no—
384C22—5841no—
385C22#1798846511yescharacteristic
386C22#1798879311yescharacteristic
387C22#1798880311yescharacteristic
388C22#1798883411yescharacteristic
389C22#1798884611yescharacteristic
390C22#1798885111yescharacteristic
391C22#1798885811yescharacteristic
392C4—21no—
393C4—21no—
394C4—2922no—
395C4—5842no—
396C4—11681no—
397C4#1798858511yescharacteristic
398C4#1798859411yescharacteristic
399C4C4×#1192401711yescharacteristic
400C4#1798868911yescharacteristic
401C4#1798874011yescharacteristic
402C4#1798875111yescharacteristic
403C4#1798883211yescharacteristic
404C4#1798884411yescharacteristic
405C2—5842no—
406C2#2260943111yescharacteristic
407C2#2260945511yescharacteristic
408C2#2260946511yescharacteristic
409C2#2260950711yescharacteristic
410C2#2261026411yescharacteristic
411C2#2261036311yescharacteristic
412C2#2261043811yescharacteristic
subgroupG/H
centerC23#14556566
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1277C2
socleC23×C73#121
prime radicalC22×D292D4
cosocleC2×C42×C73C23
hypercenter#1277D73
hypocenterC73#9223
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
3. #22610892C2semidirect
5. C2×#17988335C2semidirect
6. #22607815C2semidirect
7. C73×#1277C2semidirect
12. #17988878C4semidirect
213. #1277D73semidirect
214. C73#9223semidirect

Maximal subgroups

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

subgroupindex
C2×#179888782
C2×#179886662
#226108922
#226099322
C2×#179883352
#226078152
C73×#12772
#922373