Finite Group Databasebeta

← #32989832

Subgroups — #32989832

Order 18688.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988878C211yesmaximal, characteristic
2#22610892C211yesmaximal, characteristic, complemented
3#22610725C211yesmaximal, characteristic
4#22609932C211yesmaximal, characteristic
5C2×#17988335C211yesmaximal, characteristic, complemented
6#22607815C211yesmaximal, characteristic, complemented
7C2×#334×C73C211yesmaximal, characteristic, complemented
8C2×#14556591C2211yescharacteristic
9C2×C4×D292C2211yescharacteristic
10#17988878—22no—
11#17988878—22no—
12C2×#14556523—21no—
13C2×C4⋊#10596831C2211yescharacteristic
14C2×#14556513—21no—
15C2×#14556513—21no—
16C2×#14556503—21no—
17C2×#14556503—21no—
18#17988457C2211yescharacteristic
19#17988350—21no—
20#17988350—21no—
21#17988350—21no—
22#17988335—22no—
23C2×C4⋊C8×C73C2211yescharacteristic
24C2×C4⋊C8×C73C2211yescharacteristic
25C2×C4×C8×C73C2211yescharacteristic
26#334×C73—22no—
27C22×Dic146—21no—
28C22×D292D411yescharacteristic
29C22×C4×D73—21no—
30C22×#10596831—21no—
31#14556591—21no—
32#14556591—21no—
33#14556591—21no—
34#14556591D412yes—
35C4×D292—21no—
36C4×D292—21no—
37C4×D292—21no—
38C4×D292—21no—
39C2×C4⋊Dic73—21no—
40C2×C4⋊Dic73—21no—
41C2×C4⋊Dic73—21no—
42C2×C4⋊Dic73—21no—
43C2×C4⋊Dic73—21no—
44C2×C4⋊Dic73D411yescharacteristic
45C2×#11924005—21no—
46#14556523—24no—
47C4⋊#10596831—21no—
48C4⋊#10596831—21no—
49#14556513—24no—
50#14556513—42no—
51#14556503—42no—
52#14556503—42no—
53C22×C8×C73—21no—
54C22×C8×C73—21no—
55C22×C8×C73D411yescharacteristic
56C22×C8×C73D411yescharacteristic
57C2×C42×C73C2311yescharacteristic
58C4⋊C8×C73—21no—
59C4⋊C8×C73—21no—
60C4⋊C8×C73—21no—
61C4⋊C8×C73D412yes—
62C4×C8×C73—21no—
63C4×C8×C73—21no—
64C23×D73—21no—
65C2×Dic146—22no—
66C2×Dic146—41no—
67C2×Dic146—81no—
68C2×Dic146—81no—
69C2×D292—21no—
70C2×D292—21no—
71C2×D292—21no—
72C2×D292—21no—
73C2×D292—21no—
74C2×D292SD1612yes—
75C22×Dic73—21no—
76C22×Dic73—41no—
77C22×Dic73—41no—
78C2×C4×D73—41no—
79C2×C4×D73—41no—
80C2×C4×D73—41no—
81C2×C4×D73—41no—
82C2×C4×D73—41no—
83C2×C4×D73—41no—
84C2×#10596831—41no—
85C2×#10596831—41no—
86C2×#10596831—41no—
87C4⋊Dic73—22no—
88C4⋊Dic73—21no—
89C4⋊Dic73—22no—
90C4⋊Dic73—21no—
91C4⋊Dic73—41no—
92C4⋊Dic73—41no—
93C4⋊Dic73—41no—
94C4⋊Dic73—41no—
95C4⋊Dic73—41no—
96C4⋊Dic73—41no—
97C4⋊Dic73—81no—
98#11924005—41no—
99#11924005—41no—
100C22×C4×C73C2×D411yescharacteristic
101C22×C4×C73C2×D411yescharacteristic
102C22×C4×C73#4111yescharacteristic
103C2×C8×C73—21no—
104C2×C8×C73—21no—
105C2×C8×C73—22no—
106C2×C8×C73—21no—
107C2×C8×C73—21no—
108C2×C8×C73—21no—
109C2×C8×C73—41no—
110C2×C8×C73—41no—
111C2×C8×C73—41no—
112C2×C8×C73—41no—
113C2×C8×C73—41no—
114C2×C8×C73SD1612yes—
115C42×C73C2×D411yescharacteristic
116C42×C73C2×D411yescharacteristic
117C42×C73#4111yescharacteristic
118C42×C73#4111yescharacteristic
119C23×C73#12111yescharacteristic
120C22×D73—22no—
121C22×D73—41no—
122C22×D73—41no—
123C22×D73—41no—
124C22×D73—41no—
125C22×D73—41no—
126C22×D73—41no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73#12011yescharacteristic
130C2×C4×C73#12111yescharacteristic
131C2×C4×C73#12111yescharacteristic
132C2×C4×C73#12311yescharacteristic
133C2×C4×C73#12311yescharacteristic
134C2×C4×C73#12411yescharacteristic
135C2×C4×C73C2×SD1611yescharacteristic
136C2×C4×C73C2×SD1611yescharacteristic
137C2×C4×C73#13511yescharacteristic
138C2×C4×C73#13511yescharacteristic
139C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C73#13711yescharacteristic
141C2×C4×C73#13711yescharacteristic
142C2×C4×C73#13711yescharacteristic
143C2×Dic73—41no—
144C2×Dic73—42no—
145C2×Dic73—41no—
146C2×Dic73—41no—
147C2×Dic73—81no—
148C2×Dic73—81no—
149C2×Dic73—81no—
150C2×Dic73—81no—
151C2×Dic73—81no—
152D292—22no—
153D292—22no—
154D292—41no—
155D292—41no—
156C4×D73—81no—
157C4×D73—81no—
158C4×D73—81no—
159C4×D73—81no—
160Dic146—82no—
161Dic146—82no—
162C8×C73—22no—
163C8×C73—42no—
164C8×C73—41no—
165C8×C73—81no—
166#10596831—81no—
167C22×C73#39411yescharacteristic
168C22×C73#46111yescharacteristic
169C22×C73#46311yescharacteristic
170C22×C73#46511yescharacteristic
171C22×C73#47011yescharacteristic
172C22×C73#47111yescharacteristic
173C22×C73#47211yescharacteristic
174C2×D73—42no—
175C2×D73—42no—
176C2×D73—42no—
177C2×D73—81no—
178C2×D73—81no—
179C2×D73—81no—
180C2×D73—81no—
181C4×C73—21no—
182C4×C73—21no—
183C4×C73#44811yescharacteristic
184C4×C73#45211yescharacteristic
185C4×C73#46011yescharacteristic
186C4×C73#46411yescharacteristic
187C4×C73#46511yescharacteristic
188C4×C73#46811yescharacteristic
189C4×C73#48311yescharacteristic
190C4×C73#48511yescharacteristic
191Dic73—82no—
192Dic73—81no—
193Dic73—161no—
194#10480—731nomaximal, Sylow
195C2×C73#167111yescharacteristic
196C2×C73#167711yescharacteristic
197C2×C73#169511yescharacteristic
198C2×C73#170311yescharacteristic
199C2×C73#203511yescharacteristic
200C2×C73#205111yescharacteristic
201C2×C73#205511yescharacteristic
202D73—82no—
203C2×#476—731no—
204#1990—731no—
205#1954—731no—
206#1802—731no—
207C2×#334D7311yescharacteristic, complemented
208C2×#333—731no—
209#1296—731no—
210C73#1048011yescharacteristic, complemented, Sylow
211C2×C4⋊Q8—731no—
212C2×C4×D4—731no—
213#476—1462no—
214#476—1462no—
215C2×#106—1461no—
216C2×C4⋊C8—731no—
217C2×C4⋊C8C2×D7311yescharacteristic
218C2×C4⋊C8C2×D7311yescharacteristic
219C2×#102—1461no—
220C2×#102—1461no—
221C2×C22⋊C8—1461no—
222C2×C22⋊C8—1461no—
223C2×C4×C8C2×D7311yescharacteristic
224#384—731no—
225#340—1461no—
226#340—1461no—
227#340—1461no—
228#334—22no—
229#333—1462no—
230C22×Q8—1461no—
231C22×D4—731no—
232C23×C4—1461no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—1461no—
236C22×C8D29211yescharacteristic
237C22×C8D29211yescharacteristic
238C4⋊Q8—732no—
239C4⋊Q8—1461no—
240C4⋊Q8—1461no—
241C4⋊Q8—1461no—
242C4×D4—1461no—
243C4×D4—1461no—
244C4×D4—1461no—
245C4×D4—1461no—
246C2×C4⋊C4—731no—
247C2×C4⋊C4—1461no—
248C2×C4⋊C4—1461no—
249C2×C4⋊C4—1461no—
250C2×C4⋊C4—1461no—
251C2×C4⋊C4—1461no—
252C2×C22⋊C4—1461no—
253C2×C42C22×D7311yescharacteristic
254#106—1464no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—1461no—
259C4⋊C8—1461no—
260C4⋊C8C22⋊D7312yes—
261#102—2922no—
262#102—1464no—
263C22⋊C8—2922no—
264C22⋊C8—2922no—
265C4×C8—21no—
266C4×C8—21no—
267C24—1461no—
268C2×Q8—2921no—
269C2×Q8—5841no—
270C2×Q8—5841no—
271C2×Q8—1462no—
272C2×D4—732no—
273C2×D4—1461no—
274C2×D4—1461no—
275C2×D4—1461no—
276C2×D4—1461no—
277C2×D4—1461no—
278C22×C4—2921no—
279C22×C4—2921no—
280C22×C4—2921no—
281C22×C4—2921no—
282C22×C4—2921no—
283C22×C4—2921no—
284C22×C4—2921no—
285C22×C4—2921no—
286C22×C4—1461no—
287C22×C4D4×D7311yescharacteristic
288C22×C4C2×D29211yescharacteristic
289C22×C4#1192402911yescharacteristic
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—2921no—
302C2×C8—2921no—
303C2×C8—2921no—
304C2×C8#1192401712yes—
305C4⋊C4—2921no—
306C4⋊C4—2921no—
307C4⋊C4—2921no—
308C4⋊C4—2921no—
309C4⋊C4—2921no—
310C4⋊C4—2921no—
311C4⋊C4—5841no—
312C4⋊C4—1462no—
313C4⋊C4—1461no—
314C4⋊C4—1462no—
315C4⋊C4—1461no—
316C22⋊C4—2921no—
317C22⋊C4—2921no—
318C42D4×D7311yescharacteristic
319C42C2×C22⋊D7311yescharacteristic
320C42#1192402811yescharacteristic
321C42#1192402911yescharacteristic
322C23—2921no—
323C23—2921no—
324C23—2921no—
325C23—2921no—
326C23—2921no—
327C23—2921no—
328C23—1462no—
329C23#1455656611yescharacteristic
330Q8—5842no—
331Q8—5842no—
332D4—2921no—
333D4—2921no—
334D4—1462no—
335D4—1462no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—2921no—
339C2×C4—2922no—
340C2×C4—2921no—
341C2×C4—2921no—
342C2×C4—5841no—
343C2×C4—5841no—
344C2×C4—5841no—
345C2×C4—5841no—
346C2×C4—5841no—
347C2×C4—5841no—
348C2×C4—5841no—
349C2×C4—5841no—
350C2×C4—5841no—
351C2×C4#1455655911yescharacteristic
352C2×C4#1455656511yescharacteristic
353C2×C4#1455656611yescharacteristic
354C2×C4#1455657611yescharacteristic
355C2×C4#1455657611yescharacteristic
356C2×C4#1455658011yescharacteristic
357C2×C4SD16×D7311yescharacteristic
358C2×C4C2×#1192401711yescharacteristic
359C2×C4#1455661211yescharacteristic
360C2×C4#1455661311yescharacteristic
361C2×C4#1455661911yescharacteristic
362C2×C4#1455662011yescharacteristic
363C2×C4#1455662011yescharacteristic
364C2×C4#1455662411yescharacteristic
365C8—22no—
366C8—42no—
367C8—41no—
368C8—81no—
369C8—5841no—
370C22—2922no—
371C22—2922no—
372C22—2922no—
373C22—5841no—
374C22—5841no—
375C22—5841no—
376C22—5841no—
377C22#1798850411yescharacteristic
378C22#1798879311yescharacteristic
379C22#1798880311yescharacteristic
380C22#1798881711yescharacteristic
381C22#1798884611yescharacteristic
382C22#1798885111yescharacteristic
383C22#1798885811yescharacteristic
384C4—21no—
385C4—21no—
386C4—5842no—
387C4—5841no—
388C4—11681no—
389C4#1798872411yescharacteristic
390C4#1798874611yescharacteristic
391C4#1798878811yescharacteristic
392C4#1798881011yescharacteristic
393C4#1798881611yescharacteristic
394C4#1798883311yescharacteristic
395C4#1798891911yescharacteristic
396C4#1798893111yescharacteristic
397C2—5842no—
398C2#2260930711yescharacteristic
399C2#2260934111yescharacteristic
400C2#2260945511yescharacteristic
401C2#2260950711yescharacteristic
402C2#2261112911yescharacteristic
403C2#2261122511yescharacteristic
404C2#2261124911yescharacteristic
subgroupG/H
centerC23#14556566
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC2×#334×C73C2
socleC23×C73#121
prime radicalC22×D292D4
hypercenterC2×#334D73
hypocenterC73#10480
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
2. #22610892C2semidirect
5. C2×#17988335C2semidirect
6. #22607815C2semidirect
7. C2×#334×C73C2semidirect
207. C2×#334D73semidirect
210. C73#10480semidirect

Maximal subgroups

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

subgroupindex
C2×#179888782
#226108922
#226107252
#226099322
C2×#179883352
#226078152
C2×#334×C732
#1048073