Finite Group Databasebeta

← #32989829

Subgroups — #32989829

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×#17988874C211yesmaximal, characteristic
2#22610725C211yesmaximal, characteristic
3#22609928C211yesmaximal, characteristic
4C2×#17988338C211yesmaximal, characteristic, complemented
5C2×#17988336C211yesmaximal, characteristic, complemented
6#22607814C211yesmaximal, characteristic, complemented
7C73×#1990C211yesmaximal, characteristic, complemented
8C2×C4×D292C2211yescharacteristic
9#17988874—22no—
10#17988874—22no—
11C2×#14556522—21no—
12C2×C4⋊#10596831C2211yescharacteristic
13C2×C4⋊#10596831C2211yescharacteristic
14C2×C4⋊#10596831C2211yescharacteristic
15C2×#14556511—21no—
16C2×#14556511—21no—
17C2×#14556503—21no—
18C2×#14556503—21no—
19#17988349—21no—
20#17988338—22no—
21#17988336—22no—
22C2×C4⋊Q8×C73C2211yescharacteristic
23C2×C4×C8×C73C2211yescharacteristic
24#384×C73C2211yescharacteristic
25#340×C73—21no—
26#340×C73—21no—
27C22×D292D411yescharacteristic
28C22×C4×D73—21no—
29C22×#10596831—21no—
30C22×#10596831—21no—
31C22×#10596831—21no—
32C4×D292—21no—
33C4×D292—21no—
34C4×D292—21no—
35C4×D292—21no—
36C2×C4⋊Dic73D411yescharacteristic
37C2×#11924005—21no—
38#14556522—24no—
39C4⋊#10596831—21no—
40C4⋊#10596831—21no—
41C4⋊#10596831—21no—
42C4⋊#10596831—21no—
43C4⋊#10596831—21no—
44C4⋊#10596831D412yes—
45#14556511—24no—
46#14556511—42no—
47#14556503—42no—
48#14556503—42no—
49C22×Q8×C73—21no—
50C22×C8×C73D411yescharacteristic
51C22×C8×C73D411yescharacteristic
52C4⋊Q8×C73—21no—
53C4⋊Q8×C73—21no—
54C4⋊Q8×C73—21no—
55C4⋊Q8×C73D412yes—
56C2×C4⋊C4×C73—21no—
57C2×C4⋊C4×C73—21no—
58C2×C4⋊C4×C73—21no—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—21no—
61C2×C42×C73C2311yescharacteristic
62C4×C8×C73—21no—
63C4×C8×C73—21no—
64C23×D73—21no—
65C2×D292—21no—
66C2×D292—21no—
67C2×D292—21no—
68C2×D292—21no—
69C2×D292—21no—
70C2×D292SD1612yes—
71C22×Dic73—21no—
72C2×C4×D73—41no—
73C2×C4×D73—41no—
74C2×C4×D73—41no—
75C2×C4×D73—41no—
76C2×C4×D73—41no—
77C2×C4×D73—41no—
78C2×#10596831—22no—
79C2×#10596831—41no—
80C2×#10596831—41no—
81C2×#10596831—41no—
82C2×#10596831—41no—
83C2×#10596831—41no—
84C2×#10596831—41no—
85C2×#10596831—41no—
86C2×#10596831—41no—
87C4⋊Dic73—21no—
88C4⋊Dic73—21no—
89#11924005—41no—
90#11924005—41no—
91C2×Q8×C73—22no—
92C2×Q8×C73—41no—
93C2×Q8×C73—81no—
94C2×Q8×C73—81no—
95C22×C4×C73—41no—
96C22×C4×C73—41no—
97C22×C4×C73C2×D411yescharacteristic
98C22×C4×C73C2×D411yescharacteristic
99C22×C4×C73#4111yescharacteristic
100C2×C8×C73—21no—
101C2×C8×C73—21no—
102C2×C8×C73—21no—
103C2×C8×C73—21no—
104C2×C8×C73—21no—
105C2×C8×C73SD1612yes—
106C4⋊C4×C73—22no—
107C4⋊C4×C73—22no—
108C4⋊C4×C73—41no—
109C4⋊C4×C73—41no—
110C4⋊C4×C73—41no—
111C4⋊C4×C73—41no—
112C4⋊C4×C73—41no—
113C4⋊C4×C73—41no—
114C4⋊C4×C73—81no—
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—
127Q8×C73—82no—
128Q8×C73—82no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—42no—
132C2×C4×C73—81no—
133C2×C4×C73—81no—
134C2×C4×C73—81no—
135C2×C4×C73—81no—
136C2×C4×C73—81no—
137C2×C4×C73#12011yescharacteristic
138C2×C4×C73#12111yescharacteristic
139C2×C4×C73#12111yescharacteristic
140C2×C4×C73#12311yescharacteristic
141C2×C4×C73#12311yescharacteristic
142C2×C4×C73#12411yescharacteristic
143C2×C4×C73C2×SD1611yescharacteristic
144C2×C4×C73C2×SD1611yescharacteristic
145C2×C4×C73#13511yescharacteristic
146C2×C4×C73#13511yescharacteristic
147C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C73#13711yescharacteristic
149C2×C4×C73#13711yescharacteristic
150C2×C4×C73#13711yescharacteristic
151C2×Dic73—41no—
152C2×Dic73—41no—
153C2×Dic73—41no—
154D292—22no—
155D292—22no—
156D292—41no—
157D292—41no—
158C4×D73—81no—
159C4×D73—81no—
160C4×D73—81no—
161C4×D73—81no—
162C8×C73—22no—
163C8×C73—41no—
164#10596831—42no—
165#10596831—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—82no—
184C4×C73—161no—
185C4×C73#44811yescharacteristic
186C4×C73#45211yescharacteristic
187C4×C73#46011yescharacteristic
188C4×C73#46411yescharacteristic
189C4×C73#46511yescharacteristic
190C4×C73#46811yescharacteristic
191C4×C73#48311yescharacteristic
192C4×C73#48511yescharacteristic
193Dic73—81no—
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#1990D7311yescharacteristic, complemented
205#1954—731no—
206#1802—731no—
207C2×#334—731no—
208C2×#333—731no—
209#1296—731no—
210C73#1048011yescharacteristic, complemented, Sylow
211C2×C4⋊Q8C2×D7311yescharacteristic
212C2×C4×D4—731no—
213#476—1462no—
214#476—1462no—
215C2×#106—1461no—
216C2×C4⋊C8—731no—
217C2×C4⋊C8—731no—
218C2×C4⋊C8—731no—
219C2×#102—1461no—
220C2×#102—1461no—
221C2×C22⋊C8—1461no—
222C2×C22⋊C8—1461no—
223C2×C4×C8C2×D7311yescharacteristic
224#384C2×D7311yescharacteristic
225#340—21no—
226#340—21no—
227#340—1461no—
228#334—1462no—
229#333—1462no—
230C22×Q8—21no—
231C22×D4—731no—
232C23×C4—1461no—
233C22×C8—1461no—
234C22×C8—1461no—
235C22×C8—1461no—
236C22×C8C22⋊D7311yescharacteristic
237C22×C8C22⋊D7311yescharacteristic
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8D29212yes—
242C4×D4—1461no—
243C4×D4—1461no—
244C4×D4—1461no—
245C4×D4—1461no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—731no—
252C2×C22⋊C4—1461no—
253C2×C42C22×D7311yescharacteristic
254#106—1464no—
255C4⋊C8—732no—
256C4⋊C8—1461no—
257C4⋊C8—1461no—
258C4⋊C8—1461no—
259C4⋊C8—1461no—
260C4⋊C8—1461no—
261#102—2922no—
262#102—1464no—
263C22⋊C8—2922no—
264C22⋊C8—2922no—
265C4×C8—21no—
266C4×C8—21no—
267C24—1461no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×D4—732no—
273C2×D4—1461no—
274C2×D4—1461no—
275C2×D4—1461no—
276C2×D4—1461no—
277C2×D4—1461no—
278C22×C4—41no—
279C22×C4—41no—
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×C22⋊D7311yescharacteristic
289C22×C4#1192402711yescharacteristic
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—2921no—
296C2×C8—2921no—
297C2×C8—2921no—
298C2×C8—2921no—
299C2×C8—2921no—
300C2×C8—2921no—
301C2×C8—2921no—
302C2×C8—2921no—
303C2×C8—1462no—
304C2×C8#1192401512yes—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—81no—
314C4⋊C4—1461no—
315C4⋊C4—1461no—
316C22⋊C4—2921no—
317C22⋊C4—2921no—
318C42D4×D7311yescharacteristic
319C42C2×D29211yescharacteristic
320C42#1192402711yescharacteristic
321C42#1192402811yescharacteristic
322C23—2921no—
323C23—2921no—
324C23—2921no—
325C23—2921no—
326C23—2921no—
327C23—2921no—
328C23—1462no—
329C23#1455656211yescharacteristic
330Q8—82no—
331Q8—82no—
332D4—2921no—
333D4—2921no—
334D4—1462no—
335D4—1462no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—2921no—
345C2×C4—2921no—
346C2×C4—2921no—
347C2×C4—5841no—
348C2×C4—5841no—
349C2×C4—5841no—
350C2×C4—5841no—
351C2×C4#1455656111yescharacteristic
352C2×C4#1455656311yescharacteristic
353C2×C4#1455656511yescharacteristic
354C2×C4#1455657311yescharacteristic
355C2×C4#1455657511yescharacteristic
356C2×C4#1455657711yescharacteristic
357C2×C4SD16×D7311yescharacteristic
358C2×C4C2×#1192401511yescharacteristic
359C2×C4#1455661111yescharacteristic
360C2×C4#1455661211yescharacteristic
361C2×C4#1455661611yescharacteristic
362C2×C4#1455662211yescharacteristic
363C2×C4#1455662311yescharacteristic
364C2×C4#1455662311yescharacteristic
365C8—22no—
366C8—41no—
367C8—2922no—
368C8—5841no—
369C8—5841no—
370C22—2922no—
371C22—2922no—
372C22—2922no—
373C22—5841no—
374C22—5841no—
375C22—5841no—
376C22—5841no—
377C22#1798850311yescharacteristic
378C22#1798878911yescharacteristic
379C22#1798879911yescharacteristic
380C22#1798881311yescharacteristic
381C22#1798884211yescharacteristic
382C22#1798884711yescharacteristic
383C22#1798885411yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—161no—
388C4—5841no—
389C4#1798872711yescharacteristic
390C4#1798875211yescharacteristic
391C4#1798878711yescharacteristic
392C4#1798880911yescharacteristic
393C4#1798881811yescharacteristic
394C4#1798883511yescharacteristic
395C4#1798891611yescharacteristic
396C4#1798892611yescharacteristic
397C2—5842no—
398C2#2260930311yescharacteristic
399C2#2260933711yescharacteristic
400C2#2260945111yescharacteristic
401C2#2260950311yescharacteristic
402C2#2261112611yescharacteristic
403C2#2261122411yescharacteristic
404C2#2261124611yescharacteristic
subgroupG/H
centerC23#14556562
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1990C2
socleC23×C73#121
prime radicalC22×D292D4
hypercenter#1990D73
hypocenterC73#10480
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
4. C2×#17988338C2semidirect
5. C2×#17988336C2semidirect
6. #22607814C2semidirect
7. C73×#1990C2semidirect
204. #1990D73semidirect
210. C73#10480semidirect

Maximal subgroups

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

subgroupindex
C2×#179888742
#226107252
#226099282
C2×#179883382
C2×#179883362
#226078142
C73×#19902
#1048073