Finite Group Databasebeta

← #32975550

Subgroups — #32975550

Order 18688.

638 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988643C211yesmaximal, characteristic
2C2×#17988638C211yesmaximal, characteristic
3#22612335C211yesmaximal, characteristic
4#22607772C211yesmaximal, characteristic, complemented
5#22607771C211yesmaximal, characteristic, complemented
6#22607770C211yesmaximal, characteristic, complemented
7C73×#1279C211yesmaximal, characteristic, complemented
8C2×#14556548—21no—
9C2×C42×D73C2211yescharacteristic
10#17988643—22no—
11#17988643C414yescomplemented
12#17988638—22no—
13#17988638C414yescomplemented
14C2×C4⋊#10596831C2211yescharacteristic
15C2×#14556520—21no—
16C2×#14556503—21no—
17C2×#14556503—21no—
18C2×#14556501—21no—
19C2×C4×#10596831C2211yescharacteristic
20#17988456C412yes—
21#17988451—21no—
22#17988446—21no—
23#17988444C412yes—
24C4×#11924007—21no—
25C4×C4⋊Dic73C2211yescharacteristic
26C2×C4⋊C8×C73C2211yescharacteristic
27C2×C4×C8×C73C2211yescharacteristic
28C4×C4⋊C4×C73C2211yescharacteristic
29C22×C4×D73C2×C411yescharacteristic
30C22×C4×D73D411yescharacteristic
31C22×C4×D73D411yescharacteristic
32C22×#10596831—21no—
33C22×#10596831—21no—
34#14556548—42no—
35#14556548—42no—
36C2×C4⋊Dic73C2×C412yes—
37C2×#11924007—22no—
38C2×#11924007—22no—
39C2×#11924005—22no—
40C2×C4×Dic73—21no—
41C2×C4×Dic73C2×C411yescharacteristic
42C2×C4×Dic73D411yescharacteristic
43C2×C4×Dic73D411yescharacteristic
44C42×D73C2×C412yes—
45C42×D73C2×C412yes—
46C42×D73D412yes—
47C42×D73Q812yes—
48C4⋊#10596831—21no—
49C4⋊#10596831C2×C412yes—
50#14556520—24no—
51#14556503—24no—
52#14556503—24no—
53#14556501—24no—
54C4×#10596831—21no—
55C4×#10596831C2×C412yes—
56#14556498—22no—
57#14556498—21no—
58C22×C8×C73—21no—
59C22×C8×C73—21no—
60C2×C4⋊C4×C73C2×C412yes—
61C2×C42×C73—21no—
62C2×C42×C73C2311yescharacteristic
63C4⋊C8×C73—21no—
64C4⋊C8×C73C2×C412yes—
65C4×C8×C73—21no—
66C4×C8×C73C2×C412yes—
67#95×C73—21no—
68C23×D73C22⋊C411yescharacteristic
69C22×Dic73—22no—
70C22×Dic73C22⋊C411yescharacteristic
71C22×Dic73C22⋊C411yescharacteristic
72C22×Dic73C22⋊C411yescharacteristic
73C2×C4×D73—22no—
74C2×C4×D73—22no—
75C2×C4×D73—21no—
76C2×C4×D73—21no—
77C2×C4×D73—21no—
78C2×C4×D73—21no—
79C2×C4×D73—42no—
80C2×C4×D73C4212yes—
81C2×C4×D73C22⋊C412yes—
82C2×C4×D73C22⋊C412yes—
83C2×C4×D73C4⋊C412yes—
84C2×C4×D73C4⋊C412yes—
85C2×C4×D73C4⋊C412yes—
86C2×#10596831—22no—
87C2×#10596831—22no—
88C2×#10596831—42no—
89C2×#10596831—41no—
90C2×#10596831—41no—
91C4⋊Dic73—22no—
92C4⋊Dic73—22no—
93#11924007—44no—
94#11924007—44no—
95#11924005—44no—
96C4×Dic73—41no—
97C4×Dic73—41no—
98C4×Dic73C4212yes—
99C4×Dic73C22⋊C412yes—
100C4×Dic73C22⋊C412yes—
101C4×Dic73C4⋊C412yes—
102C4×Dic73C4⋊C412yes—
103C4×Dic73C4⋊C412yes—
104C22×C4×C73—22no—
105C22×C4×C73C22×C411yescharacteristic
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108C2×C8×C73—22no—
109C2×C8×C73—22no—
110C2×C8×C73—42no—
111C2×C8×C73—41no—
112C2×C8×C73—41no—
113C4⋊C4×C73—22no—
114C4⋊C4×C73—22no—
115C42×C73—41no—
116C42×C73—41no—
117C42×C73C22×C411yescharacteristic
118C42×C73C22×C411yescharacteristic
119C42×C73C2×D411yescharacteristic
120C42×C73C2×Q811yescharacteristic
121C23×C73C2×C22⋊C411yescharacteristic
122C22×D73—22no—
123C22×D73—21no—
124C22×D73—21no—
125C22×D73—22no—
126C22×D73#9512yes—
127C2×C4×C73—22no—
128C2×C4×C73—42no—
129C2×C4×C73—42no—
130C2×C4×C73—42no—
131C2×C4×C73C2×C4211yescharacteristic
132C2×C4×C73C2×C22⋊C411yescharacteristic
133C2×C4×C73C2×C22⋊C411yescharacteristic
134C2×C4×C73C2×C4⋊C411yescharacteristic
135C2×C4×C73C2×C4⋊C411yescharacteristic
136C2×C4×C73C2×C4⋊C411yescharacteristic
137C2×C4×C73#13112yes—
138C2×C4×C73#13112yes—
139C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C73#13711yescharacteristic
142C2×C4×C73#13711yescharacteristic
143C2×Dic73—22no—
144C2×Dic73—21no—
145C2×Dic73—21no—
146C2×Dic73—22no—
147C2×Dic73—42no—
148C2×Dic73—42no—
149C2×Dic73—42no—
150C2×Dic73#9512yes—
151C2×Dic73#9512yes—
152C2×Dic73#9512yes—
153C4×D73—22no—
154C4×D73—22no—
155C4×D73—22no—
156C4×D73—22no—
157C4×D73—24no—
158C4×D73—22no—
159C4×D73—22no—
160C4×D73—42no—
161C4×D73—42no—
162C8×C73—42no—
163C8×C73—42no—
164#10596831—42no—
165#10596831—42no—
166C22×C73C2×#9511yescharacteristic
167C22×C73#40812yes—
168C22×C73#41811yescharacteristic
169C22×C73#41911yescharacteristic
170C22×C73#42112yes—
171C2×D73—22no—
172C2×D73—22no—
173C2×D73—22no—
174C2×D73—42no—
175C2×D73—42no—
176C4×C73—21no—
177C4×C73—21no—
178C4×C73—82no—
179C4×C73#40511yescharacteristic
180C4×C73#40511yescharacteristic
181C4×C73#41711yescharacteristic
182C4×C73#41711yescharacteristic
183C4×C73#42412yes—
184C4×C73#42811yescharacteristic
185C4×C73#42811yescharacteristic
186Dic73—22no—
187Dic73—22no—
188Dic73—22no—
189Dic73—82no—
190#7900—731nomaximal, Sylow
191C2×C73#149612yes—
192C2×C73#149712yes—
193C2×C73#172911yescharacteristic
194C2×C73#173311yescharacteristic
195C2×C73#173611yescharacteristic
196D73—42no—
197C2×#432—731no—
198C2×#431—731no—
199#2285—731no—
200#1279—731no—
201#1279—731no—
202#1279—731no—
203#1279D7311yescharacteristic, complemented
204C73#790011yescharacteristic, complemented, Sylow
205C2×#117—1461no—
206C22×C42—731no—
207#432—734no—
208#432—1462no—
209#431—734no—
210#431—1462no—
211C2×C4⋊C8—731no—
212C2×C4⋊C8—1461no—
213C2×C4⋊C8C2×D7311yescharacteristic
214C2×C22⋊C8—1461no—
215C2×C22⋊C8—1461no—
216C2×#97—1461no—
217C2×C4×C8—731no—
218C2×C4×C8C2×D7311yescharacteristic
219#384—732no—
220#382—1461no—
221#380—732no—
222#380—1461no—
223C4×C4⋊C4—731no—
224C4×C4⋊C4—1461no—
225C4×C4⋊C4C2×D7311yescharacteristic
226C23×C4—731no—
227C23×C4—731no—
228C23×C4—731no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—1461no—
232C22×C8—1461no—
233#117—2922no—
234#117—2922no—
235C2×C4⋊C4—732no—
236C2×C4⋊C4—1462no—
237C2×C4⋊C4—1462no—
238C2×C4⋊C4C4×D7312yes—
239C2×C22⋊C4—1462no—
240C2×C42—21no—
241C2×C42—731no—
242C2×C42—731no—
243C2×C42—731no—
244C2×C42—732no—
245C2×C42—732no—
246C2×C42—732no—
247C2×C42—732no—
248C2×C42—1461no—
249C2×C42C22×D7311yescharacteristic
250C4⋊C8—21no—
251C4⋊C8—732no—
252C4⋊C8—1464no—
253C4⋊C8—1461no—
254C4⋊C8C4×D7312yes—
255C22⋊C8—1464no—
256C22⋊C8—1464no—
257#97—1464no—
258C4×C8—21no—
259C4×C8—732no—
260C4×C8—1461no—
261C4×C8C4×D7312yes—
262#95—21no—
263#95—1462no—
264#95—1461no—
265C24—731no—
266C22×C4—22no—
267C22×C4—2922no—
268C22×C4—732no—
269C22×C4—732no—
270C22×C4—731no—
271C22×C4—731no—
272C22×C4—731no—
273C22×C4—732no—
274C22×C4—732no—
275C22×C4—732no—
276C22×C4—732no—
277C22×C4—1462no—
278C22×C4—1462no—
279C22×C4—1462no—
280C22×C4—1461no—
281C22×C4—1461no—
282C22×C4—1461no—
283C22×C4—1461no—
284C22×C4C2×C4×D7311yescharacteristic
285C22×C4D4×D7311yescharacteristic
286C22×C4D4×D7311yescharacteristic
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—2922no—
293C2×C8—2921no—
294C2×C8—2921no—
295C2×C8—1462no—
296C2×C8—1462no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—2924no—
300C4⋊C4—2924no—
301C4⋊C4—1462no—
302C4⋊C4—1462no—
303C22⋊C4—2924no—
304C42—41no—
305C42—41no—
306C42—2921no—
307C42—2921no—
308C42—732no—
309C42—732no—
310C42—732no—
311C42—732no—
312C42—732no—
313C42—732no—
314C42C2×C4×D7311yescharacteristic
315C42C2×C4×D7311yescharacteristic
316C42D4×D7311yescharacteristic
317C42Q8×D7311yescharacteristic
318C23—732no—
319C23—1462no—
320C23—1461no—
321C23—1461no—
322C23—1462no—
323C23C22⋊C4×D7311yescharacteristic
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—2922no—
329C2×C4—2922no—
330C2×C4—2922no—
331C2×C4—2922no—
332C2×C4—2922no—
333C2×C4—732no—
334C2×C4—732no—
335C2×C4—732no—
336C2×C4—1462no—
337C2×C4—1461no—
338C2×C4—1461no—
339C2×C4—1462no—
340C2×C4—1464no—
341C2×C4—1462no—
342C2×C4—1462no—
343C2×C4—1462no—
344C2×C4—1462no—
345C2×C4—1462no—
346C2×C4—1462no—
347C2×C4C42×D7311yescharacteristic
348C2×C4C22⋊C4×D7311yescharacteristic
349C2×C4C22⋊C4×D7311yescharacteristic
350C2×C4C4⋊C4×D7311yescharacteristic
351C2×C4C4⋊C4×D7311yescharacteristic
352C2×C4C4⋊C4×D7311yescharacteristic
353C2×C4#1455659712yes—
354C2×C4#1455659912yes—
355C2×C4#1455661511yescharacteristic
356C2×C4#1455661811yescharacteristic
357C2×C4#1455662011yescharacteristic
358C2×C4#1455662311yescharacteristic
359C8—42no—
360C8—42no—
361C8—2922no—
362C8—2922no—
363C22—2922no—
364C22—2922no—
365C22—1462no—
366C22—1462no—
367C22—1462no—
368C22#95×D7311yescharacteristic
369C22#1798855212yes—
370C22#1798859011yescharacteristic
371C22#1798859411yescharacteristic
372C22#1798860212yes—
373C4—21no—
374C4—21no—
375C4—82no—
376C4—5842no—
377C4—1462no—
378C4—1462no—
379C4—1462no—
380C4#1798854111yescharacteristic
381C4#1798854411yescharacteristic
382C4#1798858511yescharacteristic
383C4#1798858711yescharacteristic
384C4#1798861212yes—
385C4#1798862611yescharacteristic
386C4#1798862811yescharacteristic
387C2—2922no—
388C2#2260839812yes—
389C2#2260840212yes—
390C2#2260962611yescharacteristic
391C2#2260963911yescharacteristic
392C2#2260964911yescharacteristic
subgroupG/H
centerC22×C4D4×D73
commutatorC2×C4×C73C2×C42
FrattiniC2×C42C22×D73
FittingC73×#1279C2
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C42×C73C23
hypercenter#1279D73
hypocenterC73#7900
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
4. #22607772C2semidirect
5. #22607771C2semidirect
6. #22607770C2semidirect
7. C73×#1279C2semidirect
11. #17988643C4semidirect
13. #17988638C4semidirect
203. #1279D73semidirect
204. C73#7900semidirect

Maximal subgroups

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

subgroupindex
C2×#179886432
C2×#179886382
#226123352
#226077722
#226077712
#226077702
C73×#12792
#790073