Finite Group Databasebeta

← #32994177

Subgroups — #32994177

Order 18688.

814 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556523C211yesmaximal, characteristic
2C2×#17988349C212yesmaximal
3C2×#17988341C211yesmaximal, characteristic
4#22608160C212yesmaximal
5C2×#340×C73C211yesmaximal, characteristic
6C23×#10596831C2211yescharacteristic
7C22×C4⋊Dic73C2212yes—
8C22×C4×Dic73C2211yescharacteristic
9C2×#14556523—24no—
10C2×#14556523—22no—
11C2×#14556498—22no—
12C2×#14556498—22no—
13#17988349—24no—
14#17988341—22no—
15C23×C8×C73C2211yescharacteristic
16C22×C4⋊C4×C73C2212yes—
17#340×C73C414yescomplemented
18C23×Dic73—22no—
19C23×Dic73D412yes—
20C22×#10596831—21no—
21C22×#10596831—22no—
22C22×#10596831—21no—
23C22×#10596831—22no—
24C22×#10596831Q812yes—
25C2×C4⋊Dic73—22no—
26C2×C4⋊Dic73—22no—
27C2×C4⋊Dic73—22no—
28C2×C4⋊Dic73—22no—
29C2×C4⋊Dic73—22no—
30C2×C4⋊Dic73D414yes—
31C2×C4×Dic73—21no—
32C2×C4×Dic73—22no—
33C2×C4×Dic73—22no—
34C2×C4×Dic73—21no—
35#14556523—28no—
36#14556498—44no—
37#14556498—44no—
38C23×C4×C73—22no—
39C23×C4×C73C2311yescharacteristic
40C22×C8×C73—22no—
41C22×C8×C73—22no—
42C22×C8×C73C2×C412yes—
43C22×C8×C73D412yes—
44C22×C8×C73Q812yes—
45C2×C4⋊C4×C73—22no—
46C2×C4⋊C4×C73—22no—
47C2×C4⋊C4×C73—22no—
48C2×C4⋊C4×C73—22no—
49C2×C4⋊C4×C73—22no—
50C2×C4⋊C4×C73C2×C414yes—
51C22×Dic73—22no—
52C22×Dic73—22no—
53C22×Dic73—22no—
54C22×Dic73—22no—
55C22×Dic73—22no—
56C22×Dic73—22no—
57C22×Dic73—24no—
58C22×Dic73—42no—
59C22×Dic73—42no—
60C22×Dic73—42no—
61C22×Dic73—42no—
62C22×Dic73—42no—
63C22×Dic73—42no—
64C22×Dic73SD1614yes—
65C2×#10596831—24no—
66C2×#10596831—22no—
67C2×#10596831—41no—
68C2×#10596831—42no—
69C2×#10596831—41no—
70C4⋊Dic73—24no—
71C4⋊Dic73—24no—
72C4⋊Dic73—42no—
73C4⋊Dic73—42no—
74C4×Dic73—41no—
75C4×Dic73—41no—
76C4×Dic73—42no—
77C24×C73C2×D411yescharacteristic
78C22×C4×C73—24no—
79C22×C4×C73—42no—
80C22×C4×C73—42no—
81C22×C4×C73—42no—
82C22×C4×C73—42no—
83C22×C4×C73—42no—
84C22×C4×C73—42no—
85C22×C4×C73C22×C411yescharacteristic
86C22×C4×C73C2×D412yes—
87C22×C4×C73C2×D411yescharacteristic
88C22×C4×C73C2×Q811yescharacteristic
89C22×C4×C73C2×Q811yescharacteristic
90C22×C4×C73#4112yes—
91C22×C4×C73#4111yescharacteristic
92C22×C4×C73#4112yes—
93C22×C4×C73#4112yes—
94C22×C4×C73#4111yescharacteristic
95C2×C8×C73—24no—
96C2×C8×C73—22no—
97C2×C8×C73—24no—
98C2×C8×C73—22no—
99C2×C8×C73C4⋊C414yes—
100C4⋊C4×C73—24no—
101C4⋊C4×C73—24no—
102C4⋊C4×C73—42no—
103C4⋊C4×C73—42no—
104C23×C73C4×D411yescharacteristic
105C23×C73#12112yes—
106C23×C73#12111yescharacteristic
107C23×C73#12312yes—
108C23×C73C4⋊Q811yescharacteristic
109C23×C73C2×SD1612yes—
110C23×C73#13512yes—
111C23×C73AGL(1,ℤ/8)12yes—
112C23×C73#13712yes—
113C2×C4×C73—44no—
114C2×C4×C73—44no—
115C2×C4×C73—44no—
116C2×C4×C73—82no—
117C2×C4×C73—82no—
118C2×C4×C73—82no—
119C2×C4×C73—82no—
120C2×C4×C73C2×C4⋊C411yescharacteristic
121C2×C4×C73#11712yes—
122C2×C4×C73C4×D412yes—
123C2×C4×C73C4×Q811yescharacteristic
124C2×C4×C73#12011yescharacteristic
125C2×C4×C73C22⋊Q812yes—
126C2×C4×C73C22⋊Q812yes—
127C2×C4×C73C22⋊Q811yescharacteristic
128C2×C4×C73#12311yescharacteristic
129C2×C4×C73#12312yes—
130C2×C4×C73#12412yes—
131C2×C4×C73#12411yescharacteristic
132C2×C4×C73#12411yescharacteristic
133C2×C4×C73#12512yes—
134C2×C4×C73#12511yescharacteristic
135C2×C4×C73#12512yes—
136C2×C4×C73#12612yes—
137C2×C4×C73#12612yes—
138C2×Dic73—24no—
139C2×Dic73—24no—
140C2×Dic73—24no—
141C2×Dic73—42no—
142C2×Dic73—42no—
143C2×Dic73—42no—
144C2×Dic73—42no—
145C2×Dic73—44no—
146C2×Dic73—44no—
147C2×Dic73—44no—
148C2×Dic73—82no—
149C2×Dic73—82no—
150C2×Dic73—82no—
151C2×Dic73—82no—
152C8×C73—24no—
153#10596831—42no—
154C22×C73#38812yes—
155C22×C73#38911yescharacteristic
156C22×C73#39411yescharacteristic
157C22×C73#39711yescharacteristic
158C22×C73#39812yes—
159C22×C73C4×SD1612yes—
160C22×C73#44111yescharacteristic
161C22×C73#44211yescharacteristic
162C22×C73#46012yes—
163C22×C73#46112yes—
164C22×C73#46312yes—
165C22×C73#46412yes—
166C22×C73#46512yes—
167C22×C73#46812yes—
168C22×C73#48112yes—
169C22×C73#48212yes—
170C22×C73#48312yes—
171C22×C73#48512yes—
172C22×C73#49811yescharacteristic
173C22×C73#49911yescharacteristic
174C22×C73#50112yes—
175C4×C73—84no—
176C4×C73#38211yescharacteristic
177C4×C73#38511yescharacteristic
178C4×C73#38712yes—
179C4×C73#39311yescharacteristic
180C4×C73#40011yescharacteristic
181C4×C73#40012yes—
182Dic73—44no—
183Dic73—84no—
184#11257—731nomaximal, Sylow
185C2×C73#144211yescharacteristic
186C2×C73#192612yes—
187C2×C73#192712yes—
188C2×C73#194111yescharacteristic
189C2×C73#194311yescharacteristic
190C2×C73#205512yes—
191C2×C73#207712yes—
192C2×C73#208412yes—
193C2×C73#208512yes—
194C22×#106—731no—
195C2×#340—732no—
196C2×#340D7311yescharacteristic
197C2×#336—731no—
198C4⋊#95—732no—
199C73#1125711yescharacteristic, complemented, Sylow
200C23×C8—731no—
201C23×C8C2×D7311yescharacteristic
202C22×C4⋊C4—732no—
203C22×C4⋊C4C2×D7312yes—
204C22×C42—731no—
205C2×#106—1464no—
206C2×#106—1462no—
207C2×#95—1462no—
208C2×#95—1462no—
209#340—1464no—
210#340Dic7314yescomplemented
211#336—1462no—
212C23×C4—22no—
213C23×C4—732no—
214C23×C4—1462no—
215C23×C4C22×D7311yescharacteristic
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—732no—
219C22×C8—1462no—
220C22×C8—1461no—
221C22×C8—1461no—
222C22×C8—1462no—
223C22×C8Dic14612yes—
224C22×C8D29212yes—
225C22×C8C2×Dic7312yes—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—734no—
232C2×C4⋊C4—1462no—
233C2×C4⋊C4—1462no—
234C2×C4⋊C4—1462no—
235C2×C4⋊C4—1462no—
236C2×C4⋊C4—1462no—
237C2×C4⋊C4C2×Dic7314yes—
238C2×C42—1461no—
239C2×C42—1462no—
240C2×C42—1462no—
241C2×C42—1461no—
242#106—1468no—
243#95—2924no—
244#95—2924no—
245C24D4×D7311yescharacteristic
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—2922no—
254C22×C4—2922no—
255C22×C4—2922no—
256C22×C4—2922no—
257C22×C4—2922no—
258C22×C4—2922no—
259C22×C4—734no—
260C22×C4—1464no—
261C22×C4—1462no—
262C22×C4—1462no—
263C22×C4—1462no—
264C22×C4—1462no—
265C22×C4—1462no—
266C22×C4—1462no—
267C22×C4C22×Dic7311yescharacteristic
268C22×C4D4×D7312yes—
269C22×C4C2×D29211yescharacteristic
270C22×C4Q8×D7311yescharacteristic
271C22×C4C2×Dic14611yescharacteristic
272C22×C4#1192402712yes—
273C22×C4#1192402711yescharacteristic
274C22×C4#1192402812yes—
275C22×C4#1192402812yes—
276C22×C4#1192402911yescharacteristic
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—22no—
281C2×C8—2921no—
282C2×C8—2922no—
283C2×C8—2921no—
284C2×C8—1464no—
285C2×C8—1462no—
286C2×C8C4⋊Dic7314yes—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—2922no—
292C4⋊C4—2922no—
293C4⋊C4—1464no—
294C4⋊C4—1464no—
295C42—2921no—
296C42—2921no—
297C42—2922no—
298C23D4×Dic7311yescharacteristic
299C23#1455656412yes—
300C23#1455656611yescharacteristic
301C23#1455657312yes—
302C23#1455659011yescharacteristic
303C23SD16×D7312yes—
304C23#1455661212yes—
305C23#1455661512yes—
306C23#1455662312yes—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—2922no—
315C2×C4—2922no—
316C2×C4—2922no—
317C2×C4—2922no—
318C2×C4—2924no—
319C2×C4—2924no—
320C2×C4—2924no—
321C2×C4—5842no—
322C2×C4—5842no—
323C2×C4—5842no—
324C2×C4—5842no—
325C2×C4—1464no—
326C2×C4—1464no—
327C2×C4—1464no—
328C2×C4C2×C4⋊Dic7311yescharacteristic
329C2×C4#1455655012yes—
330C2×C4D4×Dic7312yes—
331C2×C4Q8×Dic7311yescharacteristic
332C2×C4#1455656111yescharacteristic
333C2×C4#1455656812yes—
334C2×C4#1455656912yes—
335C2×C4#1455657111yescharacteristic
336C2×C4#1455657511yescharacteristic
337C2×C4#1455657612yes—
338C2×C4#1455657712yes—
339C2×C4#1455657911yescharacteristic
340C2×C4#1455657911yescharacteristic
341C2×C4#1455658112yes—
342C2×C4#1455658211yescharacteristic
343C2×C4#1455658312yes—
344C2×C4#1455658412yes—
345C2×C4#1455658612yes—
346C8—24no—
347C8—2922no—
348C22#1798848012yes—
349C22#1798848611yescharacteristic
350C22#1798850511yescharacteristic
351C22#1798851411yescharacteristic
352C22#1798852112yes—
353C22SD16×Dic7312yes—
354C22#1798869411yescharacteristic
355C22#1798869911yescharacteristic
356C22#1798878612yes—
357C22#1798879112yes—
358C22#1798880112yes—
359C22#1798881212yes—
360C22#1798881712yes—
361C22#1798883412yes—
362C22#1798890412yes—
363C22#1798891012yes—
364C22#1798891612yes—
365C22#1798892612yes—
366C22#1798898411yescharacteristic
367C22#1798898711yescharacteristic
368C22#1798899512yes—
369C4—84no—
370C4—2924no—
371C4—5844no—
372C4#1798845311yescharacteristic
373C4#1798846211yescharacteristic
374C4#1798847312yes—
375C4#1798850011yescharacteristic
376C4#1798852711yescharacteristic
377C4#1798852912yes—
378C2#2260818811yescharacteristic
379C2#2261056012yes—
380C2#2261056712yes—
381C2#2261066011yescharacteristic
382C2#2261067011yescharacteristic
383C2#2261125012yes—
384C2#2261136612yes—
385C2#2261141112yes—
386C2#2261141712yes—
subgroupG/H
centerC24D4×D73
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#340×C73C2
socle, prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenterC2×#340D73
hypocenterC73#11257
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
17. #340×C73C4semidirect
199. C73#11257semidirect
210. #340Dic73semidirect

Maximal subgroups

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

subgroupindex
C22×#145565232
C2×#179883492
C2×#179883412
#226081602
C2×#340×C732
#1125773