Finite Group Databasebeta

← #32978652

Subgroups — #32978652

Order 18688.

682 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×#17988438C211yesmaximal, characteristic
2#22609909C212yesmaximal
3C2×#17988312C212yesmaximal
4#22607774C211yesmaximal, characteristic
5C73×#1280C211yesmaximal, characteristic
6C22×#11924007C411yescharacteristic
7C22×#11924007C2211yescharacteristic
8C22×C4×Dic73C411yescharacteristic
9C22×C4×Dic73C411yescharacteristic
10C22×C4×Dic73C2211yescharacteristic
11C2×#14556520—22no—
12C2×C22⋊#10596831C2212yes—
13#17988438—22no—
14#17988438—22no—
15C2×#14556498C411yescharacteristic
16#17988341—22no—
17#17988312—24no—
18C2×C22⋊C8×C73C2212yes—
19C2×#95×C73C2211yescharacteristic
20C23×Dic73C2×C411yescharacteristic
21C23×Dic73C2×C411yescharacteristic
22C23×Dic73C2×C411yescharacteristic
23C23×Dic73C2×C411yescharacteristic
24C22×#10596831—22no—
25C2×#11924007—22no—
26C2×#11924007—22no—
27C2×#11924007—22no—
28C2×#11924007—22no—
29C2×#11924007C814yescomplemented
30C2×#11924007D412yes—
31C2×#11924007Q812yes—
32C2×C4×Dic73—21no—
33C2×C4×Dic73—21no—
34C2×C4×Dic73—22no—
35C2×C4×Dic73—21no—
36C2×C4×Dic73—21no—
37C2×C4×Dic73—22no—
38C2×C4×Dic73—42no—
39C2×C4×Dic73—42no—
40C2×C4×Dic73D412yes—
41C2×C4×Dic73D412yes—
42#14556520—44no—
43C22⋊#10596831—22no—
44C22⋊#10596831—22no—
45#14556498—22no—
46C23×C4×C73C2×C411yescharacteristic
47C23×C4×C73C2×C411yescharacteristic
48C23×C4×C73C2311yescharacteristic
49C22×C8×C73—22no—
50C22⋊C8×C73—22no—
51C22⋊C8×C73—22no—
52#95×C73—22no—
53C22×Dic73—21no—
54C22×Dic73—21no—
55C22×Dic73—21no—
56C22×Dic73—21no—
57C22×Dic73—21no—
58C22×Dic73—21no—
59C22×Dic73—22no—
60C22×Dic73—22no—
61C22×Dic73—21no—
62C22×Dic73—21no—
63C22×Dic73—41no—
64C22×Dic73—41no—
65C22×Dic73—41no—
66C22×Dic73—41no—
67C22×Dic73C22⋊C412yes—
68C22×Dic73C22⋊C411yescharacteristic
69C22×Dic73C22⋊C411yescharacteristic
70C22×Dic73C4⋊C412yes—
71C22×Dic73C4⋊C412yes—
72C22×Dic73C2×C812yes—
73C22×Dic73C2×C812yes—
74C22×Dic73M1612yes—
75C22×Dic73M1612yes—
76C2×#10596831—42no—
77C2×#10596831—42no—
78C2×#10596831—42no—
79#11924007—24no—
80#11924007—24no—
81#11924007—28no—
82C4×Dic73—24no—
83C4×Dic73—24no—
84C4×Dic73—24no—
85C4×Dic73—42no—
86C4×Dic73—42no—
87C4×Dic73—42no—
88C24×C73C22×C411yescharacteristic
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—41no—
94C22×C4×C73—41no—
95C22×C4×C73—41no—
96C22×C4×C73—41no—
97C22×C4×C73—41no—
98C22×C4×C73—41no—
99C22×C4×C73C2×C812yes—
100C22×C4×C73M1612yes—
101C22×C4×C73C2×D412yes—
102C22×C4×C73C2×D411yescharacteristic
103C22×C4×C73C2×Q811yescharacteristic
104C22×C4×C73#4112yes—
105C2×C8×C73—42no—
106C2×C8×C73—42no—
107C2×C8×C73—42no—
108C23×C73—21no—
109C23×C73—21no—
110C23×C73—21no—
111C23×C73—21no—
112C23×C73C2×C22⋊C411yescharacteristic
113C23×C73C2×C4⋊C411yescharacteristic
114C23×C73#11711yescharacteristic
115C23×C73C22×C811yescharacteristic
116C23×C73C2×M1611yescharacteristic
117C23×C73C2×M1611yescharacteristic
118C23×C73C2×M1611yescharacteristic
119C2×C4×C73—22no—
120C2×C4×C73—22no—
121C2×C4×C73—22no—
122C2×C4×C73—22no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—42no—
126C2×C4×C73—42no—
127C2×C4×C73—42no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—42no—
133C2×C4×C73—42no—
134C2×C4×C73#12112yes—
135C2×C4×C73C22⋊Q812yes—
136C2×Dic73—24no—
137C2×Dic73—24no—
138C2×Dic73—22no—
139C2×Dic73—22no—
140C2×Dic73—21no—
141C2×Dic73—21no—
142C2×Dic73—22no—
143C2×Dic73—22no—
144C2×Dic73—21no—
145C2×Dic73—21no—
146C2×Dic73—22no—
147C2×Dic73—22no—
148C2×Dic73—22no—
149C2×Dic73—22no—
150C2×Dic73—22no—
151C2×Dic73—42no—
152C2×Dic73—42no—
153C2×Dic73—42no—
154C2×Dic73—41no—
155C2×Dic73—41no—
156C2×Dic73—42no—
157C2×Dic73C22⋊C814yes—
158C2×Dic73#9912yes—
159C2×Dic73#10112yes—
160C2×Dic73C4⋊C814yescomplemented
161C8×C73—82no—
162#10596831—82no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73#38011yescharacteristic
176C22×C73C2×C22⋊C811yescharacteristic
177C22×C73#40711yescharacteristic
178C22×C73C2×#9911yescharacteristic
179C22×C73#41011yescharacteristic
180C22×C73C2×#10111yescharacteristic
181C22×C73#41311yescharacteristic
182C22×C73C2×C4⋊C811yescharacteristic
183C22×C73#42311yescharacteristic
184C22×C73#43111yescharacteristic
185C22×C73#43211yescharacteristic
186C4×C73—22no—
187C4×C73—22no—
188C4×C73—42no—
189C4×C73—42no—
190Dic73—24no—
191Dic73—24no—
192Dic73—22no—
193Dic73—22no—
194Dic73—42no—
195#8560—731nomaximal, Sylow
196C2×C73—21no—
197C2×C73—21no—
198C2×C73—21no—
199C2×C73—21no—
200C2×C73#145911yescharacteristic
201C2×C73#146111yescharacteristic
202C2×C73#146311yescharacteristic
203C2×C73#146411yescharacteristic
204C2×C73#179711yescharacteristic
205C2×C73#179911yescharacteristic
206C2×C73#180111yescharacteristic
207C2×C4×C4⋊C4—731no—
208#1797—732no—
209C2×#324—732no—
210#1280—731no—
211#1280D7311yescharacteristic
212C73#856011yescharacteristic, complemented, Sylow
213C22×C4⋊C4—731no—
214C22×C4⋊C4—731no—
215C22×C42—731no—
216C22×C42—731no—
217C22×C42—731no—
218C2×C4⋊C8—1462no—
219C2×C22⋊C8—732no—
220C2×C22⋊C8C2×D7312yes—
221C4×C4⋊C4—1462no—
222C4×C4⋊C4—1462no—
223C2×#95—731no—
224C2×#95C2×D7311yescharacteristic
225#336—1462no—
226#324—1464no—
227C23×C4—731no—
228C23×C4—731no—
229C23×C4—731no—
230C23×C4—731no—
231C23×C4C4×D7311yescharacteristic
232C23×C4C4×D7311yescharacteristic
233C23×C4C22×D7311yescharacteristic
234C22×C8—22no—
235C22×C8—1462no—
236C2×C4⋊C4—734no—
237C2×C4⋊C4—732no—
238C2×C4⋊C4—732no—
239C2×C4⋊C4—1462no—
240C2×C4⋊C4—1462no—
241C2×C4⋊C4—1462no—
242C2×C4⋊C4—1462no—
243C2×C42—2922no—
244C2×C42—2922no—
245C2×C42—732no—
246C2×C42—732no—
247C2×C42—1461no—
248C2×C42—1461no—
249C2×C42—1461no—
250C2×C42—1461no—
251C2×C42—1462no—
252C2×C42—1462no—
253C4⋊C8—2924no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—1462no—
257C22⋊C8—1462no—
258#95—22no—
259#95—1462no—
260C24C2×C4×D7311yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—2921no—
272C22×C4—2921no—
273C22×C4—2921no—
274C22×C4—2921no—
275C22×C4—732no—
276C22×C4—732no—
277C22×C4—732no—
278C22×C4—732no—
279C22×C4—731no—
280C22×C4—731no—
281C22×C4—732no—
282C22×C4—732no—
283C22×C4—732no—
284C22×C4—1461no—
285C22×C4—1461no—
286C22×C4—1462no—
287C22×C4—1462no—
288C22×C4—1461no—
289C22×C4—1461no—
290C22×C4—1461no—
291C22×C4—1461no—
292C22×C4—1461no—
293C22×C4—1461no—
294C22×C4C8×D7312yes—
295C22×C4#1192401112yes—
296C22×C4D4×D7312yes—
297C22×C4D4×D7311yescharacteristic
298C22×C4Q8×D7311yescharacteristic
299C22×C4#1192402812yes—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—2922no—
304C2×C8—2922no—
305C2×C8—2922no—
306C4⋊C4—1468no—
307C4⋊C4—1464no—
308C4⋊C4—1464no—
309C42—2922no—
310C42—2922no—
311C42—2922no—
312C42—1464no—
313C42—1464no—
314C42—1464no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23C22⋊C4×D7311yescharacteristic
320C23C4⋊C4×D7311yescharacteristic
321C23#1455654811yescharacteristic
322C23C2×C8×D7311yescharacteristic
323C23M16×D7311yescharacteristic
324C23M16×D7311yescharacteristic
325C23C2×#1192401111yescharacteristic
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—2922no—
342C2×C4—2921no—
343C2×C4—2921no—
344C2×C4—2922no—
345C2×C4—2922no—
346C2×C4—2922no—
347C2×C4—734no—
348C2×C4—732no—
349C2×C4—732no—
350C2×C4—734no—
351C2×C4—1462no—
352C2×C4—1464no—
353C2×C4—1464no—
354C2×C4—1462no—
355C2×C4—1462no—
356C2×C4—1462no—
357C2×C4—1462no—
358C2×C4—1462no—
359C2×C4—1461no—
360C2×C4—1461no—
361C2×C4—1462no—
362C2×C4—1461no—
363C2×C4—1462no—
364C2×C4—1461no—
365C2×C4—1462no—
366C2×C4#1455656412yes—
367C2×C4#1455656912yes—
368C8—82no—
369C8—5842no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22#1798844611yescharacteristic
383C22C22⋊C8×D7311yescharacteristic
384C22#1798854911yescharacteristic
385C22#99×D7311yescharacteristic
386C22#1798855911yescharacteristic
387C22#101×D7311yescharacteristic
388C22#1798856811yescharacteristic
389C22C4⋊C8×D7311yescharacteristic
390C22#1798860911yescharacteristic
391C22#1798863811yescharacteristic
392C22#1798864211yescharacteristic
393C4—22no—
394C4—22no—
395C4—42no—
396C4—42no—
397C4—2922no—
398C4—1464no—
399C4—1462no—
400C4—1462no—
401C4—1464no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2—21no—
406C2#2260824611yescharacteristic
407C2#2260825311yescharacteristic
408C2#2260825911yescharacteristic
409C2#2260826211yescharacteristic
410C2#2260991011yescharacteristic
411C2#2260991811yescharacteristic
412C2#2260992611yescharacteristic
subgroupG/H
centerC23C22⋊C4×D73
commutatorC23×C73C22×C8
FrattiniC23×C4C22×D73
FittingC73×#1280C2
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22×C4
cosocleC23×C4×C73C23
hypercenter#1280D73
hypocenterC73#8560
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
29. C2×#11924007C8semidirect
160. C2×Dic73C4⋊C8semidirect
212. C73#8560semidirect

Maximal subgroups

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

subgroupindex
C2×#179884382
#226099092
C2×#179883122
#226077742
C73×#12802
#856073