Finite Group Databasebeta

← #32977802

Subgroups — #32977802

Order 18688.

868 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22610239C212yesmaximal
2C2×#17988324C212yesmaximal
3C2×#17988313C211yesmaximal, characteristic
4#22608184C211yesmaximal, characteristic
5C2×#544×C73C211yesmaximal, characteristic
6C22×C4⋊Dic73C2211yescharacteristic
7C2×#14556515—22no—
8C2×C22⋊#10596831C2212yes—
9C2×#14556498—21no—
10C2×#14556498—21no—
11C2×#14556498C2211yescharacteristic
12#17988349—22no—
13#17988324—24no—
14#17988313—22no—
15#544×C73—24no—
16#544×C73—22no—
17#544×C73C414yescomplemented
18C2×C4⋊Q8×C73—22no—
19C2×#125×C73—22no—
20C2×C22⋊Q8×C73—21no—
21C2×C22⋊Q8×C73C2212yes—
22C2×#117×C73—21no—
23C22×C4⋊C4×C73C2211yescharacteristic
24C23×Dic73—21no—
25C23×Dic73—21no—
26C22×#10596831—22no—
27C2×C4⋊Dic73—22no—
28C2×C4⋊Dic73—42no—
29#14556515—44no—
30C22⋊#10596831—22no—
31C22⋊#10596831—22no—
32#14556498—22no—
33#14556498—42no—
34#14556498—42no—
35C22×Q8×C73—21no—
36C22×Q8×C73D412yes—
37C23×C4×C73D411yescharacteristic
38C23×C4×C73D411yescharacteristic
39C23×C4×C73C2311yescharacteristic
40C4⋊Q8×C73—28no—
41C4⋊Q8×C73—28no—
42#125×C73—28no—
43#125×C73—28no—
44C22⋊Q8×C73—24no—
45C22⋊Q8×C73—24no—
46C22⋊Q8×C73—22no—
47C22⋊Q8×C73—22no—
48C22⋊Q8×C73C2×C414yes—
49C22⋊Q8×C73D414yes—
50#117×C73—24no—
51#117×C73—24no—
52C2×C4⋊C4×C73—21no—
53C2×C4⋊C4×C73—21no—
54C2×C4⋊C4×C73—21no—
55C2×C4⋊C4×C73—22no—
56C2×C4⋊C4×C73—22no—
57C2×C4⋊C4×C73—22no—
58C2×C4⋊C4×C73—22no—
59C2×C4⋊C4×C73C2×C412yes—
60C2×C4⋊C4×C73D412yes—
61C2×C4⋊C4×C73D412yes—
62C2×C22⋊C4×C73—22no—
63C2×C42×C73—22no—
64C22×Dic73—22no—
65C22×Dic73—22no—
66C22×Dic73—41no—
67C22×Dic73—41no—
68C22×Dic73—41no—
69C22×Dic73—41no—
70C22×Dic73—41no—
71C22×Dic73—41no—
72C22×Dic73—41no—
73C22×Dic73—41no—
74C22×Dic73—41no—
75C22×Dic73—41no—
76C22×Dic73—41no—
77C22×Dic73—41no—
78C2×#10596831—42no—
79C2×#10596831—42no—
80C2×#10596831—42no—
81C4⋊Dic73—24no—
82C4⋊Dic73—42no—
83C24×C73C2×D411yescharacteristic
84C2×Q8×C73—24no—
85C2×Q8×C73—22no—
86C2×Q8×C73—42no—
87C2×Q8×C73—42no—
88C2×Q8×C73—81no—
89C2×Q8×C73—81no—
90C2×Q8×C73C22⋊C414yes—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—21no—
94C22×C4×C73—21no—
95C22×C4×C73—22no—
96C22×C4×C73—22no—
97C22×C4×C73—21no—
98C22×C4×C73—21no—
99C22×C4×C73—41no—
100C22×C4×C73—41no—
101C22×C4×C73—41no—
102C22×C4×C73—41no—
103C22×C4×C73C22⋊C412yes—
104C22×C4×C73C22⋊C412yes—
105C22×C4×C73SD1612yes—
106C22×C4×C73Q1612yes—
107C22×C4×C73C22×C411yescharacteristic
108C22×C4×C73C2×D412yes—
109C22×C4×C73C2×D412yes—
110C22×C4×C73C2×D411yescharacteristic
111C4⋊C4×C73—24no—
112C4⋊C4×C73—24no—
113C4⋊C4×C73—24no—
114C4⋊C4×C73—28no—
115C4⋊C4×C73—24no—
116C4⋊C4×C73—24no—
117C4⋊C4×C73—24no—
118C4⋊C4×C73—24no—
119C4⋊C4×C73—22no—
120C4⋊C4×C73C22⋊C414yes—
121C22⋊C4×C73—24no—
122C22⋊C4×C73—24no—
123C42×C73—28no—
124C23×C73—21no—
125C23×C73—21no—
126C23×C73—21no—
127C23×C73—21no—
128C23×C73C2×C22⋊C411yescharacteristic
129C23×C73#12011yescharacteristic
130C23×C73#12011yescharacteristic
131C23×C73C2×SD1611yescharacteristic
132C23×C73C2×Q1611yescharacteristic
133C23×C73#13711yescharacteristic
134C23×C73#13711yescharacteristic
135Q8×C73—44no—
136Q8×C73—82no—
137Q8×C73—84no—
138C2×C4×C73—22no—
139C2×C4×C73—22no—
140C2×C4×C73—22no—
141C2×C4×C73—22no—
142C2×C4×C73—22no—
143C2×C4×C73—22no—
144C2×C4×C73—22no—
145C2×C4×C73—22no—
146C2×C4×C73—24no—
147C2×C4×C73—24no—
148C2×C4×C73—22no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—42no—
152C2×C4×C73—42no—
153C2×C4×C73—42no—
154C2×C4×C73—41no—
155C2×C4×C73—42no—
156C2×C4×C73—41no—
157C2×C4×C73—42no—
158C2×C4×C73—42no—
159C2×C4×C73—81no—
160C2×C4×C73#10314yescomplemented
161C2×C4×C73C2×C22⋊C412yes—
162C2×C4×C73#12012yes—
163C2×Dic73—42no—
164C2×Dic73—42no—
165C2×Dic73—42no—
166C2×Dic73—42no—
167C2×Dic73—42no—
168C2×Dic73—42no—
169C2×Dic73—81no—
170C2×Dic73—81no—
171C2×Dic73—81no—
172C2×Dic73—81no—
173C2×Dic73—81no—
174C2×Dic73—81no—
175C2×Dic73—81no—
176C2×Dic73—81no—
177#10596831—82no—
178C22×C73—21no—
179C22×C73—21no—
180C22×C73—21no—
181C22×C73—21no—
182C22×C73—21no—
183C22×C73—21no—
184C22×C73—21no—
185C22×C73—21no—
186C22×C73—21no—
187C22×C73—21no—
188C22×C73—21no—
189C22×C73—21no—
190C22×C73#37911yescharacteristic
191C22×C73C2×#10311yescharacteristic
192C22×C73#41911yescharacteristic
193C22×C73#44811yescharacteristic
194C22×C73#44811yescharacteristic
195C22×C73#45111yescharacteristic
196C22×C73#45111yescharacteristic
197C22×C73#45411yescharacteristic
198C22×C73#45611yescharacteristic
199C22×C73B(4,𝔽2)11yescharacteristic
200C22×C73#45811yescharacteristic
201C4×C73—24no—
202C4×C73—22no—
203C4×C73—22no—
204C4×C73—44no—
205C4×C73—42no—
206C4×C73—44no—
207Dic73—82no—
208Dic73—82no—
209#8406—731nomaximal, Sylow
210C2×C73—21no—
211C2×C73—21no—
212C2×C73—21no—
213C2×C73—21no—
214C2×C73#159911yescharacteristic
215C2×C73#160511yescharacteristic
216C2×C73#161111yescharacteristic
217C2×C73#161711yescharacteristic
218C2×C73#178711yescharacteristic
219C2×C73#179011yescharacteristic
220C2×C73#179311yescharacteristic
221C2×#544D7311yescharacteristic
222#1872—732no—
223C2×#328—732no—
224C2×#324—731no—
225#1441—731no—
226C73#840611yescharacteristic, complemented, Sylow
227#544—24no—
228#544—22no—
229#544Dic7314yescomplemented
230C2×C4⋊Q8—22no—
231C2×#125—22no—
232C2×C22⋊Q8—21no—
233C2×C22⋊Q8C2×D7312yes—
234C2×#117—21no—
235C22×C4⋊C4—731no—
236C22×C4⋊C4C2×D7311yescharacteristic
237C2×#103—1462no—
238C2×C22⋊C8—732no—
239C2×#95—731no—
240C2×#95—1461no—
241C2×#95—1461no—
242#340—1462no—
243#328—1464no—
244#324—1462no—
245C22×Q8—21no—
246C22×Q8C22⋊D7312yes—
247C23×C4—1461no—
248C23×C4—1461no—
249C23×C4C22⋊D7311yescharacteristic
250C23×C4C22⋊D7311yescharacteristic
251C23×C4C22×D7311yescharacteristic
252C22×C8—1462no—
253C4⋊Q8—28no—
254C4⋊Q8—28no—
255#125—28no—
256#125—28no—
257C22⋊Q8—24no—
258C22⋊Q8—24no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8C2×Dic7314yes—
262C22⋊Q8C22⋊D7314yes—
263#117—24no—
264#117—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—2922no—
273C2×C4⋊C4—1462no—
274C2×C4⋊C4C2×Dic7312yes—
275C2×C4⋊C4C22⋊D7312yes—
276C2×C4⋊C4C22⋊D7312yes—
277C2×C22⋊C4—22no—
278C2×C42—22no—
279#103—2924no—
280C22⋊C8—1462no—
281C22⋊C8—1462no—
282#95—2922no—
283#95—2922no—
284#95—1462no—
285C24C2×C22⋊D7311yescharacteristic
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—81no—
291C2×Q8—81no—
292C2×Q8C22⋊Dic7314yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—2921no—
306C22×C4—2921no—
307C22×C4—2921no—
308C22×C4—2921no—
309C22×C4—2921no—
310C22×C4—2921no—
311C22×C4—2921no—
312C22×C4—2921no—
313C22×C4—2921no—
314C22×C4—2921no—
315C22×C4—2921no—
316C22×C4—2921no—
317C22×C4—1462no—
318C22×C4—1462no—
319C22×C4C22⋊Dic7312yes—
320C22×C4C22⋊Dic7312yes—
321C22×C4#1192401512yes—
322C22×C4#1192401812yes—
323C22×C4C22×Dic7311yescharacteristic
324C22×C4C2×C22⋊D7312yes—
325C22×C4C2×C22⋊D7312yes—
326C22×C4C2×C22⋊D7311yescharacteristic
327C2×C8—2922no—
328C2×C8—2922no—
329C2×C8—2922no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—2922no—
340C4⋊C4—1464no—
341C4⋊C4C22⋊Dic7314yes—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C42—28no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23C2×C22⋊Dic7311yescharacteristic
350C23#1455656011yescharacteristic
351C23#1455656011yescharacteristic
352C23C2×#1192401511yescharacteristic
353C23C2×#1192401811yescharacteristic
354C23#1455662211yescharacteristic
355C23#1455662211yescharacteristic
356Q8—44no—
357Q8—82no—
358Q8—84no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—81no—
381C2×C4—2922no—
382C2×C4—2922no—
383C2×C4—2922no—
384C2×C4—2922no—
385C2×C4—2922no—
386C2×C4—2922no—
387C2×C4—5841no—
388C2×C4—5841no—
389C2×C4—5841no—
390C2×C4—5841no—
391C2×C4—5841no—
392C2×C4—5841no—
393C2×C4—5841no—
394C2×C4—5841no—
395C2×C4#1455651514yescomplemented
396C2×C4C2×C22⋊Dic7312yes—
397C2×C4#1455656012yes—
398C8—5842no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22#1798844311yescharacteristic
412C22C2×#1455651511yescharacteristic
413C22#1798859611yescharacteristic
414C22#1798872611yescharacteristic
415C22#1798872611yescharacteristic
416C22#1798874311yescharacteristic
417C22#1798874311yescharacteristic
418C22#1798876011yescharacteristic
419C22#1798877011yescharacteristic
420C22#1798877411yescharacteristic
421C22#1798877711yescharacteristic
422C4—24no—
423C4—22no—
424C4—22no—
425C4—42no—
426C4—44no—
427C4—44no—
428C4—5842no—
429C4—5842no—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—21no—
434C2#2260887111yescharacteristic
435C2#2260890511yescharacteristic
436C2#2260893911yescharacteristic
437C2#2260897311yescharacteristic
438C2#2260986611yescharacteristic
439C2#2260987911yescharacteristic
440C2#2260989011yescharacteristic
subgroupG/H
centerC23#14556560
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#544×C73C2
socleC23×C73#120
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenterC2×#544D73
hypocenterC73#8406
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
17. #544×C73C4semidirect
160. C2×C4×C73#103semidirect
226. C73#8406semidirect
229. #544Dic73semidirect
395. C2×C4#14556515semidirect

Maximal subgroups

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

subgroupindex
#226102392
C2×#179883242
C2×#179883132
#226081842
C2×#544×C732
#840673