Finite Group Databasebeta

← #32978639

Subgroups — #32978639

Order 18688.

898 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊C4×Dic73C211yesmaximal, characteristic
2#22609909C212yesmaximal, complemented
3C2×#17988311C211yesmaximal, characteristic
4#22607773C212yesmaximal, complemented
5C2×#323×C73C211yesmaximal, characteristic
6C24×Dic73C411yescharacteristic
7C22×C22⋊Dic73C411yescharacteristic
8C22×C4×Dic73C411yescharacteristic
9C22×C4×Dic73C2211yescharacteristic
10C2×#14556520—22no—
11C2×C22⋊#10596831C2212yes—
12C22⋊C4×Dic73—22no—
13C22⋊C4×Dic73—22no—
14C2×#14556498C411yescharacteristic
15C2×#14556498C2211yescharacteristic
16#17988341—22no—
17#17988311—22no—
18C22×C22⋊C4×C73C2211yescharacteristic
19C2×C22⋊C8×C73C2212yes—
20#323×C73—22no—
21C23×Dic73—21no—
22C23×Dic73—21no—
23C23×Dic73—22no—
24C23×Dic73—42no—
25C23×Dic73—42no—
26C23×Dic73C814yescomplemented
27C23×Dic73C2×C411yescharacteristic
28C23×Dic73C2×C411yescharacteristic
29C23×Dic73C2×C411yescharacteristic
30C23×Dic73C2×C411yescharacteristic
31C22×#10596831—22no—
32C2×C22⋊Dic73—22no—
33C2×C22⋊Dic73—22no—
34C2×C22⋊Dic73—22no—
35C2×C4×Dic73—21no—
36C2×C4×Dic73—21no—
37C2×C4×Dic73—42no—
38C2×C4×Dic73—42no—
39C2×C4×Dic73D412yes—
40C2×C4×Dic73D412yes—
41#14556520—44no—
42C22⋊#10596831—22no—
43C22⋊#10596831—22no—
44#14556498—22no—
45#14556498—22no—
46C25×C73C2×C411yescharacteristic
47C23×C4×C73C2×C411yescharacteristic
48C23×C4×C73C2311yescharacteristic
49C22×C8×C73—22no—
50C2×C22⋊C4×C73—22no—
51C2×C22⋊C4×C73—22no—
52C2×C22⋊C4×C73D412yes—
53C2×C22⋊C4×C73Q812yes—
54C22⋊C8×C73—22no—
55C22⋊C8×C73—22no—
56C22×Dic73—22no—
57C22×Dic73—22no—
58C22×Dic73—21no—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—21no—
63C22×Dic73—21no—
64C22×Dic73—24no—
65C22×Dic73—24no—
66C22×Dic73—41no—
67C22×Dic73—41no—
68C22×Dic73—42no—
69C22×Dic73—42no—
70C22×Dic73—48no—
71C22×Dic73—48no—
72C22×Dic73—42no—
73C22×Dic73—42no—
74C22×Dic73—41no—
75C22×Dic73—41no—
76C22×Dic73—41no—
77C22×Dic73—41no—
78C22×Dic73C22⋊C412yes—
79C22×Dic73C22⋊C411yescharacteristic
80C22×Dic73C22⋊C411yescharacteristic
81C22×Dic73C2×C812yes—
82C22×Dic73C2×C812yes—
83C22×Dic73M1612yes—
84C22×Dic73M1612yes—
85C2×#10596831—42no—
86C2×#10596831—42no—
87C2×#10596831—42no—
88C22⋊Dic73—28no—
89C4×Dic73—24no—
90C4×Dic73—42no—
91C4×Dic73—42no—
92C4×Dic73—42no—
93C24×C73—21no—
94C24×C73—21no—
95C24×C73—21no—
96C24×C73—21no—
97C24×C73—41no—
98C24×C73—41no—
99C24×C73—41no—
100C24×C73—41no—
101C24×C73C4⋊C412yes—
102C24×C73C2×C812yes—
103C24×C73M1612yes—
104C24×C73C22×C411yescharacteristic
105C22×C4×C73—21no—
106C22×C4×C73—21no—
107C22×C4×C73—41no—
108C22×C4×C73—41no—
109C22×C4×C73—41no—
110C22×C4×C73—41no—
111C22×C4×C73C4⋊C412yes—
112C22×C4×C73C2×D412yes—
113C22×C4×C73C2×D411yescharacteristic
114C22×C4×C73C2×Q811yescharacteristic
115C22×C4×C73#4112yes—
116C2×C8×C73—42no—
117C2×C8×C73—42no—
118C2×C8×C73—42no—
119C22⋊C4×C73—24no—
120C22⋊C4×C73—24no—
121C23×C73—21no—
122C23×C73—21no—
123C23×C73—21no—
124C23×C73—21no—
125C23×C73—22no—
126C23×C73—22no—
127C23×C73—22no—
128C23×C73—22no—
129C23×C73—22no—
130C23×C73—22no—
131C23×C73—42no—
132C23×C73—42no—
133C23×C73—42no—
134C23×C73—42no—
135C23×C73—42no—
136C23×C73—42no—
137C23×C73—42no—
138C23×C73—42no—
139C23×C73—42no—
140C23×C73—42no—
141C23×C73—42no—
142C23×C73—42no—
143C23×C73—41no—
144C23×C73—41no—
145C23×C73—41no—
146C23×C73—41no—
147C23×C73C4⋊C814yescomplemented
148C23×C73C2×C22⋊C411yescharacteristic
149C23×C73C2×C4⋊C411yescharacteristic
150C23×C73#11711yescharacteristic
151C23×C73C22×C811yescharacteristic
152C23×C73C2×M1611yescharacteristic
153C23×C73C2×M1611yescharacteristic
154C23×C73C2×M1611yescharacteristic
155C2×C4×C73—22no—
156C2×C4×C73—22no—
157C2×C4×C73—22no—
158C2×C4×C73—41no—
159C2×C4×C73—41no—
160C2×C4×C73—42no—
161C2×C4×C73—42no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—42no—
165C2×C4×C73#12112yes—
166C2×C4×C73C22⋊Q812yes—
167C2×Dic73—21no—
168C2×Dic73—21no—
169C2×Dic73—24no—
170C2×Dic73—24no—
171C2×Dic73—22no—
172C2×Dic73—22no—
173C2×Dic73—22no—
174C2×Dic73—22no—
175C2×Dic73—22no—
176C2×Dic73—48no—
177C2×Dic73—42no—
178C2×Dic73—42no—
179C2×Dic73—42no—
180C2×Dic73—42no—
181C2×Dic73—41no—
182C2×Dic73—41no—
183C2×Dic73—41no—
184C2×Dic73—41no—
185C2×Dic73—48no—
186C2×Dic73—42no—
187C2×Dic73—42no—
188C2×Dic73C22⋊C812yes—
189C2×Dic73C22⋊C812yes—
190C2×Dic73#9912yes—
191C2×Dic73#10012yes—
192C8×C73—82no—
193#10596831—82no—
194C22×C73—21no—
195C22×C73—21no—
196C22×C73—21no—
197C22×C73—21no—
198C22×C73—21no—
199C22×C73—21no—
200C22×C73—21no—
201C22×C73—21no—
202C22×C73—21no—
203C22×C73—21no—
204C22×C73—21no—
205C22×C73—21no—
206C22×C73—24no—
207C22×C73—44no—
208C22×C73—44no—
209C22×C73—44no—
210C22×C73—44no—
211C22×C73—42no—
212C22×C73—42no—
213C22×C73—42no—
214C22×C73—42no—
215C22×C73—42no—
216C22×C73—42no—
217C22×C73#38011yescharacteristic
218C22×C73C2×C22⋊C811yescharacteristic
219C22×C73#40711yescharacteristic
220C22×C73C2×#9911yescharacteristic
221C22×C73#41011yescharacteristic
222C22×C73C2×#10011yescharacteristic
223C22×C73#41311yescharacteristic
224C22×C73C2×C4⋊C811yescharacteristic
225C22×C73#42311yescharacteristic
226C22×C73#43111yescharacteristic
227C22×C73#43211yescharacteristic
228C4×C73—22no—
229C4×C73—22no—
230C4×C73—42no—
231Dic73—22no—
232Dic73—42no—
233Dic73—42no—
234Dic73—42no—
235Dic73#32314yescomplemented
236#8557—731nomaximal, Sylow
237C2×C73—21no—
238C2×C73—21no—
239C2×C73—21no—
240C2×C73—21no—
241C2×C73—44no—
242C2×C73C2×#32311yescharacteristic
243C2×C73#145811yescharacteristic
244C2×C73#146511yescharacteristic
245C2×C73#146611yescharacteristic
246C2×C73#179711yescharacteristic
247C2×C73#179911yescharacteristic
248C2×C73#180011yescharacteristic
249C2×C4×C22⋊C4—731no—
250#1797—732no—
251C2×#323—731no—
252C2×#323D7311yescharacteristic
253#1280—732no—
254C73#855711yescharacteristic, complemented, Sylow
255C24×C4—731no—
256C22×C22⋊C4—731no—
257C22×C22⋊C4C2×D7311yescharacteristic
258C22×C42—731no—
259C22×C42—731no—
260C2×C4⋊C8—1462no—
261C2×C22⋊C8—732no—
262C2×C22⋊C8C2×D7312yes—
263C4×C22⋊C4—1462no—
264C4×C22⋊C4—1462no—
265C2×#95—731no—
266C2×#95—731no—
267#336—1462no—
268#323—22no—
269#323—1462no—
270C25C4×D7311yescharacteristic
271C23×C4—2922no—
272C23×C4—2922no—
273C23×C4—734no—
274C23×C4—731no—
275C23×C4—731no—
276C23×C4—731no—
277C23×C4—731no—
278C23×C4—1461no—
279C23×C4—1461no—
280C23×C4—1462no—
281C23×C4C4×D7311yescharacteristic
282C23×C4C22×D7311yescharacteristic
283C22×C8—22no—
284C22×C8—1462no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—1462no—
288C2×C22⋊C4—1462no—
289C2×C22⋊C4—1462no—
290C2×C22⋊C4Dic14612yes—
291C2×C22⋊C4C22⋊D7312yes—
292C2×C42—2922no—
293C2×C42—2922no—
294C2×C42—732no—
295C2×C42—732no—
296C2×C42—1461no—
297C2×C42—1461no—
298C4⋊C8—2924no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—1462no—
302C22⋊C8—1462no—
303#95—1462no—
304#95—1462no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24—41no—
313C24#1192400712yes—
314C24C8×D7312yes—
315C24#1192401112yes—
316C24C2×C4×D7311yescharacteristic
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—2922no—
324C22×C4—2922no—
325C22×C4—2928no—
326C22×C4—2922no—
327C22×C4—2922no—
328C22×C4—2921no—
329C22×C4—2921no—
330C22×C4—2921no—
331C22×C4—2921no—
332C22×C4—2921no—
333C22×C4—2921no—
334C22×C4—2928no—
335C22×C4—732no—
336C22×C4—732no—
337C22×C4—731no—
338C22×C4—731no—
339C22×C4—732no—
340C22×C4—732no—
341C22×C4—732no—
342C22×C4—1464no—
343C22×C4—1464no—
344C22×C4—1461no—
345C22×C4—1461no—
346C22×C4—1461no—
347C22×C4—1461no—
348C22×C4—1461no—
349C22×C4—1461no—
350C22×C4—1462no—
351C22×C4—1462no—
352C22×C4#1192400712yes—
353C22×C4D4×D7312yes—
354C22×C4C2×C22⋊D7311yescharacteristic
355C22×C4C2×Dic14611yescharacteristic
356C22×C4#1192402712yes—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—2922no—
361C2×C8—2922no—
362C2×C8—2922no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—1468no—
366C42—2922no—
367C42—2922no—
368C42—2922no—
369C42—1464no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23#1455652014yescomplemented
397C23C22⋊C4×D7311yescharacteristic
398C23C2×#1192400711yescharacteristic
399C23#1455654711yescharacteristic
400C23C2×C8×D7311yescharacteristic
401C23M16×D7311yescharacteristic
402C23M16×D7311yescharacteristic
403C23C2×#1192401111yescharacteristic
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—2922no—
415C2×C4—2922no—
416C2×C4—2922no—
417C2×C4—2922no—
418C2×C4—2921no—
419C2×C4—2921no—
420C2×C4—2921no—
421C2×C4—2921no—
422C2×C4—2928no—
423C2×C4—2928no—
424C2×C4—2922no—
425C2×C4—2922no—
426C2×C4—732no—
427C2×C4—732no—
428C2×C4—732no—
429C2×C4—732no—
430C2×C4—1462no—
431C2×C4—1462no—
432C2×C4—1462no—
433C2×C4—1462no—
434C2×C4—1462no—
435C2×C4—1464no—
436C2×C4—1464no—
437C2×C4—1461no—
438C2×C4—1461no—
439C2×C4#1455656312yes—
440C2×C4#1455656812yes—
441C8—82no—
442C8—5842no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—24no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22#1798844511yescharacteristic
467C22C22⋊C8×D7311yescharacteristic
468C22#1798854911yescharacteristic
469C22#99×D7311yescharacteristic
470C22#1798855911yescharacteristic
471C22#100×D7311yescharacteristic
472C22#1798856811yescharacteristic
473C22C2×#1455652011yescharacteristic
474C22#1798861011yescharacteristic
475C22#1798863711yescharacteristic
476C22#1798864111yescharacteristic
477C4—22no—
478C4—22no—
479C4—42no—
480C4—2922no—
481C4—2922no—
482C4—2922no—
483C4—734no—
484C4—1462no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—44no—
490C2#323×D7311yescharacteristic
491C2#2260824211yescharacteristic
492C2#2260826611yescharacteristic
493C2#2260827011yescharacteristic
494C2#2260990911yescharacteristic
495C2#2260991711yescharacteristic
496C2#2260992111yescharacteristic
subgroupG/H
centerC23C22⋊C4×D73
commutatorC23×C73C22×C8
FrattiniC23×C4C22×D73
FittingC2×#323×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC25×C73C2×C4
cosocleC23×C4×C73C23
hypercenterC2×#323D73
hypocenterC73#8557
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. #22609909C2semidirect
4. #22607773C2semidirect
26. C23×Dic73C8semidirect
147. C23×C73C4⋊C8semidirect
235. Dic73#323semidirect
254. C73#8557semidirect
396. C23#14556520semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×Dic732
#226099092
C2×#179883112
#226077732
C2×#323×C732
#855773