Finite Group Databasebeta

← #32977652

Subgroups — #32977652

Order 18688.

1034 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22610017C212yesmaximal, complemented
2#22609919C211yesmaximal, characteristic, complemented
3#22608499C212yesmaximal, complemented
4#22608249C211yesmaximal, characteristic, complemented
5C2×#547×C73C211yesmaximal, characteristic
6C2×#14556506—21no—
7C2×C22⋊#10596831C2212yes—
8#17988447C2212yes—
9#17988354—21no—
10#17988349—24no—
11#17988321—24no—
12#17988313—21no—
13#17988311—21no—
14C22×#41×C73—21no—
15#547×C73—24no—
16#547×C73—24no—
17C2×C4⋊Q8×C73C411yescharacteristic, complemented
18C2×#123×C73—21no—
19C2×#123×C73—21no—
20C2×C22⋊Q8×C73—21no—
21C2×#121×C73C411yescharacteristic, complemented
22C2×#121×C73C2211yescharacteristic
23C2×C4×D4×C73C411yescharacteristic, complemented
24C2×C4×D4×C73C2211yescharacteristic
25C22×C4⋊C4×C73C411yescharacteristic, complemented
26C22×C4⋊C4×C73C2211yescharacteristic
27C22×#10596831—22no—
28C2×C4⋊Dic73—22no—
29C2×C22⋊Dic73—22no—
30#14556506—42no—
31C22⋊#10596831—22no—
32C22⋊#10596831—22no—
33#14556498—22no—
34C2×#41×C73—28no—
35C2×#41×C73—44no—
36C2×#41×C73—42no—
37C2×#41×C73—42no—
38C22×Q8×C73—21no—
39C22×D4×C73—21no—
40C22×D4×C73—21no—
41C22×D4×C73C2×C411yescharacteristic
42C22×D4×C73D411yescharacteristic
43C23×C4×C73—21no—
44C23×C4×C73—21no—
45C23×C4×C73—21no—
46C23×C4×C73C2×C411yescharacteristic
47C23×C4×C73C2311yescharacteristic
48C4⋊Q8×C73—22no—
49C4⋊Q8×C73—22no—
50#123×C73—28no—
51#123×C73—28no—
52C22⋊Q8×C73—28no—
53#121×C73—22no—
54#121×C73—22no—
55#121×C73—22no—
56#121×C73D414yes—
57C4×D4×C73—22no—
58C4×D4×C73—22no—
59C4×D4×C73—22no—
60C4×D4×C73—22no—
61C2×C4⋊C4×C73—21no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C4⋊C4×C73—22no—
65C2×C4⋊C4×C73—22no—
66C2×C4⋊C4×C73—22no—
67C2×C4⋊C4×C73C2×C411yescharacteristic
68C2×C4⋊C4×C73C2×C411yescharacteristic
69C2×C4⋊C4×C73C2×C411yescharacteristic
70C2×C4⋊C4×C73D411yescharacteristic
71C2×C4⋊C4×C73D414yes—
72C2×C22⋊C4×C73—21no—
73C2×C22⋊C4×C73—21no—
74C2×C22⋊C4×C73—21no—
75C2×C22⋊C4×C73D411yescharacteristic
76C2×C22⋊C4×C73D411yescharacteristic
77C2×C42×C73C2×C411yescharacteristic
78C22×Dic73—42no—
79C2×#10596831—44no—
80C2×#10596831—42no—
81C4⋊Dic73—44no—
82C22⋊Dic73—42no—
83C22⋊Dic73—42no—
84C24×C73—21no—
85C24×C73C22⋊C411yescharacteristic
86C24×C73C2×D411yescharacteristic
87#41×C73—416no—
88#41×C73—416no—
89C2×Q8×C73—24no—
90C2×Q8×C73—42no—
91C2×Q8×C73—42no—
92C2×D4×C73—22no—
93C2×D4×C73—22no—
94C2×D4×C73—24no—
95C2×D4×C73—24no—
96C2×D4×C73—21no—
97C2×D4×C73—42no—
98C2×D4×C73—42no—
99C2×D4×C73—42no—
100C2×D4×C73—41no—
101C2×D4×C73—41no—
102C2×D4×C73—42no—
103C2×D4×C73C22⋊C412yes—
104C2×D4×C73D812yes—
105C2×D4×C73SD1612yes—
106C22×C4×C73—24no—
107C22×C4×C73—24no—
108C22×C4×C73—21no—
109C22×C4×C73—21no—
110C22×C4×C73—21no—
111C22×C4×C73—24no—
112C22×C4×C73—21no—
113C22×C4×C73—21no—
114C22×C4×C73—42no—
115C22×C4×C73—42no—
116C22×C4×C73—42no—
117C22×C4×C73—42no—
118C22×C4×C73—42no—
119C22×C4×C73—41no—
120C22×C4×C73—41no—
121C22×C4×C73—42no—
122C22×C4×C73—42no—
123C22×C4×C73C22⋊C411yescharacteristic
124C22×C4×C73C22⋊C411yescharacteristic
125C22×C4×C73C22⋊C411yescharacteristic
126C22×C4×C73C22×C411yescharacteristic
127C22×C4×C73C2×D412yes—
128C22×C4×C73C2×D412yes—
129C22×C4×C73C2×D411yescharacteristic
130C4⋊C4×C73—24no—
131C4⋊C4×C73—24no—
132C4⋊C4×C73—21no—
133C4⋊C4×C73—21no—
134C4⋊C4×C73—21no—
135C4⋊C4×C73—24no—
136C4⋊C4×C73—22no—
137C4⋊C4×C73C22⋊C412yes—
138C4⋊C4×C73C22⋊C412yes—
139C4⋊C4×C73C22⋊C412yes—
140C22⋊C4×C73—24no—
141C22⋊C4×C73—24no—
142C22⋊C4×C73—22no—
143C22⋊C4×C73—24no—
144C22⋊C4×C73—22no—
145C42×C73—21no—
146C42×C73—21no—
147C23×C73—21no—
148C23×C73—21no—
149C23×C73—22no—
150C23×C73—41no—
151C23×C73—41no—
152C23×C73—41no—
153C23×C73—41no—
154C23×C73—42no—
155C23×C73—42no—
156C23×C73—41no—
157C23×C73—41no—
158C23×C73#10212yes—
159C23×C73C2×C22⋊C411yescharacteristic
160C23×C73#12012yes—
161C23×C73C2×D811yescharacteristic
162C23×C73C2×SD1611yescharacteristic
163C23×C73AGL(1,ℤ/8)12yes—
164Q8×C73—48no—
165D4×C73—24no—
166D4×C73—24no—
167D4×C73—42no—
168D4×C73—48no—
169D4×C73—44no—
170D4×C73—44no—
171D4×C73—42no—
172C2×C4×C73—21no—
173C2×C4×C73—21no—
174C2×C4×C73—22no—
175C2×C4×C73—24no—
176C2×C4×C73—21no—
177C2×C4×C73—21no—
178C2×C4×C73—22no—
179C2×C4×C73—22no—
180C2×C4×C73—22no—
181C2×C4×C73—41no—
182C2×C4×C73—42no—
183C2×C4×C73—42no—
184C2×C4×C73—48no—
185C2×C4×C73—48no—
186C2×C4×C73—41no—
187C2×C4×C73—41no—
188C2×C4×C73—41no—
189C2×C4×C73—41no—
190C2×C4×C73—41no—
191C2×C4×C73—41no—
192C2×C4×C73—44no—
193C2×C4×C73—44no—
194C2×C4×C73—42no—
195C2×C4×C73—41no—
196C2×C4×C73—41no—
197C2×C4×C73#10212yes—
198C2×C4×C73C2×C22⋊C411yescharacteristic
199C2×C4×C73C2×C22⋊C411yescharacteristic
200C2×C4×C73#12012yes—
201C2×Dic73—82no—
202C2×Dic73—82no—
203C2×Dic73—82no—
204#10596831—82no—
205C22×C73—22no—
206C22×C73—21no—
207C22×C73—22no—
208C22×C73—21no—
209C22×C73—22no—
210C22×C73—41no—
211C22×C73—41no—
212C22×C73—42no—
213C22×C73—42no—
214C22×C73—42no—
215C22×C73—41no—
216C22×C73—41no—
217C22×C73—42no—
218C22×C73—42no—
219C22×C73—42no—
220C22×C73—82no—
221C22×C73—82no—
222C22×C73#37911yescharacteristic
223C22×C73C2×#10211yescharacteristic
224C22×C73#41811yescharacteristic
225C22×C73#44712yes—
226C22×C73#45012yes—
227C4×C73—22no—
228C4×C73—21no—
229C4×C73—22no—
230C4×C73—41no—
231C4×C73—41no—
232C4×C73—42no—
233C4×C73—42no—
234C4×C73—42no—
235C4×C73#41012yes—
236Dic73—162no—
237#8376—731nomaximal, Sylow
238C2×C73—22no—
239C2×C73—21no—
240C2×C73—21no—
241C2×C73—22no—
242C2×C73—42no—
243C2×C73—82no—
244C2×C73#178211yescharacteristic
245C2×C73#178611yescharacteristic
246C2×C73#179611yescharacteristic
247C2×#547D7311yescharacteristic
248#1823—732no—
249#1799—731no—
250#1519—732no—
251#1459—731no—
252C73#837611yescharacteristic, complemented, Sylow
253C22×#41—21no—
254#547—24no—
255#547—24no—
256C2×C4⋊Q8Dic7311yescharacteristic, complemented
257C2×#123—21no—
258C2×#123—21no—
259C2×C22⋊Q8—21no—
260C2×#121Dic7311yescharacteristic, complemented
261C2×#121C2×D7311yescharacteristic
262C2×C4×D4Dic7311yescharacteristic, complemented
263C2×C4×D4C2×D7311yescharacteristic
264C22×C4⋊C4Dic7311yescharacteristic, complemented
265C22×C4⋊C4C2×D7311yescharacteristic
266C2×#99—1461no—
267C2×C22⋊C8—732no—
268#380—732no—
269#342—1461no—
270#340—1464no—
271#327—1464no—
272#324—1461no—
273#323—1461no—
274C2×#41—28no—
275C2×#41—44no—
276C2×#41—42no—
277C2×#41—42no—
278C22×Q8—21no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4C2×Dic7311yescharacteristic
282C22×D4C22⋊D7311yescharacteristic
283C23×C4—21no—
284C23×C4—21no—
285C23×C4—21no—
286C23×C4C2×Dic7311yescharacteristic
287C23×C4C22×D7311yescharacteristic
288C22×C8—1462no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291#123—28no—
292#123—28no—
293C22⋊Q8—28no—
294#121—22no—
295#121—22no—
296#121—22no—
297#121C22⋊D7314yes—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—1462no—
309C2×C4⋊C4C2×Dic7311yescharacteristic
310C2×C4⋊C4C2×Dic7311yescharacteristic
311C2×C4⋊C4C2×Dic7311yescharacteristic
312C2×C4⋊C4C22⋊D7311yescharacteristic
313C2×C4⋊C4C22⋊D7314yes—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—1462no—
318C2×C22⋊C4C22⋊D7311yescharacteristic
319C2×C22⋊C4C22⋊D7311yescharacteristic
320C2×C42C2×Dic7311yescharacteristic
321#99—2922no—
322C22⋊C8—1462no—
323C22⋊C8—1462no—
324#95—1462no—
325C24—21no—
326C24C22⋊Dic7311yescharacteristic
327C24C2×C22⋊D7311yescharacteristic
328#41—416no—
329#41—416no—
330C2×Q8—24no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—21no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4C22⋊Dic7312yes—
345C2×D4#1192401312yes—
346C2×D4#1192401612yes—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—2922no—
365C22×C4C22⋊Dic7311yescharacteristic
366C22×C4C22⋊Dic7311yescharacteristic
367C22×C4C22⋊Dic7311yescharacteristic
368C22×C4C22×Dic7311yescharacteristic
369C22×C4C2×C22⋊D7312yes—
370C22×C4C2×C22⋊D7312yes—
371C22×C4C2×C22⋊D7311yescharacteristic
372C2×C8—2924no—
373C2×C8—2922no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—2924no—
382C4⋊C4C22⋊Dic7312yes—
383C4⋊C4C22⋊Dic7312yes—
384C4⋊C4C22⋊Dic7312yes—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—22no—
388C22⋊C4—24no—
389C22⋊C4—22no—
390C22⋊C4—2922no—
391C22⋊C4—2922no—
392C42—21no—
393C42—21no—
394C23—21no—
395C23—21no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—42no—
402C23—42no—
403C23—41no—
404C23—41no—
405C23#1455651212yes—
406C23C2×C22⋊Dic7311yescharacteristic
407C23#1455656012yes—
408C23C2×#1192401311yescharacteristic
409C23C2×#1192401611yescharacteristic
410C23#1455661712yes—
411Q8—48no—
412D4—24no—
413D4—24no—
414D4—42no—
415D4—48no—
416D4—44no—
417D4—44no—
418D4—42no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—48no—
431C2×C4—42no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—5842no—
445C2×C4—5842no—
446C2×C4—5842no—
447C2×C4#1455651212yes—
448C2×C4C2×C22⋊Dic7311yescharacteristic
449C2×C4C2×C22⋊Dic7311yescharacteristic
450C2×C4#1455656012yes—
451C8—5842no—
452C22—21no—
453C22—22no—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—41no—
458C22—41no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—41no—
463C22—41no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—82no—
468C22—82no—
469C22#1798844311yescharacteristic
470C22C2×#1455651211yescharacteristic
471C22#1798859211yescharacteristic
472C22#1798872112yes—
473C22#1798873812yes—
474C4—22no—
475C4—21no—
476C4—22no—
477C4—41no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—41no—
482C4—11682no—
483C4#1798856112yes—
484C2—22no—
485C2—21no—
486C2—21no—
487C2—22no—
488C2—42no—
489C2—82no—
490C2#2260984911yescharacteristic
491C2#2260986211yescharacteristic
492C2#2260990611yescharacteristic
subgroupG/H
centerC22#17988443
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
Fitting, prime radicalC2×#547×C73C2
socleC22×C73#379
cosocleC23×C4×C73C23
hypercenterC2×#547D73
hypocenterC73#8376
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #22610017C2semidirect
2. #22609919C2semidirect
3. #22608499C2semidirect
4. #22608249C2semidirect
17. C2×C4⋊Q8×C73C4semidirect
21. C2×#121×C73C4semidirect
23. C2×C4×D4×C73C4semidirect
25. C22×C4⋊C4×C73C4semidirect
252. C73#8376semidirect
256. C2×C4⋊Q8Dic73semidirect
260. C2×#121Dic73semidirect
262. C2×C4×D4Dic73semidirect
264. C22×C4⋊C4Dic73semidirect

Maximal subgroups

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

subgroupindex
#226100172
#226099192
#226084992
#226082492
C2×#547×C732
#837673