Finite Group Databasebeta

← #32989129

Subgroups — #32989129

Order 18688.

508 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22610767C211yesmaximal, characteristic
2#22608498C211yesmaximal, characteristic
3C2×#17988320C211yesmaximal, characteristic
4#22607810C211yesmaximal, characteristic, complemented
5#22607798C211yesmaximal, characteristic, complemented
6#22607779C211yesmaximal, characteristic, complemented
7C73×#2549C211yesmaximal, characteristic, complemented
8C2×#14556565C2211yescharacteristic
9C2×#14556505—21no—
10C2×#14556505—21no—
11C2×C22⋊#10596831C2211yescharacteristic
12C2×C22⋊#10596831C2211yescharacteristic
13#17988447C2211yescharacteristic
14#17988320—21no—
15#17988320—21no—
16#17988320—21no—
17#17988320—21no—
18C2×C22⋊Q8×C73—21no—
19#398×C73—21no—
20#395×C73—21no—
21#391×C73—21no—
22#389×C73—21no—
23#385×C73—21no—
24#380×C73C2211yescharacteristic
25C4×C22⋊C4×C73C2211yescharacteristic
26C2×#95×C73C2211yescharacteristic
27C22×D292Q811yescharacteristic, complemented
28C22×C22⋊D73—21no—
29C22×#10596831—21no—
30C22×#10596831—21no—
31#14556565—21no—
32#14556565—21no—
33#14556565—21no—
34#14556565—21no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73Q811yescharacteristic, complemented
37C2×C22⋊Dic73—21no—
38C2×#11924005—21no—
39#14556505—42no—
40#14556505—42no—
41C22⋊#10596831—21no—
42C22⋊#10596831—21no—
43C22⋊#10596831—21no—
44C22⋊#10596831—21no—
45#14556498—21no—
46C22×Q8×C73—21no—
47C23×C4×C73—21no—
48C23×C4×C73C2311yescharacteristic
49C22⋊Q8×C73—42no—
50C22⋊Q8×C73—42no—
51C2×C4⋊C4×C73—21no—
52C2×C4⋊C4×C73—21no—
53C2×C4⋊C4×C73—21no—
54C2×C4⋊C4×C73—21no—
55C2×C4⋊C4×C73—21no—
56C2×C4⋊C4×C73—21no—
57C2×C22⋊C4×C73—21no—
58C2×C22⋊C4×C73D411yescharacteristic
59C2×C22⋊C4×C73D411yescharacteristic
60C2×C42×C73—21no—
61#95×C73—21no—
62#95×C73—21no—
63#95×C73—21no—
64#95×C73—21no—
65#95×C73D411yescharacteristic
66#95×C73D411yescharacteristic
67#95×C73D411yescharacteristic
68#95×C73D411yescharacteristic
69C23×D73—21no—
70C2×D292—21no—
71C2×D292—21no—
72C2×D292—81no—
73C2×C22⋊D73—41no—
74C2×C22⋊D73—41no—
75C2×C22⋊D73—41no—
76C2×C22⋊D73—41no—
77C2×C22⋊D73—41no—
78C2×C22⋊D73—41no—
79C22×Dic73—21no—
80C22×Dic73—41no—
81C2×#10596831—41no—
82C2×#10596831—41no—
83C2×#10596831—81no—
84C2×#10596831—81no—
85C4⋊Dic73—21no—
86C4⋊Dic73—21no—
87C4⋊Dic73—81no—
88C22⋊Dic73—41no—
89C22⋊Dic73—41no—
90#11924005—41no—
91#11924005—41no—
92C24×C73C2×D411yescharacteristic
93C2×Q8×C73—42no—
94C2×Q8×C73—42no—
95C2×Q8×C73—81no—
96C22×C4×C73—21no—
97C22×C4×C73—21no—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73—21no—
101C22×C4×C73—21no—
102C22×C4×C73—42no—
103C22×C4×C73—41no—
104C22×C4×C73—41no—
105C22×C4×C73—81no—
106C22×C4×C73—81no—
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73C2×D411yescharacteristic
109C22×C4×C73C2×Q811yescharacteristic
110C22×C4×C73#4111yescharacteristic
111C22×C4×C73#4111yescharacteristic
112C22×C4×C73#4111yescharacteristic
113C4⋊C4×C73—42no—
114C4⋊C4×C73—42no—
115C4⋊C4×C73—42no—
116C4⋊C4×C73—42no—
117C4⋊C4×C73—81no—
118C4⋊C4×C73—81no—
119C22⋊C4×C73—21no—
120C22⋊C4×C73—21no—
121C22⋊C4×C73—41no—
122C22⋊C4×C73—41no—
123C22⋊C4×C73D812yes—
124C22⋊C4×C73SD1612yes—
125C42×C73—81no—
126C23×C73—41no—
127C23×C73—41no—
128C23×C73#12011yescharacteristic
129C23×C73C22⋊Q811yescharacteristic
130C23×C73#12411yescharacteristic
131C23×C73C2×D811yescharacteristic
132C23×C73C2×SD1611yescharacteristic
133C23×C73AGL(1,ℤ/8)11yescharacteristic
134C23×C73#13711yescharacteristic
135C22×D73—41no—
136C22×D73—41no—
137C22×D73—41no—
138C22×D73—81no—
139C22×D73—81no—
140Q8×C73—84no—
141C2×C4×C73—22no—
142C2×C4×C73—22no—
143C2×C4×C73—22no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—82no—
166C2×C4×C73—82no—
167C2×C4×C73C22⋊Q811yescharacteristic
168C2×C4×C73C22⋊Q811yescharacteristic
169C2×C4×C73#12411yescharacteristic
170C2×C4×C73#12411yescharacteristic
171C22⋊D73—81no—
172C22⋊D73—81no—
173C22⋊D73—81no—
174C22⋊D73—81no—
175C2×Dic73—41no—
176C2×Dic73—41no—
177C2×Dic73—41no—
178C2×Dic73—81no—
179C2×Dic73—81no—
180C2×Dic73—81no—
181D292—81no—
182D292—81no—
183#10596831—81no—
184#10596831—81no—
185C22×C73—21no—
186C22×C73—21no—
187C22×C73—21no—
188C22×C73—21no—
189C22×C73—21no—
190C22×C73—21no—
191C22×C73—41no—
192C22×C73—41no—
193C22×C73—41no—
194C22×C73—41no—
195C22×C73#39311yescharacteristic
196C22×C73#44711yescharacteristic
197C22×C73#44811yescharacteristic
198C22×C73#47411yescharacteristic
199C22×C73#47611yescharacteristic
200C22×C73#48611yescharacteristic
201C22×C73#48911yescharacteristic
202C2×D73—81no—
203C2×D73—81no—
204C2×D73—81no—
205C2×D73—161no—
206C2×D73—161no—
207C4×C73—21no—
208C4×C73—21no—
209C4×C73—21no—
210C4×C73—21no—
211C4×C73—42no—
212C4×C73—42no—
213C4×C73—42no—
214C4×C73—81no—
215C4×C73—81no—
216C4×C73—81no—
217Dic73—81no—
218Dic73—161no—
219#10374—731nomaximal, Sylow
220C2×C73—21no—
221C2×C73—21no—
222C2×C73—21no—
223C2×C73—21no—
224C2×C73—21no—
225C2×C73—21no—
226C2×C73#202311yescharacteristic
227C2×C73#203111yescharacteristic
228C2×C73#203211yescharacteristic
229D73—161no—
230#2549D7311yescharacteristic, complemented
231#1964—731no—
232#1519—731no—
233C2×#327—731no—
234#1294—731no—
235#1289—731no—
236#1282—731no—
237C73#1037411yescharacteristic, complemented, Sylow
238C2×C22⋊Q8—21no—
239C2×#121—731no—
240C2×#99—1461no—
241C2×#99—1461no—
242C2×C22⋊C8—731no—
243C2×C22⋊C8—731no—
244#398—21no—
245#395—21no—
246#391—21no—
247#389—21no—
248#385—21no—
249#380—731no—
250#380C2×D7311yescharacteristic
251C4×C22⋊C4C2×D7311yescharacteristic
252C2×#95C2×D7311yescharacteristic
253#327—1461no—
254#327—1461no—
255#327—1461no—
256#327—1461no—
257C22×Q8—21no—
258C22×D4—731no—
259C22×D4—1461no—
260C23×C4—21no—
261C23×C4C22×D7311yescharacteristic
262C22×C8—1461no—
263C22×C8—1461no—
264C22⋊Q8—42no—
265C22⋊Q8—42no—
266#121—1461no—
267#121—1461no—
268#121—1461no—
269#121—1461no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—731no—
277C2×C4⋊C4—1461no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—1461no—
280C2×C22⋊C4—1461no—
281C2×C22⋊C4C22⋊D7311yescharacteristic
282C2×C22⋊C4C22⋊D7311yescharacteristic
283C2×C42—21no—
284#99—2922no—
285#99—2922no—
286C22⋊C8—1461no—
287C22⋊C8—1461no—
288C22⋊C8—1461no—
289C22⋊C8—1461no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—1461no—
295#95C22⋊D7311yescharacteristic
296#95C22⋊D7311yescharacteristic
297#95C22⋊D7311yescharacteristic
298#95C22⋊D7311yescharacteristic
299C24—1461no—
300C24C2×C22⋊D7311yescharacteristic
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—81no—
304C2×D4—2921no—
305C2×D4—2921no—
306C2×D4—2921no—
307C2×D4—2921no—
308C2×D4—2921no—
309C2×D4—2921no—
310C2×D4—5841no—
311C2×D4—1461no—
312C2×D4—1461no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4—2921no—
325C22×C4—1461no—
326C22×C4C2×C22⋊D7311yescharacteristic
327C22×C4C2×C22⋊D7311yescharacteristic
328C22×C4Q8×D7311yescharacteristic
329C22×C4#1192402911yescharacteristic
330C22×C4#1192402911yescharacteristic
331C22×C4#1192402911yescharacteristic
332C2×C8—2921no—
333C2×C8—2921no—
334C2×C8—5841no—
335C2×C8—5841no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—5841no—
343C4⋊C4—1461no—
344C4⋊C4—1461no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—2921no—
350C22⋊C4—2921no—
351C22⋊C4—2921no—
352C22⋊C4—2921no—
353C22⋊C4#1192401312yes—
354C22⋊C4#1192401512yes—
355C42—81no—
356C23—41no—
357C23—41no—
358C23—2921no—
359C23—2921no—
360C23—2921no—
361C23—5841no—
362C23—5841no—
363C23#1455656011yescharacteristic
364C23#1455656711yescharacteristic
365C23#1455658011yescharacteristic
366C23C2×#1192401311yescharacteristic
367C23C2×#1192401511yescharacteristic
368C23#1455661711yescharacteristic
369C23#1455662211yescharacteristic
370Q8—84no—
371D4—5841no—
372D4—5841no—
373D4—5841no—
374D4—5841no—
375D4—5841no—
376D4—5841no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—2921no—
404C2×C4—2921no—
405C2×C4—2921no—
406C2×C4—5841no—
407C2×C4—5841no—
408C2×C4—5841no—
409C2×C4#1455656711yescharacteristic
410C2×C4#1455656711yescharacteristic
411C2×C4#1455658011yescharacteristic
412C2×C4#1455658011yescharacteristic
413C8—5841no—
414C8—5841no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—5841no—
426C22—5841no—
427C22—5841no—
428C22—11681no—
429C22—11681no—
430C22#1798849811yescharacteristic
431C22#1798872111yescharacteristic
432C22#1798872611yescharacteristic
433C22#1798886411yescharacteristic
434C22#1798887411yescharacteristic
435C22#1798893511yescharacteristic
436C22#1798894911yescharacteristic
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—5841no—
448C4—11681no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—11681no—
456C2#2261106411yescharacteristic
457C2#2261110011yescharacteristic
458C2#2261110711yescharacteristic
subgroupG/H
centerC22#17988498
commutator, cosocleC23×C4×C73C23
FrattiniC23×C4C22×D73
FittingC73×#2549C2
socleC22×C73#393
prime radicalC2×#14556565C22
hypercenter#2549D73
hypocenterC73#10374
LCSG ▷ C23×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2549

Complements

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

NH ≅ G/Ntype
4. #22607810C2semidirect
5. #22607798C2semidirect
6. #22607779C2semidirect
7. C73×#2549C2semidirect
27. C22×D292Q8semidirect
36. C2×C4⋊Dic73Q8semidirect
230. #2549D73semidirect
237. C73#10374semidirect

Maximal subgroups

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

subgroupindex
#226107672
#226084982
C2×#179883202
#226078102
#226077982
#226077792
C73×#25492
#1037473