Finite Group Databasebeta

← #33069372

Subgroups — #33069372

Order 18688.

628 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621257C212yesmaximal, complemented
2#22620510C211yesmaximal, characteristic, complemented
3#22609471C212yesmaximal, complemented
4#22609464C212yesmaximal, complemented
5#22609335C211yesmaximal, characteristic, complemented
6#22609330C211yesmaximal, characteristic, complemented
7#22609291C211yesmaximal, characteristic, complemented
8#22609286C211yesmaximal, characteristic, complemented
9C2×#17988340C211yesmaximal, characteristic, complemented
10#22608627C212yesmaximal, complemented
11C73×#2966C211yesmaximal, characteristic, complemented
12#17989137C2212yes—
13#17988992C2211yescharacteristic
14#17988964C2211yescharacteristic
15#17988934C2212yes—
16#17988884—22no—
17#17988882C2212yes—
18#17988877C2212yes—
19#17988823—21no—
20#17988823—21no—
21#17988821—22no—
22#17988818—22no—
23#17988799—22no—
24#17988787C2212yes—
25#17988782C2212yes—
26C4×D584—21no—
27C4×D584—21no—
28C2×C4⋊#10596831C2212yes—
29#17988340C2212yes—
30#17988340C2211yescharacteristic
31#17988340C2211yescharacteristic
32#17988335C2212yes—
33#17988335C2212yes—
34#17988330C2212yes—
35#17988330C2212yes—
36#529×C73C2211yescharacteristic
37#446×C73C2211yescharacteristic
38C8×Q8×C73C2211yescharacteristic
39#436×C73C2211yescharacteristic
40#435×C73—21no—
41C8×D4×C73—21no—
42C8×D4×C73C2211yescharacteristic
43#433×C73—21no—
44#432×C73C2211yescharacteristic
45#405×C73—21no—
46C2×C4×C8×C73C2211yescharacteristic
47C2×#11924028—22no—
48C2×#11924015—22no—
49C2×D584—21no—
50C2×D584—21no—
51C2×#11924013—22no—
52C22×#10596831—22no—
53#14556591C2312yes—
54#14556588C2312yes—
55#14556579—22no—
56#14556565—22no—
57#14556565—22no—
58C4×D292—22no—
59C4×D292—22no—
60#14556525—21no—
61#14556525—21no—
62#14556524—22no—
63#14556522—22no—
64C4⋊#10596831—22no—
65C4⋊#10596831C2312yes—
66C4⋊#10596831C2312yes—
67#14556515—22no—
68#14556513—22no—
69#14556513—22no—
70#14556512—22no—
71#14556511—22no—
72#14556511—22no—
73C2×M16×C73—21no—
74C22×C8×C73—21no—
75C22×C8×C73D411yescharacteristic
76C22×C8×C73D411yescharacteristic
77#126×C73—21no—
78#125×C73D411yescharacteristic
79#124×C73D411yescharacteristic
80#123×C73—21no—
81C22⋊Q8×C73D411yescharacteristic
82#121×C73D411yescharacteristic
83C4×Q8×C73C2311yescharacteristic
84C4×D4×C73—21no—
85C4×D4×C73C2311yescharacteristic
86#117×C73—21no—
87C2×C42×C73C2311yescharacteristic
88C4⋊C8×C73—21no—
89C4⋊C8×C73—21no—
90C4⋊C8×C73C2311yescharacteristic
91C4⋊C8×C73C2311yescharacteristic
92C22⋊C8×C73—21no—
93C22⋊C8×C73—21no—
94C22⋊C8×C73—21no—
95#97×C73—21no—
96#97×C73—21no—
97C4×C8×C73—21no—
98C4×C8×C73D411yescharacteristic
99C4×C8×C73D411yescharacteristic
100C4×C8×C73C2311yescharacteristic
101C4×C8×C73C2311yescharacteristic
102#11924028—84no—
103C2×Dic146—22no—
104C2×D292—22no—
105C2×D292—22no—
106C2×D292—22no—
107C2×C22⋊D73—42no—
108C2×C4×D73—42no—
109#11924015—82no—
110D584—42no—
111D584—42no—
112#11924013—82no—
113C2×#10596831—22no—
114C2×#10596831—22no—
115C2×#10596831—82no—
116C4⋊Dic73—22no—
117C4⋊Dic73—22no—
118#11924005—42no—
119#11924005—42no—
120C2×Q8×C73C2×D411yescharacteristic
121C2×D4×C73—21no—
122C2×D4×C73C2×D411yescharacteristic
123C22×C4×C73—21no—
124C22×C4×C73C2×D411yescharacteristic
125C22×C4×C73C2×D411yescharacteristic
126C22×C4×C73#4111yescharacteristic
127M16×C73—42no—
128C2×C8×C73—22no—
129C2×C8×C73—21no—
130C2×C8×C73—21no—
131C2×C8×C73—41no—
132C2×C8×C73—42no—
133C2×C8×C73C2×D411yescharacteristic
134C2×C8×C73C2×D411yescharacteristic
135C2×C8×C73C2×D411yescharacteristic
136C2×C8×C73C2×D411yescharacteristic
137C4⋊C4×C73—21no—
138C4⋊C4×C73—21no—
139C4⋊C4×C73—21no—
140C4⋊C4×C73—21no—
141C4⋊C4×C73C2×D411yescharacteristic
142C4⋊C4×C73C2×D411yescharacteristic
143C22⋊C4×C73—21no—
144C22⋊C4×C73—21no—
145C22⋊C4×C73—21no—
146C22⋊C4×C73—21no—
147C22⋊C4×C73—21no—
148C42×C73—21no—
149C42×C73C2×D411yescharacteristic
150C42×C73C2×D411yescharacteristic
151C42×C73#4111yescharacteristic
152C42×C73C2411yescharacteristic
153C23×C73—21no—
154C23×C73#12111yescharacteristic
155C22×D73—42no—
156Q8×C73—21no—
157Q8×C73—21no—
158D4×C73—21no—
159D4×C73—21no—
160D4×C73—42no—
161C2×C4×C73—22no—
162C2×C4×C73—21no—
163C2×C4×C73—21no—
164C2×C4×C73—21no—
165C2×C4×C73—21no—
166C2×C4×C73—21no—
167C2×C4×C73—21no—
168C2×C4×C73—21no—
169C2×C4×C73—41no—
170C2×C4×C73—41no—
171C2×C4×C73#12111yescharacteristic
172C2×C4×C73#12111yescharacteristic
173C2×C4×C73#12111yescharacteristic
174C2×C4×C73C22×D411yescharacteristic
175C2×C4×C73C22×D411yescharacteristic
176C2×C4×C73C2×#4111yescharacteristic
177C22⋊D73—84no—
178C2×Dic73—42no—
179D292—42no—
180D292—42no—
181D292—42no—
182D292—42no—
183D292—82no—
184C4×D73—84no—
185Dic146—82no—
186C8×C73—21no—
187C8×C73—21no—
188C8×C73—21no—
189C8×C73—41no—
190C8×C73—41no—
191C8×C73#13512yes—
192#10596831—42no—
193#10596831—42no—
194C22×C73—21no—
195C22×C73—21no—
196C22×C73—41no—
197C22×C73—41no—
198C22×C73—41no—
199C22×C73#47212yes—
200C22×C73C2×#12111yescharacteristic
201C2×D73—82no—
202C2×D73—82no—
203C2×D73—82no—
204C4×C73—21no—
205C4×C73—21no—
206C4×C73—21no—
207C4×C73—21no—
208C4×C73—41no—
209C4×C73—41no—
210C4×C73—41no—
211C4×C73C2×#13511yescharacteristic
212C4×C73#57511yescharacteristic
213C4×C73#57611yescharacteristic
214C4×C73#57811yescharacteristic
215Dic73—82no—
216#19222—731nomaximal, Sylow
217C2×C73—22no—
218C2×C73—81no—
219C2×C73#303511yescharacteristic
220C2×C73#306111yescharacteristic
221C2×C73C2×#47211yescharacteristic
222D73—162no—
223#3247—732no—
224#3169—731no—
225#2966D7311yescharacteristic, complemented
226#1698—732no—
227#1697—732no—
228#1676—731no—
229#1675—731no—
230#1668—731no—
231#1667—731no—
232C2×#335—731no—
233#1551—732no—
234C73#1922211yescharacteristic, complemented, Sylow
235#532—732no—
236#529C2×D7311yescharacteristic
237#500—731no—
238#493—731no—
239#486—732no—
240#478—1462no—
241#477—732no—
242#476—732no—
243#466—1461no—
244#466—1461no—
245#466—1462no—
246#465—1462no—
247#463—1462no—
248#460—732no—
249#459—732no—
250#446C2×D7311yescharacteristic
251C8×Q8C2×D7311yescharacteristic
252C4×D8—1461no—
253C4×D8—1461no—
254#436C2×D7311yescharacteristic
255#435—21no—
256C8×D4—21no—
257C8×D4C2×D7311yescharacteristic
258#433—21no—
259#432C2×D7311yescharacteristic
260C2×C4⋊C8—732no—
261#405—21no—
262C2×C4×C8C2×D7311yescharacteristic
263#335—731no—
264#335—731no—
265#335—732no—
266#333—732no—
267#333—732no—
268#331—732no—
269#331—732no—
270C2×#41—1462no—
271C2×SD16—1462no—
272C2×D8—1462no—
273C2×D8—1461no—
274C2×D8—1461no—
275C2×M16—21no—
276C22×C8—21no—
277C22×C8—1462no—
278C22×C8C22⋊D7311yescharacteristic
279C22×C8C22⋊D7311yescharacteristic
280C4⋊Q8—732no—
281#127—732no—
282#126—21no—
283#125D29211yescharacteristic
284#124—1462no—
285#124D29211yescharacteristic
286#123—21no—
287C22⋊Q8D29211yescharacteristic
288#121—1462no—
289#121—1462no—
290#121D29211yescharacteristic
291C4×Q8C22×D7311yescharacteristic
292C4×D4—21no—
293C4×D4—1462no—
294C4×D4—1462no—
295C4×D4C22×D7311yescharacteristic
296#117—21no—
297C2×C42C22×D7311yescharacteristic
298#107—1461no—
299#107—1461no—
300#107—1462no—
301#106—1462no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—732no—
305C4⋊C8—732no—
306C4⋊C8—1462no—
307C4⋊C8C22×D7311yescharacteristic
308C4⋊C8C22×D7311yescharacteristic
309#103—1462no—
310#102—1462no—
311#102—1462no—
312#102—1462no—
313#102—1462no—
314#102—1462no—
315C22⋊C8—21no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318#97—21no—
319#97—21no—
320C4×C8—21no—
321C4×C8C22⋊D7311yescharacteristic
322C4×C8C22⋊D7311yescharacteristic
323C4×C8C22×D7311yescharacteristic
324C4×C8C22×D7311yescharacteristic
325#41—5844no—
326C2×Q8—1462no—
327C2×Q8C2×D29211yescharacteristic
328C2×D4—21no—
329C2×D4—2922no—
330C2×D4—1462no—
331C2×D4—1462no—
332C2×D4—1462no—
333C2×D4C2×D29211yescharacteristic
334C22×C4—21no—
335C22×C4—2922no—
336C22×C4C2×C22⋊D7311yescharacteristic
337C22×C4C2×D29211yescharacteristic
338C22×C4#1192402811yescharacteristic
339SD16—5842no—
340D8—2922no—
341D8—2922no—
342D8—5842no—
343M16—42no—
344C2×C8—22no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—5842no—
350C2×C8—1462no—
351C2×C8—1462no—
352C2×C8C2×C22⋊D7311yescharacteristic
353C2×C8C2×C22⋊D7311yescharacteristic
354C2×C8C2×C22⋊D7311yescharacteristic
355C2×C8C2×C22⋊D7311yescharacteristic
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—1462no—
361C4⋊C4—1462no—
362C4⋊C4C2×D29211yescharacteristic
363C4⋊C4C2×D29211yescharacteristic
364C22⋊C4—21no—
365C22⋊C4—21no—
366C22⋊C4—21no—
367C22⋊C4—21no—
368C22⋊C4—21no—
369C22⋊C4—2922no—
370C22⋊C4—2922no—
371C42—21no—
372C42C2×C22⋊D7311yescharacteristic
373C42C2×D29211yescharacteristic
374C42#1192402811yescharacteristic
375C42C23×D7311yescharacteristic
376C23—21no—
377C23—2922no—
378C23#1455656511yescharacteristic
379Q8—21no—
380Q8—21no—
381Q8—5842no—
382D4—21no—
383D4—21no—
384D4—42no—
385D4—2922no—
386D4—2922no—
387D4—2922no—
388D4—2922no—
389D4—5842no—
390D4—5844no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—2922no—
402C2×C4—5844no—
403C2×C4#1455656511yescharacteristic
404C2×C4#1455656511yescharacteristic
405C2×C4#1455656511yescharacteristic
406C2×C4C22×C22⋊D7311yescharacteristic
407C2×C4C22×D29211yescharacteristic
408C2×C4C2×#1192402811yescharacteristic
409C8—21no—
410C8—21no—
411C8—21no—
412C8—41no—
413C8—41no—
414C8—2922no—
415C8—2922no—
416C8#1455661112yes—
417C22—21no—
418C22—21no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—5842no—
423C22—5842no—
424C22—5842no—
425C22#1798885712yes—
426C22C2×#1455656511yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—5842no—
435C4C2×#1455661111yescharacteristic
436C4#1798938511yescharacteristic
437C4#1798939211yescharacteristic
438C4#1798940411yescharacteristic
439C2—22no—
440C2—81no—
441C2—11682no—
442C2#2261938611yescharacteristic
443C2#2261960111yescharacteristic
444C2C2×#1798885711yescharacteristic
subgroupG/H
centerC22C2×#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#2966C2
socleC22×C73C2×#121
hypercenter#2966D73
hypocenterC73#19222
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
1. #22621257C2semidirect
2. #22620510C2semidirect
3. #22609471C2semidirect
4. #22609464C2semidirect
5. #22609335C2semidirect
6. #22609330C2semidirect
7. #22609291C2semidirect
8. #22609286C2semidirect
9. C2×#17988340C2semidirect
10. #22608627C2semidirect
11. C73×#2966C2semidirect
225. #2966D73semidirect
234. C73#19222semidirect

Maximal subgroups

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

subgroupindex
#226212572
#226205102
#226094712
#226094642
#226093352
#226093302
#226092912
#226092862
C2×#179883402
#226086272
C73×#29662
#1922273