Finite Group Databasebeta

← #33069346

Subgroups — #33069346

Order 18688.

562 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621106C212yesmaximal
2#22620517C211yesmaximal, characteristic, complemented
3#22609513C212yesmaximal, complemented
4#22609506C212yesmaximal, complemented
5#22609335C211yesmaximal, characteristic, complemented
6#22609330C211yesmaximal, characteristic, complemented
7#22609301C211yesmaximal, characteristic, complemented
8#22609296C211yesmaximal, characteristic, complemented
9#22608679C211yesmaximal, characteristic, complemented
10#22608642C212yesmaximal, complemented
11C73×#2966C211yesmaximal, characteristic
12#17989144C2212yes—
13#17988992C2211yescharacteristic
14#17988986—22no—
15#17988967C2211yescharacteristic
16#17988941—22no—
17#17988938C2212yes—
18#17988925—22no—
19#17988908—22no—
20#17988882C2212yes—
21#17988877C2212yes—
22#17988872C2212yes—
23#17988867C2212yes—
24#17988828—21no—
25#17988828—21no—
26C4×Dic292—21no—
27C4×Dic292—21no—
28#17988640C2212yes—
29#17988340C2211yescharacteristic
30#17988340C2211yescharacteristic
31#17988338C2212yes—
32#17988335C2212yes—
33#17988335C2212yes—
34#17988334C2212yes—
35#17988334C2212yes—
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×Dic292—21no—
48C2×Dic292—21no—
49#14556591C2312yes—
50#14556591C2312yes—
51#14556583—22no—
52#14556570—22no—
53#14556570—22no—
54C4×Dic146—22no—
55C4×Dic146—22no—
56#14556550—22no—
57#14556525—21no—
58#14556525—21no—
59#14556524—22no—
60#14556524—22no—
61#14556522—22no—
62#14556522—22no—
63C4⋊#10596831C2312yes—
64C4⋊#10596831C2312yes—
65#14556516—22no—
66#14556516—22no—
67#14556515—22no—
68#14556515—22no—
69#14556512—22no—
70#14556512—22no—
71C22⋊#10596831—22no—
72C4×#10596831—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
102C2×Dic146—22no—
103C2×Dic146—22no—
104Dic292—42no—
105Dic292—42no—
106C2×#10596831—22no—
107C2×#10596831—22no—
108C4⋊Dic73—22no—
109C4⋊Dic73—22no—
110C4⋊Dic73—22no—
111C4⋊Dic73—22no—
112#11924007—42no—
113#11924007—42no—
114C22⋊Dic73—42no—
115C4×Dic73—42no—
116C2×Q8×C73C2×D411yescharacteristic
117C2×D4×C73—21no—
118C2×D4×C73C2×D411yescharacteristic
119C22×C4×C73—21no—
120C22×C4×C73C2×D411yescharacteristic
121C22×C4×C73C2×D411yescharacteristic
122C22×C4×C73#4111yescharacteristic
123M16×C73—42no—
124C2×C8×C73—22no—
125C2×C8×C73—21no—
126C2×C8×C73—21no—
127C2×C8×C73—41no—
128C2×C8×C73—42no—
129C2×C8×C73C2×D411yescharacteristic
130C2×C8×C73C2×D411yescharacteristic
131C2×C8×C73C2×D411yescharacteristic
132C2×C8×C73C2×D411yescharacteristic
133C4⋊C4×C73—21no—
134C4⋊C4×C73—21no—
135C4⋊C4×C73—21no—
136C4⋊C4×C73—21no—
137C4⋊C4×C73C2×D411yescharacteristic
138C4⋊C4×C73C2×D411yescharacteristic
139C22⋊C4×C73—21no—
140C22⋊C4×C73—21no—
141C22⋊C4×C73—21no—
142C22⋊C4×C73—21no—
143C22⋊C4×C73—21no—
144C42×C73—21no—
145C42×C73C2×D411yescharacteristic
146C42×C73C2×D411yescharacteristic
147C42×C73#4111yescharacteristic
148C42×C73C2411yescharacteristic
149C23×C73—21no—
150C23×C73#12111yescharacteristic
151Q8×C73—21no—
152Q8×C73—21no—
153D4×C73—21no—
154D4×C73—21no—
155D4×C73—42no—
156C2×C4×C73—21no—
157C2×C4×C73—21no—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—21no—
161C2×C4×C73—21no—
162C2×C4×C73—21no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73#12111yescharacteristic
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73#12111yescharacteristic
169C2×C4×C73C22×D411yescharacteristic
170C2×C4×C73C22×D411yescharacteristic
171C2×C4×C73C2×#4111yescharacteristic
172C2×Dic73—42no—
173C2×Dic73—42no—
174Dic146—42no—
175Dic146—42no—
176Dic146—82no—
177C8×C73—21no—
178C8×C73—21no—
179C8×C73—21no—
180C8×C73—41no—
181C8×C73—41no—
182C8×C73#13512yes—
183#10596831—42no—
184#10596831—42no—
185C22×C73—21no—
186C22×C73—21no—
187C22×C73—21no—
188C22×C73—41no—
189C22×C73—41no—
190C22×C73—41no—
191C22×C73C2×#12111yescharacteristic
192C4×C73—21no—
193C4×C73—21no—
194C4×C73—21no—
195C4×C73—21no—
196C4×C73—41no—
197C4×C73—41no—
198C4×C73—41no—
199C4×C73C2×#13511yescharacteristic
200C4×C73#57511yescharacteristic
201C4×C73#57811yescharacteristic
202C4×C73#57811yescharacteristic
203Dic73—82no—
204Dic73—82no—
205#19220—731nomaximal, Sylow
206C2×C73—41no—
207C2×C73—81no—
208C2×C73#303511yescharacteristic
209C2×C73#305911yescharacteristic
210C2×C73#306711yescharacteristic
211#3233—732no—
212#3170—731no—
213#2966D7311yescharacteristic
214#1704—732no—
215#1703—732no—
216#1676—731no—
217#1675—731no—
218#1670—731no—
219#1669—731no—
220#1564—731no—
221#1554—732no—
222C73#1922011yescharacteristic, complemented, Sylow
223#533—732no—
224#529C2×D7311yescharacteristic
225#500—731no—
226#499—1462no—
227#494—731no—
228#488—1462no—
229#487—732no—
230#485—1462no—
231#482—1462no—
232#477—732no—
233#476—732no—
234#475—732no—
235#474—732no—
236#467—1461no—
237#467—1461no—
238#446C2×D7311yescharacteristic
239C8×Q8C2×D7311yescharacteristic
240C4×Q16—1461no—
241C4×Q16—1461no—
242#436C2×D7311yescharacteristic
243#435—21no—
244C8×D4—21no—
245C8×D4C2×D7311yescharacteristic
246#433—21no—
247#432C2×D7311yescharacteristic
248#431—732no—
249#405—21no—
250C2×C4×C8C2×D7311yescharacteristic
251#335—731no—
252#335—731no—
253#334—732no—
254#333—732no—
255#333—732no—
256#332—732no—
257#332—732no—
258C2×Q16—1461no—
259C2×Q16—1461no—
260C2×M16—21no—
261C22×C8—21no—
262C22×C8C22⋊D7311yescharacteristic
263C22×C8C22⋊D7311yescharacteristic
264C4⋊Q8—732no—
265C4⋊Q8—732no—
266#126—21no—
267#125—1462no—
268#125D29211yescharacteristic
269#124D29211yescharacteristic
270#123—21no—
271C22⋊Q8—1462no—
272C22⋊Q8—1462no—
273C22⋊Q8D29211yescharacteristic
274#121D29211yescharacteristic
275C4×Q8—1462no—
276C4×Q8—1462no—
277C4×Q8C22×D7311yescharacteristic
278C4×D4—21no—
279C4×D4C22×D7311yescharacteristic
280#117—21no—
281#117—1462no—
282C2×C42C22×D7311yescharacteristic
283#107—1461no—
284#107—1461no—
285#107—1462no—
286#107—1462no—
287#106—1462no—
288#106—1462no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—732no—
292C4⋊C8—732no—
293C4⋊C8C22×D7311yescharacteristic
294C4⋊C8C22×D7311yescharacteristic
295#103—1462no—
296#103—1462no—
297#103—1462no—
298#103—1462no—
299#102—1462no—
300#102—1462no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—1462no—
305#97—21no—
306#97—21no—
307C4×C8—21no—
308C4×C8—1462no—
309C4×C8C22⋊D7311yescharacteristic
310C4×C8C22⋊D7311yescharacteristic
311C4×C8C22×D7311yescharacteristic
312C4×C8C22×D7311yescharacteristic
313C2×Q8—1462no—
314C2×Q8—1462no—
315C2×Q8C2×D29211yescharacteristic
316C2×D4—21no—
317C2×D4C2×D29211yescharacteristic
318C22×C4—21no—
319C22×C4C2×C22⋊D7311yescharacteristic
320C22×C4C2×D29211yescharacteristic
321C22×C4#1192402811yescharacteristic
322Q16—2922no—
323Q16—2922no—
324M16—42no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—1462no—
331C2×C8—1462no—
332C2×C8C2×C22⋊D7311yescharacteristic
333C2×C8C2×C22⋊D7311yescharacteristic
334C2×C8C2×C22⋊D7311yescharacteristic
335C2×C8C2×C22⋊D7311yescharacteristic
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—2922no—
341C4⋊C4—2922no—
342C4⋊C4—1462no—
343C4⋊C4—1462no—
344C4⋊C4—1462no—
345C4⋊C4—1462no—
346C4⋊C4C2×D29211yescharacteristic
347C4⋊C4C2×D29211yescharacteristic
348C22⋊C4—21no—
349C22⋊C4—21no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—2922no—
354C42—21no—
355C42—2922no—
356C42C2×C22⋊D7311yescharacteristic
357C42C2×D29211yescharacteristic
358C42#1192402811yescharacteristic
359C42C23×D7311yescharacteristic
360C23—21no—
361C23#1455656511yescharacteristic
362Q8—21no—
363Q8—21no—
364Q8—2922no—
365Q8—2922no—
366Q8—5842no—
367D4—21no—
368D4—21no—
369D4—42no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—2922no—
381C2×C4—2922no—
382C2×C4#1455656511yescharacteristic
383C2×C4#1455656511yescharacteristic
384C2×C4#1455656511yescharacteristic
385C2×C4C22×C22⋊D7311yescharacteristic
386C2×C4C22×D29211yescharacteristic
387C2×C4C2×#1192402811yescharacteristic
388C8—21no—
389C8—21no—
390C8—21no—
391C8—41no—
392C8—41no—
393C8—2922no—
394C8—2922no—
395C8#1455661112yes—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22C2×#1455656511yescharacteristic
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—5842no—
411C4—5842no—
412C4C2×#1455661111yescharacteristic
413C4#1798938511yescharacteristic
414C4#1798940411yescharacteristic
415C4#1798940411yescharacteristic
416C2—41no—
417C2—81no—
418C2#2261938611yescharacteristic
419C2#2261958611yescharacteristic
420C2#2261965111yescharacteristic
subgroupG/H
centerC22C2×#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#2966C2
socleC22×C73C2×#121
prime radical#121×C73D4
hypercenter#2966D73
hypocenterC73#19220
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
2. #22620517C2semidirect
3. #22609513C2semidirect
4. #22609506C2semidirect
5. #22609335C2semidirect
6. #22609330C2semidirect
7. #22609301C2semidirect
8. #22609296C2semidirect
9. #22608679C2semidirect
10. #22608642C2semidirect
222. C73#19220semidirect

Maximal subgroups

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

subgroupindex
#226211062
#226205172
#226095132
#226095062
#226093352
#226093302
#226093012
#226092962
#226086792
#226086422
C73×#29662
#1922073