Finite Group Databasebeta

← #33069420

Subgroups — #33069420

Order 18688.

562 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621106C212yesmaximal
2#22620517C211yesmaximal, characteristic, complemented
3#22609459C211yesmaximal, characteristic, complemented
4#22609454C211yesmaximal, characteristic, complemented
5#22609425C211yesmaximal, characteristic, complemented
6#22609420C211yesmaximal, characteristic, complemented
7#22609389C212yesmaximal, complemented
8#22609382C212yesmaximal, 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#17988340C2212yes—
30#17988338C2211yescharacteristic
31#17988338C2211yescharacteristic
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—22no—
152D4×C73—22no—
153D4×C73—42no—
154C2×C4×C73—21no—
155C2×C4×C73—21no—
156C2×C4×C73—22no—
157C2×C4×C73—21no—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—41no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73#12111yescharacteristic
164C2×C4×C73#12111yescharacteristic
165C2×C4×C73#12111yescharacteristic
166C2×C4×C73C22×D411yescharacteristic
167C2×C4×C73C22×D411yescharacteristic
168C2×C4×C73C2×#4111yescharacteristic
169C2×Dic73—42no—
170C2×Dic73—42no—
171Dic146—42no—
172Dic146—42no—
173Dic146—82no—
174C8×C73—21no—
175C8×C73—21no—
176C8×C73—21no—
177C8×C73—41no—
178C8×C73—41no—
179C8×C73#13512yes—
180#10596831—42no—
181#10596831—42no—
182C22×C73—21no—
183C22×C73—21no—
184C22×C73—21no—
185C22×C73—41no—
186C22×C73—41no—
187C22×C73—41no—
188C22×C73C2×#12111yescharacteristic
189C4×C73—21no—
190C4×C73—21no—
191C4×C73—21no—
192C4×C73—21no—
193C4×C73—41no—
194C4×C73—41no—
195C4×C73—41no—
196C4×C73C2×#13511yescharacteristic
197C4×C73#57511yescharacteristic
198C4×C73#57812yes—
199Dic73—82no—
200Dic73—82no—
201#19226—731nomaximal, Sylow
202C2×C73—41no—
203C2×C73—81no—
204C2×C73#303511yescharacteristic
205C2×C73#305911yescharacteristic
206C2×C73#306711yescharacteristic
207#3233—732no—
208#3170—731no—
209#2966D7311yescharacteristic
210#1696—731no—
211#1695—731no—
212#1690—731no—
213#1689—731no—
214#1684—732no—
215#1683—732no—
216#1564—731no—
217#1554—732no—
218C73#1922611yescharacteristic, complemented, Sylow
219#533—732no—
220#529C2×D7311yescharacteristic
221#500—731no—
222#499—1462no—
223#494—731no—
224#488—1462no—
225#487—732no—
226#485—1462no—
227#482—1462no—
228#477—732no—
229#476—732no—
230#475—732no—
231#474—732no—
232#467—1461no—
233#467—1461no—
234#446C2×D7311yescharacteristic
235C8×Q8C2×D7311yescharacteristic
236C4×Q16—1461no—
237C4×Q16—1461no—
238#436C2×D7311yescharacteristic
239#435—21no—
240C8×D4—21no—
241C8×D4C2×D7311yescharacteristic
242#433—21no—
243#432C2×D7311yescharacteristic
244#431—732no—
245#405—21no—
246C2×C4×C8C2×D7311yescharacteristic
247#335—732no—
248#334—731no—
249#334—731no—
250#333—732no—
251#333—732no—
252#332—732no—
253#332—732no—
254C2×Q16—1461no—
255C2×Q16—1461no—
256C2×M16—21no—
257C22×C8—21no—
258C22×C8C22⋊D7311yescharacteristic
259C22×C8C22⋊D7311yescharacteristic
260C4⋊Q8—732no—
261C4⋊Q8—732no—
262#126—21no—
263#125—1462no—
264#125D29211yescharacteristic
265#124D29211yescharacteristic
266#123—21no—
267C22⋊Q8—1462no—
268C22⋊Q8—1462no—
269C22⋊Q8D29211yescharacteristic
270#121D29211yescharacteristic
271C4×Q8—1462no—
272C4×Q8—1462no—
273C4×Q8C22×D7311yescharacteristic
274C4×D4—21no—
275C4×D4C22×D7311yescharacteristic
276#117—21no—
277#117—1462no—
278C2×C42C22×D7311yescharacteristic
279#107—1461no—
280#107—1461no—
281#107—1462no—
282#107—1462no—
283#106—1462no—
284#106—1462no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—732no—
288C4⋊C8—732no—
289C4⋊C8C22×D7311yescharacteristic
290C4⋊C8C22×D7311yescharacteristic
291#103—1462no—
292#103—1462no—
293#103—1462no—
294#103—1462no—
295#102—1462no—
296#102—1462no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—1462no—
301#97—21no—
302#97—21no—
303C4×C8—21no—
304C4×C8—1462no—
305C4×C8C22⋊D7311yescharacteristic
306C4×C8C22⋊D7311yescharacteristic
307C4×C8C22×D7311yescharacteristic
308C4×C8C22×D7311yescharacteristic
309C2×Q8—1462no—
310C2×Q8—1462no—
311C2×Q8C2×D29211yescharacteristic
312C2×D4—21no—
313C2×D4C2×D29211yescharacteristic
314C22×C4—21no—
315C22×C4C2×C22⋊D7311yescharacteristic
316C22×C4C2×D29211yescharacteristic
317C22×C4#1192402811yescharacteristic
318Q16—2922no—
319Q16—2922no—
320M16—42no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—1462no—
327C2×C8—1462no—
328C2×C8C2×C22⋊D7311yescharacteristic
329C2×C8C2×C22⋊D7311yescharacteristic
330C2×C8C2×C22⋊D7311yescharacteristic
331C2×C8C2×C22⋊D7311yescharacteristic
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—2922no—
337C4⋊C4—2922no—
338C4⋊C4—1462no—
339C4⋊C4—1462no—
340C4⋊C4—1462no—
341C4⋊C4—1462no—
342C4⋊C4C2×D29211yescharacteristic
343C4⋊C4C2×D29211yescharacteristic
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—21no—
349C22⋊C4—2922no—
350C42—21no—
351C42—2922no—
352C42C2×C22⋊D7311yescharacteristic
353C42C2×D29211yescharacteristic
354C42#1192402811yescharacteristic
355C42C23×D7311yescharacteristic
356C23—21no—
357C23#1455656511yescharacteristic
358Q8—22no—
359Q8—2922no—
360Q8—2922no—
361Q8—5842no—
362D4—22no—
363D4—42no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—2922no—
374C2×C4—2922no—
375C2×C4#1455656511yescharacteristic
376C2×C4#1455656511yescharacteristic
377C2×C4#1455656511yescharacteristic
378C2×C4C22×C22⋊D7311yescharacteristic
379C2×C4C22×D29211yescharacteristic
380C2×C4C2×#1192402811yescharacteristic
381C8—21no—
382C8—21no—
383C8—21no—
384C8—41no—
385C8—41no—
386C8—2922no—
387C8—2922no—
388C8#1455661112yes—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22C2×#1455656511yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—5842no—
404C4—5842no—
405C4C2×#1455661111yescharacteristic
406C4#1798938511yescharacteristic
407C4#1798940412yes—
408C2—41no—
409C2—81no—
410C2#2261938611yescharacteristic
411C2#2261958611yescharacteristic
412C2#2261965111yescharacteristic
subgroupG/H
centerC22C2×#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#2966C2
socleC22×C73C2×#121
prime radical#121×C73D4
hypercenter#2966D73
hypocenterC73#19226
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. #22609459C2semidirect
4. #22609454C2semidirect
5. #22609425C2semidirect
6. #22609420C2semidirect
7. #22609389C2semidirect
8. #22609382C2semidirect
9. #22608679C2semidirect
10. #22608642C2semidirect
218. C73#19226semidirect

Maximal subgroups

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

subgroupindex
#226211062
#226205172
#226094592
#226094542
#226094252
#226094202
#226093892
#226093822
#226086792
#226086422
C73×#29662
#1922673