Finite Group Databasebeta

← #32980300

Subgroups — #32980300

Order 18688.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618499C211yesmaximal, characteristic
2#22609951C211yesmaximal, characteristic
3#22608702C211yesmaximal, characteristic, complemented
4#22608687C211yesmaximal, characteristic, complemented
5#22607833C211yesmaximal, characteristic, complemented
6#22607771C211yesmaximal, characteristic, complemented
7C73×#1476C211yesmaximal, characteristic
8C2×#14556598—21no—
9#17988614—22no—
10#17988614—21no—
11#17988611—21no—
12#17988611C2211yescharacteristic
13#17988611C2211yescharacteristic
14C2×C4⋊#10596831—21no—
15C2×C4⋊#10596831C2211yescharacteristic
16C2×#14556518—21no—
17C4×C4⋊Dic73C2211yescharacteristic
18#17988346—21no—
19#17988338—22no—
20#17988338—21no—
21#17988338—21no—
22C4×#41×C73C2211yescharacteristic
23#431×C73C2211yescharacteristic
24C2×C4×C8×C73C2211yescharacteristic
25Q8⋊C8×C73C411yescharacteristic, complemented
26Q8⋊C8×C73C411yescharacteristic, complemented
27D4⋊C8×C73C411yescharacteristic, complemented
28D4⋊C8×C73C411yescharacteristic, complemented
29#14556598—42no—
30#14556598—42no—
31C2×#11924012—22no—
32C2×#11924012—21no—
33C2×#11924012—21no—
34C22×#10596831—22no—
35C22×#10596831—21no—
36C2×C4⋊Dic73—21no—
37C2×C4×Dic73—21no—
38C4⋊#10596831—22no—
39C4⋊#10596831—22no—
40C4⋊#10596831—21no—
41C4⋊#10596831—21no—
42C4⋊#10596831—21no—
43C4⋊#10596831—21no—
44C4⋊#10596831D411yescharacteristic
45C4⋊#10596831D411yescharacteristic
46C4⋊#10596831D412yes—
47#14556518—42no—
48#14556498—21no—
49C2×#41×C73D411yescharacteristic
50C22×C8×C73—21no—
51C4×Q8×C73C2×C411yescharacteristic
52C4×Q8×C73Q811yescharacteristic
53C4×D4×C73—21no—
54C4×D4×C73—21no—
55C4×D4×C73C2×C411yescharacteristic
56C4×D4×C73Q811yescharacteristic
57#117×C73—21no—
58#117×C73D411yescharacteristic
59C2×C42×C73—21no—
60C2×C42×C73C2311yescharacteristic
61C4⋊C8×C73C2×C411yescharacteristic
62C4⋊C8×C73C2×C411yescharacteristic
63C22⋊C8×C73—21no—
64C4×C8×C73—21no—
65C4×C8×C73—21no—
66C4×C8×C73C2×C411yescharacteristic
67C4×C8×C73C2×C411yescharacteristic
68C22×Dic73—41no—
69#11924012—44no—
70#11924012—41no—
71#11924012—41no—
72#11924012—41no—
73#11924012—41no—
74C2×#10596831—21no—
75C2×#10596831—22no—
76C2×#10596831—21no—
77C2×#10596831—44no—
78C2×#10596831—41no—
79C2×#10596831—41no—
80C2×#10596831—41no—
81C4⋊Dic73—22no—
82C4⋊Dic73—41no—
83C4×Dic73—41no—
84C4×Dic73—41no—
85#41×C73—21no—
86#41×C73—21no—
87#41×C73SD1612yes—
88#41×C73SD1612yes—
89C2×Q8×C73C4⋊C411yescharacteristic
90C2×D4×C73—21no—
91C2×D4×C73C4⋊C411yescharacteristic
92C22×C4×C73—21no—
93C22×C4×C73—21no—
94C22×C4×C73—21no—
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73#4111yescharacteristic
97C22×C4×C73#4111yescharacteristic
98C2×C8×C73—21no—
99C2×C8×C73—21no—
100C2×C8×C73—21no—
101C2×C8×C73—21no—
102C2×C8×C73—42no—
103C4⋊C4×C73—21no—
104C4⋊C4×C73—21no—
105C4⋊C4×C73C4⋊C411yescharacteristic
106C4⋊C4×C73C4⋊C411yescharacteristic
107C22⋊C4×C73—21no—
108C22⋊C4×C73—21no—
109C22⋊C4×C73—21no—
110C42×C73—21no—
111C42×C73—21no—
112C42×C73—21no—
113C42×C73C22×C411yescharacteristic
114C42×C73C2×D411yescharacteristic
115C42×C73C2×D411yescharacteristic
116C42×C73C2×Q811yescharacteristic
117C23×C73—21no—
118C23×C73#12311yescharacteristic
119Q8×C73—21no—
120Q8×C73#10612yes—
121D4×C73—22no—
122D4×C73—22no—
123D4×C73—21no—
124D4×C73#10612yes—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—22no—
129C2×C4×C73—22no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—81no—
136C2×C4×C73C2×C4⋊C411yescharacteristic
137C2×C4×C73C4×D411yescharacteristic
138C2×C4×C73C4×D411yescharacteristic
139C2×C4×C73#12011yescharacteristic
140C2×C4×C73C22⋊Q811yescharacteristic
141C2×C4×C73C22⋊Q811yescharacteristic
142C2×C4×C73C2×SD1611yescharacteristic
143C2×C4×C73C2×SD1611yescharacteristic
144C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C73#13711yescharacteristic
146C2×Dic73—42no—
147C2×Dic73—81no—
148C2×Dic73—81no—
149C8×C73—22no—
150C8×C73—41no—
151C8×C73—41no—
152C8×C73—41no—
153#10596831—41no—
154#10596831—42no—
155#10596831—41no—
156#10596831—81no—
157C22×C73—21no—
158C22×C73—21no—
159C22×C73—22no—
160C22×C73—41no—
161C22×C73—41no—
162C22×C73#38511yescharacteristic
163C22×C73#48111yescharacteristic
164C22×C73#48311yescharacteristic
165C4×C73—22no—
166C4×C73—21no—
167C4×C73—21no—
168C4×C73—21no—
169C4×C73—41no—
170C4×C73—41no—
171C4×C73C2×#10611yescharacteristic
172C4×C73#42811yescharacteristic
173C4×C73#44311yescharacteristic
174C4×C73#44411yescharacteristic
175C4×C73#44811yescharacteristic
176C4×C73#45011yescharacteristic
177Dic73—82no—
178#8904—731nomaximal, Sylow
179C2×C73—21no—
180C2×C73—21no—
181C2×C73—42no—
182C2×C73#185311yescharacteristic
183C2×C73#186111yescharacteristic
184C2×C73#186611yescharacteristic
185#2906—731no—
186#1806—731no—
187#1570—731no—
188#1566—731no—
189#1476D7311yescharacteristic
190#1303—731no—
191#1279—731no—
192C73#890411yescharacteristic, complemented, Sylow
193C2×#131—1461no—
194C4×#41C2×D7311yescharacteristic
195#431C2×D7311yescharacteristic
196#424—1462no—
197#424—1461no—
198#423—731no—
199#423—731no—
200#423—1461no—
201C2×C4⋊C8—731no—
202C2×C4⋊C8—1461no—
203C2×#104—1461no—
204C2×C4×C8C2×D7311yescharacteristic
205C4×C4⋊C4—731no—
206#339—1461no—
207#334—1462no—
208#334—1461no—
209#334—1461no—
210Q8⋊C8Dic7311yescharacteristic, complemented
211Q8⋊C8Dic7311yescharacteristic, complemented
212D4⋊C8Dic7311yescharacteristic, complemented
213D4⋊C8Dic7311yescharacteristic, complemented
214C2×#41D29211yescharacteristic
215#131—2922no—
216#131—2922no—
217C2×M16—1461no—
218C2×M16—1462no—
219C2×M16—1461no—
220C22×C8—21no—
221C22×C8—1462no—
222C22×C8—1461no—
223C4×Q8Dic14611yescharacteristic
224C4×Q8C2×Dic7311yescharacteristic
225C4×D4—21no—
226C4×D4—21no—
227C4×D4Dic14611yescharacteristic
228C4×D4C2×Dic7311yescharacteristic
229#117—21no—
230#117D29211yescharacteristic
231C2×C4⋊C4—1461no—
232C2×C42—21no—
233C2×C42—1461no—
234C2×C42C22×D7311yescharacteristic
235C4⋊C8—731no—
236C4⋊C8—731no—
237C4⋊C8—732no—
238C4⋊C8—1462no—
239C4⋊C8—1462no—
240C4⋊C8—1461no—
241C4⋊C8—1461no—
242C4⋊C8—1461no—
243C4⋊C8—1461no—
244C4⋊C8C2×Dic7311yescharacteristic
245C4⋊C8C2×Dic7311yescharacteristic
246#104—2922no—
247C22⋊C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8C2×Dic7311yescharacteristic
251C4×C8C2×Dic7311yescharacteristic
252#95—1461no—
253#41—21no—
254#41—21no—
255#41#1192401712yes—
256#41#1192401712yes—
257C2×Q8C4⋊Dic7311yescharacteristic
258C2×D4—21no—
259C2×D4C4⋊Dic7311yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—2921no—
264C22×C4C2×D29211yescharacteristic
265C22×C4#1192402711yescharacteristic
266C22×C4#1192402711yescharacteristic
267M16—2924no—
268M16—2921no—
269M16—2921no—
270M16—2921no—
271M16—2921no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—42no—
277C2×C8—2924no—
278C2×C8—2921no—
279C2×C8—2921no—
280C2×C8—2921no—
281C2×C8—1461no—
282C2×C8—1461no—
283C2×C8—1462no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—2921no—
287C4⋊C4—1462no—
288C4⋊C4C4⋊Dic7311yescharacteristic
289C4⋊C4C4⋊Dic7311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—2921no—
297C42—2921no—
298C42C22×Dic7311yescharacteristic
299C42D4×D7311yescharacteristic
300C42D4×D7311yescharacteristic
301C42C2×Dic14611yescharacteristic
302C23—21no—
303C23#1455657511yescharacteristic
304Q8—21no—
305Q8#1455652312yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4#1455652312yes—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—2922no—
322C2×C4—5841no—
323C2×C4—5841no—
324C2×C4C2×C4⋊Dic7311yescharacteristic
325C2×C4D4×Dic7311yescharacteristic
326C2×C4D4×Dic7311yescharacteristic
327C2×C4#1455656111yescharacteristic
328C2×C4#1455656811yescharacteristic
329C2×C4#1455656811yescharacteristic
330C2×C4C2×#1192401711yescharacteristic
331C2×C4C2×#1192401711yescharacteristic
332C2×C4#1455661911yescharacteristic
333C2×C4#1455662411yescharacteristic
334C8—22no—
335C8—41no—
336C8—41no—
337C8—41no—
338C8—2921no—
339C8—2922no—
340C8—2921no—
341C8—5841no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22#1798846211yescharacteristic
348C22#1798890611yescharacteristic
349C22#1798891811yescharacteristic
350C4—21no—
351C4—22no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—5842no—
357C4C2×#1455652311yescharacteristic
358C4#1798862911yescharacteristic
359C4#1798870411yescharacteristic
360C4#1798870911yescharacteristic
361C4#1798872811yescharacteristic
362C4#1798874011yescharacteristic
363C2—21no—
364C2—21no—
365C2—42no—
366C2#2261012611yescharacteristic
367C2#2261017311yescharacteristic
368C2#2261020811yescharacteristic
subgroupG/H
centerC2×C4#14556561
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1476C2
socleC22×C73#385
prime radicalC2×#41×C73D4
cosocleC2×C42×C73C23
hypercenter#1476D73
hypocenterC73#8904
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
3. #22608702C2semidirect
4. #22608687C2semidirect
5. #22607833C2semidirect
6. #22607771C2semidirect
25. Q8⋊C8×C73C4semidirect
26. Q8⋊C8×C73C4semidirect
27. D4⋊C8×C73C4semidirect
28. D4⋊C8×C73C4semidirect
192. C73#8904semidirect
210. Q8⋊C8Dic73semidirect
211. Q8⋊C8Dic73semidirect
212. D4⋊C8Dic73semidirect
213. D4⋊C8Dic73semidirect

Maximal subgroups

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

subgroupindex
#226184992
#226099512
#226087022
#226086872
#226078332
#226077712
C73×#14762
#890473