Finite Group Databasebeta

← #33003160

Subgroups — #33003160

Order 18688.

512 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989022C211yesmaximal, characteristic
2C2×#17988399C211yesmaximal, characteristic, complemented
3#22611643C211yesmaximal, characteristic, complemented
4#22610797C211yesmaximal, characteristic, complemented
5#22608041C211yesmaximal, characteristic, complemented
6#22608013C211yesmaximal, characteristic, complemented
7C2×#443×C73C211yesmaximal, characteristic, complemented
8C22×D584C2211yescharacteristic
9#17989022—22no—
10#17989022C414yescomplemented
11C2×#14556534—21no—
12C2×#14556532—21no—
13C2×#14556531C2211yescharacteristic
14C2×#14556531C2211yescharacteristic
15C2×#14556525C2211yescharacteristic
16C2×#14556513—21no—
17#17988489—21no—
18#17988399—22no—
19#17988391—21no—
20#17988388—21no—
21C2×#135×C73C2211yescharacteristic
22C2×#131×C73—21no—
23#443×C73—22no—
24#443×C73—22no—
25C2×#108×C73C2211yescharacteristic
26C2×#104×C73—21no—
27C2×C4×C8×C73C2211yescharacteristic
28C22×D292—21no—
29C2×D584—21no—
30C2×D584—21no—
31C2×D584—21no—
32C2×D584—21no—
33C2×D584—21no—
34C2×D584C2×C412yes—
35C2×C4⋊Dic73—21no—
36C2×#11924005—42no—
37#14556534—24no—
38#14556532—24no—
39#14556531—21no—
40#14556531—21no—
41#14556531—21no—
42#14556531C2×C412yes—
43C2×#11924003—21no—
44C2×#11924003—21no—
45#14556525—21no—
46#14556525—21no—
47#14556513—41no—
48#14556513—41no—
49C2×#41×C73—21no—
50#135×C73—21no—
51#135×C73—21no—
52#135×C73C2×C412yes—
53#135×C73D412yes—
54C2×Q16×C73D411yescharacteristic
55C2×SD16×C73—21no—
56C2×D8×C73D411yescharacteristic
57#131×C73—24no—
58#131×C73—42no—
59C2×M16×C73—21no—
60C2×M16×C73—41no—
61C22×C8×C73—41no—
62C22×C8×C73D411yescharacteristic
63C22×C8×C73C2311yescharacteristic
64C2×C42×C73D411yescharacteristic
65#108×C73—21no—
66#108×C73D412yes—
67#104×C73—24no—
68C4×C8×C73—21no—
69C4×C8×C73—21no—
70C23×D73—41no—
71C2×D292—22no—
72C2×D292—41no—
73C2×D292—41no—
74C2×D292—41no—
75C2×D292—41no—
76C2×D292—41no—
77C22×Dic73—41no—
78D584—22no—
79D584—22no—
80D584—41no—
81D584—41no—
82C4⋊Dic73—41no—
83C4⋊Dic73—41no—
84#11924005—84no—
85#11924003—22no—
86#11924003—41no—
87#41×C73—22no—
88#41×C73—22no—
89#41×C73—41no—
90#41×C73—41no—
91C2×Q8×C73—21no—
92C2×D4×C73—21no—
93C2×D4×C73—41no—
94C22×C4×C73—21no—
95C22×C4×C73—41no—
96C22×C4×C73C2×D411yescharacteristic
97Q16×C73—21no—
98Q16×C73C22⋊C412yescomplemented
99SD16×C73—22no—
100SD16×C73—22no—
101D8×C73—21no—
102D8×C73C22⋊C412yescomplemented
103M16×C73—22no—
104M16×C73—44no—
105M16×C73—41no—
106C2×C8×C73—21no—
107C2×C8×C73—21no—
108C2×C8×C73—44no—
109C2×C8×C73—41no—
110C2×C8×C73D811yescharacteristic
111C2×C8×C73D811yescharacteristic
112C2×C8×C73C22×C411yescharacteristic
113C2×C8×C73C2×D411yescharacteristic
114C2×C8×C73C2×D411yescharacteristic
115C2×C8×C73C2×D411yescharacteristic
116C2×C8×C73#4111yescharacteristic
117C2×C8×C73#4111yescharacteristic
118C42×C73—21no—
119C42×C73—21no—
120C23×C73—41no—
121C23×C73AGL(1,ℤ/8)11yescharacteristic
122C22×D73—42no—
123C22×D73—82no—
124C22×D73—82no—
125C22×D73—81no—
126C22×D73—81no—
127Q8×C73—22no—
128Q8×C73—41no—
129D4×C73—22no—
130D4×C73—42no—
131D4×C73—42no—
132D4×C73—41no—
133C2×C4×C73—22no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—42no—
138C2×C4×C73—42no—
139C2×C4×C73C4×D411yescharacteristic
140C2×C4×C73#12011yescharacteristic
141C2×C4×C73#12111yescharacteristic
142C2×C4×C73C2×D811yescharacteristic
143C2×C4×C73#13511yescharacteristic
144C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
145C2×Dic73—82no—
146C2×Dic73—81no—
147D292—42no—
148D292—42no—
149D292—81no—
150D292—81no—
151C8×C73—21no—
152C8×C73—21no—
153C8×C73—42no—
154C8×C73C2×C22⋊C411yescharacteristic
155C8×C73C4×D411yescharacteristic
156C8×C73#12111yescharacteristic
157C8×C73#12311yescharacteristic
158C22×C73—21no—
159C22×C73—21no—
160C22×C73—42no—
161C22×C73—81no—
162C22×C73—81no—
163C22×C73#44211yescharacteristic
164C22×C73#44911yescharacteristic
165C22×C73#45911yescharacteristic
166C2×D73—84no—
167C2×D73—82no—
168C2×D73—162no—
169C2×D73—161no—
170C2×D73—161no—
171C4×C73—42no—
172C4×C73—42no—
173C4×C73#38611yescharacteristic
174C4×C73C4×D811yescharacteristic
175C4×C73#44711yescharacteristic
176C4×C73#46311yescharacteristic
177Dic73—161no—
178#13029—731nomaximal, Sylow
179C2×C73—21no—
180C2×C73—21no—
181C2×C73—82no—
182C2×C73#187911yescharacteristic
183C2×C73#221211yescharacteristic
184C2×C73#221311yescharacteristic
185D73—162no—
186C2×#509—731no—
187C2×#443D7311yescharacteristic, complemented
188C2×#361—731no—
189#2139—731no—
190#1970—731no—
191#1387—731no—
192#1376—731no—
193C73#1302911yescharacteristic, complemented, Sylow
194C2×#135C2×D7311yescharacteristic
195C22×D8—731no—
196C2×#131—21no—
197#509—734no—
198#509—1462no—
199C2×SD32—1461no—
200C2×D16—1461no—
201C2×M32—731no—
202C2×M32—731no—
203#443—22no—
204#443—22no—
205C2×#108C2×D7311yescharacteristic
206C2×#107—731no—
207C2×#104—21no—
208C2×#102—1461no—
209C2×C4×C8C2×D7311yescharacteristic
210#390—1461no—
211#361—1462no—
212#358—1461no—
213#357—1461no—
214C2×#41—21no—
215C22×D4—1461no—
216#135—21no—
217#135—21no—
218#135C4×D7312yes—
219#135C22⋊D7312yes—
220C2×Q16D29211yescharacteristic
221C2×SD16—21no—
222C2×D8—732no—
223C2×D8—1461no—
224C2×D8—1461no—
225C2×D8—1461no—
226C2×D8—1461no—
227C2×D8—1461no—
228C2×D8D29211yescharacteristic
229#131—24no—
230#131—42no—
231C2×M16—21no—
232C2×M16—41no—
233C22×C8—41no—
234C22×C8C22⋊D7311yescharacteristic
235C22×C8C22×D7311yescharacteristic
236C2×C4⋊C4—1461no—
237C2×C22⋊C4—2922no—
238C2×C42C22⋊D7311yescharacteristic
239SD32—1464no—
240D16—1464no—
241M32—732no—
242M32—1461no—
243M32—1461no—
244M32—1461no—
245C2×C16—1461no—
246C2×C16—1461no—
247#108—21no—
248#108C22⋊D7312yes—
249#107—1461no—
250#107—1461no—
251#104—24no—
252#102—2921no—
253#102—2921no—
254C4×C8—21no—
255C4×C8—21no—
256C24—2921no—
257#41—22no—
258#41—22no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—2921no—
265C2×D4—2921no—
266C2×D4—2921no—
267C2×D4—2921no—
268C2×D4—2921no—
269C2×D4—1462no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—2921no—
273C22×C4C2×C22⋊D7311yescharacteristic
274Q16—21no—
275Q16#1192400512yescomplemented
276SD16—22no—
277SD16—22no—
278D8—21no—
279D8—2921no—
280D8—2921no—
281D8—1462no—
282D8—1462no—
283D8#1192400512yescomplemented
284M16—22no—
285M16—44no—
286M16—41no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—44no—
290C2×C8—41no—
291C2×C8#1192401311yescharacteristic
292C2×C8#1192401311yescharacteristic
293C2×C8C2×C4×D7311yescharacteristic
294C2×C8C2×C22⋊D7311yescharacteristic
295C2×C8C2×C22⋊D7311yescharacteristic
296C2×C8C2×D29211yescharacteristic
297C2×C8#1192402811yescharacteristic
298C2×C8#1192402811yescharacteristic
299C4⋊C4—2921no—
300C4⋊C4—2921no—
301C22⋊C4—5844no—
302C42—21no—
303C42—21no—
304C16—2921no—
305C16—1462no—
306C23—41no—
307C23—2922no—
308C23—5842no—
309C23—5842no—
310C23—5841no—
311C23—5841no—
312C23#1455661611yescharacteristic
313Q8—22no—
314Q8—41no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—41no—
319D4—2922no—
320D4—2922no—
321D4—5841no—
322D4—5841no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—5842no—
330C2×C4—5841no—
331C2×C4C4×C22⋊D7311yescharacteristic
332C2×C4#1455656011yescharacteristic
333C2×C4#1455656511yescharacteristic
334C2×C4C2×#1192401311yescharacteristic
335C2×C4#1455661111yescharacteristic
336C2×C4#1455661711yescharacteristic
337C8—21no—
338C8—21no—
339C8—42no—
340C8C2×#1192400511yescharacteristic
341C8C4×C22⋊D7311yescharacteristic
342C8#1455656511yescharacteristic
343C8#1455657411yescharacteristic
344C22—21no—
345C22—21no—
346C22—42no—
347C22—81no—
348C22—81no—
349C22—5844no—
350C22—5842no—
351C22—11682no—
352C22—11681no—
353C22—11681no—
354C22#1798869711yescharacteristic
355C22#1798873211yescharacteristic
356C22#1798878211yescharacteristic
357C4—42no—
358C4—42no—
359C4—11681no—
360C4#1798846611yescharacteristic
361C4C4×#1192401311yescharacteristic
362C4#1798872111yescharacteristic
363C4#1798880211yescharacteristic
364C2—21no—
365C2—21no—
366C2—82no—
367C2—11682no—
368C2#2261027911yescharacteristic
369C2#2261200011yescharacteristic
370C2#2261200711yescharacteristic
subgroupG/H
centerC22#17988782
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#443×C73C2
socleC22×C73#459
prime radicalC2×#17989022C2
cosocleC22×C8×C73C23
hypercenterC2×#443D73
hypocenterC73#13029
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
2. C2×#17988399C2semidirect
3. #22611643C2semidirect
4. #22610797C2semidirect
5. #22608041C2semidirect
6. #22608013C2semidirect
7. C2×#443×C73C2semidirect
10. #17989022C4semidirect
98. Q16×C73C22⋊C4semidirect
102. D8×C73C22⋊C4semidirect
187. C2×#443D73semidirect
193. C73#13029semidirect
275. Q16#11924005semidirect
283. D8#11924005semidirect

Maximal subgroups

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

subgroupindex
C2×#179890222
C2×#179883992
#226116432
#226107972
#226080412
#226080132
C2×#443×C732
#1302973