Finite Group Databasebeta

← #33003034

Subgroups — #33003034

Order 18688.

476 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989017C211yesmaximal, characteristic
2C2×#17988394C211yesmaximal, characteristic, complemented
3C2×#17988393C211yesmaximal, characteristic, complemented
4#22610819C211yesmaximal, characteristic, complemented
5#22608024C211yesmaximal, characteristic, complemented
6#22608009C211yesmaximal, characteristic, complemented
7C2×#443×C73C211yesmaximal, characteristic, complemented
8C2×#14556613C2211yescharacteristic
9#17989017—22no—
10#17989017—22no—
11C2×#14556537C411yescharacteristic, complemented
12C2×#14556535C411yescharacteristic, complemented
13C2×#14556534C411yescharacteristic, complemented
14C2×#14556532C411yescharacteristic, complemented
15C22×#11924003C2211yescharacteristic
16C22×#11924003C2211yescharacteristic
17C2×#14556527C2211yescharacteristic
18C2×#14556519—21no—
19#17988553—21no—
20#17988394—22no—
21#17988393—22no—
22C2×#135×C73C2211yescharacteristic
23C2×#131×C73—21no—
24#443×C73—22no—
25#443×C73—22no—
26C2×#108×C73C2211yescharacteristic
27C2×#104×C73—21no—
28C2×C4×C8×C73C2211yescharacteristic
29C2×#11924028—21no—
30#14556613—21no—
31#14556613—21no—
32#14556613—21no—
33#14556613—21no—
34C2×Dic292C2×C411yescharacteristic
35C2×#11924017—21no—
36C2×D584C2×C411yescharacteristic
37C2×#11924012—21no—
38#14556537—22no—
39#14556535—22no—
40#14556534—22no—
41#14556532—22no—
42C2×#11924003—21no—
43C2×#11924003—21no—
44C2×#11924003—21no—
45C2×#11924003—21no—
46C2×#11924003—21no—
47C2×#11924003C2×C411yescharacteristic
48C2×#11924003C2×C411yescharacteristic
49#14556527—21no—
50#14556527—21no—
51#14556519—42no—
52#14556503—41no—
53#14556503—41no—
54C2×#41×C73—21no—
55#135×C73—21no—
56#135×C73—21no—
57#135×C73D412yes—
58#135×C73D412yes—
59C2×Q16×C73C2×C411yescharacteristic
60C2×SD16×C73—21no—
61C2×D8×C73C2×C411yescharacteristic
62#131×C73—24no—
63#131×C73—42no—
64C2×M16×C73—21no—
65C2×M16×C73—41no—
66C22×C8×C73—41no—
67C22×C8×C73D411yescharacteristic
68C22×C8×C73C2311yescharacteristic
69C2×C42×C73D411yescharacteristic
70#108×C73—21no—
71#108×C73D412yes—
72#104×C73—24no—
73C4×C8×C73—21no—
74C4×C8×C73—21no—
75#11924028—41no—
76#11924028—41no—
77#11924028—41no—
78#11924028—41no—
79C2×Dic146—21no—
80C2×D292—21no—
81C2×C22⋊D73—41no—
82C2×C4×D73—41no—
83Dic292—21no—
84Dic292—21no—
85#11924017—41no—
86#11924017—41no—
87D584—21no—
88D584—21no—
89#11924012—41no—
90#11924012—41no—
91C2×#10596831—41no—
92#11924003—21no—
93#11924003—21no—
94#11924003—41no—
95#41×C73—22no—
96#41×C73—22no—
97#41×C73—41no—
98#41×C73—41no—
99C2×Q8×C73—21no—
100C2×D4×C73—21no—
101C2×D4×C73—41no—
102C22×C4×C73—21no—
103C22×C4×C73—41no—
104C22×C4×C73C2×D411yescharacteristic
105Q16×C73—21no—
106Q16×C73C22⋊C412yes—
107SD16×C73—22no—
108SD16×C73—22no—
109D8×C73—21no—
110D8×C73C22⋊C412yes—
111M16×C73—22no—
112M16×C73—44no—
113M16×C73—41no—
114C2×C8×C73—21no—
115C2×C8×C73—21no—
116C2×C8×C73—44no—
117C2×C8×C73—41no—
118C2×C8×C73D811yescharacteristic
119C2×C8×C73D811yescharacteristic
120C2×C8×C73C22×C411yescharacteristic
121C2×C8×C73C2×D411yescharacteristic
122C2×C8×C73C2×D411yescharacteristic
123C2×C8×C73C2×D411yescharacteristic
124C2×C8×C73#4111yescharacteristic
125C2×C8×C73#4111yescharacteristic
126C42×C73—21no—
127C42×C73—21no—
128C23×C73—41no—
129C23×C73#13511yescharacteristic
130C22×D73—41no—
131Q8×C73—22no—
132Q8×C73—41no—
133D4×C73—22no—
134D4×C73—42no—
135D4×C73—42no—
136D4×C73—41no—
137C2×C4×C73—22no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—42no—
142C2×C4×C73—42no—
143C2×C4×C73C4×D411yescharacteristic
144C2×C4×C73#12011yescharacteristic
145C2×C4×C73#12111yescharacteristic
146C2×C4×C73C2×D811yescharacteristic
147C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
149C22⋊D73—81no—
150C22⋊D73—81no—
151C2×Dic73—41no—
152D292—41no—
153D292—41no—
154C4×D73—81no—
155C4×D73—81no—
156Dic146—41no—
157Dic146—41no—
158C8×C73—21no—
159C8×C73—21no—
160C8×C73—42no—
161C8×C73C2×C22⋊C411yescharacteristic
162C8×C73C4×D411yescharacteristic
163C8×C73#12111yescharacteristic
164C8×C73#12311yescharacteristic
165#10596831—81no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—42no—
169C22×C73—81no—
170C22×C73—81no—
171C22×C73C4×D811yescharacteristic
172C22×C73#44911yescharacteristic
173C22×C73#46311yescharacteristic
174C2×D73—81no—
175C2×D73—81no—
176C2×D73—81no—
177C4×C73—42no—
178C4×C73—42no—
179C4×C73#38611yescharacteristic
180C4×C73#44211yescharacteristic
181C4×C73#44711yescharacteristic
182C4×C73#45911yescharacteristic
183Dic73—81no—
184#13011—731nomaximal, Sylow
185C2×C73—21no—
186C2×C73—21no—
187C2×C73—82no—
188C2×C73#187911yescharacteristic
189C2×C73#217811yescharacteristic
190C2×C73#217911yescharacteristic
191D73—161no—
192C2×#508—731no—
193C2×#443D7311yescharacteristic, complemented
194C2×#359—731no—
195C2×#359—731no—
196#1974—731no—
197#1381—731no—
198#1374—731no—
199C73#1301111yescharacteristic, complemented, Sylow
200C2×#135—731no—
201C2×#135C2×D7311yescharacteristic
202C2×#131—21no—
203#508—1462no—
204#508—1462no—
205C2×Q32—731no—
206C2×SD32—731no—
207C2×SD32—731no—
208C2×D16—731no—
209C22×C16—731no—
210C22×C16—731no—
211#443—22no—
212#443—22no—
213C2×#108—731no—
214C2×#108C2×D7311yescharacteristic
215C2×#104—21no—
216C2×#104—1461no—
217#408—1461no—
218C2×C4×C8C2×D7311yescharacteristic
219#359—1462no—
220#359—1462no—
221C2×#41—21no—
222C2×#41—1461no—
223#135—21no—
224#135—21no—
225#135—1461no—
226#135—1461no—
227#135—1461no—
228#135—1461no—
229#135D29212yes—
230#135C22⋊D7312yes—
231C2×Q16—731no—
232C2×Q16C4×D7311yescharacteristic
233C2×SD16—21no—
234C2×SD16—1461no—
235C2×D8—731no—
236C2×D8C4×D7311yescharacteristic
237#131—24no—
238#131—42no—
239C2×M16—21no—
240C2×M16—41no—
241C2×M16—1461no—
242C22×C8—41no—
243C22×C8C22⋊D7311yescharacteristic
244C22×C8C22×D7311yescharacteristic
245C2×C42C22⋊D7311yescharacteristic
246Q32—1462no—
247SD32—1462no—
248SD32—1462no—
249D16—1462no—
250C2×C16—731no—
251C2×C16—731no—
252C2×C16—1461no—
253C2×C16—1461no—
254C2×C16—1461no—
255C2×C16—1461no—
256C2×C16—1461no—
257#108—21no—
258#108—1461no—
259#108—1461no—
260#108C22⋊D7312yes—
261#104—24no—
262#104—2922no—
263C22⋊C8—2921no—
264C22⋊C8—2921no—
265C4×C8—21no—
266C4×C8—21no—
267#41—22no—
268#41—22no—
269#41—41no—
270#41—41no—
271#41—2921no—
272#41—2921no—
273#41—2921no—
274#41—2921no—
275C2×Q8—21no—
276C2×Q8—1461no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—2921no—
280C2×D4—1461no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—2921no—
284C22×C4C2×C22⋊D7311yescharacteristic
285Q16—21no—
286Q16—1461no—
287Q16—1461no—
288Q16#1192400512yes—
289SD16—22no—
290SD16—22no—
291SD16—2921no—
292SD16—2921no—
293D8—21no—
294D8—1461no—
295D8—1461no—
296D8#1192400512yes—
297M16—22no—
298M16—44no—
299M16—41no—
300M16—2921no—
301M16—2921no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—44no—
305C2×C8—41no—
306C2×C8—2921no—
307C2×C8#1192401311yescharacteristic
308C2×C8#1192401311yescharacteristic
309C2×C8C2×C4×D7311yescharacteristic
310C2×C8C2×C22⋊D7311yescharacteristic
311C2×C8C2×C22⋊D7311yescharacteristic
312C2×C8C2×D29211yescharacteristic
313C2×C8#1192402811yescharacteristic
314C2×C8#1192402811yescharacteristic
315C42—21no—
316C42—21no—
317C16—2921no—
318C16—1461no—
319C16—1461no—
320C23—41no—
321C23—2921no—
322C23#1455661111yescharacteristic
323Q8—22no—
324Q8—41no—
325Q8—2921no—
326Q8—2921no—
327D4—22no—
328D4—42no—
329D4—42no—
330D4—41no—
331D4—2921no—
332D4—2921no—
333D4—5841no—
334D4—5841no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—2921no—
342C2×C4—5841no—
343C2×C4—5841no—
344C2×C4C4×C22⋊D7311yescharacteristic
345C2×C4#1455656011yescharacteristic
346C2×C4#1455656511yescharacteristic
347C2×C4C2×#1192401311yescharacteristic
348C2×C4#1455661611yescharacteristic
349C2×C4#1455661711yescharacteristic
350C8—21no—
351C8—21no—
352C8—42no—
353C8—5841no—
354C8C2×#1192400511yescharacteristic
355C8C4×C22⋊D7311yescharacteristic
356C8#1455656511yescharacteristic
357C8#1455657411yescharacteristic
358C22—21no—
359C22—21no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—5841no—
364C22—5841no—
365C22—5841no—
366C22C4×#1192401311yescharacteristic
367C22#1798873211yescharacteristic
368C22#1798880211yescharacteristic
369C4—42no—
370C4—42no—
371C4—5841no—
372C4#1798846611yescharacteristic
373C4#1798869711yescharacteristic
374C4#1798872111yescharacteristic
375C4#1798878211yescharacteristic
376C2—21no—
377C2—21no—
378C2—82no—
379C2—11681no—
380C2#2261027911yescharacteristic
381C2#2261181011yescharacteristic
382C2#2261181511yescharacteristic
subgroupG/H
centerC2×C4C2×#11924013
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#443×C73C2
socleC22×C73C4×D8
prime radicalC2×#17989017C2
cosocleC22×C8×C73C23
hypercenterC2×#443D73
hypocenterC73#13011
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
2. C2×#17988394C2semidirect
3. C2×#17988393C2semidirect
4. #22610819C2semidirect
5. #22608024C2semidirect
6. #22608009C2semidirect
7. C2×#443×C73C2semidirect
11. C2×#14556537C4semidirect
12. C2×#14556535C4semidirect
13. C2×#14556534C4semidirect
14. C2×#14556532C4semidirect
193. C2×#443D73semidirect
199. C73#13011semidirect

Maximal subgroups

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

subgroupindex
C2×#179890172
C2×#179883942
C2×#179883932
#226108192
#226080242
#226080092
C2×#443×C732
#1301173