Finite Group Databasebeta

← #33003069

Subgroups — #33003069

Order 18688.

462 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989017C211yesmaximal, characteristic
2#22611667C211yesmaximal, characteristic, complemented
3#22611658C211yesmaximal, characteristic, complemented
4#22610829C211yesmaximal, characteristic, complemented
5#22608028C211yesmaximal, characteristic, complemented
6#22608020C211yesmaximal, characteristic, complemented
7C73×#2957C211yesmaximal, characteristic, complemented
8C2×#14556613C2211yescharacteristic
9#17989017—22no—
10#17989017—22no—
11C2×#14556537C411yescharacteristic, complemented
12C2×#14556535C411yescharacteristic, complemented
13C2×#14556534C411yescharacteristic, complemented
14C2×#14556532C411yescharacteristic, complemented
15C2×#14556531C2211yescharacteristic
16C22×#11924003C2211yescharacteristic
17#17988625C2211yescharacteristic
18#17988603—21no—
19#17988560—21no—
20#17988397—22no—
21#17988393—22no—
22C2×#135×C73C2211yescharacteristic
23#568×C73—21no—
24#444×C73—22no—
25#443×C73—22no—
26C2×#108×C73C2211yescharacteristic
27#421×C73—21no—
28#405×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
37#14556550—21no—
38#14556537—22no—
39#14556535—22no—
40#14556534—22no—
41#14556532—22no—
42#14556531—21no—
43#14556531—21no—
44C2×#11924003—21no—
45C2×#11924003—21no—
46C2×#11924003—21no—
47C2×#11924003C2×C411yescharacteristic
48C2×#11924003C2×C411yescharacteristic
49#14556525—21no—
50#14556523—21no—
51#14556517—42no—
52#14556505—42no—
53C2×#41×C73—21no—
54#135×C73—21no—
55#135×C73—21no—
56#135×C73D412yes—
57#135×C73D412yes—
58C2×Q16×C73C2×C411yescharacteristic
59C2×SD16×C73—21no—
60C2×D8×C73C2×C411yescharacteristic
61#131×C73—24no—
62#131×C73—42no—
63C2×M16×C73—21no—
64C2×M16×C73—42no—
65C2×M16×C73D411yescharacteristic
66C22×C8×C73C2311yescharacteristic
67#117×C73D411yescharacteristic
68#108×C73—21no—
69#108×C73D412yes—
70#104×C73—24no—
71#97×C73—21no—
72C4×C8×C73—21no—
73#11924028—41no—
74#11924028—41no—
75#11924028—41no—
76#11924028—41no—
77C2×Dic146—21no—
78C2×D292—21no—
79C2×C22⋊D73—41no—
80C2×C4×D73—41no—
81Dic292—21no—
82Dic292—21no—
83#11924017—41no—
84#11924017—41no—
85D584—21no—
86D584—21no—
87C4⋊Dic73—41no—
88C22⋊Dic73—41no—
89C4×Dic73—41no—
90#11924003—21no—
91#11924003—21no—
92#11924003—41no—
93#41×C73—22no—
94#41×C73—22no—
95#41×C73—41no—
96#41×C73—41no—
97C2×Q8×C73—21no—
98C2×D4×C73—21no—
99C2×D4×C73—41no—
100C22×C4×C73—41no—
101C22×C4×C73C2×D411yescharacteristic
102Q16×C73—21no—
103Q16×C73C22⋊C412yes—
104SD16×C73—22no—
105SD16×C73—22no—
106D8×C73—21no—
107D8×C73C22⋊C412yes—
108M16×C73—21no—
109M16×C73—22no—
110M16×C73—44no—
111M16×C73—41no—
112M16×C73D812yes—
113C2×C8×C73—21no—
114C2×C8×C73—44no—
115C2×C8×C73—41no—
116C2×C8×C73C22×C411yescharacteristic
117C2×C8×C73C2×D411yescharacteristic
118C2×C8×C73C2×D411yescharacteristic
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73#4111yescharacteristic
121C2×C8×C73#4111yescharacteristic
122C4⋊C4×C73—21no—
123C22⋊C4×C73—21no—
124C42×C73—21no—
125C23×C73—41no—
126C23×C73AGL(1,ℤ/8)11yescharacteristic
127C22×D73—41no—
128Q8×C73—22no—
129Q8×C73—41no—
130D4×C73—22no—
131D4×C73—42no—
132D4×C73—42no—
133D4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—42no—
136C2×C4×C73—42no—
137C2×C4×C73—41no—
138C2×C4×C73C4×D411yescharacteristic
139C2×C4×C73#12011yescharacteristic
140C2×C4×C73#12111yescharacteristic
141C2×C4×C73C2×D811yescharacteristic
142C2×C4×C73#13511yescharacteristic
143C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
144C22⋊D73—81no—
145C22⋊D73—81no—
146C2×Dic73—41no—
147C2×Dic73—81no—
148D292—41no—
149D292—41no—
150C4×D73—81no—
151C4×D73—81no—
152Dic146—41no—
153Dic146—41no—
154C8×C73—22no—
155C8×C73—42no—
156C8×C73C2×C22⋊C411yescharacteristic
157C8×C73C4×D411yescharacteristic
158C8×C73#12111yescharacteristic
159C8×C73#12311yescharacteristic
160C22×C73—41no—
161C22×C73—42no—
162C22×C73—161no—
163C22×C73#44211yescharacteristic
164C22×C73#44911yescharacteristic
165C22×C73#45911yescharacteristic
166C2×D73—81no—
167C2×D73—161no—
168C4×C73—42no—
169C4×C73—81no—
170C4×C73#38611yescharacteristic
171C4×C73C4×D811yescharacteristic
172C4×C73#44711yescharacteristic
173C4×C73#46311yescharacteristic
174Dic73—81no—
175Dic73—161no—
176#13016—731nomaximal, Sylow
177C2×C73—21no—
178C2×C73—21no—
179C2×C73—21no—
180C2×C73—82no—
181C2×C73#187911yescharacteristic
182D73—161no—
183C2×#508—731no—
184#2957D7311yescharacteristic, complemented
185#2145—731no—
186#2143—731no—
187#1976—731no—
188#1382—731no—
189#1379—731no—
190C73#1301611yescharacteristic, complemented, Sylow
191C2×#135—731no—
192C2×#135C2×D7311yescharacteristic
193#568—21no—
194#508—1462no—
195#508—1462no—
196C2×Q32—731no—
197C2×SD32—731no—
198C2×SD32—731no—
199C2×D16—731no—
200C2×M32—731no—
201C22×C16—731no—
202#444—22no—
203#443—22no—
204C2×#108C2×D7311yescharacteristic
205#427—731no—
206#421—21no—
207#421—1461no—
208#410—1461no—
209#405C2×D7311yescharacteristic
210#360—1462no—
211#359—1462no—
212C2×#41—21no—
213C2×#41—1461no—
214#135—21no—
215#135—21no—
216#135—1461no—
217#135—1461no—
218#135—1461no—
219#135—1461no—
220#135D29212yes—
221#135C22⋊D7312yes—
222C2×Q16—731no—
223C2×Q16C4×D7311yescharacteristic
224C2×SD16—21no—
225C2×SD16—1461no—
226C2×D8—731no—
227C2×D8C4×D7311yescharacteristic
228#131—24no—
229#131—42no—
230C2×M16—21no—
231C2×M16—42no—
232C2×M16C22⋊D7311yescharacteristic
233C22×C8C22×D7311yescharacteristic
234#117—1461no—
235#117C22⋊D7311yescharacteristic
236Q32—1462no—
237SD32—1462no—
238SD32—1462no—
239D16—1462no—
240M32—1461no—
241M32—1461no—
242C2×C16—731no—
243C2×C16—731no—
244C2×C16—1461no—
245C2×C16—1461no—
246C2×C16—1461no—
247#108—21no—
248#108C22⋊D7312yes—
249#107—1461no—
250#106—1461no—
251#104—24no—
252#104—2922no—
253#99—2922no—
254#97—21no—
255C4×C8—21no—
256#41—22no—
257#41—22no—
258#41—41no—
259#41—41no—
260#41—2921no—
261#41—2921no—
262#41—2921no—
263#41—2921no—
264C2×Q8—21no—
265C2×Q8—1461no—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—2921no—
269C2×D4—1461no—
270C22×C4—41no—
271C22×C4—2921no—
272C22×C4C2×C22⋊D7311yescharacteristic
273Q16—21no—
274Q16—1461no—
275Q16—1461no—
276Q16#1192400512yes—
277SD16—22no—
278SD16—22no—
279SD16—2921no—
280SD16—2921no—
281D8—21no—
282D8—1461no—
283D8—1461no—
284D8#1192400512yes—
285M16—21no—
286M16—22no—
287M16—44no—
288M16—41no—
289M16#1192401312yes—
290C2×C8—21no—
291C2×C8—44no—
292C2×C8—41no—
293C2×C8C2×C4×D7311yescharacteristic
294C2×C8C2×C22⋊D7311yescharacteristic
295C2×C8C2×C22⋊D7311yescharacteristic
296C2×C8C2×D29211yescharacteristic
297C2×C8#1192402811yescharacteristic
298C2×C8#1192402811yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—2921no—
301C22⋊C4—21no—
302C22⋊C4—2921no—
303C42—21no—
304C42—2921no—
305C16—2921no—
306C16—1461no—
307C16—1461no—
308C23—41no—
309C23—2921no—
310C23#1455661611yescharacteristic
311Q8—22no—
312Q8—41no—
313Q8—2921no—
314Q8—2921no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—41no—
319D4—2921no—
320D4—2921no—
321D4—5841no—
322D4—5841no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—2921no—
328C2×C4—5841no—
329C2×C4—5841no—
330C2×C4—5841no—
331C2×C4C4×C22⋊D7311yescharacteristic
332C2×C4#1455656011yescharacteristic
333C2×C4#1455656511yescharacteristic
334C2×C4C2×#1192401311yescharacteristic
335C2×C4#1455661111yescharacteristic
336C2×C4#1455661711yescharacteristic
337C8—22no—
338C8—42no—
339C8C2×#1192400511yescharacteristic
340C8C4×C22⋊D7311yescharacteristic
341C8#1455656511yescharacteristic
342C8#1455657411yescharacteristic
343C22—41no—
344C22—42no—
345C22—161no—
346C22—5841no—
347C22—11681no—
348C22#1798869711yescharacteristic
349C22#1798873211yescharacteristic
350C22#1798878211yescharacteristic
351C4—42no—
352C4—81no—
353C4—5841no—
354C4—11681no—
355C4#1798846611yescharacteristic
356C4C4×#1192401311yescharacteristic
357C4#1798872111yescharacteristic
358C4#1798880211yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—82no—
363C2—11681no—
364C2#2261027911yescharacteristic
subgroupG/H
centerC4#17988721
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#2957C2
socleC2×C73#1879
prime radicalC2×#17989017C2
cosocleC22×C8×C73C23
hypercenter#2957D73
hypocenterC73#13016
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
2. #22611667C2semidirect
3. #22611658C2semidirect
4. #22610829C2semidirect
5. #22608028C2semidirect
6. #22608020C2semidirect
7. C73×#2957C2semidirect
11. C2×#14556537C4semidirect
12. C2×#14556535C4semidirect
13. C2×#14556534C4semidirect
14. C2×#14556532C4semidirect
184. #2957D73semidirect
190. C73#13016semidirect

Maximal subgroups

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

subgroupindex
C2×#179890172
#226116672
#226116582
#226108292
#226080282
#226080202
C73×#29572
#1301673